Modelling and stability analysis of a permanent-magnet synchronous machine taking into account the effect of cage bars

D.W. Shimmin J. Wang N. Bennett K.J. Binns

lndexing t e r m : Synchronous machines, Stability analysis, Permanent-magnet motors, Cage bars

Abstract: Several methods have been investigated for the computational and experimental determi- nation of static and dynamic parameters of permanent-magnet synchronous machines, based on a sixth-order-transient two-axis model includ- ing the effect of cage bars. In addition to reviewing and comparing established methods, some new techniques are proposed with special emphasis on the determination of stator leakage inductance and cage-bar parameters, by computation and experiment. The equivalent-circuit model devel- oped is suitable for the analysis of both steady- state and dynamic behaviour. The results from different methods are compared. The model is used to investigate the open-loop stability of the machine’s operation by means of computer simu- lation and experiment. The good agreement between simulated and experimental results for stability boundaries confirms the validity of the proposed modelling methods.

List of symbols

Rs = stator-winding resistance, Cl L d , L, = stator d- and q-axis synchronous inductance,

LlS = stator-winding synchronous inductance, H Lid, LI, = stator d- and q-axis leakage inductance, H L,, L , = stator d- and q-axis magnetising inductance,

Rkd, R , = damper-circuit d- and q-axis resistance, R Lkd, L , = damper-circuit d- and q-axis inductance, H Ldo, L, = steady state d- and q-axis synchronous induc-

+ d , +, = stator d- and q-axis flux linkage, Wb *m = flux linkage induced by permanent magnets,

Wb 6 = load angle, rad

H

H

tance, H

~

0 IEE, 1995 Paper 1719B (Pl), received 1st July 1994 K.J. Binns and D.W. Shimmin are with the Department of Electrical Engineering and Electronics, University of Liverpool, Liverpool L69 3BX, United Kingdom J. Wang is with the Department of Electrical & Electronic Engineering, University of East London, Essex RM8 2AS, United Kingdom N. Bennett is with British Steel, Shotton Works, Clwyd CH5 2NH, United Kingdom

IEE Proc.-Electr. Power Appl., Vol. 142, No. 2, March 199s

0, 0 s P

= rotor angular velocity, rad/s = stator angular frequency, rad/s = number of pole pairs

1 Introduction

There has been a rapid development in permanent- magnet-machine performance in recent years, due largely to improving material characteristics and control tech- niques. The wide field of applications includes those areas where dynamic performance is critical. In such cases, it is important to model the machine behaviour to sufficient accuracy for assessment of open-loop stability and to permit robust and reliable design of closed-loop drive systems. However, for high-field permanent-magnet syn- chronous machines a conventional fourth-order model with constant parameters is inadequate, owing to the complex magnetic flux distribution, and it is important to take account of parameter variations with load and direction. The presence of a damper circuit in many machines of this type is also of great importance.

The machine parameters can be obtained by computa- tional or experimental methods. Numerical techniques such as finite-element or boundary-integral solutions allow assessment of parameters which are difficult to determine experimentally, but the limitations of com- puter modelling must be appreciated. A number of experimental methods have been used successfully, such as flux integration [l], load-angle testing [2], zero- power-factor and steady-state short-circuit testing [2, 31. However, these techniques do not provide information on the damper circuit. Other methods usually adopted for wound-field synchronous machines involve signal injection into the field winding, and are not applicable to permanent-magnet excitation systems.

Based on a transient two-axis model including the effect of cage bars, this paper explores both experimental and computational methods for the determination of all parameters in the model, and in turn analyses the stabil- ity of open-loop operation. The objective of this dual approach is twofold: first, the experimental techniques can sometimes provide information which is very difficult or time consuming to deduce computationally, and sec- ondly it allows comparison of the techniques when a par- ticular quantity can be obtained by either approach.

The machine under consideration has an asymmetrical buried-magnetic rotor with cage bars [4], as shown in Fig. 1; the constructional details of the machine are given

137

in Table I . This machine can be used as a line-start syn- chronous motor, or as a variable-speed synchronous motor with high damping qualities. As such, the signifi- cance of the cage bars and copper wedges in affecting the

nonmagnetlc magnets steel shaft

v Fig. 1 Configuration oJthe high-jeld synchronous mnchine

Table 1 : Parameters of permanent-magnet synchronous mrchina

Parameter Value

Number of poles 4 Number of phases 3 Magnet material SmCo, Rotor diameter, mm 79 Axial length, mm 90 Rated voltage at 50 Hz, V 37.5 Rated current at 50 Hz, V 7 Stator-winding resistance Der Dhase, n 0.36

0 I I

0 1 I b

Fig. 2 synchronous machine (I D-axis circuit b Q-axis circuit

Equivalent transient two-axis model for the permanent-magnet

machine performance is of particular interest. This partic- ular topology is a ‘high-field’ configuration having the property that the flux density in the airgap is higher than that in the magnet material. Overall, this configuration exhibits a high power-to-weight ratio, with mechanically

138

robust rotor construction and good pull-in torque for synchronising. The asymmetrical magnet and cage-bar distribution produces a slightly different machine per- formance in one direction from another. The presence of magnetic material in the path of the direct-axis flux results in a smaller reactance in the direct axis than that in the quadrature axis, representing a major parameter difference between permanent-magnet machines and con- ventional wound-rotor synchronous machines. By appli- cation of Park’s transformation and using the approach of Smith [SI, the electrical dynamics of the machine can be modelled by the dq-axis equivalent circuits shown in Fig. 2, with leakage inductances L,, , L,,, magnetising inductances L,,, L,,, and where the influence of cage bars is considered by the inclusion of a damping parallel LR branch in each axis (L,,, R,, and L,, , R,,). In these circuits, the stator resistor R , can easily be determined by application of standard stator-resistance tests. Therefore the attention in the following sections is focused on deter- mination of other parameters involved in the model via computation and experiment. Note that this model does not explicitly include end-winding effects or slot harmo- nics except for L,, and L,,, but has proved useful for steady-state and dynamic modelling [ l , 3, 51.

2 C o m p u t a t i o n a l d e t e r m i n a t i o n of the p a r a m e t e r s

Owing to the asymmetrical rotor configuration and the complex flux distribution in the machine, accurate deter- mination of the machine parameters by means of compu- tation requires the use of field-analysis techniques. To meet this demand, finite-element methods are adopted.

2.1 Calculation of steady-state inductances For steady-state operation, the parallel LR branches in Fig. 2 can be removed since there are no induced cage- bar eddy currents. Thus the steady-state direct- and quadrature-axis synchronous inductances Ld, and L,, are expressed by

Ldo = LLd + Lmd

L,, = 4, + L,, (1) Determination of these inductances is achieved by use of the software package MOTORCAD which is specially designed for steady-state performance and parameter analysis of permanent-magnet machines. In the computa- tions, L,, and L,, are calculated from the magnetic energy storage. A nonlinear calculation is first performed with the magnet and stator current excitations active. The element permeabilities are then frozen while a linear calculation is performed with the magnet excitation dis- abled. The contribution from the stator currents to the total magnetic energy is evaluated from the difference between these two results. L,, and L, are obtained in a similar way from the fundamental component of the airgap flux-density distribution by performing Fourier analysis, under conditions of appropriate excitation and saturation. The stator leakage inductances L,, and L,, are then calculated using eqn. 1. The influence of saturation on the parameters is investigated by applying different excitation currents. The calculated results for both the d and q axis are shown in Fig. 3, demonstrating the varia- tion of these inductances with saturation. It is also seen from the Figure that the leakage inductances in the d and q axes are slightly different and vary slightly with saturation. In practice, these differences and their varia- tion can be included in the magnetising inductances L , and L,,, and a single value L,, is used in the later stages of the computation.

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2.2 Calculation of cage-bar inductance The calculation of cage-bar parameters is based on the assumption that, during the transient period, the relative

L”

‘\6 -6 -L -2 0 2 current,A

Fig. 3 with current L = q-axis synchronous inductance L: = d-axis synchronous inductance L,, = q-axis leakage inductance L,, = d-axis leakage inductance

Variation of calculated synchronous and leakage inductances

movement between the stator field and cage bars is not very rapid and thus the cage-bar parameters are con- sidered to be frequency independent. From the equivalent-circuit model of Fig. 2, the d- (or 4-) axis oper- ational inductance is given by

where

For two given frequencies w 1 and w 2 , the following expressions are derived:

(4)

The above expressions illustrate that, if I Ld(jo)I, IL,(jw)l, L,, and L,, have been calculated by field analysis then z,, T , ~ and consequently L,,, R,, are determined. Similar results hold for q-axis cage-bar parameters.

The physical model for this computation is considered as a two-dimensional steady-state AC problem and the PE2D software package is used for the analysis. To take the nonlinear nature of the permanent-magnet machine into account, the whole analysis is separated into two stages. First, the field excited by the permanent magnet and stator current is calculated, in which the stator current in different slots is distributed such that the resulting current space vector is aligned with the d or q axis. After the static calculation, the permanent magnet is replaced with a material of the same relative permeability while the permeabilities of other ferromagnetic materials in different elements are kept unchanged. The computa- tion of the steady-state AC problem is then restarted with

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

the permeabilities obtained in the static solution. To avoid permeability variation, the problem for the restarted computation is treated as a linear one.

The flux-linkage method is adopted to find out the operational inductances from the results of the field com- putation. By definition, I Ld(jw) I and I L,(jw) I are given hv

The d,q-axis flux linkages can be calculated by Park’s transformation:

+ $JflCOS (0 + $)} I + sin (e + F)} J

where 0 is the angle in electric degrees between the d-axis and the stator red-phase axis. l ( l B , i)y and are the flux linkages of stator red, yellow and blue phase windings, respectively, and are obtained by use of the postprocess- ing program of PE2D. If the current space vector is aligned with the d (or q) axis, I, (or I,) is equal to the peak value of the excitation current. Finally, the cage-bar parameters can be determined by combination of the above equations. The results for the q-axis parameters are presented in Fig. 4. The values of L,d, L,, and L,,

a - 0 6 0.6

0.11

0 1 0 IO 2.0 3.0 40 5 0 6.0

current,A Variation of calculated q-axis damper-circuit inductance and Fig. 4

resistance with q-axis current a q-axis damper-circuit inductance L,, b d-axis damper-circuit resistance L,,

used to determine the cage-bar parameters are those shown in Fig. 3.

3

Although the static and dynamic parameters of permanent-magnet machines may be conveniently obtained by applying appropriate computational tech- niques, the more direct approach of experimental evalu- ation is often preferred. In a permanent-magnet machine, the field excitation is constant; hence experimental parameter-estimation techniques which rely on field per- turbation cannot be implemented. Sections 3.1 and 3.2 will introduce several experimental techniques which are suitable for the parameter determination of permanent-

Experimental determination of the parameters

139

magnet machines with special attention paid to the mea- surements of stator leakage inductance and cage-bar parameters.

3.1 Static parameter measurements Several methods for the measurement of static induc- tances Ld, and L,, have been proposed, among which are the flux-integration method C2, 31, the zero-power-factor test and the load-angle test [4]. The results from these tests are compared in Fig. 5.

. 01 I - 6 - 4 - 2 0 2 L 6 8

current.A

Fig. 5 sponding axis current 4 L,, load-angle method b L ,0ux-integration method c L:, zcrc-power-factor method d L, , flux-integration method L, = d-axis synchronous inductance L, = q-axis synchronous inductance

Variation of measured synchronous inductances with corre-

For the steady-state analysis, the information on L, and L,, is sufficient, although the complete determination of the machine model shown in Fig. 2 requires the separation of L,, from Ld, or Lqo. For a permanent- magnet machine, this separation is not possible by the Potier test since the field is not adjustable. The solution can be found by examination of the self and mutual inductances of stator windings. These inductances are functions of the rotor position and have the form

LRR(&) = L, + Los + L(@J

Lyy(e,) = L, + L,, + L,, e + - ( r :)

L,de,) = L,, + L, + L,, e - - ( r :)

M1.de,)= MBY(e,)= -

The above expressions illustrate that the self inductance consists of three terms: (i) the stator leakage inductance L, which is constant, (ii) a constant magnetising induc- tance Los, and (iii) a magnetising inductance L&) which is a function of rotor position. Similarly, the mutual inductance is split into two magnetising inductances. Since the stator phase windings are distributed within a 60" phase belt, the constant part of the mutual induc- tance has approximately half the value of Los. The con-

140

stant term of both the self and mutual inductances may be determined by performing a Fourier analysis of the above inductance functions. The stator leakage induc- tance is therefore given by

The integration is carried out with the data from the self- and mutual-inductance measurements obtained by the flux-integration method, while the rotor position 0, is recorded with the aid of a positional encoder. Typical results of the red- and yellow-phase inductances LRR and MRY are shown in Fig. 6.

o t

-2 0 60 120 180 240

rotor positton, d q electrical

Fig. 6 position a Self inductance b Mutual inductance

Variation of measured self and mutual inductance with rotor

32 Cage-bar-parameter measurements To determine experimentally the parameters of the cage- bar branches of the equivalent circuits, a standstill frequency-response test is employed. To perform this test, the rotor is locked in the direct-axis position and a low- voltage sinusoidal source from a linear power amplifier is injected into the machine phase terminals. Both the mag- nitude and phase of the impedance at the machine ter- minals are measured with the aid of a digital storage oscilloscope. Saturation effects in both axes may be inves- tigated by variation of the alternating excitation currents and inclusion of a DC bias current. The measured fre- quency response of the stator impedance is normalised in terms of the winding resistor R,, with consideration of the connection factors in both the d axis (2/3) and the q axis (1/2). From Fig. 2, the normalised impedance- transfer function has the form

The variation of magnitude and phase of the normalised impedance is plotted logarithmically against frequency, and a nonlinear curve-fitting technique is used to deter-

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

mine the values of the three times constants q, & and T, . With these time constants as well as the values of R, and L,,, the other parameters in the Fig. 2 can be fully determined. Fig. 7 shows the effect of saturation on the

0.71

0.1 "'I r 01 Io

0 2 4 6 0 current, A

Fig. 7 q-axis current (I Inductance L b Resistance R Z

Variation of measured q-axis damper-circuit parameters with

quadrature-axis cage-bar parameters. Similar behaviour is observed in the direct-axis cage-bar circuit.

4 Comparison of different calculation and test methods

Computer modelling can only be as accurate as the data supplied in terms of machine geometry, construction and material properties. The static parameters (Fig. 3) obtained using two-dimensional analysis compare well with the experimental results (Fig. 5). The calculated parameters are slightly higher than the experimental values, by less than 10%. This discrepancy is believed to be due to inadequate representation of lamination losses and end-winding effects. The calculation for damper- circuit parameters is less consistent (Figs. 4 and 7). Both finite-element computation and frequency-response testing have some problems in this area. The finite- element solution was performed using the two- dimensional package PEZD. As such, the cage-bar effects only include eddy currents within the bars themselves and flowing from one pole to another, but not currents circulating between the bars within one pole. A similar diffculty with modelling the cage-bar circuits using two- dimensional finite-element algorithms has been reported previously [SI. The experimental method, on the other hand, relies on accurate estimation of gain and phase shift for small modulation signals imposed on a DC bias. Such measurements are difficult to perform at low ampli- tude and low frequency. Further, the DC bias used to represent the saturation conditions of a loaded machine only provides approximately the same flux distribution inside the motor as would occur in practice. In particular, there is no representation of the difference between the preferred and nonpreferred direction of rotation which has a significant effect on the q-axis parameters for this type of machine [l].

The flux-integration method, first developed by Jones [7] for use in wound-rotor synchronous machines, is a simple test requiring no measurement of machine load angle. This method is also applicable to the static induc- tance measurements of permanent-magnet machines, and has the advantages of simplicity and convenience [l]. The impact of saturation level on the parameters can

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

easily be investigated by this method. In practice, it is important that the machine rotor should be clamped as rigidly as possible to prevent any vibration caused by the electromagnetic force produced by the interaction between the permanent magnets and the current varia- tion. Any such vibration will lead to an extra flux-linkage change owing to the presence of the permanent magnet and consequently increase the test error. The results from this test are consistent, although the test conditions are not the same as in real operation. By contrast, since the load-angle test is performed under real operating condi- tions the test results are usually more reliable and accu- rate provided that the load angle is measured with high precision. Clearly, a disadvantage is the requirement for load-angle measurement. However if an encoder has already been mounted on a tested machine, the test can be easily conducted without the need for extra equipment. Although the zero-power-factor test and the short-circuit steady-state test also have the feature of sim- plicity, the application of these methods is limited by the fact that they can only be used to determine the demag- netising direct-axis inductance of the machine.

5

Previous work [SI, in which the effect of cage bars was not considered, has shown that permanent-magnet syn- chronous machines can be unstable under certain oper- ation conditions. For a machine with cage bars, the role of the damper circuit in stabilising machine operation is important, and neglecting its influence may lead to unac- ceptable results. With the establishment of the complete transient two-axis model shown in Fig. 2, stability analysis taking the effect of cage bars into account then becomes possible. This assessment of stability also pro- vides a method of checking the measurements of machine parameters.

Stability analysis for open-loop drive operation

5.1 Sixth-order dynamic model of the open-loop drive

From the equivalent circuits in Fig. 2, the dynamic equa- tions governing the electromechanical behaviour of the machine operation are described by

I dY dt V' = R,i,, + = 0

d6 -- dt - U, - U,

J Jdw, B p dt P

T , = - - + + U , + T ,

where J is the moment of inertia, B is the viscous coefi- cient, p is the number of pole pairs, 6 is the load angle, U, is the supply angular frequency, U, is the rotor speed in electrical radians per second and \yd,, yqs, yd,, y%, are the stator and rotor d,q-axis flux linkages, respectively.

141

They are given by:

y d s = (Lis + L m d ) i d f Lmd ikd + y m

yqs = ( L k + Lmq)iq + Lmq ikq

ydv = (Lmd + Lkd)ikd + Lmd id

yqr = (Lmq + L k q k k q + Lmq iq

where Y , denotes the flux linkage induced by the per- manent magnets on the rotor. If the inverter which drives the machine is modelled as a voltage source in series with an equivalent regulation resistance R i , from the vector diagram of a permanent-magnet machine, the d - and q-axis voltage components applied to the machine terminal are

V ds - - - V sin 6 - Ri id

V,, = V cos 6 - Ri iq (12) Substituting eqns. 1 1 and 12 into eqn. 10 and rearranging in terms of differentiated variables results in

[i] = [L]-'[A][i] + [L]-'([F] + [H]V,

6 = 0, - w, 1 3 P 2 h, = - - { ( & - Lq)id iq + 'f', i, + Lmd iq ikd 2 5

where

0

Eqn. 13 may be expressed in state-space form as

[XI =f(CXI, CUI)

CUI = [ V , w,. TLIT = Lid > i q , ikd > ikq 5 w v l T (14)

where [XI and [U] are the state-space vector and input- control vector respectively, andf([X], [U]) is a nonlinear function of [XI and [U].

5.2 Stability analysis To evaluate the stability of the open-loop drive system (inverter and machine) eqn. 14 is linearised with respect to a small perturbation [AX] about a nominal steady- state operating point [X,] :

A[X]= - A[X] [::I**

Thus the eigenvalues of the above Jacobian matrix provide complete information about the stability proper- ties of the system under given operating conditions. Sta- bility is assured if the eigenvalues have negative real parts. The ACSL simulation software package is used to perform the stability analysis. When the entire drive- system dynamics described by eqn. 14 are formulated with an ACSL program, linearisation of eqn. 14 can be undertaken automatically by a series of ACSL com- mands. In the simulation, calculation of the steady-state operating point [X,] for a given input-control vector [U] is first executed taking the effect of saturation on L d , L, and into account [SI. The computer then calcu- lates the eigenvalues of the resulting Jacobian matrix for a small perturbation about this operating point. The effect of machine-parameter variations on the drive- system stability can be investigated by examination of these eigenvalues.

The results from simulation show that the eigenvalues consist of a lightly damped complex-conjugate pair and other poles well within the left half of the s plane; hence the drive-system stability is dominated by the lightly damped pole pair. Fig. 8 shows the drive-system root

90

85 I

-10 -8 -6 -4 -2 0 real part, radls

Drive-system root locus as a function of applied load at 50 H z Fig. 8 -0- with cage bars -0- without cage bars

locus of the dominant pole pair for 0.9 lagging-power- factor operation at 50 Hz as the load torque is varied up to rated value. The result for the same operating condi- tion but in the absence of cage bars is also presented in the Figure, which suggests that the machine without cage bars is almost unstable under this operating condition. Comparison of these two loci clearly illustrates that the system damping factor is substantially improved with the use of cage bars in the rotor. It is equally true to say that the cage bars have a large effect on system stability, and neglecting this effect in stability analysis will lead to unacceptable results. Fig. 9 shows the drive-system root locus for different power-factor (lagging and leading) operations under rated load as the system frequency is varied. With the increase of excitation frequency over 25 Hz, the system-damping factor decreases while the oscillating frequency increases, illustrating that the system-stability properties deteriorate. Meanwhile the Figure demonstrates that lagging-power-factor operation has better stability performance than leading-power- factor operation although, in this particular example, the machine can still operate at 0.8 leading power factor within a limited frequency range. When operating at

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

lagging power factor, the machine is stable over a wide range of excitation frequencies from a few hertz to a few hundred hertz. However, the machine will become

100

In V . 2 80

0 a 60 > L O 5 LO

E 20

real part, radls

Fig. 9 power factor af full load -0- 0.9 lagging -A-~ 1.0 unit -0- 0.9 leading ~ ~0- 0.8 leading

Dritv-system roof locus as a lirnction 1q supply frequency and

A B

I I

b Fig. 10

factor a Phase current b Inverter DC link current

Example of low-frequency instability ai 0.9 leading power

unstable when operating under leading power factor at lower frequencies, as can be seen from Fig. 9: the domi- nant conjugated pole pair split into two real poles

IEE Proc.-Electr. Power Appl., Vol. 142, N o . 2, March 1995

moving in the opposite directions along the real axis with the decrease of excitation frequency. In this example, the lower-frequency stability boundaries are found to be 7.2 Hz, 17.6 Hz and 21.8 Hz for unit-power-factor, 0.9 and 0.8 leading-power-factor operation, respectively. Clearly, this region of unstable frequency is getting smaller as the operating power factor varies from leading to lagging, and is eventually eliminated. A similar lower- frequency instability problem occurs in a machine without cage bars, but analysis shows that high- frequency instability is more serious in such machines.

Analysis of the effect of machine-parameter variations on stability shows that the system-stability boundary is enlarged by an increase in d-axis equivalent cage-bar resistance and that an increase in inverter-regulation resistance or stator-phase resistance also tends to improve system stability. It is also apparent that this type of machine has greater stability when operating at a lagging power factor (underexcited, with high V/f ratio) than when operating at a leading power factor (overexcited, with low Vif ratio). This has implications for motors driven from a simple inverter designed for induction motors with limited control over the Vif profile.

The system stability has also been investigated experi- mentally. The machine was driven at full load by an ultrasonic sinusoidal MOSFET PWM inverter. The ter- minal voltage was adjusted to ensure that the machine operates at the same 0.9 leading power factor as that in Fig. 9. When the supply frequency decreases to 18.7 Hz, the onset of instability appears, as shown on Fig. 10. A small load disturbance will induce oscillation of the direct link current and cause unstable phase current (point A) or even cause pullout (point B). This result agrees well with the simulation results in Fig. 9, where the stability boundary for frequency variations is 17.6 Hz at the same operating condition. This result confirms the computed and measured values of the machine param- eters.

6 Conclusions

Both experimental and computational methods described in this paper can be used independently to determine the essential parameters included in the permanent-magnet- machine model. The nonlinear nature of the machine model may be investigated by these methods. The basic agreement between computed and experimental results confirms the validity of this approach. With this model, the stability analysis shows that the use of cage bars will substantially improve the machine stability, and that the machine will exhibit instability under leading-power- factor operation at low frequency. Experimental results agree well with the stability boundary found from the analysis.

7 References

1 MELLOR, P.H., CHAABAN, F., and BINNS, K.J.: ‘Estimation of parameters and performance of rare-earth permanent magnet motors avoiding measurement of load angle’, IEE Proc. E, 1991, 138, (6), pp. 322-330

2 MILLER, T.J.E.: ‘Methods for testing permanent magnet polyphase AC motors’, IEEE IAS Con/. Rec., 1981.23D, pp. 494-499

3 CHALMERS, B.J., HAMED, S.A., and BAINES, G.D.: ‘Parameters and performance of a high field permanent magnet synchronous motor for variable frequency operation’, IEE Proc. B, 1985, 132, (3). pp. 117-124

4 BINNS, K.J., and WONG, T.M.: ‘Analysis and performance of high field permanent magnet synchronous machines’, IEE Proc. B, 1984, 131, (6), pp. 252-258

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5 SMITH, R.T.: ‘Analysis of electrical machines’ (Pergamon Press, 7 JONES, C.V.: ‘The unified theory of electrical machines’ 1982) (Butterworth, 1967)

6 STURGESS, J.P.: ‘Two dimensional finite element modelling of eddy 8 MELLOR, P.H., AL-TAEE, M.A., and BINNS, KJ.: ‘Open loop sta- cumnt problems - the effect of circuit connection’. Proceedings of bility characteristics of synchronous drive incorporating high field ICEM 92, UMIST, 1992, pp. 450-454 permanent magnet motor’, IEE Proc. B, 1991,138, (4), pp. 175-184

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