Modern Techniques in Electrical Machine Parameter Identification

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I FA( : Idt'1I1 iri(';tl illll and SV.\I('JJ1 York . l lK. I!H(~)

ELECTRICAL AND HEATI:-.JC SYSTEMS

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MODERN TECHNIQUES IN ELECTRICAL MACHINE PARAMETER IDENTIFICATION D. Brook and D. MOTton /)r/)(JlllllfIIl 1I/1': II{;illl'l'rill{; Syslt'III. I .

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Abstract. A technigue is presented which allows the injection of a perturbation signal On a d.t. machine allowing standard control engineering techniques to be performed. The tests considered are (i) frequency response, (ii) cross-correlation, (iii) Fast Fourier Transform analysis, (iv) Ka1man Filter and (v) simple step response. Keywords. Automatic testing; electric drives; frequency response; correlation; Fast Fourier Transforms; Ka1man filter. INTRODUCT ION Whilst a number of conventional testing techniques are available for the determination of electrical machine parameters it has always remained a problem to determine these parameters under the actual conditions of operation (Nita and Okitsu, 1966). The techniques generally available use a series of separate tests for determination of each individual parameter or rely on a step response test with the machine initially stationary (Lord and Hwang. 1974\. As a result the parameters are often inadequate and misleading when used for analysing the machine dynamic response. Difficulties in injecting a suitable perturbation signal has so far restricted the application of control engineering identification techniques to overcome this problem.

One can identify the following requirements for a signal injection device.

2 3 4 5

The capacity to handle the power and Signal amplitude requirements. An adequate band width. Minimum change to the plant dynamics, and in this work this means minimum output impedance . An ability to carry any nominal load conditions. In this case. an ability to carry the nominal armature current . Linearity. In particular the device must have the same characteristics for an increasing perturbation as for a decreasing one.

(If only a binary signal is being used these requirements are. by and large. easier to satisfy but in this work it was necessary to consider continuous signals.)

Whilst it is of course appreciated that frequency response testing has been performed on d.c. machines, signal injection has been through a drive amplifier thereby making it difficult to isolate the machine response from that of the amplifier.

Various schemes were considered but that outlined in Fig 1 below was found to be the most satisfactory. Since the machine under test was quite small (lKW. 110V) all the components of the injection rig were available as general laboratory items and no design work. as such. was needed. However for much larger machines careful design of components would be necessary. (In this instance the power amplifier itself was large and could have been used to provide both the main supply and perturbation signal but the arrangement shown was used in anticipation of its use in much larger installations).

This paper describes a method of injecting a perturbation signal on a conventional d.c. machine. The method is based upon modulating a 100Hz carrier with the perturbation signal and injecting this via a transformer and rectifier connected in a series with the machine armature. The net result is a perturbation of the machine about its steadystate operating speed. Using this system traditional frequency response. cross-correlation. FFT analysis, Ka1man Filter and step response tests can be performed. The benefit of the technique is that tests can be performed with the machine at its actual operating condition with the result that relevant incremental parameters may be determined for different operating conditions.

The operation is quite straightforward. A modulated carrier from the low power signal generator is given adequate power and amplitude by the amplifier and then presented to the transformer, the rectified output of which becomes the perturbation signal. Neither the amplifier nor the transformer windings carry the nominal armature current. but the rectifier must have the capacity to carry it as well as the perturbation component and the current amplitudes must be such that a reversed current through it is not demanded.

SIGNAL CONVERSION An essential. but little noted, part of on-line testing is the means of signal conversion into. and out of the plant or machinery. In the work described here there wa s no essential problem in getting a signal representing motor speed; a proprietary digital tachometer was used together with a frequency to voltage converter and circuitry to "back-off" the nominal speed component. A greater problem was that of injecting an armature voltage perturbation.

The signal of the perturbation generator can be of any appropriate form and in this work included sine, gaussian and square forms. A carrier of 100Hz was chosen. arbitrarily, to be high enough to exceed with adequate margin the highest fr equency components of the perturbation but not high enough to be beyond the capability of the transformer. IX2'j

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D. Brook alld D. I\lor\oll

Using a resistive load instead of the motor and supply, the frequency response was flat well beyond the value required (10Hz). A conventional calibration graph (Pig 2) gave the output resistance as 200. Obviously this is of high order in relation to that of the armature and results have to be corrected accordingly. The input signal (perturbation) to the motor was taken from an auxiliary rectifier across the main rectifier and the output signal from the digital tacho. A potential divider across the auxiliary rectifier and an amplifier associated with the tacho gave the necessary signal levels. Both signals were passed through low pass filters to remove the carrier generated components on the one hand and to clean up the output on the other, Fig 3. FREQUENCY RESPONSE Using the signal injection circuit previously described the frequency response characteristics of the motor were established by measuring "gain" and "phase" at selected frequencies with a Solartron Frequency Response Analyser. These characteristics are presented in Figs 4 and 5. CORRELATION ANALYSIS As is well known (Eykhoff, 1974) the impulse response of a system can be obtained, under ideal conditions, from R

xy

(T) = foo g (A ) R (T-A) dx

xx

0

(l)

and when the input is white this reduces to Rxy (T)

=

Ng ( T )

(2)

This shows, of course, that the impulse response can be determined by measuring the input-output cross-correlation. In a real test infinite integration time and true white signals are not possible but provided an dequate integration time and signal bandwidth is achieved satisfactory results can be obtained. Using the method outlined previously with the perturbation generator as a source of Band Limited Noise, and a proprietary correlator arranged to measure the correlation relating armature speed variation with the armature voltage perturbation the correlogramof Fig 6 was obtained. The conditions of the test were:Perturbation bandwidth Number of samples - 16

5Hz

Fig 7 shows the form of the armature voltage perturbation and the corresponding speed variation. Repeated runs of the test gave repeatable results but no attempt was made at this stage to perform a statistical analysis. Using conventional logarithmic linearisation Fig 8 was obtained from which the time constant was found to be 883mS corresponding to a bandwidth of 0.20Hz. FAST FOURIER TRANSFORM (FFT) ANALYSIS The frequency domain equivalent of cross-correlation (Equation 1) is Sxy (w)

=

G( jw ) . SlQ{ (w)

The conditions of the tests were Perturbation bandwidth 12Hz Number of averages - 16 STEP RESPONSE By making the perturbation generator produce a square wave of long period repeated step responses could be observed on an oscilloscope as shown in Fig 10. Logarithmic linearisation produced a time constant of 0.8s. KALMAN FILTER ESTIMATION A Kalman Filter estimation technique (Mayne, 1974) essentially involves an estimation of the coefficients of the state transition model of a linear system in the presence of random gaussian inputs. The application of this technique to a d.c. motor requires that motor be perturbed about its steady-state operating speed with a gaussian noise signal. The resultant variations in motor speed and armature current are then recorded and stored sequentially. The stored data is then used to make an optimal estimate of the coefficients of the state equation. It is from these coefficients that the motor parameters may then be established. Initially an estimate of the coefficients is made and a new "up-to-date" estimate is then computed from the last estimate and the new received data sample. This recursive estimation procedure is continued until the values of the coefficient remain constant. This technique can be implemented simply if a facility is available for the perturbation of the motor armature voltage about its steady-state value. This can be achieved with the injection circuitry discussed in this paper. The variations in motor speed and armature current are then sampled and stored in a microcomputer. A computer program for the recursive Kalman Filter estimation procedure is then implemented to determine the motor parameters . Previous work carried out by the authors (Freeman, Hassan and Morton, 1985) has implemented this technique on the d.c. machine discussed in this work. The work was carried out prior to the development of the injection technique discussed in this work and the motor armature voltage was perturbed using a simple transistor power amplifier . This method of perturbation is justifiable with the motor operating on "no-load", but would be unsatisfactory for high load torques since the power amplifier would have to carry the full load current. In this respect the injection circuit considered in this paper would be satisfactory and the authors intend to implement the Kalman Filter technique using this method of perturbation.

( 3)

The motor parameters determined from the Kalman Filter test are as follows:-

( 4)

Armature Resistance Ra . ........ ... ..... . 1. 22 Armature Inductance La . .. . . ............. 0.00925H 2 Moment of Inertia J ..... . ...... .. ... . 0. 0179 kgm Coefficient of Viscou s Damping B .. 0. 0046 Nm / rad / s

which, with a white input becomes \y( w) = G( jw) .N.

Thus using Fourier Transform techniques the crossspectral density can give the system frequency response. Again proper experiment design is necessary for a real te st. Using a proprietary digital signal analyser with its own signal source replacing the perturbation generator direct computation and display of G (j w ) was effected on the rig. Fig 9 shows the results which were easily repeated. The bandwidth is seen to be 0.24Hz corresponding to a time constant of .663 ms.

Elect rical Machille Parameter Id e lltificatioll

LOAD TORQUE AND FIELD V,)L TAGE PERTl'RBATION The tests considered have been carried out on a d.c. motor with a perturbation of the armature voltage. Equally the same tests can be implemented with a perturbation of the field voltage or load torque. The field voltage could be perturbed using the circuit considered in this work and the load torque could be perturbed using a dynamometer with an electrically activated variable torque facility or by using the injection circuit to perturb the dynamometer. DISCUSSION OF RESULTS AND CONCLUSIONS The injection circuit given in this paper has provided a method by which a perturbation signal may be injected into a d.c. machine in order to carry out traditional control engineering identification techniques. From the frequency response, FFT and correlation results it was possible to determine the electromechanical time constant of the motor directly. This value and the corresponding break frequency obtained from the various tests is given in the table below . Mechanical Break Time Constant(s) Frequency(Hz) Frequency Response FFT Correlation Step Response

0.66 0.65 0.79 0.80

0.24 0.244 0.20 0.20

The motor gain constant was determined by plotting the relationship between motor speed and armature voltage and was found to be 105 rad/s / V. This along with the other data allows an equation to be established which will predict the motor response.

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At this stage little attention has been given to the quality and comparison of the various test results and the authors now intend to carry out further tests under a variety of operating conditions and with different machines . In the longer term it is proposed to develop a low-cost electrical machines identification package possibly microcomputer based. REFERENCES Nita, K, and Okitsu, H. "Armature resistance of a d.c. motor and its effect on the motor time constraint"; Elect Eng in Japan, Vo1 86, October 1966, pp 77-86. Lord, Wand Hwang J H. "Pasek's technique for determining the parameters of high per formance d.c. motors"; Proceedings of the Third Annual Symposium on Incremental Motion Control Systems and Dev1ces, OnlVers1ty of 1I11no1s, May 1974 pp 1-10 . Eykhoff, P. "Sys tem Identifi cati on: Parameters and State Estimation", ~itey, 1974. Mayne, D Q. "Optimal non-s a 10nary estimation of the parameters of a linear system with gaussian inputs", Journal of Control, Vo1 14, Jan. 1965, pp 101-112. Freeman, J M, Hassan, F Nand Morton, D. "Ka1man Filter estimation of electrical machine parameters", to be pub 1i shed, 1985

D . Brook and D . MOl'ton

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