Effects of flux level on a CSI-fed field-oriented induction motor

B.C.Ghosh S.N.Bhadra

Indexing termx Induction motor drives, Flux-saturcition ej’ects, Vector controllers

Abstract: A definite relation exists between the flux level, torque and slip speed of a vector controlled induction motor. An untuned vector controller generates an inappropriate slip frequency that changes the operating flux of the machine. This affects the electromagnetic torque linearity and dynamic performance of the drive system. For an induction motor with magnetic saturation, the ratio of flux and torque-producing components of stator current differs from unity for the maximum electromagnetic torque. With a linear model this ratio is unity. The dependence of this ratio for maximum torque output is explained for linear and nonlinear models of the machine. The torque characteristic is analysed with three aspects of magnetic state: true saturation curve, hard-limit saturation curve, and constant inductance model. Degradation in dynamic performance due to saturation for a CSI-fed system is presented.

1 Introduction

Vector-controlled induction motors arc being widely used as motion-control industrial drives throughout the world. A good number of control methodologies [l, 31, differing in inverter type, have been developed over the last two decades. This control aspect is based on the independent control of flux and torque-producing currents fed to the motor. The direct method requires additional circuits to detect the position and magnitude of flux. The popular one is the indirect method that requires a feedforward slip to control the position of the stator MMF assuming the rotor flux vector as reference. It is well known that generation of torque of all types of motor is generic in nature. Like DC machines the flux and torque-producing currents in the induction machines are in orthogonal orientation. To have fast dynamic response a vector control scheme keeps the flux magnitude constant. The torque and flux components of stator current fed to the machine must 0 IEE, 1997 IEE Proceedings online no. 19971 167 Paper rweived 2nd January 1997 B.C. Ghosh is with the Electrical and Electronic Engineering Department, BIT Khulna-9203, Bangladesh S.N. Bhadra is with the Electrical Engineering Department, IIT Kharag- pur, PIN 721302, India

correspond to the slip for a tuned controller. The deviation of rotor resistance or mutual inductance from its nominal value changes the flux level from its set value. This causes over- or under-excitation in the machine and results in degradation in performance. This aspect has been the subject matter of investigation by a large number of researchers [l-51 over the last two decades.

The performance of a field-oriented induction motor with saturation has been reported in the literature [4-9] in which the magnetising characteristic is represented either by hard-limit saturation curve [9] or by a nonlin- ear mathematical model [3, 61. Variation of flux to improve performance of the motor has been reported in [6, 91. The inductances as reported in [2] vary from 80 to 120% of their nominal values.

This paper presents the effects of magnetic flux saturation on the performance of a current-regulated field-oriented induction motor drive in analytical form. The analysis considers the influences of the excitation component of stator-current variation under steady- state condition. The magnetisation characteristic is represented by a two-term quintic-fit curve and the magnetising inductance is expressed as a polynomial for a more realistic representation of a saturated induction machine. To visualise the effect of saturation on transient performance a laboratory-type CSI-fed induction motor drive was considered. The control law was implemented in an 8086187 microprocessor and PC environment and was operated under field orientation and deorientation condition

B Y L

we: w* + W S [

OS a Fig. 1 U ) - Actual blalor ‘ A phase anis U,. - Actual rotor ‘A’ phase axis a ~ /3 - Stator fixed reference frame x - y - Rotor Gxed reference frame d - q - Synchronously rotating reference frame idq - Stator MMF vector

Relation between various co-ordinate systems of induction motor

IEE Proc.-Electr. Power Appl., Vol. 144, No. 5, September 1997 295

2 Performance equations of vector-controlled induction motor

Under the usual assumptions of no hysteresis, no eddy currents and no space harmonics a three-phase induc- tion motor can be suitably modelled in terms of d-q variables in an arbitrary synchronous reference frame [ l l ] as

R s + P L e -Lswe PLm & + p L s Lmwe

p L m -wsiLm RT+PL, - L r w s ~ Lrnwsl PLm LTwsl RT + p L r

(1) where the symbols have their usual meanings. The

rs of the induction motor in different reference frames are shown in F 1. Under steady-state conditions, the two rotor current components from eqn. I are

Z d r = RZ + Lip

The developed electromagnetic torque is given by

(4) where

Xdr = L.r%Clr f LmZds

Xqr = L r ~ q r + Lmlqs (5) The magnitudes of the components of vector current idq in the d-q axes are

% & = % d q g d s a d q cos 6 t q s = i d q g q s - - i d q sin 8

where iR is the DC link current and 8 is the angle between stator MMF and d-axis of the reference frame.

To develop the field-orientation model, the rotor flux or of the induction motor is assumed to align along

the d-axis. Under this constraining condition, Ad, = A, and A,, = 0 and the values of rotor flux linkage and slip speed are written as

( 7 )

where z, = L,/R,. The electromagnetic torque under this condition is modified as

Again use of eqns. 2 and 3 in eqn. 4 gives the general- ised torque equation under steady-state conditions as

(9)

For a closed-loop speed-controlled developed electromagn

te conditions are the and since iR2 = id:

equation for DC link current is

From eqns. 7 and 8 with the constraints T,, = TL, the steady-state rotor flux linkage of the machine is written as

This relationship always holds good inside the induc- tion motor irrespective of the speed-control scheme adopted.

I l l weed vs

CS inv. I

based pp para.

rotor flux speed command command

Fig.2 Current regulated closed-loop speed control system

3 Magnetic nonlinearity and magnetising inductance

The magnetic state of an induction motor depends on the radially directed air-gap flux and circumferentially directed leakage flux. If the teeth are not highly satu- rated the leakage flux has negligible effect on system modelling. For well-designed machines the saturation level in teeth is restricted within limits. Moreover, leak- age reactance is a small portion of the total reactance. In general, analytical techniques consider the effect of iron saturation by appropriately adjusting the value of magnetising inductance L,. The variable magnetising inductance as a function of magnetising inductance is expressed as

Lm = f ( z m a , )

1 bo + b l a m a g + b22Lag + b32kag + b32kag + b 4 2 k a g

(12) In terms of d-q-axis quantities the magnetising current is written as

%mag J ( i c i s + zdr)2 - ( i q s + z q r ) 2 (13)

L, in eqn. 12 was obtained from the experimentally determined true saturation characteristics and the two- term quintic polynomial fit of the form

(14) 5 %mag a1 Xm - a5Xm

4 Analysis of field-oriented induction motor with variable flux level

The induction motor in the d-q-axis model as developed in the previous Section was used for analysis and to deduce the graphical characteristics of the machine. The nonlinear (saturated) characteristics reflect the behaviour with the actual magnetic state of the machine. Side by side the linear (constant parameter) model is also used to visualise the discrepancies in the characteristics. The nonlinearity in flux level with excitation current (inherent in the machine) is presented in Fig. 3. A comparison of the

296

characteristics with the constant-parameter model reflects their deviation from that of the saturation model.

r

2 1 6 8 10 excitation current, A

Fig.3 ~ linear model _ _ _ _ saturated model

Ejject of excitution current on rotor jlux

4. I Relationship between torque, slip speed and flux level Fig. 4 shows the effects of slip speed on torque output and rotor flux linkage. This characteristic may be deduced from eqn. 11. The inherent characteristic of the induction motor demands a requisite value of slip frequency for field orientation based on load torque. If this slip speed is generated within the vector controller then the machine operates under the field-orientation condition. Inappropriate transmission of slip frequency by the indirect vector controller creates an over- or under-flux condition of the machine. If the slip speed is less than that required for a particular load the machine operates in the over-flux (saturated) condition. The curves illustrated in Fig. 4 are for three different values of developed torque.

P X d L

L 0

0 Y

I I

5 10 15 20 25 30 35 LO 45 50 slip speed, rad/s

Effects of slip speed und torque on,flux level selection Fig.4

4.2 Maximum torque production condition In designing a vector controller with a current control- led drive, emphasis is given to the availability of maxi- mum electromagnetic torque for a particular stator current. This maximum electromagnetic torque of a current controlled induction motor is dependent on the degree of saturation, as illustrated in Fig. 5. The ratio of id, to iqs here represents the degree of saturation for

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

a particular stator current. Torques for three specific values of stator current are plotted as a function of the ratio of the torque component to flux component of the stator current, with and without saturation. The effects of saturation on torque apparent from these curves are (i) The torque characteristic with the nonlinear (satura- tion) model is flatter than that for the linear model. (ii) The ratio of iql to id, is greater than unity and increasing for increased values of link current to pro- duce peak developed torque with a nonlinear magnetis- ing characteristic. As indicated in Fig. 5 , the peak torque for an unsaturated machine occurs at a ratio of unity. In this case i, and idr each equal 70.7% of the magnitude of the stator current vector.

50r

Ulr , I I J 0 2 L 6 8 10

Ids 'I qs

Fi .5 w i x two machine mod& __ linear model _ _ _ _ saturated model

Effects of tor ue undjlux producing currents on developed torque

6 8 10 12 11 stator current,A

Fig. 6 vuriuble input current __ current ratio _ _ _ _ maximum torque

Maximum torque production condition with suturution model and

Fig. 6 shows the relationship between the stator input current with the factor iqjidJ and the maximum torque resulting therein for the induction motor with a nonlinear model. The current controlled drives show almost linear relationship for the maximum output torque but the factor i,,lids shows nonlinearity for lower input current.

The reason for the shift towards higher values of iqsl idr for the peak torque to appear with the saturated machine may be visualised from the torque expression eqn. 9 when compared with the level flux saturation

291

characteristic in Fig. 7. Because of the slow rate of increase of the rotor flux with magnetising current beyond the knee point, a two-segment linear level flux saturation model in place of the exact saturation model (Fig. 7) may be considered to be sufficient for qualita- tive examination and reasonably good approximate quantitative assessment of the influence of saturation on the torque characteristic.

01991

i 0.6

n 3 x

t I I I

0 I , I I I

0 2 L i 6 8 10 12 ido: L.25

current, A Approximation of saturation characteristic by piecewise hard- Fig. 7

limit Saturation curve

For the rotor flux-based vector control scheme it is evident from the torque equation in eqn. 8 that the L, to L,. ratio will show slow change with degree of satura- tion if the leakage reactance is small and the magnetis- ing current is not too large. So for a given stator current i,’, T,, will take the maximum value Tern, ,ux- when the product of Ar and iqS attains maximum value.

- - L,, ido (Fig. 7). As magnetising current greater than id, is inef- fective in raising the flux level further, the best effect for given i‘, will be produced when

Maximum value of Ar is given by A ,

Hence T,,, lllilx will be

Tern,,,, = f p %&JG Lr, (16)

and in the T,, $,/ids plane, will lie on a straight line whose slope by eqns. 15 and 16 is given by

This indicates a shift in the Te,, ,ax position unlike in the constant-parameter model of the induction machine.

4.3 Effect of excitation on motor current and torque per ampere square The flux component of stator current (id,) has a vital effect on the value of input current to develop a requisite amount of torque. This can be visualised from Fig. 8. The study was based on eqn. 10 for both saturated and unsaturated machine models. It is apparent that the minimum values of stator-current demand occur at different values of excitation current with different load torques. The constraints of slip speed and parameters for the models represented in eqn. 10 are responsible for variation and its nature for the two models. If torque/A2 characteristics are drawn with variation of excitation current it takes the shape

298

as represented in Fig. 9. It is observed that the maximum values of torque per ampere square are shifted towards higher i, for higher load torques. It is also observed that the curves with saturation model are flatter than those with the linear model.

6 i”h c W L

3 U

L 0 0 ul

- c

5 01 I I I I

2 L 6 8 10

I d s Fig.8 __ unsaturated model _ _ _ ~ saturated model

Effect of excitation on input current

01 I , 2 4 6 8 10

Ids Effects of excitation current on torque per ampere square Fig.9

~ linear model saturated model _ _ ~ ~

4.4 Degradation in dynamic performance To study the effects of saturation on dynamic performance a CSI-fed vector controlled drive was selected. An i-8086/87-based controller was used to implement the control law and a PC/AT was used for parameter adaptation. In the control scheme the difference in command speed and actual speed was used to generate the command value of the torque component of current i,$. The flux component of current id, was generated from the command rotor flux and stator current. Slip speed was calculated using i,, (deemed to be actual torque producing current) in eqn. 7. The control scheme is shown in Fig. 2. The controller was equipped with a parameter adaptation scheme invoked from keyboard as and when desired. The speed response at starting with a particular load torque and reference speed (105radh) under field orientation is shown in Fig. 10 and the same for the deorientation condition is represented in Fig. 1 1. Over- flux was created by transmission of slip speed lower

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

(67%) than that required for field orientation. With vector-control system will be severely misleading if imposed field orientation the flux level does not saturation effects are ignored. Saturation degrades the change. But with deorientation the flux level changes dynamic performance of drives, especially current-fed and causes saturation for low values of slip. Very fast ones, and in most cases makes it worse than with scalar response for the field-oriented drive in comparison to controlled drives. Comments made in this paper may the saturated condition can be visualised from the be extended to all current regulated systems. characteristics. Fig. 12a shows the oscillographic record of speed and link current under the field-orientation condition when the load torque was suddenly increased (from 3.5 to 8Nm), and Fig. 12b shows the same quantities with saturation imposed. A larger dip in the speed characteristic of Fig. 126 in comparison with that in Fig. 120 establishes the degradation in performance with saturation.

c

O o o r 1’O

Io 0 2 L 6 8 10

t ime,s

Fig. 10 ~ speed

Speed characteristic under $eld-orientation condition (10 Nm)

current ~~~~

1000r 71 0

OV Io 0 2 1 6 8 10

time,s

Fig. 11 parameters

~ speed

Speed churacteristic with mismatch in controller and machine

current _ _ ~ ~

5 Conclusions

The influence of saturation on machine characteristics for tuned operation of a vector controller has been studied. It is seen from the analysis and simulation results that selection of proper flux level and excitation current is of vital importance for designing a vector controller. It is observed that the effect of saturation becomes prominent at higher values of load torque. The dependence of input current on excitation current means that for efficient operation the designer should give emphasis on the general trend of load torque to select a suitable value of excitation current. The studies explain clearly the differences between the drive characteristics with linear and saturation models. With increased rotor resistance at high load torque, predictions of the performances of a slip-regulated

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

0 10

C t ime, s 0 .-

E

10 t i m e , s

b

0

Fig. 12 Oscillograms for U sudden increase in load torque a For field orientation b For saturation and deorientation

6 References

1 GARCES, L.J.: ‘Parameter adaptation for the speed controlled static AC drive with squirrel cage induction motor’, IEEE Trans., 1980, IA-16, pp. 173-178 KRISHNAN, R., and DORAN, F.C.: ‘Study of parameter sensi- tivity in high performance inverter-fed induction motor drive sys- tems’, ZEEE Trans., 1987, IA-23, pp. 623-635

3 NORDIN, K.B., NOVOTNY, D.W., and ZINGER, D.S.: ‘The influence of motor parameter deviations in feedforward field ori- entation drive systems’, ZEEE Trans., 1985, IA-21, pp. 1009-1015

4 VAS, P., and ALAKULA, M.: ‘Field oriented control of satu- rated induction machines’, IEEE Trans. Energy Conv., 1990, 5, pp. 218-224 OJO, O., and VIPIN, M.: ‘Steady state performance evaluation of saturated field oriented induction motor’,Conference Record,

6 KHATER, F., LORENZ, R.D., NOVOTNY, D.W., and TANG, K.: ‘Selection of flux level in field oriented induction machine controllers with consideration of magnetic saturation effect’, ZEEE Trans., 1987, IA-23, pp. 276-281 LEVI, E., and VUCKOVIC, V.: ‘Field oriented control of induc- tion machines in the presence of magnetic saturation’, Electr. Machin. Power Syst., 1989, 16, pp. 133-147 LORENZ, R.D., and NOVOTNY, D.W.: ‘Saturation effects in field-oriented induction machines’, ZEEE Trans., 1990, IA-26, pp. 283-289

9 XU, X., DONCKER, R.K., and NOVOTNY, D.W.: ‘Stator flux orientation control of induction machines in the field weakening region’. Annual meeting record of IEEE-IAS, 1988, pp. 437-443

10 BHADRA, S.N.: ‘A direct method to predict instantaneous satu- ration curve from rms saturation curve’, IEEE Trans., 1982,

11 CORNELL, E.P., and LIPO, T.A.: ‘Modelling and design of con- trolled current induction motor drive systems’, ZEEE Trans., 1977, IA-13, pp. 321-330

2

5

IEEE-IAS, 1990, pp. 52-60

7

8

MAG-18, pp. 1867-1869

299

7 Appendix Stator self inductance L, 0.1594H

7. I Name plate data and machine parameters Induction machine (squirrel cage) Name plate data: 3-phase; 4001440V; 3.75kW; 50Hz; 4-pole; A -connected Coupled DC generator: 240V; 2.0kW; 10A Nominal parameters (referred to stator) Stator resistance R, 1.38 C2 Rotor resistance R, 1.5087Q Mutual inductance Lm 0.1458H

Rotor self inductance L, 0.1594H Stator leakage inductance I I 0.0135H Rotor leakage inductance Z2 0.0135H Moment of inertia J Damping coefficient B 0.005 Nm-slr Saturation characteristic fo r induction machine

0.091 kg-m2

Lrn = S ( h " ) - 3 - bo + b i zmug + b z i k , , + b3iLag + b3inzag +

bo = 0.24399; bl = +0.007228; bz = -0.0070313;

b3 = 0.0006815; b4 = -1.97313 x lo5

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