FLUX TRACKING IN INDUCTION MACHINES BY MEANS OF VOLT-AMP QUANTITIES1

PETER DALTON AND VICTOR GOSBELL Department of Electrical Engineering University of Sydney N.S.W. 2006 Australia

Abstract-It is demonstrated that an instantaneous quantity re- lated to average reactive power can be defined for machines with three phase sinusoidally distributed windings. Expressions for the flow of this quantity into the stator of an induction machine are obtained and interpreted in terms of the behaviour of flux dis- tributions within the machine. These results are used to devise an algorithm for tracking the rotor flux inside an induction ma- chine without dependence on stator or rotor resistance and with- out need for motor modifications. Application of this algorithm * I , a field oriented control system utilising a voltage sourced cur-

mt controlled inverter is investigated. Computer simulations are ;,Icsnted to support the theoretical results.

I. INTRODUCTION.

Many advanced induction motor control schemes described in the last 15 years make use of the field oriented control strategy in which the torque and rotor flux magnitudes in the motor are controlled by manipulating the components of stator current per- pendicular ant1 parallel to the rotor flux linkage vector [I]. In such schemes it is preferable to track the location of the rotor flux with- out recours? to transducers in the machine or other modifications. To this end algorithms are often used which carry out a partial simulation of the motor system in real time, so as to predict the location of the rotor flux linkage vector. These algorithms are of- ten referred to as estimators. Such algorithms proposed to date have the weakness that they require knowledge of either the rotor or stator resistance or both, and these machine parameters vary due to thermal effects. Much work has been done recently to de- vise techniques for following the slowly varying resistances so that flux tracking can be performed more accurately [4,6]. However, it is possible to construct a flux tracker for speeds less than rated without motor modifications and without dependence on winding resistance.

11. COMPLEX VECTOR NOTATION FOR THREE PHASE MACHINES.

In uniform air gap AC machine analysis a convention is often used which enables three phase quantities to be represented by a single complex number [2]. If it is true that

z , + z b + z c = 0 (1)

then it is possible to represent the three real quantities z,, z b and z, by a unique complex number z .One frequently used definition is

'The authors wish to acknowledge the Electrical Research Board for funding for this project.

Department of Electrical Engineering University of Sydney N.S.W. 2006 Australia

where

1 .a a = - z + 5 (3)

For example, voltage and current in a three phase winding can be represented by

and

( 5 )

The real quantities may be recovered by taking projections of the complex quantity in the appropriate directions,

x, = Re(x)

x, = Re(za*')

where * denotes the complex conjugate. The complex quantities thus defined may be referred to as vec-

tors as they may have a physical significance if drawn as vectors in the two dimensional plane. For example, if C,, @ b and Qc are the flux linkages of the three stator windings then the corresponding complex quantity Q will point in the direction of maximum flux density.

These complex numbers are distinct from phasors, which a r e used to describe sinusoidal quantities in single phase circuits, al- though there is some correspondence . For example, if the three phases of an AC machine carry balanced sinusoidal signals with phasors X,, x b and X , and if the angular frequency is w then the corresponding complex quantity z is given by

(9:

Unlike phasors, the complex quantities given by (2) are definec under both steady state and transient conditions.

111. INSTANTANEOUS POWER AND REACTIVE POWER.

The instantaneous power flowing into a three phase winding is

88CH2565-0/88/0000-0216$01.W 0 1988 IEEE

P = vaia + Vbib + v,i,

and it is easily shown that = (i) [Im(DQz;) -w,Re(Qz;)]

p = (i) Re(vi*) (11) The total flow into the machine will be given by

(f) q = (h) I ~ ( D Q * * ) - w,Re(\ki;) (21) provided either the currents or voltages sum to zero.

Reactive power is normally defined only for circuits carrying si- nusoidal signals and is an average rather than instantaneous quan- tity. It can be shown that if a three phase winding is excited by

Now it is easily shown that for a complex number 2

a set of balanced currents and voltages then the reactive power is given by

3 Q = (z) Im(vi*)

and if z is viewed as a two dimensional vector then this quantity is equal to exactly two times the rate at which area is swept out

(I2) by the vector. Also

However the rhs of this equation is defined under a much broader set of conditions than those for which reactive power is defined, and so by analogy with (11) it is possible to define an instantaneous Re(Dzz') = lzlDlzl (23) quantity

From this and from the rotor voltage equation (15) it follows that

Under balanced conditions q is constant and equal to Q. When both positive and negative sequences are present then q has a constant component and a component varying at twice the supply frequency. However, the constant component of q under these conditions is not, in general, equal to Q.

= U,[ - l\klDlQl - R e ( h ; ) ] ('25)

If we now restrict ourselves to induction machines and put ~2 = 0 then

G)

IV. INSTANTANEOUS q FLOW IN THE MACHINE.

Consider an AC machine with constant air gap and with the usual assumptions, including that of linear magnetic circuits, and for the moment make the further assumption that their is no flux leakage from the magnetic circuits.Then using the complex nota- tion the electrical equations of the machine are

vi = Rlil + D\k (14)

so finally

1 (f) q = (t) l *mlZDL*m + U r (-1 I*IDI*I (27) Rz So in the case of the induction machine q is the rate of sweeping

out of area by the flux vector plus a term that is non-zero only if the flux magnitude is varying and if the rotor speed is non-zero.

The result of (27) is easily generalised to an induction machine with stator and rotor leakage. The final result can be expressed in different forms depending on which flux quantities are of interest. From Appendix I

1 (;) Q = (i) l*mlZDLQm + (T) 19r112DLQr1 + where w, is the rotor speed, D denotes differentiation with respect to time, subscripts 1 and 2 denote stator and rotor quantities respectively and all the vectors are defined with respect to a fixed frame of reference. (i) IQlZlZDLQlZ + U T (k) l *Z lD1*2 l

The q flow into the stator terminals will be (28)

and into the rotor terminals

217

V. USE OF THE RESULTS TO TRACK ROTOR FLUX IN AN INDUCTION MACHINE.

From (29)

Stator voltage and current can be measured directly and so q and il can be found. Now it is usual practice to hold I Q z l constant for speeds less than rated. In field oriented controllers this can be achieved by controlling the component of stator current parallel to the rotor flux. That is il is regulated so that

Re(Mi lQ;) = J Q 2 1 z (33)

where the rhs is the desired operating flux level. Then if we put

DlQ2l = 0

(32) simplifies to

(34)

Tracking Q2 then becomes a matter of numerically solving the following ordinary differential equation (ODE)

Fig. 1.Region of stability for the new flux tracking met hod.

By finding the Jacobian of the system comprising (37) and (38) and evaluating at all operating points satisfying (33) it can be shown that the system is stable only under the conditions

and

wdw, + w s ) > 0

where

(39)

(41) W , = DLQZ-W,

is the slip frequency. The region of stability so defined is illustrated in Fig. 1.

in real time. The above approach to tracking Qz depends on a simplification

which arises when the flux magnitude is held constant. A more general approach which will track rotor flux without this restric- tion is possible. Appendix I1 derives a system of two ODES in Qz with q as an input and without dependence on R1 or Rz.

VI. STABILITY.

VII. PHYSICAL IMPLEMENTATION CONSIDERATIONS.

In adopting the flux tracking algorithm described above we are in effect solving a system of two OD&

1 DlQzl = -(Z) [lQzl- IQ21

Re(Mi lQ;) (37)

but with the restriction that the controller operates only at states where the flux magnitude is constant.

Typical field oriented controllers make use of voltage sourced current controlled inverters to supply the power to the motor and microprocessor based controllers to perform the necessary compu- tation. It is necessary to consider how the algorithm described above can be implemented on such a system.

Suppose the controller performs iterations of the flux tracking algorithm at intervals tn-l, t,, tn+l etc.. , then integrating (35) over one such interval gjves

(42) rearranging,

(43)

218

5 -

0 I

0.1 0.2 0 . 3 0.4 0.5 time/s

( 9 . 0

?

Eo U z m m L

0 L 0)

\ W

U L

U T

3 20 0 0 . L

e - P

0 ‘ I

0 0 .1 0.2 0 . 3 0.4 0 .5 time/s

I

Fig. 2.Field oriented control of an induction machine using the new flux tracking method (a) with an instantaneous change in commanded torque and (b) with the rate of change of commanded torque limited.

The integral of q flowing from the inverter in the period t, to tn+l can be measured most easily if the phase voltages of the stator are inferred from the state of the voltage sourced inverter. If the source voltage is assumed constant then calculating the integral of q over a period of time becomes a matter of integrating the phase currents with the sign of each current in the integration depending on the state of the inverter. The final result is

where the binary quantities Sa, SI, and S, describe the inverter state according to

Va,b,c = X t o r + ( -l)sa’b’cvd (45)

where &tor is the star point voltage of a star connected stator and Va is the magnitude of the DC source voltage.

This method is due to Abbondanti [3] who used it to measure average reactive power.

from (43) is equal to twice the area swept out by the stator current vector in the interval t , to tn+l. To determine this quantity it is necessary to know the path of the current vector. If the value of the reference currents generated by the field oriented controller are updated after each iteration of the flux tracking algorithm then it may be sufficient to assume that the stator current moves in a straight line between the reference current values. To insure the validity of this approach it is necessary to limit the rate of change of the reference currents so that the stator current can always attain the new reference value in the interval between successive iterations.

Fig. 2 shows the results of numerical simulations of the flux tracking algorithm. The motor simulated is a 30 kW squirrel cage induction machine with an inertial load controlled by a field ori- ented controller. The motor is initially at rest. In (a) a 100 Nm torque step is commanded at 0.1 s. As the current controller is not capable of driving the stator current to its new reference value in one iteration period (2 ms) am error in the calculated flux angle occurs. This error decays away in the usual fashion. In (b) the rate of rise of the torque command has been limitted to 20 Nm/ms so that the commanded stator current change is within the capa- bilities of the current controller.

VIII. EARLIER RESULTS.

The term If an induction machine is in a state where rotor flux magnitude

and torque are constant then the stator currents and voltages will be sinusoidal and balanced and all flux linkages and currents will rotate at the same angular frequency which will be the frequency of (46)

219

the stator currents and voltages. Let w be this angular frequency and rearrange (35). v1 = Rl i l + DQ1

where Q can be substituted for q because Q is defined, and equal to q , under these conditions.

With allowance for different notation and some algebraic manip- ulation this result is the same as that derived by different means by Abbondanti [3] and also by Garces (41. Abbondanti used the result to regulate the flux magnitude in a speed control system. Garces (and others) have used the result to implement a rotor parameter identification scheme. The above argument shows this result to be a special case of the more general result of (29).

Lip0 and Chang [5] described a flux tracking scheme which makes use of the potentials induced in two coils from the same stator phase winding. The coil potentials have components due to the stator resistance but these components are equal, as both coils carry the same current, and so may be cancelled out by taking the difference of the potentials. Now this approach can be generalised as follows. Any two coils can be used as long as the currents car- ried by the coils are known and the coil resistances are known to be equal. The potentials of the two coils are then differenced with the potential from each coil being weighted in inverse proportion to the current carried. For example an R1 independent quantity would be

am = M ( i l + i2)

911 = llil

9 1 2 = lziz

where the subscript 1 denotes leakage components. The q flow into the stator terminals will be

and into the rotor terminals

(59) X = iavb - ibv, (48)

Then from the definitions of the complex quantities V I and il and from the definition of q it can be shown that = (i) [Im(DQZi;) - w,Re(92i;)]

If we restrict ourselves to induction machines then using the result of (26)

The above shows that the flux tracking method presented in this paper corresponds to a generalisation of the method of Lip0 and Chang.

If the flux linkages are expressed in terms of the magnetising and leakage flux linkages then the total reactive power flow into the induction machine will be given by

IX. CONCLUSION.

A new instantaneous quantity q has been defined in the context of three phase AC machines. It has been shown that the value of this quantity is related to the movement of the magnetic flux link- ages within the machine. In particular, two ordinary differential equations are derived: one equation enables the position of the rotor flux linkage to be followed providing the magnitude is held constant; the other equation can be solved to gjve both position and magnitude of the rotor flux l i i g e .

A flux tracking algorithm for induction machines based on the simpler of the two results is described, practical considerations for its implementation discussed and results of simulations presented. This algorithm does not require knowledge of the rotor or stator resistance and requires no modifications to the motor.

Alternatively, eliminating the magnetising flux linkage

or

X. APPENDIX I.

For a constant air gap machine with flux leakages but linear magnetic circuits the basic equations in a stationary frame of ref- erence are [2]

220

where

XII. REFERENCES.

and

XI. APPENDIX 11.

Consider a constant air gap machine with flux leakage but with linear magnetic circuits as in Appendix I. &arranging the voltage equations gives

DQl = v1 - Rlil (68)

DQz = jw,Qz-Rziz (69)

and eliminating Ql and iz in favour of qz and il

DQz = (Lzwl - L2Rlil - L*LzoDil) /M (70)

(71) R

D Q ~ = jw,Q2 - L ( a 2 - M i l ) L2

Now taking the component of DQ2 perpendicular to any uncer- tainty due to RI from (70) gives

I m ( D Q 1 ( M i l ) * ) = LzIm(wli;) - L I L z o I m ( D i l i i ) (72)

and taking the component of D92 perpendicular to any uncer- tainty due to Rz in (71) gives

Im(Daz(Q2 - M i l ) * ) = w, [1Q2I2 - Re(QzMi;)] (73)

Now it is true that if two projections of a complex number z are taken perpendicular to two complex numbers Q and b then z may be reconstructed by

(74) b Im(za*) - a Im(zb*)

Im(ba*) z =

providing that a and b are non-zero and not parallel. This result can be applied for z = DQz, a = Mil and b = Qz - Mil . The condition on a and b now becomes a condition that il and \kZ be non-zero and that i l and Qz not be parallel, which is a reasonable condition for an induction motor under field oriented control.

so

[l] F.Blaschke, “The principle of field orientation as applied to the new Transvector closed loop control system for rotating field machines,” Siemens Review, XXXIX, No.5, pp.217-220, 1972.

[2] K. P. Kovdcs and I. Racz, Transiente Vorgange in Wechselstrom-Maschinen. Akademiai Kiado, 1959.

[3] A. Abbondanti, “A method of flux control in induc- tion motors driven by variable frequency, variable voltage supplies,” in Conf. &c. 1977 12th Annu. Meet. IEEE Ind. Appl. SOC., pp.177-187.

[4] L. Garces, “Parameter adaption for the speed controlled static AC drive with squirrel cage induction motor,” in Cod. Rec. 1979 14th AMU. Meet . IEEE Ind. Appl. Soc., pp.843-850.

[5] T. A. Lip0 and K. C. Chang, “A new approach to flux and torque sensing in induction machines,” IEEE Trans. Ind. Appl., vol.IA-22 no. 4, pp.731-737, 1986.

[6] R. Gabriel and W. Leonhard, “Microprocessor control Control of induction motor,” in Conf. Rec. IEEE Ind. Appl. Soc. ISPCC’82., pp385-396.

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