Estimation of speed, stator temperature and rotor temperature in cage induction motor drive using the extended Kalman filter algorithm

J . K. AI -Tay i e P. P. Aca rn ley

Indexing terms: Induction machines, Kalman filter, Speed estunation, Temperature estimation

Abstract: Application of the extended Kalman filter (EKF) algorithm to the estimation of speed, stator temperature and rotor temperature in induction motor drives is described. The estimation technique is based on a closed-loop observer that incorporates mathematical models of the electrical, mechanical and thermal processes occurring within the induction motor. Speed and temperature estimation is independent of the drive’s operating mode, though closed-loop estimation is possible only if stator currents are nonzero. The EKF algorithm used to perform the estimation process has been implemented using a TMS320C30 digital signal processor and experimental results demonstrate the effectiveness of the new estimation algorithm.

List of principal symbols

Ids, lqs = d-q stator currents ldn iqr = d-q rotor currents k = sample index k r = iron loss constant k lo , k20, k,, = thermal power transfer coefficients at

kl,,,, k,,, k3w = variation of thermal power transfer with

P PPI = pole number 4 = sampling time Vds, vqs B = viscous friction constant Hl = stator thermal capacity H2 = rotor thermal capacity J = total inertia K = Kalman filter gain

. .

. .

zero speed

speed = time differential operator (dldt)

= d-q stator voltages

OIEE, 1997 IEE Proceedings online no. 19971 166 Paper first received 25th September 1996 and in revised form 13th Janu- ary 1997 J.K Al-Tayie was wth and P.P. Acamley is with the Electnc Dnves and Machmes Group, Department of Electncal and Electromc Engmeenng, University of Newcastle, Newcastle upon Tyne, UK J K Al-Tayie is now wlth the Department of Electromc and Electncal Engineenng, University of Leeds, UK

L1 = stator self inductance L2 = rotor self inductance L, = stator to rotor mutual inductance PL1 = stator power loss PL2 = rotor power loss RIO = stator resistance at ambient R2, = rotor resistance at ambient T = motor torque Tl = load torque a S,, = delta function CT = coupling coefficient 8, 8, or = rotor speed U = input vector v = measurement noise vector w = system noise vector x = state vector y = output vector A = continuous-time system matrix B = continuous-time input matrix C = output matrix F = discrete-time system matrix G = discrete-time input matrix P = error covariance matrix Q R

= temperature coefficient of resistance

= stator temperature above ambient = rotor temperature above ambient

= process noise covariance matrix = measurement noise covariance matrix = estimated value of a state vector

1 Introduction

High-performance induction motor drives, giving fast torque response over the entire speed range, require a rotor-speed feedback signal, usually derived from a mechanical device such as an incremental optical encoder. The encoder makes a significant contribution to the overall drive costs, particularly for units with ratings below lOkW, and additional signal lines are needed to link the encoder and the control electronics. A recent trend in research on drive control has been the introduction of new techniques for eliminating the speed sensor requirement by estimating rotor speed from measurements of motor terminal voltages and currents using either the extended Kalman filter

301 IEE Proc -Electr Power A p p l ~ Vol 144, No 5, Seprember I997

(EKF) [l-51 or the model reference adaptive system (MRAS) [6-91. A new contribution to this effort, using the EKF, is described in the present paper.

Control and estimation strategies for induction motor drives are based on electrical equivalent circuit models of the motor. In many cases the model is the familiar steady-state equivalent circuit, but for high- performance drives a full transient model of the motor is required. Effective modelling, and therefore the effectiveness of drive control and estimation, is limited by the complexity of the physical processes occurring within the motor. Frequency dependence of the rotor electrical circuit, nonlinearity of the magnetic circuit, and temperature dependence of the stator and rotor electrical circuits all impact on the accuracy with which the motor can be modelled [lo, 111. This paper addresses the third of these effects (temperature dependence) by incorporating a thermal model of the motor in the estimation process. The frequency dependence of the rotor electrical circuit and nonlinearity of the magnetic circuit are not included. Nevertheless, the technique is applicable to an important class of induction motor drives such as those operating at rated flux with vector control, where rotor slip frequencies are a fraction of the maximum stator supply frequency and the flux levels in the motor are substantially constant.

Temperature estimation in the induction motor has been dealt with by many authors [ll-151, but most of these publications describe either a very complex lumped-parameter network or the finite-element method. A complex model would be inappropriate in this work, where estimation is being performed in real time, so a simple lumped-parameter thermal model has been developed empirically from experimental and operational observations. This model can be adapted for use with any induction machine without requiring information about the machine size and dimensions or materials’ properties: thermal parameters, such as ther- mal capacities and power transfer coefficients, can be determined by a few simple tests.

Real-time estimation of stator and rotor temperatures is described by Nestler and Sattler [14]. However this contribution is based on measurement not only of electrical quantities, but also of selected stator temperatures, using embedded thermocouples. The present work avoids the need for direct stator or rotor temperature measurement because stator current is the only variable used for the calculation of estimation error.

Estimates of stator and rotor temperature have a potentially valuable role in condition monitoring and thermal control of the induction motor. The temperature estimates are ‘bulk’ quantities which do not allow identification of local hotspots in the electrical circuits and therefore cannot compete with direct measurements from temperature sensors located in problem areas. However, the bulk temperature estimates of stator and rotor do include contributions from all sources of local heating, some of which may be remote from embedded temperature sensors.

2 induction motor model

A state-variable model of the induction motor is required for the EKF algorithm. The twin-axis stator reference frame [16, 171 is used to model the motor’s electrical behaviour, because physical measurements are

302

where 0 = L,L2 - Lm2. For the purposes of speed and temperature estimation this four-state electrical model must be augmented by appropriate models of the mechanical and thermal processes in the motor. The mechanical behaviour can be modelled by

in which the electromagnetic torque of tlie motor T can be represented in term of stator and rotor current com- ponents

T pnLm(2qsidr - i d s i q r ) (10) so the state-space equation for the rotor speed is

Pn Lrn ( i q s i d r - i d s i q , ) - -U, B - T1 - (11) J J pw, = ~

J The thermal model is derived by considering the power dissipation, heat transfer and rate of temperature rise in the stator and rotor. The stator power losses include contributions from copper losses and frequency- dependent iron losses

PL1 = (& + iZs)R1 + k,,w,2 (12) The rotor power losses are dominated by the copper loss contribution if the motor is operated at a low value of slip, so

PL2 1 (z:, + %:,)E2 (13)

stator, HI, PLI,BI

t k 1 4

Fig. 1 Structure of thermal model of induction motor

A simple representation of the assumed heat flow is given in Fig. 1. Heat flow from the rotor is either

IEE Proc.-Eleciu. Power Appl., Vol. 144, No. 5, September 1997

directly to the cooling air with heat transfer coefficient k2, or across the airgap to the stator with heat transfer coefficient k3

PL2 = k.282 + H2~6'2 + k3(0Z - 01) (14) Heat flow from the stator is directly to the cooling air, with heat transfer coefficient k ,

PL1 = k l Q l + H1p01 - k3(& - e,) (15) For an induction motor with a shaft-mounted cooling fan, the heat transfer coefficients are dependent on the rotor speed. This dependence has been modelled approximately by a set of linear relationships

k l = klO(1 + kLwW,) (16)

k.2 k . a o ( l + k . 2 w ~ , ) (17)

k3 = k O ( 1 + kSurW7.) (18) The well-known linear relationship between resistance and temperature must be taken into account for the stator and rotor resistances

RL = Rio(1 +ai&) (19)

R, Rao(1 + ~ 2 6 ' 2 ) (20) Substituting into eqns. 14 and 15 from eqns. 12, 13, 16-20 and rearranging yields the thermal state equa- tions for the stator and for the rotor

Inclusion of speed, stator temperature and rotor temperature in the state-space model produces a nonlinear seven-state system. Therefore the model must be linearised around steady-state or nominal values. In addition, the model must be discretised for use with sampled data. The linearised-discretised state equations are

;ds(k + 1) =ano;(is(k) + aoiiqs(k) + a o 2 L ( k ) + ao3;qT(k)

+ ao4G,(k) + aos81(k) + aos8z(k) + b00%Ls ( k ) + w1 (IC) ( 2 3 )

i q s ( k + 1) =a&(k) + a&(k) + a 1 2 L ( I C ) + U1:3iq,(IC)

+ U14f iT(k) + a1581 ( k ) + ~ l 6 & ( k )

+ + wz (IC) (24)

L ( k + 1) =a205ds(k) + a21.Lqs(IC) + CL225dr(k) + a235,,(k)

+ a24&(k) + a25&(IC) + a&(k)

i q r ( k + 1) = a & ( k ) + a31ips(k) + aaa&,(k) + a33iq,(IC)

+ a34'j,(k) + a3581(k) + U 3 6 & ( k )

+ bll?Ap(k) + w4(k) (26)

&,(k + 1) = a 4 0 i d s ( k ) + U4lfqs(k) + a 4 & ( k ) + U43LJk)

+ a44%(k) + w5(k) (27)

+ b l l V d s ( k ) + w3(k) (25)

IEE Pmc -Elecir. Power Appl., Vol. 144, No. 5 , September 1997

Ql(k + 1) =a502?ds(k) + a51iqs(k) + U & ( k ) + a53iqr(k)

+ a54&(k) + a558,(k) + a5082(k) + W 6 ( k )

(28)

& ( k + 1) =a602?ds(k) + ~,li,,(k) + u62;dr(k) + a63iqr(k)

(29) + at34&(k) + Us581 ( k ) + a66&(k) + W ( k )

where the coefficients aik etc. are defined in the Appendix (Section 10.1). Eqns. 23-29 define the electromagnetic, mechanical and thermal model of the induction motor, from which the estimated values of the states are derived using the extended Kalman filter. Hence the state variables, e.g. i&, include the A notation to denote their status as estimated values.

The extended Kalman filter algorithm takes account of process and measurement noise in a general nonlin- ear system

2 ( k + 1) = f ( s ( k ) , u ( k ) ) + w ( k )

9 ( k ) = C(k)@) + 44 (30)

(31 1 Eqn. 30 is the general formulation of the state eqns. 23-29, in which w(k) represents process noise. The out- put eqn. 31 is assumed to be linear, which is the case for speed and temperature estimation in the induction motor, and includes the measurement v(k). Since the stator current components id,s, i,,\ are measured, the measurement matrix C is the 2x6 constant matrix

I 1 0 0 0 0 0 0 0 1 0 0 0 0 0 c = [

The process and measurement noise vectors w(k) and v(k) are random white-gaussian variables and can be described by statistical measures (mean and variance). The means of the vectors are zero and their variances can be described by covariance matrices. The process noise covariance matrix is defined as

E { w ( ~ ) w ( ~ ) ~ } = Q S t , ( 3 3 )

E { v ( ~ ) u ( ~ ) ~ } = R6,k (34)

and the measurement noise covariance matrix is defined as

where Q is a 7x7 constant matrix and R is a 2x1 con- stant matrix. The initial value of the state vector x(0) is described in terms of mean value i ( 0 ) = E{z(O)}

= [ { 2 * d s ( 0 ) l ys(0) & ) lqr(0) &(O) e&) &(O)}l' ( 3 5 )

(36)

(37 )

and a 7x7 square error covariance matrix

P(0) = U" - W ) l [ 4 0 ) - 5$)lT) P(O) = diag[Poo p11 l'22 P33 P44 p55 P e 6 1

3 EKF estimator

The EKF estimator for speed and temperatures of the induction motor is a predictor-corrector estimator, which can be described generally as follows. The pre- diction stage is

i q k + 1) = f ( ? ( k ) , u ( k ) )

~ ( k + I ) = ~ ( k ) ~ ( k ) ~ ~ ( k ) + Q

(38)

(39) 303

The correction stage is

K(k 3.1) = P(k f l)CTICP(k + l )CT +RI-' (40)

P(k + l / k ) P(k + 1) - K(k + 1)CP(k + 1) (41)

4 Evaluation of model parameters

(43)

the coefficients of this matrix are given in the Appendix (Section 10.1). The EKF algorithm used for the combination of speed and temperature is illustrated by the flow-chart of Fig. 2 and the system schematic of Fig. 3.

set cycle time

give matrices Q.R. P(0) L acquire vds and vqs

perform f (... and calculate Jacobian matrix F(k)

acquire ids and iqS

calculate filter gain, estimation error. and correct state estimates

t

t

t

t

i

4

update R ? ( k ) and R2(k )

update nominal states

is time =cycle time

I output estimates I no

Yes

Fig. 2 temperature estwnation in induction motor

Flowchart of EKF ulgorizhm used for combination of speed and

Fig.3 mation

Structure of EKE algorithm used for speed and temperature esti-

The motor electrical parameters were calculated from the standard DC test, locked rotor test and no-load test. The parameters are tabulated in the Appendix (Section 10.2). The thermal parameters of the machine were evaluated by a sequence of four tests, which are described in the Appendix (Section 10.3).

5 implementation

The state-space model of the motor was implemented on a TMS320C30 digital signal processor system, shown schematically in Fig. 4. The motor is a standard 3 kW, three-phase squirrel cage induction machine. The motor is driven by a voltage source IGBT PWM inverter. The motor is coupled to a DC generator driv- ing a switchable-resistance loading bank.

load bank resistor

I I I 1 - 1

2- ADC cards

buffer circuits

II DSP card II I

active buffer pod

Fi .4 infction motor

Experimental set up for speed and temperature estimation in

The required signals for the estimation algorithm are digitised and processed by the data acquisition system, based on a Texas Instruments TMS320C30 DSP oper- ating at 33MHz and providing up to 33.3M floating point operations per second.

6 Estimation results

Initial tests on the speed and temperature estimation scheme were conducted with the following covariance matrix settings, determined by trial and error:

P(O)=diag[5 5 5 5 2 1 11 Q = diag[l .2 1.2 0.3 0.3 0.01

R = [O,, on,] Fig. 5 shows the measured and estimated speed and temperature. The estimator was started while the motor was running at a constant speed of 85radls and with rated load torque. Fig. 5a shows the estimated speed

IEE Pvoc -Electr Power Appl , Vol 144 No 5, September I997 304

while Fig. 5b shows the measured speed. The two speeds are in reasonable agreement and the difference is within 3.5rad/s, a figure which should be compared with motor's slip speed at rated torque of 8.4rad/s. Fig. 5c shows the estimated stator temperature and rotor temperature. The two temperatures vary at differ- ent rates because of differences in the losses, thermal capacities and power transfer coefficients between the stator and rotor. Also shown in Fig. 5c is the measured stator temperature, which is in good agreement with the estimated value. Some differences are to be expected since the measured temperature was a local value obtained from a thermocouple, while the esti- mated value relates the average temperature across the whole stator.

100r

C

0 100 200 time,min

d Speed and temperature estimates for full-load operation and dur- Fig. 5

ing subsequent cooling a Estimated speed b Measured speed c Temperature during operation d Temperature during cooling (i) Rotor estimated (ii) Stator estimated (iii) Stator measured

Fig. 5d shows the estimated stator and rotor temper- ature together with the measured stator temperature when the machine is stationary with zero stator current following a prolonged period of operation at full load. Temperature estimation continues as the machine cools, but due to the absence of current the estimates are open loop. Therefore there are differences of around 5% when the measured and estimated stator temperatures are compared.

The two traces of Fig. 6 show the estimated speed and temperatures with a speed reversal occurring after 90 min. The load torque is at its rated value for steady- state operating conditions in both directions. Fig. 6a shows the estimated speed tracking the speed reversal.

IEE Pvoc -Electr. Power Appl., Vol 144, No. 5, September 1997

The two graphs of Fig. 6b show the estimates of stator and rotor temperatures. The temperature estimates are similar to those of Fig. 5c with the exception of a small (4%) disturbance at the reversal time. This disturbance can be attributed to the EKF gain, which is dependent on the changing speed and is reflected into the estimate of the temperature.

loom

-100 a

' O r rotor

stator

1 I 0 100 200

time, min b

Fig. 6 a Estimated speed b Estimated temperature

Estimated speed and temperatures with speed reversal

VI a e d n W W

VI

a

r

U

ii Y E

,rotor

stator

1

0 100 200 300 400 time, min

b Fig. 7 a Speed

estimated - measured b Estimated temperatures

Estimated speed and temperature under heating and cooling _ _ _ _

Fig. 7 presents the estimated speed and temperature under heating and cooling circumstances. The machine was run for about 150min at full-load torque and then stopped for about 50min. Subsequently the machine

305

was restarted and run for a further 100min with rated load torque. Fig. 7a shows the measured and estimated speed, which are in very good agreement both in steady and transient operating conditions. Fig. 76 shows the estimation of the stator and rotor temperature. This Figure demonstrates that the EKF algorithm is able to track the temperature variations over a range of oper- ating conditions.

Fig. 8 demonstrates the fast transient response of the speed-estimation process. The estimated speed is shown to track the measured speed during acceleration from low speed to rated speed with a connected load inertia equal to the motor inertia.

time,s Measured mid estimated speed during acceleration fvom low to Fig. 8

rated speed - ~ - - estimated

measured

lLOr stator

mechanical and thermal models of the machine. Fig. 10 shows the effect of changing QCC which is the

element of the Q matrix describing the uncertainty in the estimation of the rotor temperature. Q66 was changed from 1 x to 2 x The rotor tempera- ture estimate rose faster than the stator temperature estimate and settled at a higher value (121°C) which is far from the temperature estimated in Fig. 5 (67°C). The speed estimate (Fig. 106) showed an error of 23%.

ro tor

0 I I I

a loor

zot O l I I

0 50 100 150 200 time,min

b Fig. 10 mation of tempevuture and speef a Estimated temperature b Estimated speed

The ejfect ofchangin Q66fvoni 1 x IP4 to 2 x IO-' on the esti-

ov 3 I

a 1 O O r

01 I

0 50 100 150 200 time,min

b Qjj from 1 x I @ to 2 x lF4 on the esti-

a Estimated temperature h Estimated speed

The purpose of the results in Figs. 9 and 10 is to illustrate the influence of the process noise elements on the estimated values of speed and temperature. It should be appreciated that the noise elements have been varied by a factor of 20 from their optimum settings to produce the gross estimation errors in the results of Figs. 9 and 10. Optimum settings of the process noise elements have been derived by minimising the error between measured and estimated speed values. Therefore a manufacturer wishing to incorporate this technique into drive-controller software would have to perform factory tests on typical converter/motor combinations with a speed sensor in place so that the required constants could be preset.

Fig. 9 shows the effect of changing Qjj, which is the process noise element responsible for uncertainty in the stator temperature estimate. Qj5 was changed from 1 x 10-5 to 2 x indicating increased uncertainty in the stator temperature estimate. The stator temperature estimate rose to a level of 110°C (compared with a cor- rect value of 61°C). The estimate of rotor temperature is affected because of the thermal interconnection between the two machine elements. The speed estimate (Fig. 9b) was badly affected, showing an error of 19% because of the interlinking between the electrical,

306

0 1 2 time,min

Fig. 11 and d$jei.uent operating speeds

estimated ~ measured

The measured and estimated speed with double speed veversal

- _ _ _

IEE Proc.-Electv. Power Appl., Vol. 144, No. 5, September 1997

Fig. 11 shows the measured and estimated speed for low-speed operation including a double-speed reversal. The estimator started when the motor was running for- ward at a speed of lOrad/s. After 20s the direction of rotation was reversed for a period of 30s. During the subsequent forward running the speed was reduced until the motor stalled at a speed of 3radls. Both the estimated and the measured speed were noisy and the estimation process was terminated when the motor stalled, with both the estimated and measured speed equal to zero.

7 Conclusions

The estimation results have shown the ability of the EKF to combine speed and temperature estimation in the cage induction motor and have demonstrated the effectiveness of the simple lumped-parameter thermal model. Parameters of the thermal model can be estab- lished using a sequence of four simple tests and the whole estimation procedure is easily and conveniently implemented by using a digital signal processor such as the TMS320C30.

Values for estimated and measured speed agree to within 40% of the rated slip speed during steady-state and transient operation, with the estimator tracking speed down to values as low as 3radIs. The quality of the speed estimate is, in itself, inadequate for slip con- trol of torque, but is potentially useful for low-cost open-loop control schemes.

Estimated and measured stator temperatures agree to within 5%,, even though the estimated value is an average value for the whole stator winding, whereas the measured temperature is derived from a thermocouple placed at a specific location adjacent to the winding. Additional validation of the temperature estimates has been obtained by observing the errors in speed estimation, which result from incorrect settings of temperature covariance parameters.

8 Acknowledgments

The authors acknowledge the financial assistance pro- vided by the UK Engineering and Physical Sciences Research Council via the Link PEDDS initiative, and the support of the industrial partners in the Concerted Action on Power Electronic Control.

References

ANDERSON, B.O., and MOORE, J.B.: ‘Optimal filtering’ (Pren- tice-Hall, 1979) JAZINSKI, A.H.: ‘Stochastic processes and filtering theory’ (Academic Press, 1970) KIM, Y., SUL, S., and PARK, M.: ‘Speed sensorless vector con- trol of induction motor using extended Kalman filter’, IEEE Trans., 1994, IA-30, ( 5 ) , pp. 1225-1233 KUBOTA, H., and MATSUSE, K.: ‘Speed sensorless field-ori- ented control of induction motor with rotor resistance adapta- tion’, IEEE Trans., 1994, IA-30, (5), pp. 1219-1224 HENNEBERGER, G., BRUNSBACH, B.J., and KLEPSCH, T.: ‘Field-oriented control of synchronous and asynchronous drives without mechanical sensors using a Kalman filter’. Proceedings of EPE, Firenze, Italy, 1991, Vol. 3, pp. 664671 TAJIMA, H., and HORI, Y.: ‘Speed sensorless field-orientation control of the induction machine’, IEEE Trans., 1993, IA-29, (l),

BOUSSAK, M., GAPOLINO, G.A., and NGUYEN PHUOC, V.T.: ‘Speed measurement in vector-controlled induction machine by adaptive method’. Proceedings of EPE, Firenze, Italy, 1991, Vol. 3, pp. 653-658 PENG, F., and FUKAO, T.: ‘Robust speed identification for speed-sensorless vector controlled of induction motors’, IEEE Trans., 1994, IA-30, (5), pp. 1234-1240

pp. 175-180

ZHEN, L., and XU, L.: ‘A mutual MRAS identification scheme for position sensorless field orientation control of induction machines’. Proceedings of IEEE Annual Meeting, USA, 1995, pp. 159-165

10 TUNGPIMOLRUT, K., PENG, F., and FUKAO, T.: ‘Robust vector control of induction motor without using stator and rotor circuit time constant’, IEEE Truns., 1994, IA-30, (5), pp. 1241- 1246

11

12

13

14

15

16

17

BOYS, J.T., and MILES, M.J.: ‘Empirical thermal model for inverter-driven cage induction motor’, IEE Pror. Electr. Power Appl., 1994, 141, (6), pp. 360--372 AL-TAYIE, J.K.: ‘Speed and temperature estimation in the brushed DC motor and cage induction using the extended Kalman filter’. PhD thesis, Department of Electrical and Elec- tronic Engineering, University of Newcastle upon Tyne, 1996 GERLANDO, A.D., and VISTOLI, 1.: ‘Improved thermal mod- elling of induction motors for design purposes’. Proceedings of IEE 6th international conference on Electricul muchines und drives, Oxford, 1993, pp. 381-386 NESTLER, H., and SATTLER, P.K.: ‘On-line estimation of temperatures in electrical machines by an observer’. Conference record of ICEM 90, Cambridge, MA, USA, 1990, Vol. 3, pp. 874-879 GUENOV, S.Y., FLOROV, P.I., and IBEH, C.C.: ‘Steady-state thermal analysis of induction motors by finite element method’. Conference record of ICEM 92, Manchester, UK, 1992, pp. 948- 952 SEN GUPTA, D.P., and LYNN, J.W.: ‘Electrical machine dynamics’ (Macmillan, 1980) JONES, C.V.: ‘The unified theory of electrical machines’ (Butter- worth, 1967)

10 Appendix

IO. 1 Coefficients of linearised-discretised state equations

a00 =

a01 =

a02 =

a03 =

a04 =

a05 =

a0G =

a10 =

all =

a12 =

a13 =

a14

a15 =

a16

a20 =

a21 =

a22 =

IEE Puoc.-Electr. Power Appl., Vol. 144, No. 5, September 1997 307

where the Z d ~ ( k ) , Iqf(k), Idr (k> , zqr (k ) , w r ( k ) , @l(k), @ 2 ( k ) represent the nominal values of the states to linearise the nonlinear system. The nominal states are the last best estimates of the states.

10.2 Motor details Brook Crompton 415V, 50Hz squirrel cage induction motor; rated current = 6.5A; rated power = 3kW; rated speed = 1420revlmin; poles = 4; rated torque = 20". Electrical equivalent circuit parameters RI 2.4252 R2 = 2.0552 L1 = 0.237H L2 = 0.237H L, = 0.23H

10.3 Evaluation of thermal parameters The thermal equations of the motor involve the nine additional parameters (k,,, k20, k30, k lw, k2w, k3w, k,,, H I , H2) which were determined by the following four tests:

R, = 95.552

10.3.7: Two locked rotor tests were carried out at supply frequencies of 6 and 8Hz to evaluate the ther- mal power transfer coefficients at zero speed (klo, k20, k30). In each case the stator and rotor temperatures, measured using embedded thermocouples, were allowed to reach steady-state values. From eqns. 21 and 22, the steady-state temperatures are related to the phase cur- rent and thermal power transfer coefficients by

0 = R i o ( l + ~!iBi)(i& + Zis) - kioBi + k3o(Bi - 6'2) (44)

0 = R20(ZiT + Z&) - k2082 - k30(81 - 8 2 ) (45) where at 6 Hz O1 = 65"C, 0, =72"C, phase current = 4.38A, and at 8Hz O1 = 98"C, 82 = 109"C, phase cur- rent = 6.1A. From these equations the thermal power transfer coefficients were calculated

k1o = 3.556WPC k2o = 1.147WI"C k3o = 1.236Wi"C

10.32 Three tests were performed at supply frequen- cies of 30, 40 and 50Hz to calculate the speed variation of the thermal power transfer coefficients (klw, k3J and the iron-loss coefficient kir. Steady-state stator and rotor temperatures were measured using embedded thermocouples. In the case of rotor temperature it was necessary to bring the motor to rest before connecting to the thermocouple. Observations of the rate of rotor temperature change during cooling were used to correct the measurements to take account of the temperature reduction caused by the delay (of approximately 30s) in taking the measurement.

The tests have satisfied the relation from eqn. 21:

0 = R l o ( I + a18i)(iz, + iis) + ki,W:

- k l o ( 1 f k l w " J ~ ) e l + k30(1 + k 3 w w ~ ) ( & -82 ) (46) the measured quantities are -- at 30Hz CO, = 89radis; el = 51°C; 6, = 58°C; phase current = 4S6A -- at 40Hz CO, = 120radls; el = 64°C; 13, = 73°C; phase current = 5.1A

IEE Proc.-Electr. Power Appl., Vol. 144, No. 5, September I997 308

at 50Hz wr = 150radk; 8, = 68°C; 13, = 78°C; phase current = 5.46 A Substituting these measured values, together with the values of previously-determined parameters, into eqn. 46, gives three equations in terms of the three unknown parameters. Hence the parameter values can

k l , = 0.0052slrad k3w = 0.0030drad

10.3.3: The only thermal power transfer coefficient left to be found is k2w. This coefficient was calculated from the previous test at 40Hz steady-state conditions

0 =R20(I + “2@2) (2 : , +2iT) - kzo(1 + k2wu,)B2

- k30(1 + k3wW7 (47)

The rotor current was calculated using the measured stator current and the electrical model of the motor.

uting into eqn. 47 the thermal parameter k2, was found.

kZw = 0.0025drad

70.3.4: The two thermal capacities H I and H2 were calculated from the variation of the stator and rotor temperatures with time during the locked rotor test. The observed thermal time constants were stator thermal time constant = 34 min rotor thermal time constant = 37 min From the thermaI time constants and the thermal power transfer coefficients already calculated, the ther- mal capacities were determined from the relation

thermal capacity = thermal time constant x thermal power transfer coefficient Hi = 4.73 kJPC EE, = 5.29kJPC

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