Transient analysis of three phase reluctance motors fed from a single phase supply

M.A. Badr A.I. Alolah

lndexing terms: Transient analysis, Three phase reluctance motors, Single phase supply

Abstract: The inherent advantages of three phase reluctance motors have called for their use in applications requiring constant speed. This has created a growing interest in this field, and con- siderable effort has been spent in analysing the steady state and dynamic behaviour of these motors. Single phase operation of three phase reluctance motors has been suggested recently as a practical possibility that extends the range of applications of such motors and adds flexibility to their modes of operation. Previous work in this direction has been very scarce and is limited so far to the study of steady-state operating conditions. The main objective of the paper is to investigate the transient process of three phase reluctance motors fed from a single phase supply. For this purpose a rigorous mathematical model has been developed. The model, which is based on the state space form of representation, is then simulated and the transient characteristics for such a mode of operation are obtained. The feasibility of the principle of operation and the validity of the mathematical model have been verified experi- mentally.

List of symbols

U, i = pu instantaneous value of voltage and

6 , I , = base value of voltage and current, respec-

H , H , = motor and load inertia in seconds, respec-

L, r, X = pu inductance, resistance and reactance,

X,, X,, = pu direct and quadrature axis magnetising

C , X , =capacitance in pF and pu capacitive reac-

T , TL = pu motor and load torque, respectively

t = time in seconds ra = pu stator resistance S = slip 6 = load angle

current, respectively

tively

tively

respectively

reactance, respectively

tance of the phase balancer, respectively

P = d/dwt

0 IEE, 1995 Paper 1702B (Pl), first received 20th January and in revised form 19th September 1994 The authors are with the College of Engineering, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia

104

Y = pu flux linkage e p0 = pu rotor speed w

Subscripts d, q kd, kq

a, b, c

= angle between the axis of phase a and d-axis

= supply frequency in rad/s

= direct and quadrature axis, respectively = direct and quadrature axis damper windings,

respectively = stator phases

S = supply

1 Introduction

Three phase reluctance motors have been receiving growing interest in the recent past. The main drive behind this is the success in developing motors with high degree of saliency, or what are called segmental type reluctance motors [l]. Such motors are characterised by considerably high pull out torques and enjoy, accord- ingly, relatively high steady state stability margins. Because of this property, and due to their inherent advantages of ruggedness, simple construction and absence of additional DC excitation arrangement, three phase reluctance motors have replaced classical synchro- nous motors in some constant speed applications.

The operation of three phase reluctance motors under balanced steady state and dynamic conditions has been well investigated [1-61. No significant effort has yet been directed towards analysing their behaviour under unbal- anced supply conditions, or when they are fed from a single phase supply [7,8]. The last point particularly is of special importance, since it adds more flexibility to the modes of operation of these motors.

The authors have successfully developed mathematical expressions, whereby the asynchronous starting torque of three phase reluctance motors fed from a single phase supply under the effect of saliency can be obtained [8]. The need arises, however, for a more rigorous handling of the transient behaviour of these motors during the start- ing process. Satisfying this requirement represents one of the main objectives of this paper. Special interest is directed to the study of the pulling into step problem as affected by the value of the balancer capacitance and the degree of saliency.

The authors would like to thank the Research Centre at the College of Engineering, King Saud University for supporting this work. The effort offered by Eng. Ali Awad, research assistant at the same centre, is also appreciated.

1EE Proc-Electr. Power Appl., Vol. 142, No. 2, March 1995

2 Mathematical modelling and analysis

2.1 Mathematical approach The system under study comprises a star connected three phase reluctance motor fed from a single phase AC supply using a single capacitor as a phase balancer as shown in Fig. 1. The analysis can be easily modified to suit a delta connected motor.

a

* - - - - - J,a----..* + t c

--- - - _ _ _ _ _ - - - - Fig. 1 System under study

The use of Parks d-q transformation for developing the mathematical models of three phase reluctance as well as synchronous motors fed from a balanced three phase AC supply has been a common practice. This is attributed to the considerable simplification obtained by its application. This is not the case with the single phase operation of three phase reluctance motors, where both mechanical and electrical asymmetry are experienced. The use of Parks transformation does not provide defi- nite advantages in this context. Therefore, it has been found more sound to write down the mathematical model of the system under study in the original phase values reference frame. This kind of representation has also the advantage that the variables dealt with are physical, and saturation effects can be easily considered, whenever needed.

2.2 Voltage equations The loop voltage and current equations for the motor circuit of Fig. 1 are

U, - v, + U, = 0

U, - v, - Ucap = 0 io + i, + i, = 0

1 . P%3p = ' c

Considering the three phase reluctance motor under study, the following voltage equations can be written:

(5 ) Cuol = PWOI + CRolCiol where

r , O O 0 0

P o l = O O O r , 0 0 0 0 rk4

It should be pointed out here that per unit quantities are used throughout this paper.

2.3 Flux linkage equations The corresponding flux linkage equations can be written as follows:

WOl = CLolCiol (6) where [Lo] is the well known inductance matrix of the synchronous machine with no field windings as given in Appendix 7 [SI. The elements of [Lo] are trigonometric functions of the angle e and are also given in Appendix 7.

2.4 Torque equation The developed electromagnetic torque can be obtained as the partial derivative of the energy stored in the mutually coupled inductive circuits with respect to the angle 8. Accordingly, the electromagnetic torque can be expressed as

a T = Pol' - CLolCiol ae (7)

This equation can be expanded to yield the following expression:

T = TR, + TR2 + T,, + T, (8) where

aLab aL, 8LbC TR2 = 2i, i, - ae + 2i, i, - ae + 2i, i, - ae

T,, = 2ikq[ia % + i, ae + i, 4 aLck ae 1 The mechanical equation of motion can be written as

T - TL = 2Hwp(pe) (9)

2.5 Flux linkage state space model Although the development of a state space current model is mathematically easier, the flux linkage state space model has certain advantages as discussed in Reference 10.

The first step in developing the flux linkage state space model is to reduce the order of the voltage and flux linkage equation by eliminating phase b quantities through the use of eqns. 1 and 2. This, in addition to the solution of eqn. 3, yields the following equation:

(10) [VI = PCYI + CRICil where

r 1

105 I E E Proc.-Electr. Power Appl., Vol. 142, No. 2, March 1995

The following flux linkage equations can also be written:

cy1 = CLlCil where

CLI =

The second step in developing the flux linkage state space model needed, is to obtain expressions for the different machine currents in terms of the machine flux linkages. This has been carried out and resulted in the following equation:

[i] = [L]-'['Y] (12) Having achieved this, the flux linkage state space repre- sentation of the three phase reluctance motor when fed from a single phase supply can be written as

pC'rl= CF3C'YI + [VI (13)

[F] = -[R][L]-' (14)

where

The overall state space model of the system under study can be obtained by augmenting the capacitor differential equation, eqn. 4, and the equation of motion, eqn. 9, to eqn. 13. This yields the following state space model:

pCXl= CAlCXl+ CBI (15)

where

CBI =

CAI =

v s

0 ! ] - TL

0

, [XI =

3 Computation results and discussions

3.1 System data and solution technique The differential equations describing the behaviour of three phase reluctance motors fed from a single phase supply, as given by the state space form of eqn. 14, have been solved numerically. For this purpose, a computer program has been developed using the fourth-order Runge-Kutta algorithm for numerical integration. Both starting process and dynamic behaviour can be investi- gated by the developed program.

Two reluctance motors with different designs have been considered. The first is a conventionally designed motor with a relatively low reactance ratio ( X d / X q = 1 .S), while the other is a segmental type motor with a high reactance ratio ( X , / X , = 3.17). Data of both motors are given in Appendix 9 [ I 13.

3.2 Low reactance ratio motor The transient behaviour of motor 1, which is a relatively low reactance ratio motor, during the period of starting against a 0.4 pu constant load torque is described in Figs. 2-5. From these Figures it can be realised that the value of the phase balancer capacitance plays an important role during the startup period. A capacitor designed for yielding maximum starting torque would allow the motor to approach synchronous speed quickly, but results in considerable torque pulsations as the motor pulls into step (Fig. 2a). The high frequency of the resulting electro- magnetic torque oscillations is attributed to the presence of a considerable negative sequence voltage component at synchronous speed.

On the other hand, providing the reluctance motor with a phase balancer capacitor designed for achieving minimum unbalance conditions at synchronous speed reduces significantly the severity of the torque pulsations near synchronism as shown in Fig. 2h. This is achieved, however, on the expense of longer startup time. Fig. 3 shows a comparison between the sliptime curves of both cases. It is advisable under such circumstances to operate the motor with two capacitors, one for starting and another for running. The transient response correspond- ing to this mode of operation is given in Fig. 4. Varia- tions of the current of phase a and the capacitor voltage versus time are displayed in Fig. 5. This Figure corre- sponds to the case of operation with a single capacitor designed for achieving minimum unbalance at synchro- nous speed.

For comparison, the torque-time and sliptime curves of this motor when operated under balanced supply con- ditions are shown in Fig. 6. The machine can synchronise easily without appreciable torque pulsations.

where A,, - A,, are given in Appendix 8. It should be pointed out that the derived model, with

minor modifications, is also suitable for analysing the motor under three phase balanced operation.

I

3.3 High reactance ratio motor Fig. 7a shows that the segmental type reluctance motor, which is a high reactance ratio motor, failed to synchro- nise when it was provided with a phase balancer capa-

I E E Proc.-Electr. Power Appl., Vol. 142, No. 2, March 1995 106

2 5 1 2 0 8 1 2

0 4 1 5

0 8 0 8

0 5 3 a 3

2 YI O L m- VI 0 4 0 0 2 .

c -0 5

0 0 0 0 -0 4 - I 5

-0 4 -2 5 -0 4 -08

0 00 025 0 50 000 0 25 0 50 0 75 t S I S

o b

Fig. 2 a X , for achieving max starting torque (X, = 0 77 pu)

Variation ofslip and electromagnetic torque ofmotor I with time b X, for achieving mm unbalance at synchronous speed (X, = 162 pu)

0

m o

0

\

I I I ' 1 0 0 02 0 4 06

t . s

Effect of phase balancer capacitance on sliptime curve of Fig. 3 motor I ~ X, for max st. torque ~ X, for minimum unbalance

2 1 6 0

8 c a - 0 0 I c L

5 - 1 6

0 00 0 25 050 t ,s

a

1 5 7 a

0 8 0 5 d

9 7 5

8

0 5

- 2 - 0 5 YI 0 4 00; :

0 0 -05 - 1 5

- 2 5 I' 1 0

0 00 0 25 0 50 t.5

-0 4-

000 0 25 0 50 t . s

Fig. 4 Vmiation of slip a d electromagnetic torque of motor I with Fig, of and ofmotor I , with during starting f ime

torque and other for securing min. unbalance at synchronous speed, switching

is provided with capaciton: one for achieving max. Motor is provided with capacitor for achieving minimum unbalance at synchro- nouss@ a ' 0

b s p k d = 0.8 pu

I E E hoc.-Electr. Power Appl., Vol. 142, No. 2, March 1995 107

citor for achieving maximum torque at starting. Under such a condition, the motor suffers from severe torque pulsations as it approaches synchronous speed. This is expected as the capacitance of the phase balancer selected to yield maximum starting torque does not guarantee a low negative sequence to positive sequence voltage ratio. The situation is considerably improved for operation with minimum unbalance phase capacitor as shown in Fig. 7b. The motor can synchronise and the high fre- quency torque pulsations are reduced. The starting tran- sient response corresponding to balanced three phase

I 2 l r O 8

ratio may be a problem. A comparison between the s l i p time curves corresponding to the different modes of oper- ation of motor 2 is shown in Fig. 9.

3.4 Effect of load inertia The inertia constant of the load driven by a reluctance motor has been found to have a significant effect on its starting transients. This can be seen from Fig. loa, which represents the starting sliptime curve of niotor 1 when provided with a phase balancer capacitor selected for achieving minimum unbalance at synchronous speed.

O 8 1

0 8

VI 0 4

0 0 0 0

I 0 00 025 0 50

1,s a

-0 2 V

0 4 ,

Fig. 6 (I vanation of ela’tromagnetic torque and slip with time

Three phase balanced operation of motor I b torque slip corve

0 8 2 0

2

L 0 4 0 0; :

0 0 -2 0

0 4 2 - 4 0 000 0 25 0 50

t ,S

a

Fig. 7 a X. for achwmg maximum starting torque (X, = 0.66 pu)

Variation of slip and electromagnetic torque of motor 2 with timt b X, for achieving

operation of this motor is displayed in Fig. 8a. This figure reveals that, under this mode of operation, this motor is unable to synchronise smoothly, although the severity of its torque pulsations near synchronous speed is reduced. The frequency of these torque pulsations is relatively low, as they are actually electromechanical oscillations gov- erned mainly by the synchronising coefficient and the inertia constant. Comparing the limit cycle displayed in Fig. 8b with that of Fig. 6b illustrates the more oscil- latory nature of this motor with respect to motor 1. This is an interesting and important result and is in full agree- ment with the conclusions in Reference 12 where, and through a simplified approach, the authors stated that pulling into step of reluctance motors with high reactance

106

0 6

O 4 3 a

0 2s

-0 2 0 2 06 IO t .s

b

--0 5

--I 0

I -04- -1 5

0 00 0 25 0 50 . s

minimum unbalance at synchronous speed (X, = 1.195 pu)

Higher load inertia results in a smoother synchronous operation but increases the startup time. The same con- clusion applies also for the case of motor 2 as shown in Fig. lob.

4 Experimental verification

The feasibility of the basic concepts presented in the paper and the validity of the mathematical model devel- oped above, have been verified experimentally. For this purpose, a star connected three phase 1 HP, 380V, 60Hz, 4-pole reluctance motor provided with a phase balancer capacitor bank has been used. The parameters of this motor have been measured using the conventional

IEE Proc.-Electr. Power Appl., Vol. 142, No. 2, March I995

techniques described in Reference 13. The measured parameters in pu are given in Appendix 8.

Fig. 1 la and b display, respectively, the measured and computed starting speed-time curves of this motor under

1 2 -

0 a-

04-

0 0 -

-0 4.

0 0 50 0 25

a

1 1.5

Fig. 8 a variation of electromagnetic torque and slip with time

Three phase balanced operation of motor 2 b torque slip curve

no load, when provided with an 18pF phase balancer capacitor and connected to a 380 V single phase supply. The capacitance value is selected to achieve minimum unbalance at synchronous speed. The corresponding

l'1

v 1 -0 4 0 0 0 4 0 8 12

S

b

"1 speed

t

0 00 0 25 0 50 t .s

Fig. 9 modes of operation

Comparison between slipltime curve of motor 2 for diflerent

X, for max. st. torque __ X, lor min. unbalance -*- balanced operation

4 i c 1 div

a

t.s

Fig. 10 Effect oflaad inertia on starting transients of mator I when it is provided with capacitor for achieving min. unbalance at synchronous speed (I H L = 0 . 0 6 4 s b H , = 0.0 s

IEE Proc.-Electr. Power Appl., Val. 142, No. 2, March I995

t.5

b

Fig. 11 V , = J(3)pu, C = 18 pF, H , = Os a measured (time scale = 0.1 s/div) b computed

Speed-time curve aftest motor fallowing application of supply

109

measured and computed variations of phase a current and capacitor voltage with time are displayed in Figs. 1 2 and 13, respectively. From Fig. 11, it can be noticed that both measured and computed speed-time curves show almost the same pattern of variations. The time to reach

1 div a

synchronous speed is almost identical. The measured and computed variations of both phase a current and capa- citor voltage have also the same pattern. The speed-time curve of the motor when coupled to an inertia load is given by Fig. 14. The effect of the load inertia constant

4 0

+ 0 C L

a - 2 0

-4 0 0 00

Fig. 12 a measured (time scale = 0.05 sldiv, current scale = 1.05 puidiv)

Variation of i. of test motor corresponding 10 operating condition ofFig. I 1 b computed

0 25 1 ,s

I 0 50

1 div a

Fig. 13 a measured (time scale = 0.05 s/div, voltage scale = 0.7 pu/div)

Variation ofv,, oftest motor corresponding to operating condition of Fig. I I b computed

0.00

speed

t

I 0 50

l 2 1 A

0 0 0 5 1,s

-IF 1 dw

b

Fig. 14 a measured (time scale = 0.2 sldiv)

Speed-time curve of test motor corresponding to operating condition of Fig. I I but with H, = 0.08 b computed

10

110 IEE Proc.-Electr. Power Appl., Vol. 142, No. 2, March I995

can be seen by comparing Fig. 14 with Fig. 1 1 . The experimental results confirm again the conclusions obtained theoretically that a higher load inertia constant results in a longer startup time and smoother synchro- nous operation.

The slight quantitative discrepancies between the com- puted and measured results can be attributed to the following factors:

(i) The neglection of saturation effect in the develop- ment mathematical model;

(ii) The approximations in measuring the motor parameters;

(iii) The variation of the rotor resistance with fre- quency ;

(iv) The dependence of the measured electrical tran- sients on the instant of supply switching.

5 Conclusions

This paper presents, as one of its objectives, a flux linkage state space model representing the transient behaviour of three phase reluctance motors fed from a single phase supply. A computer program simulating this model has been developed and used for investigating the starting process of three phase reluctance motors under this mode of operation. From the analysis given in this paper, the following can be concluded.

(a) The value of the phase balancer capacitance plays a very important role in allowing this type of motors to pull into synchronism. If the motor is to start against a high load torque, it is advisable to provide it with two capacitors for achieving maximum torque at starting and minimum unbalance at synchronous speed.

(b) High reactance ratio reluctance motors, although have high pull-out torques and so good stability margins, may face troubles during starting and be unable to self synchronise with the power supply. This conclusion applies to both balanced operation and operation from a single phase supply.

(c) In view of conclusion (b), measures should be taken during the design procedure of high reactance ratio motors to handle the pull into step problem.

(d) Load inertia plays an important role during the starting process. Higher load inertia increases the start- up time but smooths out the experienced speed oscil- lations.

The validity of the developed mathematical model has been verified experimentally.

6 References

1 LAWRENSON, P., and AGU, L.: ‘Low inertia reluctance machines’, IEE Proc., 1964,3, (12), pp. 2017-2025

2 LAWRENSON, P., and BOWES, S.: ‘Stability of reluctance machines’, IEE Proc., 1971, 118, (2), pp. 356-369

3 LAWRENSON, P., MATHUR, R.. and STEPHENSON, J.: ‘Tran- sient performance of reluctance machines’. IEE Proc., 1971, 118, (6), pp. 777-783

4 HONSINGER, V.: ‘Steady state performance of reluctance machines’, IEEE Trans., 1971, PAS-90, ( I ) , pp. 305-317

5 LIPO, T., and KRAUSE, P.: ‘Stability analysis of a reluctance syn- chronous machine’, IEEE Trans., 1967, PAS-86, pp. 825-834

6 HONSINGER, V.: ’Stability of reluctance motors’, IEEE Trans., 1972, PAS91, pp. 1536-1543

7 ALOLAH, A.I.: ‘Steady state operation of three phase reluctance motor from single phase supply’, J. IEE (India), 1989, EL-70, ( 9 , pp, 157-161

8 ALOLAH, A.I., and BADR, M.A.: ‘Starting of three-phase reluc- tance motors connected to a single phase supply’, IEEE Trans., 1992, EC-7, (2), pp. 295-301

ILE !‘roc.-Elertr. Power Appl., Vol. 142, NO. 2, March 1995

9 SARMA, M.: ’Synchronous machines’ (Gordon and Breach Science Publishers, New York, 1979)

10 VAS, P.: ‘Electrical machines and drives, a space-vector theory approach (Oxford University Press, Oxford, 1992)

I 1 OSHEBA, S., and ABDEL-KADER, F.: ‘A comparison between conventional and segmental reluctance motors’, Elec. Mach. Power Syst., 1989, 16, pp. 249-264

12 FAROUK, A., and ABO-SHADY, S.: ‘Synchronisation of reluc- tance motors’, IEEE Trans., 1981, PAS-100, (4), pp. 1885-1892

13 JAIN, G.: ‘Design, operation and testing of synchronous machines’ (Asia Publishing House, London, 1966)

7 Appendix

CL01 =

where

L,, = L,, + La, cos 20 + L,

L,, = L,, + L,, COS 2(0 + 2n/3) + L,

Lbb = Lao + La, COS 2(8 - 2 ~ 1 3 ) + L,

Lab = -[Lbo + La, COS 2(0 + n/6)]

L,, = - [ L bo + La, cos 2(8 + 5n/6)]

Lb, = -[LbO + La, cos 2(8 + n/2)]

= cos Lbkd = La,,, COS (0 - 2 ~ 1 3 )

L&d = La,,, cos (0 + 2n/3) La,, = - La,,, sin 0

Lbkq = - Lo,,, sin (0 - 2n/3)

L,,, = - Lokqo sin (0 + 2n/3)

The relations between the inductance matrix elements in the original phase values reference frame and the motor per unit parameters in the d-q reference frame are

La0 = 2LbO = ( x a d + x q ) / 3 L , = xd - Xad

= ( x a d - xq)/3 Lkkd = 2 X k k d / 3

Lkkq = 2 x k k q / 3 Lotdo = 2Xd3 Lakclo = 2xq/3

8 Appendix

The torque equation in expanded form is given by eqn. 8. By eliminating ib and through further mathematical manipulations, the following torque equation results :

(16) T = D,i, + D, ic + D, ikd + D4 ikq

D, = 6L,, [sin (20 - 120)1,, - sin (28 + 120)IC0]

D, = 6L,,[-sin (20 + 120)1,, + sin 201,,]

D, = 2Lak,,[{sin (0 - 120) - sin O } I a o

where

+ {sin (0 - 120) - sin (0 + 120)}Ic0]

D4 = 2L,k,o[{cos (8 - 120) - COS 8 } I o o + {cos (e - 120) - cos (e + 120)}1c0]

111

9 Appendix

The pu parameters of the motors are

Motor 1

r, = 0.205,

X, = 0.669,

X, = 1.202

X,, = 1.1 16 x,, = 0.583, x k d = 0.512

xk, = 0.597, R,, = 0.164

Rk, = 0.375, H = 0.032 S

HL = 0.064 s TL = 0.4 PU,

Motor 2

ro = 0.205, X , = 1.296 X , = 0.409,

X,, = 0.323, X,, = 1.210

X,, = 0.341

xkq = 0.427,

R,, = 0.453, Rk, = 0.177

H = 0.038 s

HL = 0.076 s TL = 0.5 PU,

Test motor

r. = 0.167, X, = 1.577 X, = 0.835,

X,, = 0.759,

X,, = 1.501

X, = 0.631 x k q = 0.177,

Rk, = 0.159, Rk, = 0.410

H = 0.092 s V, = 220,/(2) V (peak value)

I b = 2.04,/(2) A (peak value)

I E E Proc.-Electr. Power Appl., Vol. 142, No. 2, March 1995

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