STUDY OF A CROSS-FIELD ELECTRIC MACHINE AMPLIFIER USING AN ELECTRONIC MODEL

V. P. LAGUNOV, V. M. RIKKONEN

(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)

Cross-field electric machine amplifiers (EMAs) have found wide application in practice, in particular in automated electric drives. On the basis of EMAs, various systems of automatic control and regulation are created, in which the amplifier performs the functions of a power amplifier, a summing device, etc. Naturally, the properties of an EMA exert a great influence on the static and dynamic properties of the entire control system.

It is known that the study of the stability and quality of linear systems does not present great difficulty; however, in the study of nonlinear systems the difficulties increase many times over and may prove insurmountable for purely theoretical analysis.

There is another way of solving nonlinear problems—modeling on electronic models, which at present is one of the powerful means of analysis.

In [1] a structural and electronic model of a cross-field EMA is considered, taking into account various flexible and rigid internal feedbacks that have a noticeable influence on the dynamic properties of the amplifier.

As can be seen from Fig. 1, in constructing the model the nonlinearities of three principal characteristics of the amplifier were taken into account:

feedback \(a\) takes into account the function \(f_2(i_2)\)—the nonlinear dependence of the magnetomotive force of the reaction of the commutating currents of the transverse circuit;

feedback \(b\) takes into account the function \(f_1(i_2)\)—the nonlinear dependence of the voltage drop in the brush contact on the current in the transverse circuit;

and, finally, the nonlinearity of the no-load characteristic of the second stage is taken into account by the variable coefficient \(S\).

The variable coefficients \(\varepsilon\)—the degree of compensation—and \(\chi\)—the displacement of the transverse brushes—may vary within the limits of normal values and are taken as constant in the analysis of the properties of the EMA.

In [2] a stability boundary was constructed for the EMU-5P amplifier when it operates into an active-inductive load; the stability boundaries were constructed under a linear consideration of the amplifier parameters in the plane of two parameters: \(\varepsilon\)—the degree of compensation and \(\chi\)—the displacement of the transverse brushes.

In the present work, with the aid of a model, the influence of individual nonlinear parameters (characteristics) on the quality of transient processes is considered when the EMU-5P amplifier operates near the stability boundary.

Fig. 1. Functional diagram of an electric-machine amplifier operating into a resistive–inductive load.

With a compensation ratio \(\varepsilon = 0.75\) and a relative displacement of the brushes of the cross-field circuit \(\chi = -0.035\) (the minus sign indicates that the brushes have been displaced opposite to the direction of rotation of the armature), which corresponds to a brush shift by 0.5 commutator divisions, and assuming linearity of all parameters of the amplifier, a transient process occurs that characterizes self-excitation of oscillations (Fig. 3, a). The system is unstable.

Fig. 2. Principal nonlinearities of the cross-field amplifier and their linearization:
1) \(F_2 = f_2(I_2)\); 2) \(E_3 = f(I_2)\); 3) \(\Delta U_{\mathrm{sh}} = f_1(I_2)\)

If now the resistance of the brush contact of the cross-field circuit is considered as the nonlinear characteristic \(f_1(i_1)\), then the output quantities of the amplifier (the current of the longitudinal circuit \(i_3\) and the electromotive force at the longitudinal brushes \(e_3\)) will be represented by Fig. 3, b, i.e., they have the form of oscillations decreasing to a certain amplitude.

The system is also unstable; however, the oscillatory mode arises when voltage is applied to the control winding, i.e., this is a hard excitation mode, and the amplitude of the oscillations is limited.

If one also takes into account the nonlinearity of the no-load characteristic of the second stage (Fig. 3, c), then the transient process is characterized by damped oscillations. The system is stable.

Finally, if the nonlinearity of the reaction of the commutation currents of the cross-field circuit is additionally taken into account (Fig. 3, d), then the transient process is characterized by damped oscillations down to a certain amplitude (Table 1).

From the oscillograms presented it is evident that the linearization of the reaction of the commutation currents introduces a fictitious stability margin (transition from Fig. 3, d to Fig. 3, c). This conclusion must be kept in mind when con-

Fig. 3. Voltage curve (upper curve) and current curve (lower curve) of the load circuit at \(\varepsilon = 0.75;\ x = -0.05\).

Panels: a, b, c, d.

taking into account the electromechanical amplifier, or in analyzing the stability of the control system. Indeed, if the reaction of the commutation currents is assumed linear at \(\varepsilon = 0.75\) and \(x = -0.035\), then the operation of the amplifier (Fig. 3,v) is stable, whereas in the real amplifier (when all the basic nonlinearities are taken into account) a regime of periodic motions is observed, i.e., oscillations of a definite amplitude and frequency arise (Fig. 3,g). Such a state may be regarded as critical, when the amplifier is actually operating at the stability boundary. Conversely, linearization of the no-load characteristic (transition from Fig. 3,v to Fig. 3,b) makes the system unstable.

Table 1

It should also be noted that, when the reaction of the commutation currents and the no-load characteristic of the second stage are taken as linear (Fig. 3,b) and, correspondingly, nonlinear (Fig. 3,g), the quality of the transient process in fact does not change.

This can be explained as follows: in the real amplifier the demagnetizing effect of the commutation-current reaction is smaller than in the idealized one; this leads to an increase in the emf \(e_z\) and to compensation for its reduction due to saturation.

For this reason, the influence of the nonlinearities of these two dependences is compensated to some extent, and the character of the transient process is affected only by the nonlinearity of the resistance of the brush contact of the transverse circuit. Of course, this conclusion cannot be general. It is valid for a definite range of operation of the amplifier (\(\varepsilon\) and \(x\)) and for the method of linearization. However, it is important to establish the influence of individual nonlinearities on the character of the transient process, as well as their combined influence under a certain combination of characteristics.

Thus, by selecting nonlinear internal characteristics of the electromechanical amplifier, we can change not only its external static characteristics, but also the character of stability, which is clearly manifested at the critical parameters of the EMA adjustment.

References

1. V. P. Lagunov, V. M. Rikkonen. Electronic Model of a Cross-Field Electromechanical Amplifier with Its Internal Feedbacks, Proceedings of TPI, vol. 132, 1965.

2. V. P. Lagunov, V. M. Rikkonen. On the Question of the Stability of Operation of a Cross-Field Electromechanical Amplifier. Ministry of Higher and Secondary Specialized Education of the RSFSR, Interuniversity Collection of Papers, issue 3, Novosibirsk, 1963.