chronous machine phasor diagrams, y'ield complete steadystate performance predictions for all types of CSI -
synchronous machine drives. The effect of the inverteris shown to be to eliminate the'normal role of'thesynchronous reactance in establishing equilibrium be-tween the applied and internal voltages.
Applications to self synchronous systems employingrotor position angle or electrical phase angle feedbackas well as to open loop systems are includefd. The cir-cuit representation clearly illustrates the close anal-ogy of self synchronous drives to dc machines. The in-fluence of commutation overlap is also considered andillustrated in an example.
INTRODUCTION
Utilization of a current source inverter (CSI) 4nda synchronous machine as a load commutated ac drive'iswell known. A number of applicatiops of this combina-tion have been reported in the literature [1-3] and an-
alytical methods for deteripining performance have beenpresented [4-7]. In general, the analysis of thesesystems has involved matrix methods employing synchron-ous or rotor-referred machine quantities or has beencarried out for only the ac side of'the system usingconventional synchronous machine phasor diagrams.
As in any inverter drive, a co plete analysis re-
quires relating the dc side inverter variableq and theac side machipe variables. With voltage inverters themacnine voltage is directly related to theinvertervolt-age. Since the dc side impecance is always small in a
voltage inverter, it is often quite acceptable to neg-lect i#s effects and the machine then can'le consideredas being supplied from an ideal voltage source. Sincethis is also the case in most machine applicationsutil-izing a conventional fixed frequepcy supply, machineperformance with a'voltage inverter supply is oftenvery similar to conventional operation of the machine.
With a current inverter, however, it'is the motorcurrent that is directly relate4 to inverter current.The machine voltage is therefore determined not'solelyby the inverter supply voltage but also by the piachinecharacteristics (i.e. the machine power factor). Thisis a much different situation t4an that occurring inconventional 'machine operation and the operating char-acteristic of the drive is profoundly affected by themachine - inverter interaction'. This interaction hasbeen recognized and correctly treated by numerous au-
thors [2-7]. However, the methods employed have beensuch that the relatiqnship to conventional steady state
81 WM 191-6 A paper recommended and approved by theIEEE Rotating Machinery Committee of the IEEE Power
Engineering Society for presentation at the IEEE PES
Winter Meeting, Atlanta, Georgia, February 1-6, 1981.
Manuscript submitted S tptenber 2, 1980; made available
for printing November 14, 1980.
machine theory (equivalent circuits, phasor diagrams)has not been clear. In an effort to develop an approachwhich would make 'possible the use of conventional theory,a pair of equivalent circuits for representing voltageand current inverters has been developed [8-10]. Thesecircuits have proved useful in explaining inverter-ma-chine interactions tp students and others familiar withconventional machine theory. This paper presents theapplication of these concepts to the CSI-synchronousmachine drive. The resulting overall equivalent circuitsprovide a sipiple means for understanding the strong in-
teraction of the inverter and the machine. The abilityof the inverter to completely alter the reactance volt-age equilibrium of the machine and produce entirelynew overall characteristics is clearly shown. It shouldbe emphasized that the results expressed by the newequivalent circuits are in full agreement with the re-
sults obtained by other methods. Only the point of viewand the circuits therpselves are new.
BASIC CSI - SYNCHRONOUS MACHINE SYSTEM
The basic system to be modeled is illustrated inFig. 1. It consists of three primary components; asynchronous machine, a contrpllable dc source with a
dc link inductor, anld a variable frequency CSI. Ap-propriate controis for the synchronous machine fieldexcitationi, the firing of the CSI thyristors and thelevel of the dc source voltage are also part of thesystem. The exact nature of the controls and the or-
igin of the control signals are arbitrary -since themodels to be developed only apply to the steady stateand will describe any system with the basic. CSI-synch-ronous machine-configuration of Fig. 1.
Neglecting thyristor voltage drop and assuming thecommutation interval is negligible compared to the con-duction interval, it is easily shown that the 1200 con-duction periqd CSI can be characterized as a switchingsystem with six distinct conduction modes [8-11]. Thesemodes and the associated input-output constraints areillustrated in Fig. 2. It is clear from the mode dia-gyams that there is a complete set of constraintson thecurrents, i.e. there is a unique relationship betweenthe dc link current and the inverter output current. Itis also clear that the voltage constraints associatedwith the mode diagrams are not complete since there isalways one open-circuited phase.
Fig. 2. Conduction modes and input-output constraints
If the input current is assumed to be substantiallyconstant over one cycle and attention is confined tosteady state conditions, the output ac current is thefamiliar 1200 rectangular wave illustrated in Fig. 3.A first harmonic representation of this output currentcan be obtained by retaining only the first termr in the
FUNDAME N TALCOMPONENT
LINECURRENT
Fig. 3. Output current waveform and fundamental com-ponent for CSI.
Fourier series of the rectangular wave.relation between the dc link current I.amental line current I is
I = - I.71 1
The resultand the fu
Lingind-
(1)
As noted previously, there is not a unique relationbetween the input dc voltage V. and the ac output rmsvoltage V<. However, a constraint relation can be ob-tained from power balance by equating the input dc pow-er to the output ac power
V .I = 3 V I cos e (2)
Combining (1) and (2) yields
V. V cos e (3)71
as the inverter constraint, relation relating input volt-age to output voltage. The relation involves the loadpower factor cos, 0 and is a well known and acceptedapproximate result describing CSI performance.
Eciuations (1) and (3) can be represented in theform of an equivalent circuit by incorporating a vari-able reactive element to absorb the quadrature compo-nent of voltage excluded in (3). Fig. 4 illustratessuch a circuit where the variable reactive element isshown as a capacitor [8]. The circuit also includes theac side equivalent of the dc link resistance and theper phase rms ac side equivalent of the input dc volt-age Vdc* In effect, the inverter is modeled as a dc toac converter with a voltage ratio of (7r/3/6), a currentratio of (/6/ir) and as a source of reactive power whichalways exactly satisfies the reactive demand of the load.
82-8 Rdc
Fig. 4. Per phase fundamental component -equivalentcircuit of three phase CSI
This reactive supply capability is an inherent propertyof the inverter. Since the load is a balanced threephase load, all that is required is the ability totransfer the stored energy from phase to phase. Forinductive loads this is accomplished by the commutationof current from phase to phase via the commutation cir-cuits. With a capacitive load it occurs as a resultof the natural commutation of the currents and the ex-istence of the voltage on the open circuited phase. Inmany of the examples in this paper, the series reactiveelement is inductive because the load power factor ispurposely maintained leading to obtain load commutationcapability.
NON-SALIENT POLE SYNCHRONOUS MACHINES
The simplest case to handle is the non-salient polemachine since all that is required is to connect themachine equivalent circuit to the inverter equivalentcircuit. The salient pole machine is somewhat more dif-ficult since a machine equivalent circuit does not ex-ist. This case is, therefore, postponed until the basicanalysis of the simpler case is completed.
Complete Equivalent Circuit
Combining the equivalent circuit of Fig. 4 with theconventional equivalent circuit of a non-salient polesynchronous machine yields the complete circuit shownin Fig. 5. All of the voltages and currents in thiscircuit are rms per phase values and, since it is com-mon to all parts of the circuit, the current is takenas the reference. The machine internal voltage Eq isshown with an angle y which is measured with referenceto the current; y is therefore the internal power fac-tor angle. Although shown as a variable capacitor, theseries reactive element is always adjusted to absorbthe part of the terminal voltage in quadrature with the
CURRENT SOURCE SYNCHRONOUS
INVERTER MACHINE
3a2RdC IRs jX,
+ .I0aVC0°( Vcos eo,o vzeaVdc/Z0 EqLy
Fig. 5. Complete per phase fundamental component equiv-alent circuit for CSI driven non-salient polesynchronous machine.
2922current and in many cases will actually be inductive.The voltage ahead of this element will be V cos e ata phase angle of zero degrees with respect to the cur-rent as shown in Fig. 4.
Phasor Diagrams
Before reducing this circuit to its simpler form itis instructive to examine several phasor diagrams as anintermediate step. Fig. 6 illustrates several diagramsdrawn for a fixed value of V cos 0 and a fixed valueof internal angle y. If Rdc is small, a fixed valueV cos e corresponds to a fixed input voltage Vdc andthis is a useful way to visualize the behavior illus-trated in Fig. 6. The left hand diagram neglects stator
aVdcZoo Eq cosYZOO
Fig. 7. Reduced per phase fundamental component equiv-alent circuit for CSI driven non-salient polesynchronous machine.
for the I R losses. The resultEqIcosy
T = qn - = K4f I cos Y (4)
\ V2
VI
Eq2'Eql %
E qo
b) R included
of CSI driven non-salient poleRdc 0 Vdc = constant,
resistance Rs and shows that increased motor currentsimply causes an increased IX5 drop and reduced ter-minal voltage. In all cases the quadrature voltageV siTn would be absorbed by the series reactive ele-ment which would be inductive in this case. Note thatE would remain fixed for this idealized case.
The right hand diagram includes the effect of sta-tor resistance. In this case E must reduce in mag-
qnitude as the current increases; under the stated as-
sumptions this would require a reduction in motor speedand frequency. As before, the quadrature voltageV sin e would be entirely absorbed by the series re-
active element.
Reduced Equivalent Circuit
It is clear from the complete circuit of Fig.5 andthe phasor diagrams of Fig. 6 that, because of thecompensating effect of the series reactive element, thequadrature components of the various system voltagescan be eliminated from the equivalent circuit. Thisleads to the reduced equivalent circuit of Fig. 7 whereonly the in-phase component of each voltage has beenretained. Note that since the IX5 drop is entirely a
quadrature component, X5 disappears from the circuit.The voltage Eq cos y is the component of Eq which isin-phase with the current I. Although the circuit isan ac circuit, all the voltages and currents are in-phase and ordinary algebra can be used to relate thevarious quantities. Note the similarity of the circuitto that relating armature quantities in a dc machine.
Calculation of the developed torque is a simplematter of calculating the power input and accounting
where w is the operating frequency, n is the numberof poles, q is the number of phases and
Eq~f w (5)
is the open circuit flux produced by the field excita-tion. The definition in (5) completes the analogy tothe dc machine. It is important to emphasize that al-though the phasor diagrams of Fig.6 are drawn for fixedvalues of V cos 6 and y, the circuits of Fig. 5 andFig. 7 are completely general and apply regardless ofthe variation of the various quantities.
SALIENT POLE MACHINES
The analysis of a salient pole machine is somewhatmore involved because of the more complex relationshipsdescribing the synchronous machine. In fact, no coun-terpart to the complete circuit of Fig.5 exists becausethere is no stator equivalent circuit for the salientpole machine. However, as is demonstrated below, thereis a reduced equivalent circuit which describes a CSI-salient pole synchronous machine system.
Phasor Diagram
Fig. 8 is a phasor diagram for a salient pole ma-chine operating at an internal power factor angle y. Asa result of the existance of the two reactive voltagesjIdXd and jIqXq, the total voltage drop caused bythe current is no longer in quadrature with the current.Therefore, unlike the situation illustrated in Fig. 6a,
Xq
iId Xd
Fig. 8. Phasor diagram of CSI driven salient polemachine with Rdc =0 R = 0.
dc
II I2 I3 Vcos8c - tV31
Ia) Rs neglected
Fig. 6. Phasor diagramsmachine withy = constant
operation with fixed Vdc and fixed y does not re-sult in a fixed value of Eq. Hence, even neglectingall resistance, the speed of a salient pole machinewith fixed y will vary as the load changes unlike thecase of the non-salient pole machine. This suggeststhat there must be an equivalent resistance resultingfrom the saliency that appears in a reduced equivalentcircuit similar to Fig. 7.
2923
Three examples are included to illustrate the useof the equivalent circuits with widely different setsof constraints. The first two represent practical sys-tems which have been described in the literature. Thelast example is an open loop system included to illus-trate a third type of constraint.
Constant y Self-Synchronous System
Reduced Equivalent Circuit
The expression relating the in-phasFig. 8 is
V cos 0 = Eq cos y - IdXdcosy + I X '
Using the relations between I and Id and I,
Id =I sin y
(6)
Iq = I cosy
V cos 0 = Eq cos Y + I sin y cos y(Xq-xeor as
Xq"XdV cos Y = Eq cos y + I sin 2y(
Equations (8) and (9) clearly show the aephase voltage drops caused by saliency an(reduced equivalent circuit shown in Fig.appearing as an equivalent resistance, theis not dissipated as heat in the machineverted to mechanical power. This power i
Req = I (Xq-Xd)sin2Y
Fig. 9. Reduced per phase fundamental comi
alent circuit for CSI driven s
synchronous machine.
with themachine.from thebe
reluctance torque produced in theThe torque equation can again
power in the circuit and is reac
T = qnvEqIcosy I (Xq-Xd)sin2y
APPLICATIONS
In applying the circuits developed ining sections to specific CSI-synchronous me
the nature of the constraints imposed bycontrols utilized in the drive determinesneeded. To completely evaluate any drive rthe equivalent circuit and the phasor diagicases the equivalent circuit alone determirtorque relation and the phasor diagram isto evaluate internal variables. In otheithe circuit and phasbr diagram are neededspeed-torque relation. In every case, Ecancellation of the reactive voltage and tsuppression of the normal role of the synci
tance has a very significant effect on pern
One of the most widely used systems [1,3,6,7] em-ploys a position sensor to measure the rotor position
e voltages in and uses the resulting information to control the fir-ing of the inverter thyristors. This gives direct con-
siny (6) trol of the internal power factor angle y and causessiny (6) the machine to be self-synchronous, i.e. the frequencyq always matches the rotor speed. In practice the angle
y is often programmed to be some function of load cur-(7) rent to obtain optimum performance from the machine.
Often the field excitation is also programmed to fur-*ther enhance machine utilization. Regulating y and
d) (8) If to produce operation as close as possible to unitypower factor while ensuring self-commutation is onepossible strategy.
With independent (position feedback) control of ythe equivalent circuit of Fig. 7 or Fig. 9 clearly
(9) shows a close analogy to the dc shunt machine. The
dditional in- back emf Eq cos y adjusts to the equilibrium value re-d suggest the quired to allow the necessary motor current to exist by9. Although means of changes in Eq (since y is fixed). For a
power in Re fixed value of If this can only be accomplished bybut is con- changes in rotor speed. This is exactly the mechanismis associated that applies in a dc machine. The speed-torque curve
is therefore identical to that of the dc shunt motorexcept that the total circuit resistance in Fig. 9 canbe either positive or negative depending on the rela-tive sizes of Xd and Xq. For the conventional caseofXd > Xq, Req is negative and the total resistance maybe negative. This would lead to a speed torque curve
)E COSY100with a speed rise as the load increased. Note that the
)EqCOSYLO0 value of Req depends on sin 2 y and hence the totalcircuit resistance could be either positive or negativedepending on y.
The machine voltage and power factor can be foundby using conventional phasor diagrams like those inFigs. 6 or 8. Thus, for example, after determining thespeed and current corresponding to a particular torqueload from the equivalent circuit the phasor diagram canbe employed to find V and 6 for the calculated val-
ponent equiv- ue of I and w. Figures 10 and 11 illustrate the ex-salient pole ternal characteristics of this type of drive as evalua-
ted by the equivalent circuit approach of this paper.Note the close similarity to a dc shunt motor as is
salient pole suggested by the form of the equivalent circuit.be obtained
lily shown to Constant e Self-Synchronous System
Rather than utilize a mechanical position sensor,(10) many systems have been designed using some electrical
angle as the control variable [2,4,5]. This not onlyavoids the requirement of a position sensor but can al-so have advantages in making better use of the machine(by reducing excessive overexcitation) while ensuringself-commutation. As an illustration, consider a con-
the preced- troller set to fire the thyristors in the inverter sochie drecive as to hold the power factor angle 0 constant. Phasor
the approach diagrams illustrating this mode of operation for a non-
requires both salient pole machine are shown in Fig. 12.
ram. In somet As can be seen by examining Fig. 12, an increase
in load current from I to I requires a decreasenies the speed- l 2needed only in Eq cos y, an increase in y and an increase in Eq.
r cases both This is significantly different than the fixed y sys-I to find the tem of the preceding section. Note that unless If isiowever, the programmed to increase as the load current increases,theresu
the increase in E will cause the speed to increase.the resulting This would result in a rising speed-torque character-ronous reac- istic.formance.
I6
It/ -I I I I
0 0.2 0.4 0.6 0.8 1.0
Vdc = 0.5.I Y = 50
RS= 0.05RdC = 0.0Xd= 1.0
Xq = 0.7
0 0.2 0.4 0.6 0.8
Per Unit Torque
Fig. 10. Performance characteristics for CSI drivennon-salient pole machine operated with con-
stant Y.
The analysis of this mode of operation is againaccomplished using the equivalent circuits of Figs. 7or 9 and the corresponding phasor diagrams. However,unlike the previous case it is now necessary to use thecircuit and the phasor diagram simultaneously to eval-uate performance. For example, to evaluate an operatingpoint like that associated with I2 in Fig. 12, an ap-
proach would be:
1) From V and I2 find V cos 0 and hence Vdc I
(for the given controlled value of 0) from theequivalent circuit.
2) Find E and y from a phasor diagram likeFig. l.
3) Find the speed from Eq (or I2X5).
4) Find the torque from Eq. 4 or Eq. 10.
Thus, although the order of steps is altered and theresults are quite different, the approach to the anal-ysis is similar to the previous example. Fig. 13 is an
illustration of the characteristics of a machine whenoperated with constant power factor. Other electricalangles such as the angle between air gap voltage andcurrent have also been employed for control [2] and canbe analyzed by the method in this section.
Constant Speed Synchronous Machine
As a final example, consider an open loop systemwhere Vdc, field current and thyristor firing (fre-quency) are all independently adjusted. In this casethe speed is fixed by the input command to the inverterand the machine must operate as a conventional synchro-nous machine. However, the equivalent circuits of Figs.7 and 9 still apply; in this case E is fixed and theangle y must vary to satisfy the circuit constraints.For a given load torque or load current the equivalentcircuit determines the value of E cos y and hence y.A phasor diagram can then be utilized to find the vol-
Per Unit Torque
Fig. 11. Performance characteristics for CSI drivensalient pole machine operated with constantY.
V Cos 9
Fig. 12. Phasor diagram of CSImachine operated withVd = constant).
l V
IJR
\ -j II XS
Eqi
j I2Xs
Eq2
driven non-salient poleconstant e (Rdc = 0
tage and power factor. The phasor diagram in Fig. 14illustrates the characteristics of this mode of opera-tion. Note that for an increase in load current (ortorque) y increases and V and e decrease. Note alsothat if Eq is not larger than V cos 0, the circuit cur-rent will become large and acceptable performance isnot attainable.
EFFECT OF COMMUTATION OVERLAP
The assumption of zero commutation time in mostself-commutated CSI synchronous machine systems is an
over-simplification since it is uneconomical to design
2924
0.;
2925
the circuit of Fig. 5) and only correct for the changein phase.
To illustrate one procedure for incorporating com-mutation overlap, consider the constant y system il-lustrated in Fig. 10. Neglecting overlap, this systemoperates with a constant internal power factor angle ybecause the inverter current is assumed to respond in-stantaneously to the command of the rotor position feed-back loop. With overlap there is a-delay in this re-sponse resulting in a reduction of the actual angle yfrom the pre-set value y0. The amount of this delay isdependent on the amount of overlap which in turn de-pends upon the load current and the value of yo. Thus,an iterative procedure can be used to obtain the solu-tion by employing well-established approximate methodsfor evaluating the overlap angle.
One widely used procedure [12,13] for evaluatingcommutation overlap makes use of the commutating re-actance Xc and the voltage behind commutating reac-tance E". The solution for the commutation overlapangle ,u requires simultaneous solution of the twoequations [12] TV
[1I- IX(1-q)siny]cos- 6F E
Eq I 6Eqcos 6 =
El"/Eq2 dcxccos($--p) = Cos 0 + I
(11)
(12)
Per Unit Torque
Fig. 13. Performance characteristics for CSI drivennon-salient pole machine operated with con-stant 0.
II 12 vcosG
Fig. 14.
I VIII RsiII x5
Phasor diagram of CSI driven non-salient polemachine operated open loop (Rdc =:-0, Vdcconstant).
for a short commutation interval over all conditions ofoperation. For most systems the steady state limit isset by the allowable overlap during self-commutation.
Fortunately, incorporating overlap does not inval-idate the models developed in this paper since it isquite feasible to treat overlap as a correction to thebasic models. All that is required is to relate thefundamental component of the trapezoidal current wave
resulting from overlap to the currents in the equiva-lent circuits. The result is a change in the multiplierrelating the fundamental component to the dc link cur-
rent and a change in the phase angle between the ideal-ized rectangular current wave and the actual trapezoi-dal current wave. Usually it is sufficiently accurateto neglect the change in the amplitude relation ('a' in
whereX = Xd - Xc qX = Xq - Xc (13)
If the resulting current waveform is assumed to be atrapezoid, the new fundamental component can be shownto be of essentially the same amplitude as for theidealized rectangular wave but phase shifted by an an-gle of p/2.
To evaluate machine performance for the system ofFig. 10 including overlap, the following procedure canbe used:
1. Calculate machine performance neglecting over-lap using the equivalent circuit and phasordiagram.
2. For the condition of '1', evaluate the voltagebehind commutating reactance.
3. Solve equations 11 and 12 for ,.
4. Calculate a modified value of y from
(14)Ynew = old -p/2
5. Repeat steps 1 through 4 until y is known toacceptable accuracy. Usually only a few iter-ations are necessary.
Fig. 15 illustrates the results obtained for the systemof Fig. 10 by this procedure with Xc assumed equal toXj. Comparison of the two figures clearly shows the de-crease in y and e caused by overlap.
SUMMARY
An equivalent circuit approach to analyzing CSI fedsynchronous machines has been presented. The most sig-nificant results are:
1) Simple, single loop, equivalent circuits exist forthe fundamental component behavior of CSI fed syn-chronous machines.
a) The circuits plus the standard synchronous ma-chine phasor diagrams yield all quantities ofinterest.
b) Results are identical to other, less easilyused, analytical methods based on the funda-mental component approximation.
2926
e
0.4 0.6 0.8 1.0Ppr Uni t Tora ue
Fig. 15. Performance characteristics for CSI drivennon-salient pole machine operated with con-
stant y0 including the effects of commuta-
tion overlap
c) The circuits are applicable to all modes of
control and illustrate the close analogy ofthe CSI fed synchronous machine to the dc ma-
chine.
2) The development yields two important new propertiesof the CSI fed synchronous machine.
a) The normal role of the synchronous reactances
in establishing equilibrium between appliedand internal voltage is cancelled by the in-
verter transfer characteristic.
b) The effect of saliency can be represented byan equivalent resistance proportional to(Xq-Xd) sin 2y.
3) The main effect of commutation overlap can be in-
corporated using a simple iterative procedure.
REFERENCES
1. N. Sato, V. V. Semenov, "Adjustable Speed Drive
with a Brushless DC Motor", IEEE/IAS Transactions,Vol. IGA-7, No. 4, pp. 539-543, July, August 1971.
2. A. B. Plunkett, F. G. Turnbull, "Load Commutated
Inverter/Synchronous Motor Drive Without a Shaft
Position Sensor", Conference Record IEEE/IAS Annual
Meeting, October 1977.
3. N. Sato, "A Study of Commutatorless Motor", Elec-
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4. A. B. Plunkett, F. G. Turnbull, "System DesignMethod for a Load Commutated Inverter-SynchronousMotor Drive", Conference Record IEEE/IAS Annual
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1, IEEE Winter Power Meeting, January 1979.
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8. M. Abbas, D. W. Novotny, "Stator Referred Equiva-lent Circuits for Inverter Driven Electric Ma-chines", Conference Record, IEEE/IAS Annual Meeting,October, 1978.
9. D. W. Novotny, "Equivalent Circuit Steady StateAnalysis of Inverter Driven Electric Machines", El-electrical & Computer Engineering Dept.Report, ECE-80-19, University of Wisconsin, June 1980.
10. D. W. Novotny, "Switching Function Representationof Polyphase Inverters", Conference Record IEEE/IAS Annual Meeting, pp. 823-831, October 1975.
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