STRUCTURAL DIAGRAMS OF A TRANSVERSE-FIELD EMU OPERATING WITH AN ACTIVE-INDUCTIVE LOAD

V. P. LAGUNOV

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

In the analysis of automatic control systems, their structural diagrams acquire special importance, since they show the possible paths of signal propagation that provide interaction among the main elements of the system.

The structural diagram of a transverse-field electromachine amplifier (EMU TF) is intended to simplify the study of its characteristics both in steady-state and in transient regimes. It also makes it possible to simplify the study of electromachine automation systems with an EMU TF. In addition, the structural diagram of the amplifier makes it possible to pass easily to an electronic model of the amplifier. There are two ways of setting up problems on modeling devices:

a) setting up according to the differential equation describing the given system;

b) setting up according to the structural diagram of the system under study.

However, in the practice of modeling automatic control systems, for a number of reasons the second method has found the widest application \[1\].

In systems with an EMU TF, separately excited direct-current motors are widely used. In this case the amplifier may supply either the armature circuit of the motor or the circuit of its field winding. Cases are also frequent in which amplifiers operate on the field winding of a direct-current generator. When amplifiers operate on a field winding, the load of the latter is an active-inductive resistance.

In the present work, various variants of structural diagrams of an EMU TF are considered when it operates on an active-inductive resistance.

Figure 1 presents the structural diagram of an EMU TF constructed on the basis of the differential equations of transient processes (1), the derivation of which is given in \[2\].

Fig. 1. Block diagram of a transverse-field EMU operating with an active-inductive load

\[ e_2 = \frac{c_{12}}{r_y(1+T_1p)} \left[ U_y - r_y\frac{w_p(i_2)}{w_y} \left( \frac{c\Phi_2}{S c_{23}} \pm x i_3 \right) - r_y\frac{k_c}{w_y}c\Phi_2 - r_y\frac{w_3}{w_y}(1-\varepsilon)i_3 \right], \tag{1} \]

\[ c\Phi_2 = \frac{S c_{23}}{r_{\mathrm{я}}(1+ST_{\mathrm{я}}p)} \left[ e_2 - f_1(i_2) \mp x r_{\mathrm{я}} i_3 \right], \]

\[ i_3 = \frac{1}{R_3(1+T_{\mathrm{н}}p)} \left[ c\Phi_2 + \frac{\sigma_{\mathrm{я}}w_3p}{c}(1-\varepsilon)i_2 \right]. \]

In the system of equations (1) and in the figures below, the following notation is used:

\(r_y, r_{\mathrm{я}}, R_3\) — active resistances of the control winding, the armature, and the load circuit;

\(e_2, e_3\) — emf at the transverse and longitudinal brushes of the amplifier;

\(c\) — coefficient of proportionality between the emf \(e_2, e_3\) and the fluxes \(\Phi_1\) and \(\Phi_2\);

\(w_y, w_{\mathrm{я}}\), and \(w_{\mathrm{д}}\) — numbers of turns of the control winding, the armature, and the auxiliary pole;

\(T_y, T_k, T_{\mathrm{я}}\), and \(T_{\mathrm{н}}\) — time constants of the control winding, compensating winding, armature, and load circuit of the EMU;

\(c_{12}\) and \(c_{23}\) — gain coefficients of the first and second stages;

\(\varepsilon\) — degree of compensation of the longitudinal armature reaction of the amplifier;

\(x\) — relative displacement of the transverse-circuit brushes from the neutral;

\(\Phi_2\) — flux along the transverse axis;

\(i_2, i_3\) — currents in the transverse and longitudinal circuits.

The diagram in Fig. 1 takes into account all the principal factors affecting the operation of the transverse-field EMU. These may include the following:

1. The demagnetizing effect of losses in the armature steel.
2. Delayed commutation of the transverse-circuit current.
3. Displacement of the transverse brushes, which creates along the longitudinal axis of the machine, depending on the direction of the displacement, a demagnetizing or magnetizing flux.
4. A flux occurring in an undercompensated machine owing to the absence of equality between the fluxes of the longitudinal armature reaction and the compensating winding.

In addition, the functions \(S\), \(f_1(i_2)\), \(f_2(i_2)\) have been introduced into the diagram; these respectively take into account the nonlinearities of the no-load characteristic of the second-

of the cascade, the resistance of the brush contact, and the m.m.f. of the reaction of the commutation currents of the transverse circuit. Finally, the given diagram takes into account the influence of the displacement of the longitudinal brushes, which is possible in amplifiers in which the longitudinal and transverse brushes are rigidly connected to one another.

Thus, the structural diagram in Fig. 1 conveys the physical essence of the feedbacks sufficiently fully and clearly, and accurately reproduces the transient process of the amplifier under all possible modes of its operation. It makes it possible to pass easily to an electronic model of the amplifier, by means of which one can investigate the static and dynamic characteristics of the amplifier and determine the influence of individual internal feedbacks and parameters on these characteristics and on the stability of amplifier operation, taking their nonlinearity into account.

In most cases the structural diagram obtained for the amplifier (Fig. 1) can be considerably simplified.

In [3] it was established that the transformer coupling has

\[ \frac{c_{12}\sigma_q W_3(1-\varepsilon)p}{c r_y(1+T_1 p)}\,\sum U_y = \frac{(1-\varepsilon)T_{13}p}{1+T_1p}\,\sum U_y \]

a very small influence on the operation of the amplifier. When it is eliminated, the value of the e.m.f. \(e_3\) in transient regimes changes by no more than 2%; therefore, in constructing the structural diagram it may be neglected.

When adjusting the operation of a transverse-field EMU, usually only the transverse brushes are displaced. Even in the case when the transverse brushes can be shifted independently of the longitudinal ones, there is no need for couplings \(a\) and \(b\) (Fig. 1), which are associated with the displacement of the latter. The structural diagram of the transverse-field EMU (Fig. 1) is somewhat simplified and assumes the form shown in Fig. 2.

Fig. 2. Simplified structural diagram of a transverse-field EMU.

To facilitate the mathematical analysis of the dynamic properties of an EMU, a number of simplifying assumptions are very often adopted:

1. The no-load characteristic of the second cascade is taken to be rectilinear, which is entirely permissible in the absence of forcing. In this case the function \(S\) is taken equal to unity.

2. The resistance of the brush contact of the transverse circuit of the EMU is considered constant. In this case there is no need for the nonlinear coupling \(b\) (Fig. 1), and the value of the brush-contact resistance is taken into account by the resistance \(r_2\) (Fig. 3).

3. It is assumed that the magnetizing force of the reaction of the commutation currents of the transverse circuit depends linearly on the current of the transverse circuit. The action-

the effect of the indicated factor on the operation of the EMU in this case is taken into account by the coefficient \(w_p\), which also takes into account the shift of the transverse brushes and the demagnetizing action due to losses in the armature steel.

Under these assumptions, all processes in the transverse-field EMU can be described by a system of linear differential equations. Taking the above into account, when transforming the diagram in Fig. 1, the system of differential equations of the transverse-field EMU operating with an active-inductive load can be written in the following form:

\[ e_2 = \frac{c_{12}}{r_y(1+T_1p)} \left[ U_y - \frac{r_y w_p}{w_y c_{23}}\,e_3 - \frac{r_y w_3}{w_y}(1-\varepsilon)i_3 \right], \]

\[ e_3 = \frac{c_{23}}{r_y(1+T_2p)}\,e_2,\quad i_3 = \frac{1}{R_3(1+T_n p)}\,e_3 . \tag{2} \]

On the basis of the system of equations (2), a structural diagram of the transverse-field EMU can be constructed for a linear consideration of its parameters and with the assumptions adopted above (Fig. 3).

Fig. 3. Structural diagram of a transverse-field EMU under linear consideration of its parameters.

In conclusion, we note that the structural diagrams presented in Figs. 1 and 2 take into account all the principal interactions of the circuits of the transverse-field EMU and are suitable for an accurate analysis of its operation in steady-state and transient modes. The diagram in Fig. 3 can be recommended for an approximate assessment of the static and dynamic properties of the amplifier.

Using the structural diagrams presented, one can easily construct electronic models of the transverse-field EMU, making it possible to study its static and dynamic characteristics and, most importantly, to determine the separate influence of internal feedbacks and parameters on these characteristics and on the stability of amplifier operation under both linear and nonlinear consideration of them.

REFERENCES

1. D. V. Vasil’ev and V. G. Chuch. Calculation of Automatic Control Systems. Mashgiz, 1959.

2. V. P. Lagunov, V. M. Rikkonen. Electronic model of a transverse-field electric-machine amplifier with its internal feedbacks. Izvestiya TPI, vol. 132, 1965.

3. 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.