MATHEMATICAL MODELING OF A TRANSVERSE-FIELD EMU WITH A SMOOTH ARMATURE ON THE MN-14 ANALOG COMPUTER

G. T. KONSTANTINOV, A. I. SKOROSPEŠKIN, V. P. LAGUNOV

(Recommended by the seminar of the departments of electrical machines and general electrical engineering)

An EMU is a widely used element of electric-machine automation and tracking electric drives.

In [1—4] it is shown that, in studying automatic-control systems, it is erroneous to represent an EMU as two series-connected inertial elements without taking internal feedbacks into account. The static and dynamic characteristics of an EMU are determined primarily by its internal rigid and flexible feedbacks.

In the present work, transient processes are investigated in an EMU with a smooth armature, as a qualitatively new design [5]. The quantitative and qualitative influence of the principal feedbacks and nonlinearities on the nature and duration of transients in the amplifier with a smooth armature is analyzed in comparison with production EMUs.

Usually, when studying transients in electric-machine amplifiers, it is assumed that the magnetic system along the longitudinal and transverse axes is unsaturated, that the resistance of the brush contact of the transverse circuit is constant, and that the emfs of the commutation and eddy currents vary according to a linear law. However, as indicated in [6], such comparatively crude assumptions may lead to serious errors and incorrect conclusions, especially when considering systems operating near the boundaries of the stability region.

The use of modern nonlinear analog computing machines makes it possible to take into account all the above-mentioned assumptions, to investigate and estimate their influence on the quality of transients in an EMU. For this purpose, on an actual EMU with a smooth armature, nonlinear dependences were experimentally obtained for the magnetization characteristic of the second stage of the amplifier, the resistance of the brush contact, the emf of the transverse armature reaction, and the emf of the commutation and eddy currents on the current of the transverse circuit. The method for determining the indicated nonlinear dependences and the amplifier parameters is given in [7].

The complete system of differential equations of an EMU with a smooth armature, taking into account the principal feedbacks and nonlinearities when operating with an active-inductive load, is represented in the following form:

\[ \Phi_1 = \lambda_1 \left[i_y W_y + i_2 W_{\mathrm{nod}} + i_k W_k - i_3 W_z - f_2(i_2) - f_3(i_2) - f_4(i_2) + X W_{\mathrm{ya}} i_2 \right]; \]

\[ \Phi_2 = \lambda_2 \left\{ \left[(1-X)W_{\mathrm{ya}} + W_{\mathrm{noq}}\right]i_2 + X W_{\mathrm{ya}} i_3 \right\}\cdot S; \]

\[ U_y = i_y r_y + \sigma_y W_y p\Phi_1; \tag{1} \]

\[ e_2=C\Phi_1=i_2r_2+f_1(i_2)\cdot i_2+\sigma_y W_2 p\Phi_2+\sigma_{\mathrm{pod}}W_{\mathrm{pod}}p\Phi_1; \]

\[ e_3=C\Phi_2=i_3r_3+r_k i_k-(\sigma_y W_3-\sigma_k W_k)p\Phi_1 L_n p i_3; \]

\[ 0=\sigma_k W_k p\Phi_1+R_k i_k-r_{\mathrm{sh}}i_3, \]

where

\[ W_3=W_y-W_d;\qquad r_3=r_y+r_d+r_{\mathrm{sh}3}+r_n;\qquad r_2=r_y+r_{\mathrm{pod}}+r_{\mathrm{poq}}; \]

\[ W_2=\left(\frac{1+\alpha_i+2\beta_{\mathrm{p.o}}}{2}\right)W_y;\qquad R_k=r_k+r_{\mathrm{sh}}; \]

\(\Phi_1,\ \Phi_2\) and \(\lambda_1,\ \lambda_2\) are the resultant fluxes and magnetic permeances along the longitudinal and transverse axes of the amplifier;

\(U_y\) is the voltage applied to the control winding;

\(W_y,\ W_{\mathrm{pod}},\ W_{\mathrm{poq}},\ W_k,\ W_y\) and \(W_d\) are the numbers of turns of the control, longitudinal biasing, transverse biasing, compensating, armature, and auxiliary-pole windings;

\(i_y,\ i_k,\ i_2\) and \(i_3\) are the currents of the control winding and of the compensating, transverse, and longitudinal circuits of the amplifier;

\(r_y,\ r_{\mathrm{pod}},\ r_{\mathrm{poq}},\ r_k,\ r_d,\ r_{\mathrm{sh}},\ r_n,\ r_y,\ r_2\) and \(r_3\) are the active resistances of the control, longitudinal biasing, transverse biasing, compensating, auxiliary-pole, shunt of the compensating winding, load, armature, transverse, and longitudinal circuits of the amplifier;

\(\sigma_y,\ \sigma_y,\ \sigma_{\mathrm{pod}}\), and \(\sigma_k\) are the leakage coefficients of the control, armature, longitudinal biasing, and compensating windings;

\(C\) is the proportionality coefficient between flux and electromotive force;

\[ X=\frac{2p\beta}{\pi} \]

is the relative displacement of the brushes from the neutral line (\(\beta\) is the displacement angle, \(p\) is the number of pole pairs); a positive value of \(X\) corresponds to displacement of the brushes in the direction of rotation, and a negative value to displacement against the direction of rotation;

\(f_1(i_2),\ f_2(i_2),\ f_3(i_2)\) and \(f_4(i_2)\) are, respectively, the nonlinear dependences of the brush-contact resistance, the transverse armature reaction m.m.f., the commutation m.m.f., and the eddy currents;

\(S\) is a nonlinear function that takes into account the nonlinearity of the magnetization characteristic of the second stage of the EMA; for the straight-line section of this characteristic \(S=1\);

\(\alpha_i\) is the pole-arc coefficient;

\[ \beta_{\mathrm{p.o}}=\frac{W_{\mathrm{poq}}}{W_y} \]

is the relative number of turns of the transverse biasing winding.

The system of differential equations (1) is nonlinear. The solution of such systems by analytical, frequency, graphical, and graph-analytical methods, as is known, is either impossible or very cumbersome and difficult. All this naturally leads to the investigation of such systems by methods of mathematical modeling using nonlinear analog computing machines.

On the basis of the system of differential equations (1), Fig. 1 shows an electronic model of the EMA operating with an active-inductive load.

This model makes it possible to investigate the load mode and the no-load mode. In the latter case the coefficients 71, 72, 73, and 74 are set equal to zero. When the brushes are shifted in the direction of rotation, switch \(P\) is placed in position 2.

In the patching scheme for modeling the EMA, three inte—

integrating amplifiers 2, 4, and 7, which reflect the inertia of the EMA along the longitudinal and transverse axes and the inertia of the load circuit. For simplicity, the load is assumed linear; if it is necessary to take account of the nonlinearity of the load circuit, a nonlinear block should be included in the feedback circuit of integrating amplifier 7, reproducing, for example, the nonlinearity of the generator magnetization characteristic.

A nonlinear block 41 is included in the feedback circuit of integrating amplifier 4; it carries out a piecewise-linear approximation of the nonlinear dependence of the brush-contact resistance of the transverse circuit.

The current \(i_2\) is reproduced by a circuit including summator 5, summing amplifier 6, multiplication blocks 51, 52, and nonlinear block 42. The use of this block makes it possible to take account of the nonlinearity of the no-load characteristic of the second stage and the change in the time constant of the transverse circuit of the EMA as the magnetic system becomes saturated.

The control-current circuit includes integrating amplifier 2 and nonlinear blocks 43, 44, and 45, which carry out a piecewise-linear approximation of the nonlinear dependences of the m.m.f. of the transverse armature reaction, commutation, and eddy currents. These nonlinearity blocks, together with summator 9, inverter 8, and voltage dividers 131÷134 of summator 13, reproduce the internal current feedback of the transverse circuit \(i_2\). The control-current circuit also includes a chain consisting of inverters 3, 10, 12, summator 11, and integrating amplifier 7, reflecting the presence of a dynamic decompensation current.

The accuracy with which transient processes are reproduced on the model was checked by comparing oscillograms obtained on the actual EMA and on its model, for no-load, active-load, and active-inductive-load conditions. In all the indicated modes the character of the transient processes remains the same. Therefore, Fig. 2 gives oscillograms of the change in voltage on the model (curve a) and on the actual amplifier (curve b) only for the active-inductive-load condition. As can be seen from the oscillograms shown in Fig. 2, the agreement of the transient processes in the actual EMA and on its model is very close.

Without dwelling on the method of determining the scale coefficients and the transmission coefficients of the operational amplifiers of the model, let us consider the influence of the principal nonlinearities and feedbacks on the character and duration of the transient processes in an amplifier with a smooth armature.

In studying the transverse reaction, the reaction of commutation and eddy currents in transverse-field electric-machine amplifiers, it was established that in amplifiers with a smooth armature the total m.m.f. of these reactions amounts to about 20 percent of the control m.m.f., whereas in series EMAs it amounts to (50–60 percent) [7].

This could not fail to have a positive effect on the dynamic characteristics and stability of the amplifier with a smooth armature. Indeed, as studies on the model have shown, in an amplifier with a smooth armature the limits of the region of stable operation are substantially widened. When a series EMA and an EMA with a smooth armature operate on one and the same active-inductive load, the latter operates stably with unity compensation and with a displacement of the transverse brushes opposite to the direction of rotation by three commutator divisions. Or, with the brushes set in the neutral position, it permits overcompensation of about 20 percent, whereas in series EMAs the permissible displacement of the transverse brushes does not exceed half a commutator division, and the overcompensation is 2–3 percent. In Fig. 3, a, b, and c oscillograms are presented that illustrate, respectively, the influence of the degree of overcompensation and of the displacement of the transverse brushes ...

Fig. 1. Electronic model of an electric machine amplifier with a smooth armature, with the parameters considered linearly, during operation into a resistive-inductive load.

Fig. 2. Oscillograms of the voltage at the EMU armature, recorded on a loop oscillograph:
a — on the model, b — on a real EMU with a smooth armature

of the transverse brushes on the transient process of the amplifier with a smooth armature. Here, as in all subsequent figures, oscillograms were recorded of the change in voltage at the EMU armature (upper curve) and oscillograms of the change in control current (lower curve). The oscillograms were obtained on the electronic indicator I-10. The marks occurring at intervals of 1 sec are clearly visible on the curves.

As can be seen from the oscillograms shown in Fig. 3, a, b, and c, as the degree of overcompensation is increased and the transverse brushes are shifted opposite to the direction of rotation, the character of the transient process remains aperiodic, but its duration increases substantially. Overshoot and oscillation are absent.

In investigating the effect of the longitudinal demagnetizing winding, used to increase the amplification factor, on the transient process in an EMU with a smooth armature, it was established that switching on the longitudinal demagnetizing winding leads to a considerable increase in the transient-process time without changing its character. At the same time, the increase in the m.m.f. of the longitudinal demagnetizing winding is limited by the conditions of self-excitation. For this amplifier, on the basis of these conditions, the maximum permissible number of turns of the longitudinal demagnetizing winding is 24. In Fig. 4, a and b, two oscillograms are presented of the change in voltage at the EMU armature, respectively with the longitudinal demagnetizing winding absent and with it switched on with the number of turns equal to 24. As can be seen from the oscillograms, the character of the transient process did not change, but the time increased by approximately one and a half times.

Investigation of the possibility of representing an EMU with a smooth armature as a linear system by replacing nonlinear connections (nonlinear blocks 41—45) with linear ones showed that, with accuracy sufficient for practice, the amplifier with a smooth armature may be regarded as a linear system. Proof of this is provided by the oscillograms in Fig. 5. Here 1 is the oscillogram of the voltage at the EMU armature under a nonlinear treatment, and 2 under a linear treatment of the system. This is explained by the fact that in an EMU with a smooth armature, owing to its design features, the magnitude of the air gap increases considerably; consequently, the control m.m.f. increases and the inductance of the armature winding decreases substantially.

In a linear treatment of the amplifier with a smooth armature, consid-

a

b

c

Fig. 3. Oscillograms of the voltage at the EMU armature:
a — unity compensation, brushes in neutral; b — brushes in neutral, overcompensation 20%; c — unity compensation, transverse brushes displaced opposite to the direction of rotation by three commutator divisions.

a               b

Fig. 4. Oscillograms of the EMU armature voltage:
a — without a longitudinal biasing winding; b — the number of turns of the longitudinal biasing winding is 24

Fig. 5

...the electronic model is substantially simplified (Fig. 6). The circuit contains no nonlinearities, multiplication blocks, or servo amplifier.

Conclusions

1. The use of nonlinear analog computing machines makes it possible to represent and investigate all feedbacks and nonlinearities of the EMU most fully and in a physically correct manner.

2. In an EMU with a smooth armature, the limits of the region of stable operation are significantly widened when the degree of compensation is varied and the brushes are shifted.

3. The use of a longitudinal biasing winding does not change the character of the transient process, but significantly increases its duration.

4. An EMU with a smooth armature may, with accuracy sufficient for practice, be regarded as a linear system, in contrast to series-produced EMUs.

References

1. A. G. Iosif’yan, B. M. Kagan. Fundamentals of the Servo Drive, GEI, 1954.

2. N. M. Yakimenko. The electric-machine amplifier as an element of a controlled system. Elektrichestvo, No. 9, 1948.

Fig. 6. Electronic model of an EMA with a smooth armature under a linear treatment of the parameters, operating into a resistive-inductive load.

1. E. M. Ettinger, Yu. R. Reingold. Experimental investigation of the dynamic properties of an EMU and determination of its parameters. Elektrichestvo, No. 3, 1956.

2. V. G. Vasil'ev, V. P. Lomakin. Structural diagrams and electronic models of systems containing transverse-field electric-machine amplifiers. Elektromekhanika, No. 4, 1960.

3. A. I. Skorospeshkin, Yu. A. Stepanov. Transverse-field EMU with a smooth armature. Izvuz — Elektromekhanika, No. 10, 1964.

4. V. P. Lagunov. On the question of investigating the stability of operation of a transverse-field EMU, taking into account the nonlinearity of its parameters. Dissertation, Tomsk, 1965.

5. A. I. Skorospeshkin, B. I. Kostylev, G. G. Konstantinov. Transverse armature reaction, reaction of commutation and eddy currents in a transverse-field EMU. In the present collection.