ELECTRONIC MODEL OF A CROSS-FIELD ELECTRIC MACHINE AMPLIFIER WITH ITS INTERNAL FEEDBACK LOOPS

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) are widely used in modern automatic-control systems. The course of transient processes in these systems is determined to a significant degree by the static and dynamic properties of the EMA, on which, in turn, its internal rigid and flexible feedback loops exert a strong influence.

The rational use of these feedback loops makes it possible in some cases to avoid special stabilizing devices, while changing the coefficients of individual feedback loops permits the rate of the transient process at the amplifier output to be varied over wide limits.

For this reason it is often important to investigate the separate influence of the internal feedback loops, especially the flexible ones, on the transient processes of the amplifier.

In the case where the EMA is used in a system in which strong forcing is provided, it is necessary to take into account the nonlinearity of the magnetization curve of the second stage of the amplifier.

In view of the foregoing, the structural diagram and electronic model of an EMA, in which the internal feedback loops are taken into account and in which it is also possible to investigate their separate influence on the dynamic properties of the EMA, are of considerable interest. In addition, this diagram and model must take into account the nonlinearity of the individual loops and parameters of the amplifier. To obtain the corresponding structural diagram and model, it is necessary to consider the differential equation of the transient processes of the EMA, taking into account the factors listed above.

When compiling the general differential equation of the EMA for any operating conditions, there is no need to use the mutual-inductance coefficients between windings. The derivation is simpler and more transparent if, in formulating these equations, one operates with the resultant fluxes acting along the longitudinal and transverse axes. Let us consider these fluxes.

Flux along the longitudinal axis. An EMA differs substantially from ordinary direct-current generators in that the magnetomotive force (m.m.f.) of the control winding and the resistance of the short-circuited transverse circuit are relatively very small. Therefore, the modes of ma—

of the aforementioned windings, such factors have a great influence as do not influence, or hardly influence, the characteristics of ordinary generators.

These may include the following:

1. The demagnetizing action of losses in the armature steel.
2. Delayed commutation of the current of the transverse circuit.
3. Displacement of the transverse brushes, which creates along the longitudinal axis of the machine, depending on the direction of displacement, a demagnetizing or magnetizing flux.

Fig. 1. Schematic diagram of an EMU.

1. The flux present in an undercompensated machine due to the absence of equality between the fluxes of the longitudinal armature reaction and the compensating winding.

For an unsaturated magnetic system it is usually assumed that the m.m.f. due to losses in the steel, acting against the m.m.f. of the control winding, is proportional to the current of the transverse circuit \([1]\). This proposition is valid in the case where proportionality is preserved between the current \(i_2\) and the transverse flux. When, however, saturation of the magnetic system of the second amplification stage is present and the brushes are displaced from the geometrical neutral, this proposition becomes invalid. In these cases—

it is more correct to assume that \(F_c\), due to losses in the steel, is proportional to the flux of the transverse circuit [2]

\[ F_c = k_c' \Phi_2 = k_c e_3', \]

where \(k_c'\) and \(k_c\) are the coefficients of proportionality between \(F_c\) and the flux of the transverse axis \(\Phi_2\) and the e.m.f. at the longitudinal brushes \(e_3'\).

The demagnetizing effect of the currents of delayed commutation in the sections commutated by the short-circuited brushes of the amplifier has been investigated theoretically and experimentally by many authors. Most of them, in the analytical calculation of this effect, use the expression given in [5]. According to this expression, \(F_z\) is a nonlinear function with respect to the current of the transverse circuit \(i_2\). This nonlinearity is introduced mainly by the nonlinear dependence of the transient resistance of the brush contact on the magnitude of the current of the transverse circuit. It is especially sharply expressed in the region of small values of the transverse current.

Taking the foregoing into account, the longitudinal flux may be expressed by the equation

\[ \Phi_1 = \lambda_1 \left[w_y i_y + w_k i_k + x w_{\text{я}} i_2 - k_c' \Phi_2 - f_2(i_2)i_2 - (w_{\text{я}} - w_{\text{д}})i_3\right], \tag{1} \]

where \(\Phi_1\), \(\lambda_1\) are the magnetic flux and magnetic conductance along the longitudinal axis;

\(w_y, w_k, w_{\text{я}}\) and \(w_{\text{д}}\) are the numbers of turns of the control, compensating, armature, and auxiliary-pole windings;

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

is the relative displacement of the brushes of the transverse circuit from the neutral ( \(\beta\) is the angle of displacement, \(p\) is the number of pole pairs);

\(f_2(i_2)i_2\) is the magnetizing force due to the reaction of the commutation currents in the transverse circuit;

\(k_c'\Phi_2\) is the magnetizing force due to losses in the armature steel.

Flux in the transverse axis. In the case when the brushes of the transverse circuit are in the neutral position and when the transverse magnetic circuit is unsaturated, the magnetic flux is directly proportional to the transverse current. When the brushes of the transverse circuit are displaced from the neutral position, the value of the transverse magnetic flux decreases by an amount proportional to \(x\). In many types of amplifiers the transverse and longitudinal brushes are closely connected with each other; for this reason, when the transverse brushes are displaced, the longitudinal brushes are displaced as well. In this case a transverse component of the longitudinal flux appears, proportional to the magnitude of the relative displacement of the brushes and to the current \(i_3\).

In the presence of saturation of the transverse magnetic circuit, the proportionality between the currents \(i_2, i_3\) and the resultant flux of the transverse axis is disturbed.

The expression for the transverse flux, taking into account the above-listed factors, will have the following form:

\[ \Phi_2 = S\lambda_2\left[(1 - x)w_{\text{я}}i_2 + xw_{\text{я}}i_3\right], \tag{2} \]

where \(S\) is a function expressing the nonlinearity of the no-load characteristic of the second amplification stage \(I_3 = f(i_2)\); for the linear unsaturated part of the characteristic \(S = 1\);

\(\lambda_2\) is the value of the magnetic conductance of the transverse axis for an unsaturated magnetic system.

The emf equilibrium equations take the following form:

for the control winding

\[ U_y=r_y i_y+\sigma_y w_y p\Phi_1; \tag{3} \]

for the transverse circuit of the amplifier

\[ e_2=c\Phi_1=r_y i_2+f_1(i_2)i_2+\sigma_y w_y p\Phi_1; \tag{4} \]

for the longitudinal circuit of the amplifier

\[ e'_3=c\Phi_2=r_3 i_3-(\sigma_y w_3-\sigma_k w_k)p\Phi_1+r_k i_k+L_H p i_3; \tag{5} \]

for the circuit consisting of the compensating winding—the shunting resistance

\[ \sigma_k w_k p\Phi_1+R_k i_k-r_{\mathrm{sh}} i_3=0. \tag{6} \]

In equations (3)—(6) the following notation is used:

\[ \sigma_y,\ \sigma_y \text{ and } \sigma_k \]

are the leakage coefficients of the control, armature, and compensating windings;

\[ e'_3 \]

is the emf induced by the resultant flux \(\Phi_2\) at the longitudinal brushes;

\[ c \]

is the coefficient of proportionality between the emfs \(e_2,\ e'_3\) and the fluxes \(\Phi_1\) and \(\Phi_2\);

\[ r_3=r_y+r_d+r_k+r_{\mathrm{sh}3} \]

is the sum of the active resistances of the armature windings, the commutating pole, the compensating winding, and the transition resistance of the brush contact along the longitudinal axis;

\[ R_k=r_k+r_{\mathrm{sh}} \]

is the total resistance of the circuit consisting of the compensating winding—the shunting resistance;

\[ f_1(i_2)i_2 \]

is the voltage drop under the brushes of the transverse circuit.

In [4, 5] it is shown that one of the principal nonlinearities that substantially affect the operation of the amplifier is the nonlinearity of the resistance of the brush contact \(f_1(i_2)\); for this reason, the indicated function \(f_1(i_2)\) has been introduced into equation (4). Taking expressions (1)—(6) into account, the system of equations describing the transient process of a cross-field electric-machine amplifier when operating into an active-inductive load takes the following form:

\[ \left. \begin{aligned} e_2={}& \frac{c_{12}}{r_y(1+T_1p)} \left[ U_y-r_y\frac{w_p(i_2)}{w_y} \left( \frac{c\Phi_2}{Sc_{23}}+x i_3 \right) \right.\\ &\left. {}-r_y\frac{k_c}{w_y}c\Phi_2 -r_y\frac{w_3}{w_y}(1-\varepsilon)i_3 \right], \\[6pt] c\Phi_2={}& \frac{Sc_{23}}{r_y(1+ST_y p)} \left[ e_2-f_1(i_2)i_2+x r_y i_3 \right], \\[6pt] i_3={}& \frac{1}{R_3(1+T_Hp)} \left[ c\Phi_2+\frac{\sigma_y w_3 p}{c}(1-\varepsilon)i_2 \right]. \end{aligned} \right\} \tag{7} \]

In the system of equations (7) the following additional notation is used:

\[ c_{12}\ \text{and}\ c_{23} \]

are the coefficients of proportionality between \(e_2,\ e_3\) and the currents \(i_y\) and \(i_2\);

\[ \varepsilon=\frac{r_{\mathrm{sh}}}{R_k}\cdot\frac{w_k}{w_3} \]

is the degree of compensation of the longitudinal armature reaction of the EMA;

Fig. 2. Structural diagram of the EMA when operating with an active-inductive load.

Visible labels in the diagram include: \(U_y\), \(\Sigma U_y\), \(e_2\), \(c\Phi_2\), \(e_3\), \(i_2\), \(i_3\), \(-\dfrac{i_y}{W_y}\), \(f_2(i_2)\), \(-z_y\dfrac{W_a}{W_y}x\), \(-z_y\dfrac{m_c}{W_y}\), \(\dfrac{C_{12}}{z_y(1+T p)}\), \(\dfrac{C_{26}\alpha W_3(1-\varepsilon)p}{z_y C(1+T_1p)}\), \(\dfrac{sC_{23}}{z_a(1+sT_a p)}\), \(\dfrac{1}{R_3(1+T_n p)}\), \(-z_a x\), \(-z_y\dfrac{W_3}{W_y}(1-\varepsilon)\), \(f_1(i_2)\), \(+\!x\), and \(\dfrac{S^{-1}}{C_{23}}\).

Fig. 3. Schematic of the electronic model of the EMA when operating into a resistive–inductive load.

\[ R_3 = r_3 + \frac{r_{\mathrm{н}} r_{\mathrm{ш}}}{R_{\mathrm{к}}} + r_{\mathrm{н}} \]
— the total active resistance of the load circuit.

\[ T_{\mathrm{н}} = \frac{L_{\mathrm{н}}}{R_3} \]
— the time constant of the EMU load circuit;

\[ w_{\mathrm{p}}(i_2) = \pm x w_{\mathrm{я}} + f_2(i_2); \]

\[ w_3 = w_{\mathrm{я}} - w_{\mathrm{д}}. \]

On the basis of the system of equations (7), a block diagram of the EMU can be constructed. As is seen from Fig. 2, the main path of the control-signal passage consists of series-connected elements: conversion of the control signal into the emf of the second stage \(e_2\); conversion of the emf \(e_2\) into the emf on the longitudinal brushes of the amplifier \(e'_3 = c\phi_2\), and conversion of \(e'_3\) into the load current \(i_3\). In addition, the block diagram contains a number of rigid and flexible feedbacks, the physical essence of which is evident from Fig. 2. Using the system of equations (7), or the block diagram shown in Fig. 2, one can construct an electronic model of the EMU when operating into an active-inductive load, which is presented in Fig. 3. This model makes it possible to investigate both the load and the no-load operating conditions of the amplifier. In the latter case the switch \(P_4\) is open. Switches \(P_1\), \(P_2\), and \(P_3\) are set to position 1 when the transverse brushes are shifted in the direction of rotation and to position 2 when shifted against the direction of rotation.

Three principal nonlinearities have been introduced into the model, which can substantially affect the operating conditions of the amplifier. With the aid of this model one can investigate the influence of these nonlinearities on the operation of the amplifier both in static and in dynamic regimes, and determine the limits of permissible linearization of nonlinear parameters on the basis of the quality of the transient process. The solution of the questions enumerated above, as is known, is practically impossible by analytical methods; therefore, the use of the model in analogous cases is the most rational method of investigation. Thus, the proposed EMU model makes it possible to reproduce, with considerable accuracy and in a physically clear manner, the static and dynamic characteristics of the amplifier and to determine the influence of individual internal feedbacks and parameters on these characteristics and on the stability of the amplifier operation, taking their nonlinearity into account.

REFERENCES

1. N. M. Yakimenko. “An electric-machine amplifier as an element of a regulating system.” Elektrichestvo, No. 9, 1948.

2. F. A. Goryainov. Electric-Machine Amplifiers. Gosenergoizdat, 1962.

3. P. S. Sergeev (ed.). Design of Electrical Machines. Gosenergoizdat, 1956.

4. V. V. Rudakov. Electric-Machine Amplifiers in Automation Systems. Gosenergoizdat, 1961.

5. V. P. Lagunov. “The influence of the nonlinearity of brush-contact resistance on the self-excitation process of a cross-field electric-machine amplifier at no load.” Izv. TPI, vol. 117, Tomsk State University Press, 1963.