Influence of the Nonlinearity of Brush Contact Resistance on the No-Load Self-Excitation Process of a Cross-Field Electric Machine Amplifier

V. I. LAGUNOV

(Presented by the scientific seminar of the Electromechanical Faculty)

A number of sources indicate that, when the parameters of a cross-field electric machine amplifier are considered linearly (Fig. 1), its self-excitation in the no-load mode occurs when the brushes are shifted from the transverse axis opposite to the direction of rotation by an angle \(\beta > \beta_k\), where the angle \(\beta_k\) corresponds to such a brush shift at which a magnetizing component of the transverse armature reaction arises in the amplifier, exactly compensating the influence of the demagnetizing action of eddy currents and hysteresis in the armature core, the voltage drop, and the reaction of commutation currents in the transverse armature circuit.

Naturally, with such a linear treatment of the amplifier parameters, one can speak only of soft self-excitation of the amplifier, i.e., of one that occurs under the influence of an infinitely small external action. Moreover, the increase in the longitudinal e.m.f. \(e_{\text{з}}\) occurs up to the moment when the magnetic system of the amplifier reaches complete saturation; at this point the self-excitation process ceases. Thus, under a linear

Fig. 1. Circuit diagram of a cross-field electric machine amplifier. \(У\) — control winding; \(К\) — compensating winding; \(r_{\text{ш}}\) — shunting rheostat for the compensating winding.

when considering the amplifier parameters, depending on the value of the displacement angle $\beta$, there may exist two types of transients that differ in quality:

1. For $\beta < \beta_k$—the transient is stable.
2. For $\beta > \beta_k$—an unstable transient, or a process of soft self-excitation.

The aim of the present work is to analyze the influence of the nonlinearity of the brush-contact resistance on the process of self-excitation.

One of the principal nonlinearities that substantially affect the operation of the amplifier is the nonlinearity of the brush-contact resistance, which determines the transitional voltage drop $\Delta u = f(j)$. Since exactly the same brushes are installed in the cross circuit as in the longitudinal one, they operate at current densities reduced by a factor of three to four, causing the amplifier to operate in the region of substantial nonlinearity of the brush-contact resistance. It should also be noted that the brush-contact resistance substantially affects the total resistance of the transverse armature circuit, giving it a nonlinear character, since it is quite comparable with the resistance of the armature winding.

In addition, the transient resistance of the brush contact to a considerable degree determines the magnitude of the additional commutation current, which creates the magnetizing force of the reaction of the commutation currents [1], $F_z = f_z(i_z)$, imparting to it a nonlinear character, especially in the region of small currents $i_z$ (Fig. 2). In turn, the magnetizing force $F_z$

Fig. 2. Curves of the transient voltage drop $\Delta u$ per pair of brushes and of the magnetizing force of the reaction of the commutation currents $F_z$ in the transverse circuit of the amplifier.

—has a significant influence on the balance of the magnetizing forces of the longitudinal axis.

Thus, the nonlinearity of the brush contact of the transverse circuit of the amplifier substantially affects its operation and must be taken into account when investigating transient processes, including the process of self-excitation.

In studying the self-excitation process, it is very important to determine the possibility of the occurrence of this process, and therefore it is necessary to describe analytically, with sufficient accuracy, the dynamic properties of the amplifier, especially at the beginning of this process. For this reason, first of all, it is necessary to take into account the nonlinearity of the brush contact of the transverse armature circuit.

At the same time, without great error one may assume the magnetic system along the longitudinal and transverse axes of the amplifier to be unsaturated and consider the winding parameters constant.

Under the assumptions made, the dynamic properties of the amplifier in the no-load mode are described by the following system of equations:

\[ \left. \begin{aligned} \Phi_1 &= \lambda_1 w_y i_y + \lambda_1 w_k i_k - \lambda_1 w_c i_2 + \lambda_1 x w_a i_2 - \lambda_1 f_2(i_2),\\ \Phi_2 &= \lambda_2 w_a i_2,\\ \sigma_k w_k \frac{d\Phi_1}{dt} + R_k i_k &= 0,\\ c\Phi_1 - r_a i_2 - f_1(i_2) - \sigma_a w_a \frac{d\Phi_2}{dt} &= 0,\\ U_y - r_y i_y - \sigma_y w_y \frac{d\Phi_1}{dt} &= 0,\\ e_3 &= c\Phi_2, \end{aligned} \right\} \tag{1} \]

where \(\lambda_1, \lambda_2\) are the magnetic permeances along the longitudinal and transverse axes;

\(w_c i_2\) is the magnetomotive force due to losses in the armature iron;

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

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

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

\(f_1(i_2)\) is the contact voltage drop in the brush contact of the transverse circuit;

\(\sigma_y, \sigma_k, \sigma_a\) are the leakage coefficients of the control, compensating, and armature windings;

\(c\) is the coefficient of proportionality between the e.m.f. and the flux.

The system of equations (1) can be reduced to a system of two equations:

\[ \left. \begin{aligned} \frac{de_2}{dt} &= \frac{\kappa_y}{T_y+T_k} U_y - \frac{1}{T_y+T_k} e_2 - \frac{1}{T_y+T_k} \left[ A e_3 + B\beta e_3 + C\varphi_2(e_3) \right],\\[6pt] \frac{de_3}{dt} &= \frac{\kappa_{2a}}{T_a} e_2 - \frac{1}{T_a} \left[ e_3+\kappa_{2a}\varphi_1(e_3) \right], \end{aligned} \right\} \tag{2} \]

where

\[ \kappa_y=\frac{w_y\lambda_1 c}{r_y} \]

is the voltage transfer coefficient of the first stage of the amplifier;

\[ T_k=\frac{L_k}{R_k} \]

is the time constant of the compensating-winding circuit

\[ (R_k=r_k+r_m); \]

\[ T_y=\frac{L_y}{y_y} \]

is the time constant of the control-winding circuit.

The remaining coefficients are equal to:

\[ A=\frac{\lambda_1 w_c}{\lambda_2 w_{\mathrm{я}}};\qquad B=\frac{\lambda_1\cdot 2}{\lambda_2\cdot \pi};\qquad C=\lambda_1 c; \]

\[ K_{2\mathrm{я}}=\frac{\lambda_2 w_{\mathrm{я}} C}{r_{\mathrm{я}}};\qquad T_{\mathrm{я}}=\frac{L_{\mathrm{я}}}{r_{\mathrm{я}}}. \]

In system (2), instead of the functions \(f_1(i_2)\) and \(f_2(i_2)\), a transition has been made to the functions \(\varphi_1(e_3)\) and \(\varphi_2(e_3)\), since the e.m.f. \(e_3\) is linearly related to the current \(i_2\). The indicated functions for the EMU-5P amplifier are presented in Fig. 2. In order to investigate the self-excitation processes that may arise in this dynamical system, let us consider its free motion. In this case its dynamical properties are described by the system of differential equations (3)

\[ \left. \begin{aligned} \frac{de_2}{dt} &= -\frac{1}{T_y+T_k}\,e_2 -\frac{1}{T_y+T_k}\left[Ae_3+B\varphi_2 e_3+C\varphi_2(e_3)\right] = P(e_2,e_3),\\[6pt] \frac{de_3}{dt} &= \frac{\kappa_{2\mathrm{я}}}{T_{\mathrm{я}}}\,e_2 -\frac{1}{T_{\mathrm{я}}}\left[e_3+\kappa_{2\mathrm{я}}\varphi_1(e_3)\right] = Q(e_2,e_3). \end{aligned} \right\} \tag{3} \]

Thus, the behavior of the amplifier under no-load conditions is equivalent to motion in autonomous second-order dynamical systems, which is represented by two nonlinear first-order differential equations.

For the mathematical study of the motions of such dynamical systems, the system of differential equations (3) alone is still insufficient; it is also necessary to know the manifold of possible states of the system under consideration—in other words, its phase space, the points of which correspond one-to-one and continuously to the states of the system [2].

If one of equations (3) is divided by the other, time can be eliminated and one nonlinear first-order equation (4) obtained:

\[ \frac{de_3}{de_2} = \frac{Q(e_2,e_3)}{P(e_2,e_3)} = f(e_2,e_3). \tag{4} \]

The solution of this equation, \(F(e_2,e_3)=C\) (\(C\) is the constant of integration), gives a family of its integral curves, i.e., such curves in the plane \(e_2,e_3\) which at each of their points have a tangent slope determined by equation (4).

Owing to the analyticity of the functions \(P(e_2,e_3)\) and \(Q(e_2,e_3)\), through every point of the plane \(e_2,e_3\) there passes one and only one integral curve of equation (4), except, perhaps, for the singular points of this equation, where it loses its meaning. In the case considered by us, the singular points are only those points at which the functions \(P(e_2,e_3)=0\) and \(Q(e_2,e_3)=0\), i.e., only the equilibrium states of system (3). At them the integral curves may intersect. Hence, in order to find the equilibrium states on the phase plane, it is necessary to find those points of the phase plane where the phase velocity is equal to zero; in other words, it is necessary to find the points of intersection of the curves

\[ P(e_2,e_3)=0,\qquad Q(e_2,e_3)=0. \tag{5} \]

Thus, the equilibrium states of the dynamical system under consideration are the singular points of the family of integral curves on the phase plane.

Fig. 3. Functions \(P(e_2,e_3)=0\) and \(Q(e_2,e_3)=0\) with the brushes shifted by the angle \(\beta=-6^\circ45'\).

Fig. 4. Functions \(P(e_2,e_3)=0\) and \(Q(e_2,e_3)=0\) with the brushes shifted by the angle \(\beta=-4^\circ45'\).

Figures 3 and 4 present the curves (5) for the EMU-5P amplifier (its parameters are indicated in the note) when the brushes of the transverse circuit are shifted against the direction of rotation of the armature, respectively by the angles \(\beta = 6^\circ 45'\) and \(\beta = 4^\circ 45'\). The special points in Fig. 3 are the origin \(O\), and in Fig. 4 the points \(O\), \(A\), and \(B\).

To investigate the stability of the equilibrium states, we shall use the analytical method developed by Poincaré and Lyapunov [2]. Its essence is as follows. Suppose we are interested in the stability of the equilibrium state at the point \(e_{20}, e_{30}\) (\(e_{20}, e_{30}\) are the coordinates of the special points). Since, in analyzing stability, we are interested in the nature of the motions in the presence of certain deviations from the equilibrium state, for convenience of exposition we introduce, instead of the variables \(e_2, e_3\), new dependent variables \(\xi, \eta\), defining them as displacements relative to the equilibrium position in the phase plane:

\[ e_3 = e_{30} + \eta,\qquad e_2 = e_{20} + \xi . \tag{6} \]

Passing from the variables \(e_2, e_3\) to the variables \(\xi, \eta\) in equations (3), we obtain

\[ \left. \begin{aligned} \frac{d\xi}{dt} &= -\frac{1}{T_y+T_k}(e_{20}+\xi) -\frac{1}{T_y+T_k}\left[A(e_{30}+\eta)+B\varphi_1(e_{30}+\eta)+C\varphi_2(e_{30}+\eta)\right],\\[4pt] \frac{d\eta}{dt} &=\frac{k_{2я}}{T_{я}}(e_{20}+\xi) -\frac{1}{T_{я}}\left[(e_{30}+\eta)+k_{2я}\varphi_1(e_{30}+\eta)\right]. \end{aligned} \right\} \tag{7} \]

Expanding \(\varphi_1(e_{30}+\eta)\) and \(\varphi_2(e_{30}+\eta)\) in a Taylor series and, restricting ourselves to the first two terms of the series, we obtain

\[ \varphi_1(e_{30}+\eta) = \varphi_1(e_{30}) +\eta\frac{d[\varphi_1(e_{30})]}{dt} +\ldots = \varphi_1(e_{30})+\rho_1\eta, \]

\[ \varphi_2(e_{30}+\eta) = \varphi_2(e_{30}) +\eta\frac{d[\varphi_2(e_{30})]}{dt} +\ldots = \varphi_2(e_{30})+\rho_2\eta, \]

where

\[ \rho_1=\frac{d[\varphi_1(e_{30})]}{dt} \]

is the tangent of the angle of inclination of the characteristic \(\varphi_1(e_3)\) at the point corresponding to the equilibrium state;

\[ \rho_2=\frac{d[\varphi_2(e_{30})]}{dt} \]

is the tangent of the angle of inclination of the characteristic \(\varphi_2(e_3)\) at the point corresponding to the equilibrium state.

Taking into account that \(e_{20}\) and \(e_{30}\) satisfy condition (5) and that all terms containing them will sum to zero, we obtain a system of linear equations with constant coefficients, the so-called system of equations of the first approximation (8):

\[ \frac{d\xi}{dt}=a\xi+b\eta,\qquad \frac{d\eta}{dt}=c\xi+d\eta, \tag{8} \]

where

\[ a=-\frac{1}{T_y+T_k};\qquad b=-\frac{A+B\rho_1+C\rho_2}{T_y+T_k}; \]

\[ d=-\frac{1-k_{2я}\rho_1}{T_{я}};\qquad c=\frac{k_{2я}}{T_{я}}. \]

The solution of the system of equations (8) will be known if the roots of the characteristic equation (9) are known.

or

\[ \lambda^2-(a+d)\lambda+(ad-bc)=0, \]

\[ \lambda^2+\sigma\lambda+\Delta=0. \tag{9} \]

Lyapunov showed that, in the case where both roots of this equation have real parts different from zero, the investigation of the equations of the first approximation, obtained by discarding the nonlinear terms, always gives the correct answer to the question of the stability of the equilibrium state in system (2).

Namely, if both roots have a negative real part and if, consequently, all solutions of the equations of the first approximation decay, then the equilibrium state will be stable; but if at least one root has a positive real part, i.e., if the system of equations of the first approximation has increasing solutions, then the equilibrium state is unstable [2].

In systems similar to the one under consideration, the following equilibrium states are possible:

1. \(\Delta>0,\ \sigma^2-4\Delta>0\). The roots of the characteristic equation (9) are real and of the same sign. The equilibrium state is a node (stable for \(\sigma>0\)).

2. \(\Delta<0\). The roots of the characteristic equation are real and of opposite signs. The equilibrium state is a saddle.

3. \(\Delta>0,\ \sigma^2-4\Delta<0,\ \sigma\ne0\). The roots of (9) are complex conjugates. The equilibrium state is a focus (stable for \(\sigma>0\)).

Turning to the equilibrium states 1–3 listed above and taking into account that in the system under consideration \(\sigma>0\) under all conditions, one may conclude that the process of self-excitation occurs only in the case when \(\Delta<0\). Analysis of the expression for \(\Delta\) shows that it can become negative when the brushes of the transverse circuit are shifted opposite to the direction of rotation of the armature (in the expression for \(\Delta\), in this case, a minus sign is placed before the coefficient \(B\)).

Thus, \(\Delta=0\) is the limiting condition for the occurrence of the self-excitation process in the system. From the expression \(\Delta=0\) one can determine the critical angle \(\beta'_k\) (10), above which, under certain conditions, the process of self-excitation may occur in the system:

\[ \beta'_k=\frac{1}{B}\left[\frac{1}{k_{2я}}+A+\rho_1+C\rho_2\right]. \tag{10} \]

It is evident that \(\beta_k\), for a particular amplifier, depends entirely on \(\rho_1\) and \(\rho_2\). Assigning various values to \(\rho_1\) and \(\rho_2\) according to Figs. 2 and 5 (the values of \(\rho_1\) and \(\rho_2\) are interrelated), one can find the critical angle \(\beta'_k\) for various values of \(\rho_1\) and \(\rho_2\) and construct the graph \(\beta'_k=f_1(\rho_1)\). Fig. 5 makes it possible simply to determine the regions of stable and unstable equilibrium states for a specified angle of rotation of the brushes of the transverse circuit \(\beta\). In Fig. 5 the region of stable equilibrium states can be divided into two parts, one (II) of which has the equilibrium state in the form of a stable node, and the other (I)—in the form of a stable focus. The boundary of these two parts is the curve \(\beta''_k=f_2(\rho_1)\), found from the expression \(\sigma^2-4\Delta=0\).

\[ \beta''_k=\frac{1}{B}\left[A+C\rho_2-\frac{\left(1+aT_{\!я}+k_{2я}\rho_1\right)^2}{4acT_{\!я}^{\,2}}\right]. \tag{11} \]

Thus, with different rotations of the brushes of the transverse circuit

against the direction of rotation, three types of equilibrium states are possible in the amplifier (Fig. 5):

1. Stable foci (region I).
2. Stable nodes (region II).
3. Saddles (region III).

As a result of the analysis carried out above, the qualitative aspect of the singular points that may exist in the amplifier when the brushes of the transverse circuit are shifted against the direction of rotation of the armature was clarified.

For a complete representation of all possible motions in the dynamic system under consideration, it is also necessary to know its phase space. For this, one must have a family of integral curves, whose equation is obtained by solving (4).

But up to now there have been no regular methods for integrating nonlinear differential equations in the general form, nor, at the same time, rigorous methods for constructing the phase portrait of the nonlinear dynamic system under study. Therefore, for the investigation of analogous systems the simplest approach is often the method of approximate graphical integration, i.e., the method of constructing an approximate phase portrait of the given dynamic system. The most suitable technique of approximate graphical integration for our purposes is the method of isoclines.

Fig. 5. Diagram of equilibrium states as a function of the brush shift angle of the amplifier.

The curves \(f(e_2, e_3)=\Theta=\mathrm{const}\) (4) on the phase plane are the geometric locus of such points through which all the integral curves sought by us pass at one and the same angle to the abscissa axis, namely at an angle whose tangent is equal to \(\Theta\). Therefore the curves \(f(e_2,e_3)=\Theta\) are called isoclines (curves of equal inclination). By assigning \(\Theta\) different numerical values, one can construct on the phase plane a family of isoclines of the integral curves being sought. In Figs. 6 and 7 the phase plane is presented in the region of the singular points \(O\), \(A\), and \(B\), shown in Fig. 4. From consideration of Figs. 6 and 7 it may be concluded that, in the case of brush shift ...

Fig. 6. Phase diagram with two singular points: \(O\)—a stable node, \(A\)—a saddle.

Fig. 7. Phase diagram with a singular point \(B\) of the stable-node type.

of the transverse axis by an angle \(\beta = 4^\circ 45'\), we obtain the following singular points: \(O\) and \(B\) are stable nodes, and \(A\) is a saddle. Figure 8 presents an oscillogram for the case under consideration. It shows that, when a pulse of amplitude \(i_y = 0.9\,ma\) and duration \(0.18\,sec\) is applied, the process of self-excitation does not arise (Fig. 8, curve 1), and the system returns to its initial state (to the stable node \(O\) in Fig. 6).

When a pulse of the same duration but with amplitude \(i_y = 1.2\,ma\) is applied, the representative point along the corresponding phase trajectory will move toward the stable node \(B\) (Fig. 7), asymptotically approaching it, and the process of self-excitation will occur in the amplifier (Fig. 8, curve 2).

Thus, if the pulse has sufficiently large amplitude and duration, it will transfer the amplifier from the equilibrium state \(O\) to the equilibrium state \(B\).

Consideration of the phase portraits (Figs. 6 and 7) and the diagram (Fig. 5) makes it possible to conclude that, in the zone of brush shift from \(\beta_{k1}\) to \(\beta_{k2}\), we have a case of hard self-excitation, i.e., a self-excitation such as may arise when a pulse of definite amplitude and duration is applied to the system.

Fig. 8. Oscillogram of the transient operating condition of the amplifier. Curve 1—at a pulse \(i_y = 0.9\,ma\). Curve 2—at a pulse \(i_y = 1.2\,ma\).

From consideration of Fig. 5 it follows:

1. For \(\beta < \beta_{k1}\), only stable equilibrium states exist in the system.

2. For \(\beta > \beta_{k1}\), there may be both stable and unstable equilibrium states.

3. For \(\beta_{k2} > \beta > \beta_{k1}\), a regime of hard self-excitation may exist.

4. For \(\beta > \beta_{k2}\), there is a regime of soft self-excitation. Moreover, for \(\beta > \beta_{k2}\), the self-excitation process proceeds until the magnetic system of the amplifier reaches complete saturation.

From the analysis carried out it is evident that the self-excitation process is determined to a large extent by the magnitude of the brush-contact resistance, which depends on many factors. These include: the grade of the brushes, the temperature of the commutator, vibration, atmospheric pressure, etc. [3]. Thus, for example, an amplifier adjusted in the cold state (dashed lines in Figs. 2 and 5) for the maximum gain coefficient, by shifting the brushes of the transverse axis opposite to the direction of rotation of the armature, close to the critical value, may, upon heating, pass into a regime of hard or even soft self-excitation, since

section II-a of stable region II, upon heating, passes into unstable region III (Fig. 5).

Thus, taking account of only one factor (temperature) affecting the brush-contact resistance indicates that the dynamic properties of the system, including the process of self-excitation, are thereby substantially altered.

The foregoing examination of the self-excitation processes of cross-field electric-machine amplifiers, taking into account the nonlinearity of the brush contact, makes it possible to draw the following conclusions:

1. Taking account of the nonlinearity of the brush-contact resistance of a cross-field electric-machine amplifier makes it possible to investigate qualitatively the nature of the self-excitation process and to reveal the presence of hard self-excitation, which exists for a definite brush displacement lying within the interval $\beta_{k1} \div \beta_{k2}$.

2. For a brush displacement $\beta > \beta_{k2}$, the amplifier operates in the region of soft self-excitation.

3. The nature of the self-excitation process, for a definite brush displacement, is strongly affected by the state of the brush contact of the transverse armature circuit.

4. With an increase in the nonlinearity of the brush contact, the critical angle of soft and hard self-excitation increases, and conversely.

5. In view of the fact that the nonlinearity of the brush-contact resistance does not remain constant when the commutator temperature changes, the critical angle $\beta_k$ also changes. Thus, as the commutator temperature increases, the nonlinearity of the brush contact decreases, and the critical angle $\beta_k$ also decreases; in other words, an amplifier that operates stably at a low temperature may lose stability and pass into a self-excitation mode.

Note. The experimental verification of the mathematical investigation of self-excitation processes was carried out on an EMU-5P amplifier with nominal data $P = 0.7$ kW, $U = 115$ V, $n = 5000$ rpm. The values of the coefficients used in the calculation are given in the table below.

REFERENCES

1. N. V. Vinogradov, F. A. Goryainov, and P. S. Sergeev. Design of Electrical Machines. GEI, 1956.

2. A. A. Andreev, A. A. Vitt, and S. E. Khaikin. Theory of Oscillations. Moscow, 1959.

3. M. D. Belkin and S. G. Shtykhnov. Brushes for Electrical Machines: Their Manufacture and Application. GEI, 1952.

Received by the editors
in June 1962.