The operation, mainly in a steady state, of a half controlled single-phase bridge circuit with a finite inductive load, has been treated analytically. Some operation features of particular bridge circuits are discussed

Fukumori, Kooichi

Himei, Toyoji

Inoue, Jyunichi

Okayama University Scientific Achievement Repository

MEMOIRS OF THE SCHOOL OF ENGINEERING, OKAYAMA UNIVERSITY, Vol. 1, No.1, MARCH, 1966Analysis of a Half-Controlled Single-Phase BridgeCircuit with Free-Wheeling Diode (1)Toyoji HIMEl, Jyunichi INOUE(Department of Electrical Engineering)andKooichi FUKUMORI(Kurashiki Technical High School)(Received November 20, 1965)The operation, mainly in a steady state, of a half-controlled single-phasebridge circuit with a finite inductive load, has been treated analytically. Someoperation features of particular bridge circuits are discussed.Fig. 1. SCR half-controlled single-phase bridgecircuit with series RL load.(1)(2)SCRRi it 1"Df XSRIn the interval during which D, is conductingOC:::::cpC:::::a,Ri, + X di, =0dffIn the interval during which D) is noncon-ducting aC:::::cpC:::::rr,E · R' X diomsm~= zo+ d~SCRrectifiers are negligible,2) the leakage currents of SCRs and rectifiersare negligible,3) the turn on and turn off time of SCRs isnegligible,4) in the interval when the free-wheelingdiode D, is conducting, SCRs are nonconduct-ing, and5) the wave form of the supply voltage issinusoidal.The operation of the circuit is divided intotwo modes, depending on whether the free-wheeling diode D, is conducting or not. Thenaccording to the modes, the circuit equations arederived as follows.§ 2. Analysis of the idealized circuit with-out source impedanceSThe circuit, shown in Fig. 1, consists of twoSCRs. two rectifiers, a-c supply, an inductiveload and a free·wheeling diode. In order tosimplify the analysis, it is assumed that1) the forward voltage drops of SCRs and§ 1. IntroductionRecently silicon-controlled rectifiers (SCRs)have become in use in m3ny application fields,for instance, in rectifier circuits. There is awide variety in rectifier circuits, among whicha half-controlled single-phase bridge circuit isparticularly familiar and has been analyzed inmany references.1,2However, certain features of the operationhave not yet been analyzed satisfactorily, sincethere are many important factors which have tobe taken into account, that is, inductive load,free-wheeling diode, source impedance, reverseemf, forward voltage drops of SCRs and recti-fiers, etc.Take, the case of inductive load, for instance.Hitherto, it has frequently been assumed thatits inductance is infinite, to simplify the analy-sis of the steady state, but it is inadequate forsome practical purposes. Each factor has itsown characteristic and has to be dealt withcarefully according to the case. In this paper,three cases are discussed, that is, the idealizedcircuit of a finite inductive load with and with-out source impedance, the idealized circuit offinite inductive load with reverse emf.78Analysis of Half-Controlled Bridge with F. W. D. 79Solving Eqs. 1 and 2 for il and taking bound-ary conditions into account, the followingequations are obtained, for O::;:cp;:Saand for a::;:cp::;:nREm l-e-x"I f = - RnR l-e- x"xsin O{SinO + sin(iJ-a) e-~("-")} (6)where lz. 10 and If are the average valuescorresponding to it, i o and if respectively.10 and If were calculated numerically fromEqs. 5 and 6, which are shown in Fig. 3 (a) and(b), as functions of firing angle, with RIX asparameter. The dots in the above figures repre-fill ; If' vw • 377R ; 37.1J'l.j( I 01:1. 75,' • .3 778.8S. o.n.E•.j If 1 vw '371R ; 37.7 J1X ; 00, 7S.11 I 37.7/8.8S, OJl.60 9Q I2QFIRING ANGLE C( (DE(iRfES)'b'30It ... ,.,~'.0.~.-----,,~o----;,,:---::9Q----:::--=~~iiiiio.-..l-_'81F/RIN4- ANGLE(OJsent measured points. The departure betweencalculation and measurement, about two orthree percentage, seems to be due to neglectingthe source impedance. Conversely the depar-ture could be reduced by taking the source im-pedance into account.Now, the transient response of the circuitwill be obtained by changing the phase of firingangles. It is assumed that XI R is considerablylarge, and that the curve, shown in Fig. 4, isvalid over the transient interval.If the instantaneous values of the load currentat phase angles nn and nn +a are denoted by IbnFig. 3. Calculated and measured average value versusfiring angle, with R/X as parameter. (a) Averagevalues of the current through bridge circuit. (b)Average values of the current through free-wheel.ing diode.(5)(4)360/80 270PHASe (DEGRE[S)90~- .... , Em; 14/ V'e lJJ ; 371~ Rj377 Jl)( ; '37.7' j'I~: ,/ ~~~o.:r~}~!~/~~~120'are also marked in the same figure. They seemto be in good agreement with calculated curve.Of course, it is obvious that the ripple compo-nent of the load current is smaller than that ofthe circuit in which the effect of the free-wheeling diode is not able to be expected.The average value of the current which flowsthrough the load and the free-wheeling diode,can be derived by solving Eqs. 3 and 4.Fig. 2. Calculated and measured load current versusphase angle, with firing angle as parameter.where a is the firing angle of SCRs. io is theinstantaneous value of the current which flowsthrough the bridge circuit. if is the instantane-ous value of the current which flows throughthe free-wheeling diode. i 2 is the instantaneousvalue of the load current and (1 = arctan (XIR).Instantaneous values of the load current werecalculated numerically from Eqs. 3 and 4. InFig. 2, the wave forms of the load currents arerepresented as functions of firing angle, withRIX as parameter. Some experimental points80 T. HIMEl, J. INOUE and K. FUKUMORI (Vol. 1,Ib3 I. I.I.I./.: I..I., I"1.3I"0 0( TC 7[tf/. 27[ 2Hc(Fig. 4. Theoretical wave form of load current in transient interval. Ian andhn are boundary values at the phase angles mT+a: and mT respectively.using a field winding of a syn-chronous motor as load. Theyare both straight lines and soverifies Eq.7. This is the casethat the inductance of the loadis much larger than that ofthe source impedance, namelythat the time constants forboth increasing and decreasingtransients, have about thesame value. But, it happensoccasionally that the differencebetween them can not be neg-lected.(9) Fig. 6. The circuit with inductive load containingreverse emf.§ 3. Analysis of the idealized circuit withinductive load containing the reverseemfThe circuit, shown in Fig. 6, is that the re-verse emf E b, which is symbolized by a battery,is inserted into the load impedance of Fig. l.The wave form of the load current assumedSRSC RThen, Eq. 9 is derived as the average valuei mean = ~R (1 + cosa)( )-1iC<p-a)- Ia-Ia1 e xand Ian respectively, the load current duringthe transient interval is expressed as follows.la-Ian Ib-Ibn e-~ (7)Ia- Ia(n-1) I b - Ibcn-11Eq. 7 shows that I an and Ibn are located on theexponential curve which have the time constantL/R. The average value of the dash,hatchedarea in Fig. 4, is expressed by Eq. 8 and isconstant for any half cycle.1,41= ~R(1+COSa)-Ia (8)to be that shown in Fig. 7. q>l and q>z are phaseangles at which the value of the supply voltageequals to the reverse emf, and also q>1<q>Z.The operation is roughly divided into twomodes; continuous load current and discontin-uous load current. Furthermore, the latter isdivided into two parts depending on whetherthe current through free-wheeling diode flowsor not. The circuit equations can easily be de-rived. Here, the resultant equations are given.First, the case when the load current is con,tinuous. For OS::q>;;Sa10 fOTIME (ll rio Seto"dJFig. 5. Transient'response.' (A) for "increasing",(B) for "decreasing". The field winding of asynchronous motor was used as load.E . . -1iC1C-a). _ _-",-- SlOO-slO(a-O)e xt,- vR2+ X2 _li"1-e xX e-~<P _Eb (10)RFig. 5 shows measured values of the case1966) Analysis of Half-Controlled Bridge with F. W. D. 81,,,,,,.............'}:o./90 180r:,RINq. A"q.I.E ~ (DE(i.REES)1.0From Eq.10,cos of Sin8e~c"-")-sin(a-o)}Rex" -1> E b (15)EmThe critical points were calculat-ed numerically from Eq.15 andshown in Fig. 8, as functions offiring angle, with R/X as para-meter. The range of the con-tinuous load currents is the do-main below curves correspond-ing to each parameter.Next, the critical point of fir-3.0, ,~1.0II)DO!------;"''::'·-----,:Il'''O·---......::!~/...=-FIRIN(j AN"L£ d. (OECiREESJFig. 9. Critical condition for current throughfree-wheeling diode being present, functions offiring angle. Parameter is R/X.Fig. 8. Critical condition for continuous load current,as functions of firing angle. Parameter is R/X.ing angles where the current through the free-wheeling diode is present, will be derived. Inthis case, the value of the load current at thephase angle a, is zero, and io at rp2, has to beconducting. Substituting above conditions intoEq.12 givesThe calculated values, from Eq. 16, are markedin Fig. 9, as functions of firing angle, with(12)~Fig. 7. Theoretical wave form of load current with inductive load con-taining reverse emf. (1) for "continuous", (2) and (3) for" discontin-uous" lp 1 and lp2 are phase angle at which supply voltage is equal to Eb.io= VR::X 2 { sin(lfi- O) -sin(a-{})e-~("'-")}-~b{1_e-~C"'-")} (13)where if in Eq. 12 has meaning only for if 20,if not so, if must be taken to be zero. In orderthat SCRs can be fired initially, the supply volt-age must be larger than the reverse emf. Thecondition for this is given by Eq. 14.sin a > :: (14)Furthermore, when the circuit is operating,the range, in which SCRs can be fired, is influ-enced by the m~des of load currents. To clarifythis, the critical condition of firing angles willbe derived.At first, the critical point of firing angles forthe continuous load current will be derived Inthis case, the current through the free-wheelingdiode will not vanish before the SCR is fired inthe succeeding half cycle. This means that theboundary value fa of Fig. 7 is not zero at rp=a.. Em {. ( )to vR2+ X 2 sm lfi-{}11sin{} e-x" -sin(a-{}) -~c",-..;} E b+ Re x --1-e-x" R(11)and second, the case when the load current isdiscontinuous. For O:S::rp:S::a. _[ Em {. . ( ) _R C,._ ..)}tf- VR2+ X2 smO-sm a-O e x+E b -!l.c,.- ..)] _!I", E bRex Xex- Rand for a::Srp::Srr82 T. HIMEl, J. INOUE and K. FUKUMORI (Vol. 1,RjX as parameter. Eq. 16 is satisfied in thedomain below each curve. In Fig. 10, the threecritical conditions are shown with RjX as para-meter. From inspection of Fig. 10, in order thatfrom that of Fig. 1 by the added source im-pedance. The operation of the circuit, expressedin Fig. 12, is divided into three modesFig. 12. Theoretical wave form of load current whensource impedance is not negligible. la, Ib and Ieare boundary values when phase angles are a, 0and -p, respectively. p is overlapping angle.1(-/1o A,,,/o 1<:.0-----:Jl7----e-..,---J,,-ffJ-==::=.,,~o--'~50,--------2l/,,,::-FIlUN(i AN4lE ,( (DE(fREE"S)Fig. 10. Critical conditions which determines theoperation of the circuit with inductive load con-taining reverse emf, as functions of firing angle.(1) and (2) refer to Figs. 8 and 9, respectively.(3) is supply voltage. (4) indicates the value ofEbl Em.\1\I \\ /r, \ r./\ /\ /\ I-,/ ......-',/ ,/ ~/ \//// \~\\\\\\\\the SCR will be fired initially, the phase of fir-ing angles must be between AD; at the phaseof firing angles between AB, the load currentil is continuous; between B-C, i l is discontinu-ous but the current through the free-wheelingdiode if flows; between CoD, if does not flow atall; and between D-E, the SCR is not fired atall.depending on values of io and If. From inspec-tion of Fig. 11 and 12, however, the circuitequation results as the following.For -{j::::; rp::::; aX dil + Ril= 0 (17)dqJand for a::::; rp::::; rr - {j(18)(19)(20)(21)Ie = Em sin [1rwhere {j is the overlapping angle.Solving Eqs. 17 and 18 gives, for - (j;;;;;'rp;;;;;'aX dil (R )'e= dqJ + +r tlThe starting point of the overlapping intervalis decided by considering the potential of thecircuit. Namely,i l = Emsin[1e-~(q>+f3)rand for a;;;;;'rp::::;rr-{jR§ 4. Analysis of the idealized circuit takingthe source impedance (resistive) intoaccount4When the source impedance, assumed to beresistive, is not negligible, it has to be con-sidered that the voltage drop takes place as theproduct of the resistance times the currentthrough the bridge circuit. As the instantaneousvalue of the supply voltage becomes small nearthe end of a half cycle, the overlapping angle,in which i o and if flow together, takes place.The circuit, shown in Fig 11, is only differentFig. 11. The circuit when source impedance (re-sistive) is not negligible.where (Jt = arctan Xj(R+r), I a and Ie are bound-ary values and expressed by Eqs. 22 and 23.1966) Analysis of Half-Controlled Bridge with F. W. D. 83(22)(23)ParameterThe overlapping angle is given by Eqs. 19 and 23sin p _ cosel sinC9+el)-sin(a-el)e-~("-"'-I3)-r- - R+r 1 _R+r" ~r_("'+I3)-e x exlQ"UJ 30 [:------'----,''-----...J<b:zc(""~ 20~ r--------..L.----,L...J'"~ 101--------'--_°Dt===$:Q~==~6(1===:::9LO===f::;:2iJ~==~;IS~O~~~/~80-::--FI RINCf ANIHE II- COEC,. REES)Fig. 13. Overlapping angles as functions of firing angle.is r/R.(24)From Eq. 24, the overlapping angles, shownin Fig. 13, were calculated numerically, as func-tions of firing angle, with r/ R as parameter.In the interval when i o flows, the time constantof the circuit is L/(R+r). On the other hand,in the interval when if flows, the time constantL/R. Then it may be said that the larger thesum of the firing angle and the overlappingangle, the longer the response time of the cir-cuit in transient interval. It is certain thattransient phenomena are of interest, but willnot be dealt with in this paper.§ 5. ConclusionThe operation, mainly in the steady state, ofa half-controlled single-phase bridge circuit witha finite inductive load, is treated analytically.The representation of the load current in tran-sient interval, agrees closely with the measure-ment. The critical conditions for the continuousload current and for the current through thefree-wheeling diode being present, were calcu-lated. This influences on the characteristics ofthe circuit and also restricts the phase of firingangles. It is very useful to take the sourceimpedance resistive, because the overlappingangles can be calculated easily.References1) R. Murray: Silicon Controlled Rectifier Desig-ners' Handbook, P.lO-18, 19 (Westinghouse Elect-ric Co., 1965)2) J. Schaefer: Rectifier Circuits; Theory andDesign, P.203 (John Wiley, 1965)3) T. Himei, J. Inoue and K. Fukumori; Annual Meet-ing of 1. E. E. of Japan, Kansai District, Nov. (1964) 134.4) T. Himei, J. Inoue and K. Fukumori; AnnualMeeting of 1. E. E. of Japan, Chugoku District,Nov. (1964) 13.

Analysis of a Half-Controlled Single-Phase Bridge Circuit with Free-Wheeling Diode (1)

Abstract

Similar works

Full text

Available Versions

Okayama University Scientific Achievement Repository