Contains reports on eight research projects

Cerrillo, Manuel V.

Chien, C. K.

Chu, L. J.

Goldman, Stanford

Lally, P. M.

Loewenthal, M.

Macnee, A. B.

Mueller, H.

Parke, N. G.

Parshad, R.

Redheffer, R. M.

Scott, R. E.

Smullin, Louis D.

Vivian, W. E.

Wallman, Henry

Contains reports on seven research projects

DSpace@MIT

IX. MISCELLANEOUS PROBLEMSA. ELECTRONIC DIFFERENTIAL ANALYZERStaff: Professor H. WallmanA. B. MacneeThe work of the past quarter has been divided principally into twoparts: (1) engineering and development of the various basic units used in thesolution of ordinary differential equations with constant coefficients and(2) preliminary investigations of the solution of some non-linear differentialequations.Linear Differential Equations. Some time was spent in the investigation ofthe relative merits, in general, of d-c coupled amplifiers versus d-c restoreda-c coupled amplifiers. It has been concluded that where the time necessaryfor d-c restoration (or clamping) is available, this method is the moresatisfactory with regards to stability, warm-up time, and economy of tubesand components.An amplifier of this type having an amplification of 1500 has beendesigned and tested. Three of these units with suitable feedback connectionshave been used to solve the equationd2 y + k k2y =O (1)dtwhere the k's are fixed numbers. Particular emphasis was placed on thestability of the solution displays on the cathode-ray tube screen; this ledto considerable engineering in the initial condition and gate-generators.The equation2.+ W 2y = 0 (2)dt2 0has been studied in some detail to determine the restrictions imposed by thefinite bandwidth and gain of the amplifiers and integrators. The functionof frequency-obtained from this equation by the Laplace transform has poleson the imaginary axis in the complex frequency plane. It is particularlysensitive to bandwidth and gain limitations. These limitations introducerespectively negative and positive damping of the exact solution, which isan undamped sine wave. The bandwidth restriction is particularly interesting.If the solution amplitude is to change less than 1 per cent over the solu-tion display time, then2 w .01 (3)2 W1where T = display time,w = natural frequency of the solution,wl= upper half-power frequency of the amplifier.-63-Il-L-C- l  I~-~I__ _ _II_In particular if the solution period is 1/120 second and the amplifier band-width is 400 kc/sec, the maximum permissible value of fo= uo/2T becomes392 cps. Thus for this case the amplifier bandwidth must be 1000 times thenatural frequency of the solution.Non-Linear Differential Equations. The three units discussed above havebeen used together with a function generator to solve equations of the formd t k i ± k2y = F(y). (4)dt 2No particular difficulties have been encountered in solving these equations.Solutions of two particular equations of this type are shown inFigs. IX-2 and IX-3. Figure IX-2 is a triple exposure photograph of threedifferent solutions of the equationd2  2 (5)dt 2with y0 held fixed and Yo varied. Equation (5) describes the motion of aparticle in a potential field. A typical potential of this type is shownin Fig. IX-1.Fig. IX-1. A cubic potential curve.In Fig. IX-2 for one initial value the particle oscillates stablyin the potential well between yl and y2. The other two exposures show casesfor which the particle is shot off with sufficient initial kinetic energy tocarry it over the potential peak at yl after which it drops off rapidlytoward minus infinity.Figure IX-3 displays three solutions of the equationd 2 = - sin ydt-64-~_~.~___~rurr~--rurr~--1-- -- - ^ -.------ ~"Y-~IP~ ' -~-~~ --with yo held fixed and yo varied. This is the exact equation for the motionof a physical pendulum. The manner in which the period of oscillation in-creases with increasing amplitude can be clearly seen.Fig. IX-2. Plot of y versus t for fixed y0 andvariable o, where y is the solutionof the equation d2y/dt2 = y .Fig. IX-3. Plot of y versusable, wher y isequation d y/dtt for Y0 fixedy o vari-the solution of the= - sin y.-65-Finally Van der Pol's famous non-linear differential equationd2 x A -3B x2] + x = 0dt dtdt(7)has been set up and solved1. A typical solution of Van der Polls equationis given in Fig. IX-4.Fig. IX-4. A plot of x versus t for Van der Pol's equation.Figure IX-5 is another solution of the same equation for the caseof A and B both of the order of magnitude unity. Figure IX-6 is a photo-graph of the build-up to the phase-space limit cycle for this case. A con-siderable amount of mathematical work has been done on the characteristics of2limit cycles of this type1. This equation is most easily handled by the analyzer if it is firstconverted by the change of variable x = d and one integration todtd y B(d -Z )] A + y = 0.dt2  dt dtdt2. N. Minorsky, "Non-linear Mechanics", Edwards Brothers, Ann Arbor,Michigan, pp 113-115, pp 131-133.-66-Fig. IX-5. A plot of x versus t for Van der Pol's equation (low Q, case)Fig. IX-6 Build-up to limit cycle in the solution of Van der Pol's equation.-67-IX. B. Al AUTOIMATIC IMPEDANCE ANALYZERStaff: Professor H. WallmanR. E. ScottSome progress has been made during the last quarter upon the develop-ment of an automatic impedance and gain-function analyzer.Theory. The electrolytic tank used by Linvill and by Hansen and Lundstromfor the analysis of impedance functions was described in the QuarterlyProgress Report of January 15, 1948. The method depends upon the mathe-matical analogy between potential problems and complex variable theory. Itis well known that any physically realizable impedance function can be repre-sented as the ratio of two polynomials in the complex frequency X. This istrue for driving-point impedances and for transfer impedances.Z() = ( - l) (  - ) ..- 2)(X - X .......where Z(A) = the impedance function,X = the complex frequency variable,hi = zeros of the function for i odd,Ni = poles of the function for i even.logeZ(X) = logelZ(X) I + i arg Z(A)loglZ(() = logej X - N1 + loge l - N31- logel - logl- - 41From potential theory the voltage drop from theX in an infinite conducting plane is+ ........origin to any pointV(X) = A loge ( - l )(  A3 ) .........X 2 (N X 4).........where A is a constant depending on the magnitude of the currenentering and leaving the plane, and on theconductivity;Xi = point where a current enters the plane for i even= point where a current leaves the plane for i odd.(the currents at each point are of equal magnitude)The similarity of these two equations is at once apparent and it isseen that by measuring the voltage along the imaginary axis, the equivalent-68-_- ~IU---.~l--l--__- - Y- ^X^impedance function is evaluated at real frequencies. In order to exploitthe method to the full it is desirable to have the potential along theimaginary axis displayed on a cathode-ray tube so that the effects ofchanges in the positions of the poles and zeros may be observed immediately.If such a device could be produced, it would be extremely useful for synthe-sizing impedances and filters. Arbitrary conditions could be imposed uponthe poles and zeros with great ease; for example, they might be confinedto the negative real axis. This restriction would limit the network toR, C elements.Experimental Approach. The first step in producing a practical device isto replace the electrolytic tank by a sheet of uniformly conducting paper.Teledeltos Type H (made by Western Union) has been found satisfactory. Itis uniform to better than one per cent and will carry a current of 8 mawithout leaving a mark at a sharp electrode. In the actual experiment thecurrent is limited to one-tenth of this value. Currents of this magnitudeallow voltages of about twenty volts maximum to be measured along theimaginary axis. In the experiment direct currents are used from currentsources with open-circuit voltages of plus and minus 300 volts. In orderto prevent arcing when the probes are removed from the paper, it isnecessary to provide a limiter diode at each probe to keep the open-circuitvoltage below 45 volts.The voltages along the imaginary axis are picked up on 15 probesand commutated mechanically. The resultant voltage is fed to the X platesof a cathode-ray tube. For an even spacing of the probes and a linearsweep, the voltage observed on the screen is the logarithm of the magni-tude of the impedance, plotted against the frequency.Qualitatively the device was successful. Its small size, however,made it rather inaccurate, and the mechanical commutator was a source ofconstant trouble. It was decided that it would be desirable to use at leasta hundred pick-up probes in the final device and that an electronic commu-tator might be advisable. For this reason a prototype electronic commutatorwas constructed.The Electronic Commutator. An electronic commutator was built which wouldswitch four lines. It consists of a matrix of multivibrators operating atmultiple frequencies and a 6AS6 gate tube in each line. In the presentcircuit two multivibrators are used and their outputs are combined in fourways to produce gate pulses to operate the 6AS6 tubes. To switch n lines,the number of tubes which would be required (exclusive of power supplies) isn + 2 log 2n .The circuit operates satisfactorily at frequencies from 1000 to10,000 cycles per second with an adjacent channel rejection factor ofabout 50. This would probably be sufficiently good for the applicationproposed.-69-C -YILllrC--------- - _ ~__ -__._I__~L-~-- I(~CILIV 11III~-- ~ i.As an alternative to the electronic commutator which has been de-scribed on the preceding page, a cyclophon tube might be used. These tubeshave 25 separate plates and can be used to commutate 25 lines. Such atube has been received from Federal Telecommunications Laboratories andit is being tested for our applications at the present time.IX. C. PROTON-VELOCITY METERStaff: L. D. SmullinP. LallyWork has begun on a microwave proton-velocity meter to be used onthe one-million volt Van de Graaff generator now under construction in theLaboratory for Nuclear Science and Engineering.The proposed method uses the H2 beam that is separated from theproton beam magnetically, as the beam in a "klystron amplifier". That is,the beam is passed through a bunching cavity, driven at 3000 Mc/sec by anexternal stabilized oscillator and amplifier. The velocity-modulated beamthen flows through a drift tube about 2 meters long to a catcher cavity,where it induces 3000-Mc/sec oscillations. The phase of this inducedcurrent is compared with that of the exciting current in a conventionalcircuit. Changes in beam velocity will be observed as changes in phase.It is intended to use the phase information to control an electron currentflowing up the Van de Graaff machine to regulate the voltage. Responsetime of 10 to 100 lisec, and regulation to one part in 104 are hoped forwith this system. Figure IX-7 shows a block diagram of the complete system.BUNCHER CATCHERFig. IX-7. Block diagram of proton-velocity meter.-70-r --I .-~- ---.- )*II.^-------^~l---P 31P LI~ L ----^~~~- n-nllI- r -The stabilized klystron and amplifier are well under way and work is begin-ning on the "synchronous local oscillator" and on power supplies for thei-f strips in the phase detector circuit.With about 30 watts of driving power at the bunching, it is hopedto get an output power of about 3x10 8 watts/(Ia)2 of beam current. Thisshould make it possible to measure phase to within about ± 5 degrees withwide-band receivers (2-5 Mc/sec wide). If slower response times arepermissible, greater receiver sensitivity and accuracy can be obtained.IX. D. ELECTRONIC POTENTIAL MAPPINGStaff: Dr. S. GoldmanC. K. ChienW. E. VivianThe new scanning pickup tube using electrostatic focus and deflec-tion, which was specially constructed for small spot size, was tested andfound to have a very small spot size as expected. There was considerablyless interaction between pickup grids than was found with the earliermagnetic-type tube, and it appeared that this tube would be more suitablefor use in every way. Unfortunately there was an internal open lead toone of the pickup grids, so that the tube will have to be repaired beforeit is satisfactory for displaying potential maps.An alternative switching system to the cathode-ray tube types has alsobeen developed using gate circuits. This has been tried and appears tobe superior to the use of an electron-beam switching tube.During the past quarter, actual displays of the potential distri-bution on the surface of the left front of the chest have been made. Thesetests showed clearly which areas of the chest were active during the differentparts of the heart-beat cycle and showed marked differences in the heartactivity of a subject after exercise and the same subject during prolongedrelaxation.A detailed account of the circuits and equipment and of the tech-nique and results will be given in the next progress report.E. PHYSICAL LIMITATIONS OF R-F RADIATING SYSTEMSStaff: Professor L. J. ChuDr. M. V. CerrilloDr. M. LoewenthalA report on the results obtained thus far is in preparation.-71-IX. F. MATHEMATICAL PROBLEMSStaff: R. M. RedhefferFurther work has been done on the problem of separating Laplace'sequation. The case in which the solution has the form R(u,v,w)X(u)Y(v)Z(w)has been solved completely. It appears that there are 26 different co-ordi-nate systems, though most of them are merely inversions of the well-knowncases in which R=l. Investigation of the case in which the solutions havethe form S(u,v)R(u,v,w)Z(w) has been nearly completed. Results willbe reported in the near future.G. THE SCATTERING OF ELECTROMAGNETIC WAVES. MATRIX METHODSStaff: Professor H. MuellerN. G. ParkePrior Work in Field. At the inception of this research there were two alge-braic methods which had been developed to attack the general problem of scat-tering of radiation by matter. One of the algebras, due to Jones, had as itselements the electric vectors. The other algebra, due to Mueller, had asits elements the Stokest vectors, (I,M,C,S) where I,M,C,S are four intensitiemeasured with the aid of a polarizer and wave plate. There also existedWiener's generalized harmonic analysis which provides the general techniquefor handling the statistical aspects of the radiation.Generalized Optical Algebra, We have succeeded in constructing, on the basisof modern algebraic techniques, a general wave algebra in which Jones' alge-bra and Wiener's generalized harmonic analysis play an essential role andwhich leads to a generalization of Mueller's algebra which is capable ofhandling, from a uniform standpoint, the problem of scattering from crystals,liquids, and electrolytes. We have a statistical description of the Stokes'vector which gives a new picture of line broadening and a new insight intothe phenomenon of partial polarization.The algebra uses the tensor notation with the added generaliza-tion that a continous variable, here time, is used as an integration index.This leads to a generalization of the summation convention to include the"convolution" and Wiener's "correlation" processes. To simplify the manipula-tion, we write 0f(t-t )g(t o ) for f(t-t )g(t )dt (convolution),and Tf(t+t o )g*(t o )  for Lim f(t+to)g*(t o )dt o  (correlation).T-oo-T-72-~.,.-I~~-- -.. -~- ---YIIYI ~-1U--- -~------.~II. ~- -111-1-In this notation, the repetition of to underlined once means "convolution"and the repetition of to underlined twice means "correlation". The summa-tion convention on subscripts and superscripts is the familiar one. Thenet result is a smooth-working algebra into which one can readily trans-late the physical problem and which then allows rapid and meaningful manip-ulation to obtain the desired solution. This technique was to generalizeMueller's matrices and arrive at the general law for their additionM = N Mawhere the form of N depends upon the statistical interrelation of theindividual scattering centers, Maa is the Mueller matrix of scatteringcenter a, and M is the "relative" Mueller matrix for scattering centersa and p.We believe that the applicability of this algebra is broaderthan the original problem field it was designed to solve. It appears thatit can be extended to handle microwave problems involving radiation definedby random functions. The basic elements of the algebra would, in this case,be the characteristic sets of functions for the guides and cavities involved.-73-

Miscellaneous Problems

IX. MISCELLANEOUS PROBLEMSA. ELECTRONIC DIFFERENTIAL ANALYZERStaff: Professor H. WallmanA. B. MacneeDuring the past quarter the electronic differential analyzer hasbeen expanded to permit the solution of higher-order equations. Linearequations with constant coefficients of orders through four have beensolved; in addition, a number of linear equations with variable coefficientshave been investigated. A few examples are given below.Linear Differential Equations with Constant Coefficients. The computingunits now available consist of four integrators, two adders, and one multi-plier or function generator. No particular difficulties have been encoun-tered in the solution of linear, constant-coefficient differential equationsof order through four. The solution of a typical fourth-order differentialequation whose characteristic equation has two pairs of complex conjugateroots is shown in Fig. IX-1. In this example one pair of roots has negativereal parts and one pair has positive. Thus one term in the solution shownis increasing with time while the other damps out with time.Fig. IX-1. Solution of a fourth-order differentialequation with constant coefficients.Linear Differential Equations with Variable Coefficients. A number of linearequations with variable coefficients of'the form+ F(t)y = 0 (1)dt2-80-have been solved. A block diagram of the differential analyzer setupfor the solution of this equation is shown in Fig. IX-2.-yFig. IX-2. Block diagram of electronic differential analyzer setupfor solution of the equation d2y/dt2 + F(t)y = 0A solution of Eq. (1) for the case F(t) = t2 is shown in part(a) of Fig. IX-3. For this equation the analytic solution happens to beknown and is 1 ( t 2y = t2 J 2)a graph of the analytically obtained solution is given in partfig TX-3~(a)(b)(b) ofCALCULATED SOLUTION OF d2 - 2yI 2 Yy -f J -)4V5 VFig. X1-3. Plot of y versus t for the equation *Y + t2y = 0, (a) asobserved on the differential analyzer and (b) as calcu-lated from y = r J 1 /4(t/2).-81-r% -/\F(t) = w2 (l + c sin wmt),0 (2)Eq. (1) becomes the well-known Mathieu equation, particular solutions ofwhich are the Mathieu functions. Figure IX-4 shows plots of a typicalMathieu function versus time as (a) observed on the cathode-ray tube,and(b) photographed from the Jahnke and Emde Tables.(a)(b)Fig. IX-4. Observed and calculated solutions of the Mathieuequation, y + w2 (1 + c sin Wmt) = 0.B. AN IMPEDANCE FUNCTION ANALYZERStaff: Professor H. WallmanR. E. ScottThis project is concerned with the design, construction, andtesting of a machine for the analysis of impedance functions in the com-plex frequency domain. The device will consist of a conducting sheet ofTeledeltos paper to represent the complex plane, a set of probes carry-ing plus and minus direct currents to represent the poles and zeros, anda means of displaying on a cathode-ray tube the voltage along the imagin-ary axis of the plane. This voltage can be made to represent either-82-(a) the logarithm of the amplitude of the impedance function, or(b) the phase function of the network.The machine can be used in several ways:(1) It can be used to synthesize a desired curve of the loga-rithm of the magnitude of the impedance vs. frequency. To accomplishthis the desired curve is plotted on the face of the cathode-ray tube andthe poles and zeros are moved around in the plane until the desired curveis obtained within the required degree of approximation.(2) In these transient-synthesis cases in which the desiredphase-vs.-frequency curve as well as the logarithm of the amplitude-vs.-frequency curve are known,the whole factored complex form of the systemfunction can be synthesized.(3) The machine can be used to analyze a given transfer ordriving point impedance if the poles and zeros are known. The magnitudeand the phase-vs.-frequency curves can then be plotted directly by themachine.(4) The machine introduces a new and helpful method of think-ing about networks in general.Theory. The analogy between the potential in a conducting plane and thelogarithm of the magnitude of an impedance function was discussed in thelast progress report. For this analogy to be exact, it is necessary tohave an infinite plane. It is possible to make corrections for the finitesize of the plane but a more useful procedure is based on the propertiesof conformal mapping. Because of symmetry, it is necessary to use only ahalf plane. This half plane can be mapped conformally upon a.circularregion as in the Smith Chart, or upon a rectangular strip by means of alogarithmic transformation. In the latter case the strip remains infinitebut the log-frequency scale on it corresponds closely to the frequencyspectrum, and this correspondence is useful in a practical problem.-Determination of the phase-vs.-frequency curve depends on thefact that the phase and the logarithm of the amplitude of an impedancefunction are conjugate potential functions. The rate of change of phasewith respect to frequency is therefore equal to the rate of change oflogarithm of the amplitude with respect to the real variable. This rateof change is approximately obtained from two parallel sets of probesalong the imaginary axis. A simple summation then gives the phase.Experiment. Three models of the device have been built to date. Thefirst two have been described in the last progress report. The third isan improved version of the first two. It consists of a circular plane of-83-~- -~~--C -crl - --se~---------~~11111I~---L-PI--~-X-- _ 1-~1i.1.il*- -.- __conducting Teledeltos paper of 16-inch diameter. One hundred pick-upprobes are spaced uniformly along the imaginary axis. The voltages at,-these probes are fed to a mechanical commutator, and this procedureallows each voltage to be displayed in turn upon the face of a cathode-ray tube. The sequence of one hundred voltages is displayed in approxi-mately one twenty-fifth of a second. With a linear sweep this displayssatisfactory plots of the logarithm of the amplitude of the impedance vs.the frequency. No attempt has been made as yet to remove the restric-tion on the finite size of the plane or to obtain the phase-vs.-frequencycurves.C. PROTON-VELOCITY METERStaff: L. D. SmullinP. LallySince the last progress report the stabilized klystron and Klys-tron amplifier have been completed. The 30-Mc/sec, i-f amplifier with abalanced waveguide mixer has been tested, and a reaction ratio of about25 db between a signal entering the local-oscillator arm and one enteringthe signal-arm has been obtained. Since the device is to operate only onefrequency, it is hoped that a stable rejection ratio of 30 to 40 db can beachieved.Currently, the major effort is being placed on the developmentof a diode (silicon crystal) modulator to produce either phase- or ampli-tude-modulation, to be used in the comparison arm as the "synchronouslocal oscillator".A crystal-controlled microwave frequency calibrator is beingassembled as a piece of test equipment. It consists of a 2C51 50-Mc/seccrystal oscillator-tripler putting 150 Mc/sec into a 6AK5 buffer amplifier,which in turn drives a silicon crystal. The harmonic output of the cry-stal is of the order of 10- 7 watts at 300 Mc/sec, which is strong enoughto be used for calibration.D. ELECTRONIC POTENTIAL MAPPINGStaff: Dr. S. GoldmanC. K. ChienW. E. VivianThe gate circuit commutator mentioned in the last progressreport has been working very well and is now being used exclusively.Studies are being made of the area distribution of potentialacross the surface of the chest for interpretation of the action of thehuman heart. The frame frequency (frequency of scanning a complete-84--rmx~- -L~ L-~--~ C~-r .-11-~---41111-1- 1I~--^LII I..-Y..III 1--1~-1 ~~l_~_llipicture) which is usually used is around 250 per second. Motion picturesof the activity have been taken. Since these pictures show only 15frames per second some of the rapid activity is lost in photography. Asa sample, the accompanying photographs, (see Fig IX-5) show four consecu-tive frames taken from the reel around the time of the QRS interval dur-ing the heart beat. The lower right-hand quadrant of the pictures cor-responds to the location of the left ventricle. The difference in theintensity-area distribution in the region in successive frames, picturesthe activity of the right ventricle during the QRS interval. The inter-pretation of the oscilloscope patterns has just begun.Construction of equipment for the display of skull potentialpatterns is also in progress.A comprehensive report on the circuits used is in progress andshould be available at the time of the next quarterly report.Fig. IX-5. Frames from motion picture reel of the heart beat.E. PHYSICAL LIMITATIONS OF R-F RADIATING SYSTEMSStaff: Professor L. J. ChuDr. M. V. CerrilloDr. M. LoewenthalA report entitled "Physical Limitationsof OmnidirectionalAntennas", RLE Technical Report No. 64, has been published. This reporthas also been submitted to the Journal of Applied Physics to be consideredfor publication. A second report on the physical limitations of pencil-beam antennas is in preparation.F. MATHEMATICAL PROBLEMSStaff: R. M. RedhefferAs noted in the last progress report, the co-ordinates' forwhich Laplace's equation separates when the solution has the form RXYZ, Rbeing fixed, are essentially the well-known cases or their inversions,and the wave equation so separates only in the standard cases giving R = 1.We have now found the condition for partial separation, the solu-tion with co-ordinates (u,v,w) being R(u,v,w)S(u,v)Z(w), for both the wave-85-- 11 7I I I _U N =I Nand Laplace equations, and have shown that co-ordinates for partial orcomplete separation of the Laplace equation have the group property underinversion.We have investigated in detail the difference between separationof the equation and of the solution, which questions appear not to beclearly treated in the literature. Separation of the equation in one stepis proved essentially equivalent to existence of a two-parameter familyRXabYaZb plus aX/aa = aX/ab; equivalent to RXbYab Zb; and the general caseRXabYabZ ab for which the equation itself does not separate, leads tostill other conditions. In each case we have the necessary and sufficientconditions on R and the linear element of the space, which conditions showhow to get R by inspection.This work was done in collaboration with N. Levinson, and a jointpaper by Levinson, Bogert,and Redheffer has been submitted to the Quarterlyof Applied Mathematics.G. SCATTERING OF ELECTROMAGNETIC WAVES. MATRIX METHODSStaff: Professor H. MuellerDr. N. G. ParkeThis research, last described in the Quarterly Progress Report,April 15, 1948, has been completed. A summary of this work will appearas RLE Technical Report No. 70, "Matrix Algebra of Electromagnetic Waves",dealing with the mathematical aspects of an algebraic technique developedduring optical research, but which appears to be generally applicable toany class of linear systems for which there exists a transfer function.The algebra incorporates Wiener's generalized harmonic analysis in anatural manner and thus allows a uniform treatment of the statisticalaspects of waves and of the instruments through which they pass. In par-ticular it furnishes a basis for answering questions of coherence anddepolarization, i.e., the statistical aspects of superposition.H. A D-C AMPLIFIERStaff: R. ParshadA low-drift d-c anplifier has been built using two separateB-supply voltages and two triodes or pentodes. Figure IX-6 shows theconnections for the pentode case. An improvement of greater than 50:1in stabilit y can be realized by this connection as compared with con-ventional single-supply circuits. The voltage gain obtained was 2,200.-86--- -~---~~- l-~--lrr~--~- --sl LpllX I ~-I*YY---~~--- ~~~ - -- ~~~ aBIB ,B, POWER SUPPLIESVOLTAGE GAIN 2200Fig. IX-6. High-gain d-c amplifier.-87-T_L~L~n~~alXI~~ XQL~ IXI_--~LII~L.Bi

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Miscellaneous Problems

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