IEEE TRANSACIONS ON INDUSTRIAL ELECTRONICS AND CONTROL INSTRUMENTATION, VOL. IECI-28, NO. 2 MAY 1981

Transient Analysis of Chopper-Fed DC Series MotorG. K. DUBEY AND WILLIAM SHEPHERD, SENIOR MEMBER, IEEE

Abstract-The paper deals with the transient analysis of a de seriesmotor, controlled by a chopper with time-ratio control (TRC) orcurrent-limit (CLC) control with square-wave output voltage. Threemethods of transient analysis are derived for TRC control andcompared with regard to accuracy and usefulness. The methodsrequire much less computation time than is necessary for a step-by-step solution of the system differential equations, for different in-tervals of the chopper cycle. Two of the methods can be used forderiving transfer functions of the time-ratio chopper-controlled deseries motor for small-signal perturbations around a steady-statepoint.Two methods of transient analysis are described for CLC control.

One of these methods can be employed for deriving a transfer functionof the current-limit-controlled dc series motor for small-signalperturbations about a steady-state operating point.

LIST OF SYMBOLSRatio rms/average armature current in

steady-state, A.Instantaneous armature current, A.ia(t = 0) for conduction, freewheeling inter-

vals, A.Average value of armature current, A.Steady-state value of iav' A.Small increment in av' A.Instantaneous armature current during con-

duction, freewheehng intervals, A.Initial value of current in duty interval ofnth chopping cycle, A.

Particular value of i,,,, A.Effective or root mean square armature

current, A.Value of irms for steady-state operation, A.Laplace variable.Time, s.Instantaneous speed, rad/s.Average speed, rad/s.Average speed during steady-state operation,

rad/s.Initial value of speed in duty interval of nthchopping cycle, rad/s.

Small increment ofw, rad/s.Viscous friction coefficient, N'm/rad/s.Constants.Denominator equation in the variable s.

Manuscript received July 20, 1977; revised November 19, 1980.G. K. Dubey was with the Postgraduate School of Studies in Elec-

trical and Electronic Engineering, University of Bradford, Bradford,West Yorkshire BD7 IDP, England. He is now with the Indian In-stitute of Technology, Kanpur, India.

W. Shepherd is with the Postgraduate School of Studies in Elec-trical and Electronic Engineering, University of Bradford, Bradford,West Yorkshire BD7 11P, England.

EiKjKw

K,K

KoK0

Km£MafM4f0

RR1RlRTTaTM1Tm21 L

ATLVSV= Vay

a,#/60

Induced armature EMF, V.Polar moment of inertia of motor plus load,kgm2

Motor voltage constants, V/rad/s.Motor constant, N-m/A.Value of K at steady-state operating point,

V/rad/s.Td/WaV, N-mfrad/s.Motor constant in (16).Total inductance of armature circuit, H.Constant Kfiav, V's/A-rad.Value of Maf for steady-state operation,V-s/A-rad.

Total resistance of armature circuit, S.R +MafoWav.Composite resistances in (65), (66).Periodic time of chopper, s.Armature circuit time constant L/R, s.Mechanical time constant J/B, s.Time constant JR/(BR + K), s.Load torque, N'm.Value of TL in steady-state operation, Nm.Small increment of T1, N-m.Source (direct) voltage, V.Average voltage at motor terminals over achopping cycle, V.

Constants in the Appendix, s-Ratio Vav/V.Value of 6 for steady-state operation.

INTRODUCTIONT,HE THYRISTOR chopper is now widely used for the con-

trol of dc series motors. The output voltage of the choppermay be controlled either by using time-ratio control (TRC) orcurrent-limit control (CLC). In TRC, the on-off time ratio ofthe chopper is controlled. The chopper may then be operatedat a constant frequency or at a constant on (or off) time withvariable frequency. In CLC, the load current is controlledbetween specified maximum and minimum values. When theload current reaches its specified maximum value, the chopperdisconnects the supply from the load and reconnects it whenthe current reaches its specified minimum value.

The main problem in the analysis of the chopper-fed dcseries motor for steady-state and transient conditions arisesdue to the nonlinear relation between the armature-inducedvoltage and the armature current. Because of this, the differ-ential equations that describe the operation of the motor inthe different modes of a chopper cycle are nonlnear. Accuratesolution of these equations can only be obtained numericallyby the use of considerable computation time.

Steady-state analysis of a dc series motor fed by a chopper

0018-9421]/81/0500-0146$00.75 1981 IEEE

ja

ial1 ia2

iavJayo

idl (n)

to

lrms

irmsotw

WayWavo

WdI(fl)

Away,BcDcl,,-c22D(s)

146

DUBEY AND SHEPHERD: CHOPPER-FED DC SERIES MOTOR

with TRC and square-wave output voltage has been describedin [1]-[3]. This method has been extended to take intoaccount the effect of the commutation interval [4], [5] . Non-linearity of the field inductance has also been considered [6] .

More general analytical methods, applicable to choppers ofany configuration, are described in [7] -[10]. These methodsemploy numerical techniques and require large computationtimes. Steady-state analysis of a chopper-fed dc series motorwith CLC has been described in [1] and [11].

The present paper is concerned with the transient analysisof a dc series motor fed by a chopper with either TRC or CLC.The paper mostly describes new methods developed by theauthors but includes a review of the few methods previouslyreported in the literature.

II. TRANSIENT ANALYSIS OF THE CHOPPER-FEDDC SERIES MOTOR WITH TIME-RATIO

CONTROL (TRC)

A. Analytical Description of Chopper OperatingModesCertain initial assumptions are made that turn out to be

experimentally justified.1) Thyristors and diodes are ideal switches.2) The resistances and inductances of the motor armature

and field are constant.3) The chopper-output voltage is a square wave or can be

approximated by a square wave.4) For a series motor, the induced voltage is a function of

the armature current. Therefore, unlike a shunt motor, discon-tinuous conduction occurs in only a very narrow region of itsoperation. Conduction can therefore be assumed to be con-tinuous without any loss of accuracy.

5) Eddy-current loss is neglected.A chopper with square-wave output voltage normally oper-

ates in two modes; namely, the duty interval and the free-wheeling interval, during a chopping cycle. The performanceof a dc series motor is described by the following differentialequations, referring to Fig. 1:

Duty Interval (O < t (2 6T):

didL- =V-Rid-K(id)w. (1)

dt

Also from the dynamics of motor plus load

dwJ- = K(id)id-Bw-TL. (2)

dt

Freewheeling Interval (6 T5c t < T):

difL d = Kif-K(if)w (3)

dt

dwJ-= K(if)if-Bw-TL. (4)

dt

In (l)-(4), the terminology K(id) and K(if) denotes that Kis a function of the instantaneous value of the motor current,so that the equations represent a set of simultaneous, nonlineardifferential equations.

+ I

- T

chopperi-- I

firin9circult

filter

fieldwinding

fwdarmature

(a)

v

E=K(id8)l

(b) (c)Fig. 1. Representation of a chopper-controlled dc series motor. (a)

Basic schematic diagram. (b) Equivalent circuit of duty interval. (c)Equivalent circuit of freewheeling interval.

B. Transient Analysis

One approach to the calculation of the transient response isto solve (1)44) numerically for each interval of the choppingcycle by using the final conditions of the previous interval asthe initial condition for each successive chopping interval. Thisapproach has the advantage that it permits the calculation oftransient response exactly for the assumed model. The necessarycomputation time is, however, quite large, particularly forsystems with slow transient response.

Three new methods- which require much less computationtime are described. These methods are straightforward and twoof them can be used for deriving a small-signal transfer func-tion of the pulse-controlled dc series motor around a steady-,state point. This can then be usefully employed in dynamicresponse analysis of a close-loop system incorporating a pulse-controlled dc series motor.

Since the input to the motor consists of discrete pulses, thepulse-controlled dc motor is essentially a discrete data system.The basic approach of the methods described here is to obtainan equivalent continuous model of a pulse-controlled dc seriesmotor using certain mathematical approximations.

In the methods of transient analysis proposed, the motorconstant K is assumed to be a function of the average value iavof armature current rather than the instantaneous value as in(1 ){4)

K= f(iav). (5)

This approximation has been shown to be valid for steady-state analysis [1] -[6] .

During a given chopper cycle iav is fixed so that K is alsofixed. The value of K for a given iav is obtained from the mag-netization characteristic of the motor. Thus with this assump-tion

K(id) = K(if)= f(iav) = K.

147

(6)

IEEE TRANSACTIONS ON INDUSTRLAL ELECTRONICS AND CONTROL INSTRUMENTATION, VOL. IECI-28, NO. 2, MAY 1981

ing to the value of current under consideration and the origin.Thus

K Mafiavwhere

Mafo = Koio

In Fig. 2(c) the magnetization characteristic is approximatedby a straight line which passes through the point under consid-eration and has the same slope as that of magnetizationcharacteristic at that point. Thus

K=Co +Cliav. (10)

The approximations of Fig. 2(a)-(c) are termed Approxi-mations I, II, and III, respectively.

For most applications the chopping period is usually smallcompared with the mechanical time constant of the motor.This assumption is made throughout. Three separate methodsof analysis are now described, involving different approxi-mations of the instantaneous current and/or speed.

Method I: From (1) and (3) taking the averages for a givenchopping cycle

Vav = 6V= Riav + KWav(b)

approx IIIK- CO+ C. iav

C, -A II

S\,e CtX)

Ii{y

0Rsy ~~~I

// I L II ~~I L II

i 0,~ ~ '0

(C)) Oz

Fig. 2. Various approximations of magnetization curve for small in-crements around a point. (a) Approximation I. (b) ApproximationII. (c) Approximation III.

For any small increment Ai in the average value of the cur-

rent, the magnetization characteristic can be approximated bythe three approaches shown in Fig. 2 for two values of iav, i.e.,io and i02.

In Fig. 2(a) it is assumed that Ko remains constant for a

small increment of current. Thus

K = Ko. (7)In Fig. 2(b), the magnetization characteristic is approxi-

mated by a straight line passing through the point correspond-

or

6V-Kwaviav = (11)

The average voltage across the inductance is neglected. Thisis very small for small chopping periods (compared to themechanical time constant) and low inductance values.

Again, from (2) and (4)

dwavJ t Kiav-Bwav- TLdtav

(12)

The above equations, in fact, assume that the electrical timeconstant of the motor is negligible. Thus at any speed, thearmature current has the same value as when time motor is inthe steady state. Substituting for iav from (11) into (12) andrearranging gives

dWav +K2 +BR 6KV)

dt VR ) av- R -L) (13)

Equation (13) is a first-order nonlinear differential equationthat can be solved by piecewise-linear approximation.

Values of iav and wav at the beginning of the transient willbe known. A suitable iteration interval is chosen. The value ofK is assumed to be constant during this iteration interval and istaken to be equal to its value corresponding to values of iav atthe beginning of the interval, i.e., the approximation of Fig.2(a) is used. This makes (13) linear for this interval. The value

approx I, K= Ko

lIIlIIlIIlIIlII

K

(a)

(8)

(9)

COZ

co,

148

DUBEY AND SHEPHERD: CHOPPER-FED DC SERIES MOTOR

of way at the end of this interval is obtained from (13). Acorresponding value of iav at the end of the interval is thenobtained from (11).

If (1)-(4) are solved by the basic point-by-point method (toget the transient response), the numerical computations mustbe done for small increments of time in each interval (mode)of the chopping cycle, thus requiring a large number of com-putations for each cycle. In the approximate method previouslygiven, (13) can be solved by a piecewiselinear approximationby choosing the time of each iteration as large as ten choppingcycles without introducing appreciable error. The number ofcomputations and the computation time are then reduced by alarge amount.

One can similarly derive expressions using the approxima-tions of Fig. 2(b) and (c). These, however, give nonlineardifferential equations which cannot be solved by piecewise-linear approximation. They could be solved numerically usingthe Runga-Kutta or Predictor corrector methods. This how-ever, increases the computation time.

Transient responses for small-signal perturbations around asteady-state operating point, using Approximation I, can alsobe obtained from (13) as follows:

Let SO, Wavo, iavo, and TLO be the values at any steadystate. Let the incremental changes in these variables due to adisturbance be denoted by AS, Awav, Aiav, and ATL. Assuminga value ofK =f(zavo) = Ko = constant for small increments iniav from (13), then

d(Wavo + Awav) /_wRB)ditR

KoV=(i0 + A6) ]R (TL + ATL).R

Separating the transient part gives

(14)

ATIAS)

A Mavls)

Fig. 3. Small-signal perturbation block diagram of TRC chopper-feddc series motor, using Method I, Approximation I.

AWav(S) TM2/JATL (s) I +STm2

(18)

A block diagram of a chopper-fed dc series motor for smallperturbations is shown in Fig. 3. This is a first-order approxi-mation to the system which is, in fact, of second order.

Similarly, one can derive small-signal perturbation transferfunctions using Approximations II and III, using (1 1) and (12).

Method II: The following equations are derived in theAppendix:

Wdl (n + 1) =Cl 3idl(n) + Cl 4Wdl (n) + Cl 5

idl(n+l±) =Cl6idl(n)+Cl7Wdl(n)+Cl8(19)

(20)Wdl(n + 2) =Cl wdl(n +±) + C2owdl (n) + C21 (21)

idl(n + 2) = Cl 9idl (n + 1) + C20idl (n) + C22 * (22)

Since the chopping period is always much smaller than themechanical time constant of the niotor

Wdl(n +2)-Wdl(n + 1) ddwdl(n + 1)T dt

Wd(l±n 1)- Wdl (n) dwa1(n)

T

(23)

(24)dt

cd(Awav) 1Ko +RB\J +dt R /

Aw = K0 - -ALAav RDifferentiating (24) and rearranging the terms

dwdl(n + 1) d2Wdl(n) dwdl (n)cit, dt1=T- dt+dt d2 dt

Tm 7d(AWav) + Awav=Km As _ Tm 2 AT

where

Tm2 =K 2 +Rd and Km- 2V

16)K0 RB JR

A small-signal transfer function relating Away to AS can beobtained from (15) when ATL = 0

AWav(s) Km

/AS(s) 1+sTM2

Similarly, a transfer function relating Awav to ATL can beobtained from (15) by letting AS= 0

(15) From (23)-(25)

Wdl (n + 1) = T d ) + Wdl (n)dit

(26)

Wdl (n + 2) = T wd(n) 2T dtWdl(n)

(27)

Substituting from (26) and (27) into (21) and rearrangingthe terms gives

2d2wd(n) dWd1(n)T +(2-Cl)Tdt+ (1- Cl 9 C20o)Wdl1(n) C21 1-° (28)

or (25)

149

E TRANSACONS ON INDUSTRIAL ELECTRONICS AND CONTROL INSTRUMENTATION, VOL. ECI-28, NO. 2, MAY 1981

This leads to the differential equation

2(2 dwaT2 - + (2-Cj)Tddt2 dt

+ (1 -Cg-C20)wav-C21iO.Similarly, one can get

d2i diavT2 "'avT 2~+(2-Clg)T ddt2 dt+(1 -C19 -C20)iav C22= 0.

relating controlled variables w and iav with control variable6. Method III described in the following subsection removesthis drawback.

Method III: The approach of Method III is similar to thatof Method II, but uses different approximations. From (1)-(4)

(29) and (6) for the nth chopping period

(30)For a given value of 6, (29) and (30) are nonlinear second-

order differential equations. They may be solved by a piecewise-linear approximation as follows:

Values of iav and wav at the beginning of the transient willbe known. A suitable iteration interval is chosen. The value ofK is assumed to be constant during this iteration interval andits value is taken to be equal to its value corresponding tovalue of iav at the beginning of the interval(Approximation I).This makes (29) and (30) second-order linear differentialequations for this interval. They can, therefore, be solved ifthe initial conditions are known. Initial values of wav and iavwill always be known. Values of

dt t=O

diand av

dt t=

are calculated as follows: From (19) and (24)

dwd 1 (n)

dtCl3idl(n)+ (Cl4 -l )Wdl(n)+Cl 5

T

I; idl(n+I)T

di(n)1dl (n)

IFV (n+I)T Kdt--

L L T JnT T

(n+)T

-T i(h) dt .

I fwdl(n+1)T Wd(n) (n)

ITL (nl )T-J(n±UT dt .

JT nT

Therefore

id l (n 1) )-Idl (n)-

I

[v Kwav(n)Riav(n)] (34)T

Wdl(n + 1) Wdl(n) _-1 EVn-WVn LWdI(fl+)WdI(fl -_ [Kiav(nl) -Bwav(fl) - TL I'7 TI

dwav

dt t=o

since

Cl 3iavl + (Cl4l)Wavl + CI5(31)

T

didi(n) idl(n + l)-idl(n)

dt T

From (20)

did l (n) (Cl 6 - )idl (n)-Cl 7wdl (n) +Cl8dt T

(32)

(35)Because the chopping period is usually very small compared

with the mechanical time constant of the motor, fluctuationsin w and i are very small compared with their absolute valuesand, therefore

'dl (n + 1)-idl (n) diav(n)T dt

Wdl(n + 1)-Wdl(n) dWav(n)TI

T dt (36)

diav

dt t

(C16 )liavl -C1 7Wavl +C1 8

T

where 1av, and wavy are the values at the beginning of theiteration interval.

Method II leads to second-order differential equations for asecond-order system and therefore permits the calculation oftransient response with better accuracy than Method I. Thecomputation time is again much less than that of the basic step-by-step method though more than Method I. Method II, how-ever, has the important drawback that the control variable 6does not appear linearly in (29) and (30). Therefore, onecannot derive a transfer function (for small-signal transients)

Substituting the approximate relations (36) into (34) and (35)(33) gives the following general relations:

diav 1d= [V-KWav-Riav]dt L

dWav 1

v-I[Kiav-BWav-TL ].dt J

(37)

(38)

These nonlinear equations can be solved using any of thethree approximations given by (7), (8), and (10) and Fig. 2(a),(b), and (c).

Using Approximation I, i.e., assuming K to be constant for

J(n+I)Tw(n) dt

nT

1 K (n+l)T Bi(n)dt--

JL_ T JT

J (n+ 1 )Tw(n)dt

nT

J

150

DUBEY AND SHEPHERD: CHOPPER-FED DC SERIES MOTOR

(S)

Fig. 4. Small-signal perturbation block diagram of TCR chopper-feddc series motor, using Method III, Approximation III.

a small interval of time, one gets, from (37) and (38), byseparating the variables

d2iav (i T.\\diav±I. 8BV+KTLTa dt2 T(1 +JRa Z av=

(39)

Aiav(s) - KI(S)As(s) D(s)

and for AS = 0 from the same equations

AWav(s) KW(S)ATL(s) D(s)

Aiav(S) KA(S)ATL (s) D(s)

where

Kw = Ko V/JL K1 =K0/JL

KI(s) = (Js + B)V/JL and K (s) = (R + Ls)/JL

d2wav TaTa + 1±+ -

dt2 Tm

dwav 1Lay

Tm2

and

D(s) = s2 + (JR + BL)s/JL + (BR + Ko2)/JL.KS V-RTL

JR (40)

These are second-order nonlinear differential equations andcan be solved by piecewise-linear approximation as explainedabove.

Approximations II and III can also be employed to solve(37) and (38). Using Approximation II (i.e., substitution from(9) into (37) and (38)) gives

dia 1

t (L V -MafiavWavR-Rav) (41)dt L

dwav 1(Mafiav- Bwav TL)- (42)

These nonlinear differential equations can be solved numericallyusing the Runga-Kutta method or Predictor-Corrector meth-ods which both require more computation time than usingpiecewise-linear approximations. Equations of a similar natureare obtained when Approximation III is employed.

The methods of transient analysis for small perturbationsaround a steady-state operating point will now be presentedemploying all the three approximations.

a) Approximation I: The value ofK is assumed unchanged(equal to Ko, its value at the steady-state point) for small per-turbations. From (37) and (38) for small increments denotedby A

VA6(s) = (R + LS)Aiav(s) + KoAwav(s)

ATL(S) = -(Js + B)Awav(s) + KoA iav(S).

For small-size motors D(s) will have real roots. For medium-and large-size motors the roots can be complex, particularlywhen an additional external inductance is included to keep themotor current ripple within tolerable limits.

b) Approximation II: Equations (37) and (38) can bemore appropriately written as [16]

Riav +±L MafWaviav= V (51)dt

dWavJ- +8Bwav + TL =Majfrms

dt(52)

where irms= rms value of the armature current. Let the formfactor at any steady-state point be denoted by fo, then

1rm so

iavo

Now

irms2 (rmso +± Airms)2iav2 (iavo + Aiav)2

Z 2/1 + trms2

'rmso 1 + 21rm so (rm so 2

2 iav lavovavo

iavo

(53)

(43)

(44)

Equations (43) and (44) give the block diagram for thesystem shown in Fig. 4.

For ATL = 0, from (43) and (44),

Aw --) Kw_ay(S) K* (45)A6(s) D(s)

=f02. (54)

From (52) and (54)

J +BWav + TL =Maffo2iav2.dt

(55)

Substituting the values of the variables after small perturba-tions into (51) and (55), separating steady-state and time-

(46)

(47)

(48)

(49)

151

LKo

Ire

1EEE TRANSACTIONS ON INDUSTRLAL ELECTRONICS AND CONTROL INSTRUMENTATION, VOL. IECI-28, NO. 2, MAY 1981

ATL (s)

Fig. 5. Small-signal perturbation block diagram of TRC chopper-con-trolled dc series motor, using Method III, Approximation II.

varying terms, and neglecting the products and squares ofincremental time-varying terms gives

L dAiav4 R1AiL +R1~ai + MafoiavoAwav=VAdt a (56)

AE (s) I

Fig. 6. Small-signal perturbation block diagram of TRC chopper-con-trolled dc series motor, using Method III, Approximation III.

L?wavJ Wv + BAwav + ATL = R2 Aiavdt

where

(66)

dAWav Mff2I ±-BAwav-iafOfOlavOAiav+ATL =0 (57)dt

where Rl = R + Mafowavo- The following transfer functionsare obtained from these equations:

Awav(S) 2MafOfO2iavO VIJLAs (s) D(s)

AWav(s) (R 1 + Ls)/IJLATL (S) D(s)

Aiav(s) V(Js + B)/IJLAs (s) D(s)

Aiav(s) MafoiavolJLATL (S) D(s)

(58)

(59)

(60)

(61)

where

R1 =R+CiWavo R2 =CO + 2CI iav f0

The corresponding block diagram is shown in Fig. 6. The fol-lowing transfer functions are obtained from these equations:

Awav(s) R2 VIJLAA (s) D(S)

Awav(s) (R 1 + LS)/IJLATL (S) D(S)

A\iav(s) V(Js± B)/JLAll(s) D(s)

Alav(s) (CO + Ci iayv)/IL'\TL (S) D(s)

(67)

(68)

(69)

(70)

where

12+ +SBL BRI +(Co Cliavo)R2D(s)-=s± sIJL /JL_____ R1+2M 2f 2lav 2D(s)-=s2 + ( +R J) s +R 2afo to ia

(62)

A block diagram is shown in Fig. 5.Equations (58)461) are the same as those obtained by

Bhadra [17] using a different approach.c) Approximation III: The performance equations will

now be

dlaRiav + L ±+ Ci waviav + Cowav =l V (63)dt

dwavJ av +BWay + TL = Coiav + Cif02ay2* (64)

dt Bwv 64

Substituting values of the variables after small perturbationsinto (63) and (64), separating the steady-state and time-varyingterms, and neglecting the products and squares of incrementaltime-varying terms gives

AlavL ~+RiAjiav ±(Co±+Cl iayo),Awavy=V4l (65)dt

Method III has the advantage that it gives not only a second-order approximation for calculating transients under large per-turbations, but also permits the derivation of second-ordertransfer functions for small-signal perturbations.

C. Comparison of Transient Analysis MethodsA 220-V, 2.2-A, 0.375-kW dc series motor was chosen for

this study. Chopper period T was chosen to have the typicalvalue of 0.005 s. In order to examine the accuracy of themethods for transients due to large disturbances it was decidedto compare their results with those obtained by the step-by-step calculation. Some starting transients calculated by thesemethods for the motor with coupled generator are shown inFig. 7. In these calculations, it was assumed that the motorstarts only after its developed torque becomes greater than theload torque.

Since Method I is only a first-order approximation of asecond-order system, considerable error is introduced, partic-ularly in the current-time curves. The motor under study isonly a small motor and, therefore, its electrical time constantis comparatively small compared with a large-size motor. Theerror, therefore, would be much larger in the case of largersize motors. Methods II and III provide quite a close approxi-

152

AT, ls)

DUBEY AND SHEPHERD: CHOPPER-FED DC SERIES MOTOR

Poi n t by - poi nt

o Q O Method I

x x x Method II

* ** Method III2-4

method

8 1.2 1.6 2.0 2.4 0 0.4 0-8 1-2 1.6Time, secs-_ Time, secs.

Fig. 7. Starting transient of TRC chopper-controlled dc series motor.

mation both for -speed-time and current-time curves. MethodIII, however, has the advantage over Method II in that it canbe used to derive the transfer functions of the motor for smallperturbations around a steady-state point. All three methodsrequire much less computation time than the point-by-pointmethod. The computation time required by the three methodshas the following order: Method I minimum and Method IImaximum.

Small perturbation transient responses calculated by MethodI with Approximation I, and by Method III using Approxima-tions I, II, and III for input perturbations of A6 = 0.05 andload torque perturbation of 0.05 per unit on the above motor(excluding coupled generator) are shown in Figs. 8 and 9. AsMethod II with Approximation III is the most accurate itsresults are used to evaluate the accuracy of the other threemethods. The error in the case of the current-time curves islarge both for Method I with Approximation I and Method IIIwith Approximation I. Furthermore, Method III with Approxi-mation I shows an oscillatory response. Calculated results ofMethod III with Approximation III were compared with thoseobtained experimentally in Figs. 10 and 11 for a 2.5-kV motor.There was satisfactory agreement.

The small-signal perturbation transient responses obtainedby Bhadra [17] of a medium-size motor of 14 kW, by a first-order approximation (Method I) and Approximation II, byMethod III with Approximation II and by a step-by-stepnumerical computation are shown in Fig. 12. These responsesshow that only Method III (second-order approximation)should be employed with medium- and large-size motors.

III. TRANSIENT ANALYSIS OF THE CHOPPER-FEDDC SERIES MOTOR WITH CURRENT-LIMIT

CONTROL (CLC)Let the specified maximum and minimum current limits be

denoted by ia2 i'l . Then the duty interval will consist of theduration (not known) in which the current changes from ia1to 4a2 and the performance will be described by nonlineardifferential equations (1) and (2). The freewheeling intervalwill be of duration (not known) in which the current changes

006

0*04

0-02-

x x x

1/

_I

0 0.4 0 8time (sec.)

(a)

5

ciCL

._

012

0.08-

004

o

12 1 6

xXIII'I

'X ,

X A0 04 08 1.2t me (sec.)

(b)

Fig. 8. Transient response of TRC chopper-fed dc series motor for aninput perturbation. AS = 0.05, iavo = 1 pu, wavo = 0.3352 pu. (a)Variation of Awav. (b) Variation of Aiav.

Method I with Approximation Ix**: Method II with Approximation I-.--: Method III with Approximation II---: Method III with Approximation III.

from ia2 to i01 and the performance will be described by thenonlinear differential equations (3) and (4).

153

154 TRANSACTIONS ON INDUSTRLAL ELECTRONICS AND CONTROL INSTRUMENTATION, VOL. IECI-28, NO. 2, MAY 1981

-0 04 calculated

x -Kexperimental

(a)

time (sec.)

(a)

0 04 0 8 1 2time (sec.)

time (sec.)

(b)Fig. 10. Experimental and calculated (Method III, Approximation III)

transient response for small-signal perturbation in input voltage. (a)Cu-rrent I. (b) Speed w.

(b)Fig. 9. Transient response of TRC chopper-controlled dc series motor

for a load torque perturbation of 0.05 pu. iav = 1 pu, wavo = 0.3362pu. (a) Variation of Awav. (b) Variation of Aiav.

Method I, Approximation Ix**: Method II, Approximation I4.: Method III, Approximation II

Method III, Approximation III.

Method I: One approach to the calculation of transientresponse is to solve (1)44) numerically for each interval of thechopping cycle, using the final conditions of the previousinterval as the inital conditions for each successive choppinginterval. During each interval equations are solved numericallyas follows [19]:

From (1) and (2), for the duty interval

LdidV-Rid-K(i4,)w

K(id)id -Bw TLdw- dt. (72)

During a duty interval, the current varies between i01 andia2. This current variation is- divided into a suitable number ofequal incremental current intervals (current increments), say n,which fixes the value of did. Initial values of id and w for anyincremental-current interval are known from the calculationsof the previous incremental-current interval. A value of K(id) isassumed to be constant for each incremental-current interval,equal to its value at initial value id of the interval.

For any incremental-current interval, dt is calculated from

2 0 oo=e_6 g -

calculate

-x-X-X- experimeW 1(2a

0.8

0-4

0 10 20time (sec.)(a)

time (sec.)O . * .

edintal

\10 20

= _ i

-4

-A3 -12

-16

(b)

Fig. 11. Experimental and calculated (Method III, Approximation III)transient response for small perturbation in load torque. (a) CurrentAI. (b) Speed Aw.

(71), then increment dw is obtained from (72). Values of w,id, and t at the end of this incremental-current interval arenow obtained and these then form the initial conditions forthe next incremental-current interval.

The same approach is used for the solution of the corre-sponding equations valid for the freewheeling interval.

Average values of speed and current for any chopping cyclemay be simultaneously calculated from the following equa-

._

154

DUBEY AND SHEPHERD: CHOPPER-FED DC SERIES MOTOR

'.A\I .-\Ss.

I-y ---

02 04 06 08 0.0 0-2 0-4 0-6 0-8 1.0

t im e (sec.)

(a)

0 0-2 04time (se c.)

lav [Jo (a+ a2ra t dt

ST

Equation (75) shows that iav is nearly constant at all speedsif the assumption of linear variation of current is true. Asshown later, iav in fact turns out to be nearly constant at allspeeds.

The value of K corresponding to iav =(a 1 + ia2 )/2 can beobtained from the magnetization characteristic of the motor.Since, from (75), iav is constant and independent of speed,Kis also constant and independent of speed.

Average torque Tav = Kiav = constant.

From (2), (4), (5), and (76)

(b)Fig. 12. Calculated small-signal transient response of a medium-size

series motor. (a) After 5-percent perturbation of input voltage. (b)After 5-percent perturbation of load torque.

- : Method III, Approximation II

-.-.: Method I, Approximation II

---: step-by-step numerical integration.

tions:

2n 2n

iav= 2 i,dDt 2:dtr=l1 r=l1

2n 2n

Wav= wdt zfdt.r=1 r= 1

This approach has the advantage that it perniits acci

calculation of the transient response for the assumed m(However, the calculations require to be done numericstep-by-step in each interval of chopping cycle, and sequentbetween different intervals of chopping cycles. The necescomputation time is, therefore, quite large, particularlysystems with slow transient response. A simple method wrequires much less computation time is described below.method can also be used to obtain a small-signal transfer ftion.

Method II. The basic approach of this method is also tctain an equivalent continuous mathematical model u

certain approximations. In addition to the approximatior(5) and (6), the following additional approximation is used

Since the chopper period is usually small compared toarmature circuit time constant, the variation of the cur

between the limits ia 1 and ia2 is along the initial partexponential curves which can be assumed to be linear

[20]

dwavJ +Bwav= (Tav -TL)

dt (77)

Any of the three approximations described by (7), (8), and(10) will give the same equation.

Equation (77) is a first-order, linear, differential equationand can be solved easily so that values of wav for differentvalues of t can be calculated.

For duty intervals with ia t=O = ia 1 from (1)

(78)(73) =a -Kav (1et/Ta)+iale t/Taia

R

( e

(74) and for freewheeling intervals, with ia It 0-ia2, from (3)

ia =- R (1 -e-tTa)+i e-t/Ta

Now from (78) and (79)

V-KWav

'a2 =- V

(79)

V-VKwav) TIT

R

/ Kwav e- (1-6)TITa Kwav

ta1 - \la22 R / R

and from (80) and (81)

T= Ta(X + Y)

T6= x

T

where

(80)

(81)

(82)

(83)

0

EC -4z

< -8

5 0

S 40

< 3-0

20

I*0 -is

\(1-6)T 'a

a2 + t' dt

~~~(Il-6)T

ial +±a22 (75)

(76)

155

156EE TRANSACTIONS ON INDUSTRAL ELECTRONICS AND CONTROL INSTRUMENTATION, VOL. ECI-28, NO. 2, MAY 1981

lgV Ri, I1 Kwa,V -Ria2 -Kwav

i

lgRia 1 +Kwav

1[0

063 3046

0-4(84)

Once the wav/t relation is obtained from (77) the T/t and b/trelations may be calculated from (82) and (83).

In Method I, calculations are done for small increments ofcurrent in each chopping cycle, thereby requiring a large numberof computations for each cycle. In this present Method II,solution of a hnear differential equation requires much lesscomputation time.

George [201 has described a method of obtaining transientresponse for small-signal perturbation around a steady-stateoperating point using Approximation I.

Substituting values of the variables after small perturbationsinto (77) and separating the steady-state and transient termsgives

dAwav±+BAwav=ATaV-ATL (85)

Since for a given setting of i'a1 and , the motor torque isindependent of speed and constant, steady-state stable opera-ting speed will not be obtained if the load torque is also con-stant and independent of speed. It is, therefore, assumed thatload torque is passive and proportional to speed, i.e., TLKLWav thus

ATL= KAWY . (86)

From (75)

ia v + iai +A AI (87)2 2

where AI/2 is the ripple in the armature current.The usual control strategy for CLC is to maintain AI

constant and obtain variations in torque and speed by chang-ing the value of ia - Therefore, from (37),

Aiav = Aial (88)

Using Approximation I and (88)

/ATav = KAiav= KAia4 (89)

From (85), (86), and (89)

±+ (B + KL)AWav =KoAia idt

which gives

AWay(s) Ko/JAia1 (s) s + (B +KL)J

(90)

02

A-of,

0

v~~~~~~~~

0 2 4 6 8 10 12 14time (sec.)

(a)

(b)Fig. 13. Starting transient response of a do series motor fed by a

chopper with CLC. ial = 0.6 pu, ia = 1.0 pu.

Method I,:Method II.

Similarly, one can derive transfer functions using Approxi-mations II and III, following the method developed above.A 220-V, 2.2-A, 0.375-kW, dc series motor was chosen to

compare the results of Methods I and II. Starting transientswere calculated by these methods for various combinations ofvalues of ia1 and 'a2- In these transients, it was assumed thatthe motor starts only after its developed torque becomesgreater than the load torque. Transient response for 'a2 1 puand ial -0.6 pu are shown in Fig. 13(a) and (b) [191. Plotsof Way and i ainst t are gven in Fig. 1 3(a) while those ofTITa and 6 against t are given in Fig. 13(b). Since Method I isbased on numerical solution of the performance equationswithout any approximations this was chosen as a referencefor ascertaining the accuracy of Method IL. As seen from Fig.13(a) and (b), the results obtained by Method II are in closeagreement with that of Method I.

Starting transients calculated by Method II and those ob-tained experimentally on a 220-V 2.5-kW, 15-A dc seriesmotor fed by a chopper with CLC are shown in Fig. 14 [20].An eddy-current brake was used to load the motor. There is afairly satisfactory agreement between calculated and experi-mental curves.

IV. CONCLUSIONSThe chopper-fed de series motor is essentially a sampled

data system. It can be approximated by an equivalent con-tinuous model for predicting transient response, involvingvariations of average speed and current with time, due to

(91) change in either duty ratio or load torque TL. In the paper,three approaches for these approximations are described for

156

DUBEY AND SHEPHERD: CHOPPER-FED DC SERIES MOTOR

'3

0. 8

calculated-; Sx - x-x experimental

0.4 -

A 0-2

O 1 0 2.0 3.0time (sec.)

Fig. 14. Starting transients in a dc series motor fed by a chopper withCLC.

*x*: experimental-: calculated (Method II).

chopper with TRC control and one for chopper with CLCcontrol.

The main problem in the analysis of chopper-fed dc seriesmotor is that the relation between armature-induced voltageand the armature current is nonlinear. Furthermore, themotor-armature current fluctuates between maximum andminimum limits during a chopping cycle causing similarvariations in armature-induced EMF. These problems can beovercome by assuming the motor-induced EMF constant K tobe a function of average armature current rather than itsinstantaneous values. This assumption permits the use of thesteady-state magnetization characteristics of the machine. Thisassumption is also satisfactory for the derivation of an equiva-lent continuous model that can be used to provide variation ofaverage speed and current with time. At this stage, we get theperformance equations which are nonlinear. These can besolved by piecewise-linear approximation or other numericalmethods of solution of nonlinear differential equations. Totalcomputation time with these methods is much shorter thanthe point-by-point solution of the nonlinear differential equa-tions valid for each mode of chopper cycle.

For small perturbations around a steady-state operatingpoint, the motor can be modeled by approximating themagnetization characteristics using three approaches, with thethird one being the most accurate. A transfer function can alsobe derived which can be usefully employed for stability anal-ysis when the chopper-fed dc series motor is used in a closed-loop system.

Among the three methods described for obtaining an equiv-alent continuous model of the TRC chopper-fed dc seriesmotor, Method III is the best. First it gives a second-ordermodel and, therefore, good accuracy and second, it can alsobe used to derive a transfer function for small perturbationsaround an operating point.

The equivalent continuous model for a CLC chopper-fed dcseries motor consists of a first-order linear differential equa-tion, which along with other performance equations allows thecalculation of variation of speed, chopper period, and dutyratio with time. The approach also permits the derivation of atransfer function for small perturbations around a steady-stateoperating point.

APPENDIX

If Wdl (n), ida(n), W'd1(n), idl (n), Wfl (n), if, (n), w'fl (n),and i'fl (n) denote initial values of w, i, dw/dt, and di/dt,respectively, for the duty and freewheeling intervals of any nthchopping cycle, then using the approximation of (6)

kidl (n) -Bwdl(n) - TLWdIl (n)= i

- V -Ridl (n) -Kwdi(n)

L

Wf '(n)

(Al)

(A2)

tA3)KAif (n) Bwfi (n) - TL

i

ifI'(n)-Rif I (n) - Kwfi (n)i 1 (n)= .L (A4)

Since the initial conditions of a particular interval are equal tothe final conditions of a previous interval, then

Wd2 (n) Wfl (n) id2 (n) = ifl (n)

Wdl(n + l)=Wf2(n) idl(n±J)-=if2(n)*

(A5)

(A6)

Subscript 2 in these equations refers to the final conditions ofa relevant mode.

Using (Al) to (A4) by elimination ofvariables, the following.equations are obtained:

d2Wd / TaATa + 1+

dt2 TM I

d2id I Ta "Ta 1+ +

dt2 Tm1

d2wTa dt2

d2ia dt2

+ (+ Ta

Tmi

TMTa

dwd d K1dt Tm 2 Tm2

~lLd W K2

-wf+ --n d-K3

dt Tm 2 Tm 2

dif +f K4dt Tm2 Tm2

where

L iTa- Tm= Tm2 =JR/(BR+K)RB

KT= Tm2(KV-RTL)/JR

K2=Tm2(BV+KTL)/JR K3=-Tm2TL/JK4 Tm2KTLIR.

(A7)

(A8)

(A9)

(AIO)

(All)

Using (Al) to (AIO) one can derive the following recursiverelations:

Wfl(n) = Clidl(n) + C2Wdl(n)+ C3

ifl(n) C4idl(n) -Cswdl(n)+ C6

(A12)

(A13)

157

158.E TRANSACTIONS ON INDUSIRIAL ELECTRONICS AND CONTROL INSTRUMENTATION, VOL. IECI-28, NO. 2, MAY 1981

Wdl (n + 1) = C7ifI (n) + C8Wfl (n) + C9

idl(n +±1) =C1oifl(n)-C1l wfl(n)+ C1l2-

(Al 4)

(A15)

For a givern value of 6, parameters C1 to C12 are functions ofK and are given by the following equations:

C1-=JoeSTsin BT

-a_ T -Ja - 1C2=e sinOT+1 cosin6T

C3 =K- T [ sin ]8T±KlBcosf3T

e-o,T [Lct-RIC4= 0 LL sinf36T+j3cos0t5Tj

C5 = e-°tSTsin0TfL

e-oSTvV- ULK21C6 =K2 + [VcL sinS6T- 3K1 cos06T]

C7K

e- a(l -S)T sin ,B(1-)T

+Bcos,B(l-)T

e= l--- [Jc-0-sin ,B(1-6)TC8= 0 L

+±cos (1- )T

e-oe(l-6 )TCg = K3-

TL +JaX3[TL-JO-K3 sin 3(l -S)T + OK3 cos ,B(1- )T

Cio= - [L sin(l-6)T+ Ocos (1-5)T

C11 =- e -6)T sin 0(1-6 )T

K4e-a(1-6 )TC12 =K4- 0

* [a sin B(1-6 )T +O cos ,B(1- )T] (A16)

where

(±+TalTm1)2Ta

/(4Ta/Tn 2)(1 ± Ta/Tm1)2±1 2Ta (Al7)

From (A12) to (A15)

Wdl(n + 1) =C13id1(n)+C14Wdn,(f)+C15

idl(n+1) =Cl6idl(n)-Cl7wdl(n)+C18

and

(A18)

(A19)

Wdl(n +2)=C13idl(n + 1)+C14Wdl(n + 1)+C15

(A20)id, (n + 2) =Cl6idl(n + l)-Cl7wdl(n + 1) +C18

(A21)where

C13-C4C7+C1C8 C14=C2C8-C5C7C15 =C6C7+C3C8+C9 C16=C4C1I---C1C11C1 7 =CSC10 + C2C1 1 C1O-C6C1l0-C3C11 + C12-

(A22)

From (A18) and (A19)

Wdl(n +2)-CI9wl(n+ 1) +C20W(n)±+ C21

idl(n+2) =Cl9idl(n±+)+C20idl(n)+C22where

C19 = C14 + C16

C20 =-(C13C17 +C14C16)

C21 =Cl3Cl8-Cl5Cl6+C15C22 =-(Cl5C7 ±Cl4C 8 -C18).

(A23)

(A24)

(A25)

In the above derivations, K has been assumed to be fixed.This implies that the change in iav in two chopping cycles isvery small and therefore the change in K is so small that it canbe neglected.

REFERENCES[I] G. K. Dubey and W. Shepherd, "Analysis of dc series motor

controlled by power pulses," Proc. Inst. Elec. Eng., vol. 122, no.12, pp. 1397-1398, 1975.

[2] ' "Analysis of dc series motor controlled by power pulses,"Res. Rep. 197, Postgraduate School of Electrical and ElectronicEngineering, University of Bradford, England, June 1975.

[3] K. A. Krishnamurthy, G. K. Dubey, and G. N. Revankar,"Analysis of ac chopper fed dc series motor," Inst. Eng. (India),vol. 59, pp. 1-8, Aug. 1978.

158

IEEE TRANSACTIONS ON INDUSTRLAL ELECTRONICS AND CONTROL INSTRUMENTATION, VOL. IECI-28, NO. 2, MAY 1981

[41 G. K. Dubey, "Analytical methods for performance calculation ofchopper controlled dc series motor,' Inst. Eng. (India), vol. 58,pp. 69-74, Oct. 1977.

[5] D. B. Ranade and G. K. Dubey, "A chopper for control of dctraction motor," J. Elec. Machines Electromech., vol. 4, no. 4, pp.299-319, 1979.

[61 S. N. Bhadra, S. Sahaderan, N. K. De, and A. K. Chattopadhyay,"The effects of commutating capacitor on the regenerative brakingof a thyristor-chopper controlled dc series motor," in Proc. Conf.on DC Technology, Feb. 6-8, 1978, pp. 534-544.

[71 P. D. Damle and G. K. Dubey, "Analysis of chopper-fed dc seriesmotor," IEEE Trans. Id. Electron. Contr. Instrum., vol. IECI-23,pp. 92-97, Feb. 1976.

[81 P. D. Damle and G. K. Dubey, "A digital computer program forchopper-fed dc motors," IEEE Trans. Ind. Electron. Contr. In-strum., vol. IECI-22, pp. 408-412, Aug. 1975.

[91 B. Mellitt and M. H. Rashid, "Analysis of chopper circuits bycomputer-based piecewise-linear technique,'" Proc. Inst. Elec.Eng., vol. 121, no. 3, pp. 173-178, 1974.

[10] B. W. Williams, "Complete state-space digital computer simu-lation of chopper-fed dc motors,'" IEEE Trans. Ind. Electron.Contr. Instrum., vol. IECI-25, no. 3, pp. 255-260, Aug. 1978.

[I1] P. W. Franklin, "Theory of dc motor controlled by power pulses,"IEEE Trans. Power App. Syst., vol. PAS-91, pp. 249-255, 1972.

[121 G. K. Dubey and W. Shepherd, "Analysis and performance of dcmotor controlled power pulses-Part I-Steady state operation andPart Il-Transient operation," Rep. 204, Postgraduate School of

Electrical and Electronic Engineering, University of Bradford,England, 1975.

[131 , "Transient analysis of dc motor controlled by powerpulses," Proc. Inst. Elec. Eng., vol. 124, no. 3, pp. 229-230,Mar. 1977.

[14] F. E. Edwin and G. K. Dubey, "Transient analysis of converter-controlled dc separately excited motor," J. Elec. MachinesElectromech., vol. 2, pp. 325-402, 1978.

[15] A. Yanase, "Starting characteristic of chopper-controlled dc seriesmotor with nonlinear magnetisation curve," Elec. Eng. Japan, vol.97, no. 1, pp. 94-101, 1977.

[161 M. Ramamoorty and B. Ilango, "The transient response of a

thyristor-controlled series motor," IEEE Trans. Power App. Syst.,vol. PAS-90, pp. 289-297, 1971.

[171 S. N. Bhadra, "Transient response of thyristor controlled dc seriesmotor under small disturbances," J. Inst. Eng. (India), Elec. Eng.Div., vol. 56, pp. 192-197, Feb. 1976.

[18] D. B. Ranade, "Chopper control of dc traction motors," M. Tech.thesis, IIT Bombay, India, 1977.

[19] G. K. Dubey, "Transient analysis of dc series motor fed by achopper with current-limit control," J. Inst. Eng. (India), Elec.Eng. Div., vol. 59, pp. 115-118, Oct. 1978.

[201 V. George, "DC series motor fed by a chopper with current-limitcontrol," M. Tech. thesis, IIT Bombay, India, 1978.

[21] T. Fujimaki, K. Ohniwa, and 0. Miyashita, "Simulation of thechopper controlled dc series motor," in Proc. IFAC Conf., pp.609-618, 1977.

Some Investigations on the Operation of Fully ControlledThyristor Converters with Sequence Control

S. PALANICHAMY, MEMBER, IEEE, AND V. SUBBIAH

Abstract-Operating diagram is developed for the sequence controlof two single-phase fully controlled thyristor converters. Possiblemodes of operation can be identified with the help of this diagram.Minimum inductance necessary for continuous current operation ofthe sequence-controlled converters is evaluated.

I. INTRODUCTIONW HEN phase-controlled thyristor converters with symmet-

rical triggering operate at delayed trigger angles to obtaindc output voltage less than the maximum, the supply powerfactor assumes low values. It has been shown that the powerfactor improvement can be obtained by resorting to forced

Manuscript received February 13, 1980. This work was supportedunder Grant from the University Grants Commission, Government ofIndia.

The authors are with the Department of Electrical Engineering, PSGCollege of Technology, Coimbatore, India.

commutation [1], [21. But then the advantage of naturalcommutation gets lost. Moreover, the above approach will notbe of avail in the case of fully controlled converters which willbe operated as rectifiers as well as line commutated inverters.A well-known method, which is devoid of the drawbacks re-ferred to, is the so-called "asymmetrical triggering" [3], [4].

For converters comprising of two (or more) commutatinggroups in series, the power factor can be improved by theasymmetrical triggering of the commutating groups. Essen-tially, one group is maintained at full advance or full retard sothat its reactive power requirement is minimized, while theother group is phase-controlled to obtain the required dc out-put voltage. This technique is not recommended for single-bridge converters [4] , [5]. However, it is acceptable when ap-plied to two or more bridge converters connected in series.This is commonly referred to as "sequence control" [3], [4].In view of the fact that the sequence control of more than two

0018-9421/81/0500-0159$00.75 © 1981 IEEE

159