A GENERALISED OPERATIONAL EQUIVALENT CIRCUIT OF INDUCTION MACHINES FOR TRANSIENT/DYNAMIC STUDIES UNDER DIFFERENT OPERATING CONDITIONS

A GENERALISED OPERATIONAL EQUIVALENT CIRCUIT OF INDUCTION MACHINES FOR TRANSIENT/DYNAMIC STUDIES UNDER DIFFERENT OPERATING CONDITIONS S. S. Murthy Department of Electrical Engineering Indian Institute of Technology New Delhi-11 16. (India) Abstract. The paper derives a generalised operational equivalent circuit for induction machines starting from fundamental principles in any arbitrary rotating reference frame. Directly measurable parameters are used in this equivalent circuit From this generalised model, the equations for any reference frame are derivable as particular cases. The model presented is versatile as it can be used for any operating condition and any asymmetrical connection or switching and for both 3-phase and 2-phase wound machines. The equivalent circuit provides space phase vectors and the relevant expressions can be used to develop software for on line control of motors through field orientations. Different possible modes are considered in this paper to derive relevant equation from the general model to illustrate its potential. 1. INTRODUCTION Induction machine modelling has continuously attracted the attention of researchers not only because such machines are made and used in largest numbers but also due to their varied modes of operation both under steady and dynamic states. Increased use of Power Electronic controllers with such machines makes appropriate modelling and parameter identification crucial both for working control strategies and performance prediction. Induction machines operate in both motoring and generating modes. Till recently motoring operation was almost universal. But recent exploitation of renewable energy systems such as wind and small hydro has led to use of grid connected induction generators driven by wind and hydro turbines. While steady state operation of 3-phase motors at fixed balanced voltage, frequency and load is common, operations at unbalanced voltages and asymmetrical winding connection (1-phase, 2-phase, 3-phase) are often prevalent needing appropriate models to analyse. The power devices used for motor control often cause asymmetrical switching of windings. Transient studies under run-up, reswitching, braking, load change and short circuit conditions too require suitable analytical models. Modelling under induction generator operation driven by wind and hydro turbines has recently assumed importance. Field oriented or vector control requires dynamic modelling under different reference frames for real time transformations. Machines with asymmetrical windings need special equations. This paper presents a generalised model with an operational equivalent circuit from which specific equations for different operating modes can be derived by substituting appropriate constraints. The concepts of generalised rotating field theory and symmetrical component theory have been used effectively for steady state operation under unbalanced voltages or asymmetrical connections [1,2,3]. Mathematically, though not physically, analogous concepts of instantaneous symmetrical components and operational equivalent circuits have been shown to be effective in analysing transients under similar conditions [4-9]. But these were confined to a stationary reference frame with symmetrical stator and rotor windings. The concepts of space vectors of voltage, current and flux-linkages [1,11] used in field orientation and field acceleration method of control of motors have a direct relation to instantaneous symmetrical components representing dynamically changing complex quantities which can be denoted by voltage, current and flux linkage vectors. Further, the equivalent circuit is an easily understandable concept for a student working on induction machines, wherein the parameters can be determined by test or design, It is possible to incorporate non linear variation of these parameters under different operating modes dynamically in a program. For example studies on capacitor self excitation require simulation of non-linear magnetising reactance. This is true in any situation involving flux control. It has been experienced by the author that operational equivalent circuit leads to model equations under asymmetrical operation more elegantly compared to alternative methods using direct d-q variables. The generalised operational equivalent circuit presented in the paper relates to an arbitrary rotating reference frame such that specific equation for any reference frame can be evolved as a particular case. The concept may be extended for asymmetrical 2-phase or 3-phase winding configurations, which may also include zero sequence circuit. 2. DERIVATION OF GENERALISED EQUIVALENT CIRCUIT A symmetrical induction machine can be considered to have 3 phase symmetrical stator windings sa, sb, sc and equivalent short circuited 3 phase symmetrical rotor windings ra, rb, re (Fig. 1) or 2 phase symmetrical windings sa, sb and -783-2795- 622

(b) Fig.1 Winding Orientation of a Symmetrical Induction Machine (a) 3-phase ra, rb at quadrature. Let R» R, be resistance of each stator and rotor winding and L^ L, be self inductances. Let M^, M,,, be mutual inductance between any two windings in stator and rotor respectively. Let M be the mutual inductance between a stator and a rotor winding when they are aligned. Let the rotor speed be u, elect, rad/s. Let us consider an arbitrarily rotating reference frame at «o elect, rad/s on which d and q axes at quadrature are fixed (Fig, 2). If 6 is the angle between sa and ra axis at any instant, we may assume that all inductances expect those between stator and rotor windings are independent of e. Neglecting space harmonics the mutual inductance between stator and rotor windings may be considered to be varying sinusoidally with. Transformations are resorted in machine theory to make the inductance parameters independent of 6 in the transformed plane. In the arbitrarily rotating frame both stator and rotor quantities are transformed to s d, s q, s o and r d, r q> r o windings using the transformation matrix, [C] where ICJ (1) (b) 2-phase Transformed Windings on Arbitrary Rotating Frame where the time derivative operator p= If we denote [V], P'], [R«], [L'] as the corresponding matrices in the transformed plane it is possible to show that where an (2) (3) (3a) Starting from the original [R], [L] & [G] matrices and using the transformation of (1) we get [R'], [L'J & [G'] as given in Appendix-I, from which eq. (3) leads to the following transformed equations. cos8- -dnd JS) -^(6 A -L 3 3,/2 (la) (4b) (4c) c«<e.-e) 42 3 J i (lb) (4d) Here e o = w o t and = «, t Denoting [V], [I], [R], [L] as voltage, current, resistance and inductance matrices and [G] as derivative of [L] with, the basic volt-ampere equation is (4d) (4Q 623

Following relations on inductances are valid L K - M sm - L,, + (3/2) L^ K - M ra - L,r + (3/2) L^ where L^, Lj r are stator and rotor leakage inductance per phase all referred to stator turns, L^ is the mutual inductance of all windings referred to stator turns. At the base radian frequency eo, following reactance can be defined Stator leakage reactance/phase, x,,=<i> L^ Rotor leakage reactance/phase (ref. to stator) x Jr "<«> r Same holds true for currents and rotor voltages. Since negative sequence quantities are complex conjugates of positive sequence ones, only the latter would suffice to define the system. Defining operator />'..«_ p.u. speed of rotor v= -. p.u. speed of rotating frame v a = Magnetizing reactance x,,, = u> (3/2) L ra The reactances x b, x, r and x^, are the standard parameters of the per phase equivalent circuit of an induction machine defined at the base frequency. Referring all the quantities to stator turns, eq. (4) now simplifies to eq. (5) through (6) yields, (7a) x,. x_ x. (5a) (7b) The above equations can be represented by the operational equation circuit of Fig. 3, with the following notations X ls CO (< >.-< > (» -«,) (5b) (5c) <5d) v, i R, x v, v p' < > : voltage, current. : Resistance, reactance : p.u. rotor speed and speed of rotating frame : operator, (1/co) (d/dt) : base radian frequency Subscripts : s, r : stator, rotor 1, m : leakage, magnetising Superscripts : +, - : positive, negative sequence. The general equivalent circuit is derivable from symmetrical 3-phase or 2-phase machine and only the transformation matrices [C ffl ] and [CJ of {1) would differ as also the inductance parameters. For negative sequence quantities eqs. (7a) & (7b) can be modified by replacing the superscript " + ' by "-" and the operator j by -j. Same changes apply for Fig. 3 to obtain the negative sequence operational circuit. It is possible to write a zero sequence operational equivalent circuit using zero sequence voltage equation of eq. (5) as shown in Fig. 4 (5f) General torque expression is 2.1 Generalised Operational equivalent Circuit Using eq. (5) we may attempt to derive a generalised operational equivalent circuit, by effecting so called instantaneous symmetrical component transformation, by defining [4-7] n 1 1 -/ 1 ; 1 (6) Iq. terms of transformed quantities On substitution from eqs. (1) (8) Referring all terms to stator turns, general torque 624

"Is"» rp Fig.3 Generalised Operational Equivalent Circuit expression in terms of sequence quantities is (o) stator (b) Roto' fig.4 Zero Sequence Operational Equivalent Circuit R* x ls p> "Ir"' 3. MODEL FRAMES WITH PARTICULAR (9) REFERENCE In practice we need model equations in different frames; for example the d-q axes fixed to stator, rotor or rotating at synchronous speed. The operational equivalent circuit for each frame can be obtained by simple substitution illustrated below. 3.1 Stationary reference frame Here > o =, e o =, v o =, the matrices [CJ, [CJ of eq. [1] are simplified such that [CJ becomes a constant matrix leaving only [C J dependent on e. With these substitution, the operational equivalent circuit is as in Fig.5, from which the relevant volt-ampere and torque expression can be obtained. 3.2 Rotor reference frame Here d-q axes are fixed to rotor and & ~Q, &> o = &> r, v o = v. The matrices [CJ, [CJ of eq. [1] are simplified such that [CJ becomes a constant matrix leaving only [CJ dependent on 8. With these substitution the operational equivalent circuit is as in Fig. 6, from which the relevant voltampere and torque expressions can be obtained. 33 Synchronously rotating reference frame Here &> o = to, v o = 1. Here both [CJ and [CJ of eq. (1) are 6 dependent. Defining p.u. slip s = 1 - v, and with the above substitution the operational equivalent circuit is as in Fig. 7, from which the relevant volt-ampere and torque expression can be obtained. 4. SOME IMPORTANT INTERPRETATIONS OF THE EQUIVALENT CIRCUIT Before looking at some special applications of the general circuit, it is important to identify and interpret some special features. With due idealisation, it contains measurable resistance and reactance parameters, whose nonlinear variation with operating conditions such as temperature, flux, current and frequency can be accounted for. The terms v,*, i,* and i r * are complex numbers often changing with time. They can also be interpreted as space phasors adoptable for field orientation or field acceleration through real time modelling [1,11]. From the general Fig.5 Operational Equivalent Circuit for Stator Reference Frame Fig.6 Operational Equivalent Circuit for Rotor Reference Frame Fig.7 Operational Equivalent Circuit for Synchronously Rotating Reference Frame equivalent circuit appropriate model equations for developing the software for.field orientation/acceleration can be developed along with relevant transformations for the chosen reference frame. The current vectors i, + and i + r respectively represent net stator and rotor fields. By controlling the real and imaginary components of i, + independently, the stator flux can be oriented. The resultant flux is caused by V = i, + + ir + The magnitude of i, +, i/, i^,* and angle between them will have a bearing on torque or power development as per eq.(9). The above space phasors'are either rotating or stationary under steady state, depending on the frame chosen. They rotate at synchronous speed (each with fixed magnitude) with stator frame and slip speed with rotor frame. They are stationary at synchronously rotating frame. Further detailing of these concepts is beyond the scope of this paper. Equations (1) & (6) help in obtaining space phasors from original machine voltages and currents. 625

5. APPLICATION OF THE GENERAL EQUIVALENT CIRCUIT TO PARTICULAR CASES The general operational equivalent circuit has been found to be useful in deriving model equations for almost all practical cases. Only a few cases are presented here for illustration. 5.1 Transients ander motoring, generating and braking Transients in induction motors and generators can be estimated through suitable modelling. The operational equivalent circuit with stationary frame can be employed. For the given voltage and torque conditions currents and speed can be determined by deriving the expressions for derivatives of currents p'[i] and speed as given in Appendix-II where subscripts x and y denote real and imaginary parts of complex quantities. By choosing suitable initial conditions of [i] and v, transient response with time can be computed at different time steps using Runge-Kutta method. Motor transients under run-up, reswitcaing and load change can be obtained by setting suitable initial conditions and effecting the transient input change. Typical transients in induction generator relate to aided starting, (k is started as motor with additional torque provided by the prime mover) change of input power and switching the generator to the grid. Transient under counter current braking or d.c. injection are obtained by injecting respective voltage or current at the operating speed v. Throughout the transient process resistance and reactance parameters need to be suitably chosen or changed if required. 52 SYMMETRICAL INDUCTION MOTOR WITH 1- PHASE SUPPLY There are several practical situations wherein a symmetrical induction motor (3-phase or 2-phase) is connected to a 1-phase supply. Dynamic models are obtainable from the general equivalent circuit as follows. sb! sb Fig.8 2-Phase Motor With 1-Phase Supply \ = Z, i, If the zero sequence component is present v. = Z, i. From eqs. (1, 11, 12) we get V =( 1) 2 which can be shown by the equivalent circuit of Fig.9 (12b) (12c) 13) 52.1 2-phase symmetrical induction motor with 1-phase supply across one winding Fig.8 shows a 2-phase machine' with symmetrical btator windings sa and sb (with symmetrical rotor). The sa winding is connected to a single phase supply, such that the applied voltage v = v M Further i,,, =. In terms of sequence voltage v s following relations hold v ( v v / > Since i^, =, and the (1) Fig.9 Operational Equivalent Circuit With 1-Phase Supply 3-phase induction motor with 1-phase supply across one winding Fig. 1 shows a 3-phase star connected induction motor (with symmetrical rotor) with 1-phase supply across one winding. Here (14) '«'» r-» (11) From the operational circuit of Fig. 4, we may write V=W (12) where Z, + = operational positive sequence impedance. Similarly we may write for negative sequence (15) Note that zero sequence component is also present here. Combining eqs. (12, 14, 15), we get 626

(17) Rg.lL Fig.1 3-Phase Motor With 1-Phase Supply ^ - j - ^. (16) which can be shown by the equivalent circuit of Further v/3 (18) Combining (12,. 17, 18), we get V = (Z, + + Z>) I,, leading to the operational circuit of Fig. 13. There are several practical situations when such a single-phasing operation occurs such as sudden disconnection of a supply line and in-line-thyristor switching. The modelling of such systems using the operational circuits has been reported[6]. 52.4 3-phase delta connected motor with 1-phase supply Similar to star connected motor, modelling and analysis of a delta connected motor with 1-phase supply becomes necessary e.g. sudden line disconnection or use of inline thyristor for voltage control. For the connection of Fig. 14 following terminal relations hold v sb which yield Fig. 11 Operational Equivalent Circuit of 3-Phase Motor With 1-Phase Supply Logically equation (13) & (16) and circuits of Figs.8 & 1 must be same. While stator leakage impedance matches by including the zero sequence impedance, air gap impedances appear to differ. It is important to note that the air gap impedance parameters x,,,, x^ and x,, in Fig. 9 and Fig.ll are not numerically same, being referred to 2-phase and 3-phase systems respectively, The parameters of the 2- phase system would be (2/3) times those of the 3-phase system, as the transformation matrices of eq. (1) differ. 5.23 3-phase star connected induction motor with 1- phase supply across 2-windings For the connection shown in Fig. 12, where 1-phase supply is connected across both sa and sb phases. -, v K 1 Leading to the operational circuit of Fig. 15. 5.2.5 3-phase motor with 1-phase supply with phase converter It is possible to operate a 3-phase star or delta connected motor with 1-phase supply using a static phase conyertor, which is normally a capacitor across two of the terminals. Dynamic modelling and analysis can be elegantly handled using operational equivalent circuits. If a capacitor of reactance x,. at base frequency is connected across the terminals a, c of Fig. 12, the following terminal relations hold v - V M 1- v* = Methodology of modelling and analysis has been detailed in an earlier paper [7], using the equivalent circuit, for both star and delta connected cases. 627

S3 Analysis under capacitor self excitation Induction machines driven at any speed experience self-excitation and operate under generating or braking modes with sufficient terminal capacitors. Under balanced conditions the operational equivalent circuit of Fig. 3 can be used. It is important to note that x m in Z, + is a variable parameter changing from unsaturated to saturated value as the voltage builds up. The- loop equation 'so need to be solved with suitable initial conditions, to obtain the response. For determining the voltage build-up at a fixed speed as a self excited induction generator this equation should be solved numerically keeping v constant and choosing a small initial value of current. The value of x,,, has to be continuously changed during integration. To analyse under capacitor braking, the same equations have to be solved but choosing suitable initial values before braking and allowing the speed to change due to mechanical factors and the braking torque. Capacitor - excited generator or braking action is prevalent [12] even with a single capacitor connected across two of its terminals. This is similar to replacing the voltage source with a capacitor in Fig. 12 which leads to the circuit of Fig. 16, with the operational impedance xjp' replacing v, such that Fig.13 Equivalent Circuit for Connection of Fig.12 "5b Hg.14 Delta Connected Motor on 1-Phase Supply This equation can be solved for both generating and braking modes but employing suitable value of x^, in each iteration. T Z<f Fig.12 1-Phase Supply Across 2-Phases 5.4 Other cases with stator frame of reference A few other cases can be cited wherein the operational circuit with stator frame of reference may be tried. Two-phase induction motor with 1-phase supply is one such case with a capacitor for phase shift. In this paper the cases with asymmetrical windings have not been considered wherein the windings may have unequal turns or arbitrary phase displacements. It is hoped that the concepts enunciated in this paper may be extended to such cases too similar to steady state operation with gene~alised rotating field theory [1]. The transformation through the general operational circuit can be employed for real time-on line modelling for inverter fed motor control with field orientation. Necessary equation to effect intelligent control can be evolved. Fig.15 Equivalent Circuit for Delta Connected Motor with 1- Phase Supply P' Z*(P) Fig.16 Equivalent Circuit With Capacitor Self-Excitation 628

5J5 Modelling with rotor unbalance Unbalance in rotor is prevalent mostly in slip ring motors due to asymmetrical switching or unbalanced external impedances connected across rotor terminals. Power electronmtconvertors in rotor circuits for slip energy control may often result in rotor unbalance. Start-up of slip ring motor with unbalanced external starter resistors is another practical case. The modelling through operational circuit in the rotor frame of reference can be effected in such cases especially under asymmetrical/unbalanced conditions. -** Only one case of single-phasing in rotor circuit when onw of the three star connected rotor phases is open circuited is considered here, while other cases can be taken up for further study. The corresponding rotor circuit is shown in Fig. 17. The operational circuit of Kg. 5 with rotor frame can be written with voltage sources across rotor terminals as in Fig.18 and v r ' can be written from Fig. 5 and its complement as V + xj'a:*o M x w p%+q (22b) Denoting the complex terms i, + and i," in terms of real and imaginary parts as Z-i.+K (23a) and substituting in (21) through (2), (22), (23), we get (24) writing the stator voltage equation from the stator current loop of Fig. 5 can be simplified as Fig.17 1-Phasing in Rotor V*V<**«>V v <V*J l >^ V^v V«(26) Fig.18 Equivalent Circuit for Fig.17 Following relations holds true for Fig. 17 = V 3 rt> T, which yield (19) (2a) (2b) Equations (24-26) are the governing volt-ampere equations with single-phasing in rotor. Combined with appropriate torque equation, total dynamics can be studied through numerical methods. [6-8] 6. CONCLUSIONS The paper presents a generalised operational equivalent circuit for induction machines for different possible operating conditions specially when external asymmetries are involved. The general circuit for an arbitrarily rotating reference frame yields specific models for a particular frame with simple substitution. Several practical cases including stator and rotor asymmetries can be elegantly handled using this general circuit as demonstrated in the paper. Considerable scope exists towards varied applications of the proposed model and also for extending the same for machines involving internal winding asymmetries. The non-linear parameters can also be easily accounted for a dynamic simulation. The circuit can also be used to explain field orientation. 7. REFERENCES 1. J. B. Brown and C. S. Jha, " Generalised Rotating Field Theory of Polyphase Induction Motors and its Relationship with Symmetrical Component Theory ", Proc. IEE, Vol. 19A, pp 59-69, Feb. 1962. 2. J. E. Brown and C S. Jha," The Starting of a 3-phase Induction Motor Connected to a 1-phase Supply System ", Proc. IEE, Vol. 16A, pp Ig3-19, April 1959. 629

3. C. S. Jha, " The Starting of Single-phase Induction Motors having Asymmetrical Stator Windings not in Quadrature ", Proc. IEE, Vol. _ 19A, pp 47-58, Feb. 1962 4. J. E. Brown, P. Vas and S. S. Murthy, * Instantaneous Operational Equivalent Circuits of 3-phase Motors having Current Displacement Rotors ", Proc. International Conference on Electrical Machines, Athens, Sept. 198. 5. S. S. Murthy, Bhim Singh and A. K. Tandon, " Dynamic Models for Transient Analysis of Induction Machines with Asymmetrical Winding Connections *, Electrical Machines and Elcctromechanics (USA) Vol. 6, No. 6, 1981, pp 479-492. 6. S. & Murthy and GJF. Berg, " A New Approach to Dynamic Modelling and Transient Analysis of SCR Controlled Induction Motors *, IEEE Trans, on Power Apparatus & System, No. 9 SepL 1982, pp 314-315. 7. S. S. Morthy G. J. Berg, Bhim Singh, C. S. Jha and B. P. Singh, ' Transient Analysis of a 3-phase Induction Motor with Single-phase Supply ", (paper No, 82wm 228-5), IEEE Trans, on Power Apparatus & Systems, VoL 12, Jan. 1983, pp 28-37. 8. S. S. Murthy and G. J. Berg, " Transient Analysis of Induction Motor with Plugging, Operation ", Canadian Electrical Engineering Journal, VoL 9(1), 1984, pp 22-28. 9. S. S. Murthy and G. J. Berg, " Improved Dynamic Model for the Transient Analysis of SCR Controlled Induction Motors ", J.I.E (India), VoL 66, Oct. 1985, pp 57-65. 1. Sakae Yama Mura," A.C. Motors for High Performance Applications ", Marcel Deckker, 1986. 11. P. Vas," Vector Control of A.C Machines ", Clarendon Press, Oxford, 199. 11 S.& Murthy, G. J. Berg, C. S. Jha, and A. K. Tandon, " A Novel Method of Multistage Dynamic Braking of Three Phass Induction Motors ", IEEE Trans, on Industry Applications, April 1984. pp. 329-334. Appeedis-I The matrices R', L' and G' of eq. (3a) on substitution yields Appendix-II In Fig. 5 v, +, i, +, i, + are complex quantities and can be written as Relevant loop equations for derivative of current as [6,7] where x, x m V *r x_ x. The time of derivative of speed is [6.7] and i/ yield the time R, R, R, R r M^ 2<*H where T e is developed electromagnetic torque from eq. 9, T L is the load torque and H is the inertia constant. -M L n -M -M -~M -( 63