Stator Flux Estimation Based on Voltage Model

In DTC and FOC induction motor (IM) drive systems, it is important to accurately estimate either the stator or the rotor flux linkages. Inaccurate estimations of the flux linkages will deteriorate the performance of the drive systems which is mainly caused by the phase error between the actual and the estimated fluxes.

(i) In FOC, an incorrect stator or rotor flux position will result in a coupling between the field and the torque components of the stator current [1].

(ii) In DTC, an incorrect estimation of the stator flux position (and magnitude) will result in a wrong voltage

vectors selection in order to control the torque and flux [2]. Most of the methods used to estimate the stator flux in high performance induction motor drive systems are based on the voltage and/or current models. The performances of these two models are discussed in detail in [3]. Here, in this note, only estimation method which is based on voltage model will be discussed. The stator flux based on voltage model is obtained by integrating the stator flux in the stator voltage equation of the IM:

( )∫ −=ψ dtRiv ssss (1)

Under sinusoidal steady state condition, this can be written as:

e

ssss

j

RIV

ω

−=Ψ , (2)

where ωe is the sinusoidal frequency. The major problem associated with this model is its sensitivity to offset and drift errors. This can be seen from the frequency response of an integrator which basically has infinite DC gain. To go around this problem, it is customary to replace the integrator with a low-pass filter. Low-pass filter will effectively blocks the DC offset and low frequency noise, hence avoids the saturation and drift in the integrator. With low pass filter, (2) becomes

ce

sss's

j

RIV

ω+ω

−=Ψ (3)

The estimation of the stator flux based on a pure integrator and a low pass filter can be visualized using a phasor diagram as shown in Figure 1.

Figure 1 Steady state phaso diagram for stator flux estimation based on a pure integrator and a LP filter with cutoff

frequency of ωc

From (2) and (3), we can write

φ∠ω+ω

ω=

Ψ

Ψ

2c

2e

e

s

's (4)

where )/(tan2/ ce1 ωω−π=φ −

For example, the magnitude and phase plot of (4) with ωc = 10 rad/s is shown in Figure 2.

Figure 2 Magnitude and phase frequency response of the estimated to the actual stator flux ratio Figure 2 shows that the operating frequency has to be as higher as possible than the cutoff frequency (in this case 10 rad/s) in order to ensure negligible phase error in the stator flux estimation; this however will reduce the effectiveness of the LP filter to tackle the drift and offset problems. To be able to place the cutoff near the operating frequency, the magnitude and phase compensation has to be performed. It should be noted that the magnitude and phase compensations designed based on (4) only valid under steady state condition [2]. Thus from (2) and (3), we can obtain the compensation needed,

'sc

'sese jj Ψω+Ψω=Ψω (5)

⇒ 's

e

c'ss j Ψ

ω

ω−Ψ=Ψ (6)

In d-q coordinates (6) can be written as:

e

c'sq

'sdsd

ω

ωψ+ψ=ψ (7)

e

c'sd

'sqsq

ω

ωψ−ψ=ψ (8)

The stator flux estimation based on voltage model also suffers from the stator resistance variation. The problem tends to amplify at lower frequency when the term IsRs drops is relatively large compared to the stator voltage. Several methods have been proposed to overcome the problem, for example by using hybrid flux estimator [4], adaptive compensator [5] and fuzzy estimator [6].

References

[1] N. Kamarudin, D.W. Novotny and D.S. Zinger, “The Influence of Motor Parameter Deviations in Feedforward Field Orientation Drive Systems”, IEEE Trans. Ind. Applicat., vol. IA-21, pp. 1009–1015, July. 1985.

[2] N.R.N. Idris and A.H.M. Yatim, “An Improved Stator Flux Estimation in Steady-StateOperation for Direct

Torque Control ofInduction Machiness,” IEEE Trans. Ind. Applicat., vol. 38, pp. 110 - 116, Jan./Feb. 2002. [3] P. L. Jansen and R. D. Lorenz, “A physically insightful approach to the design and accuracy assessment of

flux observers for field oriented induction machine drives,” IEEE Trans. Ind. Applicat., vol. 30, pp. 101–110, Jan./Feb. 1994.

[4] T.G. Habetler, F. Profumo, G. Griva, M. Pastorelli and A. Bettini, “Stator resistance tuning in stator flux field-

oriented drive using an instantaneous hybrid flux estimator”, IEEE Trans Power Electronics, Vol. 13, No.1, pp 125-133., 1998

[5] B.S. Lee and R. Krishnan, “Adaptive stator resistance compensator for high performance direct torque

controlled induction motor drives,” 33rd

IEEE IAS annual meeting, 1998. [6] S Mir, M. Elbuluk and D.S. Zinger, “PI and fuzzy estimators for tuning the stator resistance in direct torque

control of induction machines,” 25th IEEE Annual Power Electronics Specialists Conference (PESC), 1994.