July 1965 Report 4 FLUX SWITCHING IN MULTIPATH CORES Prepared [or: JET PROPULSION LABORATORY 4800 OAK GROVE DRIVE PASADENA, CALIFORNIA CONTRACT 950943 UNDER NAS7-100 (Extension of work under Contract 950095 under NASw-6) By: D. NITZAN V.W. HESTERMAN SR1 Project 5094 Approved: D. R. BROWN, MANAGER COMPUTER TECHNIQUES LABORATORY J. D. NOE, EXECUTIVE DIRECTOR ENGINEERING SCIENCES AND INDUSTRIAL DEVELOPMENT work was performed for the Jet Propulsion Laboraio_, 'California Institt,.te of Technology, sponsored by the National Aeronautics and Space Administradon unth_ Contract NAS7.100. G7 Copy IVo............... https://ntrs.nasa.gov/search.jsp?R=19660006519 2020-03-16T21:36:56+00:00Z ABSTRACT In the total b(t) waveform of a ferrite core switched by mmf, F(t), from negatfve remanence, ¢ = -¢r, three components are distinguished: elastic, decaying inelastic, and main inelastic. The model for the elastic component, be, is b_ = £F, where e is a function of F, _, and the previous switching. The model for the decaying inelastic component, _, is bi = ki(F - F_) _ exp[-(t - Ti)(F - Fi)/Ci] , where ki, Fi, _, and C_ are switching parameters. The model for the main inelastic component, _.a' is b.a = bp{1 - [(2¢ + Cr -- Cd)/(_r + Cd )]2}' where bp is the peak value of b and Cd is the _ value on the static _(F) curve, both of which are given as functions of F. If F is large (compared with the coercive mmf, Fc), then b i + b.. % bp{1 - [(2¢ + _s - Cd)/(¢s + Ca )]2), where Cs is saturation flux. A computer program is written and applied in com- puting b(t) and its components for a thin toroidal ferrite core. The results agree well with experimental _(t) waveforms obtained by using F(t) with different rise times (T _ 0.1 #sec and T _ 0.02 _sec) and different amplitudes (seven to eight values, varyfng from 2/3 tc more than twice F ). A small delay (of the order of 0.15 T ) between computed and experimental _e(t) shows that an improved model for the elastic is gbe + b e = cF, where _ is a constant proportional to the viscous damping. The parameters Li and C i were found to be affected by T : as T r decreases, k i increases (slightly) and C i decreases. The flux change involved in obtaining the static ¢(F) curve for low F values is much larger than f_¢idt; this flux change is ascribed to a very slow switching te component which is either part of bi or b.. with threshold lower than F 0. Numerical analyses and computer programs are given for three additional magnetic circuits: a loaded core, a core-diode-transistor binary counter, and a loaded, saturable, three-leg core. In each case, the basic problem is to solve a set of first-order nonlinear differential equations together with a transcendental solution for some of the time variables. If the load is inductive, the loaded-core program provides more exact results that agree better with experimental data than those obtained previously. The iii binary-counter program includes computation vs. time of three currents and F, 6, and _b of each of two coupled cores during the fast-switching mode of operation of a single stage in the counter. The computed results are essentially identical with results computed by a more complex and more exact method of solution of differential equations (Runge-Kutta and Adams) and agree quite well with experimental data. The last program in- cludes three types of computation for flux division: time variables (F, 6, and _b of each leg), flux-division ratio D vs. drive amplitude for different loads, and D vs. leg-length ratio for different loads and drive amplitude. The agreement between computed and measured D vs. N[ is satis- factory except for very low values of drive mmf. The switching properties of a thin-ring core in response to a ramp drive, F = kt, have been investigated for k varying from0.1 to 10 amp-turn/_sec. Experimental _p(k) and tp(k) curves are compared with computed curves based on the parabolic model _p = L(F-Fo)V [1 - (2_+_, --(pd)2/(_# + _d)2]. The experimental _p(k) curve fell entirely below the computed curve when step-F parameters were used in the computation. The computed _p(k) curve could be made to agree with the experimental curve over the entire range of k by using a lower value of k, k r. The computed and experimental tp(k) curves could be made to agree at any k value by using a lower value of F0," F"0r. However, exact agreement could be obtained at only one k value because these t (k) curves cross each other. In spite of this problem, p the above model can be used for practical applications if k does not vary over a wide range. A study was made of the effect of temperature (in the range -50°C to +75°C) on the switching properties of two square-loop ferrite toroids (a thin ring and a thick ring). Temperature coefficients have been found for the parameters of the model described above. The static q6(F) curves for a partially set state changed with temperature by approximately the same percentage as the major static qb(F) curves. As the temperature in- creased, the _v(F) curves for a step-F drive shifted to lower F values but their slope and curvature were hardly affected. The _v(k) curves for a ramp-F drive for different temperatures cross each other. The tp(k) curves were shifted downward by an increase in temperature, but were unaffected in slope and curvature. These effects are mostly due to tem- tt perature variations in FOr and k r. iv CONTENTS ABSTBACT ................................... LIST OF ILLUSTRATIONS ............................ LIST OF TABLES ................................ LIST OF SYMBOLS ............................... ACKNOWLEDGMENTS ............................... PREFACE ................................... I ELASTIC AND INELASTIC FLUX-SWITCHING COMPONENTS ............. A. Introduction ............................ B. Flux-Switching Models ........................ 1. Elastic ¢ Spike ......................... 2. Inelastic Decaying ¢ ...................... 3. Inelastic Main ¢ ........................ 4. Discussion ........................... 5. Summary ............................. C. Computer Programs for ¢ Components ................. 1. Computation of b e and ¢' 2. Computation of _i and _i °l 3. Computation of ¢=Q and _=a ................... 4. Summary ............................. D. Experimental Verification ...................... i. Experiment ........................... 2. Computation ........................... 3. Results ............................. 4. Discussion ........................... 5. Summary ............................. E. Conclusions ............................. II OOMPUTATION OF FLUX SWITCHING IN MAGNETIC CIBCUITS ........... A. Loaded Core ............................. 1. Introduction .......................... 2. Inductive Load ......................... 3. Noninductive Load ........................ 4. Experimental and Computed ¢(t) and iL(t) of Inductively Loaded Core 5. Summary ............................. B. Core-Diode-Transistor Binary Counter ................ i. Operation ............................ 2. Analysis of Mode I ....................... iii vii ix xi xix xxi 1 1 4 4 6 11 12 15 15 16 17 17 18 18 19 22 26 26 41 42 45 45 45 47 47 48 49 49 49 56 CONTENTS 3. Transcendental Solution of Currents ................ 4. Computation Using a Simple Method ................. 5. Computation Using Bunge-Kutta and Adams Methods .......... 6. Discussion ............................ 7. Summary .............................. C. Flux Division in a Loaded Saturable Core ............... lo 2. 3. 4. 5. 6. 7. D. Introduction ........................... Experiment ............................. Computational Analysis ...................... Computer Program ......................... Experimental Verification ..................... Discussion ............................ Summary .............................. Conclusions .............................. 58 61 66 68 72 72 72 73 74 77 8l 86 88 89 III VABIATIONS OF FLUX-SWITCHING PABAMETEBS ..................... 91 A. Bamp-F Switching ........................... 91 1. Introduction ........................... 91 2. Algebraic Calculation of.¢p and tp ................ 92 3. Numerical Computation of Cp and tp ................. 96 4. Experiment . ........................... 99 5. Parameter Corrections ....................... 102 6. Summary .............................. 106 B. Effects of Temperature ......................... 106 1. Introduction ........................... 106 2. Static ¢(F) ............................ 107 3. Step-F Switching ......................... 118 4. Bamp-F Switching ......................... 123 5. Summary .............................. 130 C. Conclusions .............................. 130 APPENDIX A ................................... 133 APPENDIX B ................................... 135 APPENDIX C ................................... 137 APPENDIX D ................................... 139 APPENDIX E ................................... 141 APPENDIX F ................................... 147 APPENDIX G ................................... 153 APPENDIX H ................................... 159 APPENDIX I ................................... 167 BEFEBENCES ................................... 177 INDEX ..................................... 179 vi ILLUSTRATIONS Fig. 1 Fig. 2 Fig. 3 Fig. 4 Fig. 5 Fig. 6 Fig. 7 Fig. 8 Fig. 9 Fig. 10 Fig. i! Fig. 12 Fig. 13 Fig. 14 Fig. 15 Fig. 16 Fig. 17 Fig. 18 Fig. 19 Fig. 20 Fig. 21 Fig. 22 $(t) Oscillogram of Interrupted-F,,Switching in a Thin Toroidal Ferrite Core Using F > F 0 ....... : .............. 1 F(t) and _ v _ O_ci 1 lo_rams of Interrupted-F,Switching in a Thin_Toroidal Ferrite Core Using f < F 0 ................. 3 Effects of F, qb, and Previous Switching on Elastic- Switching Coefficient _ ........................... 5 Effect of T r on q3i(t) Waveform for a Given F D ................ 9 Approximate Variation of F i with f ...................... 11 $(_b) Oscillograms of Step-F Switching of a Thin Ferrite Ring ......... 13 Resemblance Between $(t) Waveforms Derived from Two Models for Inelastic qb ............................... 14 Ringing in $(t) for F(t) with a Short Rise Time ............... 19 Approximate Drive-Current Function ....................... 23 Experimental (Solid Line) and Computed (Dashed Line) F(t) and qb(t) Waveforms of Unloaded Core E-6 During Beginning of Switching, Using F(t) with T r of About 0.1 /zsec .............. 27 Experimental (Solid Line) and Computed (Dashed Line) F(t) and $(t) Waveforms of UnIoaded Core E-6 During Beginning of Switching, Using F(t) with T r of About 0.02 /zsec ............. 31 Multiple Exposure of $(t) Waveforms of Cure E-6 Corresponding to F(t) Pulses of Different Amplitudes .................... 37 Computed $_ _. Sma and $ vs. t of Core E-6 During the Beginning of S_itching .......................... 38 Computed Sip us. FD of Core E-6 with Tr as a Parameter ............ 40 Experimental and Computed _(t) and iL(t) Waveforms of Core J-1 whxch is Inductively Loaded and Switched by Step-F Drive ........... 50 Experimental and Computed _(t) and iL(t) Waveforms of Core J-1 whxch is Inductively Loaded and Switched by Remp-F Drive ........... 51 A Single Stage of a Core-Diode-Transistor Binary Counter ............ 52 Incomplete Flux Switching in Four Modes of Operation of a Core-Diode-Transistor Binary Counter .................. 54 Equivalent Circuit for Mode I of a Single Stage in a Core-Diode-Transistor Binary Counter ..................... 57 Experimental and Computed Current and Voltage Waveforms in Mode I of a Core-Diode-Transistor Binary Counter .... for V = 27 Volts ............................... 65 Experimental and C_mputed Current and Voltage Waveforms in Mode I of a Core-Diode-Transistor Binary Counter for V = 8.6 Volts .............................. 66 Superpositipn of _(t) Waveforms Computed by a Simple Method and _(t) Waveforms Computed by Runge-Kutta and Adams Methods .............................. 69 vii ILLUSTRATIONS Fig. 23 Fig. 24 Fig. 25 Fig. 26 Fig. 27 Fig. 28 Fig. 29 Fig. 30 Fig. 31 Fig. 32 Fig. 33 Fig. 34 Fig. 35 Fig. 36 Fig. 37 Fig. 38 Fig. 39 Fig. 40 Fig. 41 Fig. 42 Fig. 43 Fig. 44 Fig. 45 Fig. 46 Fig. 47 Fig. 48 Fig. 49 Fig. 50 Fig. 51 Fig. 52 Fig. 53 Fig. 54 Fig. 55 Variations of ¢1 vs. F.1, and _b2 vs. F2 During Mode I of a Core-Diode-Translstor Binary Counter .................. Flux Division in a Loaded Saturable Three-Leg Core ............. Experimental (Solid Line) and Computed (Dotted Line) Ni(t) and q_(t) Waveforms of Core S (14/l 3 = 1.83), Driven Unloaded by MMF NI of 2.0 amp-turns in a Flux-Division Experiment .......................... Measured and Computed Flux-Division Ratio vs. MMF with Load as a Parameter for Core S (14/l 3 = 1.83) ............. Computed Plots of Flux-Division Ratio vs. Leg-Length Ratio with Load as a Parameter and NI of 10 and 100 amp-turns for a Core Identical with Core S Except for the Length of Leg 4 .................................. _p(k) for Ramp F(t) kt (k) for Ramp F(t) p (k) for Ramp F(t) using _r and FOr .................... tl H (ktp - F O) vs. k for Ramp F using)t r and F 0 ................. Definitions of Measured Quantities for Ramp-F Switching ........... Experimental vs. Computed ¢(t) for Ramp F with )t and F,' Adjusted to Make _p and tp Agree ...... . .r . . . ur .......... (ktp - F" ................Or ) vs k for Ramp F using )k and F"• r Or Static ¢(F) Curves vs. Temperature of Core I-4 ............... Static qb(F) Curves vs. Temperature of Core K-1 ............... Static ¢(F) Parameters vs. Temperature for Core I-4 ............. Static ¢(F) Parameters vs. Temperature for Core K-1 ............. Wing Sharpness of Static ¢(F) Curves vs. Temperature ............ Static ¢(+F) vs. Temperature of Partially-Set Core I-4 ........... Static ¢(-F) vs. Temperature of Partially-Set Core I-4 ........... Static ¢(+F) vs. Temperature of Partially-Set Core K-1 ........... Static ¢(-F) vs. Temperature of Partially-Set Core K-1 ........... Major and Partially-Set Static ¢(F) of Core I-4 at 75 ° and 0°C ....... Major and Partially-Set Static ¢(F) of Core K-1 at 75 ° and -50°C ...... qb (F) vs. Temperature of Core I-4 ...................... p Cp(F) vs. Temperature of Core K-1 (F) Parameters vs. Temperature for Core I-4 ................ p _p(F) Parameters vs. Temperature for Core K-1 ................ (k) Curves of Core I-4 vs. Temperature with Ramp F(t) ........... p _p(k) Curves of Core K-1 vs. Temperature with Ramp F(t) to(k) Curves of Core I-4 vs. Temperature with Ramp F(t) ........... tp(k) Curves of Core K-1 vs. Temperature with Ramp F(t) ........... Variations of 2t r andX with Temperature for Core I-4 ............ Variations of)t r and)t with Temperature for Core K-1 ............ 71 73 83 84 85 95 96 97 98 100 101 104 109 110 112 113 114 115 115 116 116 117 118 120 121 122 123 124 125 126 127 129 129 viii TABLES Table I Table II Leg Dimension and Switching Parameters of Core S ............ Dimensions, Switching Parameters and Temperature Coefficients of Cores I-4 and K-1 ................... 82 108 ix LIST OF SYMBOLS Symbo_______lll A B B r B, b C C C i E k F F 8 F c F D FDB End F ex F_ f o Definition Cross-sectional area of a leg Flux density Maximum residual B Saturation B Time rate of change of B Decaying inelastic Capacitance tanh -1 (_r/_s) in derivation of @p(k) and tp(k) for ramp-F switching Parameter in a model for decaying .o inelastic Parameter in a model for a junction diode MMF MMF at boundary between nonlinear and linear regions: of _p(F) Coercive F Amplitude of drive MMF Boundary value for F D below _hich Tip = T r Static F threshold Excess F above threshold MMF threshold for decaying inelastic _i MMF threshold obtained by extrapo- lating linear Cp(F) to F axis xi Reference Rep. 1, Fig..19 Rep. 2, Fig. 2 Eq. (13), p. 10 Eq. (101), p. 93; Eq. (102), p. 94 Eq. (5), p. 7 Eq. (42), p. 46 Rep. 3, Fig. 3 Rep. 3, Fig. 1 Rep. 3, Fig. 6 Eq. (12), p. 9 Rep. 3, Fig. 1 Eq. (17), p. 10 Rep. 3, Fig. 3; Eq. (19), p. 11 SYMBOLS Symbol tl Fo tl Fo r Foi F12 F23 F H H a H B H ¢ H i [ H H q H o tt H o IC Definition Dynamic F threshold tt F 0 of ramp-F switching Asymptotic value for F i Boundary between first and second regions of computed static _(F) curve, i.e., Hql i Boundary between second and third regions of computed static qb(F)curve, i.e.,Hl q o Time rate of change of F Magnetic field Parameter in the hyperbolic model for static B(H) H at boundary between linear and non- linear regions of Bp(H) Coercive H H threshold for decaying inelastic Parameter in the hyperbolic model for static B(H) Parameter in the hyperbolic model for static B(H) H threshold, obtained by extrapolating linear B (H) to H axis P Dynamic H threshold Average H Height of a leg (or core) Drive-current amplitude in flux- division experiment Reference Rep. 3, Fig. 3; Eq. (19), p. i] p. 96; Eq. (112), p. i02 Fig. 5, p. ii Rep. 2, pp. 6 and 28 Bep. 2, pp. 6 and 28 Rep. 2, Eq. (10) and Fig. 2 Rep. 2, Fig. 15 Rep. 2, Eq. (14 Eq. (15), p. 10 Rep. 2, Eq. (11 and Fig. 2 and Fig. 2 Rep. 2, Eq. (11 and Fig. 2 Rep. 2, Eq. (24) and Fig. 15 Bep. 2, Eq. (83) and Fig. 15 Bep. i, Fig. 19 Fig. 24, p. 73 Asymptotic value of collector current Fig. 17, p. 52 in binary counter xii SYgBOLS Symbol ID I D I o i C i D i d i L i L i S i sr k L l l. t l 0 M i N N C Definition Inside diameter of a toroid Amplitude of drive current Initial value of i in a binary $ counter Saturation current in a model for a junction diode Instantaneous drive current in flux- division experiment Collector current in a binary counter Drive current Loop current via diode in a binary counter Current source in a binary counter Load current Drive current in a binary counter during STORE count Drive current in a binary counter during CARRY count Index of iteration in a numerical solution of variables for each nth At, e.s., _(i)' _._J)' q._J)" q.(i)' n(i-1)' etc. Slope of r:amp F(t) Inductance Average leg length Length of the short edge of a leg Length of the long edge of a leg Material parameter in a model for decaying inelastic B Number of turns N of load winding Reference Rep. 3, Fig. 6 Fig. 17, p. 52; Eq. (72), p. 62 Eq. (42), p. 46 Eq. (93), p. 77 Fig. 17, p. 52 Rep. 3, Fig. 14 Fig. 19, p. 57 Fig. 19, p. 57 Rep. 3, Fig. 14 Fig. 17, p. 52; Fig. 19, p. 57 Fig. 17, p. 52 Eq. (106), p. 99; Fig. 32, p. 100 Rep. 3, Fig. 5 Rep. 3, Fig. 5 Rep. 3, Fig. 5 Eq. (16), p. 10 Rep. 3, Fig. 14 xiii Symbol N D NL NV n OD P q R R d RL r i F o S p T r i T tp T n r TrB SYMBOLS Definition N of drive winding N of load winding in flux-division experiment Negligible Value of _E Index number for a time element At and the associated variables, e.g., _n' _n' etc. Outside diameter of a toroid Ratio of computed _ [using static _(F)-limited _arabo_ic model] and experimental _p for ramp-F drive Electric charge Load current, i L = dq/dt Resistance Forward resistance of a diode Load resistance Inside radius of a toroid Outside radius of a toroid Maximum slope of drive current Duration of flux switching in flux- division experiment Time when _i begins to rise when _i reaches peakTime Abbreviation for (13 - t" 10 8 ) in i L function in binary counter Rise time of a drive pulse Boundary value for T r below which Tip = T r Reference Rep. 3, Fig. 14 Fig. 24, p. 73 Eq. 108), p. 102 Eq. 43), p. 46 Eq. 43), p. 46 Eq. (42), p. 46 Bep. 3, Fig. 14 Fig. 9, p. 23 Fig. 24, p. 73 Fig. 4, p. 9 Fig. 4, p. 9 Eq. (74), p. 64 Fig. 9, p. 23 Eq. (10), p. 8; Fig. 4, p. 9 xiv Symbol t t= tp U V Vd V 2 VR i-L W Y &t /_t SY_OLS Definition Time Time when drive current reaches maximum slope Peak time of Drive-current parameter Supply voltage in a binary counter Forward voltage across a junction diode Minimum allowed V in binary counter Abbreviation for (¢, - ¢,)l[(to - Zi)no] Abbreviation for [(¢s + Cr)Hq]/[( lo - li)Hn] Voltage across R 1 and L in binary counter Width of a leg General time variable Time rate of change of y Small time increment used in compu- tation of time variables The time interval used to measure the slope of a ramp-F drive Change of ¢ /375 associated with obtaining a point on static _(F) curve, i.e., _d + _r a¢ due to ¢i A¢ i for t _ m Elastic Zh¢ Reference Fig. 9, p. 23 Rep. 3, Fig. 1 Fig. 9, p.23; Eq. (34), p. 24 Fig. 17, p. 52 Eq. [56(b)], p. 58 p. 55 Rep. 3, Eq. (28) Rep. 3, Eq. (29) Eq. (49), p. 56 e.g., Fig. 24, p. 73 Eqs. (70) and (71), p. 61 Eqs. (70) and (71), p. 61 e.g., Eq. (29), p. 22 Fig. 32, p. 100 Eq. (41), p. 41 Eq. (38), p. 39 Eq. (39), p. 39 Eq. (37), p. 39 X¥ SYMBOLS Symbol E K K . I L L r I) P Pi Pp P 7- $ ¢ Definition A_b for t _ _ Constant proportional t9 viscous damping in a model for CE Reference Eq. (40), p. 39 Eq. (35), p. 35 Correction to i in iterative solution p. 63 for i Coefficient of elastic Coefficient of inelastic Peak Flux form factor in parabolic model for main ¢ Coefficient of main inelastic B in nonlinear B (H) region P Coefficient of decaying inelastic Bi Coefficient of main inelastic $ in nonlinear _p(F) region Coefficient of decaying inelastic $i k for ramp-F switching Power coefficient of main inelastic in nonlinear _v(F) region Power coefficient of decaying inelastic _i Coefficient of inelastic Coefficient of decaying inelastic $i Peak p Average p Switching time Magnetic flux Eqs. (2) and (3),p.4; Fig. 3, p. 5 Rep. 2, Eq. (24) p. 76; Rep. 2, Eq. (82) Eq. (20), p. 11 p. 76; Rep. 2, p. 38 Eq. (14), p. 10 Eq. (19), p. 11 Eq. (5), p. 7 p. 96; Eq. (109), p. 102 Eq. (19), p. 11 Eq. (5), p. 7 Rep. 1, Eq. (38) Eq. (.4), p. 6 Eq. (19), p. 11 Rep. 2, pp. 17-19 Eq. (90), p. 76 xvi SYMBOLS Symbol _d _r # Cd q6ip qb_a % % & Definition value on a static _(F) curve Maximum residual Saturation d%/dF Time rate of change of Decaying inelastic _ component Peak _i o Main inelastic _ component Peak Elastic component of _/OF _,/_F _. o/bE d$_/dr _. o/_ Reference Rep. 3, Fig. 1 Rep. 3, Fig. 1 Rep. 3, Eq. (31), (32), or (35) p. 6 Eq. (11), p. 8; Fig. 4, p. 9 p. 11 Eq. (19), p. 11 p. 4 p. 16 Eq. (27), p. 17 p. 17 Rep. 3, Eq. (37) or (39) Eq. (25), p. 16 Eq. (28), p. 18 xvii ACKNOWLEDGMENTS The assistance of D. F. Fraser in computer programming is appreciated. We wish to thank Alan J. DeYilbiss for his comments relative to the analysis of the core-diode- transistor binary counter. xix PREFACE This project report, Contract 950943 under NAS7-100, Stanford Research Institute Project 5094, is an extension of work under a previous project, Contract 950095 under NASw-6, SRI Project 3696. From an engineering viewpoint, the modeling of the terminal proper- ties of ferrimagnetic or ferromagnetic core materials is useful. There are different ways to study these terminal properties. One extreme way is to study each property inas many core materials as possible before pro- ceeding with the next property. Another extreme way is to study all the properties of one given core material before proceeding to the next core material. The present investigation lies between the extremes, although it is closer to the latter. In the past, we have studied the major characteristics of switching from a hard state (_ = --_r ) and from a cer- tain type of soft state (1 1 < Cr) in a limited number of ferrite core materials. Among the materials studied, two magnesium-manganese ferrite materials have been investigated relatively thoroughly. This report describes the initial elastic and inelastic _ spikes of one core material, the properties of ramp-F switching in three core materials, and the tem- perature effect on step-F switching, ramp-F switching and static _(F) curves in two core materials. Additional properties of these materials need to be investigated, but other materials also need investigation. In Report 3, the switching model was used in computational analyses* of magnetic circuits (unloaded core, loaded core, and core-diode shift register). The agreement with experimental data has encouraged us to extend this application to other magnetic circuits. Consequently, in this report, computational analyses and experimental verification are given for the initial _(t) spikes of an unloaded core, for a loaded core (using an improved algorithm), for a core-diode-transistor binary counter, and for flux division in a loaded saturable three-leg core. By "computational analysis" we mean a numerical analysis which is programmed and run on a digital computer. xxi I ELASTIC AND INELASTIC FLUX-SWITCHING COMPONENTS A. INTRODUCTION In this section, improved switching models are proposed and verified experimentally for three components of _: the elastic _ spike, be, occur- ring while F changes in time; the decaying inelastic ¢ components, _t, which falls exponentially after reaching peak during the rise of F; and the main inelastic _ component, _,_, which is bell-shaped and, if F is not too low, accounts for most of the flux switching. The first and third components are well known, and were discussed in detail and applied in the previous three reports 1'2'3. (hereinafter referred to as Reports 1, 2, and 3). The decaying _ component, _i' was introduced in Report 3 (using the symbol _pi instead of ¢i), where an attempt was made to model this component. An additional experimental verification for the existence of ¢i is given in Fig. 1 by showing _(t) waveforms of a thin toroidal ferrite core (Core E-6, Report 3, p. 23). These _(t) waveforms resulted from the A__ O.OI v/turn 7- -_ _-- I vsec TA-'?415-1 FIG. I ¢(t) OSCILLOGRAM OF INTERRUPTED-F SWITCHING t! IN A THIN TOROIDAL FERRITE CORE USING F > F 0 Core E-6: OD/ID = 1.06; F'_= 0.95 amp-turn; Fc = 0.9 amp-turn; F = 1.17 amp-turn. * References are listed at the end of the report. 1 interruption of a step-F switching with low F (F = 1.3 F c, where F c is tile coercive mmf). The three fast-switching _ spikes (the second of which is negative) |lave been retouched because tile original traces were too faint for photographic reproduction. The elastic _ spikes shown occur during the rise and fall of the first F pulse and during the rise of the second F pulse. The two F pulses have the same amplitude. During the first F pulse and in the beginning of the second F pulse, the total c_(t) is decaying despite the rise of the main c_(t) component. The differ- ence between the total qb(t) and the main c_(t) component is the decaying component, <_i" Our conclusion that $i is due to inelastic domain-wall motion (cf. Report 3, p. 12) is verified by the following three obser- vations: 1) The relaxation time of _i is much longer than typical relaxation time of rotation of magnetization. 4 2) Referring to the first F pulse, the area under _i (following the rise time) is much larger than the area under the negative decaying _ component (following the fall time). The latter is elastic in nature, and presumably results from the excess of the number of walls moving backward over the number of walls moving forward toward energy minima when F is suddenly interrupted. This excess in number of wails is due to the slower average velocity of the walls passing between energy-valley and energy-peak positions compared with walls moving between energy-peak and energy-valley positions. 3) The _i waveform that follows the positive _ spike of the second F pulse continues to decay smoothly from its value at the end of the first F pulse. This behavior is characteristic only of domain-wall motion. Experimental F(t) and _(t) oscillograms of interrupted-F switching are shown in Fig. 2 for the same conditions as in Fig. 1, except that It H F = 0.8 amp-turn. Since F < F 0 (F 0 = 0.95 amp-turn), the main @(t) com- ponent is not present, and _5(t) = _5(t) + &i(t)- Flux-switching models have been proposed for _e, _i, and &,. in Report 3 (the subscript ma is not added in Reports 1, 2, and 3). In tile course of further investigation of _i' it was found that certain modifi- cations should be incorporated into the models for both _i and _a in order to obtain better agreement with experimental data. These modified models are given next. For completeness, we shall also summarize briefly the model for _bE. L! m 0._ __u_i_ F1 /I ,5 millivolts/turn 0.2/J.se¢ FIG. 2 TA-S094-1 F(t) AND _(t) OSClLLOGRAMS OF INTERRUPTED-F SWITCHING i! IN A THIN TOROIDAI_ FERRITE CORE USING F < F0 Core E-6: OD/ID = 1.06; F(_' = 0.95 amp-turn; Fc= 0.9 amp-turn; F = 0.8 amp-turn. 3 B. FLUX-SWITCHING MODELS Consider a thin core (or leg) which is driven by F(t). Based on the- discussion above, the total @ is expressed as Semiempirical models for _e' _i' and _,a are reviewed and modified as follows. 1. ELASTIC _ SPIKE Following Eq. (32) in Report i (p. 23), _E = _F , (2) where c is a function of F, ¢, and the history of previous switching. For a core in saturation, e(F) is given by Eq. (34), Report 1, and is plotted in Figs. 26 and 27 of Report 1 [for convenience, it is replotted in Fig. 3(a)]. For _ = -¢r and if IFI is not much larger than Fc, then for a toroidal core [cf. Eq. (7), Report 3, p. 10], r° (3) = 27-r(ro - ri)Ha In where Cs is saturation flux, Cr is maximum residual flux, H a is a material parameter, and r ° and r i are outside and inside radii of the toroid. The effects of _ and the previous switching (to reach _) on _ are shown schematically in Fig. 3(b). For I¢] > ¢_, _(¢) may be derived from e(F), shown in Fig. 3(a), and the static ¢(F) curve (not shown). For -¢_ _ _ _ ¢_, e(¢) peaks near ¢ = 0; for a given _ value, the faster the previous switching from -¢: to _ is, the higher is e. 5 More investigation in the area of flux switching from a partially-set state is needed before we can propose a model for _(¢, switching history). &IE 0 (,a) (vs. F IN NEGATIVE SATURATION F -_]m,. MEDIUM SLOW _PREVIOUS SWITCHING FROM -_r TO -_r O _r (b)E vs. (_ WITH PREVIOUS SWITCHING AS A PARAMETER TA-5094-2 FIG. 3 EFFECTS OF F, ¢_, AND PREVIOUS SWITCHING ON ELASTIC-SWITCHING COEFFICIENT S 2. INELASTIC DECAYING a. INTRODUCTION In I_eport 3, Eq. (8) (p.ll), a semiempirical model was propos,'d for $i resulting from a drive of constant amplitude, F o, and rise tim(-" "['r" According to this model, . -(t-r )(ro-_)/c _ (4) _i = pi(F D _ Fd)e r where Pi is switching resistance per turn squared, F_ is threshold, and C is a constant of proportionality of the decay time constant. Equation (4) was based [cf. Report 3, pp. 11-12] on the hypothesis that _ results from the motions of those domain walls that do not collide with each other, at least in the beginning of switching, each terminating when the wall is obstructed by an inhomogeneity (energy hill of a high slope). The decay- ing waveform of _i stems from the random distributions of the distance to the obstructing hill and the average slope of nonobstructing hills. The higher is the excess of F over a threshold F_, the faster the motions of these walls are. Hence, _i is proportional and the decay time constant is inversely proportional to (F - F_). b. LIMITATIONS OF THE PREVIOUS MODEL The limitations of the model for _i in Report 3 (pp. 11-12) are as follows: (1) The model does not account for the rise of $i from zero to its peak value at t = T (while f rises from F i to, say, 95 percent of Fo). In evaluating the sum _ + _i in Fig. ¢ of Report 3, _i is neglected during t < T , whereas_E is neglected from t = T until F = F 0. (2) The model is not valid if the rise time, T , is rela- tively long. (3) The model is applicable only if the drive mmf, F, is of constant amplitude, F D. (4) Further investigation of a given ferrite material has shown that if Fo/F _ is not much larger than unity, then _i is proportional not to F D - F_ but rather to (Fo - F_)Vii , where F i < F_ and v i > 1. This behavior is similar to that of the main _, and may be attributed to the increase in the number of nucleation centers with the excess mmf. 6 6 C. MODIFIED MODEL Let T i be the time at which F reaches the value F t . The above- mentioned drawbacks are overcome if the model of Eq. (4) is modified to the following expression (Mlichis valid if F _ F i and thus for t _ TL): . -( t-T i ) (F-F i ) /C i$_ = Li(F - Fi )ute (5) Here, Li is a constant of proportionality [replacing p_ in Eq. (4), in analogy with the expressions _(F - F0 )_ and pp(F - F o) for _p in the model for the main 6, Report 3, Eq. (2)] and F has an arbitrary waveform with an arbitrary rise time, T r. From t = 0 to t = Ti, ¢i = 0. During T i < t T B, then the expression given by Eq. (9) is larger than r r . Since for t > rr, _i(t) in Eq. [(8(b)] decays exponen- tially, _i(t) reaches a discontinuous peak at t = T, and so Tip = T. We may thus conclude that, for a given F o value, if T > T B, Eq. (10), then Tip < r r and Eq. (9) is valid; but if T <_ rrB, then Tip = T r. The two cases, designated by Subscripts (1) and (2), are shown in Fig. 4. In both cases _(t) is discontinuous at t = r r, but in Case (2) the discontinuity and the peak of _Pi(t) coincide. In each case, the peak value of _i' _ip' may be determined by substituting t = T_p into Eq. [8(a)]. Thus, ¢- (CizJiFD_ viI2 ki_':e;-\z 1 ] if T r _ TrB [Case (1)] u. T (FD-F i )2/(FoC i ) [Case (2)] ki(F o - F i ) 'e r if T r _ T B (11) We conclude from Eqs. (7) through (11) that, for a given value of F a, the larger T r is, the larger are T i and Tip and the smaller is _ip" In the limit of Case (1), as Tr _ _, _i _ O. In the limit of Case (2), as r r _ O, @ip _ Xi(FD - Fi )vi', hence, _i rises instantaneously to _ip at t = 0 and decays exponentialIy thereafter in accordance with Eq. [8(b)] in which T i = O. F0 F2 ¸ /SF/ -_i, Eq.(So) _" 0 0 Tt(2) Ti(I) T|p(2)=Tr(2) TrB T_p(i ) Tr(i) t -----),- T&--5094-3 FIG. 4 EFFECT OF T r ON (_(t) WAVEFORM FOR A GIVEN F D A similar analysis may be carried out for a given value of T r and a variable Fo: From Eq. (9) we find the value FOB = F i + + F, + such that if F 0 _ FDB, then Tip _ T r, but if F 9 _ FBB, then Tip = T r. e. EFFECTS OF GEOMETRY ON SWITCHING PARAMETERS In analogy with the material inelastic-switching properties of the main _ of a thin core (or leg) of cross-sectional area A and average length, l (cf. Report 2, pp. 8, 37-40), Eq. (5) may be converted from _i(F, t) into Bi(H,t), where B i = Ppi/A and H = F/l. Thus, in analogy tl = K(H - H0)" of Eq. (83) in Report 2with the relation Bp B i = Ki(H --Hi)ui- e , (13) where K i, H i , and M i are material parameters (H reaches the threshold value H i at t = Ti). Following the geometrical relations expressed by Eqs. (91) and (92) of Report 2 (p. 40), 12 . k i = KiA/l _ (14) and F i = Hil (15) Assuming that the decay time constant is determined solely by the material- switching behavior, C i = Mil (16) Equations (14) through (16) are useful in calculation of _i of legs (or cores) of the same material, but of different geometry, e.g., in calcula- tion of flux division in a saturable three-leg core, as we shall see later. f. VARIATION OF F i We have presumed that _i is generated by the motion of noncol- liding wails which are finally obstructed by randomly distributed centers of imperfections (energy hills of high slopes). In line with our hypo- thesis that the average slope of the nonobstructing hills is randomly distributed, the slope of the first hill is also randomly distributed. As a result, we may expect the threshold F i to increase from near zero to some finite asympotic value, Foi. Physically, this means that a small percentage of the wails are initially at a barely stable state (i.e., a very small applied F will displace them inelastieally) and that most of the wails will break free if F Z Foi" A plot of F i vs. F is shown in Fig. 5. As an approximation, F i vs. F may be described by a tanh function, i.e., F i = Foi tanh (F/Foi) (17) 10 I Foi Fi O 7 //_'J" i _-- Foi tanh (F/Foi) I I I I I I 0 Foi ZFOi I I F ,,_.]_,. 3Foi FIG. 5 APPROXIMATE VARIATION OF F i WITH F I I 4Foi TA-50c3A-4 Note that the slope of Fi(F) at F = 0 is unity, as expected from the dis- cussion above. Thus, for F values of F0i, 2Foi , and 3F0i , F t reaches 76.16 percent, 96.4 percent and 99.5 percent of F0i, respectively. With F defined as in Eq. (17), T i = 0 because F = F i at t = 0. The relation between the value of F0i and the values of other threshold parameters will be discussed later (p. 36). 3. INELASTIC MAIN a. INTRODUCTION A model for the main _ was proposed and applied in Report 3. Following Eqs. (1), (2) and (4) of Report 3, where and ¢.,, = _,__ , (18) 0 . if F < F 0 it _Sp = L(F - f0) v if f o < F < F B (19) pp (F - F o) if Fn < F 2¢ + q5 - _d) 2 (20) 11 aud where _d is the _ value on the static _(F) curve. Based on experi- mental verification in Report 3, we concluded [cf. Report 3, p. 33] that the above model is satisfactory, except if F is low, i.e., around the coercive mmf or lower. b. MODIFIED _(_) Typical @(4) oscillogram of step-F switching in a thin ferrite core (Core E-6, Beport 3, p. 23) are shown in Fig. 6 for three amplitude values: F D 1.2, 1.4, and 1.8 amp-turns. In the beginning of switching, as @ rises by a small _q5 above --_r' $(_) is due primarily to _e and $i" Beyond this region of @, $(4) is due primarily to @i and _ . As _ in- creases, the contribution of $i to the total _ diminishes to a negligible amount, and $(4) is essentially $,°(4). An extrapolation of $,°(4) to the axis is dash-lined in each case of Fig. 6. Each $,o(_) extrapolation intersects the _ axis at _ y -_, rather than _ _ -@ . As a result of this observation, we shall modify N(@) of Eq. (20) by replacing _ by _, i.e., .24 + 4r -- 4e) _ (21) Note that $ is the total flux, i.e., obtained by integration of the three components of $, Eq. (i). 4. DIscussioN The experimental data given in Fig. 6 were known before the previous D(_) function, Eq. (20), was proposed. One may then ask why we chose _(_) given by Eq. (20) instead of N(_) given by Eq. (21). The answer lies in the.solution to the differential equation of $_o in the absence of the de- caying component, $i" In order to simplify the explanation of this point, let us assume an ideal step-F switching (F rises to FD in a zero rise time for which $ , Eq. (19), is constant. We shall first neglect the elastic P $ component, Se" If _ is identified with the main component, _ (i.e., the flux due to time integration of $_ only), then the resulting solution of the differential equation expressed by Eq. (18) in which N(_) is given by Eq. (21) is absurd: The initial value of $ is zero and the switching nime required to change _ from --_r to zero is infinite [cf. Report 1, Eq. (40) and Fig. 30, pp. 27-28]. In order to overcome this difficulty, 12 (a) FD --1.2 amp-turn (b) Fo: 1.4omp-turn (c) Fo : 1.8 amp-turn FIG. 6 _(_) OSCILLOGRhJ_S OF STEP-F SWITCHING OF A THIN Drive: 0.05 #sec rise time; variable amplitude F D. Core: E-6; OD/ID -- 1.06; F c = 0.9. amp-turn. Added dashed lines are extrapolated q_.mavs. _. Scale = 1.04 maxwell/major div.; _ scale: (a) 3.3 mv/turn, (b) 6.9 mv/turn, (c) 13.8 mv/turn. TA-S094-5 FERRITE RING 13 we approximated _(¢) by the expression given in Eq. (20). However, this difficulty in obtaining the proper rate of increase of _ from -¢r does not. exist with our present model because of the presence of the additional, decaying, _ component, whose initial value is finite, i.e., Li(F 0 - F i) Under this condition, we are allowed to use the more exact _(_) function given by Eq. (21). In the discussion above we have neglected the elastic component, @e" If we include _e, then, in the absence of _i' _ = _,, + be' and as F rises from zero to F D, ¢ increases by the amount eF D. Although the use of _(¢) of Eq. (21) will not Iead to an infinite switching time, the initial value of @m_ at _ = -¢r + e_is extremely low, and the resulting computed @ wave- form will have littie resemblance to the observed ¢ waveform. Therefore, even if bE is incIuded but _i is not included in the total 6, we have to use the _(@) function given in Eq. (20) and not the one given in Eq. (21). If F o is larger than the F value at the upper knee of the static ¢(F) curve, then the sum _i + @,,' in which _,_ is calculated by using N(4) of Eq. (21), is approximated quite well by _,_ calculated by using _(¢) of Eq. (20), as is illustrated in Fig. 7. The resemblance between the two _(t) waveforms in Figs. 7(a) and 7(b) justifies the practice of using the approximation @p{1-[(2¢+4s-¢a)/(4s +_d )]2} _ _ +¢p{1- [(2¢+4,-¢d)/(4r +4a)] 2} (22) 0 tp t "--_" 0 tp t "--_ TC*5094 -6 FIG. 7 RESEMBLANCE BE.TWEEN ¢(t) WAVEFORMS DERIVED FROM TWO MODELS FOR INELASTIC 6 14 m • • if the drive mmf is large enough to switch ¢ to the saturation region hbove the knee. However, if F is low (e.g., F _ Fc), then such an approxi- II marion may become quite poor. Furthermore, if F i < F < F 0, then bi > 0, whereas b.. = 0 regardless of which N(¢) function is used because, following Eq. (19), b v = 0. In general, if the instantaneous _ is of interest, the elastic com- ponent of _, _e' should be included, However, if inelasticA¢ is to be calculated, then it is justified to neglect be. 5. SUMMARY Three components are distinguished in the total b(t) waveform of a ferrite core: elastic 6 spike, 6e; inelastic decaying _, _i; and the bell-shaped main inelastic 6, _n." The following semiempirical switching models are proposed for the three components of 6: For the elastic 6, be = ¢F, where e is a function of F, ¢, and the previous switching; for ¢ = -¢r and F not much larger than Fc, e = {(¢s - dPr)/[27v(ro - ri)Ha]} In (ro/ri) For the decaying inelastic 6, 6 i ki(F Fi) _ -(t-ri (F-Fi)/Ci= -- te , where _i is a proportionality factor, F i is the threshold, v is a power coef- ficient, T i is the time of beginning of bi switching (when F reaches Fi) , and Ci is a constant proportional to the decay time constant. For the maih inelastic b, 6. a = _p{l - [(2¢ + Cr -- Cd)/(¢r + Cd )]2)' where, for given F, bp is the peak value of 6, Cd is the _ value on the static ¢(F) curve and Cr is the maximum residual flux. If F is low (around F or lower), 6i should be distinguished from _.a, but ifFis large, then bi + _.. may be approximated quite well by = 6p{1 - [(2¢ + ¢, - ¢d)/(¢_ + Ca )]2} in which ¢_ is saturatibn flux. C. COMPUTER PROGRAMS FOR b COMPONENTS In Report 3, a computer program for only the main component of 6 was provided. It was felt then (cf. Report 3, p. 34) that more needed to be known about 6e and 6i before these components were incorporated into the 15 over-all computer program. After studying @i in more detail, its incor- poration has been undertaken. (It should be emphasized that our present" information is based on the study of one core only; more core materials will be investigated in the future.) If a core is switched unloaded, then F(t) and F(t) are given. How- ever, if a loaded core is switched, then the time variables, including F and F, are solved for transcendentally. If Newton's method (cf. Beport 3, p. 42) is used in this solution, then the value of @' = d@/dF needs to be known. Following Eq. (1), • I _5' = _ + _bI + _b a , (23) where +_ = d@_/dr, +i = d+i/dF, and _a = d+.a/dF. Each of these @' com- ponents will be computed in the corresponding _ PBOCEDUBE where the component itseIf is being computed. 1. COMPUTATION OF @_ AND @_ Computation of _e is based on Eq. (2), i.e., _e = eF. A computer program for computing _e and _ is given in Appendix A in a form of PBOCEDUBE @e(F, At, NV, _). The elastic switching parameter, e, is gIobal, i.e., declared throughout the program; it may be evaluated by using Eq. (3), The input parameters are F, At, and NV, the latter standing for a negligible value of _¢; @_ on the other hand, is an output parameter. In the case of a loaded core, F at t = t n = nat is approximated by differences rather than differentials, i.e., F n - Fn_ 1 F At (24) Since only F n is solved for (the values of F _ 1 and At are given), : d( r)/dY, hence, $ _ : 6 ,/A t ( 2 5 ) For practical consideration, if _e < NV, we shall assume that _¢ = 0 and, therefore, also _' = O. 6 16 2, COMPUTATION OF _i AND _ Based on Eq. (5), _i is computed as a function of F, t, and T . A computer program for computing _ and _ = a ,/aF is given in appendix B in the form of PROCEDURE _i(F, t, T_, _i). The core parameters h_, F_, v i and C i are assumed to be global parameters. The excess mmf, F - F i, is computed each time _i(F, t, T i, _i) PROCEDURE is called, and as soon as F - F i > O, T i is identified with the corresponding t value. Using Eq. (5), 0 1 v. -(t-Ti) (F-Fi)/Ci t k_(F - F i ) _e if t < T i if t > T i Differentiation of Eq. (26) with respect to F gives (26) to( 11 t - T i- if t > Ti (27) For practical consideration, if _i is negligible, e.g., _i < 0.001 ki(F - Fi), then we assume that _i = 0 and _ = 0 3 COMPUTATION OF @.a AND @' Computation of the main component of _, _,a is based on Eqs. (18), 19), and (21). The computer program for computing _,, and _a = a&.o/ar "# is given in Appendix C in the form of PROCEDURE _.,(F, _, _d, _,,)' in which F and _ are input parameters and _d and @_o are output parameters. This PROCEDURE is identical with _(F, if, _a, _') PROCEDURE given in Report 3 (Appendix A, pp. 133-135) except for two modifications: First, _(_) follows Eq. (21) instead of Eq. (20) and second, F12, F_3, V 1, and V 2 [cf. Report 3, Eqs. (26) through (29), pp. 18-19] are treated as global core parameters instead of being computed once inside the _ PROCEDURE. Following the first modification, _ replaces _ in evaluation of _ [cf. Report 3, Eq. (41), p. 20]. For practical consideration, if _d - _ _ 0"001_r' then we assume that _,a = 0 and therefore, also @_, = 0. 17 We have shown [cf. Eq. (22)] that if P is not low, (+i + +,o) may be replaced by _a alone if we repiace 7(95) of Eq. (21) by 7(95) of Eq. (20), Under this condition, PHIS should replace PHIR in Lines PHDINA42 through PB_TMA45 of the _,a(F, 95, 95d' +_o) PROCEDURE, Appendix C. In Appendix D, two computer PROCEDUREs of different output parameters are given for this case: _(F, 95, 95d, _') and _(F, 95, _', _*). The first PROCEDURE is the same as the one used in Appendix A of Report 3 (p. 133), and its output furnishes the values of +, 95d, and _'. The output of the second PROCEDURE includes +, +' and _*, where, as shown on p. 43 of Report 3, Another difference, which is minor, is that the core parameters F12, F23, V I and V2 are arbitrarily treated as global parameters in the &(F, _, _', @*) PROCEDURE, but not in the @(F, 95, 95d, @') PROCEDURE. 4. SUMMARY Computer programs for the three components of @ and _' = d_'/dF are given in PBOCEDURE forms in Appendices A, B, and C. PROCEDURE @e(F, At, NV, _)is based on Eqs. (2) and (25); PROCEDUBE £_(F, t, T_, _i) is based on Eqs. (26) and (27); and PROCEDURE _.(f, 95, 95d' _.) is based on Eqs. (18), (19), and (21). Two computer PROCEDUREs that differ in their output parameters, _(F, 95, 95a, _') and @(F, 95, _', @*), are given in Appendix D for the case in which the sum _i + _. is approximated by /(_ + _)_2: Lqs. (18) through (20). D. EXPERIMENTAL VERIFICATION The experimenta[ study of the _ components consisted of clearing and setting a Lhin ferrite core (Core E-6, Report 3, p. 23), and photographing the waveforms of @(t) and f(t) during the beginning of the SET pulse. Variations in the SET pulse included two rise-time values, each with seven or eight cifferent amplitude values. A computer program was written and, with nhe proper core and circuit parameters, used to compute the @(t) waveforms. Experimental and computed results were then compared and analyzed. 18 1. EXPERIMENT a. INTRODUCTION In order to investigate the decaying component, _i, meaningfully, the main component, _,_, should be as small as possible, certainly not much larger than _i" This condition can be achieved if the rise time of F(t) is short. On the other hand, a short rise time generates a high- frequency ringing in the core windings (due to stray capacitance), causing a distortion in _(t). Another difficulty in this type of investigation stems from the variations of _ if the rising portion of F(t) is noL smooth. b. RISE TIME Two values of rise time were used: one was around 0.1 #sec, and the other around 0.02 #sec. No difficulties were encountered using the longer rise time. However, some ringing was present in the case of T r _ 0.02 _sec. A typical example is shown in Fig. 8, where the waveform of the observed &(t) has been traced. It can be seen that subtraction of a decaying _(t) ringing of ahigh frequency (about 100 megacycles) from the observed _(t) oscillogram results in a smoother _(t) waveform. The latter should be considered to be the actual _(t) waveform. t $ t I0 mv/t I I I I I I I ,,,, 5 nsec TA-50(J4-7 FIG. 8 RINGING IN _(t) FOR F(t) WITH A SHORT RISE TIME F D = 1.50 amp-turn; T r = 19 nanoseconds. 19 A mercury-relay pulser, with its capability of providing current pulses up to 40-ampere amplitude with less than 0.7-nanosecond rise time, is very attractive for studying high-F @_ and _i" Unfortunately, an attempt to use the mercury relay has been unsuccessful so far because of an exces- sive ringing in the _(t) waveforms. This problem should be pursued further in the future. c. TESTED CORE Oscillograms of _(t) were recorded only for one core, the thin ferrite toroid (OD/ID = 1.06) which is referred to as Core E-6 in Report 3. The dimensions and switching parameters of this core may be found in Report 3, p. 23. The nominal composition of the core material (commer- cially known as Telemeter Magnetics T-5) is +++0[Mgo. 3 2Zno. 1 oMno. 5 8] ++ [Mno. 2 6Feo. 7 4] 2 4 The _(t) waveforms of additional cores of different material and larger OD/ID ratio were observed and appeared to behave in a manner similar to that of the _(t) of Core E-6. d. CORE HOLDER The same coaxial core holder was used as described on p. 85 of Report 3. The sense winding was increased to 20 turns of No. 48 copper wire. The negative clear winding consisted of 10 turns distributed around the circumference of the core. The drive winding was modified so as to have a single turn made of six No. 48 copper wires. The pulses with 0.1-_sec rise time were applied to this single-turn 6-conductor winding. In this case the center conductor of the 50-ohm transmission line was not used. The pulses with 0.02-_sec rise time were applied via the center conductor of the 50-ohm line. e. PULSE SEQUENCE The pulse sequence was that discussed in Report 3, p. 83: first, a positive CLEAR pulse; second, a negative CLEAR pulse; and third, the (positive) SET pulse during which measurements were made. 2O f. EQUIPMENT Four different current pulsers were used in this experiment: (I) The SET pulses with 0.1-_sec rise time* were generated by paralleling several high-impedance transistor current drivers (Digital Equipment Corp., Model 62). Maximum amplitude was 5.0 amperes. (2) The SET pulses with 0.02-_sec rise time* were generated by a tube current driver (Hewlett-Packa'rd, Model 214A). Maximum amplitude was 2.0 amperes. (3) The negative CLEAR pulse was generated by paral- leling five tube drivers (Digital Equipment Corp., Model 50; 0.1-_sec rise time). Maximum amplitude was around 7 amperes. (4) The positive CLEAR pulse (1.4 ampere × i0 turns) was generated by a single tube driver (Digital Equipment Corp., Model 51; 0.1-_sec rise time) for the cases of T _ 0. l _sec and by two trans- r istor drivers (Digital Equipment Corp., Model 63; 0.0S-_sec rise time) in parallel for the cases of T r = 0.02 _sec. Two oscilloscopes were used. The waveforms corresponding to 0.02-_sec rise time were recorded on a Hewlett-Packard 185A sampling oscilloscope, having a response time of about 0.5 nsec. The waveforms corresponding to 0.1-_sec rise time were recorded on a Tektronix 545 oscilloscope with a Type K plug-in unit, resulting in a combined response time of 13 nsec. The response of the Tektronix oscilloscope together with the plug-in unit was checked with a Tektronix Hodel 108 mercury pulser in order to make sure that no overshoot was obtained and that the response time was short enough. The delay experienced by the F(t) pulse with 0.02-_sec rise time (between the time when _ was measured and the time when F was measured) was 3.5 nsec. This time delay was corrected by shifting the oscilloscope trace of F(t) with relation to the _(t) trace and by photographing the two traces separately. The temperature of the core was automatically maintained at 30°C by means of a thermistor probe and an electrical heater imbedded in the outer conductor of the coaxial core holder. * The rise time Tr is twice the time that it takes the current pulse to reach half of its amplitude as de- fined later in Fig. g. 21 2. COMPUTATION a. METHOD OF COMPUTATION Computation of _ of an unloaded core may be performed by the same simple predictor-corrector method used in Report 3 [Eqs. (42), (43), and (44), p. 25], except that _ now includes three components [$,, Eq. (2); _i' Eq. (5); and _._, Eqs. (18), (19), and (21)] instead of one [&_ , Eqs. (18), (19), and (20)]. Following this method. <_n at t = t = n_t (At is a short time interval compared with the switching time, ?s) is first predicted from the relation 4_n = _,-2 + 2Gt$, _ 1 (29) This is followed by an iterati_,e comou_:at,_,n of _'_. and _.,.. Since f and f are both a function of time, $; may be formally expressed as $, = @e(t ) + $i(tn) + @_(t_,4_) (30 ]'he expression for the corrected c_ is _ = _ + 0.5At,S, + " (31. Equations (30) and (31) are used repeatedly until proper convergence of _ and _ is achieved. DRIVF CUI_RFN'I :, _ . TIML As shown in Fig. 9, the waveiorm ol the drive current, which ;s applied to the unloaded c-ore, is approximated bv the following functions: 1) -- ut - 2 - ut - 3 if 0 < t < t 2 ,2 _ -- -- < [/@ [i + tanh u(t - t )] _f t _;:_! ;_;;:T;ii_!!!:llill:iiili:l!i _ :.....:_ i ;F- i;-,-,--+-+- =; _ ::i:2 _O.2Fsec ii -+-_+ + -4-+-+-+- --+_--H- ee-H- _4L]2 _+_ _+--+---- H¢ TA-5094-16 (h) F D = 2.4 amp-turns F. scale = 1.0 (for i) and 0._5 (for ii) amp-turn/major division q_scale = 12.5 (for i) and 37.5 (for ii) millivolts/turn/major division Fig. 10 Concluded 3O ,, _illi i:i ii i: +_+ +i+ + ii'_ _+ .-_-_.-_.,-.i 'rJl .... I lll_l[i_l_llllll "_:'_] il ...... IlIlIIlL_ IIlll IIIIIIIAI_]I 11|11 1 l l l [ l_ l l I I FI_ ] 1 1 III _ llIll Illll ......... II ..... I1111!i]iiiiiiiii _ll:llIll][lllll] lllllillllllIllllll IIT]I_III]IIIIIIII] [IZ:IilIIII]]IIIIll il l._llll []iilllT}llllllll llt!lllllllJllllJl]l!F!l_i]lllllllll __ + : i i i i i i _ , i 1 .... 1...... t111 lllll] I i I i i i i i +] ..... II .....; kll T ] I [ I I I l I I I I III ]llllJlllll_ lilllll|l',', I i (a) F D = 0.6 amp-turn ii F scale = 0.1 amp-tum/maiordivision + scale - 5.0 millivohs/tum/maior division -F _-t+ -+ I ..... iIITIII l!llt '+ [_,i ' ' 'III _ i -s4_;. I _ _iLLL + T-- TA-_JOI4- I ? !iJ ++i+ I :P: :::: ::: :::: III II +i i ..... !! 1 i!!+ iiii i ill ilil Ill II11 J iii IIII II11 Ill II11 I Ill 1111 i Ill Till [Ill Illi I . :I , ;J iii! II I]:] "!I if! I iiii I i l i ii t,_o_-. (b) F D = 0.8 amp-turn F scale = 0.25 amp-turn/major division + scale - 5.0 millivohs/turn/maior division FIG. 11 EXPERIMEN.TAL (SOLID LINE) AND COMPUTED (DASHED LINE) F(t) AND qS(t) WAVEFORMS OF UNLOADED CORE E-6 DURING BEGINNING OF SWITCHING, USING F(t) WITH T r OF ABOUT 0.02 /_sec 31 Ilitl][l[iLl[[llll .... -H-I+F}lilllll,,,,r__i:_ ][ITILTIll]Illlllm _r[I 11111111111IIrll_Elll IIl[ll_lll[[lll_[l[ll lll[ll_llll[Ir_l[lll] I]ll[llllllllEllllllll !!!!!L!!!!!!Lv?!!!!!!L I[ [[llllll111111 II"II ...............|l iiE_llEilliLIJJ II|lrln|_nlnLllIliiii iiiiiiiiiiiii:::ii::::::: ::::A_LIIr[_II LI:::II::_: [lll Larlllll[I ]1 ... :::: :::::::::::::::::::::: iiiiiiiiilliiiiiiil iiil.l '''Illlll []lll ,,, .,,4e_F:21Z_ :: i !! z i EJ i_ :: ;;iii :ii ,,,,z [ II lllll [I lllll '' I]lllil lllIl II lUll| II IIIll I1 lll[I II IIll-I II III1| I1 I!lll I111 II Illl II Ill[ IllltlllllllllIIIIllllllllll I - , 1 _ ...................... LJ i i El J,[[ I l I I [ ] [ [ ] [ __ ]_l ] J III I [ I I I I II[ITI I IFTIII I .... I .......... II[Jlll II 11 []]ll Illl]ll I IIII111 Illll Illlllll] [ I [-] I I I I I ! 1 I I I ] I I I I l I I I III I 1 I I I I l I l I ......... Ill .................. 1t ......... fill Illlll I IIII | I I I I I I I I I I I I I I I I I I 1 I I T l I I ............................. I] ....... Ill [[rl II1[[ ] [ ]111 i iii iiiiii1111 I1 iii1 i iiillll[ [ I Ill II I[i i i111 i i iii i i [l_ll IIIIIIllltlllllllll I [ [ MI I ] [ I I I I I I I I I I [IA II Ill l I I I I I I I l I I I I .......... Ill ........ I ..................... lii Ililfl[ll t/ IIIIIIll]llllllllllllllllll ......... II111 ............................. i i i Ipt i i i i i illl ill ii illlllll Illlll [llll•llll I ......... Ill .................................. I I IT I I I I 11111111111111111111111111111 l_llllllil I 111111111111111111111111 lllllllll[lll]lllllllllllllllllllllllllllllll I_11 I I III i i iii 11 iii illlllllllllllll llllllll l_l_ll i i [ [ I I I II1 III1 I I Illlllllll Illl I Illtlllli llllll[llll]l]l]llllllllllllllllllllllllLlll[ III111111111111111111111 11- IIlltl[ll II I ll'J I l lllllllll II III IIIll I I II1.1 III I IIII I I III I III I I Illll I IIIll Illlllll I _r : "2_L 2 : .R s;r: ....i !; -_'T .i._/! I 1 -rl 22_ i i i ; --_ _ 5nsec _ :/44- i; 4i % -tI4-H/..H_,:_ T r ;-: 7CkkN-+ __ _'I'_'__;: '* .]ii :!;j_j }_ i (d) FD l 1.0 amp-turn F scale - 0.25 amp-turn/major division scale - 5.0 millivolts/turn/major division Fig. 11 Continued 32 il, !K II !! ii i-i. _-VFI-FIq']IIITI-ITTI-lqlqlIItltjI Ii ....................... Ill:,]ll_]lll_ llllllllllllllllllllllllll Illll|llfllJ_lllLlll]lllllllllllllllll _ IIIIIIILIIIII_II ! llllllllll I I II lllllllItllilmlllllllllllllll l I "" llll_r m • ,--111111111111111lliltllliii+ lllllllllllllllllI_llllllllllllI i IIiiiiiit1111111111 i iiiiiiitt i l l ii|l[l!i _11111|1111111 i I | lllllJll _l_ii]iiili_ iilliiiiiiiilliilll I llllll]llllilllltllllillllllJll[llll i IIIIIIIII!t111111111111111111111 I I IIIIIIIIIII]1 Illlll] ill|llllil)l I I IIIIl[ll]llllllllllli _ llllllllllll I I ]llllllllllllllllJllll_llllllllllll i I IIIIlJIIIIIIlIIIIIIIIB]IilllIIIIIIIII I : ]III_II[IIIIIIIIIIIIILIlIIIIIIII]II I IIII llllllllllllllilllIlllllllllll I I III l]lllllIlllllllJllllllllllll|llllllI]l I llll]l]IlllliJil]lllllliIILll'l i i ..................... ,,, ............ Jllll lllllll Bllll]llllllllllllll_lllll--:_i.Jlll Jl J ]llm]llllllllI]l]i]llllllllll_LJl _ IIIIIIIITIIlIIIIIlIIIlIIIIIIIlII_--I_]]I JllTlllllllll_l]lllllll1111111111r I I T_ lllll _ llilil!Z!llllllallllllllllllllll i ; llllll-lllllll_!l]l]llllllllllllllll]lll i _lilllllllillll!lillllllll I I ]IllI lllllllll] IIIIIIIIIIIIII I l ]III I-ILIlI]IIII[III]IIII ] I | _l_lllllnlllillUlllnXllllllllllllllUllnllll] :lmlllllllllillllllllllllllll]llillllllll --,,,,_,,,,,,,_,,,,,,,,,,,,,,,,_,,,,,,,,IIN i (e) F D = 1.18 _p-turn F scale = (_ scale - 0.25 amp-turn/major division ' 7!!;I ]1]1)ii]1 I|l[lllll Iii111111 i;ii_ilil ]IIIIIIII 'II]IIIII !!]!1!!! !!!i!!!! iiiiiiii !]l!!!!! iiiiiill II...... iiiiiiii !!]!?!!! 5.0 millivolts/turn/maior division I! II I I I I :I ÷H-,_ lllll lllUl i i,ii] iii ]]!! _1111 !!!!! : II .... :' II iiiilf: I 1: ,,,,ii_i,_ illl llll" ilil , .... ff iiii *° ,i1! , ,ii......fi! .i. "'_: :i ;;:] i ] : 'II _ !i] _,iiI ft_ w4q ,,_, ,TT,r; ......!] i:i! (fl F D - 1.5 amp-turn F scale - 0.5 amp-turn/major division _H,H !'!11111 7!iliill i]111111 ':ilil;!!!!_ _iiiii _iiiiii !!]]! '-1_1 iI,r!il iiiiiii ] iiii]i iIllllll IIIIll III!Jll i III111I1?1111 I1:11] _'_' _ II,_4-4-_.IIIII[l l]:1 1_ 4" -'_'-_-4+e _ iiiiiii ! II'i]I i iii!iiii:__ _iii_iiiii! 5 ,,,. ,i , scale - 10.0 millivolts/turn/majordivision Fig. 11 Continued 33 I iii!i 17]_[ _ 5 nsec :_ i!_ '_._.. |[11 !!E i!, [|] II [ILml I ], Illltl I lllrttl ...... _iiIIIII111111! iJ_ll]llllllllI[lll B'F I I I I I I I L_--I-H-_ [IIIIII[II [Illl[]l [lll iiLiii',i!!ii',iiiiiiiii!lllI, H_t-H%t-H444-fH4+H+4-_ lYlITTI[EI rllallml |I_ l[l]llllllLilll! I _÷ ltllt',il FIIII;I 11111111 III .... t[1111.', t111t:=I IIIII1,1 IIIIII1:1 ]11111:1 IIIll_ ;I Ill ....t It I Ili ..... [1111111 IIllll;] IT1/iirl JJ_LL I ttt!l _L 1,1[[1 , lllllIl i if! !LLI LLJI / ................ H4fl'+H-t- llllll]llil]l l|llllllllll IIIIIIIIIIIII IIIIII IIIIllllll]ll Iiill]l I:[11:][]1111115II111111111111!1111 !]llllll]l[ll] _ll]lll[Xlll IIIIIIIIIIIIIIII IIiII ]lT[l!l[lllllYl_ Illll]l]l J1ill!]llllll[llllllllllllfllllll _lll[:l]tlllllllllIIIIIIIIl[]l]lfll : ; II11111111 ' lllltlllillil II IIIIII 1 I]11i[11]11G IIIIIlll t111111IIlllllll11111111 ]IIIIT/IIII ' ..... I .................. H4@I-H ...... '_+11tt ........ _I_t_t! _tfii_t_ _ I I I { i - @H-H4-H÷H-rf_ffH-_ t-_t-H _O.02H.sec I +I : ;ii L__L; '_ ..... i t _;,_lT_'_r , _[_,,,i : It,_H_ II!;i,_ill!7!li!l III1 ftHi 1111"_1!111 r ,, _I i, I::II i!!!I _ ..... l!'_'"'ll,,_,,,, , i ....i IIIIIItl_;ll_ ....... IIIII 1 ....... ,,, i!! ,Ilrl:II I[] II II_LIlIlII!II]:[ Illll II1_I1_11: Ill[ ii IIIII IIIII 111lllllll:;I lllll IltllII,,l,![I!, t: ll:_,ll] _t_111_]]:1! [ (g) FD = 2.0 amp-turns F scale = 0.5 amp-turn/major division scale = 10.0 millivolts/turn/majordivision Fig. 11 Concluded 34 The model for the elastic component of b, b_ = £F, turns out to ,be too simple. It is based on the assumption that the elastic motion of domain walls and elastic rotation of magnetization encounter only pull- back (or stiffness) forces that are proportional to the relatively small displacement and small angle of rotation of magnetization. There is, however, a short delay between eF and the actual bE. This delay is prob- ably caused by viscous damping that is proportional to be. Inclusion of the viscous damping will result in a differential equation of the form _%_ +b_ : _F , (35) where _ is a constant proportional to the viscous damping. The model for _i' Eq. (5), appears to agree quite well with ex- perimental b(t) during the beginning of switching. It should be emphasized that only one core (Core E-6) has so far been investigated thoroughly. The b(t) waveforms of other tested cores were similar to those of Core E-6; however, these waveforms have not yet been analyzed quantitatively. Clearly, such an investigation, although costly, should be performed with other materials in order to substantiate the proposed _i model. b. VARIATIONS IN SWITCHING PARAMETERS We have seen that in order to obtain a satisfactory agreement between observed b(t) waveforms and the model for bi , the switching parameters k i and C i had to depend on the rise time, Tr, of F(t). A slightly larger k i and a smaller C i correspond to a shorter T r. This dependence of k i and Ci on Tr is very complex, and may, perhaps, be ex- plained qualitatively as follows. If T r is short, domains in some regions would expand only locally when F(t) = F D, and thus contribute a bi com- ponent of a short duration. If T r is long, such local domain expansions are given no chance to occur: before F reaches the local threshold values, flux reversal in these regions is caused by the applied F plus the magnetic poles brought about by oncoming unobstructed domain walls. The shorter T r is, the larger is the number of nucleation centers around which short- lived, local, domain-wallmotions take place, and thus the larger is the mean displacement time of these walls. This argument may explain why ki increases and C i decreases as T r decreases. 35 It is interesting to note that v i = v. Whether this is a coin- cidence or not remains to be determined after additional core materials are investigated. Some light may be shed on this problem by attempting to explain the physical origin of v. Conger and Essig hypothesized 6 that the threshold field for domain-wall motion in thin films is distributed randomly over the range between the coercive force and the anisotropy field. As a result, the number of walls, in addition to the wall velocity, increases with H, and the nonlinearity of i/_ s vs. H is accounted for. = K(H - ,,)v [Report 2 38In a similar way, by assuming that Bp H 0 , p. , Eq. (83)I, an effective H 0, denoted by H_, that increases with H between H i and H B [Report 2, p. 38, Fig. 153 was obtained. The motion of locally obstructed domain walls is governed by the same rule. If the value of v is determined by the distribution function of the threshold field, and if the same distribution function is applicable to F i one would expect v ' i and v to be equal. We have seen that there is an analogy between the parameters k, It F0, and v of _ma and the parameters Ki, Yi, and v i of _i" Let us carry this analogy a little further. Referring to Eq. (19), _p = pp(F - FO), It where F 0 > F 0, if F B ! F. Hence, we expect that for high F values, com- pared with those in Figs. l0 and ll, _ will become proportional to (F - F_), where F_ > Foi. This is in agreement with Eq. (4) and with the fact that the ratio Fo/F _ = 1.45/0.95 = i. 52 is close to the ratio F_/Fo_ = 0.775/0.55 = 1.41 (cf. Report 3, p. 29). In order to verify this conclusion, adrive current with higher F o values and shorter rise time than in Fig. ii should be used in the experiment (if T r is not decreased, _ma will mask _i). With our present laboratory equipment, this could be achieved by using a mercury-contact switch and a transmission line. As explained previously, when such an attempt was made, the signal-to-noise ratio was too low to be analyzed meaningfully. We hope to overcome this circuit problem in the future. C. VARIATION OF _(t) WITH F D In Fig. 10, individual _(t) waveforms are shown for the different values of F O. The effect of the magnitude of F o on the waveform of _(t) is demonstrated qualitatively in Fig. 12 by showing a multiple exposure of experimental _(t) oscillograms. The only difference between Fig. 12(a) and Fig. 12(b) is the time scale and the duration of switching. In either figure, the _(t) oscillograms correspond to F(t) drives of the same rise 36 2.5miilivolts 1o1 (b) FIG. 12 MULTIPLE EXPOSURE OF ¢(t) WAVEFORMS OF CORE E-6 CORRESPONDING TO F(t) PULSES OF DIFFERENT AMPLITUDES time and the following F D values: 0.6, 0.8, 0.9, 1.0, 1.1, and 1.2 amp- turn; the F(t) oscillogram shown above the _(t) oscillograms corresponds to F D = 1.2 amp-turn. d. COMPUTED COMPONENTS OF _(t) The _(t) waveforms shown in Figs. 10 and 11 are those of the total 6. The relative magnitudes of the three components of _(t) depend on F(t). For example, computed _(t) and its components are shown in Fig. 13 for the case of F o = 2.0 amp-turn and T r = 0.024 _sec. This in- formation about be(t) , _i(t), and _,.(t) is a part of the computer output (cf. Appendix E). e. COMPUTED _ip AND Tip vs. FD AND T r In Sec. IB-2(d), we have calculated ¢i(t) [Eqs. (8)], and its peak amplitude _ip[Eq. (11)] for the case of F(t) with a ramp rise 37 3.0 A 2.O _' 1,0 0 I I I , , I tm Tr I I I I I I I I I I I I i, 6O 5O 4O E IO I I I I I I _ma o I I I I 0 0.01 O.OZ 0.03 0.04 0.05 0.06 0.07 0.08 0.09 O.lO t (/4,$ec) TC-5094-_3 FIG. 13 COMPUTED (_, _{, (_ma AND (_ vs. t OF CORE E-6 DURING THE BEGINNING OF SWITCHING 38 followed by a constant amplitude, Eq. (6). Similar computation was per- formed for the drive current expressed by Eqs. (32) as part of the com- puted _(t) in Figs. 10 and 11. Plots of ¢ip vs. F o are shown in Fig. 14 for three values of rise time: T r _ 0.1 _sec, T r _ 0.02 _sec, and T r = 0. The small variations around a smooth curve in each of the first two cases stem from the small variations in the actual values of T . For the case , . = k (F D - Fi) _ [Eq. (5)] whereof T = 0 Ttp = 0 and ni p i F i = F0i tanh (FD/F0i) [Eq. (17)] and k i = 0.014 ohm/turn2"3amp °'3 e. ADDITIONAL LOW ¢i COMPONENT sl If F < F 0, then _,o = 0 and so = _¢ + _i (36) Suppose that F is a rectangular pulse of amplitude F D and duration T. Let us examine the amount of A_ contributed by b e and _i by the time t = T, before F begins to fall. Since b e = :F, F D( ACe = I ¢dF = cF D (37) Following Eq. (5), T i _ v -1 -T(F-Fi )/C i A¢ i = id t = kiCi(F D _ Fi ) i [1 - e ] (38) As T _ _, A¢_ _A¢i(w), where v -1 Aqbi(w ) = kiCi(F 0 _ Fi ) i (39) AddingA_ andA_i(w ), the total A_ due to a step F, F = F D, is _i-I A¢(_) = :F D + kiCi(F D - F i) (40) Let us calculateA¢(m) of Core E-6, assuming an F D value which t! is less than F 0 = 0.95 amp-turn, e.g., F D = 0.9 amp-turn. The core parameters are as follows: _ = 0.3895 m/_hy/turn 2 vi = 1.3i y 39 25 2O 15 t. :D u) o ._ E ._= I0 Ill,lllll' I 0 0.5 1.0 1.5 2.0 FD (amp-turns) FIG. 14 COMPUTED _p vs. F D OF CORE E-6 WITH Tr AS A PARAMETER 40 X = 0.014 ohm/turn2"3amp °'3 C i = 0.145 amp-turn-#sec (assumed to be samet as for T r _ 0.02 #sec), and Ft _ 0.55 amp-turn. Substituting these values into Eqs. (37), (39) and (40), we find that D_ e = 0.034 maxwell, A_i(_) = 0.148 maxwell andA_(_) = 0.182 maxwell. Now consider the static _(F) curve. Starting from ¢ = -¢r' tile total flux change involved in reaching a point (F,@a) on this curve is Aqb a = qbe + q5 (41) For Core E-6, F = F 0 = 0.9 amp-turn corresponds to SSj -- -0.12 maxwell, and since _br = 3.45 maxwells, A_b d = 3.33 maxwells. ComparingAq5 a andA_b(_), we find that (for Core E-6) AqSd/AqS(_ ) = 3.3/0. t82 = 18.3. We conclude from this result that the model for qbi given in Eq. (5) does not account for the very slow flux switching involved in reaching a point on the static _5(F) curve. This conclusion is in agreement with the _(t) waveforms in Sets (a-ii), (b-ii) and (c-ii) of Figs. 10 and 11. One can detect that after the computed _b has decayed to a negligible value, a very low experimental _ continues to exist. Physically, such a low _ results from the motion of few domain walls that are obstructed far from their original position rather than locally. If this interpretation is valid, this very low _ may be considered as an additional component of @i whose amplitude is much smaller than lJ ki(F _ Fi ) i and whose time constant is so much larger than Ci/(F - F i that the product of its amplitude and its time constant is much larger than w --1 L_Ci(F _ Fi ) i An alternative source of this very low @ to be considered it is the main _ component, @,,, with F o decreasing for decreasing F [similarly to F i , Eq. (17)]. Future investigation of this very low _i component should be based on correlation between measured qb(F) curves obtained by rectangular F pulses of different duration T and calculatedAq5 d -A_(_). 5. SUMMARY Verification of the models for the three components of _(t) was made by comparing experimental _(t) and computed @(t) = @e(t)+ _i(t) + _,_(t) during the beginning of the flux switching. The tested core was a thin magnesium-manganese-zinc ferrite ring of OD/ID = 1.06 (Core E-6, Report 3, p. 23; F = 0.9 amp-turn). The comparison was made for mmf pulses of different ampIitude, Fo, and rise time, T : eight values of F o (ranging from 0.6 to 2.4 amp-turns) for T _ 0.1 _sec and seven values of F D 41 (ranging from 0.6 to 2.0 amp-turns) for T r _ 0.02 #sec. Each case was repeated twice, using different time scales. The core and circuit param- eters were fed into a computer program which computed _¢(t), _i(t), _,a(t') and _(t). The resulting _(t) waveforms of the above 2 × 15 cases were machine plotted, and are compared with the experimental oscillograms in Figs. 10 and 11. From this comparison, the following observations are made relative to Core E-6: (1) There is a small delay (of the order of 0.15 Tr) between computed eF(t) and experimental _e(t) due, probably, to viscous damping; An improved model for elastic switching is S'qb E + _e = cF, where S is a constant proportional to the viscous damping. (2) The parameters k and C depend on T : as T decreases, i i r r k i increases (slightly) and CI decr. eases. - =k (F-F) / agrees(3) The analogy between _p =k(F Fo )p and_ip i i with the analogy between _p = pp(F-F o) and_ip = ps(F - F_) that was proposed in Report 3. Furthermore v = u ' t t! and the ratio Fo/F o is close to the ratio F°d/Foi. It is expected, therefore, that Eq. (4) is valid for high values of F o. This may be verified by applying current pulses of high amplitude and very short rise time, such as obtained by using a mercury-relay pulser. (4) The flu_ changeh_b i(w) = fo _Pi dt is much smaller than the flux change, A_b a = qbd + qb , invoived in obtaining a point on the static _b(F) curve. In order to account for the latter, another term should be added to the ex- pression for _i, Eq. (5). This term is expected to be of the same form as the existing expression, except for a much lower ampIitude and a very much longer time constant. Alternatively, most ofAq_ d may be accounted for by _,a it if F 0 becomes lower for low F values. E. CONCLUSIONS For the type of ferrite material investigated so far, the $(t) wave- form resulting from a constant-amplitude mmf of a finite rise time consists of three components: elastic, decaying inelastic, and main (bell-shaped) inelastic. 42 The elastic b component is described by the model _ = 6F. This model results in a small delay between experimental and computed _(t). This delay implies that an improved model for _ is of the form _ + b e = :F, where _ is a constant proportional to the viscous damping. The decaying _ component, hi, is described by the model _i = k (F, - F )vi: exp [-(t - Ti)(F - Fi)/C _], where k i, F0_ , v_, and C i are switching parameters and T i is the time F reaches F i. The amount of A¢ associated with reaching a point on the static ¢(F) "curve for F < F0 is much larger than m_,dt. In order to account for this A¢, the existing _ model needs to be modified either by adding a component of the same form as bi , except of a much lower amplitude and of a much ## longer time constant, or by lowering F0 for low values of F, as was done for F i . The main component, b,,, may be expressed by the model ¢.o = Cp_(_), where bp is the peak b, and _(¢) = 1 - [(2¢ + 6 r - _a)/(_r + Ca) ]2, and where, for a given F value, _d is the ¢ value on the static ¢(F) curve. If F is large (compared with the coercive mmf, Fc), computation of _i + b.a may be approximated by the _.a model alone if Cs replaces Cr in the expression for 7(¢)- The elastic component, be , may be neglected if only inelastica¢ is of concern. Further investigation is needed in the following areas: (1) Improving the model for be by including the effect of viscous damping. (2) Studying the effect of ¢ and switching history on e. (3) Extending the investigation of _i to high values of F. (4) Investigating the very slow b component associated with the static ¢(F) curve. (5) Repeating the investigation of the _ components on other types of magnetic materials. 43 II COMPUTATION OF FLUX SWITCHING IN MAGNETIC CIRCUITS In Report 3, the switching model for the main component of _ was applied (using a computer program) and verified experimentally for three magnetic circuits: an unloaded core switched by step, ramp, triangular and trapezoidal mmfs; a loaded core driven by step and ramp drives; and a core-diode shift register. In this report, computer programs are provided and the results are compared with experimental data for the following four magnetic circuits: (1) Unloaded core driven by F(t) of constant-amplitude, _, and rise-time, T --computation vs. time of _e, ,, _=_, _, and @. [See See. I-D-2-c.) (2) A core loaded by an inductive load (R-L, R-L-C, and R-L-C-diode) and driven by step and ramp mmfwcomputa- tion vs. time of F, _, @, and the load current, using a more exact algorithm than in Report 3. (3) A core-diode-transistor binary counter--computation (vs. time) of three currents and of F, _ and @ of each of two cores. (4) A loaded, saturable, three-leg core--computation vs. time of F, _, and ¢ of each leg, and computation of division of flux between two legs in parallel as a function of drive current amplitude, load, and relative leg dimensions. Item (1) is treated in Sec. I-D; Items (2), (3), and (4) are treated in the following sections. A. LOADED CORE 1. INTRODUCTION In Report 3, flux switching in a loaded core was analyzed (Report 3, pp. 39-55) and a computer program was provided (Report 3, pp. 149-153) for computation of _, @, and iL (the load current) vs. time. The load consisted of six combinations of a resistance, an inductance, a capaci- tance, and a diode: R, R-diode, R-L, R-C, R-L-C, and R-L-C-diode. Computed and measured waveforms of _(t) and it(t) were compared for 45 step and ramp drives. The algorithm used to compute these variables was the same regardless of the type of load. This resulted in accumulation of error in the cases of inductive load (R-L, R-L-C, and R-L-C-diode). Our main objective now is to improve the accuracy of computation for these cases by using a more exact algorithm. An additional objective is to explain why an iterative method of solving a transcendental equation (such as Newton's method) must be applied, as was done in Report 3, if t = 0. Let us summarize briefly the analysis in Report 3 (Fig. 14, p. 39, and pp. 41-45). A drive current tD is applied in N O turns to a core which is coupled by Neturns to a load, The loadis composed of a resistance B L, an inductance L, a capacitance C, and a diode. The diode voltage is expressed as gd = iLBd + Ekln 1 + , (42) where R d is a forward resistance, E k is a voltage constant and I 0 is saturation current. Since F = NDi D - Nci L and i D is a given function of t, _(F,@) may be written as _(t, iL,_). Letting R L + R d = R and foiLdt = q, the load loop equation is -- -- + E k ln 1 + - N _b( t,_b, i LC q + BiL + L dt = 0 (43) In Beport 3, i L was solved for transcendentally, usxng Newton's method of successive iterations. In each jth iteration, corrected values of @ and i L = q were used to correct the values of _ and q of the pre- vious iteration, using Eq. (31) and its equivalence for q. Although this method worked well for all load cases, some error was accumulated at each nth At as a result of approximating (diL/dt _ by [iL, - iL( -1)]/At and (d2iL/dt2)n by [(diL/dt) . - (diL/dt),_l]/At [cf. Beport 3, Eqs. (67), (69), and (70), pp. 44-45]. If L / 0, this error may be reduced by using a different method of evaluating iL, in each iteration. We thus distin- guish between two cases: L / 0 and L = 0. 46 2. INDUCTIVE LOAD If L _ 0, there is no need for a transcendental solution of i L . The switching function _(t, _, i L) together with the second-order differential equation, Eq. (43), may be written as a set of three simultaneous differen- tial equations of the first order: d_ = _( t, V, i,} [44(a}] dt dq - iz [44(b)] dt d--_ = -L c_( t,_p, i L) - -_ q - Ri L - E, In 1 + [44(c}] In these equations, t is an independent variable, and @, q, and i L are dependent variables. Note that, as required by the various methods of numerical solutions of differential equations, 7 differentials appear on the left side of the equality signs only. The numerical solution of Eqs. (44) may be achieved by various well-known methods, such as Adam's, Bunge-Kutta's, Milne's, etc. The simple predictor-corrector method given by Eqs. (29) and (31), which was also used in Report 3, will suffice in this case. In every jth iteration, instead of solving for iL_ transcen- dentally [cf. Report 3, Eqs. (69) through (72), pp. 44-45], we first evaluate (diL/dt) using Eq. [44(_)], and then compute iLn from an expres- sion analogous to Eq. (31), i.e., it, , = iL{,,_l) + 0.5 At[Cdiz/dt)', + (diz/dt)._ x] " (45) Other than the above evaluation if iLn, the algorithm for computing the various variables if L _ 0 is the same as in Report 3. 3. NONINDUCTIVE LOAD If L = 0, then the loop equation, Eq. [44(c)], becomes 1 (iL)Nc_b(t,dp.i L) --_ q - Bit - E_ln 1 + -- =Io 0 (46) 47 Regardless of whether a diode is present in the load or not, dq/dt = i L cannot be expressed explicitly as a function of t, _, and q because _ is an implicit function of i L . Hence, we are unable to reduce Eqs. (44) into a set of two simultaneous differential equations with derivatives to the left of the equality signs and functions of the variables (t,_, and q) only on the right side. For this reason, we are unable to use any of the conventional methods of numerical solution of differential equations directly. Instead, we substitute qn = qn-1 + 0.5 At[iLn + iL(,_I)] (47) into Eq. (46), obtain an implicit equation in iLn , and solve for iLn transcendentally. This explains the need for Newton's method in the algorithms described in Beport 3. We could, of course, use other methods We prefer however to usefor the transcendental solution of iL . , , Newton's method because it is very simple and because it is characterized by quadratic convergence, 8 i.e., the error at the jth iteration is pro- portional to the square of the error at the (j-l)th iteration (both errors are fractional if the condition for convergence is satisfied), thus result- ing in a relatively fast convergence. The pitfall to watch for when using Newton's method is when f' [c/. Eqs. (58) and (60) in Report 3] becomes much smaller than unity. 4. EXPERIMENTAL AND COMPUTED _(t)AND iL(t) OF INDUCTIVELY LOADED CORE Experimental and computed _(t) and it(t) waveforms of a loaded core driven by step and ramp drives were compared in Figs. 15 and 16 of Report 3 (pp. 48 through 51). The tested core was Core J-1 (Lockheed 145SC1, i45-mil OD, 90-mil ID) whose switching parameters (cf. Beport 3, p. 23) are as follows: l = 7.1.8 mm, l = 11.58 mm, qb r = 31.0 maxwells qbs = 33.48 maxwells, H a = 250 amp-turns/m, Hq = 26.0 amp-turns/m, JJ H = 22.5 amp-turns/m, F 0 = 0.27 amp-turn, v = 1.43, n k = 1.64ohm/turn2"43amp °'43 F 0 = 0 55 amp-turn, /)p = 2 27 ohms/turn 2 and F B = 1.2 amp-turn. We have recomputed the $(t) and i L t) waveforms for the cases in which L # 0, using the modified algorithm described above. The correspondingly modified computer program is gxven in Appendix F. The results were machine plotted and are compared in 48 Fig. 15 for step-F drive and in Fig. 16 for ramp-F drive with the same experimental oscillograms as in Report 3. It is quite clear that the agreement is better than in Figs. 15 and 16 of Report 3. Note, however, that the computed peak _ is now lower than the experimental peak _. This result is of no surprise. In fact, it was expected (but not found) in Report 3 (p. 47). Just as k and pp of ramp-F switching are smaller than step-F k and pp, one expects _ and pp of switching under a monotonically decreasing F(t) to be larger than step-F values. Such is the condition in the case of a loaded core switched by step-F or not-too-steep ramp-F drives. The more inductive the load is, the steeper is the decrease of F(t) as _ varies from -¢r to slightly above zero. This can be seen in Figs. 17 and 18 of Report 3 (pp. 52 and 53). Since step-F values of k and pp were used to compute the curves in Fig. 15, the resulting value of peak _ is lower than the experimental value. 5. SUMMARY A modification is made in the algorithm given in Report 3 ( p. 41-43) for computing _(t) and iL(t) of a loaded core switched by an arbitrary drive. If L _ 0, conventional methods are used to solve three simultaneous differential equations of the first order. If L = 0, the loop equation is implicit in i e and Newton's method (or any other comparable method) must be applied in the numerical solution of i L, @, and _. The corre- sponding computer program is given in Appendix F. The results for L _ 0 are in a better agreement with experimental oscillograms than in Report 3. B. CORE- DIODE-TRANSISTOR BINARY COUNTER 1. OPERATION a. TWO*COUNT FLUX SWITCHING The circuit diagram for a single stage of a core-diode tran- sistor binary counter is shown in Fig. 1Z* Note that Cores 1' and 2' belong to the next stage. The circuit operation is described briefly as follows (circuit components and number of turns of windings used in this description are defined in Fig. 17): Initially, Cores 1 and 2 are in a CLEAR state. The counting operation is divided into four modes, * This binary counter was developed by Alan J. DeVilbiss of Jet Propulsion Laboratory, Pasadena, California, for a future Mariner spacecraft. 49 EXPERIMENTAL PUTED I V, (a) NDI D = 1.54 amp-turn; Nc = 2;R L = 1.58D, , L = 1.00 p.hy tt.,_-EXPERIMENTAL 1 • . i L • ii t / i !, _--_COMPUTED t I/ I I _ ' !i i t"I ,i't (b) NDI D = 2.28 amp-turns; Nc _ 2; RL = 0.131_Z, L = 0.38 p.hy C = 0.253 _f TC~5094-27 (c) ND I D = 1.80 amp-turn; Nc = 2; RL = 0.131,O, , L = 0.38 p.hy C = 0.253 p.f, 1N3604 Diode FIG. 15 EXPERIMENTAL AND COMPUTED _(t) AND iL(t) WAVEFORMS OF CORE J-I WHICH IS INDUCTIVELY LOADED AND SWITCHED BY STEP-F DRIVE SO / EXPERIMENTAL !i ' i " 1 1 . , I EXPERIMENTAL _ (o) NDI D _ 3.40 amp-turns; T r - 1.60 /_sec Nc = 2; RL = 1.58_. , L -- 1.00 /_hy EX PER} MENTA'L',bI_ ,_" C6MPUTE'D ...... __,' '_ _; I t_....... : ""''L::::t ;!, t :::t:' : : i i_! i _ ] i _._T'_ J . . "_':TT _ ; : _ : : ] i _ I [ ":F XPERIMENTAL- Vi : ;-:-,T___?..... _,_.._,.. .... (b) NDI D = 4.55 amp-turns; 1"r = 1.40 /zsec Nc = 2; RL = 0.131_ , L = 0.38 /_hy C = 0.253 /_f iL T B-!_d:ll4- 211 (c) NDI D = 3.40 omlPtums; T r = 1.40 M.sec N c - 2; R L - 0.131_ , L - 0.38 /_hy C - 0.253 p.f, 1N3604 Diode FIG. 16 EXPERIMENTAL AND COMPUTED _(t) AND iL(t ) WAVEFORMS OF CORE J-1 WHICH IS INDUCTIVELY LOADED AND SWITCHED BY RAMP-F DRIVE 51 Nsi(_ CORE Ns2 COUNT (( _e_OE U\ (isr(MODE 21) PREVIOUS_ STAGE I i c ( MODES IT AND !"9") "_ _ NCL No, NBI -y CORE 2 ic i L dz !Ri ic --VIA CORE I" AND TWO TRANSFLUXORS (o) CIRCUIT DIAGRAM R2 Ns(/ CORE I' NEXT CORE 2' STAGE 0 t --'," OTr t_ (b) DRIVE CIRCUITS TC-5094-E9 FIG. 17 A SINGLE STAGE OF A CORE-DIODE-TRANSISTOR BINARY COUNTER where Modes I and I[ correspond to a STORE count, and Modes III and IV correspond to a CARRY count: Mode I--An exponentially decaying current pulse, is, sets Cores 1 and 2 simultaneously. The flux switching is relatively fast. Diode d 1 is conducting, and the transistor is maintained in an OFF state. Mode II--An exponentially rising current pulse i¢ clears Core 1 at a relatively slow switching rate, while Core 2 remains in a SET state. Diode d 1 is conducting, and the transistor is maintained in the OFF state. Mode III--An exponentially decaying current pulse, isr, tends to set both cores, but since Core 2 is already set, it only sets Core 1. The voltage NBI_ 1 brings the transistor into the ON state, and the resulting collector current i c helps current i to set Core 1, thus maintaining 52 the transistor in the ON state (positive feedback, characteristic of block- ing oscillator performance). Current _c' as it builds up exponentially through an B-L circuit (diode d 2 is blocked), switches relatively slowly three additional cores that arenot shown in Fig. 17: it clears (or tend_ to clear) Core 1", which is similar to Core 1, except two stages ahead (similarly, current ic, which clears Core 1 in Mode II, is generated two stages behind); it clears a transfluxor core which is used for nondestruc- tive readout of the binary state of the stage under discussion; and it sets the transfluxor core of the next stage. Toward the end of switching of Core 1, current isr has decayed to a negligible value, current i c is approaching its asymptotic value, I e ,zip: '/_'1 (to be exact, V stands for the supply voltage minus the collector-emitter voltage), and the transistor base current tends to decrease. Because of the decrease in i and the Sr base current and the increase in ic, at a certain instant the net mmf of Core 2 becomes high enough to clear Core 2. As in the case of Core 1, a positive feedback maintains the transistor in the ON state until Core 2 reaches a CLEAB state. The switching of both Core 1 and Core 2 in Mode II1 is relatively slow because it is done sequentially. Upon termination of the switching in Core 2, the transistor turns off, Diode d 2 starts to conduct current i L via B 1 and L, and the stored energy in the inductor L is dissipated in resistances B 1 and B 2 and in switching Core 1' and Core 2' (if the latter is initially not fully set) by current pulse i S or in swftching Core 1' alone (if Core 2' is initially fully set) by current pulse i Note that the current pulses i and i driving the sr" s $r stage under discussion in Modes I and III, respectively, have been gen- erated in a similar fashion in the previous stage. Mode IV--A current pulse i e clears Core 1 at a relatively slow switching rate, while Core 2 remains in a CLEAB state and the transistor is in the OFF state. b. RANGE OF SUPPLY VOLTAGE As the supply voltage, V, decreases below a certain value, the binary counter continues to function properly even though the cores are not fully set. A further decrease in V will cause a further decrease in _b 1 and _b 2, and it becomes necessary to determine how low _b 1 and _b 2 (or V) can be before the binary counter starts to malfunction. Let us examine incomplete flux switching during a cycle of two counts (four modes of operation), as shown in Fig. 18. In Mode I, switching is incomplete 53 f INITIAL _1 AND _Z/ i _-_r FINAL _# • I'XCrtlTIAI F ---_- (o) MODE I (b) MODE Tr FINAL 41_ ,f_ F2)_IN _ \_r42( l,_ j o A411ttA FINAL_2J TRANSISTOR TURNS ON ,, _ INITIAL 41_f4 _ _fl(Fl) F --" $1(FI) _) _I(FI) FINAL 41_ND _2 -'J"_r - F .-.-e,. (c) MODE "m" (d) MODE "m" •rc-5og4 -3o FIG. 18 INCOMPLETE FLUX SWITCHING IN FOUR MODES OF OPERATION OF A CORE-DIODE-TRANSISTOR BINARY COUNTER 54' because i s either decays too fast or has a too-low amplitude or both "(flux switching is mmf limited). In Mode II, Core 1 is cleared to -¢r (current i c is high for a long enough period to complete the required amount of switching). In the beginning of Mode III, to be referred to as Submode III-A, both cores start switching toward positive saturation as in Mode I, until Core 2 reaches positive saturation, and the transistor turns on. Core 1 completes its switching to +_r' and Core 2 is then cleared to -_r because the transistor stays saturated as long as non- negligible flux switching takes place in at least one of the cores. In Mode IV, Core i is fully switched to -_r because i c is high for a long enough period. We see that the binary counter operates properly, even though only partial switching takes place in Mode I. How low can _b 1 and Z_b 2 be in Mode I? To answer this question, let us refer to Submode III-A. In order to turn the transistor on, we must assure that Core 2 stops switching before Core 1 stops switching, i.e.. that Core 2 reaches positive saturation before i s decays to the stop-switching threshold value of Core 1. Let _2I and _k_2iiiA denote /_b 2 in Mode I and Sub- mode III-A, respectively (ideally,'Z_biiii A = 0). If both Z_bii and _b2iii A are mmf-limited, and both are assumed to be governed by nearly the same switching parameters, then _b2i _ _2IIIA _ _r' because both result from the same insufficient drive current, i s. Since we do not want _2I ÷ _2IIIA to be less than 2¢r , we conclude that _2I must exceed _r" Let us examine _k_2IiiA more carefully. In Submode III-A, Core 2 Is is initially in a partially switched state, and therefore both _ and F 0 are smaller (e.g., by 30%) than if initially 42 = -_. These two factors are opposing: the decrease in k requires a larger _2I' whereas the decrease in F_ allows _bii to be smaller. Which factor predominates depends on the magnitude of the average net mmf of Core 2 during Sub- t! mode III-A relative to F 0. We must also remember that because of the "wing" in the static _(F) curve, _2I ÷ _2IIIA may be slightly smaller than 2_r. It is thus quite clear that exact calculation of the lower limit of _2I iS extremely complex. As an approximation, we shall main- tain our previous conclusion that in order to turn the transistor on in Mode III, we require that _b2I > _r" This criterion may be used to determine the minimum value of V, Vmi n. 55 Since the fast flux switching in Mode I determines the bottom boundary of the supply voltage, and since the slow flux switching in the. other modes of operation can be analyzed manually by assuming that the net mmf follows the static _(F) curve [cf. Figs. 18(b), (c), and (d)]--we shall proceed with the analysis of Mode I only. 2. ANALYSIS OF MODE I a. SIMPLIFYING ASSUMPTIONS Before analyzing Mode I, let us refer to Fig. 17 and examine the termination of Mode III. When Core 2 is cleared to negative satura- tion and _2 approaches zero, the transistor starts to pull out of satura- tion because of insufficient base current. As a result, i c starts to drop and a positive elastic _2 is generated. The voltage N82_2 , which is clamped by Diode dl, acts as a reverse bias on the transistor, and thus causes the latter to turn off faster than if the bias voltage were zero. As soon as the drop in i c is such that di e -L - i R (48) dt c I , Diode d 2 starts to conduct current iL . Equaticr, (48) also holds true when the transistor is completely turned off. The variation of ic during the transient period in which the transistor is pulled out of saturation depends on the base curre_nt and the collector current. A solution for i c during this period is complex and is beyond the scope of this work. We shall, therefore, simplify the problem by assuming that during the above transient period, im(t) , as it rises from zero to I c (the current value that L tends to maintain), is given and that Eq.(48) holds, i.e., the voltage across R I and L is di L u R = B 1 i L + L -- (49)1 -L dt b. BASIC EQUATIONS Mode I begins as soon as the transistor of the previolts stage starts to pull out of saturation. Based on the simplifying assumptions above and since the forward voltage drop across Diode d 2 is much smaller 56 than Rli L + L(diL/dt), the equivalent circuit for Mode I is as shown in Fig. 19. The total resistance of the windings with Ns1 and Ns2 turns is denoted by R 3 and that of the windings with NBI and NB2 turns is denoted by R 4. ÷ Ra + L R 2 ÷ /NBI iL iL--iS iS _k ' id NS2_,_ _N82 . c - \\ J/ ", dl ÷ id TA- 5094- 31 FIG. 19 EQUIVALENT CIRCUIT FOR MODE T OF A SINGLE STAGE IN A CORE-DIODE-TRANSISTOR BINARY COUNTER There are seven equations with seven unknowns. The seven equations include two mmf equations (one for each core), two _(F,¢) equations (one for each core), and three loop equations. The unknowns are the following time variables: i L (beyond its _ise time), i s (the current driving the cores), i_ (the load current in Diode dl) , Fl(the net mmf of Core 1), F 2 (the net mmf of Core 2), _1 (the flux of Core 1), and ¢2 (the flux of Core 2). By inspection of Fig. 19, the mmf equations are and F I = N, lis + Ns.li a F 2 = Ns2i s - NB2i d (50) (51) 57 Using Eqs. (2) and (18) through (20), @1 = EF1 + @(Fl'_bl) and _2 = EF2 + @(F2'_2) ' where _(F,4) : _p{1 - [(24 + @s - @d)/(@s + be)] 2}. used as an approximation for _,,(F,_) + _i(F,t,Ti) , Eq. (22). tion of Fig. 19, the three loop equations are and di L L-- dt + iLB 1 + isB 3 + Nsl_bl + Ns2_b2 = 0 isB 3 + N i_1 + N z_2 - ( i m - i )B 2 = 0 where [cf. Eq. (42)] idR 4 + V d + Nnl@ 1 - NB2_ 2 = 0 , V a = lab d + E k In + is the forward voltage drop across Diode d 1. We now wish to apply the models for inelastic and elastic switching in the numerical solutions of Eqs. (50) through (56). 3. TRANSCENDENTAL SOLUTION OF CURRENTS (52) (53) Note that _(F,_) is By inspec- (54) (55) [56(a)] [56(b)] We wish to solve Eqs. (50) through (56) at every nth At. For the sake of conveniellce, the subscript n will be omitted entirely and the subscripts (n-l), (n-2), etc., will be replaced by the subscripts (-1), (-2), etc. In other words, at every nth At, we identify n with zero. Since the functions F 1 vs. t and F 2 vs. t are not known a priori, we shall evaluate F 1 and F 2 as ratios of differences rather than dif- (57) ferentials, i.e., /_'i = [F1 - FI(-1) ],/_t 58 and k2 -_ [F2 - F_¢_I_]/At (58) ° Note that FI(_I) and F2(_1) are known quantities which were evaluated while solving for the time variables of the previous At. Substitution of Eqs. (50) and (57) into Eq. (52) and substitution of Eqs. (51) and Eq. (58) into Eq. (53) results in two simultaneous nonlinear differential equations which are formally expressed as and _x = fl(i,,ia,¢l ) (59) _2 = f2(is,id,¢2 ) (60) Substitution of Eqs. (59) and (60) into Eq. (54) gives di t dt L LR1 +" isR3 +Nslfl(is' id'qbl) +Ns2f2(is'id'qb2 (61) Note that Eqs. (59), (60), and (61) form a set of three simultaneous differential equations of the first order. If Eqs. (55) and (56) did not ir, clude _1 and _2' they could be solved for i s and i a. Knowing i s and id, we could then use conventional methods in order to solve Eqs. (59) throug}, (61), which form a set of three simultaneous differential equations with differentials on the left side and functions of unknown time variables (_,1,_2 , and iL) on the right side of the equality signs. However, since Eqs. (55) and (56) do include _1 and _2' we have to modify the conventional methods of solving differential equations by incorporating a transcendental solution for i s and ia. The latter will be done next, using Newton's method for solvinz two simultaneous implicit equations. 9 Following Eqs. (56) and (55), we are looking for the roots i a and i s of the equations f = NB2_ 2 - NBI_bl - id(R d ÷ R 4 ) - E k In 1 + = 0 (62) 59 and g = ns2_ 2 + Nsl_ 1 + i (R 2 + B 3) - imB 2 = 0 (63) Following Eqs. (50) and (51), 3F1/3i a 3F1/'3i s = N 1 , and _F2/_i s = Ns2. f and g are as follows: = NBl,3F2/3ia = -NB2 , Hence, the partial derivatives of N22_p_ + N_l_b' 1 + Ba + R, + + Ii d , (64) _g 3i $ N 2 ", N 2 ", _2_2 + _1_1 + B2 + B 3 ) (65) and 3f bg 3i 3i d = Ns2NB2_' 2 - NsINsI_' I (66) The corrections to be added to i d and i are negligible) are 9 _i d = _ f at each iteration (until both $ (67) and 8i,_ = _- f 3i a g (68) where D = - (69) Under certain conditions, the convergence to the correct solution of i a (or i s ) is oscillatory and slow. This can be detected by noticing that 8i_ (or 8is) alternates sign between two consecutive iterations and that its absolute value decreases by only a small percentage. In many cases, this situation may be remedied by correcting i a (or i s ) by half of 8i a (or 8is), as was done in Report 3 [Eq. (58a), p. 42]. 6O The transcendental solution of i_ and i s described above will be • incorporated into the numerical solution of the time variables, using two types of numerical solutions of differential equations, a simple method and a more complex one (Runge-Kutta's followed by Adams'). This duplication is designed in order to check the accuracy of using a simple method in this type of problem by comparing the results with the ones obta:ined by using a more complex and exact method. . COMPUTATION USING A SIMPLE METHOD a. METHOD OF SOLUTION Simple predictor-corrector equations, similar to Eqs. (29) and (31), are used. Letting y stand for 41,¢2 , and iL, these are: Y = Y(-2) + 2Aty(-1) (70) for prediction, and, after y is evaluated from the differential equation, Y = Y(-t) + 0.5 At[3; + _{_1)] (71) for correction. b. OUTLINE FOR COMPUTER PROGRAM The computer program for the core-diode-transistor binary counter, using a simple method of solution, is given in Appendix G. The program is written in ALGOL-60 1.anguage, and has the following outline: (1) Declare all identifiers (core parameters, circuit parameters, variables, and mis- cellaneous), output lists and formats. (2) Declare PROCEDUREs _(F, At,NV,_), Appendix A, and _(F,¢,_,_*), Appendix D-2. (3) Set values of core and circuit parameters. (4) Set initial values of variables. 61 (5) (6) (7) 7- Compute a rough approximation for the value of _ using the relations* $# Is R 2 +0.6 Pp(N2sl + N 2$2 ) and S 2_r(g21 + N22 ) 0.6 NB2P p _s(N$1NB1 + Ns2NB2) - F_(NB1 '+ N82)] Compute I c = V/B 1. Print heading (core parameters, circuit parameters, and output variables) and initial values of output variables. For every nth &t during switching time, do the following: (a) Lower the &t index of the time variables computed previously by one. (b) Set &t = _ /lO00 during the rise time, Tr, of iL(t) and At = Ts/500 for t _ T r . (c) If t _ Tr, compute i L from a given empirical expression. (d) Compute first approximations for iL(if t > Tr) , _1' and _2' using Eq.(70), and for i _ and id, using the relations i _ 2i -i and i _ 2i -i s s (-1) s (-2) d d(-1) d(-2)" (e) Until a specified convergence condition is achieved, compute the following steps in a loop : fl[Eq' (50)]; _'2 [Eq. (51)]; F l[Eq. (57)]; F 2[Eq. (58)]; _mal = _(Fl,qbl,_'mal,_ _) [Appendix D-2]; _P¢ I = _P¢ (ki'At' NV'_P'(I ) [Appendix A]; +4';1 _al ¢i (72 (73) Equations (72) and (73) have been derived by using average, constant values of R, P-_= 0.6 in a rough calculation of the switching time for &fib1 = 2¢r. PP' 62 (8) t . _' = _p(F2,qb2,c_..2,_ 2) [Appendix I)-2]" a2 _: = _(k2, At, NV, q_2)[Appendix A]; • , +, _2 = ma2 e2' If t > T r, diL/dt [Eq. (54)] and t L [Eq. (71)]; #,:[Eq. (71)]; q_2[Eq" (71)]; Vd[E q. (56b) if NB2_b2 > NBI@I; otherwise, NB2_P2 NBI@I]; ] [Eq. (62)]; g [Eq. (63)]; 3f/_id[Eq. (64)]; _g/$is[Eq. (65)]; _f/_is[Eq. (66)]; _g/_id = -_f/_i •$ w D [Eq. (69)]; 3i d[Eq. (67)]; 3is[Eq. (68)]. Add _i d to previous id, but if the sign of _i d is different from the previous one, add only 0.5 _i d" add Si to previous i but if the sign of $is is different from the previous one, add only 0.5 3i,. Repeat the above steps if either Sidl > 0.0001 lidl or [3t I > 0.0001 li,l, but no more than 19 times. Count the number of times this convergence condition fails to be satisfied. Bepeat Steps (a) through (e) unless _l = 0 and _2 = 0 and ¢1 > -0.9 er" Print output (t, i L, i, Fa,¢i,_l,F2,¢2 _2, id, Vd, _,1,_, number of iterations, cumulative number of convergence failures), say, once every 20th At. 63 C. EXPERIMENTAL VERIFICATION A single stage of a core-diode-transistor binary counter, Fig. 17, was built using the following components: CoFe Lockheed 100SC1 ferrite core (OD = 100 mils, ID = 70 mils, h = 30 miIs) whose switching parameters at room temperature (25°C) are as follows: l i = 5.59 mm; l = 7.98 mm; qb r = 6.25 maxwells; qbs = 7.00 max-O wells; H = 290 amp-turns/m; Hq = 42.7 amp-turns/m; H = 38.0 amp-turns/m; v = 1.207; n t! k = 0.64 ohm/turn2"2°7amp°'2°7", F 0 = 0.35 amp-turn; pp = 0.948 ohm/turn2; F 0 -- 0.805 amp-turn; F B 3 0 amp-turn. Circuit Number of turns: N 1 = 11; Ns2 = 12; Ns1 = 16; Nn2 = 20; N¢1 = Nc2 = 12. Resistances (ohms): B I = 107.36; R 2 = 199.55; R 3 = 0.34; R 4 = 0.53. Inductance: L = 0.202 millihenry. Diodes d 1 and d2: FD 643; E k = 0.0578 volt; I 0 = 0.0615 microampere; R d = 0.1 ohm. Transistor: 2N1613. Drtve Supply voltage (minus collector-emitter voltage drop): V = 27 volts; 8.6 volts. Rise time of it: T r = 0.13 microsecond. Using a library computer program for polynomial curve fitting, the following empirical expressions for iL(t) during 0 < t 2 T r were fit to the rising portion of iL(t_ waveform (V = 27 volts) from i L = 0 (at t = 0) to i L = 0.252 ampere (at t = 0.13 _sec), in which I c = V/R1, t is in seconds, and T = 13 - t 108 • i L = 1.02 • 1019t8/3 if 0 < t < 0.02" 10 "6 4" 106t - 0.05 if 0.02" 10 .6 < t < 0.06" 10 -6 0.252- 0.01(0.00947 + T{-0.3169 + T_ [1. 729261 + T(-0.575947 + T • 0.073769)]}) if 0.06" 10 -6 < t < 0.13" 10 .6 (74) 64 The above core, circuit, and drive parameters were inserted "into the computer program, and the results were machine plotted. Experi- mental oscillograms of iL(t), is(t), _l(t), _2(t), i_(t), Vd(t ) were superimposed on computed waveforms. These are compared in Fig. 20 for Y = 27.0 volts and in Fig. 21 for Y = 8.6 volts. OJomp !t) i I?: i OJomp m -t--: _.:: ::: ......... -7-- .. .* .... *. --t-_..t. ..I,. .I Is : ....... I"' " ;,-'COMPUTED • / " i" .... ' ' _L_ 0.25 volt/turn :::::::::::::::::: :::::::: :.: -4, :_!::!t ::::l:!:!l!ii!t:;!!l!:;i':::: ": I " : 4._--EXPERIMENTAL 1 : I : _ : : ..... ...... t_ i ..... _ .... ; .... =---_-.-1-:&Zcoi_'PUTb)!:1_;::1:::: - - - q--4- ---4L-;4;--: : :.1 :: ;it_.. It. ::- _iiit-=ti ii-:-!_!li!iiii ii!itliiili!ii ! it¸ I- .... I ...... >-EXPERIMEN'I;AL :1 " : :1 " -i .:t i:,,_. t_ _-1!::il:: i:L!:i:li i i! pJ!:!-._tii!_:i!if:i!!]i!?: volt/turn _ :._....... !!!!]:!!!J: : .... i .. i .... | .... i.* ,i. i ..* . iiiii]i:i:t!!::z!!i:l!!iili : 1 0.5 volt/turn t ' :.L_ L,-4-:-i ,, :Vd: ........ -'t' _ .... _ i :i._.fl__:1--ii 14i:__. I [l]]]]11I]11 r]_ ,_T_tl _Itt, .,. ::_l::::tii:;t:i_ =]_!I ]=;I:J:f"' _. l] 1 I] | I l ] llIIll' i | I I1 : 11 I_ J!l ....milliomp _ "iii [AJ' Ill mlilllllllll [11111111 Illlllllll[ill]llilllllJ I1 t!11 _ll IIIIIIIIIILJL.L_COMPUTED _'R_J ] ]A"I .......... I_'IL[_IIIIIII]IIII l_lll_m¢_llll|llllll tlitltl H+H--H ]l_llll FIG. 20 EXPERIMENTAL AND COMPUTED CURRENT AND VOLTAGE WAVEFORMS IN MODE I OF A CORE-DIODE-TRANSISTOR BINARY COUNTER FOR Y = 27 VOLTS Time scale = 0.1 vsec/moior div. 65 0.02 omp :i L ....... i :::i .... ,.':i ':, :" ................... ii ..... ,. , .... l+,..l ......... .'.Z ................ : .......... :::, _..,6 .... I ...... 0.02 omp is ...... r4 '_'' '';'t .... ! .... .... f........ __: l ' • [ I .... EXPERIMENTAL"., . I .... l ......... i .... 0 .... ,,,*.,,...i • I :Ii!!:I!:)I!:!!I!:: 0.05 volt/turr ::i :/_ i:::I!I:__"_:'_EXPERIMENTAL:t::: .... i.. i . , ....... ,.. ..,.o ......... ,. oo -f--m volt/turn 0.5 volt/turn !-_: EXPERIMENTAL" :: :._ ""_C0MPUTEO ::" [ _-._'-_ :.EXPERIMENTAl TC - 5094 - 33 FIG. 21 EXPERIMENTAL AND COMPUTED CURRENT AND VOLTAGE WAVEFORMS IN MODE ]" OF A CORE-DIODE-TRANSISTOR BINARY COUNTER FOR V = 8.6 VOLTS Time scale = 0.2/zsec/major div. 5. COMPUTATION USING RUNGE-KUTT_ AND ADAMS METHODS a. METHOD OF SOLUTION The computation described in the previous section is based on a simple method of solving a set of differential equations. One may, then, doubt the accuracy of the computation. In order to check this accuracy, a second computer program was written, using conventional and more complex methods of solution: the Runge-Kutta method for starting the computation in the first four At, and the Adams method thereafter. The description of these methods may be found in several textbooks. 7'9 66 b. CONPUTER PROGRAM The computer program for the core-diode-transistor binary counter, using the Runge-Kutta and Adams methods, is given in Appendix H. The program language is ALGOL-60. The outline of this program is similar to the outline of the program using simple methods of solution, Part B-4-b, except for the following. The differential equations and the transcen- dental solutions for ia and i s are declared in a special PROCEDURE, F(X, Y, DX) in which X stands for the independent variable (t), Y stands for an array of the dependent variables (_I' ¢2, and iL) , and DX stands for an array of the derivatives (_:. _2' and dit/dt). The computer program for the Runge-Kutta method is a slightly modified version of a library PROCEDURE RKSTARTS (K, NF, X1, H, Y, YPB, F), m where K is the number of differential equations (here, K = 3), NF is the number of At steps for which the computation is repeated (with the use of Adams method, NF = 4), X1 is the initial value of the independent variable (here, X1 = t o = 0), O is the step size (here, H= At), Y and YPR are identifiers for two (NF + 1) • K arrays for storing the values of the dependent variables and their derivatives, respectively, (at t = 0, At, 2At, ...,NF • At), and F is the PROCEDURE F(X, Y, DX). The computer pro- gram for the Adams method is also a slightly modified version of a library PROCEDURE,I:ADAMS (X, Y, YPBIME, N, EU, EL, EPS, H, HMIN), where X is the independent variable (t), Y and YPBIME are arrays similar to those used above for the dependent variables and their derivatives, N is the number of differential equations (here, N = 3), EU and EL are upper and lower bounds, respectively, of relative error, EPS is the minimum absolute value for any dependent variable to be used as a reference for evaluation of the relative error, H is the step size (At) that depends on EU and EL (if any equation error exceeds EU, H is halved, and if all errors are below EL, H is doubled), and HMIN is the minimum value that H may be reduced to. It can be seen in Appendix H that At = _s/500, where _ is evaluated from Eqs. (72) and (73), and min(At) = At/lO. Also, during 0 5 t 5 T r , EU = 0.001 and EL = 0.0001, whereas if t > T , EU = 0.005 and EL = 0.001. For t > T, diL/dt was solved for, using Eq. (54), but for t _ T r, diL/dt was derived by differentiating Eq. (74): 67 di t I dt O. 252 2.72" 1019t s/3 if 4" 106 if 10s{-0. 316910 + T [3.458522 + T(-1.727841 + T' 0.295076)]} if O< t < 0.02.10 -6 0.02" 10 -6 < t < 0.06" 10 .6 " 0.06" 10 -6 < t < 0.13" 10 -6 where t is measured in seconds and T = 13 - t l0 s (75) C. I_ESULTS The computed results came out very close to those obtained by the simple method of solution. The largest differences were in the third place after the decimal point, and could hardly be detected in the machine plots of the results. This is demonstrated in Fig. 22 by superimposing computed _l(t) and _2(t) waveforms for V = 27 and V = 8.6 volts obtained by the two methods of solution. The agreement between any other pair of computed waveforms is either similar to or better than that shown in Fig. 22. 6. DIscussION a. VALIDITY OF COMPUTATION The agreement between the computed and the experimental results is satisfactory in general. In the case of ia(t), the agreement is poor because i d is a nonlinear function of the voltage across the diode, Vd, and because V d is the difference between two voltages, Ns2_ 2 and NBI_I, whose magnitudes are close to each other. Any small relative error in b e or _l or both (if of opposite signs) will result in a magnified rela- tive error in i_ if V d is above the "bias" voltage, i.e., above the knee of the forward characteristic of the diode. The computation of the time variables during the rise time of iL(t) is limited by two drawbacks: First, the collector current, ic, is ignored by assuming that L(dic/dt) + _cR: = O, Eq. (48), while the tran- sistor pulls out of saturation. Second, empirical expressions for iL(t) are fit to experimental data. Both drawbacks could be overcome by incor- porating a dynamic switching model for the transistor. This was not done because of the added complexity resulting from the use of this model and 68 r- c i. =i o tO OJ d J_ I- I I i i i I i I I I I I I I ---.- -.--O.IFsec I I I I I I I I tI I I I I I I I I --_ 0.2/_sec l I I I I I I I I I I I I I I I I I I I (a) V=27volts (b) V=8.6volts TC -50e4-_ FIG. 22 SUPERPOSITION OF _(t) WAVEFORMS COMPUTED BY A SIMPLE METHOD AND q_(t) WAVEFORMS COMPUTED BY RUNGE-KUTTA AND ADAMS METHODS because such a model is beyond the scope of our work. Instead of fitting an exact function for iL(t) , we could have assumed a sinusoidal function or a ramp function between t = 0 and t = T r. Although such alternatives do not depend on curve fitting, the rise time still has to be furnished and the results of computation include larger errors. Another alterna- tive with an even larger error is to ignore the switching time of the transistor and assume that T r = O. 69 b. VARIATIONS OF _I(FI) and _2(F2) DUaING SWITCHING TIME Having computed FI, _i' F2' and _2 vs. t, we are able to deter- mine the variations of _1 vs. F1 and _2 vs. F 2 during the switching time by treating t as a parameter. These variations were machine plotted for the cases of V = 27 volts and V = 8.6 volts, and are shown superimposed on manually plotted static @(F) curves in Fig. 23. It can be seen in Fig. 23 that the flux switching is complete for g = 27 volts, but for V = 8.6 volts, the flux switching is only partial. Based on the criterion that the lower boundary of V, VAin, is determined by _92I = Cr (el. Sec. II-B-l-b), it can be seen that V = 8.6 volts is close to Vmi ". By repeating the computation with other values of V it was found that VAi n = 7.1 volts. This value agrees with experimental observations. c. METHODS OF COMPUTATION The results of computation using the Runge-Kutta and the Adams methods of solution were practically identical with the results obtained by using the simple method of computation. Hence, the results of either method may be trusted to be essentially the true solutions, i.e., the computational errors are negligible. Although library computer PROCEDUREs are available for both the Runge-Kutta and the Adams methods of solution, the application of these PROCEDUREs is not easier (and may even be harder) than writing the whole computer program from scratch and using the simple predictor- corrector method. This is so because most of the computation steps using the simple method, which revolves around the transcendental solu- tion of i d and is, as well as the differential equations, must be pro- vided in the F(X, Y, DX) PROCEDURE. In addition, for the same specified accuracy, the computation time (including compiling time) using the simple method is much shorter (approximately by a factor of 4) than the one using the Runge-Kutta and the Adams methods. Since the results of the two methods of computation are essentially the same, and since the simple method requires much less computer time for the same accuracy, we conclude that, for problems of this type, the simple predictor-corrector method of solution is preferred. We expect this to be true for even more complicated magnetic circuit problems. 7O IO 5 i° 5 I - ! @_" F2T /..@,,,_.F -- I ) J EXPERIMENTAL o o -- I _ COMPUTED ! i =l= m I 2 3 4 F (amp-turns) (O) V= 27 volts _0 I } I I ] I I I I STATIC _(F) CURVE 5 W ]= 0 O E -8- --5 -- -I0 0 EXPERIMENTAL o o .... .9..- 0COMPUTED o fol"_o oP °/ J 2vs. F2 ! ! ! ! I { I I I I I 1 l l 0.2 0.4 0.6 O.B 1.0 F (Drop-turns) (b) V= 8.6 volts FIG. 23 VARIATIONS OF _1 vs. F 1 AND _2 vs. F 2 DURING MODE I OF A CORE-DIODE-TRANSISTOR BINARY COUNTER T1 _. SUMMARY The operation of a core-diode-transistor binary counter is described by dividing a two-count period into four modes of operation; Mode I, dur- ing which both cores switch simultaneously relatively fast; and Modes II, III, and IV, during which one core at a time switches relatively slowly. A computational analysis of Mode I is provided for evaluating five time variables (¢i, 42, it' is' and i a) by solving five implicit equations, three of which are differential. The computation involves Newton's method for the transcendental solution of i s and ia, in addition to a predictor- corrector method for the solution of a set of differential equations. Two methods of solution are applied, a simple one and the Runge-Kutta and Adams method. The results are practically identical, and the agreement with experimental data is satisfactory. The minimum supply voltage is computed, and its value is in agreement with experimental observation. C. FLUX DIVISION IN A LOADED SATURABLE CORE 1. INTaODUCTION The problem of flux division in a multipath core was described in Report 2 (pp. 24-48; 83-95). Calculation of flux division in a loaded saturable three-leg core, which is initially in a CLEAR state, was carried out. The calculation was based on the simple switching model = _p[1 - (_/_s)2], which is valid only if F is large enough to switch from _ = -_ to _ = +_ . Despite the simplicity of this model (relative to the more exact models we now employ), the calculation involved a large amount of algebraic manipulations and, except for very high drive current, instantaneous values of F had to be approximated by time-averaged values ofF. The agreement with experimental data for low drive-current amplitude was not satisfactory because of the limitations of the model. In addition, it was found later 12 that the experimental data, which had been recorded at a temperature of 29 ± 2°C, was very sensitive to temperature, and as a result O had to be recorded again at a temperature that was tightly controlled at 29 ±0.5 C. Our objective now is to improve the computation of flux division de- scribed in Report 2 by overcoming most of the above-mentioned drawbacks. SpecificalIy, we shall employ our latest switching models for _a and _i, we shall avoid the struggle against a complex algebra by resorting to a numerical solution of the basic equations, we shall compute instantaneous 72 (rather than average) values of the time variables, and we shall compare the results with the more reliable experimental data 12 (taken at 29 ± 0.5°C). 2. EXPEBINENT The core geometry, the circuit, and the drive current involved in the flux-division experiment were given in Fig. i0 of Beport 2. For the convenience of the reader, this figure is reproduced in Fig. 24. A three-leg saturable ferrite core, whose main leg (Leg m) has twice the flux capacity of each of the other legs (Legs 3 and 4), is initially in a CLEAR state, i.e., _= = -2_ r and @3 = ¢¢ = -¢r" With Leg 4 loaded by a resistance R t across N L turns, a SET drive mmf of constant amplitude, NI, is applied to Leg m, and its duration, T, is adjusted so that _b = 2_. We wish to compute the variations of F(t), _(t), and _(t) in each leg during the switching time and then compute the flux division ratio, D = _b3/_b 4, for various magnitudes of drive current, load, and leg-length ratio, l¢/l 3. W m LEG m \,r W 3-"ID I I ! I • / 4J " W 5 # W4 " W m /2 ill - 3llml-lil FIG. 24 FLUX DIVISION IN A LOADED SATURABLE THREE-LEG CORE 73 3. COMPUTATIONAL ANALYSIS a. BASIC EQUATIONS Because inelastic flux division is of interest, we shall neglect the elastic component of _, _e" However, since NI may be relatively low, we have to distinguish between _i' Eq. (5), and _.a, Eqs. (18), (19), and (21). Only if NI is large enough to switch all legs relatively fast can we combine _i and _a into _ expressed by Eqs. (18) through (20). Designating _a, _i, _' 4, F, and T_ of Leg m, Leg 3, and Leg 4 by Sub- scripts m, 3, and 4, respectively, three differential equations are written formally as follows: _. = _..,,(F.,4m) + SPi,,(Fm,t,Ti.), (76) and _3 = $_s(F3,43) + _i3(Fa,t,Ti3 ), _4 = _.a4(F4'44 ) + _i4(F4't'Ti¢ ) (77) (78) By inspection of Fig. (24), F 3 = F 4 + (N_/RL)_ 4, (79) and F =.NI - F 3, (80) &. = &s + _4 (81) Equations (76) through (8i) contain six dependent variables: 4_, 43, 44, F, F a, and F 4. We now wish to solve for these variables numerically. b. METHOD OF COMPUTATION For a given value of t, the values of Tia , Ti3, and Ti4 are known, and _4' Eq. (78), is a function of F 4 and 44 . Therefore, foIlow- ing Eqs. (79) and (80), F 3 and F are also functions of F 4 and 44 . When the latter are substituted into Eqs. (76) and (77), we obtain a set of three differential equations of the form 74 ¸ and (82) (83) _4 = f4(F,,_4 ) (84) Any of the conventional methods of solution of _., _3, and _4 must be supplemented by the transcendental solution of F4, which is based on substitution of Eqs. (82) through (84) into Eq. (81). Thus, we are looking for the root F 4 of the equation f = _3 + @4 - _- = 0 (85) Following Eqs. (79) and (80), BF, bF 3 NL2 , - -- = 1 +--B, (86) Differentiation of Eq. (85) with respect to F_, and substitution of Eq. (86) give _f 3F 4 - f, = (_' + _3 ) + -- c_ + _b4 (87) RL Following Newton's method, the iterative correction for F 4 is -f/f'. However, if the sign of this correction alternates, the correction will be only -0.5f/f' C. SWITCHING PAHAMETERS The switching parameters of the three legs differ from each other because of differences in leg dimensions (width and length). These parameters are evaluated using the following material parameters (cf. pp. 37- 41 of Report 2, and Sec. I-B-2-e in this report): B r, B s, H a, Hq, H, tl l-lo, v, K, H o, H B, _p" _'i' Ki, M i, and Hoi. 75 In analogy to Eqs. (24) and (25) of Report 3 (pp. 17-18), the continuity of Bp vs. H (cf. Fig. 15 in Report 2) at H =H B requires the following relationships: II vH o - H o H B - (88) _' - 1 and II ¢p = Kv(H B - N0 )_-I (89) Following Eqs. (88), (89), (91), and (92) of Report 2 (pp. 39-40), Eqs. (26) through (29) of Report 3 (pp. 18-19) and Eqs. (14) through (16) of this report, the switching parameters of a leg are expressed by the following functions of material parameters and leg dimensions II = II .[li, lo, l = (l i + Io)/2, and A = hw]: Cr = ABr; Cs = AB; F o IHo, Fs = lHs; F o = lHo; pp = _pA/l; k = KA/(IV); F12 = Hqli; F23 = Hqlo; v 1 = (_- _r)/[H.(zo - /i)]; v 2 = (_= + _,)Hq/[_ (Zo - Z_)]; . k., = KiA/(l ')', C i = Mil', and Foi = Hoili . These expressions are used to compute the switching parameters of Leg _, Leg 3, and Leg 4. d. ESTIMATION OF 7- s FOR DETERMINATION OF At In most of the magnetic circuit computer programs described so far, we have determined At as a given fraction of an estimated switching time, _, e.g., At = _/200. The value of 7- was calculated roughly using fo - - _ 6pp "basically the relation 2q5r = SpF dt, where p = 0. is the average II value of p, and F = F - F 0 (actually, F -- F - F0, but in order to ob- II It tain a positive value of _s if F 0 < F < F 0, F 0 was replaced by F0). Sees for example, Eqs. (46), (47), and (90) in Report 3(pp. 26 and 63), and I{ Eqs. (72) and (73) in this report. As long as F is not close to F0, such an estimation of _-s, crude as it is, is accepted because the length of At is not crucial. However, if F is close to F0", the resulting At becomes much - " . In order to get a closer estimatetoo short because pp (F F O) >> _p tI , = L(F - Fo) v should be used instead offor T s the relation _p ° _p = pp (F -Fo). Hence, r -- (90) O. 3L(F - F0) _ II Equation (90) should be used if F 0 < F < F 8, but may also be used if F is not much larger than F B. 76 Suppose that Leg 4 is so long that _bt = _3" In this case, the switching path is composed of Legs m and 3 in series. If NI is not much larger than F"0_ + F03", then H or H 3 is not much larger than tto," and the difference between H and H s (cf. Report 2, Fig. 16, p. 43) may be neglected. Under this condition, the relation _ = _ = _3 leads to an approximate expression for K of Legs m and 3 in series: 1 = (91) + _ Substitutions of Eq. (91), F = F + F 3 = NI, and _r = dPr./2 into Eq. (90) give /1 +,_ q5 /XI/_ X1/_ _ (92) •s 0.6 I - F0. - F o e. DRIVE CURRENT In view of the fact that the drive current has a finite rise time, Tr, we shall replace NI in Eq. (80) by Ni, where "(, - cos 77 if t < T rNi = (93) NI if t >_ T r 4. CONPUTER PROGRAU a. _ PROCEDUREs The switching parameters of the three legs are different (because of the difference in dimensions). Hence, six separate PROCEDUREs are needed in order to compute _i and _.a of each leg. In order to save program writing, DEFINE declarations 13 are made only once for the steps involved in the _i(F,t,T i ) PROCEDURE, Appendix B, and the _.a(F,_,@d,_') PROCEDURE, Appendix C. Each of these declarations is used in a correspond- ing _ PROCEDURE of a given leg after its identifiers have been identified 77 (by using DEFINE declarations) with the corresponding identifiers of the given leg. For example, the _im(Fm,t,Tim) PROCEDURE is generated by identifying F with F, @m with 4, etc., and by calling the identifier of the DEFINE declaration in which the steps involved in the _i(F,t,Ti) PROCEDURE are included [alternatively, we may call the _i(F,t,T i) PRO- CEDUBE itself, but this results in a longer running time of the computer]. There are other ways to achieve the same goal. For example, the switching parameters may be declared as formal input parameters of the ¢ PROCEDURE (in addition to the existing ones). If the _ PROCEDURE of each leg is called (in the main program) more than twice, say, this alternative is too lengthy. Another alternative is to fill three arrays, one for each leg, with the switching parameters, and to replace each switching parameter in the declaration of the _ PROCEDURE by the corre- sponding address in the array. The main drawback of this alternative is that it is hard to associate by inspection a given array address with the name of the corresponding switching parameter. b. MULTIPLE OUTPUT Three types of output may be needed from the computer program of flux division: (1) Flux switching in each leg vs. time for given values of N_/R L, 14/l 3, and NI; (2) D vs. NI, with N_/R L as a parameter, for a given value of 14/13; and (3) D vs. 14/l 3 , with N_/B L as a parameter, for a given value of NI. Which of these output types is to be executed depends on the value l, 2, or 3 assigned to an identifier SW in the program. C. PROGRAM OUTLINE The computer program for flux division is given in Appendix I, using the language ALGOL-60. The program outline is as follows. (1) Declare all identifiers (material parameters, leg dimensions, switching parameters, circuit parameters, variables, and miscellaneous), output lists and formats, 78 (2) Declare the definitions of _i (Appendix B) and _.. (Appendix C). Declare the following PROCEDUREs: _i qb" _i3' _. 3' _i4, _..4' and tanh (X1)."De_[_e 'mm" asal0-3(in order to maintain MKS units while writing leg dimen- sions in mm). (3) Set SW to 1, 2, or 3, depending on the type of output required. (4) Set the values of the material parameters. (5) Set and compute the dimensions and switching parameters of Legs m and 3. Set and compute w4' A4' _r4' and ¢s4" (6) If SW = 3, print heading for type-3 output. (7) For each of several values of S = 14/l 3 , do the following: (A) Complete computation of the dimensions and switching parameters of Leg 4. (B) If SW = 2, print heading for type-2 output. (C) Set the value of the rise time, Tr, of the drive current. (D) For each of several values of N_/R t, do the following: 1. For each of several values of NI, do the following: (i) (ii) (iii) (iv) If SW = 1, print heading for type-I output. Set the initial conditions (t = O; F = F 3 = F 4 = 0; ¢. = 0). Compute rs' using Eq. (92). Set At = "r /200. For each At during switching time, do the following (a) Compute t = t(.1} + At. Compute Ni, using Eq. (93). (b) Lower the At index of previously computed ¢., _,, ¢3' _3' ¢4' _4' and F 4 . 79 (v (vi ' Vi I (viii (ix) rX_ (c) Use the relation y = 2Y(_l) - Y(-2) to predict a first approximation for Y = _' q_3' #54' and F 4 (d) Until a specified convergence condi- tion is achieved, compute the follow- ing steps in a loop; for each leg, use the relation F i = F 0_ tanh (F/F Oi )' Eqo (17), call the corresponding _a and _i PBOCEDUREs, and correct q5 according to qb = qS(_l)+ At[_+_(_1_/2 , Eq. " 31 ;: F4(previous F 4 plus 5F4) ; Fi4; _4 = _,_4 + _i4; F3[Eq" (79)]; Fs3 -_" _ _ _3' F [Eq. (80)]_3 V ma3 i3' " Fi,; _, = _,a, + _i,; _b ; f[Eq. (85)]; ", = @' + ' ; = + • _am i_ 3 Ma3 i3' @4' : c_'m a 4 * _"' ; f'[Eq.4 (87)] bF 4 = -/,/f'(if $F 4 changes sign, SF 4 = -0.5 fir'). Bepeat the above steps if ifi > 0.0001 cP_,/_r , but no more than 19 times. Count the number of times this convergence condition fails to be satisfied. If SW = 1, print type-1 output t, _p,. _., F, _3" _3' F3' _4' q34' F4' Ni, number of iterations) once, say, every fifth At. If _b > -0.99 &r,' terminate switching when either _ _ 0 or _, < 0.0001 volt/turn. Conmuce tn_- net flux changes during the switching time: _63 = qb3 + qb 3 and Compute flux-division ratio, D=Z_b3/_b 4. If SW = 2, store the value of D in a one-dimensional array (at a location corresponding to the value of N[). If SW = 3, store the value of D in a 8O (E) e two-dimensional array (at a location corresponding to the values of N_/R L and 14/13). If SW = 2, print type-2 output (N_/R L, D vs. NI, and the cumulative number of convergence failures). If SW = 3, store the values of N_/R L and the cumulative number of convergence failures in the above two-dimensional array. If SW = 3, print type-3 output (for each value of N_/R£, D vs. 14/I 3 and the cumulative number of con- vergence failures). 5. EXPERIMENTAL VERIFICATION a. CORE The saturable core used in the flux-division experiment was referred to as Core S in Report 2 (Fig. 22, p 62). The core was cut ultrasonically from a 1/2-inch-diameter ferrite disk whose material is the same as that of Core E-6: Telemeter Magnetics, T-5, of nominal composition [Mg0.32Zn0.1Mn0.$a]++[Mn0.26Fe0.74 ]+++02v4" The material parameters of Core S at T = 29°C are given as follows: Br= 0.23 wb/m _ ; B s = 0.2484 wb/m 2", H a = 310 amp-turns/m; H = 35.0 amp-turns/m; It Hn = 30.11) amp-turns/m; H 0 = 40.0 amp-turns/m; v = 1.30; K= 3400 ohms/turn z3 amp°'3m°'7; H 0 = 61.0 amp-turns/m; //8 = 131.0 amp-turns/m [Eq. (88)]; _ = 17,105 ohms/turn2m[Eq. (89)]', v i = 1.3; K i = 592 ohms/turn 2"3 amp °" 3m°'7; M i = 10.7 amp-turns-/zsec/m; and Hoi = 24.8 amp-turns/m. The dimensions and computed switching parameters of each leg are given in Table I. b. CIRCUIT DRIVE Four values of N_/R L (in turns2/ohm) were tested in the experi- ment (Fig. 24): 0 (no load), 1.000 (N t = 1; R t = 1,Q0 ohm), 3.962 (N L = 2; R t = 1.01 ohm), and 9.524 (N L = 2; R L = 0.42 ohm). The temperature was kept at 29 ± 0.5°C. The drive-current rise time was near 0.08 _sec and the mmf amplitude NI was varied between 1.1 amp-turn and 4.0 amp-turns. 81 Table I LEG DIMENSIONS AND SWITCHING PARAMETERS OF CORE S LEG LEG m LEG 3 LEG 4 Dimensions h (mm) 1.31 1.31 1.31 l (nun) 14.363 4.310 7.887 i l ° (mm) 19.151 5.108 9.348 l (mm) 16.757 4.709 8.617 w (mm) 1.016 0.508 0.508 A (man2 ) Switching parameters Cr (maxwelis) qbs (maxwel Is) _k (ohms/turn 2" 3ampl" 3 II F 0 Camp-turn) /op (ohms/turn 2 ) F 0 (amp-turn) I F B (amp-turns) (ohm/turn 2" 3amp 1"3 ) t C i (amp-turn-/zsec) 1.33 0,665 0.665 30.612 15.306 15.306 33.061 16.531 16.531 0.921 2.398 1.093 0.670 0.188 0.345 1,359 2.417 1.321 1.022 0.287 0.526 2.195 0.617 1.129 0.i60 0.417 0.190 0.179 0.050 Foi Camp-turn) 0.356 0.107 O. 092 0.196 c. EXPERIMENTAL AND COMPUTED _( t ) I'¢AVEEORMS A comparison is made in Fig. 25 between experimental and computed waveforms of _m(t), _3(t), and &4(t) for Nf/R e = 0 and NI = 2.0 amp-turns. The computation was performed on a Burroughs B-5500 computer. The com- puted waveforms were drawn manually, using type-1 output results. The ringing in the $(t) oscillograms was caused by the ringing in the drive MMF Ni(t), which is also shown in Fig. 25. 82 FIG. 25 EXPERIMENTAL (SOLID LINE) AND COMPUTED (DOTTED LINE) Ni(t) AND ¢(t) WAVEFORMS OF CORE S (14/l 3 = 1.83), DRIVEN UNLOADED BY IvWiF NI OF 2.0 AMP-TURNS IN A FLUX-DIVISION EXPERIMENT Time scale = 0.2 #sec/maior div.; Ni scale = 1.0 amp-turn/major div.; q_mscale = 0.2 volt/turn/major div.; _3 scale = 0.25 volt/turn/major div.; _4 scale = 0.05 volt/turn/major div. d. MEASURED AND COMPUTED D vs. NI WITH N2/RLL AS A PARAMETER A comparison is made in Fig. 26 between measured and computed D vs. NI for N[/R L = 0, 1.000, 3.962, and 9.524 turns2/ohm. The computed curves were drawn manually, using type-2 output results. e. COMPUTED D us. I 4/I 3 WITH ,V_/R L AS A PARAMETER Computed curves of D vs. 14/l 3 are shown in Fig. 27 for four values of N_/R L (0,5,10, and 25 turn2/ohm) and two values of NI (10 and 100 amp-turns). The core is assumed to be identical with Core S, except for the length of Leg 4. The plots were drawn manually, using type-3 output results. 83 3O 2O IO I CALCULATED ------ MEASURED I I 1 NZ/RL= 9.524 (NL= 2 ; RL= 0.42£ L .,.B-- NL2/RL = 3.962 (NL= 2; RL= 1.01_ (NL= I; RL= I.OO.O, ) LOAD FIG. 26 MEASURED AND COMPUTED FLUX-DIVISION RATIO vs. MMF WITH LOAD AS A PARAMETER FOR CORE S_(/4/l 3 = 1.83) 84 I00 90 I I I I I ..--o--.--o-NZ = I00 amp-turns ----_---x-- NZ = I0 amp-turns I I/ 1 I ,," / / t / 7 / / i / ix // I I / /-- / / / / / / / / / / / / / / 80 -- // / /x /-_/ / / / / // I // Y 60 -- j/ _ // // // -- ," / / / / .I*" _ i/ x/" .// I" / .p/ .,/ // I / ./ / / NL25/RL= ,. I- ,ll /I I i/` I sj /// 20 I0 -- 0 0 I 2 3 4 5 6 7 8 9 I0 l,/t 3--" TC-5094-3D FIG. 27 COMPUTED PLOTS OF FLUX-DIVISION RATIO vs. LEG-LENGTH RATIO WITH LOAD AS A PARAMETER AND NI OF 10 AND 100 AMP-TURNS FOR A CORE IDENTICAL WITH CORE S EXCEPT FOR THE LENGTH OF LEG 4 85 6. DIscussioN a. _C t) WAVEFORMS The agreement between experimental and computed am(t), _3(t), and _4(t) in Fig. 25 is very satisfactory. The slight disagreement may be attributed to the difference between the actual waveform of the drive current and the assumed function for i(t). The experimental initial spikes are higher than the computed ones. This may be attributed to the omission of the _e component from the computation. The frequency of ringing in the Ni(t)is the same as in the _(t) of each leg. We thus conclude that the _(t) ringing is not pickup hut actual variations in _(t). Elimination of the drive-current ringing would prob- ably result in elimination of the _(t) ringing. b. D vs° NI WITH Nf/R L AS A PARAMETER The agreement between experimental and computed D vs. NI for the different loads in Fig. 26 is much better than the agreement in Report 2 (Fig. 36, p. 88) in the entire range of NI, and is better than the agreement in Ref. 12 for low values of NI.* Let us distinguish be- tween two regions of NI: NI between 1.1 and 2.1 amp-turns, and Nf above 2.1 amp-turns. In the region of high NI, the agreement is Oetter than in Report 2, but about the same as in Ref. 12. This may be attributed to two factors. First, the validity of the experimental data in Report 2 is in doubt because the temperature was not regulated tightly enough (29 ± 2°C com- pared with 29 ± 0.5°C in Ref. 12). Second, the switching model used in Report 2 and Ref. 12, _ = _p [1 - (_/_s)2]j is valid only for high F values. This can be seen by letting _d = _s in _(_) of Eq. (20) (which may be used to compute _i + _,a if F is high enough). Consider now the region of low NI. Although there is room for further improvement in the agreement between computed and experimental results in Fig. 26, it should be noted that both reach peak at NI in the neighborhood of 1.3 amp-turns. In contrast, in Report 2 and Ref. 12, For the same values of N /R L, the computed D vs. NI curves in Report 2 and Ref. 12 are identical, but the experimental curves are different due to difference in temperature tolerances. 86 computed D approaches infinity, and the actual behavior could be explained "only qualitatively. Infinite D resulted from the use of the model = _p[1 - (¢/_)2], which is invalid because the flux switching is incomplete. When this model was later replaced by the model = _p(1 - [(2qb ÷ @_ - @d)/(% + _d)32}, Eqs. (18) through (20), peaking D vs. NI curves, similar to the ones in Fig. 26 but of higher va!ues, were obtained. The separation between _,a and _i done here has resulted in a still better agreement (lower peaks)• A still further improvement would probably result if the slow-switching component of _i were added, because _i has an effect of increasing A¢ 4 and thus decreasing D. It should be emphasized that examining the validity of a switching model by comparing experimental and computed D is a very severe test if D is high (in which case Aq54 is small)• A slight error in L#54, which is negligible in the absolute sense, will have a large effect on D. [On the other hand, in general, comparing _b is a less severe test than comparing _(t)]. C. D vs. 14/1 3 WITH N_/R L AND NI AS PARAMETERS Let us examine the computed plot s of D vs. 14/l 3 in Fig. 27. For each load case, the plots corresponding to NI of 10 amp-turns and of 100 amp-turns intersect. The values of 14/l 3 corresponding to the intersection points increase from 1 to 2 as N_/R t increases from zero to 25. This phenomenon is quite complex and may be explained qualitatively by examining two cases of 14/l 3 values, a large one (e.g.,5) and a small one (e.g., 1). Since both Leg 3 and Leg 4 switch during the same switching time T (cf. Fig. 24), the higher the ratio _p_/_p4 (or Bp3/Bp4, since A 3 = A 4 ) is, the larger is _-qb3/_4 = D. Consider the case of large 14/13first. Because l 4 >> l 3 and because of the load on Leg 4, H3 >> H_ for either NI = 10 amp-turns or NI = 100 amp-turns, where H designates a time average value of H(t). As NI increases from 10 to 100 amp-turns the ratio B /Bp decreases ' p3 4 because of the nonlinearity in the B vs. ft curve (H is assumed smaller P than HB; see Fig. 15, p. 38, in Report 2). Hence, as NI increases, D decreases, which agrees with the results in Fig. 27. Note, however, that the decrease in BF3/Bp4 is counteracted by the load current which, in comparison with the no-load case, lowers H 4. This explains why, for 87 a given value of 14/l 3 in Fig. 27, the percentage decrease in D (as NI increases from i0 amp-turns to 100 amp-turns) is smaller the higher the value of N_/R m is. = > H only because of the load onIn the case of 14/l 3 1, H3 4 Leg 4. For tile same load, therefore, H4 is higher than in the case of a large _4/_3, and may exceed H s if NI is high enough. Hence, as N1 increases from 10 amp-turns to 100 amp-turns, the effect of the nonlinearity of B vs. H on decreasing D is overcome by the effect of the loading current P on increasing D. We see that, as NI increases, there are two opposing effects on D: an increase in D due to the load current, and a decrease in D due to the nonlinearity of B (fl). If 14/13 = 1 the effect of the load cur- p rent predominates and D increases with NI. As 14/13 is increased, H4 becomes smaller and the nonlinearity of B (H) becomes more effective, P until beyond a certain value of t4/I 3, the latter overcomes the effect of the load current and D decreases with N1. Extrapolation of the intersection points of the curves corre- sponding to lO and i00 amp-turn indicates that for Core S (14/I 3 = 1.83), D decreases as NI increases if N_/B m < 20. This is in agreement with the plot of D vs. NI in Fig. 26 and the case of N_/B L = 40 in Report 2, Fig. 36(5) (p. 88), where D increases as N1 increases above 3 amp-turns. Note, however, that the experimental data in Fig. 26 show that D increases with N1 for a value of N_/B t lower than 20. 7. SUMMABY An analysis is presented for computing flux-division ratio, D, in a saturable core having three legs: Leg m, Leg 3, and Leg 4. Leg m is driven by mmf Nf, and Leg 4 is loaded by a resistance B L across N L turns. The analysis is based on six equations: two mmf equations, one junction equation (Z_5 = 0), and three differential equations (_ = _ + _i for each leg). A numerical solution for _(t) and F(t) is obtained by incor-- porating a transcendental solution for F 4 into a simple predictor-corrector method. A computer program is provided for computing three types of output: time variables, D vs, NI ana N_/R L tot given 14/13, and D vs. /l 3 and 2[ 4 NL/B t for given NI. Computed @(t) waveiorms for NI = 2.0 amp- turns and D vs. Nf for N_/R L = 0, 1.000, 3.962, and 9.524 turns2/ohm are 88 compared with experimental data from an ultrasonically cut ferrite core "(Core S, Report 2). The agreement between computed and experimental re- sults is considerably better than the agreement in Report 2. D. CONCLUSIONS The complex nonlinearity of exact flux-switching functions makes it impossible to complete the analysis of magnetic circuits algebraically. Even if these functions are simplified, the amount of mathematical work involved is very often a burden. These two factors dictate the use of numerical methods in order to analyze (and design) magnetic circuits. Typically, the problem is to solve a set of simultaneous first-order differential equations that include variables that must be solved for transcendentally. For the transcendental solution of these variables, Newton's method (with a slight modification) has been found to be very effective. The set of differential equations may be solved by a number of well-known methods. In the case of the binary counter (which is not a simple circuit), the results obtained by using a simple predictor-corrector method are essentially the same as the results obtained by a more complex and more exact method (Runge-Kutta and Adams); on the other hand, the com- putation time using the simple method was much shorter. The simple method has also yielded results that agree quite well with experimental data for other types of magnetic circuits (unloaded core, loaded core, core-diode shift register, and a loaded saturable three-leg core). We thus find the simple predictor-corrector method to be accurate enough for the computational analysis of magnetic circuits similar to the ones above and to be relatively inexpensive. The agreements between computed and experimental waveforms of current and voltage in the circuits analyzed so far are encouraging. Future work along this line includes the following areas: (1) Computational analyses (including computer programs and experimental verification) of additional, more compli- cated magnetic circuits. (2) Computer programs for determination of core parameters from experimental data. (3) Computer programs for design of magnetic circuits. (4) Application of on-line computers in analysis and design of magnetic circuits. 89 III VARIATIONS OF FLUX-SWITCiIING PARAMETERS A. RAMP-F SWITCHING 1. INTRODUCTION It was noted in Report 3 (pp. 35 and 129) that the peak switching voltage of Core E-6 with a ramp-F drive was considerably lower than that computed from the parabolic model using step-F parameters. An investi- gation has been carried out to determine the extent of this effect as a function of the slope, k, of the ramp F. This effect is probably closely related to the effects of partial setting discussed in Report 3, but the nature of the relationship is not yet understood. The _(t) comparison in Report 3 [Fig. ll(a)] showed that the shape of the _(t) waveform was accurately given by the model for a given slope of F provided that several parameters were appropriately adjusted. The problem is thus primarily one of determining the values of the switching parameters as a function of k. One additional experimental check of the shape of the _(t) waveform will be given in Sec. III-A-4. There are two motives for pursuing this study: (1) to determine if the parabolic model can be used for practical applications where a core is driven by an F(t) which is approximately ramp-shaped, and (2) to learn more about the general effects of nonconstant F(t) switching so that, eventually, a new model can be developed which will work for any F(t). The investigation was carried out by measuring _p and tp for ramp-F switching for a number of values of k (the slope of the ramp). These data were then compared to computed curves to determine if a new, differ- ent, set of values of the switching parameters could be used in the model for all values of k. The experimental _p and tp data were plotted vs. k and compared to curves computed from the parabolic model. Curves were computed in two different ways: (1) using a simple version of the parabolic model and solving for _p and tp algebraically, and (2) using the _d parabolic model and the digital computer. The algebraic method proved useful as an intuitive aid to understanding the results and also provided an accurate 91 method for determining ramp-F corrected parameter values without resorting to trial and error. These two methods Of computing _p and tp will be dis- cussed in Parts A-2 and A-3, respectively. Next, the experimental results of Core E-6 will be briefly discussed. The results of introducing new pa- rameter values into the computations will be discussed in Part A-5. Finally, conclusions will be made in Part A-6. Experimental information for two other cores will be given in Sec. III-B-4 where the effects of temperature are treated. 2. ALGEBRAIC CALCULATION OF _p AND tp The values of _p and tp for ramp-F switching cannot be algebraically calculated using the _d parabolic model. However, they can be algebrai- cally calculated with the following version of the parabolic model: = ?_(F - F0 )v - (94 Although this model is less accurate than the q5d parabolic model, it is sufficiently accurate to aid intuitively in understanding the phenomeno- logical aspects of ramp-F switching. It will be assumed for this calculation that the entire "¢p(F) curve et can be adequately described with one set of values for )k, F 0 and v. This simplifies the calculation considerably and is quite accurate for some cores (e.g., Core E-6) in the range of F over which the experiments were performed. The computer calculation of _p and tp is easily capable of separating the So(F) curve into two regions when it is necessary to do so. The object now is to solve for Sp and tp as a function of k (the slope Jl of the applied ramp-F), k, F 0 and v. The "qb(t) function can be obtained from Eq. (94) by separation of variables. Defining t o as the time when switching begins, i.e., q5 = -q5 r at t = to, and solving for _(t) results in _b )_(F Fo)V sech 2 k ")u 1 (95) = - " (F - F 0 dt'- tanh- qbs t o 92 Substituting the ramp function F = kt (96) into the integral of Eq. (95) and using the relation t o = F_/k results in tt t (kt - Fo )2"+1(F- Fo)Vdt ' : k(v + 1) t O (97) Using this integral in Eq. (95) gives us the desired _(t) function, { }L(kt - Fo ) _'+1 Cpr"¢ = L(F - Fo )v sech 2 - tanh -1 (98) _k(v + 1) q_s ' for ramp-F switching. The solution for t is then obtained in the usual P way by setting d_/dt = 0. This gives the following transcendental function: (kt P I ,t )_+ 1 L(ktp - F o - F_) _+1 tanh Csk(Y + 1) Cr } "kCs - tanh- 1 ¢":'_ = 2---_- (99) It turns out that the argument of the tanh function is usually very small (e.g., < 0.2) so that the approximation tanh x = x (100) can be used to simplify Eq. (99). This results in a quadratic equation which can be solved for ktp, thus giving where ktp = 2L C + 2 + + F o (101) _ + 1 C _ r tanh- 93 The function for _p can be obtained by letting t = tp and _ = _p in Eq. (98), and using ktp from Eq. (101). We thereby obtain 2L _+i 1 2+___ 6 sech2 _ y + 1 12) n Note that F 0 does not appear in Eq. (102). In Eq. (i01) it appears only as an added term. Thus, two cores which have identical parameters I# and dimensions except for a difference in F 0 will both give the same values tt for _p. However, the peak of the large F 0 core will occur later by an ii ii amount 1/k (F01arse - F0small). At low values of k this is not precisely i! true because of the inaccuracies of this model. The F 0 term in Eq. (101), if divided by-k, is equal to the time required for F to reach Fg, at which time the core begins switching. The time from the onset of switching u until the peaking of _ , is independent of F 0 for this model. The three parameters of primary interest in Eqs. (101) and (102) are k, L, and F_. If everything else is combined into the quantities A and B, these equations can be written as and u (_)l/(u+l)ktp - F o = A (103) _p = BL(_) vl_u+l) (104) Equations (103) and (104) are plotted as a dashed line in Figs. 28 and 29. These curves will be compared with computer calculated curves in Part A-3. The curve is plotted as ktp vs. k rather than tp vs. k to remove the 1/k dependence so that the remaining effects can be more easily judged. it If (ktp - F O) of Eq. (103) is plotted vs. k on log-log paper, a straight line results. The slope of this line is equal to 1/(v + 1). Vertical displacement of the line corresponds to changes in L and the coefficient A which contains v, qSs, and tanh -1 qbr/_b s. Likewise, Eq. (104) gives a straight line on log-log paper. In this case, the slope is equal to v/(v + 1), and vertical displacement of the line corresponds to changes in L and the coefficient B which also contains v, q5 and tanh -1 dPt/dP s. 94 :1 r_ o 0.4 0.3 0.2 0.1 I I I I t 1 I COMPUTED FROM Eq.(102) USING STEP-F PARAMETERS COMPUTED FROM _d PARABOLIC MODEL USING STEP-F PARAMETERS 0 I 2 3 4 FIG. 28 XX X X x EXPERIMENTAL CORE E-6 I I 1 7 8 9 I I 5 6 k (omp-lurns/p. $ec) _p(k) FOR RAMP F(t) 95 i I I o Q. .I¢ I I I I I I I I I I11" COMPUTEDFROM Eq.(lOI) _I _x USING STEP-F PARAMETERS,_ _x---_x._ _'_" _._,,_ "COMPUTED FROM _a PARABOLIC _,,_ MODEL USING STEP-F xf,"_ / PARAMETERS E×PER,.ENTAL /_/f CORE E-6 / / X l o I I I I I I I I I I 0 I 2 3 4 5 6 7 8 9 I0 I k (emp-t urns//=sec) TA-5094-41 FIG. 29 ktp(k) FOR RAMP F(t) The experimental data can be conveniently analyzed on log-log scales because nearly straight lines are obtained. This makes for easier inter- polation and simplifies the comparisons with results computed from a model. Equations (103) and (104) will be used in Part A-5 to calculate l/ It new values of L and F o (e.g., _r and FOr ) which are applicable specifi- cally for ramp-F switching. 3. NUMERICAL COMPUTATION OF _p AND tp The _a parabolic model was used with the digital computer to compute more accurate _p(k) and kip(k) curves than those obtained via the alge- braic calculations just discussed. The _(t) was computed in the vicinity of the maximum _ (e.g., -0.375 r < _ < + 0.3_ r) for each of a number of k values. The values of _p and kip were then determined from the _(t) data and values plotted vs. k. The computer program, which did not include 96 any of the initial spike models, is discussed in some detail on pp. 16-21 of Report 3. The equation of this model which corresponds to Eq. (94) is This model requires a function (see p. 19 of Report 3) for _a(F). Equa- tions for this purpose contain parameters (e.g., D r, _,, Hq, H , and H=) that must be determined before this model can be used. These parameters are determined from the static ¢(F) curve. Computation of _(k) and k_k) was made for Core E-6 using the same step-F parameters as were used in the algebraic calculation above; only one region in _v(F) was used. The results are plotted as a solid curve in the linear graphs of Figs. 28 and 29 and also on the log-log graphs of Figs. 30 and 31. Only one region in _p(F) was also used for Cores I-4 I.O 0.5 0.2 0.1 0 Ii, .._. 0.05 0.02 0.01 0.1 COMPUTED FROM Eq.(lO2) USING STEP-F PARAMET: h =0.069 %'=0.95 =0.0507 Fo'; =0.56 kr= 0.0507 F_' = 0.95 •STEP-F PARAMETERS "_ k = 0.069 F_' =0.95 I , , I,,,,I 0.2 0.5 I 2 5 I0 20 k (amp- t urns//J.sec) FIG. 30 Cp(k) FOR RAMP F(t) USING h AND F'_, + EXPERIMENTAL CORE E-6 COMPUTED FROM _" (J_dPARABOLIC MODEL 50 97 I0 i I ' ' I ' ' ' '1 I ' ' I ' ' ' CORE E-6 1 COMPUTED FROM Eq. ( I01 ) USING STEP-F PARAMi X= 0.069 F_)/ = 0.95 STEP-F PARAMETERS ")'r = 0.0507"_ o.9s { ;oo 9 /F:,'=o.gs j COMPUTED FROM PARABOLIC MODEL TB-5094-43 FIG. 31 (ktp-F;) vs. k FOR RAMP F USING Xr AND F_ and K-1 to be discussed in Part B-¢ because the experimental _p(F) data for step-F drive could be adequately described with only one region. Two regions can easily be handled by the present computer program if necessary. A comparison of the _p(k) curve in Fig. 30 calculated from Eq. (102) and the curve obtained using the computer reveals several interesting points. The curve from the computer is curved downward for low values of k, whereas the algebraic calculation results in a straight line. This curvature is the result of _d being a function of F in the _d parabolic model. The computer calculated curves were consistently lower for all 98 values of k, although as k increased, the percentage difference decreased. For Core E-6 at k = 10, the difference is 4.3 percent; at k = 100, it is 1.2 percent. 4. EXPERIMENT The equipment used for the ramp-F experiments was essentially the same as that described in Sec. I-D-1. The transistor pulser, which con- sisted of four current drivers in parallel, was modified to increase the maximum rise time to about 50 _sec. A capacitor decade was inserted in the internal pulse-shaping circuit of each of these four drivers. This also increased the fall time, but that is of little consequence if switching is completed during the rise time. The pulse sequence consisted of a positive SET pulse followed by a negative CLEAR pulse, a positive CLEAR pulse, and a negative CLEAR pulse (see p. 83 of Report 3 for details ofthis method of clearing). The nega- tive CLEAR pulses were greater than 7 amp-turns for Cores E-6 and K-I, and 15 amp-turns for Core I-4. The constant portion of the SET pulse was kept short to minimize dissipation in the output transistors of the pulser. A projected graticule was used on the oscilloscope so that pa- rallax could be eliminated from the experiment. Core E-5 was used for most of the ramp-F experiments. Cores I-4 and K-1 were tested with a ramp-F drive at various temperatures (Sec. III-B-4). The measurement of _p, tp and k is illustrated by Fig. 32. First, the peak in _ was aligned with a vertical graticule line. Then, _p and F a were measured with a chopper and a voltage reference. Next, At and F b were measured. The value of k was calculated from and kt from P F b - F k = (106) At ktp = Fo (107) For all except the low values of k, the F(t) waveform had a nonlinearity near its beginning. The above method of measurement was used so that k and tp would be determined from that portion of the ramp which existed during most of the switching prior to the peak in _(t). The linearity and slope of the ramp after the peak in _(t) is of no concern since it will not affect the determination of tp and _p. 99 --4------------- At _ .q_ tp F t FIG. 32 DEFINITIONS OF MEASURED QUANTITIES FOR RAMP-F SWITCHING The comparison of the experimental and computed _(t) waveform was done at a relatively low value of k (k = 0.837 amp-turn/_sec) to comple- ment rather than duplicate the comparison made in Report 3 (k = 3.57 amp-turn/_sec). The experimental &(t) waveform and its corre- sponding F(t) waveform are shown in Fig. 33. The numerically computed _(t) waveform is also included so that the shape of the waveforms can be nl compared. The values of N r and For used in the computation were determined from the _p(k) and ktv(k) curves so that &p and tp would match the experi- t! mental values at k = 0.837. The values were Nr = 0.0507 andF 0 = 0.800 amp-turns. el Note that once Nr and For are correctly determined, very good agreement is obtained. The problem is one of determining how well a single set of values " _p(V}for N r and For [also Ppr, _r, and _r if two regions are used for ] will serve for all values of k. 100 F4 0.02 volt/turn 1.0 amp-turn ___ 0.5Fsec TA-5094-45 FIG. 33 EXPERIMENTAL vs. COMPUTED _(t) FOR RAMP F WITH _ AND F_r ADJUSTED TO MAKE _ AND tp AGREE Core: E-6; Temp. = 30°C; k : 0.837 amp-turn/_ec; ,_ : 0.049; F" = 0.80 amp-turn; _, : 1.30Or Experimental _p(k) and ktp(k) points are included in Figs. 28 through 31. Note that the experimental _p(k) data are below the numerically calculated curves (for step-F parameters) for all values of k. This is in agreement with the one point observation on p. 35 of Report 3. Thus _ needs to be decreased for all values of k. Note that the experimental _#(k) curve has a shape very similar to the numerically computed curve, i.e., nearly a straight line on the log-log plot but with a slight downward curvature at low values of k. It appears that a vertical shift downward of the numerically computed curve would give good agreement for the entire curve. This vertical shift is obtained by varying L as indicated by Eq. (104). Strangely enough, the ktp(k) experimental and computed curves are in quite good agreement except for low values of k(e.g., k < 1). This good agreement may be somewhat accidental, since a correction in k to give agreement in _p(k) will throw the ktp(k) curve out of agreement. There- st fore, F 0 has to be corrected to bring ktp(k) back into agreement [cf. Eqs. (103) and (104)]. This will be discussed in more detail in Part A-5. I01 5. PARAMETER COR_ECTIONS i! The values of k and F 0 need to be corrected to make the computed _pp(k) and ktp(k) curves agree with experimental ramp-F data. The corrected values can be calculated on the basis of Eqs. (103) and (104) and from the experimental and the numerically computed data. Define P as _p computed [using the _d parabolic model] p _ (108) experimental The value of (_v computed) can be computed using step-F parameters, or any approximate values. Both values of _p must be at the same value of k. Denote ramp-F corrected values by the subscript r. From the k relationship of Eq. 104) we obtain k k - (109) r pv+l where k is the value used in determining (_v computed). To derive a cor- ot responding equation for F" first solve Eq. (103) for F 0 for ramp-F-Or corrected parameters. This results in /k\ F" = kt - A I_-| Or Pexp :/(v_l) (110) where tp is the experimental value. exp Eq. (103) but using step-F values (or the approximate values) for k and F"0' A = kt -F Pealc A is then also determined from (111) where t is the calculated value of t . Substituting this equation Pcalc P into Eq. (110) and making use of Eq. (109) results in t! F" = kt - P(kt - Fo) (112) 0r PQxp Pcalc 102 Equations (109) and (112) were used to get the following ramp-F values for Core E-6 at k = 6 amp-turns/tzsec: = 0.0507 r F" = 0.56 amp- turnsOr The numerically computed _p(k) and ktp(k) curves using this value of _ tt but the old value of F 0 are included in Figs. 30 and 31. Note that the _p(k) is in relatively good agreement for all k values, but that the original agreement of the ktp(k) curve has now been ruined. To correct this, F_ must be decreased. The calculation was repeated using both Xr II and For. The _p(k) curve is included in Fig. 30. The kt'_(F) curve hae been plotted in a new figure (Fig. 34) because the vertical axis contains n F 0. The "_p(k) curve has been altered primarily in the low-k region as a tt result of the change in F 0. In this low-k region it has gone from slightly II above to slightly below the experimental data. Hence, decreasing F 0 in- creases the curvature of the _p(k) curve if the ffd model is used. This II slight variation in curvature could be used to determine F o very roughly. However, a value thus determined would be larger than the value 0.56 which is required to make t agree at k = 6 amp-turns/_sec. The computed ktp(k) curve (Fig. 34) now crosses the experimental curve at k = 6 where the value tt of For was determined. Unfortunately this ktp(k) curve cannot be made to agree for all k values. If v is adjusted to make the ktp(k) curves agree for all k values, which would require a significant change in v then the agreement in the _p(k) curves would be ruined because of a change in the slope of the computed curve. Thus, adjusting v would not help, but would considerably complicate matters. So far in this discussion, very little has been said about determining ramp-F values for pp, F0, and F B. In some cores the _(F) curve may re- quire use of these parameters. It was noted, in preliminary computations, that a small deviation from the experimental _p(k) curves at high k values could be traced to the inclusion of pp, F 0 and F_ in the computer program. A careful examination of the experimental _p(F) data for Core E-6 revealed that the _p(F) curve was better described if Pp, F 0 and F B were not intro- duced. If the experimental data had been extended to higher F values, perhaps pp, F 0 and F B would have been necessary to properly describe the _p(F) curve. It was therefore impossible to make experimental observations on the effect of ramp-F switching on pp, F 0 and F 8. It is possible, 103 I0.0 5.0 2.0 w c i. :3 7 Q. E 0 1.0 -=.o I O. J¢ 0.5 0.2 I I ' ' I ' ' ' ' I I ' ' I ' ' ' '_ I CORE E-6 x EXPERIMENTAL COMPUTED FROM Eq.(lOI) j_x_= x_" ' US,NG k: = 0.0507 _ /_<_COMPUTED F_OM*. o., I , , I , , ,,i I , i I I , ,, O.I 0.2 0.5 I 2 5 IO k (amp- turn/H.sec) TB-5094-46 FIG. 34 (ktp- F_)r) vs. k FOR RAMP F USING _r AND F'6r however, to make some reasonable assumptions that will allow a determina- tion of ramp-F values for .op, F 0 and F B (i.e., Ppr' For' and FI_ r) from it the values of kr and For. First, assume that the effect of ramp-F switch- ing is to reduce the ordinate of the _Sp(F) curve the same percentage for all F values. Then _p will be reduced by the same factor as is ),. Therefore _r Pp r = PP A (113) 104 Second, assume that the other effect of ramp-F switching is to shift the entire _p(F) curve leftward. Then F 0 and F 8 will be reduced by the same ## value as F 0. Thus For and FBr are given by and II II For = F 0 - (F 0 - For) (114) FBr -- F_ (F" - "r- F 0 ) (115) The equations of continuity of the _p(F) curve as given on pp. 17 and 18 of Report 3 are preserved by these assumptions, it should be emphasized that Eqs. (113), (114), and (115) have no experimental verification. The physical mechanisms responsible for the effects of ramp-F switch- ing are not presently understood. They are probably closely related to the effects of partial setting (cf. p. 129 of Report 3). It is not sur- prising that good agreement cannot be obtained for all values of k by tt adjusting only )_ and F 0. See, for example, the effects that partial setting has on tp as observed in Figs. 34(a) and 35 of Report 3, and also on the static ¢(F) curves as observed in Figs. 30 of Report 3. Similar effects may also occur in ramp-F switching. This needs further investigation. In practical applications where F(t) can be approximated by a ramp, tl it is certainly better to make an adjustment in the value of _ and F 0 rather than ignore the effects of ramp-F switching altogether. If a new tl value for F o is determined at a value of k in the middIe of the range of interest, quite good results should be obtained. The experimental deter- mination of L r and F"Or can be done with measurements at only a few k values if the results are plotted with log-log scales, since nearly straight lines result. When only ramp-F values of parameters are needed, the _(F) measure- tl ment can be eliminated and Kr' For and _ determined directly from experi- mental _p(k) and ktp(k) curves. First, a straight line is drawn tangent to the log-log _p(k) curve at the high-k end. From the slope of this straight line, y can be determined according to Eq. (104). Next, _ can be calculated from one point on this straight line by using Eq. (102). II Finally, F 0 can be determined from an appropriate point of the experimental n ktp(k) data by using Eq. (101). These approximate values of k and F 0 can then be used to compute _p(k) and tp(k) using the _d parabolic model. 105 These computed curves, together with Eqs. (109) and (112) can be used to determine final values of k and F"Or" 6. SUMMARY The parabolic switching model "q5 = L(F - F o) " - was used for ramp-F drive to derive equations for _pp(k) and tp(k), where k tt is the slope of the ramp. It was shown that _p is independent of F 0 for this model. These equations are useful as an intuitive aid for analyzing H the experimental results, and in determining corrected values for k and F 0 for ramp-F switching. The numerical computation of _p(k), and tp(k) with the _ba parabolic model and a digital computer was discussed and the re- sults compared to those obtained analytically with the simpler form of the parabolic model given above. The values were nearly equal for high values of k (e.g., _p values differed by 4.3 percent at k = 10 amp-turn/_zsec) but differed considerably at low values of k. The details of the experimental measurements were discussed and the _p(k) and tp(k) curves for Core E-6 were compared to the computed curves using step-F parameters. The experi- mental _pp(k) curve was significantly lower (e.g., 15 percent at k = 1 amp-turn/#sec). The experimental tp(k) curve was in close agreement with the computed curve except for low values of k. The algebraic equa- tions for _pp(k) and tp(k) were used to derive simple relationships for ¢t computing ramp-F corrected values for k and F 0. These values were used in the _d parabolic model to again compute _pp(k) and tp(k). The _p(k) curves were now in good agreement for all k values. The tp(k) curves were inexact agreement at only one k value, which is the point at which the two curves crossed each other. The agreement in the vicinity of this crossing point was good enough for many practical applications. B. EFFECTS OF TEMPERATURE i. INTnODUCTION In a practical magnetic circuit, operation is influenced by variations in temperature. Analysis of these circuits by use of a switching model therefore requires a knowledge of the variation of the core parameters vs. 106 temperature. It is also necessary to verify that the model which has been used at room temperature is also valid for a wide temperature range. In addition, it is helpful for qualitative studies of circuit operation to know the general trends and approximate magnitude of the effects of varying temperature. The effects of temperature variations were determined by measuring the core properties at each of several temperatures from -50°C to +75°C. First, the static ¢(F) curves were measured, starting fr0m both a hard remanent state and a partially set state. Second, the _p(F) curve for step-F switch- ing was measured at each temperature. Third, the _p(k) and ktp(k) curves for ramp-F switching were measured at each temperature. The core parameters were determined from these various curves and plotted vs. temperature. These temperature effects were measured for two cores, Core I-4, and Core K-1. The dimensions of these cores are given in Table II. Core I-4 is a thin ring which was ultrasonicallycut from a disc of Indiana General 5209 ferrite. It is the same core used in Report 3 for studying the effects of partial setting. The disc from which Core I-4 was cut was one of a batch of 10 discs which were magnetically tested for uniformity. Core K-1 is a Lockheed 100SC1 switch core. It is from a group of six cores which were magnetically tested for uniformity. The temperature of the core was controlled by a commercial temperature test chamber which used electrical heating and expansion of CO 2 for cooling. The inaccuracy of the temperature is less than ±1.5°C at all temperatures. 2. STATIC @(F) The static @(F) curves were measured starting from both a hard remanent state and a partially set state for each temperature. The partially set state was obtained by switching the core from -@r to _ = 0 by a 1-_sec rectangular pulse. The static _(F) curve from this partially set state was then measured for both a positive and a negative polarity of F. The model for the static ¢(F) curves Which we have been using (see pp. 3-7 and pp. 74, 75 of Report 2) was fitted to each hard-state experi- mental static @(F) curve. The values of the parameters thereby determined were plotted vs. temperature. However, this ¢(F) model is not applicable for partially set ¢(F) curves. An appropriate model has not yet been 107 "Fable lI DIMENSIONS, SWITCHING PARAMETERS, AND TEMPERATUBE COEFFICIENTS OF CORES I-4 AND K-1 DIMENSION r 0 (ram) r i (tam) ro/r i A (mm) w (mm) l (._n) [o (mill) PARAMETER _r (maxwel[_) _s (maxwells) H a (amp-turns/meter) Hq (amp-turns/meter) ttn (amp-turlls/taeter) A (ohm/turnZ*lamp v-l) tl F 0 (amp-turn) p v+l '_-i A r (ohm/turn amp ) TEMPERATURE COEFFICIENT a_r (i/°C) %_ (1/°C) _Hq (I/°C) aH. (I/°C) _A (I/°C) av_ (1/°C) _v (1/°C) %r (i/°C) CORE I-4 Indiana General 5209 Ultrasonically Cut 30 + 0.5°C 3.78 3.43 1.10 0.848 0.35 21.$5 23.77 5.97 6.55 230.0 43.1 37.0 0.134 1.28 1.19 0. 119 -0.0030 -0.0025 -0.0077 -0.0095 -0.001 -0.0084 *0.0008 -0.003 CORE K-1 Lockheed 100SC1 Commercial 24.6 ± 0.5°C 1.27 (50 mils) 0.89 (35 mils) 1.43 0.76 (30 tails) 0.3_ 5.59 7.95 0.43 7.10 2C+0.0 37.3 11.4 0.64 0.35 1.21 0. 477 -0.003o -0.0035 -0.._095 -0.0097 +0.0017 -0.0065 -0.0002 +0.001 108 obtained for partially set states. Therefore, the effects of temperature upon these partially set _(F) curves will be judged by looking at the curves themselves. The pulse sequence consisted of five rectangular pulses: a positive PARTIAL-SET pulse, a positive or negative TEST pulse, a negative CLEAR pulse, a positive CLEAR pulse, and finally another negative CLEAR pulse (see p. 84 of Report 3). The width of the SET pulse was 6 msec. The first CLEAR purse followed closely the end of the SET pulse. The flux switched by the PARTIAL-SET and the TEST pulses was measured at the time of the first CLEAR pulse by means of a flux reference (sec Appendix F of Report 2). The peak flux, not remanent flux, was measured for all the _(F) curves of this report. The negative CLEAR pulses were greater than 15 amp-turns for Core I-4 and 7.0 amp-turns for Core K-1 (duration = 10_sec). The general effects of temperature on the hard-state static _(F) is shown in Fig. 35 for Core I-4 and Fig. 36 for Core K-1. It was found that the _(F) model could be fit quite well for all the temperatures for Core I-4 (e.g., less than 3 percent error except right at the threshold 2 O 0 O -2 -4 -6 -8 0 o x/ J x -50oc 0 I 2 3 4 5 F (amp-turns) TB'-5094-47 FIG. 35 STATIC ¢(F) CURVES vs. TEMPERATURE OF CORE I-4 109 86 2 o E "0- -2 -4 -6 -8 I I" I '1 I [ I I I [ I Xv X /._50" xl , ; _ / / I I 1 I / / / X / / / / 1 1 I 1 I 0.5 0.6 0.7 0.8 0.9 F"(amp-turns) 0.1 0.2 0.3 0.4 t I 1.0 I.I 1.2 TB-5094-48 FIG. 36 STATIC _(F) CURVES vs. TEMPERATURE OF CORE K-1 where the experimental curve has a more rounded corner). The irregulari- ties in the -50°C curve for Core ]-4 are, of course smoothed out in the computed curve, but the percentage error due to these irregularities is less than 3 percent. Experimental be(E) data was not included in these measurements. Therefore, H a can only be very roughly determined by using the portzon of the _(F) curves between F = 0 and F = Fth. The value of H a previously determined at room temperature was adequate for all temperatures except -50°C for both cores. At -50°C H had to be drastically increased (e.g., froth about 200 to 900 amp-turns) to prevent the computed curve from rising above the experimental curve for 0 < F < Fth. This increase in H a at lower temperatures means that the _(F) Curves at -_r are flatter at lower temperatures. This is qualitatively consistent with permeability measurements which generally show an increase in permeability as temperature increases and measurements of squareness ratio which decreases with an in- crease in temperature. The increase in _, _, and F as temperature de- creases are well known effects. 110 The static _(F) curves of Core K-1 cannot be as well described by the model as those for Core I-4. The experimental _(F) curves have a relatively steep section just above the threshold, which the computed curve does not exhibit. Thus the computed curve has a lower Fth than does the experimental curve. Perhaps the inner radius of this type of core has a higher H threshold than the parts of the core with larger radius. The static ¢(F) model assumes uniform material properties throughout the volume of the core. In practice, this problem is not serious because the threshold in the switching model is determined by It F0, which is generally above Fth anyway. In other respects these ¢(F) curves were quite well described by the model. The worst error, not including the threshold problem, is about 7.5 percent, and that is over a very small range of F near F = 0.21 amp-turns for the 75°C curve. The values of Hq, H , ¢_ and ¢_ are plotted in Figs. 37 and 38 for Cores I-4 and K-l, respectively. These values were used in the computa- tions of _p(k) for a ramp-F drive in Part B-4. Since these curves are fairly linear, a temperature coefficient can be used to correct the pa- rameters for variations in temperature. The values of these temperature coefficients were determined at 30°C and are given in Table II. These values are the primary objective of these measurements. It appears as if H for Core I-4 might go to zero at a temperature somewhat above 100°C but below the Curie temperature. This is possible since H is merely the vertical asymptote for the hyperbolic static B(H) curve. The ratio _/_ is nearly constant at 0.90 for Core K-l, but for Core I-4 has a maximum of 0.92 at about 30°C and a lower value above and below this temperature. At 75°C it is 0.86. The sharpness of the wing of the static B(H) curve can be judged by the ratio Hq/H which is _ 1. For Hq/H near unity, a very sharp wing is obtained and the side of the B(H) curve is very steep. This is usually a desirable characteristic. As Hq/H increases the wing becomes more rounded and the side less steep. This Hq/H n ratio is shown in Fig. 39. It must be remembered that Core K-1 has a steep _(F) curve just above Fth which is not properly described by the static _(F) model. Therefore Core K-1 is actually a little better than indicated by the ratio Hq/H as far as steepness is concerned. The core geometry has been accounted for in determining Hq and H n so that Hq and H n are a measure of the average material properties rather than the core properties. 111 I I I IO -50 -25 0 25 50 75 TEMPERATURE (°C) TA-5094-49 FIG. 37 STATIC _(F) PARAMETERS vs. TEMPERATURE FOR CORE I-4 112 42 I I I I o 0 25 50 75 I00 TEMPERATURE ("C) 1.0 0.9 0.8 0.7_ 0.6 o I FIG. 38 STATIC _(F) PARAMETERS vs. TEMPERATURE FOR CORE K-1 30 20 -- I0 -- 0 8O 70 60 E 50 CL E 0 I 4O TA-5Og4-SO 113 z.o I I I I i 1.5 Hq H"_" 1.0 0.5 I F 0 -50 ._.-CORE K-I -- _'_-- CORE I-4 I I I [ I -25 0 25 50 75 I00 TEMPERATURE (°C) TA- 5_4 - 51 FIG. 39 WING SHARPNESS OF STATIC _(F) CURVES vs. TEMPERATURE The static _(F) curves for partially set states were obtained with a l-_sec PARTIAL-SET pulse except for Core I-4 at -50°C. For this case, the threshold had increased considerably and the current driver used for partial setting was unable to switch the core to ¢ = 0 in i _sec. There- fore, the width had to be increased to 1.6 _sec. In general, the ampli- tude of the l-_sec PARTIAL-SET pulse was adjusted so that the core would be partially set from _ = -_r to ¢ = 0. The static ¢(F) curve was obtained for both +F and -F. The curve for -F will be shown in the first rather than the fourth quadrant for easy comparison with the +F curves. The static _(F) curves for +F and -F for various temperatures are shown in Figs. 40 and 41, respectively for Core I-4 and in Figs. 42 and 43 for Core K-1. Qualitatively, the general character of these partially set curves is preserved throughout the temperature range. Note that the wing of each +F curve is more rounded than the threshold. This characteristic is reversed for the -F curves, where the threshold is more rounded than the wing. This property is enhanced for very slow partial setting. See, for 114 6o E 2 0 0 r_50oc (PARTIAL-SET PULSE = 1.6 _sec) I I I 2 5 4 F (amp-turns) FIG. 40 STATIC _(+F) vs. TEMPERATURE OF PARTIALLY-SET CORE I-4 PARTIAL-SET pulse = ] /Lsec (except at 50°C) 6 I I I I I - r-,oo /_AL-SET PULSE = Io6/u.,sec) 0 I 2 5 4 5 -F (amp-turns) TA-5094-53 FIG. 41 STATIC ¢(-F) vs. TEMPERATURE OF PARTIALLY-SET CORE I-4 PARTIAL--SET pulse = 1/zsec (except at-50°C) 115 6-$ 3= ,<4 0 E "9- 0 0 75oC 0. I o/ °/ c_'-50 o 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 F (omp-turns) FIG. 42 STATIC _(+F) vs. TEMPERATURE OF PARTIALLY-SET CORE K-1 PARTIAL-SET pulse = 1/zsec 5 6 _4 E 2 I I I I ] I I 1 I I -- _I_'--" -_ i_...- _ ____._.....--_-- - -- - _..'__--T _ 75-C/f' 7 / o - 7 i'+ I t-5o 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 -F (omp-turns) 1 & I I.I 1.2 TB-5094-55 FIG. 43 STATIC q_(-F) vs. TEMPERATURE OF PARTIALLY-SET CORE K-1 PARTIAL-SET pulse = 1 /zsec 116 example, Figs. 30(c) and (d) on page 91 of Report 3. The physical reason for this property can be explained as follows. During very slow partial setting, the regions of the core which have a lower H c switch rapidly and higher H c regions switch more slowly. When switching is continued in the case of a positive TEST pulse the remaining unswitched regions are mostly of a high H which tends to give a relatively long extensive wing in the _(F) curve as all the high H c regions are finally switched at high F. In contrast, when the F is negative (i.e., opposite in polarity to the PARTIAL-SET pulse) the TEST pulse switches the partially set flux back again to -_r" Since this involves mostly the lower threshold regions a relatively sharp wing is obtained because few of the higher threshold regions had been partially set. This physical explanation is supported by the fact that this property is most evident for very slow partial set- ting. Fast partial setting sw.itches all regions of the core nearly equally, thereby reducing the differences between the +F and -F curves of _(F). A better comparison of the positive and negative partially set curves and the major static _(F) curves can be made in Fig. 44 for Core I-4 and in Fig. 45 for Core K-1. These figures include curves for one high and I I I I I I I _ l =.=,=I......__=.=.,,,,=k_==,,_-_ o - • l l -e- -2- / L----T---.T.-" o o.I 0.2 0.3 ! I | 75"C - I O*C _ I ! I I I I 1 I I I I 0.4 0.5 0.6 0.7 0.8 0.9 1.0 I.I 1.2 F(omp-turns) FIG. 44 MAJOR AND PARTIALLY-SET STATIC ¢(F) OF CORE I-4 AT 75 ° AND O°C PARTIAL--SET pulse = ] #sec 117 4I I 1 I _ I I ] I I [ I 1! I i I 1 1 0.5 0.6 0.7 0.8 0.9 1.0 I.I 1.2 F (clmp- turns) TB-5094-57 FIG. 45 MAJOR AND PARTIALLY-SET STATIC _(F) OF CORE K-1 AT 75 ° AND -50°C PARTIAL-SET pulse = 1 #sec one low temperature (75°C and 0°C for Core I-4 and 75°C and -50°C for Core K-l). The 0°C curve was used rather than the -50°C curve for Core I-4 to avoid the one case in which a 1.6 #sec-duration PARTIAL-SET pulse was used. The effects of partial setting upon the static _(F) curves seems to be very much the same for the entire temperature range. 3. STEP-F SWITCHING The effects of temperature on step-F switching will be studied by means of the _p(F)curves. The primary goal is to obtain temperature co- H efficients for X, F0, and _'. Experimental _p(F) data were measured for Core I-4 and Core K-I at each of several temperatures in the range -50°C 11 to +75°C. These data were used to determine X, Y 0 and _ versus tempera- Lure. No attempt was made to determine pp, F o and F B versus temperature because these values cannot be accurately determined for the range of F used 118 for the two cores tested. No _p(F) vs. temperature curves were taken for partially set states. This should also be investigated in the future. The pulse sequence consisted of a SET pulse followed by a negative CLEAR pulse, a positive CLEAR pulse, and another negative CLEAR pulse. This CLEAR pulse sequence is used to completely remove all history effects (see p. 83 of Beport 3}. The SET pulse had a 60-nsec rise time (10% to 90% points) and a .naximum amplitude of 5 amp-turns. It was supplied by four Digital Equipment Corp. Model 62 current drivers in parallel. The negative clear pulses were greater than 15 amp-turns for Core I-4 and 7.0 amp-turns for Core K-i. The _p(F) and _p(k) data for Core I-4 at 30°C were taken before the core had been cooled to -50°C. The other data were taken afterward. Dropping the temperature to -50°C and returning to room temperature seemed to alter the core's magnetic properties a small amount. Thus the 30°C _p(F) and _p(k) curves are not completely consistent with the rest of the data and have therefore been omitted. The exact cause of this small effect has not been determined. The static _(F) curves at 30°C were taken both before and after the first -50°C excursion. The _p(F) curves for different temperatures are shown in Fig. 46 for Core I-4 and in Fig. 47 for Core K-1. It is readily apparent in these figures that the only major effect on _p(F) of increasing temperature is I! II the decrease in F 0. The decrease in F 0 is to be expected since it is com- monly known that Hcdecreases as temperature increases. This is illustrated II by the static _(F) curves of Figs. 35 and 36. The values of F 0, 4, and v were determined for each temperature by plotting the _p(F) data on log-log paper. These values are shown vs. temperature in Fig. 48 for Core I-4 and in Fig. 49 for Core K-1. Note that _ is nearly constant with temperature. The value of k is also nearly constant for Core I-4, but increases somewhat with temperature for Core K-1. The value of _ depends upon a number of physical quantities so that it is not surprising if its temperature be- havior differs for different cores. If we compare the parabolic model to the model of Menyuk and Goodenough, _ then _ will be determined by the fol- lowing quantities: the viscous damping parameter fl, the square of the saturation magnetization, M_, the density of domain nucleations, and which accounts for the statistical variation in the directions of the easy axes of individual ferrite grains. 119 I 75 ° 50 ° / X _ 50°C 0.3 C 0.2 0 0°1 0 0 X I / I X / 2 3 4 F (amp-turns) FIG. 46 _p(F) vs. TEMPERATURE OF CORE I-4 ;J TB-5094-58 120 I°1 f.O 0.9 0.8 0.7 0.6 In o o.s o_ "-8- 0.4 0.5 0.2 0.1 0 0 75oq / X X / / X 0.5 1.0 F (amp-turns) 1.5 FIG. 47 _'p(F) vs. TEMPERATURE OF CORE K-1 xO°/ 121 2.5 T Q. E 0 _.o i.O I I I I I I 2.0 x____ F;' 1.5 I I I -25 0 25 TEMPERATURE (°C) 1 0 -5( 0.5 X x. 50 75 1 0.15 0.1 X 0.05 0 I00 TA-5094-60 FIG. 48 _p(F) PARAMETERS vs. TEMPERATURE FOR CORE I-4 122 1.5 I I I I I _,,/] I.O 0.5 B 0 - 50 G 13 n // 0----- X X_. 10 Fo I I I I I -25 0 25 50 75 TEMPERATURE (*C) 0.6 - 0.5 ie c 0.4 -- I O. E 0 0.3 -- 0.2 -- 0.1 0 I00 TA-50¢J4-61 FIG. 49 _'p(F) PARAMETERS vs. TEMPERATURE FOR CORE K-1 _. RA.P-F SWITCHING The switching properties of a core with ramp-F excitation have been discussed in Sec. III-A for room temperature. It was shown that _ needs to be altered (lowered for Core E-6) from its step-F value in order that _p(k) can be properly described by the parabolic model. It was also shown that the experimental ktp(k) curve has a somewhat lower slope than the com- puted curve, so that exact agreement can only be obtained at one k value. We now wish to determine what effect temperature variations have on these properties. Cores I-4 and K-1 were both investigated over the temperature range -50°C to +75°C. The equipment and the experimental technique are as described in Sec. III-A-4. The amplitude and duration of the CLEAR pulses are given in Sec. III-B-2. 123 Co The effects of temperature on _p(k) is shown in Fig. 50 for Core 1-4 and in Fig. 51 for Core K-1. At first glance these curves seem to vary haphazardly, but this is not actually the case. The general trend is for the lowest temperature curve to be on top at high k values, and on the bottom for low k values, so that each curve crosses all the others in going from low-k to high-k values. The only exception is that the curves for Core I-4 are not extended to low-enough k values for this crossing to be completed. Equation (102) for _p(k) was derived by using the parabolic model with no static ¢(F) limiting included [cf., Eq. (94) with Eq. (105)]. Equation (102) results in a straight line (see Fig, 30) when _p(k) is plotted with log-log scales. The slope of this line was shown (see Fig. 30) to be dependent only upon v. Thus, if v is constant with temperature, as it practically is in Core K-1 for step-F switching, parallel lines would result so that no crossing would be obtained. However, if _a is introduced into the model as shown in Eq. (105), then log _p(log k) curves downward at low k values. This makes crossing of the curves possible even if v is con- stant with temperature. Curves of _(k) using the _a parabolic model and step-F parameters were computed for Core K-1. The resulting curves showed I.O 0.5 -- 0.2 -- O.I -- 0.05 -- 0.02 -- 0.01 0.01 I I I I I Llll I I I I I IIII I 1 I I Ell[i 0.02 0.05 0.I 0.2 0.5 I 2 5 I0 k (amp-turns/p.sec) 20 TB-5094-62 FIG. 50 _p(k) CURVES OF CORE I-4 vs. TEMPERATURE WITH RAMP F(t) 124 A C t. O w 1.0 0.5 0.2 0.1 0,05 0.02 0.01 m m w m m m m D 24.6 ° 50°C -50 ° ).01 I I I I I Illl I I i I I _III I I I I I Itll 0.02 0.05 0.1 0.2 0.5 I 2 5 I0 k ( am p - turns//J.sec ) m m 20 TB'-5094-&3 FIG. 51 _'p(k) CURVES OF CORE K-1 vs. TEMPERATURE WITH RAMP F(t) no tendency for crossing except for the -50°C curve which crossed all the other three curves (i.e., for 24.6 ° , +50 ° , and +75°C). Therefore, most, but not all, of the tendency for the experimental curves to cross is due to the parameters (namely _ and F_) varying from their step-F values. It t/ was noted in Sec. III-A-5 that decreasing F 0 increases the curvature in tl the computed log-log plot of _p(k) Thus the reduced F 0 values F"• _ _ Or 7 required for ramp-F switching will increase the tendency, for the "@p(k) curves to cross. The slight increase in v with temperature for Core I-4 makez the temperature variations more difficult to analyze. The effects of temperature upon tp can be examined from log-log plots of ktp(k). These curves are shown in Figs. 52 and 53 for Core I-4 and K-1 respectively. The major change with temperature is seen to be a vertical displacement. Examination of Eq. (101) or Eq. (103) shows that this can t! be caused by variations in _, F0, p, qb , and qb r. Only one of these pa- rt rameters, F 0 can be altered without much affecting the "qSp(k) curves. Equation (102) for "qSp is independent of F' 0. Thus, for ramp-F switching, /t as for step-F switching, it is F 0 which varies most significantly with 125 50 == :L ,_= 2O I0 0.5 0.2 0.02 50 = 75°C \ -50 ° \ \ i i i i[iJll 0.2 0.5 I k (omp-turns/p, sec) I I I I IIit 2 5 I0 T8-5094-64 FIG. 52 tp(k) CURVES OF CORE I-4 vs. TEMPERATURE WITH RAMP F(t) 126 20.0 ' ' I ' ' ''I I ' ' I' ' ' 'I I ' ' I ' ' ''I I0.0 5.0 B 2.0 = o. 1.0 -- 0.5 0.2 0.1 0.0; ! I 75 = C, COMPUTED USING _'r = 0.428 Fort'= 0.11 ornp-turn$ I , ,,,I I i = I I,,il 1 , = I ! ,!,1 0.05 O.I 0.2 05 I 2 5 tO k (omp-turn//_sec) TB-5094-65 FIG. 53 tp(k) CURVES OF CORE K-1 vs. TEMPERATURE WITH RAMP F(t) temperature. This would be expected. However, there is still the problem of getting the experimental and computed ktp(k) curves to agree for all k values. This was discussed for Core E-6 at room temperature in Sec. III-A-5 (see Fig. 34). To illustrate this problem for Core K-l, a computed curve is included in Fig. 53 for 75°C. It is shown as a dashed line. The value of Lr for this curve was determined using Eq. (109). The 0t value of For as determined from Eq. (112) for k = 6 is approximately ts -0.24 amp-turns which seems unreasonable. Since a negative For seems un- t! reasonable, a positive value equal to 1/2 F 0 was used. This resulted in good agreement for "qbv(k). Note, however, that the computed tv(k) curve in Fig. 53 crosses the corresponding experimental curve at about k = 0.45. The computed curve at k = 6 is considerably above the experimental curve. 127 ii Thus For would have to be reduced a lot more to give agreement at k = 6. Ii A new computation of For using Eq. (112) and the value of tp talc which was computed by using k = 0 428 and F" =r " 0r 0.11 gives the same negative value, -0.24, previously determined. Thus it does seem to be necessary to have a negative value of F" in order to have t agree at k = 6. TheOr p 75°C curve for Core K-1 is the only one which resulted in a negative value for F""Or, however, the values at +50°C and 24.6°C were also unreason- ably low. This problem has not yet been resolved, but is probably closely related to the general disagreement between the computed and experimental tp(k) curves. This requires further investigation. Because of the problem It concerning tp(k) and For, no meaningful plot of F"Or VS. temperature can be given. The value of L r computed for Core I-4 at 75°C from Eq. (109) was 0.114; only slightly reduced from the step-F value of 0.121. Using this value of L and F" = " = (k) gave very goodr 0r F0 0.800 amp-turns to compute tp agreement with the experimental curve for 75°C. Thus Core I-4 behaves much differently than Cores E-6 and K-1 in this respect. The agreement for other temperatures was not as good as at 75°C, but much better than any of the t (k) curves for Core K-1. P Values of k r were determined for each temperature for both Cores I-4 and K-1. This was done by using Eq. (109) and the values of _p talc and tp calc computed from step-F parameters. These values of kr were verified II by using them to compute _p(k) as discussed in Sec. III-A-3. Since For l! influences _p(k) only slightly, the values for F 0 were used for Core I-4 II and the values 1/2 F 0 were used for Core K-1. In all cases the computed "_p(k) was in good agreement with the experimental curves. These values of k r are plotted vs. temperature in Fig. 54 for Core I-4 and Fig. 55 for Core K-1. The step-F value of k is also plotted for comparison. The k r for Core K-1 is much lower than X over the entire temperature range, as was expected. The small number and the scatter of the data points makes it impossible to say for sure whether k actually varies, as shown by the smooth curve, or whether it is approximately constant with temperature. About all we can assume is that the X r temperature coefficient is roughly zero at room temperature. The L curve for Core I-4 is surprising in two respects. It is not very much lower than k above 0°C and it is significantly larger than k at -50°C. If there is any close relationship between the effects of ramp-F 128 0.20 I I I I I k 0.10 0.05 0 -50 1, I I I I / -25 0 25 50 75 I O0 TEMPERATURE-- °C T&-5Og,4-U FIG. 54 VARIATIONS OF _r AND _. WITH TEMPERATURE FOR CORE I-4 0.8 0,6 _. 0.4 0.2 I I 1 I I """""__ _ Xr I! I I I I0 ; -50 -25 0 25 50 7,5 I00 TEMPERATURE -- =C TA--5094-67 FIG. 55 VARIATIONS OF A AND A WITH TEMPERATURE FOR CORE K-1 I" 129 switching and the effects of partial setting, as has been assumed on the" basis of other experiments, then the ramp-F switching properties of Core I-4' certainly make this relationship difficult to understand. It must be noted that the ratio of k r to k is much lower for Cores K-1 and J-1 (discussed in Report 3) than for Cores I-4 and E-6. Cores K-1 and J-1 have two things in common which are not common to Cores I-4 and E-6: They are both Lockheed switch cores (Lockheed No. 100SC1 and 145SC1 re- spectively) and they both have a relatively Iarge OD/ID ratio compared to Cores I-4 and E-6. It cannot yet be stated whether either of these is related to the lower kr/k ratio for these cores than for Cores I-4 and E-6. 5. SUMMARY The effects of temperature variations on the switching properties were investigated experimentally over the range -50°C to +75°C. This was done for static _(F) curves, _p(F) curves for step-F switching, and _p(k) and t (k) curves for ramp-F switching. The static _(F) curves were taken for p both hard remanent and partially set states. The core parameters were determined from these data and plotted vs. 11 temperature. The parameters Hq, H n, and F 0 decreased very markedly with temperature, as expected. The value of q5r and 95s decreased significantly but not as much as Hq and H n. The values of k and v were not very much altered by varying the temperature. Temperature coefficients are given in Table II. Three families of experimental static qb(F) curves with tempera- ture as a parameter were given: (i) starting from --qbr; (2) starting from q6 = 0 with a positive F; and (3) starting from qb = 0 with a negative f. The core was partially set with a l-¢zsec retangular current pulse. Families of "qbp(F), _p(k) and tp(k) were also included. C. CONCLUSIONS The flux-switching properties of square-loop ferrite cores, in re- sponse to a ramp-F drive, have been investigated as a function of the slope, k, of the ramp. Three cores were tested in the experiments. It was found that the static @d parabolic model could adequately describe ta the swi%ching over a limited range of k, provided that k and F 0 are given new values k r and F_r. Generally these values need to be lower than k and F_ for step-f switching, but Core I-4 is an exception. There is no need to alter the step-F value of v. A comparison between experiment 130 and computation for thin-ring Core E-6 showed that the shape of _(t) computed by the model is accurate if appropriate values of S and F"r Or are used. The experimental and computed _p(k) curves could be made to agree over the entire range of k studied (i.e., k = 0.1 to I0 amp-turns//zsec) by using an appropriate value of 4. The computed _p(k) curve depends upon F_'r only as a second-order effect. This second-order dependence is a result of including _d(F) in the model. The computed tp(k) curves exhibited a tendency to cross the experi- mental curves. This crossing point can be fixed at any k value by appro- priate adjustment of Fd_. Thus, in a practical application, Fd_ should be chosen to make tp(k) agree with experiment near the center of the range of k values involved. nt The determination of correct values for _ and F 0 for a new core r r requires that experimental measurements be made with a ramp-F drive. However, only a few experimental points need to be taken. The effects of temperature upon the switching properties have been experimentally determined for two cores, thin-ring Core I-4 and Core K-1 (see Table II for information on these cores). The temperature range -50°C to +75°C was investigated. In general, the properties changed with temperature in only a quantitative way. The experiments indicate that the parabolic model should be as useful at any temperature in this range as it is at room temperature. The values of the parameters Hq, Hn, _r' it _s, F0' 4, v, and Lr have been plotted vs. temperature. The most sig- nificant effect of increasing temperature is the decrease in the static sl _(F) parameters Hq and Hn, and in the threshold, F 0, of the _p(F) curve. The values of _r and _s also decrease with temperature but less severely tt than Hq, H , and F 0. The values of k and v are not strongly dependent upon temperature. The small variation in k with increasing temperature for Core I-4 was, if anything, downward. Core K-l, on the other hand, exhibited a measurable increase in k with an increase in temperature (about 0.16 percent per °C at 30°C). The effects of temperature upon the static _(F) curves starting from a partially set state are difficult to describe quantitatively, since we do not yet have equations to describe these curves. The general shape, relative to the major static ¢(F) curves, was largely unchanged with tem- perature (see Figs. 44 and 45). 131 The qbp(k) curves for ramp-F switching were not very strongly in- fluenced by changes in temperature. Presumably this is a result of _bp(k) It being nearly independent of F 0 which is the major temperature-dependent parameter. The tp(k) curves are shifted downward by increasing temperature. The shape is relatively unchanged in other aspects. This downward shift t! is probably a result of the decrease in F 0 with temperature, although the tendency for the computed tp(k) curves to cross the experimental curves is not yet understood. An additional problem with tp(k) showed up at +50° and +75°C. This is the apparent negative, or very low positive, It value for For. The fact that the OD/ID ratio of the core has not been accounted for in the model [except for _d(F)] may be responsible for some t¢ of this trouble. These problems with tp(k) and F 0 require further investigation. The ramp-F switching properties of Core I-4, in relation to step-F switching properties, are different than for the other three cores which have been tested (this includes Cores E-6, K-1 and a one-point check on Core J-i in Report 3). The relationship of parameter variations for ramp-F switching and switching from a partially set state is not yet understood; although, a close relationship must exist. Except for Core I-4, _ and F_ are reduced from step-F values for both cases. Core I-4, on the other hand, exhibited a large reduction in L for a partially set state (see p. 113 of Report 3) but only a small reduction for ramp-F switching (even an increase at -50°C). This relationship appears to require rather extensive work before it can be clearly under- stood. However, the presently used switching model can be used in many practical applications where F(t) is approximately a ramp function. 132 APPENDIX A COMPUTER PROGRAM FOR PROCEDURE _e (F,At ,NV,_p_ ) Language: ALGOL 60. A tProgram Description: Computes _ and _ for givcn values of F, A t, and NV (negligible value of _ below which _e is assumed to be zero). Switching parameter E is global. Identifiers: * Identifier Symbol . Identifier S__ymbo1 DELT A t PHI DOTE be EPS e PHIDOTEPRIME _ FDOT F PHI DTE _E NV NV Program: REAL PROCEDURE PHIDOTE (FDOTe DELT, NV, PHIDOTEPRIME)! PHDOTE0I COiqMENT: THIS PROCEDURE COMPUTES PHIDOTE AND FHIDOTF_.PRIME VS. FOOT PHOOTE02 AND DELT. PHIDOTE IS ASSUMED ZERO IF BELOW NV. THE PROCEDURE USES GLOBAL PARAMETER EPS.! VALUE FDOT, DELTe NV! REAL FDOTe DELT, NVp PHIDOTEPRIME! BEGIN REAL PHIDTEI PHIDTE • EPS x FDOTI PHZDOTE • IF PHIDTE > NV THEN PHIDTE ELSE O! PHIOOTEPRIME • IF PHIDTE ) NV THEN EPS / OELT ELSE 01 END OF PHIDOTE! PI4OOTE03 PHDOTE0_ PHOOTE05 PHDOTE06 PHDOTE07 PItDOTEOB Fq.IDOTEO? PHOOTEI0 PHDOTE11 Listed in alphabetic order. 133 APPENDIX B I COMPUTER PROGRAM FOR PROCEDURE _ _ (F, t, r i, _ i ) Language: ALGOL 60 _., Program Description: Comput, es 3,, T i, and w i for given values of F and t. Switching parameters _-i, z_i' Ci and F i are global. Identifiers: Identifier Symbol Identifier Symbol CI C i PHI DOTI _ i I F F PHIDOTIPRIME _i FDEX F ¢ x PHI DTI _ FI F_ T t LAMBDAI k i TI T NUI 7; i Program: REAL PROCEDURE PHIDOTI {Ft Tt TIp PHIDOTIPRIME}! COMMENT: THIS PROCEDURE COMPUTES PHIDOTI AND PHIDOTIPRIME VS, Ft T • AND TI. THE PROCEDURE USES GLOBAL PARAMETERS LANBDAIp NUIe CI• AND FI.; VALUE F, TI REAL F, T• TIt PHIDOTIPRINEi BEGIN REAL FDEXe PHIDTI! FDEX * F - FI; IF TI _ 0 THEN PHIDTI _ LANBDAI x FDEX * NUI x EXP ((TI - T) x FOEX / CI) ELSE BEGIN PHIDTI • O! IF FDEX ) 0 THEN PHOOTIOI PHDOTI02 PHDOTI03 PHDOTI04 PHDOTI05 PHDOTI06 PHDOTI07 PHDOTI08 PHDOTI09 PHI)OTIIO PHDOTIII PHDOTII2 BEGIN TI * T! PHIDTI * LAMBDAI x FDEX * NUI x EXP ((TI - T) PHDOTI13 x FDEX / CI) ENDS END! PHIDOTI * IF PHIDTI > 0,0010 x LAHBDAI x FDEX THEN PHIDTZ ELSE O! PHIDOTIPRINE * IF PHIDTI ) 0•0010 x LAMBDAI x FDEX THEN PHIDTI x (NUI / FDEX - (T - TI) / CI) ELSE O! END PHIDOTI! PHi)OTII_ PHDOTI15 PHDOTI16 PHDOTI17 PHOOTI18 PH;OTI19 PHDOTI20 PHDOTI21 135 APPENDIX C COMPUTER PROGRAM FOR PROCEDURE _p,_(F,q_,_d,_p'la) Language: ALGOL 60 Program Description: Computes _.=, qba, and _' for given values of Fandgb. t! Switching parameters li, l o, _Dr, (_$, Ha, Hq, Ha, _., Fo, v, pp, Fo, FB, F12, F23 , V 1 and V 2 are global. Identifiers: Identifier Symbol Identifier Symbol F F PHI q6 FB F B PHID _bd F0 F 0 PHIDOTMA _.. u FOPP F 0` PHIDOTMAPRIME _' mo F12 F_ PHIBOTP _bp I F23 F23 PHIBOTPPRIME _p HA H PHIDPRIME qb'a HN H PHIR _ r PttlS qbsHQ H q LAMBDA L ROP p p LI l _ V1 V 1 LO l o V2 V 2 NU Program: REAL PROCEDURE PHIDOTMA (Fe PHIe PHIDe PHIDOTMAPRIME)! PHDTMA01 COMMENT: THIS PROCEDURE COMPUTES PHIDOTMAe PHIDe _ND PHIDOTMAPRIME PMDTMA02 FOR GIVEN VALUES OF F AND PHI. GLOBAL CORE PARAMETERS ARE: LIpLOe PHDTMA03 PHIRePHISpHApHGwHNeLAMBDAeFOPPpNUeROPeFOtFB_F12eF23eV1eAND V2,! PHDTMAOq REAL Fp PHIl PHID_ PHIDOTMAPRIMEJ PHOTMA05 BEGIN REAL PHIDPRIMEp PHIDOTPt PHIDOTPPRIME! PHDTMA06 COMMENT: COMPUTE PHID AND PHIDPRIME VS, F,! PHDTMA07 IF F $ F12 THEN PHDTMA08 BEGIN PHID * V1 x F x LN ((F - HA x LO) / (F - HA x LI)) - PMIR PHDTMA09 ; PHDTMAIO PHIDPRIME * V1 x (LN ((F - HA x LO) / (F - HA x LI)) + F x ( PMDTMAll I / (F - HA x LO) - I / (F - HA x LI))) PHDTMA12 ENDI PHDTNA13 137 IF FI2 < F AND F S F23 THEN PHDTMAI; BEGIN PHID • V2 x (F / HQ - LI + F x (1 / HN - I / HQ) x LN ((1 PHDTMAI5 - HN / He) / (1 - HN x LI / F))) - PHIR! PHDTMAI6 PHIDPRIME * V2 x (1 / HQ + (1 / HN - I / H@) x (LN (F x (1 - PHDTMA|? HN / He) / (F - HN x LI)) - HN x LI / (F - HN x LI))) PHDTMAi8 ENDI PHDTMAI9 IF F23 < F THEN PHDTMA20 BEGIN PHID _ V2 x (LO - LI + F x (1 / HN - _ / HG) x LN ((F - PHDTMA21 HN x LO) / (F - HN x LI))) - PHIR! PHDTMA22 PHIDPRIME * V2 x (1 / HN - I / H@) x (LN ((F - HN x LO) / (F PHDTMA23 - HN x LI)) + F x HN x (LO - LI) / ((F - HN x LO) x (F - HN PHDTMA2; x LI))) PHOTMA25 END; PHDTMA26 COMMENT: COMPUTE PHIDOTP AND PHIDOTPPRIME VS, Fo! PHDTMA27 IF F $ FOPP THEN PHDTNA28 BEGIN PHIDOTP _ O! PHDTMA29 PHIDOTPPRIME • 0 PHDTMA30 END; PHDTMA31 IF FOPP < F AND F S FB THEN PHDTMA32 BEGIN PHIOOTP * LAMBOA x (F - FOPP) * NU! PHDTMA33 PHIDOTPPRIME * LAMBOA x NU x (F - FOPP} * (NU - 1) PHDTMA3; END; PHDTMA35 IF FB < F THEN PHDTMA36 BEGIN PHIDOTP * ROP x (F - FO)I PHDTMA37 PHIDOTPPRIME • ROP PHDTMA58 ENDI PHDTMA39 COMMENT: COMPUTE PHIDOTMA AND PHIDOTMAPRIME°! PHDTMA_O PHIOOTMA _ IF PHID - PHI > 0.001 x PHIR THEN PHIDOTP x (1 - ((2 PHDTMA_I x PHI + PHIR - PHID) / (PHIR + PHID)) * 2) ELSE O! PHDTMAq2 PHIOOTMAPRIME * IF PHID - PHI ) O,O010XPHIR THEN (I - ((2 x PHI PHDTMAW3 + PHIR - PHIO} / (PHIR + PHID)) * 2) x PHIDOTPPRIME + W x PHIDOTPPHOTMAWW x (2 x PHI + PHIR - PHID) x (PHI + PHIR| x PHIDPRIME / (PHIR + PHDTMAW5 PHID) * 3 ELSE 0 ! PHDTMA_6 END PHIDOTMAI PHDTMA_7 138 APPENDIX D COMPUTER PROGRAMS FOR _ _ _ + _. 1. PROCEDURE _(F, _b, gbd, <_') Language: ALGOL 60 Program Description: Computes _, qSd, and _'for given values of Fandq5. Switching parameters li, l and F B are global. Identifiers: o' Dr, _, H a, Hq, Hn, _' F'_, v, PP' Fo' (1) Analytrcal identifgers Same as in Appendix C, except for PHIDOT (_) and PHIDOTPRIME (_'). (2) Auxiliary identifiers Description Boolean variable Label Identifier BONE OK Program: REAL PROCEDURE PHIDOT(FePHItPHIDePHIDOTPRIME)! PHIDOT01 COMMENT: THIS PROCEDURE COMPUTES PHIDOTp PHIDe AND PHIDOTPRIME PHIDOT02 FOR GIVEN VALUES OF F AND PHI, CORE PARAMETERSp WHICH MUST BE PHIDOT03 SUPPLIED FROM OUTSIDE THE PROCEDUREp ARE: PHIDOTO_ LIp LOp PHIRe PHXSp HAp HQp HNw LAMBDAp FOPPp NUp ROPp FOe FB ! PHIDOT05 REAL F, PHIp PHIDo PHIDOTPRINE! BEGIN REAL PHIDPRIMEp PHIDOTPp PHIDOTPPRIME ! OWN REAL Fl2t F23t Vii V2 I LABEL OK/ OMN BOOLEAN DONE! COMMENT: COMPUTE F12e F23e VIe AND V2 ONLY ONCE.! IF DONE THEN GO TO OK! DONE _ TRUES FI2 _ HQxLI ! F23 * HGxLO ! Vl * (PHZS-PHIR)/((LO-LZ)XHA); V2 • (PHZS+PHXR)XHO/((LO-LI)xHN)! OK: PHIOOT06 PHIDOT07 PHIDOT08 PHIOOT09 PHIOOTIO PHIDOTll PHIOOTI2 PHIDOTI3 PHIOOTI_ PHIOOT15 PHIDOTI6 PHZOOT17 PHIDOT18 PHIDOTI9 139 Lines PHDTMA07 through PHDTMA39 of PROCEDURE &, _(F,_,#_ d,&' ), APPENDIX C COMMENT: COMPUTE PHIDOT AND PHIDOTPRIME.I PHIDOT * IF PHID-PHI ) OoOOIXPHIR THEN_PHIDOTpx(z-((2XPHI+PHIS -PHID)/(PHIS+PHID))*2) ELSE 0 ! PHIDOTPRINE * (I-((2xpHI+PHIS-PHID)/(PHIS+PHID))*2)XpHIDOTPPRIME +_xPHIDOJPx(2xPHI+PHIS-PHID)x(PHI+PHIS)xPHIDPRIME/(PHIS+PHID)I3 END PHIDOT! PHIDOT57 PHIDOT58 PHIDOT59 PHIDOT60 PHIDOT61 PHIDOT62 . PROCEDURE _(F, ¢, &', _*) Language: ALGOL 60 Program Description: Computes _, _', and _* for given values of Fand ¢. Switching parameters li, lo, Cr' Cs' Ha, Hq, H , X, Y_, _, pp, Fo, F B, F_ , F23, V 1 , and V 2 are global. Identifiers: Same as in Appendix C, except for PHIDOT (_) and PHIDOTPRIME (_'). Program: REAL PROCEDURE PHIDOT (F, PHIt PHIDOTPRIMEp PHIDOTSTAR)! COMMENT: THIS PROCEDURE COMPUTES THE MAIN PHIDOTp PHIDOTPRIME AND PHIDOTSTR FOR GIVEN VALUES OF F AND PHI. THE PROCEDURE USES THE FOLLOWING GLOBAL PARAMETERS: LIp LOp PHIRt PHISp HAt Her HNe LAMBDA, FOPPw NUp ROP_ FOw FBt F12w F23w VltAND V2! VALUE Fw PHIl REAL Fw PHIt PHIDOTPRIMEw PHIDOTSTARt BEGIN REAL PHIDPRIMEe PHIDp PHIDOTPw PHIDOTPPRIME! ILines PHDTMA07 through PHDTMA39ofPROCEDURE _ ,(F,¢,¢d,_>' ), APPENDIX C PHIDOT01 PHIDOT02 PHIDOT03 PHIDOTOq PHIDOT05 PHIDOT06 PHIDOT07 PHIOOTO8 COMMENT: COMPUTE PHIDOTp PHIDOTPRIME AND PHIDOTSTAR.! PHIDOT_I PHIDOT * IF PHID - PHI > 0.001 x PHIR THEN PHIDOTP x (1 - ((2 x PHIDOT_2 PHI + PHIS - PHID) / (PHIS + PHID)) * 2) ELSE O! PHIDOT_3 PHIDOTPRIME * IF PHID - PHI > 0,0010 x PHIR THEN (1 - ((2 x PHI + PHIDOT_4 PHIS - PHID I / (PHIS + PHID)) * 2) x PHIDOTPPRIME + q x PHIDOTP x ( PHIDOTq5 2 x PHI + PHIS - PHID) x (PHI + PHIS) x PHIDPRIME / (PHIS + PHID) PHIDOTq6 • 3 ELSE OI PHIDOT_7 PHIDOTSTAR * - ; x PHIDOTP x (2 x PHI + PHIS - PHID) / (PHIS + PHIDOT_8 PHID) * 21 PHIDOT;9 END PHIDOT! PHIDOTSO 140 APPENDIX E COMPUTER PROGRAM FOR _., _,, _... AND _ OF UNLOADED CORE Language: ALGOL 60 " _ _' _' _a andFversus t forProgram Description: uomputes i_, _, _i' w.., given core parameters and drive function. Identifiers: (1) Analytical identifiers Identifier Symbol Identifier Symbol CAPID I o ND No CI C i NU v DELT A t NUI v i EPS E PHIC q5 F F PHIC1 q5 n--I FB F B PHIC2 _b 2 FDOTC PHIDC _ d FI F i PHIDOTC _b F0 F o PHIBOTC1 _.- 1 F0I F o i PHIDOTEC _E ttFOPO Fo OHIDOTEPR F12 F m PHIDOTIC _i F23 F23 PHIBOTIPR _'_ HA H . PHIDOTMAC _.. PHIDOTMAPR _b."'HQ H q HN H a PHIR qb, ID i D PHIS q5 LAMBDA k ROP p p LAMBDA I k i SP S p LI I i T t LO 1 o TAUS T, 141 Identifier Symbol TF T/ TII 7h TIC T i TM t TR T r (2) Auxiliary identifiers Identifier A1 A2 Abbreviation Abbreviation for Identifier Symbol TS T s U u V1 V 1 V2 V 2 CIRCUITPABAMETERH CIRCUITPARAMETERL CIBCUITPARAMETERS CORENAME COBEPABAMETERH COREPARAMETERL COBEPARAMETERS COUNT CTS CO DELPHI C FSCALE GUESS K LOOP NV OUTPUTFORMAT OUTPUTHEADING OUTPUTVARIABLES Description /(t ) for Sp t m S /t t _ m Format for the list CIRCUITPARAMETERL. List of drive parameters (Io, ND, Tr, u, tm, and Sp). AtList of drive parameters (Io, t , S v, , and plot scales for t, F, _, and type) for input-data cards. Name of core. Format for the list COREPARAMETERL. List of core parameters. List of core parameters for input-data cards. Index number of At. Index number of iteration. _(j=l) - _.(j=o)" _n(j) -- _n(j- 1)" F scale (for plotting). Label of location where prediction is made. Index number of At for plotting. Label of location where iterative computation begins. Negligible value of _E" Format for the list OUTPUTVARIABLES. Format for output-column heading. List of results (t, io, _e, _i, _,a, @, @, @d, F, and Jmax ) • 142 Identifier P PLOTE PSCALE QUIT QUITC_T STARTC_T STARTCORE SWI TCHI NG _ETA5 _CALE Description Boolean variable determining type of output (for plotting). Label (for plotting). scale (for plotting). Label of location where computation terminates. Label of location where computation of given circuit parameters terminates. Label of location where computation for given drive begins. Label of location where computation for given core begins. Label of location where computation starts for each At. [¢.(i=s) - _.(j=4)]/[¢.cj=]) - _._=0)] Time scale (for plotting). Program: ELASTIC AND INELASTIC INITIAL PHIDOT SPIKES OF AN UNLOADED CORE. BE6IN CONiqF_T: DECLARATION OF CORE PARANETERSel ALPHA CORIENAIIE ! REAL LIt LOp PHIR* PHXSt HAo He, HN, LANBI)A, FOPP* NU* ROF, FO, FB • Vl* V2• F22* F23• El>S• LAMBDAI, NUI* CI• FOX* FI! CONNENT: DECLARATION OF CIRCUIT PARANETERS. I REAL CAPID, NO• TR• Up TN• SP• TS, TAUSe THe TF, Ale A21 COMNENT: DECLARATION OF VARZABLESe! REAL T• DELT, TICe ;De F, FDOTCe PHIDOTCe PHIDOTCI, PHIDOTMAC• PHIDOTEC• PHIDOTIC• PHIDOTNAPR* PHZOOTEPR* PHIDOTZPR* PHIC* PHIC2• PHIC2• DELPHIC. CO• THETAS. PHIl)C; CONNENT: DECLARATION OF NISCELLANEOUS.; REAL NV• TSCALE• FSCALE, PSCALE 1 INTEr_IR LINES• COUNT, CTS, K! BOOLEAN P ! LABEL STARTCORE. STARTCKT• SIITCHIN6. 6UESS.LOOP*QUITCKT.PLOTEeQUITI REAL ARRAY PHIDTA* GOTAe TIN/A [0 , 500]1 CONNENT= DECLARATION OF FILE. LISTS. AND FORNATS. I FILE CR 0 (2• 20)1 FILE FI I (2, 15)1 LIST COREPARANETERS (CORENAN[. LI. LO. PHIR* PHIS, HA. He. HN. LANBOA. FOPP• NU• ROP• FO* FB* LANBOAI. NUI* CI* FOI)! LIST COREPARANETERL (CORENANE. LI x 183. LO x 183. PHIR x 188. PHIS x 188, HA, HO, HN, LANBDA* FOPP* NUt ROP. FO* FB* [PS x 189. LANBDAI• NU|• CI x 186, FOIl! LZST ¢IRCUITPARANETERS(CAPID,TN,SP•D[LT,TSCALE,FSCALIr,PSCALE,p) I LIST CIRCUITPARANETERL (CAPID, ND, TR x 286, U x 1i_'6, TN x 186, SP x 18-6)8 001 00:' 003 OOq 005 006 007 008 009 010 011 012 013 Olq 015 016 017 018 029 0=0 0=)1 0=)= 0=)3 02q 0=)5 0=6 0=)? 0=)8 029 143 LIST OUTPUTVARIABLES (T x IQbe IDp PHIDOTECp PHIDOTICe PHIDOTMACw PHIDOTC, PHIC x IgBp PHIDC x IgBt Fp CTS)I FORMAT COREPARAMETERH ("CORE "e A6p X_ "LI="_ F8.3_ X_, "LO="p F8.3, X_p "PHIR="e F8o3, X_p "PHIS:N_ FB°3p X_e "HA:"e FBe3p X_p "He ="w FB.3t X_ -HN="e F8.3 / X15e "LAMBDA="e FB.Se X_ "FOPP="e F8.3, X;t "NU="p F8.3, X;_ "ROP="_ FB.3p X4P "FO="p FBe3s X_ "FB="w FBo3 / XI5e "EPS="o FB.;p "B-9", X_e "LAMBDAI="e FBe5e X;, "NUI=", FB.3, X;, "CI=", FBo3t "G-b"p X;t "FOI= "P F8.3 /}! FORMAT CIRCUITPARAMETERH ("DRIVE PARAMETERS"_ XBe "CAPIO="p FBe;p X;e "ND:"e I2w X_e "TR="e FB.3e X_e "U="p F6,2e "B6"e X_e "TM="e FB.3, X;e "SP=", F6.2_ "G6 m /)! FORMAT OUTPUTHEADING (/ X6e "T"p Xllp "ID"_ X5_ "PHIDOTEC"e X;p "PHI OOTIC"e X_ "PHIDOTMAC", X_, "PHIDOTC"_ X8, "PHIC"e XT_ "PHZDC"_ xge "F"_ xge "CTS" /)! FORMAT OUTPUTFORMAT (F9.3, F12.3, ; F12.5_ 2 F12,2_ F12.3_ I10)! COMMENT: OECLARATION OF PROCEDURES°; PROCEDURE _p¢([_,At,NV,_5'), APPENDIX A PROCEDURE _i(F, t,T_,_5'_), APPENDIX B PROCEDURE _.a(F,qb._ba,_'a). APPENDIX C 030 031 032 033 03¢ 035 036 037 038 039 0_0 0_1 0_2 0_3 0_ 0_5 REAL PROCEDURE TANH (Xl)! VALUE XI; REAL Xl! BEGIN REAL Y! Y • EXP (Xl + Xl); TANH _ (Y - 1.0) / (Y + 1.0) END TANH; REAL PROCEDURE SECH (X); VALUE X! REAL X! BEGIN REAL U! U * EXP (X)I SECH • 2 / (U + 1 / U) END SECH; COMMENT: READ INPUT " DATA CARDS_ COMPUTE CORE AND CIRCUIT PARAMETERS_ AND PRINT HEADINGo! STARTCORE : READ (CR_ /_ COREPARAMETERS) [GUIT]! STARTCKT : READ (CR_ /_ CZRCUITPARAMETERS) [GUITCKT]; Vl * (PHIS - PHZR) / ((LO - LI) x HA)! V2 • (PHIS + PHIR) x HG / ((LO - LI) x HN); FIZ * HG x LII F23 * HG x LOt EPS * VI x LN (LO / LI); NV • 0.001 x EPS x SP! NO • I; TR • 2 x TM; U • 2 x SP / CAPID; A1 * (5P - CAPID / TM) / TM * 2; A2 * (1.5 x CAPID / TM - SP) / TH; TS * 900 x OELT; TAUS * PHIR / (0.3 x ROP x NO x CAPID)I _RITE (FI (PAGE]); _RITE (FI_ COREPARAMETERHe COREPARAMETERL); _RITE (FI_ ClRCUITPARAMETERH_ CIRCUITPARAMETERL); _RITE (FI_ OUTPUTHEADING); COMMENT: INITIALIZE VARIABLES AND PARAMETERS. I TANHO001 TANHO002 TANHO003 TANHO00_ TANHO005 TANNO006 TANHO007 SECHO001 SECHO002 SECHO003 SECHO00; SECHO005 SECHO006 SECHO007 0;6 0;7 O;S 0;9 O50 051 052 053 05_ 055 056 05"/ 05B 059 060 061 062 063 06_ O65 066 067 144 T _ TIC * 0! ID*F*O! PHIC2 * PHICl * PHIC * PHIDC • -PHIR # PHIDOTCI • PHIDOTC * PHIDOTEC *PHIDOTIC * PHIDOTNAC * 0 ! LINES *10; K *10! COUNT * 0; WRITE|FI.0UTPUTFORMAT.OUTPUTVARIABLES) ! COMMENT: COMPUTE VARIABLES DURING SWITCHING TIME.! SWITCHING: T _ T + DELT! COUNT * COUNT + 1! ID • IF T < TM THEN T * 2 x (AI x T + A2} ELSE CAPID x (1 + TANH (U x (T - TM))) / 21 F * NO x IO! FI • FOI x TANH (F/FOI)! FOOTC * IF T < TN THEN NO x T x (3 x A1 x T + 2 x A2) ELSE CAPID x NO x U x SECH {U x IT - TN)) • 2 / 2l CTS * 0! GUESS: PHIC • PHIC2 ÷ 2 x DELT x PHIDOTCI; LOOP: CTS * CTS ÷ 1l PHIDOTNAC * PHIDOTMA (F. PHIC. PHIDC. PHIDOTNAPR)I PHIDOTEC * PHIDOTE (FDOTCp OELTe NV, PHIDOTEPR)! PHIDOTIC * PHIDOTI (F. T. TIC, PHIDOTIPR); PHIOOTC * PHIDOTMAC + PHIDOTEC + PHIDOTIC; DELPHIC * PHIC1 + DELT x (PHIDOTC + PHIDOTC1) / 2 - PHICI PHIC * PHIC ÷ DELPHIC; IF CTS = 1 THEN C0 * DELPHIC; IF CTS = 5 THEN BEGIN THETA5 * DELPHIC / C0; IF 0.9 < ABS (THETA5) THEN GO TO GUESS; DELPHIC _ 0 END; IF ABS (DELPHIC) > O.001x PHIR x DELT / TAUS AND CTS ¢ 6 THEN 60 TO LOOP; PHIC2 • PHIC_i PHICI * PHIC; PHIDOTCI * PHIDOTC! COMMENT: PRINT OUTPUTo; IF P THEN K * IF T ( TR THEN 2 ELSE 10 ! IF COUNT MOD K = 0 THEN BEGIN IF LINES MOD SO = 0 THEN BEGIN WRITE (FI [PAGE]); WRITE (FI. OUTPUTHEADIN6) END; WRITE (F1, OUTPUTFORHAT. OUTPUTVARIABLES)! LINES * LINES + 1 END; IF T $ TS THEN GO TO SWITCHING ; GO TO STARTCKT ; GUITCKT : CLOSE (CR. SAVE); GO TO STARTCORE; @UIT: END. 068 069 070 071 072 073 07_ 075 076 07? 078 079 080 OBl 082 083 OB; 085 086 087 088 089 090 09i 092 093 09_ 095 096 O9'7 og8 099 100 101 102 103 101_ 105 106 107 108 109 110 111 112 113 115 116 117 118 119 120 121 A sample of input data: "E-6 ", 22.19_-3, 23.5¢g-3, 3.;5Q-8, 3.726_-8, 310.0, 35.0' 30.0, 0°069, 0°95, 1o30. 0o1132. 1oq5, 3o12,0o01200,1.300,0o2_5g-6,0o5500, 0.60, 0o0592_-6. 10.1596,0.;g-9, 2596, 5, _OOe O, 0.60. 0.0592_-6. 10.1596,2B-9, 5g6p 5, _00. 1. 145 APPENDIX F _ FOR LOADED CORE Language: ALGOL 60 Program Description: Computes i v , _, _, 4>d' F, _, and Vd versus t for given core, circuit, and drive parameters. Identifiers: (1) Analytical identifiers Identifier Symbol Identifier Symbol C C NU u CAPID I o PHIC 4> DELQD /X_ PHIC1 4> n-1 DELT At PHIC2 4> n-2 DPHDTDFC _' PH I DC _d ED e _ PHIDOTC EK E k PHIDOTC1 _. - 1 F F PHIR 4>_ FB F 8 PHI S 4> FJ f,, (j) Q q FJPR f" (j) Q1 q,-1 F0 F o QD ti FOPP F 0 QD1 qn - 1 HA H QD2 _/n- 2 HN H QDD n HQ //q QDD1 q _ - 1 ID i R RD IO I o BD R d L L RL R L LAMBDA k BOP P v LI l. T t t LO l TAUS _r O S NC N c TR T r ND ND 147 (2) Auxiliary identifiers Identifier CIRCUITPARAMETERH CIRCUITPARAMETERL CIRCUITPARAMETERS CORENAME COBEPARAMETERH COBEPARAMETERL COREPARAMETERS COUNT CTS GUESS LINES LOOP OUTPUTFORMAT OUTPUTHEADI NG OUTPUTVARIABLES P Pl QUIT S START SWITCHING XSCALE YSCALE ZSCALE Description Format for the list CIRCUITPARAMETERL. List of circuit parameters (No, RL, L, C, B a, I o, Ek, Io, No, and Tr). List of circuit parameters (same as above) for input-data cards. Core name, e.g. 3-1. Format for the list COREPARAMETERL. List of core parameters (Core name, l i lo, _r' dPs' t! Ha, Hq, H n, h., Fo, 9, pp, Fo, and F/_). List of core parameters (same as above for input-data cards. Index number of nth At during switching. Index number of jth iteration for each nth At. Label of location where initial approximation of q5 is made for each nth At. Index number of automatically plotted set of output. Index number of printed line Label of location from where iterative computation is repeated for each nth At. Format for the list OUTPUTVARIABLES. Format for output column heading. List of results it, i D, _, _b, _d' F, q, e d +i q Rd, J..x)- 1/_) if "qj # 0, zero otherwise. 1/(qi-1 -qn-1) if qj-1 / qn-l' zero otherwise. Label of location where computation terminates 1/C if C is finite, zero otherwise. Label of location where computation starts for given core and circuit parameters. Label of location where computation starts for each nth At. Time scale, used in automatic plotting of resulting waveforms. scale, used in automatic plotting of _(t). scale, used in automatic plotting of q(t). 148 Program: LOADED CORE BEGIN COMMENT: DECLARE CORE PARAMETERS• CIRCUIT PARAMETERS• VARIABLES • AND AUXILIARY IDENTIFIERSol REAL LI• LOt PHIR, PHISe HAp HQ• HN• LANBDAw FOPP• NU• ROPp FO• FB ! REAL RL• L• C, RO• IO• TRt CAP;O• EK, TAUS, R• Sp P, PI! REAL Tt OELT• PHIC, PHICI, PHIC2, PHIOOTC• PHIOOTCI, ZOt F, et glt GD• QDI• QD2• QDD, QDDI• DELQDp PHIDC, DPHDTDFCe DEPHZDOTC, DEED, GO, FJ• FJPR• XSCALE, YSCALE, ZSCAI.Et 6• 61! INTEGER LINES• CTS• COUNT, NC, NO, K! ALPHA CORENAIE. ! LABEL START, SWITCHING, GUESS, LOOP• GUZT! COMMENT: DECLARE INPUT / OUTPUT.! FILE IN CR 0(2, 10)! FILE OUT FI q(2• 15)I LIST COREPARANETERS(CORENANE, LI• LO• PHIR, PHIS, HA, HQe HNe LAMBDA, FOPP• NU• ROPe FO, FB)! LIST COREPARANETERL(CORENANE• LI x 1B3• LO x IB3• PHIR x IBBe Pills x 108• HA• HQ, HN, LANBDA• FOPP• NU, ROP, FO• FB)! LIST CIRCUITPARAMETERS(NC, RL, L, C, RD, ZQ, EKe CAPID• NO, TR)! LIST CIRCUITPARANETERL(NC, RL, L x 106• C x 10b, RD, ]0 x IQ6• EK, CAPID, ND• TR x IQ6)! LIST OUTPUTVARIABLES(T x 10b• ID, PHIOOTC, PHIC x 188, PHIDC x IB8 ' • F, GD• ED + GD x RD• CTS)! FORMAT COREPARAMETERH("CORE o, Ab, X_, °LI=°, FBe3e X;• °LO=W, F8.3• X_• °PHIR=°• F8.3, X_• NPHIS=°, F8.3• X_, °HA=m• F8.3, X_p "HG=°p F8.3p X4, °HN=°• F8.3 / XISe ULANBDA= N, F8.5, X_, NFOPP=W, F8o3, X_p _NU= I, F8.3• X_, "ROP= °e FBe3, X_, °FO=We F8o3, X_e °FB="t F8e3 /)! FORMAT CIRCUITPARANETERH(WCIRCUIT% X8, "NC= w, Z2, X_e WRL=Op F8.3 • X_• WL=°• F8.3, X_• °C=°, F8.3, X_• "RD =w, FB.3p X_w mZO=Np F8,3 , X;e °EK=°, F8.5 / °PARAMETERS°, X5• °CAPID=Ot F8.3, X4e •NO=We 12, X_, NTR:°, FB.3 /)! FORMAT OUTPUTHEADIN6(/ X6, "T N, Xll_ wIO", X7e mPHIDOTCWp XTe mPHZCW• XT, _PHIDC °e X9, WF°e X11, °QD°, XlOe _VD _, X9_ "CTS" /)1 FORMAT OUTPUTFORMAT(Fg.3, 2 F12.3• 2 F12.2• 3 F12o3e I10)! COMMENT: PROCEDURES USED BY THIS PROGRAM ARE MIN, MAXe AND PHIDOT. ! REAL PROCEDURE MIN(A,B)! VALUE A•B! REAL A,B! BEGIN NIN * IF A < B THEN A ELSE B END MZN ! REAL PROCEDURE MAX(AeB)I VALUE AeB! REAL AeB! BEGIN MAX * IF A > B TH_N A ELSE B END MAX I MZN NIN MIN MAX MAX MAX 001 002 003 00_ 005 006 O0"t 008 009 010 011 012 013 01_ 015 016 017 018 019 020 0='1 022 0='3 02_ 0='5 026 02? 0='8 0='9 030 031 03=' 033 03_ 035 036 037 038 01 02 03 01 O2 O3 PROCEDUP,E _(F,qb,¢d,_'), APPENDIX D-1 ] COMMENT: READ INPUT - DATA CARDS AND PRINT HEADINGS.! READ¢CR, /, COREPARANETERS)! START: READ(CR, /t CIRCUITPARAMETERS)[GUIT]! _RITE(FI[PAGE])! _RITE(FI_ COREPARANETERH, COREPARANETERL)! _RZTE¢FI_ CIRCUITPARAMETERHe CIRCUITPARAMETERL)! MRITE(FI, OUTPUTH£ADIN6)! 149 039 0_0 0;1 0_2 0_3 01_1_ 01_5 L|NES * 6! COUNT*O! K * O! COMMENT: %N%TZALIZE VARIABLES AND PARAMETERS,! PHXC2 * PH|CI * - PHIRI GO2 * GO1 * GOD1 * QI * PHZOOTC1 * O! S * IF C < lg10 THEN I / C ELSE OI R * RL ÷ ROt TAUS * %F TR S lg-7 THEN 2 x PHIR /(0.3 x ROP x(ND x CAPZD - FOPP) ) ELSE SGRT(PHIR x TR /(0.15 x ROP x NO x CAP%0))I DELT * TAUS / 2001 T * TR x FOPP /(NO x CAPZO)I COMMENT: COMPUTE VARIABLES DURING SWITCHING TIME.! SWITCHING: T * T + DELTI COUNT * COUNT + 1l 10 * CAPZO x MZN(T / TRp 1)! CTS * Ol GUESS: PHIC * PHIC2 + 2 x DELT x PHZDOTCI! QD * QD2 + 2 x DELT x GDDll IF I0 < 1810 THEN QD * MAX(QOe 0)1 G * el + DELT x(eO +eOl) / 2! LOOP: G1 * G! CTS • CTS + 11 F * NO x |D - NC x GO! OEPHIOOTC * PHIOOT(FP PHIC, PHIDCe DPHOTOFC) - PHZOOTC! PHIOOTC * PHIDOTC + DEPHZDOTCI PHIC * PHIC1 + DELT X(PHZOOTC + PHIDOTcz) / 2! EO * EK x LN(GD / IO + 1)! IF L _ 0 THEN BEGIN GOD *(NC x PHIOOTC - Q / C - R x QD - EO) / L! DEeD * GO1 + DELT x(QDD ÷ GDD1) / 2 - aD! ENDI IF L = 0 THEN BEG%N GOD *(gO - GO1) / OELTI FJ * R x GD + EO + S x e - NC x PHZDOTC! ZF GDD • 0 THEN P * I / GDD ELSE P * O! FJPR * R + EK /(QD ÷ %0) + S x QO x OELT / 2 + DPHDTOFC x NC * 21 DEQD * - Fd / FJPRI ENDI QD * GD + DEED! |F ZO < 1g10 THEN QD * MAX(QDe O)l G * R x QD + ED + S x Q + L x GOD - NC x PH%DOTC! IF SIGN(G) ¢ SIGN(G1) THEN BEGIN QD * QD - 0,5 x DEGD! G • 6 - 0,5 x OEQD x RI END! Q * el ÷ DELT x(QD + QDI) / 2! IF(ABS(OEQD) > 0.001 x QO AND CTS < 10) THEN GO TO LOOP! PHIC2 * PHICll PHICI * PHIC! PHIOOTC1 • PHIDOTCI G02 * COl! QD1 * GD! el * el QDDI * QDO! COMMENT PRINT OUTPUT ! ZF COUNT MOO 20 = 0 THEN BEGIN IF LINES NOD 50 = 0 THEN BES%N MRZTE(FI[PAGE])! MRITE(FIe OUTPUTHEAD%N6) ENDI WRITE(FIe OUTPUTFORMATe OUTPUTVARIABLES)! LINES * LINES + I 0_6 0_7 01$8 0_9 OSO 052 053 05_ 055 O56 057 058 O59 060 061 062 063 06; O65 066 067 068 069 070 071 072 073 071l 075 076 OT? 078 079 080 081 082 083 08_ 085 086 087 . O88 O89 O9O 091 092 093 09_ 095 096 O9? 098 099 I00 I01 I02 103 I05 106 I07 I08 I09 150 (NDt IF PHIDOTC • 0 THEN GO TO SUITCHIN6/ GO TO START! GUIT: END. 110 111 112 113 A sample of input data: Nd--I "eT.18_-3tllo5BG-3p31,0G-8,33.48G-8,250.Oe26.0p22.50el.6qt O,27wlo_3_2,27pO,55,1,20p 2_l.580eloOO_"6*l.000G20_O.OO_l.0G20,OeOOOOploSJl.,l,0+lOB-6t 2eOo131eOo38G-6eO.253G-6eOeOO,leOG20eOoOOOOe2o2811eO. lOG-6e 2pOo131eOe381D-6eO.253G-6tOoT_e2oTG-6wOeO833,1.80,lpOelOB-6_ S--RL S--RLC S--RLCD 151 APPENDIX G COMPUTER PROGRAM FOR CORE-DIODE-TRANSISTOR BINARY COUNTER USING A SIMPLE METIlOD OF SOLUTION Language: ALGOL 60 Pro6ram Description: Computes i L , is' FI' CPl' _1' F2' qb2' _2' id' Vd, _1' and _e2 versus t. Identifiers: (1) Analytical identifiers Identifier Symbol Identi fier Symbol CAPIS I • GJ D D GJM1 DELID _ i d GPRID DELIS _ i • GPRIS DELT A t HA EK E HNk EPS e HQ FB F B IC FJ f(j) ID FJMI f (j- 1) IDM1 FPRID _f/Bi d IDM2 FPRIS 3f/_i • IL F0 F o I LDOT II F0 PP F o I LDOTM 1 F1 F 1 ILM1 F1DOT F 1 ILM2 F1M1 F IS I(-I) F2 F 2 ISM1 F2DOT F ISM2 2 F2M1 F2 (- 1 ) I0 F12 F L12 F23 F 23 LAMBDA 153 g(/) g(j-1) _g/)i d H H n H q I C i d id(-1) i d(-2) i L dit/dt di L dr (- 1 ) i t(-l) i L(-2 i $ i s(-1 s(-2 I o L k Identifier Symbol Identifier Symbol LI l LO l O NB1 N_I NB2 N _2 NS1 N sl NS2 N s2 NU v PHIDOTE1 6E1 PHIDOTE2 _e2 PHIIX)TEPB1 6_1 PHIBOTEPR2 6_2 PHIDOTMA1 6 al PHIDOTMA2 6 2 PHIDOTMAPR1 6' aal PHIDOTMAPR2 6' m.2 ! PHIDOTPR1 61 PHIBOTPB2 _; PHIOOTSTR1 6_ PHIBOTSTR2 6_ PHIDOT1 _1 PHIDOT1M1 61(_i ) PHIDOT2 62 PHIDOT2M1 _2(-1) PHIMA1 ¢_.1 PHIMA2 4.. 2 PHIR PHIS PHI 1 PHI 1M1 PHI 1M2 PHI 2 PHI 2M1 PHI2M2 RD FlOP R1 R2 R3 R4 T TAUS TIN TI 1 TI 2 TR V VD V1 V2 4 r 4, 41 41(-1 41c-2 92 42(-1 42(-2 R d ,o p R 1 R 2 R 3 R 4 t 7- $ T Ttl Ti2 T r V V d V 1 V 2 (2) Auxiliary identifiers Identi fief Description CF CORE COUNT CTS EXIT FMT1 FMT2 FMT3 FMT4 Number of convergence failures. Core name. Index number of At. Index of jth iteration. Label of location where computation Format for the list LIST1. Format for the list LIST2. Format for the list LIST3. Format for output-column heading. terminates. 154 Identifier GUESS LINES LIST1 LIST2 LIST3 LOOP NV SWITCHING Program : Description Label location where predictions are made. Index number of printed line. List of core parameters. List of circuit parameters. List of results (t, i L, is, FI' _I' _1' F2' qb2, _2, ia, Va, _el, _2, Jm.,,' CF). Label of location where iterative computation begins. Negiigibie value of q_E" Label of location where computation starts for each _t. BINARY COUNTER• USIN6 SIMPLE INTEGRATION METHOD TO COMPUTE CURRENTS AND VOLTA6ES VS. TIME. BEGIN COMMENT: DECLARATION OF CORE PARAMETERS.! ALPHA COREI REAL LI• LO• PHIR, PHISe HAp HQe HN, FOPP, FBe FOp ROPe NUe LAMBDAe F12o F23e Vie V2• EPS ! COMMENT: DECLARATION OF CIRCUIT PARAt_[TERS.! REAL RI, 42, R3e R_e Le NSIe NS_e NBle NB2, EKe IO, RD, TR, ICe V, CAPIS• TAUS! COMMENT: DECLARATION OF VARIABLES.! REAL T, OELTe Tile TI2e ILe %LM1, %LM2, %Se %SMIe %SM2, %De %DMle IDM2e VD, ILDOTe ILDOTMI, Fle FIMle F2• F2Mle FIDOTe F2DOTe PHIDOTle PHIOOTMAI, PHIDOTIMI, PHIDOT2e PHIDOTMA2e PHIOOTE1, PHIDOTEPRIe PHIDOTE2e PHIDOTEPR2e PHIOOT2MI, PHIDOTPRIe PHIDOTMAPRIe PHIDOTSTRle PHIDOTPR2e PHIDOTMAPR2e PHIDOTSTR2• PHIIe PHIMAlePHIIMIe PHIIM2• PHI2, PHIMA2e PHI2MIe PHI2M2e FJe FOMIe FPRIS• FPRIO• GO• 6ON1, 6PRISe 6PRIDe De DELID, DELIS! COMMENT: MISCELLANEOUS DECLARATIONS.! REAL TIN, NV ; INTEGER LINESe COUNT• CTSe CF! LABEL SWITCHIN6• 6UESSe LOOPe EXIT/ COMMENT: DECLARATION OF FILEeOUTPUT LISTS AND FORMATS.! FILE OWT _ (2e 15)1 LIST LIST1 (COREe LI x 103e LO x 103e PHIR x 1BBe PNIS x 108e HAe He, HNe LAI_OAe FOPPe NU, ROPe FOe FBe EPSxI09)! LIST LIST2 (RI, R2, R3e R_e L x 1G3• NSI, NS2, NBI, NB2e EKe IO x 106 e hoe IC, V)! LIST LIST3 (T x IG6e ILe ISe Fie PHI1 x 108e PHIOOTIe F2e PHI2 x 108e PHIDOT2e IDe roe PHIOOTEIe PHIDOTE2e CTSe CF)! FORMAT FMTI (Xl• MCORE", A7, .X_e "LI(MM)="e FBo3e X;e nLO(MM)=m, F8.3 • X_e "PHIR(MAXWELLS)=_e FBo3e X_e "PHIS(MAXWELLS)=_e F8.3 / X16e WHA(A MP-TURNS/M)=Ne FB.3e X_e "HQ(AMP-TURNS/M)= N, F8e3, X_, NHN(AMP-TURNS/M)= m• FBo3e X;e NLAMBDA=n• F8.5 / X16, "FOPP(AMP-TURNS)="e FB.3e X;, "NU="; F8.3, X_e NRHOP(OHMS/TURN SOUARED)=Ne FB.3e X;e "FO(AMP-TURNS)=We FB.3 / Xlbe "FB(AMP-TURNS)=We FB.3e X;, "EPS(MILLIMIC ROHENRYS/TURN SQUAREO)="eFB.5/)! 001 002 003 OOq 005 006 OOT 008 009 010 011 012 013 01_ 015 016 017 " 018 019 020 021 0=)2 023 02_ 025 026 027 028 029 030 031 032 O33 03_ 035 036 037 038 155 FORMAT FMT2 (Xlp "CIRCUIT _, XB* "RI(OHMS):Ne FBo3e X_e "R2(OHMS):"e F8o3* X_e "R3(OHMS)=ne F8.3* X;, "R_(OHMS)=", F6,3 / X16, NL(MILLIHENRY )=Nw FSoSe X;e mNSI="e I_* X_e MNS2="e I_* X_e mNBl="e I_e X_, "NEl2=me l_e X_, NEK(VOLT)=", F8o5 / X16, NIO(MICROAMP)=Wo F8o5* X_, mRD(OHM)=ne FS,3e X_p wlC(AMP)=We FB*3* X_p NV(VOLTS)=mp F8.3 /)l FORMAT FMT3 (XI* FS.3e 3 FB.3e F9.3* 2 F8,3* Fg.3e FS.3e Fg._* F9.3, 2 Fg,_, 2 IS)l FORMAT FMT_ (X3* WT", X7, "ILN* X6* HISNe X6* NFIm, X5* mPHIl", X3, wPH IOOTXN, X_, RF2N, IS, mPHI2m, X3, NPHIOOT2Np X_* nION, IT* NVO_, X_ NPH IOTE1 ", X2, wPHIDTE2 ", X3, "CT$"* X2* _CF N /)! COMMENT: OECLARATION OF PROCEOURESt! 039 0_0 Oa;1 0_2 0_3 0_ 0_5 0_6 0_? 0_8 0_9 A / /m PROCEDURE _¢(F,At,NV,_), APPENDIX A PROCEDURE _(F,_,_',&*), APPENDIX D-2 COMMENT: INITIALIZE CORE PARAMETERS.! CORE • _1005C1_! He * _2.7; LAMBDA _ 0,6_01 LI • 5.59G-31 LO • 7.98G-3; PH|R • bo25G-8; PHIS * 7.00G-8; HA • 290.0 ; HN • 38.0! FOPP • 0.35; F6 • 3.001 FO • 0.605; ROP • 0.9¢8; NU • 1.207! F12 • HG x LI; F23 • HG x LO! Vl • (PHIS - PH;R) / ((LO - LI) x HA); V2 • (PHI$ + PHIR) x HG / ((LO - LI) x NN); EPS • VlXLN(LO/LI) ; COMMENT: IN|TIALIZE CIRCUIT PARAMETERS.! TR _ 0,1308-61 RI • 107.361 R2 • 199.551 R3 • 0.3_0! R_ * 0.53! L • 0,2028"3; NSI • 11! NS2 * 121 NBI • 161 NB2 • 20.0; EK • 0.0578! I0 • 0,0615G-6! RO * 0.1001 NV • 0,0001-1 V • 27.01 COMMENT: ESTABLISH INITIAL CONOITIONS,! T • O; IL * IS_2 * ISMI * IS * IOM2 • 10M2 * ID * OI FIMI • FI * F2MI • F2 • O; PHIOOTI * PH;OOTMAI • PHIDOTEI * PHXOOT2 • PHIOOTMA2 * PHIOOT(2 • O; VO • O; PHIl • PHI2 • - PHIRI 156, 050 051 O52 O53 05_ O55 O56 05_ O58 O59 060 061 062 063 06_ 065 066 067 068 069 07o 071 072 073 07_ OT5 076 O7"/ O78 079 080 081 O82 O83 08_ Oe5 086 087 088 O89 09O 091 092 _CAPIS * V / RI x R2 / (R2 • 0.6 x ROP x (NS1 • 2 • NS2 • 2))1 TAUS * 2.0 x PHIR x (NB1 s 2 + N82 • 2) / (NB2 x 0.6 x ROP x (CAPIS x (NSI x NB2 ÷ NS2 x I_Bl) - FOPP x (NB1 • NB2))); IC * V / Rll CF • O; WRITE |OWT [PAGE]) I MRZTE (OMTe FMTIB LIST1); MRITE (OMTt FMT2e LIST2)I WRITE (OWT, FMT_); WRITE (OWT, FMT3w LIST3)I LINES • 12l COUNT•O; COMMENT: COMPUTE VARIABLES DURING SMITCHIN6 TIME. ; SMITCHING: BE6IN ILM2 • ILMll ILNI * ILl PHIII*_ * PHIIM1; PHI1#1 • PHIl; PHI2;_. • PHI2MI; PHI2N1 • PHI2; PHIDOTIMI • PHIDOTI; PHIDOT2M1 • PHIDOT2; ISM2 * ISMI; ISN1 • IS; IDN2 * IDNI; IDN1 • ID; ILDOTNI • ILDOT; FIN1 • F1; F2MI • F2; DELT * IF T ( TR THEN TAUS/IO00 ELSE TAUS/500 ; T * T • OELT; COUNT • COUNT • 1; CTS • O; GUESS: IF T £ TR THEN IL • IC x (IF T < 0.0 THEN 0 ELSE IF T < 0.02Q-6 THEN 1.02B19 x T • 2.66667 ELSE IF T ( 0.06i)-6 THEN qoOG6 x • - 0.05 ELSE IF • ( 0.13Q-6 THEN 0.252 - 0.001 x (0.009_70 • ( TIN • 13 - 188 x T) x |- 0.316910 ÷ TIN x |1.729261 + TIN x (- 0.5759_7 + TIN x 0.073769)))) ELSE 0.252) / 0.252 ELSE IL • ]LM2 • 2.0 x OELT x ILDOTMI; PHI1 • PHIIM2 • 2.0 x DELT x PHIDOT1MI; PHI2 • PHI2M2 • 2.0 x DELT x PHIDOT2MI; IS • 2.0 x ISM1 - ISM2; ID • 2.0 x IDM1 - IDM2; LOOP: FJMI • FJ; GJMI • 6d; CTS • CTS • I; FI • NSI x IS • NBI x IO; F2 • NS2 x IS - NB2 x ID; FIDOT • (FI - FINI) / DI[LT; F2OOT • {F2 - F2MI) / D4[LT; PHIDOTMAI • PHIOOT (FI, PHI1, PHIDOTMAPRle PHIOOTSTRI); PHIDOTEI * PHIDOTI[ (FIDOTpDELT,NVePHIOOTEPR1) ; PHIDOT1 • PHIDO•MA1 + PHIDOTE1; PNIDOTPR1 • PHIOOTMAPR1 + PHIDOTEPRI; PHIOOTMA2 • PHIDOT (F2e PHI2÷ PHIDOTNAPR2, PHIDOTSTR2); PHIDOTE2 * PHIDOTE (F2DOT,OELTpNVePHIDOTEPR2) ; PHIDOT2 • PHIDOTNA2 + PHIOOTE2; PHIOOTPR2 • PHIDOTMAPR2 + PNIDOTEPR2; IF • • TR THEN BEGIN ILDOT • (- IL x R1 - NSI x PHIDOT1 - NS2 x PHIDOT2 - IS x R3 ) / L; IL • ILMI • 0.5 x OELT x (ILDOTMI + ILDOT); END; 093 09_ 095 096 097 098 099 100 101 102 103 10_ 105 106 107 108 109 110 111 112 113 115 116 117 118 119 120 121 122 123 124 125 126 12"/ 128 129 130 131 132 133 135 136 137 138 139 1_0 lql lq2 1q5 1_6 lq7 l_B 1;9 150 151 152 153 15_ 155 156 157 PHIl • PHIIN1 + 0.5 x DELT x (PHIDOTIMI ÷ PHIDOTI); PHIE • PHI2N1 + 0.5 x DELT x 4PHIDOT2MI ÷ PHIDOTE); VD _ IF NB2 x PHIDOT2 > NBI x PHIDOTI THEN ID x RD + EK x LN (ID/IO + 1,0 ) ELSE NB2 x PHIDOT2 - NBI x PHIDOTI; Fd * NB2 x PHIDOT2 - NBI x PHIDOT1 - VD - ID x R_; Gd _ NS2 x PHIDOT2 + NSI x PHIDOT1 ÷ IS x R3 - R2 x (IL - IS)I FPRID * - (NB2 = 2 x PHIDOTPR2 + NBI = 2 x PHIDOTPRI + RD + R_ ÷ EK / (ID + IO)); GPRIS * NS2 * 2 x PHIDOTPR2 ÷ NSI • 2 x PHIDOTPR1 + R2 + R3; FPRIS * IF NB2 x PHIDOT2 > NBI x PHIDOT1 THEN NS2 x NB2 x PHIDOTPR2 - NSI x NBI x PHIDOTPRI ELSE 0 ; 6PRID * - FPRIS; D _ FPRID x GPRIS - FPRIS x GPRID; IF D • 0 THEN BEGIN DELID * (- Fd x GPRIS + Gd x FPRIS) / D; DELIS _ (FJ x 6PRID - Gd x FPRID) / D; END ELSE DELID - DELIS * O; ID - ID + DELID; IS * IS + DELIS; IF SIGN (FJ) ¢ SIGN (FJMI) THEN IO _ ID - 0,5 x DELIO; IF ID S 0 THEN ID * 0 ; IF SIGN (GJ) ¢ SIGN (6JMI) THEN IS _ IS - 0.5 x DELIS; IF (ABS (DELID) > 0.0001 x ABS (ID) OR ABS (DELIS)) 0,0001 x ABS ( IS)) AND CTS < 20 THEN GO TO LOOP; IF CTS = 20 THEN CF * CF ÷ 1; CONMENT: PRINT OUTPUTo; IF COUNT HOD 20 = 0 THEN BEGIN IF LINES NOD 50 = 0 THEN BEGIN WRITE (OWT [PAGE]); WRITE (O_T, FMT_); ENDt WRITE (OWT, FNT3, LIST3); LINES * LINES ÷ I; CF • O; END; IF PHIOOTHA2 = 0 ANO PHIDOTMAI = 0 AND PHI1 ) -OogxPHIR THEN GO TO EXIT ELSE GO TO SWITCHING; END; EXIT: END. • 157 158 159 160 161 162 163 16_ 165 166 167 168 169 170 171 172 173 17_ 175 176 177 178 179 180 181 182 183 1B_ 185 186 187 188 1B9 190 191 192 193 19q 195 158 APPENDIX H COMPUTER PROGRAM FOR CORE-DIODE-TRANSISTOR BINARY COUNTER USING THE RUNGE-KUTFA AND ADAMS METHODS OF SOLUTION Language: ALGOL 60 Program Description: Computes i L, is" FI" c/)1" _1' F2, 02, _2, ia, Vd, _1, and _e2versus t. Identifiers: (1) Analytical identifiers Iden ti fi er Symbo 1 Iden t i fie r Symbo 1 CAlaIS I s ID i d DELT h t I DM1 i d(-1) DELTMIN At .is IBM2 i dr-2 ) DELX Ax IL i L DELXMI Ax (_ ]) IS i _, EK E _ ISM1 i s(-l) EPS e I SM2 i s(-2) FB F e I0 I o FDI F ] L L FD2 F2 LAMBDA F0 F o LI l . t ii F0 PP F o LO l o F1 F NB11 N e ] FIMI FI (-i) NB2 Ne2 F2 F 2 NS1 N • ] F2M1 F2 (- 1 ) NS2 N2 F12 F m NU v F2 3 F 23 PHIDOTE1 _E ] HA H _ PHIDOTE2 _ 2 # HN H n PHIDOTEPRI _ I HQ H q PHI DOTEPR2 _b_ IC I _ PHIDOTMAI _b mal 159 I den t i f i e r Symbo 1 I de n t i fie r Sym bo 1 PHIDOTMA2 _. a 2 BOP pp PHIDOTMAPR1 _' R1 R 1 real PHIDOTMAPR2 _' R2 R 2 ma2 PHIDOTPR1 _'1 R3 R 3 PHIDOTPB2 _ R4 B 4 PHIDOTSTR1 _; T t PHIDOTSTR2 _ TAUS _- PHIDOT 1 _l TIN T PHIDOT2 _2 TR T_ PHIMA1 q5 _l TI1 Ti 1 PHIMA2 q5 _ _ TI2 Ti 2 PHIR 9r V V PHIS Cs VD Va PHI1 q51 V1 V 1 PH 12 q52 V2 V 2 RD R d XM1 x ( _ 11 (2) Identifier CORE COUNT EL EU EXIT FCOUNT FMT1 FMT2 FMT3 FMT4 H LINES LIST1 LIST2 Auxiliary identifiers Description Core name. Index number of At. Lower limit of error in ADAMS PROCEDURE Upper limit of error in ADAMS PROCEDURE Label of location where computation terminates. Index number of jth iteration in F PROCEDURE. Format for the list LIST1. Format for the list LIST2. Format for the list LIST3. Format for output-column heading. Array for h(=Ax =A t) to be used in RK and ADAMS PROCEDUREs. Index for At to be used in RK and ADAMS PROCEDUREs. Index number of printed line. List of core parameters. List of circuit parameters. 160 Identifier Description LIST3 NV SWITCHING X Y YPR List of results (t, i L r_, FI, qbl, _1, F2' 92, _2, i_, Vj, _, _2, J._., At). Negligible value of _. Label of location where co.,putation start_ for each At. Array for storing x = t. Array for storing a dependent variable. Array for storing the time derivitive of a dependent variable. Program: BINARY COUNTER, USING RUNGE-KUTTA AND ADAMS METHODS TO COMPUTE CURRENTS AND VOLTAGES VS. TIME. BEGIN COMMENT: DECLARATION OF CORE PARAMETERSo! ALPHA CORE! REAL LI, LO, PHIR, PHIS, HAp HGt HNe FOPPe FB, FOe ROPt NUt LAMBDA, F12w F23, VI, V2, EPS! COMMENT: DECLARATION OF CIRCUIT PARAMETERS.; REAL RI, R2, R3, R_, L, NSI, NS2, NBlt NB2, EK, IOt RD, TR, IC, V, CAPIS, TAUS; COMMENT: DECLARATION OF VARIABLES.! REAL T, DELT, XN1, OELXN1, DELX, TZI, TI2, IL, ISM2, ISMI* IS, IDN2, IDH1, ID, VD, FIM1, F1, F2MI, F2, FDI, FD2, PHIDOT1, PHIDOTHAlt PHIDOTEIt PHIDOT2, PHIDOTMA2, PHIDOTE2, PHIDOTPR1, PHIDOTNAPRI, PHIDOTSTR1, PHIDOTEPRlr PHIDOTPR2, PH|DOTNAPR2, PHIDOTSTR2, PHIDOTEPR2, PHI1, PHIMA1, PHI2, PHIHA2! COMMENT: NISCELLANEOUS DECLARATIONSo! REAL EU, EL, DELTMINt TINt NV ! INTEGER I, LINES, COUNT, FCOUNT! REAL ARRAY H, X [0 : 5]' Yt YPR [0 : 5, 0 : 311 LABEL SWITCHING, EXIT! COMMENT: DECLARATION OF FILE,OUTPUT LISTS AND FORMATS.! FILE OWT _ (2, 15)! LIST LIST1 (CORE, LI x 1_3t LO x 1G3, PHIR x lg8t PHIS x 1Q8, HA, HG, HN, LAMBDA, FOPP, NU, ROP, FO, FB, EPSxIQg)! LIST LIST2 (R1, R2t R3, R_, L x 1Q3, NS1, NS2w NB1, NB2, EK, IO x IQ6 , RD, IC, V)! LIST LIST3 (T x 1_6, IL, IS, FI, PHI1 x 1G8, PHIDOT1, F2, PHI2 x IQ8, PHIDOT2, ID, VO, PHIDOTE1, PHIOOTE2, FCOUNTtOELTxlQg) ! FORMAT FMT1 (Xl, "CORE"t A7, X_, "LI(NM)= N, F8.3, X;, "LO(MN)=", F8.3 , X_, "PHIR(MAXWELLS)=', FB°3, X4p MPHISIMAXWELLS)=', FB°3 / X16, "HA(A NP-TURNS/N)=', F8.3, X_, "He(AMP-TURNS/N)=', F8.3, X_, "HN(ANP-TURNS/M)= ", F8.3, X_, "LAMBDA=", F8.5 / X16, "FOPP(ANP-TURNS)=", FBe3, X_, "NU=", F8.3, X_t "RHOP(OHMS/TURN SGUARED)=', F8.3, Xq, "FO(ANP-TURNS)='w F8.3 / X16, "FB(ANP-TURNS)=", F8°3, X_, "EPS(NZLLIMIC ROHENRYS/TURN SQUARED)=',F8.5/)! FORMAT FMT2 (Xl, "CIRCUIT"p XG, "RI(OHNS)=', F8.3, X_, "R2(OHMS)=', F8°3, X;, "R3(OHMS)=", F8.3, X_, "R_(OHNS)=', F8.3 / XlBp "L(NILLIHENRY }=', F8.5, X_, "NSI=", I_, X_, "NS2= m, I_, Xq, "NBI=', I_, X_, "NB2=", I_, X;, "EKIVOLT)=", F8-5 / X16, "IO(MICROAMP)=", F8.5, X_, "RD(OHM)=", F8,3, X_, "IC(AMP)='p FG.3P X_p "V(VOLTS)="p F8.3 /)1 FORMAT FMT3 (Xl, F5.3, 3 F8o3, F9°3, 2 F8.3, F9,3, FB.3w Fg°_, F9°3, 2 Fgo_, I3, FT._) ! 001 002 003 001_ 005 006 007 OOB 009 010 011 012 013 01_ 015 016 017 018 019 020 021 022 023 02_ 025 O26 027 028 029 030 031 032 O33 031_ 035 036 03? 038 039 0_0 0_1 0_2 01_3 161 FORMAT FMT; (X3' "T"p XTp "IL"p XG, "IS", XGp "FINp XSp "PHII_e X3p NPH XDOTX", X_t "F2np XSt "PHI2", X3t "PHXDOT2"p X_, MID", X7, _VO", X_, mPH IDTEI", X2_ "PHID•E2"," C•S DEL•"/)I COMMENT: DECLARATION OF PROCEDURES.! 0;5 0_6 " 0;? PROCEDURE APPENDIXA PROCEDURE APPENDIXD-2 PROCEDURE F (X, Y, OX)! VALUE Xi REAL X! ARRAY Y [O]t DX [03! COMMENT: THIS PROCEDURE USES THE FOLLOWING GLOBAL IDENTIFIERS: TRt RX , R2, Lt NS1, NS2t NB1, NB2t EKt IOt ROt ICt CAPISt FCOUNT, Tt XMX, DELXMI, DELX, NV, ISM2, ISMlp ISw IDM2t IOMI, ID, VO_ FINIp F£, F2Mlw F2p FDlp FD2, PHIDOTlt PHIDOTMAI, PHIDOTE1, PHIDOT2p PHIDOTMA2, PHIDOTE2t PHIDOTPRIp PHIDOTNAPRIt PHIDOTSTRIt PHIDO•EPRI, PHIOOTPR2p PHIDOTNAPR2t PHIDOTSTR2t PHIDOTEPR2! BEGIN REAL Up V, UISp UIOt VISt VIO, NUMSp NUNOt DEEISp DEEIDp OENOMI LABEL ITERAT! IF X _ XNI •HEN BEGIN DELXM1 * DELXI DELX * X - XMI! IF DELXMI = 0 THEN DELXNI * DELX! ISM2 • ISMI! ISMI • IS! XDM2 • IDMI! IOMI • IO! FIMI • FI# F2MI • F2! IS • ISMI + (ISMI - ISM2) x DELX / DELXMI! XD • IDMI + (IDM1 - IDM2) x DELX / DELXMII END! FCOUNT * 08 ITERAT : FCOUN• * FCOUN• + 1! FI * NSI x IS + NBI x ID! F2 • N$2 x IS - NB2 x ID! FDI * (FI - FIN1) / DELX; FD2 * (F2 - F2M1) / DELX; PHIDOTMAI * PHIDOT (Flt Y [lit PHIDOTMAPR1, PHIDOTSTRI); PHIDOTE1 * PHIDOTE (FD1, DELXt NVp PHIDOTEPRI) ! DX [1] * PHIDOTI * PHIDOTMAI + PHIDOTEII PHIDOTPRI * PHIDOTMAPR1 + PHIDOTEPRII PHIDOTMA2 • PHIDOT (F2s Y [23t PHIDOTMAPR2_ PHIDO•STR2)! PHIDOTE2 * PHIDOTE (FD2t DELXt NVp PHIDOTEPR2) I DX [2] * PHIDOT2 • PHIDOTMA2 + PHIDO•E2! PHIDO•PR2 • PHIDOTMAPR2 + PHIDOTEPR2! IF T $ TR THEN DX [33 •IC x (IF • < 0.00 THEN 0 ELSE IF T < 0.02_-6 THEN 2.72Q19 x T * 1.66667 ELSE IF T < 0.06G-6 THEN _°0_6 ELSE 1.0_5 x (- 0.316910 + (TIN * 13.0 - 1.0B8 x T) x (3.W58522 + TIN x (- 1.7278_1 + TIN x 0.295076)))) / 0°252 ELSE DX [3] * - ( NS2 x PHIDOT2 + NSI x PHIDOTI + Y [3] x RI) / L! VD * IF NB2 x PHIDOT2 > NBI x PHIDOTI THEN ID x RD + EK x LN (ID/IO + 1°0 ) ELSE NB2 x PHIOOT2 - NBI x PHIDOTIIF U * NSI x PHIDOTI + NS2 x PHIDOT2 + IS x R3 - (Y [3] - IS) x R21 F V * NBI x PHIDOTI - NB2 x PHIDOT2 + IO x RW ÷ VDI F UIS * NSI * 2 x PHIDOTPRI + NS2 * 2 x PHIDOTPR2 + R2 + R31 F VID * NBI * 2 x PHIDOTPR1 + NB2 * 2 x PHIDOTPR2 + RD + EK / (ID + F IO) + R_! F F 001 F 002 F 003 F 00; F 005 F 006 F 007 F 008 F 009 F 010 F 011 F 012 F 013 F 01_ F 015 F 016 F 017 F 018 F 019 F 020 F 021 F 022 F 023 F 02_ F 025 F 026 F 027 F O28 F 029 F 030 F 031 F 032 F 033 F 03_ F 035 F 036 F 037 F O38 F 03g F 0_0 F 0_1 F 0_2 F 0_3 F 0_; F 0_5 0;6 0;8 0;9 O5O 051 162 UID • VIS • IF NB2 x PHIDOT2 > NBI x PHIDOT1 THEN F 052 NS1 x NB1 x PHIDOTPR1 - NS2 x MB2 x PHIDOTPR2 ELSE 0 ! F 053 DENOM • UIS x rid - UID x VZS! F 05_ IF OENOM : 0 THEN DENOM • 1B10; F 055 NUNS • V x UID - U x VID! F 056 NUND • U x VlS - V x UIS! F 057 IF SIGN (NUNS / DENOM) • S%6N (DEEZS) THEN DEEZS • 0.5 x (NUNS / F 058 DENOM) ELSE DEEZS • NUNS / DENOM; F 059 IF 516N (NUMO / DENOM) $ SIGN (DEEZD) THEN DEEZD • 0.5 x (NUMD / F 060 DENOM) ELSE DEEZD • NUMD / DENOM! F 061 ZS • 15 + DEEZSI F 062 ZD • ZD + DEEZDI F 063 %F 1D S 0 THEN ZD * 0 ! F 06_ ZF FCOUNT < 10 AND (ABS (DEEZS) > 0.0001 x ABS (15) OR ABS (I)[EZD) F 065 ) 0.0001 x ABS (ZD)) THEN GO TO ZTERAT! F 066 XMl • X! F 067 END OF F! F 068 PROCEDURE RKSTARTS (Kw NFp Xle He Yp YPRe F)t RK 001 VALU( Ke NFe Ht RK 002 REAL X2e HI RK 003 ZNTEGER K, NF! RK 00_ ARRAY YP YPR lOP 03! RK 005 PROCEDURE F! RK 006 BEGZN ZNTEEER Ze dl RK 007 ARRAY DX, TEMPY, K1, K2, K3, K_p KS, K6 [0 : K]! RK 008 FOR Z • 0 STEP 1UNTZL (NF - 1) DO RK 009 BEGZN RK 010 BEGZN REAL X! RK 011 X • H x I • Xll RK 012 F (Xe Y [1, *], OX)! RK 013 FOR d • I STEP I UNTZL K DO RK 01_ BEGZN KI [d] • OX [J] x Ht RK 015 TEMPY [J] • K1 [J] / 3.0 + Y [Ze J] RK 016 END! RK 017 F (H / 3.0 • Xe TEMPY, OX)! RK 018 FOR d • I STEP I UNTIL K DO RK 019 BEGZN K2 [d] • OX [J] x HI RK 020 TEMPY [J] • (K2 [J] x 6.0 • K1 [J] x _.0) / 25.0 • Y [;, RK 021 J] RK 022 END! RK 023 F ((H x 2.0) / 5.0 • X, TEMPY, DX)! RK 02_ FOR d • I STEP I UNTIL K DO RK 025 BEGZN K3 [J] • DX £J] x HI RK 026 TEMPY [U] • (K3 [J] x 15,0 - K2 [d] x 12.0 • K1 [J]) / RK 027 _.0 • Y [I, J] RK 028 END! RK 02g F (H • X, TEMPYt DX)! RK 030 FOR d • 1 STEP I UNTZL K DO RK 031 BEGIN K; [J] • DX [J] x HI RK 032 TEMPY [J] • (K_ [J] x 8.0 - K3 [J] x 50.0 + K2 [J] x 90.0 RK 033 • K1 [d] x 6.0) / 81.0 • Y [Z, J3 RK 03_ END! RK 035 F ((H x 2.0) / 3.0 • X, TEMPYe OX)l RK 036 FOR d • I STEP I UNTZL K O0 RK 037 BEGZN K5 [U] • OX [J] x HI RK 038 TEMPY [J3 • (K_ rJ3 x 8.0 • K3 [J3 x 10.0 • K2 [J3 x 36.0 RK • K1 [U] x 6.0) / 75.0 • Y [Iw J] RK END ! RK F ((H x ;.0) / 5.0 • Xe TEMPY, OX); RK FOR U • I STEP I UNTZL K O0 RK BEGIN K6 [J] • OX [d] x HI RK Y [I • 1, J3 • (KI [J] x 23.0 • K3 [d] x 125.0 - K5 [J] x RK 81.0 • K6 [U] x 125.0) / 192.0 • Y [%, J] RK 039 Oq_0 0_1 Oq2 Oq3 Oq.q. 0_5 Ot;6 163 END ! END XBLOCK; X [I + 1] • X [I] + HI F (X [I + lip Y [I + le *]e YPR [I + lp .])! T t. X [I + 111 PHI1 * Y [I + le 1]! PHI2 * Y IX + le 2]! IL " Y [Z + 1_ 3]; PHZDOTI _" YPR [I + 1, 1]; PHZDOT2 * YPR [Z + le 2]; FI * NSI x IS + NBI x ZD! F2 * NS2 x IS - NB2 x ID! VO * ZD x RD + EK x LN (ZD / ZO + 1)1 WRITE (OWTP FMT3, LIST3)/ LINES * LINES + 1! COUNT _" COUNT + I; END END PROCEDURE RKSTARTSJ RK RK RK RK RK RK RK RK RK RK RK RK RK RK RK RK RK RK 0;7 0;8 0_9 050 051 052 053 05_ 055 056 057 058 059 060 061 062 O63 06; PROCEDURE ADAMS (X, Y, YPRIME, Np EUe EL, EPS, He HMIN)I VALUE N! ARRAY Y lop O]p YPRIME [0, 0], X [O]e H GO]/ REAL EU, ELi EPS, HMIN; INTEGER N! BEGIN INTEGER [p d, Q! ALPHA B! REAL TEMP, KPo KCe KKP YCe El ADAMSO01 ADAMSO02 ADAMSO03 ADAMSO0_ ADAMSO05 ADAMSO06 ADAMSOOT ADAMSOO8 REAL ARRAY U [0 : 3], P [0 : ;], C [0 : ;]e K [0 : 5], YP [0 : 20] ADAMSO09 p FP [0 : 20]; ADAMS010 LABEL AWAY I ADAMS011 FOR I * 2 STEP I UNTIL _ DO H[I] q" X [I] - X [I - 1]; ADAMS012 H [5] • H [_]! ADAMSOI3 IF B ;t 0 THEN H [5] * 2 x H [5]; ADAMS01_ B * O! ADAMS015 FOR I • 3 STEP I UNTIL 5 DO K [I]e. H [I - 1] / H [I]l ADAMS016 AWAY: FOR I * I STEP I UNTIL 3 DO ADAMS017 BEGIN U Eli * O! ADAMSOIB FOR d 4. 5 - I STEP I UNTIL ; DO U EI] • K [,J ÷ 1] x (1 + U [I]) ADAMS019 END ! ADAMS020 P [1] • - (3 + ; x (U [1] + U [2]) + 6 x U [1] x U [2]) / (12 x U [ ADAMS021 3] x (U [3] - U [1]) x (U [3] - U [2]))1 DDAMS022 P [2] • - (3 + ; x (U [1] + U [3]) + 6 x U [1] x U [3]) / (12 x U [ ADAMS023 2] x (U [2] - U [1]) x (U [2] - U [3]))! ADAMSO2q. P [3] • - (3 + ; x (U [2] ÷ U [3]) + 6 x U [2] x U [3]) / (12 x U [ ADAMS025 1] x (U [1] - U [2]) x (U [1] - U [3]))! ADAMS026 P [;] • I - (P [1] + P [2] + P [3])1 AOAMS027 C [1] * (1 + 2 x U [1]) / (12 x U [2] x (U [2] + 1) x (U [2] - U [ ADAMSO2B 1 ]) ) ! ADAMS029 C [23 • (1 + 2 x U [2]) / (12 x U [1] x (U [1] + 1) x (U [1] - U [ ADAMS030 2 ]) ) ! ADAMS031 C [;] * (3 + ; x (U [1] + U [2]) + 6 x U [1] x U [2]) / (12 x (U [ ADAMS032 1] + 1) x (U [2] + 1))I ADAMS033 C [33 • 1 - (C [1] + C [2] + C [;]); ADAMS03_ KP q" 0.2 - (P [3] x (U [1] * ;) + P [2] x (U [2] * _) + P [I] x (U ADAMS035 [33 * _))1 KC * 0.2 - (C [_] + C [2] x U [1] * ; + C [1] x (U [2] * _))! KK e. KC / (KP -, KC)! FOR d • I STEP I UNTIL N DO BEGIN YP [,J] _" Y [;e J] + H [5] x (p [_] x YPRIME [_p J] + P [3] x YPRZME [3e ,.J] + P [2] x YPRIME [2e ,J] + P [1] x YPRIME [ls d])l Y [5, d] t. yp [,j] END ! ADAMS036 ADAMS037 ADAMS038 ADAMS039 ADAMSO_O ADAMS041 ADAMSOn2 ADAMSOn3 164 • * O! ADAMSOqq FOR d *. 1 STEP 1 UNTIL N DO ADAMS045 BEGIN I *" 5! ADAMSOq6 X [5] _" X [_] + H [5]! ADAMSOq? IF d = 1 THEN F (X [5], Y [5e *], FP)! ADANS048 YC q- Y [_, ,J] ÷ H [5] x (C [_] x FP [d] 4. C [3] x YPRIME [_, J] ADAMS049 + C [2] x YPRIME [3, d] + C [1] x YPRIME [2e d])! YPRIME [5, ,J]* FP [d]! E +" KK x (YC - YP [J])l Y [5, J] * YC + El TEMP *. ABS |YC)! IF TENP < EPS THEN TEMP * EPS! IF ABS (E) > EU x TEMP AND H [5] I HNIN x 2,0 THEN BEGIN H [5] * 0.5 x H [5]! K [5] * 2 x K [5]! GO TO AMAY ENDi TEMP *. AB$ (¥C)1 IF TENP < EPS THEN TEMP * EPSI IF ABS (E) < EL x TEMP THEN G * G ÷ 2 END ! IF G = N THEN B * 11 END ADAMS! ADAMS050 ADAMS051 ADAMS052 ADAMS053 ADAMSOSq, ADAMS055 ADAMS056 ADAMS05? ADANSOSB ADAMS059 ADAMS060 ADAMS061 ADAMS062 ADAMS063 ADANS06_ ADAMS065 ADAMSO66 STREAM PROCEDURE TRANSFER (N, A, B)I VALUE N! BEGIN SI * A! DI • BI DS • N WDS! END TRANSFER! TRNSFR01 TRNSFRO2 TRNSFR03 TRNSFR04 TRNSFR05 TRNSFR06 COMMENT: INITIALIZE CORE PARAMETERS*! CORE * mlOOSClW! HG * _2.7; LAMBDA * O.b_O! LI • 5.59Q-31 LO • 7.988-3! PHIR • 6.25Q-81 PHIS • 7.00_-8! HA * 290*0 ! HN • 38,0! FOPP * 0.35! FB • 3,00! FO • O.B05! ROP • 0.9;Bl NU • 1.207l F12 * HG x LIi F23 * HQ x LOt Vl * (PHIS - PHIR) / ((LO - LI) x HA)! V2 * (PHIS + PHIR) x HQ / ((LO - LI) x HN)! EPS * VIXLN(LO/LI) ! COMMENT: INITIALIZE CIRCUIT PARAMETERS.! TR * 0.130g-6; R1 * 107.36! R2 * 199.551 R3 * 0.3401 R_ • 0.53! L * 0.202Q-3l NS1 * 11! NS2 * 12! NB1 * 1hi N82 * 20.0t EK * 0.05781 I0 • 0.06_58-61 048 049 O50 051 052 O53 054 O55 056 057 058 059 " 060 061 062 053 064 065 066 067 068 069 070 071 072 073 07q 075 076 077 078 079 080 165 RD * 0.100; V e. 27.,0 ; NV " 0.0001 I COMMENT: ESTABLISH INITIAL CONDITIONS.I T * O; IL _" ISM2 * ISMI * IS * IDM2 _' IOM1 * ID 4. OI FIMI *" F1 * F2M1 *" P2 * OI PHIDOT1 " PHIDOTMA1 " PHIDOTE1 * PHIDOT2 '_ PHIDOTMA2 _" PHIDOTE2 _" O; VO .,. O; PHI1 _" PHI2 * - PHIR; CAPIS ,. V / RI x R2 / (R2 + 0.6 x ROP x (NS1 * 2 + NS2 * 2))1 TAUS ,, 2.0 x PHIR x (NBI * 2 + N82 * 2) / (NB2 x 0.6 x ROP x (CAPZS x (NSI x NB2 + NS2 x NBI) - FOPP x (NB1 + NB2))); I[ ,_ V / RI; WRITE (OWT [PAGE])t WRITE (OWT, FMTI, LIST1); WRITE (OWT, FMT2, LIST2); WRITE (OWTe FMT_); WRITE (OWT, FMT3, LIST3); LINES " 121 COUNT * O; X [0] ," O; DELT ,. TAUS / 5001 DELTMIN ,, DELT / 10; EU * 0.0011 EL _, 0.00011 Y top 13 4" Y [0, 23 _" - PHIR; Y [0, 3] * O; XM1 _" - DELT; RKSTARTS (3, q,, O.Oe DELT, Y, YPR, F); SWITCHING: COUNT 4. COUNT + 11 IF T $ TR THEN ADAMS (X, Y, YPRe 3, EU, EL, 0.1 x PHIRe He DELTMIN) ELSE ADAMS (X, Yf YPR, 3, 5,0 x EU, 10.0 x EL, 0.1 x PHIR, Hp DELTMIN ); DELT ,. H [511 T '," X t5]; PHI1 * Y [5, I]; PHI2 " Y [5, 2]; IL • Y [5, 331 PHIDOT1 _" YPR [5, 1]; PHIDOT2 ,- YPR [5, 2]; COMMENT: PRINT OUTPUT. ; IF COUNT MOO 20 = 0 THEN BEGIN IF LINES NOD 50 = 0 THEN BEGIN WRITE (OWT [PAGE]); WRITE (OWT, FMTq.); END; WRITE (OWT, FMT5, LIST5); LINES ,. LINES + 11 END; TRANSFER (_, X [2], X [I]); FOR I " 2 STEP I UNTIL 5 DO BEGIN TRANSFER (3, Y [I, 1], Y [I- 1, 1]); TRANSFER (3, YPR tie 1], YPR [I - 1, 1]); END; IF PHIDOTNA2 = 0 AND PHIDOTMAI = 0 AND PHI1 > -0.gxPHIR THEN GO TO EXIT ELSE GO TO SWITCHING; EXIT_ END. 081 082 083 08It 085 086 087 088 089 090 091 092 093 09_ 095 096 097 098 099 100 101 102 103 104 105 106 107 108 109 110 111 112 113 11_ 115 116 11"1 118 119 120 121 122 123 12_ 125 126 127 128 129 150 151 132 135 13e_ 135 136 137 138 166 APPENDIX I COMPUTER PROGRAM FOR FLUX DIVISION IN A LOADED SATURABLE CORE Language: ALGOL 60 Program Description: Computes three types of output: (1) _., _b. F. _3, ¢P3, F3, _4, _b4, F4, and Ni versus t. (2) D vs. NI and N_/R L for given 14/l 3. (3) D vs. l 4/l 3 and N_/R L for given NI. Identifiers: (I) Identifier Analytical identifiers AM A3 A4 BR BS CIM CI3 CI4 D D:F4 DELF4 DELPHIM DELPH 13 DELPHI 4 DELT DELTAPHI3 DELTAPHI 4 FBM FB3 Symbo 1 A A3 A4 Br B $ Cs3 C_4 D _F 4(j) F 4 - F4{_1 ) At _3 A¢ 4 FB. F_3 Identifier FB4 FIM FI3 FI4 FJ FJPR FM FOIM F013 50 I4 F0M F0 3 F04 FOPP M FOPP 3 FOPP4 F12M F123 F124 Symbol FB4 F. tim Fi3 Fi¢ f f' F Foi3 Foi¢ Fo,. Fo3 Fo4 F_ It Fo 4 /;12 F123 F124 167 Identifier F23M F233 F234 F3 F4 F4M1 H HA HB HN HQ H0 HOPP H0I KAPPA KAPPAI LAMBDA I M LAMBDAI 3 LAMBDAI 4 LAMBDAM LAMBDA3 LAMBDA4 LIM LI3 LI4 LOM L03 L04 LM L3 L4 MI NI NIV NNR NU NUI Symbol F23m F233 F 234 F 3 F 4 F 4(-1) h //B H ,q H o H'£ K K i ki3 k m 3 L4 lira li4 [om l o3 I o4 [ [ 3 [ 4 M. t Nt S /RL Pi 168 Identifier PHIDM PHID3 PHID4 PHIDOTIPRM PHIDOTIPR3 PHIDOTIPR4 PHIDOTM PHI DOTMAPI_ PHIDOTMAPR3 PHIDOTMAPR4 PHIDOTMM1 PHI DOT3M 1 PHI DOT4M 1 PHIDOTPBM PHIDOTPR3 PHIDOTPR4 PHIDOT3 PHIDOT4 PHIM PHIMM 1 PHII_ PHIR3 PHIR4 PHI SM PHIS3 PHIS4 PHI 3 PHI 3M1 PHI4 PHI 4M 1 ROPM ROP3 ROP 4 S T TAUS TIM Symbol _l) d m dPd3 dPd4 _m(-1) _3(-1) _4(-1) gb.(- 1) qSr. q_sm q5_4 ¢3 q/)3 (- 1 ¢4 _)4 (- I P pln Pp3 Pp4 Ig/l3 t T s Ti_ Identifier TR TI 3 TI4 V1M V13 V14 V2M (2) Identifier ANS2 ANS3 CF CFS COUNT CT FMT1 FMT2 FMT3 GUESS I INI ILOAD IS J LEGPARF LEGPARL LINES LOOP MATPARF MATPABL Symbol Identifier Symbol r Ti3 Ti4 V13 V14 V2= V23 V24 ZETAP W3 W4 V23 V2 4 _p W a W 3 W 4 Auxiliary identifiers Description Array for storing D vs. NI for given N_/R L and l 4/l 3" Array for storing D vs. l 4 /l 3 for given N[/R L and NI. Number of convergence failures for switching time. Cumulative number of convergence failures. Index number of At. Index of jth iteration. Format for the list OUT1. Format for the list OUT2. Format for the list OUT3. Label for location where predictions are made. Index number (general). Index for NI loop. Index for N_/R L loop. Index for 14/l 3 loop. Index number (general). Format for the list LEGPARL List of leg parameters Index number of printed line. Label of location where iterative computation begins. Format for the list MATPARL. List of material parameters. 169 Identifier Description OUT1 SW SWITCHING List of type-1 results. Number of output type (1, 2, or 3). Label of location where computation starts for each At. (3) DEFINE and PROCEDURE identifiers Identifier Symbol Identi fief Symbol PttDOTI _ PHIDOTIM _. PHI DOTI 3 _5 i a PHIDOTI4 _i4 PHIDOTMAM _._ PHIDOTMA3 _a3 PHIDOTMA4 _m.4 Program: FLUX DIVISION. SW=I: FLUX SWITCHING IN EACH LEG VS. TIME FOR GIVEN LOADt L;/L3t AND DRIVE! SW=2: D VS. DRIVE AND LOAD FOR GIVEN L;/L3I SW=3: D VSo Lq/L3 AND LOAD FOR GIVEN DRIVEe BEGIN COMMENT: DECLARATION OF MATERIAL PARAMETERS, I REAL BRe BSt HAt HGe HNe HOPPe NUt KAPPAe HOe HBt ZETAPw KAPPAIr NUI* MIw HOIt COMMENT: DECLARATION OF LEG DIMENSIONS, I REAL WM, ANt LINt LOMt LMp W3t A3, LI3t LO3t L3t Wqt A_t LI_t LO_t Lqt He SI COMMENT: DECLARATION OF SWITCHING PARAMETERSe! REAL PHIRMp PHISNt FOPPHe FBNt FOM_ ROPMt LAMBDAHe FI2Mt F23Me VIM t V2Mt LAMBDAIMp C|Mp FOIMt FIMt PHIR3p PHIS3t FOPP3t FB3t FO3t ROP3t LAMBDA3, F123t F233t V13, V23p LAMBDAI3e CI3t FOI3t Fl3t PHIR;, PHIS;t FOPP;t FB;t Fn;t ROPq, LAMBDA_t F12;t F23;e Vlq_ V24 , LAMBDAIqt Clqt FOI;t FI_! COMMENT: DECLARATION OF CIRCUIT PARAMETERSo! REAL NNRt TRe Nit NIVt TAUSI COMMENT: DECLARATION OF VARIABLES,! REAL Tt DELTe FJp FUPRt Dt TINt FMt PHIDOTNt PHZDOTMMIt PHIDOTPRMt PHIDOTMAPRNt PHIDOTIPRNt PHIMw PHIMMIe DELPHIMt PHIDMe TZ3t F3w PHIDOT3s PHIDOT3NIe PHIDOTPR3e PHIDOTMAPR3t PHIDOTZPR3e PHI3t PHI3Mlw DELPHI3t PHID3e DELTAPHI3p TZqw F_e PHIDOTqt PHIDOTqMIt PHIDOTPR_P PHIDOTMAPR_t PHIDOTIPRqp PHI;t PHI;MI_ DELPHIqt PHIDqp DELTAPHIqt FqMl_ DELF_t DFq! COMMENT: DECLARATION OF MISCELLANEOUS,! INTEGER CFSs CF, CTt COUNTt INIt ILOADe ISt LINES_ SWt It dt K! LABEL SWITCHINGt LOOPt GUESSt REAL ARRAY ANS2[O : 20It ANS3[O : 20t 0 : 20]! COMMENT; DECLARATION OF OUTPUT LISTS AND FORMATS! FILE FI q(2, 15)1 LIST MATPARL(BRt BS_ HAt HGt HNt HOPPe NUt KAPPAt HOt HBt ZETAPt KAPPAIt NUIt HI x 1_6t HOI)i LIST LEGPARL(LIN x l_3t LOM x IQ3t PHIRM x IQ8_ PHISM x 1_8t LAMBDANs FOPPM, ROPMt FONt FBMe FI2Mt LAMBDAIMt CIM x 1_6, FOZMt LI3 x 1_3t L03 x 1G3e PHIR3 x 1_8t PHIS3 x 1QBt LAMBDA3t FOPP3t ROP3t FO3e FB3t F123e LAMBDAI3e CI3 x 1_6t FOI3e LI_ x l_3t LO_ x 001 002 003 OOe; 005 006 007 008 009 010 011 012 013 01; 015 016 01"I 0IS 019 020 021 022 023 02_ 025 026 02"1 028 029 030 031 032 033 03_ O35 036 037 O38 170 IG3t PHIRq x 1G8, PHIS_ x lgGo LANBDA;t FOPPq, ROPqt FOqo FBq, FI2;p LANBDAI_, CZ_ x IGbp FOZ_)I LIST OUTI(T x 186p PHIDOTNp PHZN x 188, FM, PHIDOT3, PHI3 x IG8, F3, PHIDOT_, PHI_ x IG8, F;, NIV, CT), OUT2(NNRw FOR J • 1 STEP 1 UNTIL 8 DO ANS2[J], CFS), OUT3(FOR I • 1 STEP I UNTIL 6 DO FOR J • 0 STEP I UNTIL 9 DO ANS3[Z, d])! FORMAT NATPARF(X_O, "M A T E R ! A L P A R A M E T E R Sm / "BR=", F6o3, X2, NBS=Nt F6o3, X2, mHA=", F5ol, X2, WHg=", F_°lp X2 , "FIN: m, F_.Ip X2, "HOPP:", F_.I, X2, "NU=", F_.I, X2, "KAPPA=W, F7.1, X2, "HO=", F_.Is X2, "FIB=", F5°1, X2, "ZETAP= N, FGol / mKAPPAI=_, F6°1, X2, wNUZ=", F_o2, X2, NMI="_ FSo2_ "8-6", X2p "HOI:", F_°I /)! FORNAT LE6PARF(X_O, "L E G P A R A N E T E R S" / nLIN=", F6o3p X2, WLOM:-W, F6o3, X2, wPHIRN=Np F6.3, X2, "PH;St4:.", F6.3, X2, "LANBOAN="p FSo3, X2, "FOPPN=", F5o3, X2, "ROPlq= w, FSo3, X2, NFOM=m, FS°3P X2, "FBN= _, FSo3 / "FI2N= N, FSo3, X2p WLANBDAZN-:", F5.3, X2, _CIM="t FSo3w X2, "FOIN=", FSo3 / "L;3=", F6o3, X2, "LO3=", F6o3, X2, "PHIR3=Wt F6o3, X2p "PHZS3=", F6°3, X2_ "LAMBDA3=", FS.3, X2, "FOPP3:", FS.3, X2, "ROP3 :w, F5o3, X2, "FO3=_s FS°3s X2t "FB3=", F5o3 / "F123= m, F5.3, X2, "LAMBOAZ3=", F5.3_ X2p "CI3=", FS.3t X2t "FOX3=", F5o3 / "LI_=", F6°3, X2, "LO_=", F6o3, X2, "PHIR_=", F6_3, X2, "PHIS_=", F6.3, X2, "LAMBDA_=", F5.3, X2, "FOPP_="_ FSo3, X2, "ROPe=", F5.3_ X2, "FO_=", F5.3, X2, "FB;=", F5o3 / "FI2_=", F5,3, X2, "LANBOAIW=", F5.3t X2, "CIW=", FSo3t X2t "FOI;=% FSo3 /)! FORMAT FNTI(11FIO.;_ I6), FMT2(9 FII._, Ib), FNT3(9 Fll.q, I6), TOP(X7, "T"_ XS, _PHIDOTN"t XWt "PHIN_ XG, "FN", X5, "PHIDOT3", XW_ "PHI3", XG, "F3", X5_ "PHIDOT_", X_, "PHI_ _, XG, "F_", X7, "NIV", X6 _CT"), HEADI(X20o "FLUX SWITCHING IN EACH LEG FOR NL*2/RL ="e F6o3, X5, "S =", F_.2, X5, _NI :", FW.2 / / / X7, "T _, X5_ "PHIDOTN"_ X_, _PHIN", XG, "FN", XS, "PHIDOT3", XW, "PHI3", XG, "F3 e, XS_ _PHIDOTW _, X_, _PHI; _, X8_ "FW", X7, "NZV", X6, "CT"), HEAD2(X35, _0 VS. NI AND LOAD FOR S ="o Fq.2 / / / X;, "NL*2/RL _, X_O, "NI _, XSO, "CFS" / / XlO, 8 Fll,l)s H[AO3(X35, "D VSe S ANO LOAD FOR NI =", F;oO / / / X_, "NL*2/RL _, X_O, "S", XSOt "CFS" / / XlO_ 8 Fl1,1)! COMN(NT: DECLARATION OF DEFINITIONS AND PROCEDURES.! DEFINE PHDOTI = 039 OatO 0_1 O_t2 0_3 0_5 Oq6 0_8 0_9 O5O 051 052 053 05_ 055 O56 057 058 O59 O60 061 062 063 06_ 065 066 067 O68 069 070 071 072 073 07_ 075 076 Lines PHDOTI07 _hrough PHDOTI21 of I PROCEDURE _(F,t,Ti,_), APPENDIX B # ! DEFINE PtqDOTNA: Lines PHDTMA06 _hrough PHDTMA47 of PROCEDURE - - -- -._ (F,_,_a,6"), APPENDIX C 171 # ! REAL PROCEDURE PHIDOTIN! BEGIN DEFINE F = FM #t TI = TIN #e PHIDOTIPRIME = PHIDOTIPRM _p PHIDOTI = PHIDOTIM #p LAMBDAI = LAMBDAIN #_ CI = CIM #p FI = FIN #! ,PHDOTI END PHIDOTIM! PHDOTIM1 PHDOTIM2 PHDOTIM3 PHDOTIM_ PHDOTIM5 PHDOTIM6 REAL PROCEDURE PHIDOTHAM! PHDTHAMI BEGIN DEFINE F = FM #, PHI = PHIM #p PHID = PHIDM #, PHIDOTMAPRIME PHDTMAM2 = PHIDOTMAPRM #p PHIDOTMA = PHIDOTMAM #_ LI = LIM #p LO = LOM # PHDTMAM3 p PHIR = PHIRN Up PHIS = PHISM #p LAMBDA = LANBDAM #p FOPP = PHDTMAM_ FOPPM #w ROP = ROPM #w FO = FOM #e FB = FBM #_ FI2 = F12M #, PHDTNAM5 F23 = F23N #, V1 =VIM #e V2 = V2M #! PHDTMAM6 PHDOTMA PHDTNAM7 END PHIDOTMAM! PHDTNAN8 REAL PROCEDURE PHIDOTI3! BEGIN DEFINE F = F3 #, TI = TI3 #p PHIDOTIPRIME = PHIDOTIPR3 #p PHIDOTI = PHIDOTI3 #_ LAMBDAI = LAMBDAI3 #p CI = CI3 Up FI = FI3 #! PHDOTI END PHIDOTI3J PHDOTI31 PHDOTI32 PHDOTI33 PHDOTI3; PHDOTI35 PHDOTI36 REAL PROCEDURE PHIDOTMA3! BEGIN DEFINE F = F3 #, PHI = PHI3 #p PHID = PHID3 #, PHIDOTMAPRIME = PHIDOTMAPR3 #w PHIDOTMA = PHIDOTMA3 #p LI = LI3 #p LO = L03 # p PHIR = PHIR3 #p PHI$ = PHIS3 #p LAMBDA = LANBDA3 #e FOPP = FOPP3 #_ ROP = ROP3 #t FO = F03 #t FB = FB3 #_ F12 : F123 #_ F23 = F233 #, Vl = V13 #, V2 = V23 #! PHDOTNA END PHIDOTMA3! PHDTMA31 PHDTMA32 PHDTMA33 PHDTMA3_ PHDTNA35 PHDTMA36 PHDTMA3? PHDTMA38 REAL PROCEDURE PHIDOTIt; BEGIN DEFINE F = Ft #w TI = Tit #P PHIDOTIPRIME = PHIDOTIPR_ #w PHIDOTI = PHIDOTI_ #p LAMBDAI = LAMBDAIt #_ CI = CI4 #P FI = Fit #I PHDOTI END PHIDOTI_! PHDOTIql PHDOTI_2 PHDOTI_3 PHDOTI;t PHDOTI_5 PHDOTI_6 REAL PROCEDURE PHIDOTMAt! BEGIN DEFINE F = Ft #p PHI = PHIt #, PHID = PHID; #t PHIDOTMAPRINE = PHIDOTMAPR; #p PHIDOTMA = PHIDOTMA; #p LI : Lit Cp LO : LO; # , PHIR = PHIRt #w PHI$ = PHISt _t LAMBDA = LANBDA_ #t FOPP = FOPPt #w ROP ='ROPt #p FO = FOr #p FB = FB; #p FI2 = F12t #_ F23 = F23t #p V1 =Vlt #p V2 = V2_ #! PHDOTNA END PHIDOTMAt! PHDTHAql PHDTMA_2 • PHDTMA_3 PHDTMA;_ PHDTNA_5 PHDTMA_6 PHDTHA_7 PHDTNA_B REAL PROCEDURE TANH (Xl)l VALUE Xl! REAL Xll BEGIN REAL Y; Y • EXP (Xl + Xl); TANH * (Y - 1,0) / (Y + 1,0) END TANH; TANHO001 TANHO002 TANHO003 TANHO00_ TANHO005 TANHO006 TANHO007 DEFINE MM = x 0.001 #! COMMENT: TYPE OF OUTPUT,! $_ * 21 COMMENT: MATERIAL PARAMETERS,I BR * 0,23001 BS * 1.08 x BRI OT7 078 079 OeO OBI 082 172 HA * 310! He * 35.0! HN * 30.08 fl0PP * _0.0! NU -.1.30! KAPPA * 34001 H0 * 61.0! HB _'(HO x I/U - HOPP) /(NU - 1)t ZETAP • KAPPA x NU x(HB - HOPP) *(NU - 2)t KAPPA; e, 5921 _K;I * 1.31 M1 * 10,7G-6! HOI * 2q,8t CO_NT: CORE DZNEI_ZOI_ ANO SWZTCHZNG PARAMETERS.! H * 1,31 MM! tIM • 1.016 MMI AM *" H x WMI L][N • 1q.,..363 MMI LOtq • 19.151 14NI LM *(LIM ÷ LOM) / 21 PHIRI4 * AM x BRI PHISM * AM x BS! FOPPM * LM x HOPP! FBM _" LM x I'_! FOM e. LM x HOI ROPM • ZETAP x AM / LMI LAI_BDAM * KAPPA x AM / LM * NUt FI2M • I'_ x L][MI F23M • He x LOM! VlM *(PttISH - PHIRM) /(HA xCLOR - LIM))I V2M q'IPHISM + PHYRM) x Ite /(kiN x(LOM - LZM))! LAMBDAIM • KAPPA][ x AM / LM •NUII CIM * HI[ x I..M8 FO'rM *" HO][ x L3[MI W3 * 0,508 l_q; A3 * H x W31 LI3 * 4.310 MM; L03 * 5.108 MM; L3 *(LI3 + L03) / 21 PHXR3 q" A3 x BR! PHIS3 • A3 x BSI FOPP3 * L3 x HOPP; FB3 * L3 x lt81 F03 • L3 x HO; ROP3 • ZETAP x A3 / L3! LAMBOA3 * KAPPA x A3 / L3 s NUt F123 * He x LI31 F233 • HG x L03! V13 ',(Pt11S3 - FqtlR3) /(HA x(L03 - LI3))I V23 •(PHIS3 + 1:_4ZR3) x He /(HN x(L03 - LI3))I LAMBOAZ3 • KAPPAZ x A3 / L3 • NOII CI3 • 14][ x L3! F013 • HOI x LI31 wq. *" 0.508 MMI A4 * H x wq.t PHZRq _" A; x BRI PHIS4 * Aq x BS! |F SW = 3 T_N BEG%N WRZTE(FIe MATPARF, MATPARL)! WRITE(F1* HEAO3e lO0.Op FOR S • I STEP 2 UNTIL 9 O0 S)I LINES * 71 ENDS 1S * O! FOR S * 1.83 DO 083 08q 085 086 087 088 089 090 091 092 093 O_ 095 096 097 098 099 100 101 102 103 10_ 105 106 107 108 109 210 111 112 113 llq 115 116 117 118 119 120 121 122 223 12q. 125 126 1=:7 1=:8 2:)9 130 131 135: 133 235 136 137 138 139 lq2 173 BEGIN IS • IS + 11 LI; • S x LI3I LO_ • S x L031 L_ •(LI; + LO;] / 2l FOPP; • L; x HOPP; FB; • L; x HBI F0; • L; x H0; ROP; • ZETAP x A; I L;; LANBDA; • KAPPA x A; I L; s NU; F12; • HQ x LI;; F23; • HQ x LO;I Vl; •(PHIS; - PHIR;) /(HA x(LO; - LI;))I V2; •(PHIS; + PHIR;) x HG I(HN x(LO; - LI;)); LANBDAI; • KAPPA! x A; I L; • NUI; Cl; • HI x L_; FOI; • HOI x LI;; IF SW = 2 •HEN BEGIN WRITE(FI[PAGE]); WRITE(F1, NATPARF, NATPARL); WRITE(FIp LEGPARF, LEGPARL)I WRITE(F1, HEAD2, Sp FOR NI • 1.1p 1o3, 1o5, lo7, 2.0, 3,0, ;oO, 5o0 DO HI); LINES • 16; END; CONNENT: CIRCUIT PARANETERS.; TR • 0.08G-6; ILOAD • O; FOR NNR • Ot 1o0, 3o962p 9052; DO BEGIN ILOAD • ILOAO + 1l CFS • O; INI • O; FOR NI • 1.1p 1.3, lo5, 1.7p 2o0, 3o0, ;,0_ 5,0 DO BEGIN IN! • INI + I; IF SW = 1 THEN BEGIN WRITE(FI[PAGE]); WRITE(F1, MATPARF, MATPARL); WRITE(F1, LEGPARFp LEGPARL); WRITE(Flp HEAD1, NNRp S, NI); LINES • 16; END; CONNENT: INITIAL CONDITIONS.; T • O; PHIN • - PHIRN; PHI3 • - PHIR3; PHI; • - PHIR;; FN • F3 • F; • O; CF • COUNT • O; PHIDOTN • PHIDOT3 • PHIDOT; • O; COMMENT: COMPUTATION OF SWITCHINGo; TAUS • 1.65 x PHIRN x ((I / LAMBOAN _(1 / NU) + I / LANBDA3 • (1 / NU)) /(NI - FOPPM ° FOPP3)) * NU; OELT • TAUS / 200; SWITCHING: CT • O; • • T + DELT; NIV • NI x(IF • < TR THEN(I - COS(3ol;159265; x • / TR)) 1 2 ELSE 1); PHINMI • PHIN; PHIDOTHN1 • PHIDOTH; PHI3M1 • PHI3; PHIDOT3NI • PHIDOT3I PHI;M1 • PHI;; PHIDOT;H1 • PHIDOT;; FIN! • F;; GUESS: PHIH • PHIHN1 + DELPHIN; lq7 1;8 1;9 150 151 152 153 15; 155 156 157 158 159 160 161 162 163 16; 165 166 167 168 169 170 171 172 173 17; 175 176 177 178 179 180 181 182 183 18; 185 186 187 188 189 190 191 192 193 19; 195 196 197 198 199 200 201 202 2O3 20; 205 206 20? 2O8 209 210 174 PHI3 * PHI3N1 ÷ DELPHI3! PHI_ * PHI_N1 + OELPHI_! F_ * F_N1 + DELFqt LOOP: CT * CT + 11 F_ * F_ + OF_! FI_ * FOI_ x TANH(F_ / FOIl)! PHIDOT_ • PHIDOTNA_ + PHIDOTI_! OELPHZ_ * DELT x(pHIOOT4 ÷ PHZDOT_M1) / 21 PHX4 • PHX_N1 ÷ OELPHI_; F3 * F_ ÷ NNR x PHIDOT_! FI3 * FO]3 x TANH(F3 / F013)1 PHXOOT3 * PHIDOTMA3 + PHIDOTX3! DELPHI3 * OELT x(PHIOOT3 + PHIOOT3M1) / 2! PHI3 * PHI3M1 ÷ OELPHI31 FN * NXV - F3! FIN * FOIN x TANH(FN / FOIM) I PHXDOTM * PHXDOTMAN + PHIDOTXN! OELPHIN * OELT x(PHIOOTN + PHXDOTMM1) / 21 PHIN * PHINN1 + OELPHIN! DELF_ • F; - F_NI! FJ * PHIDOT_ + PHZDOT3 - PHIOOTN! PHIDOTPRN • PHIDOTXPRN + PHXDOTNAPRNI PHIDOTPR3 • PHXDOTXPR3 + PHIDOTNAPR3! PHIDOTPR_ * PHXDOTIPR_ + PHXDOTNAPR_I FJPR _(PHZDOTPRM + PHIDOTPR3) x(l + NNR x PHIDOTPR_) ÷ PHIDOTPR_I IF FJPR = 0 THEN FJPR _ 1i-201 OF; _ XF SZ6N(DF_) = SZ6N(- FJ / FJPR) THEN - FJ / FJPR ELSE - 0,5 x FJ I FJPRI XF ABS(FJ) > 0,0001 x PHIRM / TAUS AND CT < 20 THEN 60 TO LOOPI XF CT : 20 THEN CF * CF + 11 IF COUNT NOD 5 = 0 ANO SW = I THEN BE61N WRITE(F1, FNT1, OUT1)1 LINES * LINES + 11 IF LXNES NOD 50 = 0 THEN BE61N WRITE(FI[PA6E])I WRITE(F1, TOP) I END! ENDI COUNT * COUNT * 1t ZF PHZN < 0 AND PHZDOTM > 0,0001 OR PH%N < - 0,99 x PHXRM THEN 60 TO SMXTCH|N6! DELTAPHI3 * PH%3 + PHIR3! OELTAPHZ_ * PHI; + PHIR4! IF DELTAPHI_ = 0 THEN D * 1B20 ELSE D * DELTAPHZ3 / OELTAPHZ_! CFS * CFS + CF! IF SW = 2 THEN ANS2[INI] * DI ZF SW = 3 THEN ANS3[ILOAD, IS] • 0! END NILOOPI IF SW = 2 THEN WRZTE(F1, FNT2_ OUT=))I ZF SW = 3 THEN BE6XN ANS3[XLOAD_ O] * NNR! ANS3[ILOAD_ 9] * ANS3[ILOADe 9] + CFS ENOI END LOADLOOPI END SLOOPI IF SW = 3 THEN WRZTE(FI_ FMT3_ OUT3)! END. =)11 212 213 21_ =)15 216 217 218 219 220 =)21 =)22 223 =)=)_ 225 =)=)6 =)=)7 228 =)29 230 =)31 232 =)33 =)35 =)36 =)37 =)38 =)39 2_0 =)q=) =)q7 =)_t9 =)50 =)51 =)5=) =)53 " =)55 =)56 =)57 =)58 259 =)60 =)61 =)6=) =)63 =)6_t =)65 =)66 267 =)68 =)69 =)7O 175 REFERENCES 1. D. Nitzan, "Flux Switching in Multipath Cores, " Report 1, for Jet Propulsion Laboratory, Contract 950095 under NASw-6, SRI Project 3696, Stanford Research Institute, Menlo Pa_k, California (November 1961). 2. D. Nitzan and V. W. Hesterman, "Flux Switching in Multipath Cores," Report 2, for Jet Propulsion Laboratory, Contract 950095 under NASw-6, SRI Project 3696, Stanford Research Institute, Menlo Park, Cahtornaa tNovem,er ...... 3. D. Nitzan and V. W. Hesterman, "Flux Switching in Multipath Cores, " Report 3, for Jet Propulsion Laboratory, Contract 950095 under NASw-6, SRI Project 3696, Stanford Research Institute, Menlo Park, California (June 1964). 4. N. Cushman and D. Park, "Reversal of a Loaded Ferromagnetic Core, " IRE Trans. on Component Parts, vol. CP-7, pp. 117-124 (December 1960). 5. R. W. McKay, "The Reversible Component of Magnetization," J. Appl. Phys., Supplement to vol. 30.4, pp. 56S-57S (April 1959). 6. R. L. Con_er and F. C. Essig, "Magnetization Reversal in Thin Films at Low Fields," J. Appl. Phys., vol. 28, pp. 855-858 (August 1957). 7. W. E. Milne, Numerical Solutron of Differential Equations (John Wiley & Sons, Inc., New York, 1950). 8. P. Henrici, Elements of Numerical Analysis (John Wiley & Sons, Inc., New York, 1963), Chapter 4. 9. J. B. Scarborough, Numerical Mathematical Analysis (The Johns Hopkins Press, 1950), Chapter 9. 10. A. F. Pixley and Anne G. Macek, "A B5000 ALGOL-60 Runge-Kutta-Nystr¢m Starting Procedure for Use with Milne Type Predictor-Connectors in Solving Systems of Ordinary Differential Equations," Burroughs Technical Bulletin MBS°II3 (Augustl, 1963). ii. I_ ADAMS (X, Y, YPRIME, N, EU, EL, EPS, H, HMIN, DEBIV) in the computer-program library at Stanford Research Institute, Menlo Park, California. 12. D. Nitzan, "Flux Division in a Saturable Multipath Core, " IEEE Trans. on Electronic Computers, vol. EC-13, pp. 272-277 (June 1964). 13. "Burroughs B5500 Information Processing System Extended AD3(_ Reference Manual, _ Burroughs Corporation, Detroit, Michigan (1964). 14. N. Menyuk and J. B. Goodenough, "Magnetic Materials for Digital°Computer Components. I. A Theory of Flux Reversal in Polycrystalline Ferromagnetics," J. Appl. Phys., vol. 26, pp. 8-18 (January 1955). 177 INDEX Adams method of solving differential equations, 67, 164-165 Binary counter, core-diode-transistor, 49-72 analysis of Mode I, 56-61 basxc equations, 56-58 equivalent circuit, 57 simplifying assumptions, 56 transcendental solution of currents, 58-60 circuit, 52, 64 computation of Mode I variables, 61-71 comparing results of two methods, 68, 69, 70 limitations of, 68-69 Runge-Kutta and Adams method, 66-68 computer program, 67, 159-166 simple method, 61-66 computer program, 61-64, 153-158 results, 64-66 variations of &I(F1) and _2(F2), 70, 71 conditions for proper operation, 55 drive currents, 52-64 operation, modes of, 49, 52-54 Chopper to measure F(t), 99 Circuit data, experimental: core-diode-transistor binary counter, 64 flux division, 81 initial ¢(t), 25 pulse: amphtude, 99, 109, 119 duration, 109 sequence, 99, 109, 119 Coefficient, temperature--see Temperature coefficient Comparison between computed and experimental results: initial ¢(t), 26-34 loaded core, 49o51 Components of 6, 4 computed, 37 experimental verification o_ 18-21, 20-34 very low, 39, 41 Composition of Core E-6 material, 20 Computation of: core-diode-transistor binary counter, Mode I, 61-71 data: circuit, 64 core IOOSCI, 64 Bunge-Kutta and Adams method, 66-68 computer program67,159-166 simple method, 61-66 computer program, 61-64, 153-158 results, 64-66 flux division, 72-89 approximate ¢s' 76-77 basic equations, 74 computerprogram, 77-78, 167-175 D vs. 14/l 3 and load, 83, 85, 87 drive current, 77 effect of leg dimensions on switching parameters, 76 experimental verification, 81-88 Dvs. NI and load, 83-84, 86 ¢(t) waveforms, 82-83, 86 method of computation, 74-75 initial 6(t) of unloaded core, 22-34 comparison with experimental data, 26-34 computer program, 24, 141-146 core and circuit parameters, 25 drive current, 22-24 results, 26-34 loaded core, 45-49 computer program, 147-151 core data, 48 error in Bep. 3, 46 inductive load, 47 Newton's method, 48 noninductive load, 47-48 _o(k) and tp(k), 96 Computed: core-diode-transistor binary counter, Mode I: minimum voltage, 70 time variables, 64-66 ¢1(F1 ) and ¢2(F2), 70, 71 flux division: D vs. 14/l 3 and load, 83, 85, 87 D vs. NI and load, 83-84, 86 ¢(t) waveforms, 82-83, 86 initial _(t) of unloaded core: peak ¢ip vs. FD and T, 40-41 _(t I components, 37 ¢(t) waveforms, 26-34 loaded core, ¢(t) and iL(t), 49-51 Computer program: core-diode-transistor binary counter: simple method, 61-64, 153-158 Bunge-Kutta and Adams method, 67, 159-166 elastic _, 6E, 16,133 flux division, 7%81, 167-175 inelastic decaying ¢, ¢i' 17, 135 initial _(t), 24, 141-146 loaded core, 147-151 main ¢, ¢ , 17-18, 137-138 Cooling for measuring temperature effects, 107 179 Core: E-6: composition, 20 initial _(t) results, 26-34 initial _(t) test, 18-20 ramp-F(t) switching, 99 switching parameters, 25 I-4: dimensions, 108 switching parameters, 108 J-l: dimensions, 48 switching parameters, 48 K-I: dimensions, 108 switching parameters, 108 S: dimensions, 82 switching parameters, 81-82 100SCI: dimensions, 64 switching parameters, 64 Core-diode-transistor binary counter--see Binary counter, core-diode-transistor Core holder, 20 Correction in ramp-F(t) switching parameters: F_, i01, 102, 105, 131 l, I01, 102, 104, 128, 131 v, 103 Counter, binaryT-see Binary counter, core-diode- transistor Crossing of curves: kt (k), 103, 106 p t (k), 127, 128, 130 (k), 124 p Current drive--see Drive current Current drivers in: initial _(t) test, 21 ramp-F(t) switching, 99 INDEX Damping, viscous: for wall motion, 119 of b E, 35 Differential equations, numerical solutions of, comparing results of two methods, 68, 69, 70 Bunge-Kutta and Adams methods, 66-68, 159-166 simple method, 46-48, 61-66 Dimensions of cores--see Cores, dimensions Dimensions of le G , effects on switching parameters-- see swltching parameter effects of geometry on Domain nucleations, 119 Drive current: binary counter, 52, 64-66 flux division, 73, 77 initial ¢(t), 22-24, 26-34 ramp F(t) switching, 99,100 Driver in initial _(t) test, 21 Duration of CI2AB pulses, 109 Easy axes, statistical variation of, 119 Elastic ¢, 4-5 future investigation, 43 computed CE(t), 37 computer program, 16, 133 conclusions about model, 42-43 viscous damping, 35 Equipment in: initial _(t) test, 21 ramp F(t) experiments, 99 Error in previous computation, 46 Error in static _(F), 110, iii Experiment, flux division, 73 D vs. NI and load, 83-84, 86 (t) waveforms, 82-83, 86 Experimental waveforms: binary counter, 64-66 loaded core, ¢(t) and iL(t) , 49-51 ramp-F(t), 99 _(t) for ramp-F(t), 100, 101 unloaded thin ring, _(t), 26-34, 36-37 Experiment: : ramp-F(t) switching, 99, 123 static _(F)vs. temperature, 107 _p(F) vs. temperature, 118 _(t) of unloaded thin ring, 18-21, 26-34 Equipment, 20-21 results, 26-34 ringing of _(t), 19-20 Fall time for ramp-F(t) switching, 99 Flux change: due to _, 39 due to _i' 39, 41 _¢d' 41 Flux division, computation of, 72-89 analysis, 74-77 basic equations, 74 calculation of switching parameters, 75-76 drive current, 77 estimation of zs' 76-77 method of computation, 74-75 computer program, 77-81, 167-175 multiple output, 78 program outllne, 78-81 _UBEs, 77-78 experiment, 73 experimental verification, 81-88 clrcult data, 81 core S: material parameters, 81 switching parameters, 82 _180 INDEX Flux division, computation of--continued D vs. 14/13and load, 83, 85, 87 results: D vs. NI and load, 83-84, 86 ¢(t) waveforms, 82-83, 86 limitations of previous calculation, 72 Flux reference, 109 Flux switching: components, 4, 37 in core diode-transistor binary counter: partial, 53-55, 70-71 complete, 70-71 in£errupted-F experiment, 1-3 Flux switching models, 4-15 attalogy between ¢i and_a' 36 etastic, 4-5 inelastic, decaying, 6-11 inelastic, main, 11-12 use of, in ramp-F switching, 91, 92, 98, 106, 119, 123, 124 Future investigation: computer programs for magnetic circuits, 89 flux-switching models, 43 ramp-F(t) switching effects, 105, 128, 132 Geometry e_fect 9n switching paraLaeters--see OWltchxng parameters, etIects oi geometry on Grains in ferrite, 119 Graticule of oscilloscope, 99 Heating of transistors, 107 History effects, removal of, 119 Hyperbolic B(H), 111 Inductive load, computation of, 47 results, 49-51 Inelastic ¢: approximate model for ¢i + _a' 14 computer program, 139-140 modified models for _i' 7 computer program, 17, 135 modified model for --

Flux switching in multipath cores

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