IEEE Transactions on Power Systems, Vol. PWRS-1, No. 4, November 1986

A BASIS-TRANSFORMED STATE SPACE FORMULATION FOR THE ANALYSIS OF CONTROLLEDRECTIFIERS UNDER IDEAL AND NON-IDEAL STEADY STATE CONDITIONS

137

R. W.-Y. Cheung J.D. Lavers, Member, IEEEDepartment of Electrical Engineering

University of TorontoToronto M5S 1A4

CANADAABSTRACT

This paper presents a general method for thLi analysis ofcontrolled rectifiers under ideal and non-ideal system conditions.Both 6- and 12-pulse rectifiers are considered with a generalload being represented by a series resistance-inductance togetherwith a dc emf. The formulation of the analysis, which employsa basis transformed state space technique, is detailed. Thistransformation considerably simplifies the analysis for rectifiersoperating under ideal conditions. Moreover, the effects of non-ideal conditions (i.e. supply unbalances, input filtering, real gat-ing circuits etc.) can be directly included in the analysis. Themethod is particularly well suited for implementation on smallmicrocomputer systems. Examples illustrating the flexibility andnumerical robustness of the method are given and comparisonsare made with experimental data.

INTRODUCTIONState space formulations for the steady state analysis of

power converter and inverter circuits are usually based on theassumption that the circuit is symmetrical (i.e. balanced) andthat the operating conditions are ideal. The effects of non-idealconditions and system unbalance are usually neglected since theconventional state variable approaches [1-10] would then resultin excessively complex formulations that lack generality. Thispaper describes a basis transformation that significantlysimplifies the state space analysis of power electronic circuits.The formulation leads to a highly modular software structurethat results in a numerically efficient, robust and direct solutionof the system equations in the steady state. To illustrate thebasis transformed formulation, the analysis of 6- and 12-pulsecontrolled rectifiers, typical of those used in hvdc and dc drives,is considered. In particular, non-ideal operating conditions thatcan significantly influence circuit performance are included inthe analysis. These include an ac source that can be unbalanced,an input filter (symmetric or unsymmetric) and the actualcharacteristics of the gating circuit, including the fact that thegating synchronization may not be set relative to an ideal vol-tage source.

It is standard practice today to undertake the analysis ofpower electronic circuits using the state space technique. Toobtain the steady state performance characteristics of the circuitunder analysis, one of two general approaches can be taken:

(a) The state differential equations can be integrated, one byone, through a real or quasi transient, until the steadystate is reached.

(b) The set of state equations, subject to appropriate terminaland symmetry constraints, can be cast in a form that canbe solved simultaneously and directly for the steady state.

86 WM 055-8 A paper recommended and approvedby the IEEE Power System Engineering Committee ofthe IEEE Power Engineering Society for presentationat the IEEE/PES 1986 Winter Meeting, New York, NewYork, February 2 - 7, 1986. Manuscript submittedAugust 28, 1985; made available for printingDecember 4 1985.

The latter approach is preferable so that the expensive transientcomputation can be avoided. Recognizing this, analysesdescribed in the published literature [1-10] largely concentrateon determining cost effective methods of computing the steadystate directly. Despite the fact that a broad range of state spacebased formulations have been described in the literature, nonemay be termed optimal or even particularly well suited for theanalysis of power electronic circuits when the system conditionsare non-ideal. Certainly, none of the existing formulations leadto efficient computation and fast execution on a small computersystem.

The purpose of this paper is to describe a simple and costeffective method for the analysis of power electronic circuits,with controlled rectifiers being used to illustrate the formulationof the analysis. A feature of the method is the fact that non-ideal system conditions can be included in the analysis. Themethod exploits a uniquely simple basis transformation that notonly enhances the flexibility of the state space formulation butalso simplifies the numerical solution procedure and reducescomputational costs considerably. The features of the methodare illustrated by considering 6- and 12-pulse controlledrectifiers. In the case of 6-pulse rectifiers, results are predictedfor examples where the supply is unbalanced, an input filter isincluded and the gating control is not ideal. The latter case isparticularly important since unbalanced operation can result.Experimental data are used to verify the results of the analysis.

CONTROLLED RECTIFIER SYSTEMS

The method of analysis that is described in this paper isused to predict the steady state performance characteristics ofcontrolled rectifier circuits. The system shown in Figure 1 illus-trates many of the features that should be included in theanalysis. This system is a typical 6-pulse controlled rectifier thatis connected to a 3-phase supply that is not necessarily balanced.The supply and input transformer can both contribute to a lineimpedance that often cannot be neglected; the net line induc-tance determines the overlap angle for the commutation of theindividual thyristors in the bridge. Practical rectifier systemsinclude an input filter to limit current harmonics being fed backto the line. An input filter can significantly influence therectifier performance since in many installations, the gating syn-chronization for the thyristors can only be taken from therectifier side of the filter. Consequently, synchronization is rela-tive to distorted voltages and depending on the characteristics ofthe gating circuit, rectifier operation can be unbalanced. Predic-tion of this unbalance is important when evaluating overall sys-tem performance. For the purpose of generality, the rectifier isshown feeding a general load being represented by a seriesresistance-inductance together with a dc emf in Figure 1; this inno way limits the generality of the method.

The method of analysis that is described in this paper isused to predict the steady state performance of a rectifier sys-tem similar to that shown in Figure 1, especially when theoperation is unbalanced and the gating characteristics are notideal. Of particular interest is the unbalance that can occurwhen the gating synchronization is taken on the rectifier side ofthe filter. While balanced and unbalanced operation of the 6-pulse rectifier is considered, application of the method to 12-pulse systems is limited in this paper to balanced configurations.

0885-8950/86/1 100-0137$01.00( 1986 IEEE

138

source transformer filter irettier Cc motor

Fig. 1 A 6-pulse controlled rectifier with input filter, sourceimpedance and a general load.

FORMULATION OF ANALYSIS FOR IDEAL SYSTEMS

Ideal Symmetrical Systems

For the purposes of this paper, a system is termed idealand symmetrical if the ac supply is balanced, the filter com-ponents in all three branches are equal and the controlledrectifier is operated in a symmetrical sequence. The symmetri-cal steady state operation of the thyristors in a 6-pulse bridge isillustrated in Figure 2. There are twelve states in a period ofsymmetrical operation wherein any two successive states extendfor an interval of r/3 radians and all overlap angles are equal.When this symmetrical property is utilized, it is only necessaryto select any two successive states out of the possible twelvewhen formulating the analysis. For the purpose of illustration,states 1 and 2 in Figure 2 are selected for the analysis that fol-lows.

vab vbc ca

... .. . . ~ ..S~ ..........s{'/ "-ms -

*1 :2:3 :4 5 6 :7 8 9 10:11:12: States

:Q4 Q4 Ql1 Ql:Ql: QlQl Q4 Q4.Q4 Q4:Q4 ThyristorQ6 Q6.Q6.Q6 Q6 Q3.Q3 Q3:Q3 Q3 Q6 Q6: Conducting

:Q5 :Q5:Q5Q5 Q5 Q2:Q2:Q2 Q2 Q2. Q5Q51Q5Q; ; ;1;Q2 ;03. .o4; ;o5; 0(36: ; Overlap

I Angles

0 atl ...t

Fig. 2 Symmetrical steady state operation of a 6-pulse con-trolled rectifier.

State Space Modeling

In state space analysis, each state is modelled by a stateequation which consists of a set of first order ordinarydifferential equations. The state equation can be obtained bymanipulating loop and node equations written from the stateequivalent circuit. The state equation is usually expressed in thefollowing general form:

Dx (wt) = Bi x(wtt) + C-i (wt)

where D - the differential operator dId wtB, C - constant coefficient matrices

xT - the state variable vectoru - the iisput forcing function vector

The above equation may be solved using the step-by-step numer-

ical integration method. However this approach is expensive andtedious when only the steady state is required. A more attrac-tive alternative to direct integration is to exploit the formalsolution to (1):

Y(Wt) = e ' t e.(o,r)+f| e(-' )B.C (X )dX (2)

.t,

where x(wt0) is the initial state vector and the second term is aparticular integral.

For steady state solution, the initial state vector must bedetermined. The conventional method for computing the initialvector directly (without having the transient computation) is tosolve a set of equations in the form of (2) simultaneously. Thedifficulty with this solution method lies primarily with the par-ticular integral that appears in (2). As is illustrated in AppendixI, a nontrivial amount of manipulation is required even whensystem under analysis involves only two states. The complexityof the solution formulation using this approach increases rapidlyas the number of states to be considered increases. Implementa-tion of this form of solution within a general simulation packageis therefore cumbersome and if implemented, the overallsoftware architecture will not be clean. It should be noted thatif a formulation does incorporate the particular integral, theintegral can be numerically evaluated by using an appropriatealgorithm or alternatively, the method outlined by Ooi et al in[6] and [9] can be used.

Basis Transformation

The utility of the formal solution (2) can be considerablyenhanced by performing a basis transformation and augmentingthe state vector [11]. It will be noted that the particular integralin (2) arises because of the forcing function in (1). In order toeliminate this potential difficulty, consider an additional matrixequation of the form:

Dui(wt) = F *ii(wt) (3)

which relates the derivative of the input vector to the vectoritself. It is possible to determine F for all common forcingfunctions [11]. A basis transformation is performed by combin-ing (1) and (3). The transformed state equation then becomes:

Dy(wt) = A y(7wt) (4)

where

y = [ u]' and X [=0 =

The transformed state equation has a solution of the form:

y(wt ) = e -wt° y(wto) (5)

which does not involve a particular integral.

The basis transformed solution equation (5) has severaladvantages. Firstly, since no particular integral is involved,solutions for all states exhibit a common simple form, differingonly by individual matrices A. Therefore a software packagethat simulates a variety of circuit configurations will have a veryclean and well defined structure. This significantly enhances theimplementation of interactive features. Secondly, the lack of aparticular integral means that steady state solutions to problemsinvolving more than two states can be handled with relativeease. This greatly simplifies the formulation for the analysis forcircuits operating under non-ideal or unsymmetrical conditions.Thirdly, the need to numerically evaluate the particular integralhas been avoided, albeit at the cost of expanding the statematrix. These features are particularly attractive when consid-ering the implementation of a simulation method on a smallcomputer system. Details of the basis transformation are fullydiscussed elsewhere [11].

InitializationThe initial vector y(t,) in (5) must be determined before

the steady state solution can be computed. In order to compute

d

139

the initial vector, it is firstly expressed as an explicit function ofthe input vector U(wt0). As mentioned above, state I and state 2shown in Fig. 2 are selected for the formulation. Thecorresponding formal solutions have the form:

y (@t ) =c * Y (t ) to :5 wt s5 wt' +0 (6)

and

Y(wt) = ,2 .(0tw) +t0+O wt:. wto+w/3 (7)

where 0 is the overlap angle; AX and A2 are state matrices forstates 1 and 2, respectively, and can be obtained in a fashionsimilar to that illustrated in Appendix II. Note that in the caseof ideal operation being considered in this section, the six over-lap angles shown in Figure 2 are all equal; this single overlapangle is denoted by 0. Substituting wt = wto +0 and0t =oto +i /3 into (6) and (7) respectively and combining theresults, the following expression can be obtained:

j(wto + w/3) = e (w /3-)X2 .eOI .y(w:o) (8)

As is characteristic of the steady state, there always exists asymmetrical relationship between state vectors in the first andthe last of selected states. The relationship may be expressed as

7(wt +'f/3) = T y(wt) (9)

where T is termed the transposition matrix (see an example inAppendix II). Equations (8) and (9) are combined to yield anexpression for the initial vector:

{T-_ ( e/3-092 }'Iy-(wt ) e0 (10)

Obviously, a comparison of (A5) from Appendix I and (10)above shows the efficiency of the transformed state spaceapproach. Clearly, the formulation for more than two statesusing the conventional approach will become increasingly com-plex. On the other hand, if the transformed state space approachis applied, the formulation is simple since only exponentialterms corresponding to the additional states need be added in(10). This not only simplifies the formulation, but also enhancescomputational performance when the method is implemented insoftware.

Since y = [x u]t, equation (10) may be rearranged as:

y(t ) = P (0) * u (0)t) (11)

where P is partitioned from the matrix {} in (10). Finally, theinitial vector is expressed as a function of the overlap angle 0and can be solved in the following two ways.

Newton-Raphson Iteration

If the switches in the rectifier are gated at a particularangle a(= wt. -'rr/3), then the overlap angle 0 which is circuitdependent is unknown. Therefore, the initial vector j(wt):) can-not be computed directly from (11) and some numerical itera-tion methods are needed. To date, the Newton-Raphson Methodis generally accepted as the most appropriate method for thistype of computation. The N-R iterative algorithm may beexpressed by the following equation:

ok+1 = ok 2A0_ _ f (0k)f (0k +A6)-f (k -AO)

(12)

where the iteration function f must be determined. Note thatthe differentiation of the iteration function f has been imple-mented by linearizing about 0k*. Both Newton-Raphson as wellas Secant iteration methods have been used to determine theoverlap angles. Both methods have been found to convergewithin 3 to 5 iterations for the circuits that have been con-sidered in this paper.

As is shown in Fig. 2, thyristor QI starts to conduct at woand therefore the current iQ1 is zero at wto. Since any branchcurrent can be expressed as a combination of the state variables(see an example in Appendix II), iQj may be expressed as:

iQ ()to ) =- ( ) O (13)

The iteration function f (0) can therefore be obtained from (11)and (13):

f (0) = CI P(0) i(Wto) -_ 0 (14)

The iteration steps when If I S E, where e is the error toler-ance. Generally for e = 10-4 on per unit basis, four iterationsare typical.

Direct Computation

In the preceding formulation of the state equations, thegating angle a is a naturally occurring independent variablewhile the overlap angle 0 appears naturally as a dependent vari-able. Both a and 0 are linked to one another. If a numericalcomputation is to be performed for a specific value of a, aniterative solution for the corresponding overlap angle(s) and ini-tial vector is required, as was shown in the previous section. Onthe other hand, if the overlap angle 0 is specified, the initial vec-tor can be directly computed from (11) and no iteration isrequired. Using this approach, the computation can easily beperformed for a broad range of circuit conditions with a seriesof overlap angles. From an analysis and design point of view, itdoes not matter whether the input parameter is a or 0 provideda broad enough range of operating conditions are covered. Thismethod of direct computation is an immediate consequence ofthe basis transformation that was used in formulating the prob-lem. It cannot be used if the conventional formulation thatincorporates the particular integral(s) is used.

Solution for a Complete Period

After the initial vector has been computed using either ofthe above two methods, the solution for the selected states canbe computed from (6) and (7). The solution for the wholeperiod can then be obtained by repetitively using (9).

FORMULATION OF ANALYSIS FOR NON-IDEAL SYSTEMS

Non-Ideal and Unsymmetrical Conditions

There are several conditions that can result in non-ideal orunsymmetrical operation of rectifier systems. These includeunbalance in the electrical supply, unsymmetry in the filter, gat-ing reference voltages that are distorted (reference voltagestaken from the rectifier side of the filter, for example) and gat-ing circuits that result in unsymmetry in the gating signals. It isa relatively straight forward matter to extend the basistransformed state space analysis to cover these cases. Theextension simply requires the implementation of various itera-tion loops that can be classified as follows:

(a) Single dimensioned, double loop iteration (Class A).

(b) Multi-dimensioned, single loop iteration (Class B).

(c) Multi-dimensioned, double loop iteration (Class C).

140

Class A Iteration

This class of iteration arises when the circuit conditions(i.e. the electrical supply and input filter) are balanced and sym-metric but the gating reference voltage is distorted (i.e. a singlereference voltage is used and is not taken relative to an ideal,three phase voltage source). This case often arises in the case ofdc drive systems where it may be convenient to obtain the gat-ing reference voltage from the input transformer secondary [12].In a multiple drive installation that incorporates a commonfilter, the reference voltage may of necessity be taken from therectifier side of the filter. The gating reference is therefore notfixed but rather, is circuit and load dependent. If the rectifier isgated at an angle a, then:

to,, y +a+ir/3 (15)

where y is the angle between the ideal source voltage v,b andthe actual voltage at the reference. This angle is unknown anddependent on load and operation conditions. This class of prob-lem can easily be handled with one more iteration loop in addi-tion to the iteration that was detailed in the previous section(see (12)).

The additional iteration loop can be formulated by recog-nizing that the gating control strategy is normally such that thedelay angle Q is referred to the zero-crossing of the referencevoltage. The reference voltage vtR can always be expressed as acombination of the state variables. Moreover, vR equals to zeroat wt = y. Therefore

tR = C2.Y(Y) = 0 (16)Again, y can be determined by a Newton-Raphson algorithm:

yk+ = __ 2A _ . g (2Ak (17)g (9 +Ay)-g (e -_AY)

It follows from (16) that the iteration function g is:

g(O) = C2 y(y) 0° (18)

where y(y) is obtained after the first iteration loop (see (12)) iscomputed.

Class B Iteration

This class of iteration typically occurs when the electricalsupply is unbalanced, the input filter is unsymmetric or the gat-ing signals are unsymmetric but the gating reference voltage isknown and fixed. Under these non-ideal conditions, the system isno longer symmetrical and the overlap angles shown in Fig. 2are not equal. Therefore the circuit performance can not berepresented by choosing only two states as would be the casewith a fully symmetrical system. More states must be included inthe formulation. With the application of basis transformationtechniques, the formulations for symmetrical and unsymmetricalsystems are basically similar. The only difference is that thedimension (but not the structure) of the formulation increaseswhen the system is unsymmetric.

Since more states must be considered, equation (10)becomes

If lis1 .eclZ Jy(t) = 0 (19)

where P and Ai are the state interval and state matrix for statei, respectilvely. The scalar Newton-Raphson equation (12)becomes vectorial:

k -= p - |1- | (20)

where the iteration functions in f are determined by the zero-crossings of thyristor currents as was the case when deriving(14).

Class C Iteration

This class of iteration contains two sub-sets. The firstsub-set (Class Cl) is a combination of Classes A and B wherethe non-ideal conditions include an unbalanced supply, anunsymmetric filter and a single, distorted gating reference vol-tage. The gating signals, however, are symmetric and phase dis-placed by 'w/3 radians. Two iteration loops are required for thissub-set; the inner loop is governed by (20) and the outer loop by(17).

The second sub-set (Class C2) includes all of the non-idealconditions of Class Cl. In addition, multiple gating referencevoltages (one per phase) are used to synchronize the gating sig-nals. The reference voltages are taken from an arbitrary pointin the circuit. This, therefore, is the most general of the classes.The formulation for this sub-set is the same as Class Cl exceptthat a vector form of (17) must be used:

~k+1 .j-k 2 (21)

NUMERICAL EXAMPLES AND EXPERIMENTALVERIFICATIONS

An interactive package has been developed for the analysisof controlled rectifier systems. The package has a highly modu-lar software structure and incorporates the basis transformedformulation that has been discussed in this paper. The numeri-cal examples that are given in this section were all computedusing a DEC MICRO PDPII. Performance of the simulationpackage on this relatively small microcomputer system is morethan adequate. At present, the package is used on a routinebasis for graduate research purposes. Numerical aspects of thesimulation package are described elsewhere [11,13].

The purpose of the examples that are included in this sec-tion is primarily to illustrate the broad range of operating condi-tions that can be considered when the basis transformation isused. For the most part, the system shown in Figure 1 is used asthe basis for the examples. The per unit system used in thesimulation examples is defined by:

Vbm peak rated phase voltage,Ib peak rated line current,Zbe vba/ baw

Data for the system in Fig. I are:

R = Rs +Rt +Rf = 0.015p.u. , L LL, +Lt+Lf = 0.5089p.u.

Rf - 0.Oll0p.u., Lf - = 0.3657p.u. , CfX - 0.0954p.u.

For R-L load: Rd = 1.993p.u. , Ld = 0,0641p.u., Vd = 0.0p.u.

For motor load: Rd 0.1451p.u., Ld = 0.6786p.u. eVd = 1.255p.u.

The various matrices required within the basis transformedformulation for the analysis of the rectifier system shown in Fig.1 are not detailed in this paper. The formulation of typicalmatrices is illustrated in Appendix II and the same techniquescan be used to derive the matrices required in any of the exam-ples that are included in this section. Full details of matrix for-mation are also provided elsewhere [11]. In the case of the

141

simulation package that was used to produce the numericalresults that included in this section, the assembly of the systemmatrices was automatically performed by a topologically basedalgorithm. This algorithm allows for efficient and convenientmatrix assembly and incorporates interactive, graphics basedsoftware.

As a first example, the rectifier of Fig. 1 was assumed to beunder ideal, balanced conditions. The gating circuit was assumedto be ideal and the gating reference signal was taken withrespect to the system input voltage. The per unit system parame-ters given above were used. This is the simplest mode of opera-tion to analyze. The steady state waveforms (the rectifier outputcurrent, supply input currents and filter capacitor voltages) forthe rectifier feeding a motor load are shown in Fig. 3. Inpredicting these waveforms, method of Direct Computationdescribed in a previous section was used. It was assumed thatthe overlap angle 0 was a simulation input parameter having avalue of 20°.The corresponding gating angle a obtained from thesimulation was 310. This is the fastest simulation since no itera-tion was required. The computation time required on theMICRO PDPII was 50 seconds. To put this time into perspec-tive, it is approximately 10 times more than on a VAX 780under normal loading and 100 times more than on a large main-frame such as an IBM 3033.

Similar waveforms for the rectifier feeding a RL load aregiven in the following three examples. Several iteration methodsdescribed in the previous sections were used. In predicting thewaveforms shown in Fig. 4, it was assumed that the rectifier sys-tem was under ideal conditions and the thyristors were gated atan a of 30°. Three iterations lead to determine the overlap angle0 of 11°.The computation time required in the order of 120seconds on the MICRO PDPII.

Fig. 5 shows the waveforms for the case where the rectifiersystem operates under relatively extreme input voltage unbal-ance. The resulting unbalance in the output current, particu-larly, is evident. The Phase B input voltage was set to 0.50 p.u.This value was chosen more to illustrate the numerical robust-ness of the formulation than to pick a practical level of unbal-ance. For a gating angle a of 300, the computed waveforms andthe corresponding experimental waveforms are shown in Fig. 5.This is a case of Class B iteration and five iterations wererequired to determine the overlap angles (01 and 04 of 11.40, 02and 05 of 7.4°, 03 and 06 Of 19.80). The computation timerequired on the MICRO PDPII was 250 seconds.

The steady state waveforms predicted by the simulationwhen the gatings were not ideal, are shown in Fig. 6. In this casethe gating synchronizations were taken on the rectifier side ofthe filter. Moreover, the gating circuit was such that the result-ing gating signals (i.e. the values of a) were not symmetrical.For this particular example, the values of a were: a1 = 52.30, a2= 122.10, a3 = 172.30, a4 = 242.10, as = 292.30 and a6 = 362.10.These would correspond to overlap angles: 01, 03 and 0s of 2.1°;02, 04 and 06 of 2.70. All other circuit parameters were assumedto be symmetrical with per unit values as detailed above. This isa case of Class C iteration, and five outer-loop iterations wererequired. In each outer-loop iteration, there were approxi-mately four inner-loop iterations.

It will be noted in Fig. 6 that the unsymmetrical gatingsresult in a pronounced unsymmetry in the various waveforms.The corresponding experimental waveforms are also given. Inpredicting these waveforms, the actual characteristics of the gat-ing circuit [12] were incorporated in the simulation. In particu-lar, while a delay angle for the gating signals was set as an inputparameter, the actual gating angles with respect to the idealreference, together with the corresponding overlap angles, weregoverned by both the load conditions and the gating circuitcharacteristics. This is a particularly important example sincethe system performance will be incorrectly predicted unless it ispossible to incorporate the details of gating circuit performance

f

c

vW

a'

f-0-

C)

4.,

0

IA4.,1c4'f0-0.)0

a.

4,;103-

I .

.0 .. ..0 45 90 135 180 225 270 315 360

....

0 DO':

(1 45 90 135 180 2'5 27i 3i5 360

2 o itt-- --- . .s(@B*¢.e¢*.ets-.. .. .. .. .. ..7. .. .. . ... .. . . .. ..... .. ... .. ..

2 .,i.P ... .,,: --- ., ..l.,....1i.0-::.. ......' ; / ... ....

a, -,........ -.._,.... ....................

(* !-; -

WWL I c,O: ................... '-'.>'i.".;-).

-2.00 .... ;0 45 90 135 180 ?25 270 315 360

Fig. 3 6-pulse rectifier feeding a motor load under ideal con-ditions.

. .5 j . . ..;;. .. . . ;.. . / .. .;.;i e; . , ,:....,...,,.........1 .('('.~~. .. ...

It 45 9(1 135 186. 225 27. 315. 36(

.i . E.i .. .'E..-. . '. . . . .. '. E.,.;...... ....... ............

O .. .. . ..; .. . ... .. . ...... ":..... . ..

-1 I. .-1 .0'0} . -....... ................. ........................... .......0t 45 9(l 135 182'225 270 15 36

a.0 .. ...

1.20

45 0 15 10 25.2 35 3

Cl 45 9W0 1s35 ]580 225 27ll 3515 3560

Fig. 4 6-pulse rectifier feeding a RL load under ideal condi-tions.

4.,cGP0:-f-

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142

in the simulation. It is believed that the basis transformed for-mulation provides an extremely convenient means of achievingthis end. Note that in this example, if an ideal gating circuit wasused (regardless of where the gating reference is taken), thewaveforms would be symmetrical.

As a final example, the waveforms predicted for a 12-pulserectifier system and the corresponding experimental waveformare shown in Fig. 7. The configuration of the 12-pulse systemused in predicting these waveforms was similar to that shown inFig. 1, except wherein the 6-pulse rectifier was replaced by a 12-pulse rectifier together with the appropriate transformers, and

the load was assumed to be an ideal current sink. The 12-pulsesystem was assumed to be under symmetric, balanced conditions.The per unit system parameters used in the simulation were:

R = 0.03p.u., L = 0.2941p.u., I(current sink) = 0.7553p.u.

Rf ' = 0.02p.u., Lf ' = 0.2224p.u., Cf ' = 0.0113p.u.

Application of the basis transformed state space formula-tion to the analysis of systems that can give rise to subharmonicoscillations (i.e. certain dc drive configurations, cycloconverters,HVDC) is presently being examined. Clearly, the steady stateformulation that has been described in this paper is notappropriate for stability problems involving subharmonic oscilla-tions.

COMPUTATIONAL EFFICIENCY

It is believed that the basis transformed solution is compu-tationally superior to other state space formulations that areavailable, particularly those that include the particular integral.

1,,,....................................... ....................................

0.75 ... ............... ................

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ri 45 SC9 135 180f 225 270 315 360

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C' 4S 90 135 18P ?Z5 270 315 36Cl

F[g. Sa Output and input currents for the case of unbalancedvoltage supply.

.4

4,

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CD

k4

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0 45 90 135 1 ei ;22 r 27(0 315 360.................*...... .... .................... ...............

1 t ,r z ' t, ~~~~......... ..i__,....... ..

Li .0 s

- 1 . .... 30 45 90 135 180 225 270 315 36

Fig. 5b Filter current and capacitor voltages for the case ofunbalanced voltage supply.

.J.\,.~~~~~~~~~~~~~~~~~~~~~~~~. .0.60,~~I

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h.4-c

4'4:Cc

. ...........................Zi 31D5' ' 3604 4 135 1X 0.25

0 .60.....6,,,,, , , ,, , ,,-----

otls..., A ....... ... .. .... g........

0.45 .*..03 .. .....

.....

0.15.... . 1,~~~~~~~~~.. ..'.1,.

0.00 ,~ ~~~~...... .... .

-0.30.-0 4.3

0i 45 . 90. I135 180i Z25 '0 ~ 6

Fig. 6a Output and input currents resulting from non-idealgatings.

143

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Q 45 901 135 18(1 225 27t( .i5 36c240.

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-4.00

Cl 45 90 135 18c0 225 270 315 36. 0

21 D ......... . ..... ... ....

1 0(1 - , r ; , ! - .*--- ..... ............ ...................

-1 (ict'

-2C, .................

45 90 135 225 270 315 3680

Fig. 6b Filter current and capacitor voltages resulting fromnon-ideal gatings.

This is an impression that has been gained by applying themethod to a broad range of power converter circuits. Unfor-tunately, good, well documented bench marks against which a

method can be tested are sadly lacking in the literature.

With regard to the 6-pulse rectifier under balanced condi-tions and including an input filter, steady state solutions were

typically obtained within 50 seconds on a MICRO PDP11/23.Two to three iterations were required to obtain convergence.While a similar problem was solved by Ooi et al [6], no solutiontimes were reported although convergence was also obtained in3-4 iterations. The convergence rates of the basis transformedsolution and the solution reported by Berube and Ooi [9] for the12-pulse rectifier are comparable. Again, no solution times were

reported in [9].

One possible measure of the computational efficiencyobtained with the basis transformed formulation can be gainedby comparing solution times for the induction motor transientconsidered by Murthy and Berg [14]. The best solution timeobtained by Murthy and Berg was 17.0 seconds on an unspecifiedcomputer. A Runge Kutta-Verner integration was used to obtainthe transient solution. Using a basis transformed formulationfor this transient problem, the solution time on the MICROPDP1V/23 was 251 seconds and 1.14 seconds on an IBM 3033 [13].Based on this comparison, the authors have assumed that atleast a 100:1 improvement in solution times would be obtained ifthe MICRO PDP11/23 solution times reported in this paper were

referred to a mainframe such as the IBM 3033.CONCLUSIONS

The basis transformed state space formulation for theanalysis of controlled rectifier circuits and systems has beendescribed in this paper. 'The transformation results in the elimi-nation of the particular integral that normally appears in the

Fig. 7 12-pulse rectifier feeding a current sink under idealconditions.

formal solution of the state equations. Consequently, the for-

mulation lends itself to a highly modular, matrix oriented imple-mentation that is very efficient in directly computing the steadystate performance of relatively complex circuits and systems.Numerical examples in which the method is applied to the

analysis of ideal and non-ideal rectifier systems have been

presented to illustrate the advantages of the formulation. A 6-

pulse rectifier system with input filter has been used as a basis

for the examples. It has been shown that the formulation con-

veniently handles input voltage unbalance, actual gating circuitcharacteristics and gating reference signals other than ideal

input voltages. An example of 12-pulse rectifier system under

balanced operation has also been included. Experimentalverifications of the predicted results have been provided.

REFERENCES

[1] K.R. Rao, V.V. Sastry, 'Current-fed induction motoranalysis using boundary-value approach', IEEE Trans.Ind. Elec. & Control Instr., IECI-24, pp. 178-182, 1977.

[2] V.V. Sastry, N. Prabhakaran, 'Digital simulation ofvoltage-controlled arc motor drives', Electric Machine &Electromechanics, Vol. 4, pp. 233-253, 1979.

[3] P.C. Krause, TA. Lipo, 'Analysis and simplified represen-tations of rectifier-inverter induction motor drives", IEEETrans. Power App. & Syst., PAS-88, pp. 588-596, 1969.

[4] T.A. Lipo, F.G. Turnbull, "Analysis and comparison oftwo types of square-wave inverter drives", IEEE Trans.Ind. AppI., IA-11, pp. 137-147, 1975.

'IA

4-'

.1I-

0

(1IV; .

C.

III

Ii "Illz

144

[5] E. El-Bidweihy, 'Steady-state analysis of static powerconverters', IEEE Trans. Ind. Appl., IA-18, 1982.

[6] B.T. Ooi, N. Menemenlis, H.L. Nakra, 'Fast steady-statesolution for HVDC analysis', IEEE Trans. Power App. &Syst., PAS-99, pp. 2453-2459, 1980.

[7] D.R. Kohli, PJ. Purohit, 'Steady-state analysis of phase-controlled single phase induction motor", ElectricMachine & Electromechanics, Vol. 5, pp. 473-484, 1980.

[8] V.V. Sastry, K.R. Rao, 'A numerical approach for theanalysis of inverter-fed induction motor schemes", Elec-tric Machine & Electromechanics, Vol. 3, pp. 157-170,1979.

[9] G.R. Berube, B.T. Ooi, "Fast periodic solution fortwelve-pulse converter", IEEE Trans. Power App. &Syst., PAS-102, pp. 1994-2003, 1983.

[10] L.X. Le, GJ. Berg, 'Steady-state performance analysis ofSCR controlled induction motors: a closed form solution',IEEE Trans. Power App. & Syst., PAS-103, pp. 601-611,1984.

[11] R.W.-Y. Cheung, "The Basis Transformed State SpaceApproach for the Analysis of Power Semiconductor Cir-cuits", M.A.Sc. Thesis, University of Toronto, 1983.

[12] E.B. Shahrodi, "Six-Pulse Bridge AC/DC Converters withInput Filter", Ph.D. Thesis, University of Toronto, 1983.

[13] J.D. Lavers, R.W.-Y. Cheung, "A software package forthe steady state and dynamic simulation of inductionmotor drives", IEEE PES Summer Meeting, SM 312-4,1985.

[14] S.S. Murthy, GJ. Berg, "A New Approach to DynamicModeling and Transient Analysis of SCR_ControlledInduction Motors, IEEE Trans. Power App. & Syst.,PAS-101, pp. 3141-3150, 1982.

APPENDIX I

A Conventional Approach for the Formulation of Initial Vector

Assume that a simple circuit has only two states in aperiodic steady state operation and the corresponding equationsare:

D T(wt) =B_j x(wt) +17w(Wt) otO!o5tso)t (Al)

and

_f( ot ) = e ("t 2.i(t) + f e( )82C2'U()d A

,t(A4)

(otf. wt 1; wt,+2f

Substitute wt =wt and wt = wt,, +2.fr into (A3) and (A4) respec-tively and combine the resulting equations and then utilize thesteady state property (which is x(wt0 +2±ffr) = x )t0 )). Finally theexpression of the initial vector is obtained:

=t)

(6v +2lr l)92. (Wt lst)l] x (A5)

wt1 wt +2w

c("a, +21r-adz)B.f c(adkW1.C i.u(X)dAX+ | ,¢ (a' E2e2-= A@t ~~~~~~~f (w1)L .C2.ii(k)dxj

APPENDIX II

FormulatIon of Typical Matrices

Fig. 8 shows an equivalent circuit for the rectifier systemof Fig. I when thyristors Q1, Q6 and Qs are conducting (state 1)and the filter is ignored. Write loop equations:

v R ia +XaDi (Rd +Rb)ib (Xd+Xb)Db + v (A6)

v = Ra ia + Xa Dia + Rc(ia +ib) + XcD (i +ib) (A7)

Note that in (A6)-(A7), as well as elsewhere in this paper, theoperator D is defined as being the total differential; i.e. withrespect to wt where w is the angular frequency of the supply.As a result of this definition, reactances rather than inductancesappear in the state equations and matrices. Ifu = vab, = V,,,sinwt and u2 = V,. cosWt then combine (A6) and(A7) into matrix form:

Dy(wt) = A-y(wt) (A9)

where state vector y- [aU. ib vd u I u14 and state matrix A:

Formal solutions for (Al) and (A2) are respectively,

at

X (lSt ) =-eC(w I X (<to) + .r"e(0 -A-CIW(A)d Awt.

(A3)

and

van i R

/r-N + a a

Fig. 8 A typical equivalent circuit rectifier.

a d~~~~~

Xa[ +X x v , L1-Ra-R c b±R

0 0 0 0 0

0 0 0 0 10 0 0 -1 0

(A10)

If the circuit is three-phase symmetrical, thenia 4.w /3) ib(Jt) and ib(Wt4+T/3) = ia(t) ia(Wt)T erefore

,f(wt+,r/3) T-r (t)

where the transposition matrix T [ I and x

From Fig. 8, thyristor current iQ1 equals to i,, then

i2 I C15Y

where C1 -[1 0 0 0 0].

(All)

(A)t0 :5 (1)t !' ()t I