The waveform of the load current has been analyzed theoretically on the single-phase series R-C circuit controlled by the back to back SCR pair. And then, the fluctuation of the barmonics calculated by the use of the electronic digital computer, has been discussed as for the firing angles of SCRs and the phase angle of load

Himei, Toyoji

Inoue, Jyunichi

Kondo, Tadao

Nakanishi, Senichiro

Okayama University Digital Information Repository

Analysis of Waveform of the R-C Load-Current Controlled by the back to back SCR Pair

Okayama University Scientific Achievement Repository

MEMOIRS OF THE SCHOOL OF ENGINEERING,OKAYAMA UNIVERSITY, Vol. 3, No. I, SAPTEMBER, 1968Analysis of Waveform of the R-C Load-CurrentControlled by the back to back SCR Pair *Toyoji HIMEl, Jyunichi INOUEDepartment of Electrical EngineeringandTadao KONDO, Senichiro NAKANISHITsuyama Technical College(Received March 15, 1968)The waveform of the load current has been analyzed theoretically on thesingle-phase series R-C circuit controlled by the back to back SCR pair. Andthen, the fluctuation of the l:armonics calculated by the use of the electronicdigital computer, has been discussed as for the firing angles of SCRs and thephase angle of load.Fig. I. Single-phase series R-C load circuit controlledby the back-to-back SCR pair.well known, switch on by a gate signal whenthe anode-cathode voltage is positive. But,when the anode-cathode voltage is negative, itdoes not even if a gate signal is provided. Andit is turned off either by reducing the currentto some value less than the holding current, orby applying the reverse anode-cathode voltage.I t is noted that these are the different pointsfrom conventional electro-magnetic switches.To illustrate the behavior of this circuit, thewtRe= E sinwtI mJ-I\:~ --,-C________-'IL.--.J+Fig. 2. Theoretical waveforms on series R-C load.\"\ /_\~- ---- '. -1~----\ '\ i I, Iy ,\ . II :/" ! ,{I,\\, '''-.::/-r 1"--_/ i* Presented at the Annual Meetingof 1. E. E. of Japan, Nov. (966)120, ApI. (1967) 777.§ 2. Circuit and its BehaviorFig. I shows a single-phase cir-cuit with series R-C load whichis controlled by adjusting thefiring angles of SCRs connectedback to back. A SUR does, as§ 1. IntroductionA Silicon Controlled Rectfier (SCR) is oneof the results in recent development of thesemiconductor technology. The excellent fea-tures have caused to enlarge its applicationfield. Accordingly, it has been required tocontrol various loads, that is, not only induc-tive but also capacitive. The control circuitswith SCRs involve many troublesome prob-lems to investigate. Among them, one of thecommon problems is on the distortion of loadcurrent waveform, that is, the higher harmon-ics with unnegligible power.The analysis of the circuit controlled by theback to back SCR pair has been alreadyreported on the series R-L load. 1) However,on the series R-C load, it has not yet beenseen, although the higher harmonics havemore influential effect.In this paper, the harmonic content vs. firingangles and the current ratio vs. firing anglesare theoretically calculated and shown bymeans of the normalized formu-las.151152 T. HIMEl, J. INOUE, T. KONDO and S. NAKANISHI (Vol. 3,waveforms of the source voltage e, the loadcurrent i and the capacitor voltage l'c aretraced out in Fig. 2. It shows that in spite ofthe negative source voltage, owing to the ex-istence of capacitor voltage, the voltage acrossthe SCR is possible to be positive, and there-fore the conduction of the SCR is also possible.§ 3. Analysis of CircuitIn order to simplify the analysis of Fig. I, itis assumed thatI) the forward voltage drop of SCRs isnegligible,2) the leakage current of SCRs is negligibe, /'/' 3) the turn-on and turn-off time of SCRs isnegligible,4) the source impedance is negligible, and5) the waveform of the supply voltage issinusoidal.Under these assumptions, the circuit equa-tion of the current i, in the conducting periodof the SCR, is as follows.Ri-l-+~idt=Emsin«(vt+o) (1)where 0 is the firing angle.In steady state, the solution of Eq. (1) isgiven byi = -i'" [{sin «(t)to+ 0) -sin ¢ cos (0 -I- ¢)}ewttan¢ +cos',b sin «(vt+ ¢ + J)J (0 ~t ~to) (2)where to and ¢ are defined by the equations.{sin «(t)to+ 0) -sin ¢ cos (0 + ¢)}e wtotan¢ -I-COS ¢sin «(t)to-l- ¢> + 0) = 0 (3)J J I. --300 - 30 60 90 1 0 150 CA.)"tFig. 3. Waveforms of the calculated an:l measuredon series R-C load.-1 1 (4)d.=tan -~ \, (t)GRThe calculated by using Eq. (2) and theobserved waveforms are shown in Fig. 3,where values of R, C and ¢ are as follows.R=63.6(0,) G=24(aF) ¢ =60°It is found that the former agrees with thelatter very closely except the small deviationin the vicinity of the peak point.§ 5. ConclusionThe waveform of the load current has beenanalyzed theoretically on the single-phaseseries R-C circuit controlled by the back toback SCR pair. The fluctuation of harmonicsin the load current controlled by adjustingfiring angles of SCRs has been traced out,and it has been shown clearly how the harmo.nic content increases as for the firing angleand the phase angle of load.Current Ratio = ? (5)Where In denotes the magnitudes of the steadystate rms current harmonics and I means theresultant rms current.Fig. 5 shows the harmonic contents versusfiring angles curves with phase angle as para-meter, where the harmonic content is design-ated byHarmonic Content = In (6)1From the inspection into these figures, ingeneral the harmonic contents are inclined togro w larger as the firing angle increases, andalso it is valid with regard to absolute valuesof ¢. The current ratio to the fundamentalvalue of the load current will decrease as firingangles increase. But, the current ratio to thehigher harmonics will not behave itself uni.formly as for firing angles. For example, inFig.4(a), the current ratio of the 3rd harmo-nics is larger than that of the 5th harmonics at0=90° but is smaller at 0= 120°.-- The measured-- - -- The calculatedf-- 3-ZLL!cr:nc~~'"J§ 4. HarmonicsAs above mentiond, the measured and thecalculated are so close that the higher harmo-nics of waveforms can be calculated from Eq.(2) theoretically by means of Fourier Seriesgiven in Appendix. The resultants calculatedby the aid of the electronic digital computerare plotted in Figs. 4 and 5. In Fig. 4, thecurrent ratio is shown as functions of firingangle, with phase angle ¢ as parameter. Thecurrent ratio is defined as Eq. (5).1968) Series RC Circuit Controlled by SCRs 153j180N~ I lo 3A 5o 7• 9o 3A 5o "7• 9N)( IFIRING ANGLE (DEGREE8(d) ¢ =80 0FIRING ANGLE (DEGREES)(b) ~=60°~oo L-I--=-'cl _---.J.I-,--.l------L _---Jo 30 60 90 120 150 180O~L-__~.LI_-L LL _"o 30 GO 90120150~II.Ot--lI1.0-llJCr:0::=:JUoI--qCr:of -qCr:'"\IoN>< Io 3'" 5o 7• 9o C)'" 5[: 7• 9150FIR I NG A NGLE (DEGREES)(a) 9>=30 0I t30 60FI RI NG ANGLE(DEGREES)(e) ~=70°1.0o1--qCr: r~ O.5~~ Ia ~Fig. 4. Current ratio vs. firing angle.N ~• 310 5-->< \0"-"- 7..:: I'"c "-a 9 cI-- I--Z ZW wl- I--Z Z0 0U 5 UsU UZ Z ~0 0~ ~0:: 0::q «I :r:00I J30 150 180 00FIRI NG A NGLE (DEGREES)(a) 16=30°N• 3>< 5" 7o 9I I30 60 150 180F I RI NG ANGLE (DEGREES)(b) ~=60°(Vol. 3,L_ I150 180. FIRING ANGLE (O~GREE~(d) 9=80°L-_-LI_----l-I 1- -- j:JJ 60 90 i '20I-10---'-.c~l- IZ ~WI-Z >-,o ;u5t"-ufz0::2:0::<1Iat0T. HIMEl, J. INOUE, T. KONDO and S. NAKANISHIN• 3)( 5t:. 7o 9L-~l-_:!:-----=:!:-----:--!-=-----;;L-J30 60 90 150 180FIRING ANGLE(DEGREE~(c) p=70°Fig. 5. Harmonic content vs. firing angle.154-lOt-"-cI-ZwI-Z0u 5!::!z0::2:0::<1,I00ReferenceAppendixFourier Series oj the Load CurrentThe load current in one cycle defined asfollows.1) W. SHEPHERD, "Steady-state analysis of the seriesresistance-inductance circuit controlled by silicon-controlled rectifiers," IEEE Trans. on Industryand General Application, Vol.IGA-l, pp.259-265,July/August 1965.(9)(8)(11 )(10)1 ):".an = - z (wt) cosnwtdwtIT 01 ~~"bn = --;;;:- i (wt) sinnwtdwt" 0ao 1 ~~".- -=. - z (wt) dwt2 2rr 0Current i has zero time average value so aois zero. For fundamental frequency, the co-efficients a, b are found to beA= -~m{sin(X-f-j)-sin(PcOS(;+(f»}B= -~-'" cos (PThe Fourier for i has the general formi (wt) = -~"- + El(a/sos nwt + b"sin nwt)where(7)+Bsin(wt+ P+8) I"" "+'"0, "wherex = wto(13)(12)2 [{ '( - tanep 1.) tan¢} { 1 (aI1=- A e-"',an" 1 0 cosx+ 1 2epsmx +1 2 +Bcos(¢+O)+-2 xIT +tan-c/J +tan +tan ¢sin2x) . }]+-2- sm (c/>+O)2 [{ (-1 -tan"- .) I}bI = __ A e-""an¢ cosx+ 'I' Slnx +------1 IT 1+tan2¢ 1+tan2¢ 1+tan2¢{ I ( sin2x ) sin2x . }J+B 2 x--2- cos (c/J+o)+-2- sm (¢+o)Current i has only the odd Fourier harmonics. The higher harmonic coefficients are given by2 [ { e:-""an<!> ( ) tancjJ}aI = - A 2 2 - tan¢cosnx + nsinnx + 2 2n IT n + tan ¢ n + tan 4/1968) Series RC Circuit Controlled by SCRs 155B{-1 -1 2}]+ 2 1+ n cos {(l + n) X + ,j, + {j} + 1_ n cos {(l - n) X + ¢ + (J} + 1_ n~ cos(¢ +(J) (14)2 [ { e-"'tan<!> n }bI =~ A 2---2- (tan¢cosnx+nsinnx) + 2 0"IT n +tan (I) n +tan-¢+ ~ {I ~ n sin {(l- n) X + ¢ + f}} - 1~ n sin{(1 + n) x + jJ + fJ} - 1~n2 sin (1J + {j)}] (15)The magnitude of the harmonic component is given byCn = Vi a2 I +b2In n (16)

Analysis of Waveform of the R-C Load-Current Controlled by the back to back SCR Pair

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