# Extended Sensitivity Models of D.C. Motors at the Undetermined Random Deteriorations

Jaracz, Kazimierz

Dr eng., Institute of Technology, WSP Kraków, Podchorążych 2, 30-084 Kraków

## Summary

The paper deals with the formulation of the method of realization of the mathematical description of the sensitivity function of state variable of a d.c. motor to the parameters undetermined by random disturbances.

## 1. Introduction

The possibility of examining the driving motor operation at the disturbance of supply voltage is based on the extended model of the motor. Owning to that the matrices of both state and input matrices are the functions of principle parameters describing the disturbance, they are based on the approximation of the autocovariance torque in the internal of stacionarity. This method in the theory of stochastic control is known as the method of expanding of the control object [4].

By enlarging the definition of the sensitivity function of motor current and its angular velocity (owing to the variation of deterioration function parameters) and their influence on the random functions, one receives the model, which enables the determination of behaviour of motor state variables (at the every interval of observation) on the variation of deterioration.

The method of construction of the motor sensitivity may be presented as follows:
 (i) it is necessary to determine the approximate form of the autocovariance time of voltage deterioration momentum, (ii) to construct the forming filter, which will change the white noise into coloured one, described by the approximated form of the autocovariance momentum, (iii) to determin the enlarged motor model, with the hipermatrix coefficients being the functions of parameters of both deterioration and the drive system, (iv) to define the averaged sensitivity function of current and angular velocity to the variance of parameters of probabilistic deterioration, (v) to build-up the accompanying sensitivity motor model, (vi) to calculate the averaged functions of motor sensitivity of discussed motor state variables aided by microcomputer.

As the approximation form of the autocovariance deterioration torque in its stationary, interval in practice, one of the forms of standard expressions, determining the probabilistic characteristic second momentum of the coloured noise with the dominant frequency , i.e. the expression of the form may be applied.
 , (1)
 In it the - variance, - irregularity constant, - oscillatary constant, - time shift.

All above is compatible with the empiric results of measurements and approximation of the above function for the voltage deterioration of the drive of the rolling mill [2]. The coloured noise neceived form the white noise as the result of synthesis by the forming filter (of the state [x1, x2]T) of the intensivity S will be described by the equation [4]:
 , (2)
 where [b1, b2]T - input matrix of the filter, w( ) - white noise.

## 2. Extended Model of a D.C. Motor

Applying the basic model of the d.c. motor basing on the usually applied form:
 , (3)

where M = La -; the index f values concerns the parameters and values of the excitation circuit, a - parameters and values of the armature circuit. Symbol L denotes the inductivity, R denotes the resistance, u - voltages, i - currents, - angular velocity, J - effective intertia moment reduced to the motor shaft, ms - load torque.

Owning to that, that extended model of the d.c. motor is as follows:
 , (4)

where
 , (5)

## 3. The Associated Extended Model of a D.C. Motor Sensitivity

We will start with the extension of the definition of the sensitivity of the trajectory introduced by Cruz'ea on the random stationary processes. In the disturbance conditions the averaged random sensitivity functions of the armature current and angular velocity will influence the variance of the disturbance vector parametery will have the form, respectivity:
 , (6)
 , i = 1, 2 (7)

where E - average value operator, p = [p1,p2]T = [, ]T

By differentiating the vector-hipermatrix state equation (4) in relation to pi, with the continuity conditions respected the model of the sensitivity of the discussed model of motor to variation of the coefficients of irregularity and oscillatory :
 , i = 1, 2 (8)
 , i = 1, 2 (9)

The Exemplary Results of Calculations.

The averaged sensitivity functions were calculated by means of SIPRO 3.4 language oriented on to the function graphs.

The calculations have been realized for a d.c. motor driving the long profile rolling-mill. The parameters of the motorfare as follows:
 Normal power rating PN = 6 MW Normal armature voltage UaN =1350 V Normal armature current IaN =4750 A Normal velocity n [65,165] Excitation current If [50,210] A Excitation voltage UfN = 220 V Armature circuit Ra = 0,024 Armature circuit inductivity La = 0,067 mH Excitation circuit resistance Rf = 0,66 Extation circuit inductivity Lf = 2,8 H Effective inertia J = 95410 kgm2 Number of excitation winding turns Zf = 972 Dissipation coefficient =1,2 Motor voltage constant ce = 174,46 Motor torque constatnt cm =174,46 The main flux = 0,43 Wb Electromechanical time constant Tcm = 0,186 s

The disturbances and filtr parameters:

= 47,62V

= 144 s-1

= 1783 s-1

b2 = 16,9

= 0,059

=4,5 10-5

The examples of calculation results are presented in Fig. 1.

Fig. 1. The transients of sensitivity functions of armature current of a d.c. motor, (a) and (b)

Fig. 1a present the averaged function of sensitivity of armature current on the variation of the irregularity; 2b - as before, but on the variation of oscillatory coefficient.

## 4. Recapitulation

In the paper the methods of construction of a sensitivity model of selected electric machine were presented, on the variation of random disturbances of parameters.

In order to that it is necessary to obtain the approximation of the author covariance momentum of the disturbances. As the approximation model the representative form of the autocovarince function of the discussed the noise component appearing in the armature supply voltage.

The approximation may differ from the above mentioned, but regarding in a general form, the characteristic parameter of the random disturbance.

The extended model of the machine regards these parameters of disturbances and enables the construction of associated averaged sensitivity model of the discussed machine. The microcomputer aided calculations enables receiving the quantitive evaluation the variation of current and angular velocity parameters of the motors (Fig.1-a and 1-b).

## References

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FARANA, R. 1996. Univerzální simulační program SIPRO 3.4. Uživatelská příručka. 1. vyd. Ostrava : KAKI, 1996. 116 s. ISBN 80-02-01087-6.

JARACZ, K. 1994. Random Disturbances and Expanded Model of Electric Drive of Rolling Mill. Proc. of the EPE'94 Conf. "Electric Drive Design and Applications" 19-20 Oct. EPFL Lausanne, pp. 611-G14.

JARACZ, K. 1998. Extended models of d.c. machines sensitivity at the undeterministic disturbances. (in Polish), WN WSP, pr. monogr Nr 238. Kraków.

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