ON THE SELF-EXCITATION OF DIRECT CURRENT GENERATORS

L. I. Gandzha

(Presented to the Scientific-Methodological Seminar of the EMF, TPI)

According to the method of connecting the field winding, two types of self-excited generators are distinguished: with series and with parallel excitation. In a generator of the first type (Fig. 1,a), the excitation process begins, owing to the e.m.f. \(e_0\) due to residual magnetism, from the moment a load is connected to it. In a generator of the second type, with the load disconnected from the armature (Fig. 1,b), the excitation process also begins owing to \(e_0\) from the moment its field winding is connected to the armature. In both cases the latter is connected in series with the armature, and both named elements are traversed by the same current; it is therefore natural that the self-excitation process in both cases will be described by one differential equation, and the physics of the excitation process of both generators will be one and the same. Below we shall examine in detail the phenomena connected with the excitation process of a generator of the first type, since the generator with series excitation is the one that chiefly determines the self-oscillatory properties of the “generator—motor” system. The conclusions that we obtain in considering the properties of the system in Fig. 1,a can also be extended to the system in Fig. 1,b.

Fig. 1.

Let us consider in more detail the process of excitation of a generator with series excitation when it is switched on to an ohmic resistance, i.e., to a load incapable of accumulating energy of any kind. The conclusions that we obtain in this case will prove necessary when considering self-oscillations in the simple “generator—motor” system.

1. Differential equation of the generator self-excitation process. With reference to the circuit of Fig. 1,a, introduce the following notation:

\[ e_z = e_z(I) \]

is the e.m.f. of the generator, which is a nonlinear function of the excitation current, equal to the load current, and is determined from the magnetization curve (Fig. 2);

\[ R = r_a + r_v + r_n \]

is the total resistance of the power circuit, where \(r_a\), \(r_v\), and \(r_n\) are, respectively, the resistances of the generator armature, its field winding, and the load; allowing for the voltage drop in the brush contact, which depends on the load current, \(R\) is a quan-

variable and determines the nonlinear volt-ampere characteristic \(u=RI=u(I)\), representing the voltage drop in the power circuit;

\[ L(I)=L_a(I)+L_v(I) \]

respectively, are the coefficients of self-induction of the power circuit, of the generator armature winding, and of its field winding, which are also functions of the load current.

Adopting the usual assumptions (absence of eddy currents, armature reaction, etc.), we shall have the following differential equation describing the transient process of self-excitation of the generator:

\[ e_g(I)=R(I)\cdot I+L(I)\frac{dI}{dt}=u(I)+L(I)\frac{dI}{dt}. \tag{1} \]

Equation (1) is a nonlinear differential equation of the first order. The system of Fig. 1,b is described by the same equation (1), with \(I\) in it replaced by \(i_v\).

2. Equilibrium states of the generator. Solving (1) with respect to the derivative, we obtain:

\[ \frac{dI}{dt}=\frac{e_g(I)-u(I)}{L(I)}. \tag{2} \]

After the transient process is completed, the system will reach an equilibrium state, the current will become equal to the steady-state current \(I_{\mathrm{st}}\), and the derivative

\[ \frac{dI_{\mathrm{st}}}{dt} \]

will become equal to zero. Since the system under consideration is described by a first-order differential equation, for the steady process (or for equilibrium states) it is necessary and sufficient to set equal to zero only one first derivative \(\frac{dI}{dt}\). Thus, putting \(I=I_{\mathrm{st}}\), from (2) we obtain:

\[ \frac{dI_{\mathrm{st}}}{dt} = \frac{e_g(I_{\mathrm{st}})-u(I_{\mathrm{st}})}{L(I_{\mathrm{st}})} =0, \tag{3} \]

whence it follows that the necessary and sufficient condition for equilibrium of the system of Fig. 1,a is the condition

\[ e_g(I_{\mathrm{st}})=u(I_{\mathrm{st}}). \tag{4} \]

Fig. 2.

The latter means that the system reaches equilibrium states at the points of intersection of the curves \(e_g(I)\) and \(u(I)\) (Fig. 2). Bearing in mind that the generator can be magnetized both to positive and to negative values of the emf, with corresponding directions of current in the field winding, we obtain two points \(A_1\) and \(A'_1\), corresponding to equilibrium states under the condition that the curve \(e_g(I)\) lies above the curve \(u(I)\) in the region of current values from \(0\) to \(I_{\mathrm{st}}\) (Fig. 3).

3. Stability of the equilibrium states of the generator. Since equations (1) and (2) are nonlinear, the investigation of the stability of the equilibrium states of the system under consideration must be carried out by Lyapunov’s method. For this purpose we first put

\[ I=I_{\mathrm{st}}+\Delta I, \tag{5} \]

where \(\Delta I\) is the deviation of the current from its steady-state (or equilibrium) value. Since it follows from (2) that \(\dfrac{dI}{dt}\) is a function of the current, putting

\[ \frac{dI}{dt}=f(I) \tag{6} \]

and substituting (5) into (6), we shall have:

\[ \frac{d(\Delta I)}{dt}=f(I_{\mathrm{st}}+\Delta I), \tag{7} \]

and expanding \(\dfrac{d(\Delta I)}{dt}\) in a Taylor series in powers of \(\Delta I\), we obtain:

\[ \frac{d(\Delta I)}{dt} = f(I_{\mathrm{st}}) + f'(I_{\mathrm{st}})\Delta I + \frac{1}{1\cdot 2}f''(I_{\mathrm{st}})(\Delta I)^2. \tag{8} \]

Following Lyapunov, we discard in (8) the terms containing \(\Delta I\) in powers higher than the first; bearing in mind, moreover, that in accordance with (6) and (3) \(f(I_{\mathrm{st}})=0\), we obtain the linear equation of the first approximation in the form:

Fig. 3.

\[ \frac{d(\Delta I)}{dt}=f'(I_{\mathrm{st}})\Delta I. \tag{9} \]

To determine \(f'(I_{\mathrm{st}})\), we find \(f'(I)\) on the basis of (6) and (2). We shall have:

\[ f'(I)= \frac{ \left[e_2'(I)-u'(I)\right]L(I)-L'(I)\left[e_2(I)-u(I)\right] }{ \left[L(I)\right]^2 }, \]

and substituting \(I=I_{\mathrm{st}}\) and taking (4) into account, we obtain:

\[ f'(I_{\mathrm{st}}) = \frac{ e_2'(I_{\mathrm{st}})-u'(I_{\mathrm{st}}) }{ L(I_{\mathrm{st}}) }. \tag{10} \]

Substituting (10) into (9), we finally obtain:

\[ \frac{d(\Delta I)}{dt} - \frac{ e_2'(I_{\mathrm{st}})-u'(I_{\mathrm{st}}) }{ L(I_{\mathrm{st}}) } \Delta I =0. \tag{11} \]

The characteristic equation for the linear differential equation (11) has the form

\[ p-\frac{e_z'(I_{\mathrm{st}})-u'(I_{\mathrm{st}})}{L(I_{\mathrm{st}})}=0, \tag{12} \]

whence the single root of this equation is determined:

\[ p=\frac{e_z'(I_{\mathrm{st}})-u'(I_{\mathrm{st}})}{L(I_{\mathrm{st}})}. \tag{13} \]

Since, in accordance with Fig. 3, \(|e_z'(I_{\mathrm{st}})|<|u'(I_{\mathrm{st}})|\), it follows on the basis of (13) that the root \(p<0\), which, according to Lyapunov, is a sign of a stable equilibrium state. Since equation (11) corresponds to the solution \(\Delta I=Ce^{pt}\), where \(C\) is a constant of integration, then for \(p<0\) the deviation \(\Delta I\) tends to zero as \(t\to\infty\), i.e., the system tends to its former equilibrium state.

Thus, on the basis of (13), the necessary and sufficient condition for stability of the equilibrium state of the system under consideration is the condition

\[ e_z'(I_{\mathrm{st}})-u'(I_{\mathrm{st}})<0, \tag{14} \]

and, in accordance with this, the points \(A_1\) and \(A_1'\) (Fig. 3) correspond to states of stable equilibrium.

4. Representation of the self-excitation process in the phase plane

As is known, a phase plane is the plane on which the states of a system and their changes are represented. The phase plane is the plane \(\frac{dx}{dt}=f(x)\). Introducing the notation for the derivative of the current with respect to time, \(\frac{dI}{dt}=I_t'\), we obtain, on the basis of (1):

\[ \left. \begin{aligned} e_z(I)&=u(I)+L(I)I_t',\\ I_t'&=\frac{dI}{dt}, \end{aligned} \right\} \tag{14a} \]

whence

\[ I_t'=\frac{e_z(I)-u(I)}{L(I)}, \tag{15} \]

and thus the phase trajectory \(I_t'=f(I)\) can be constructed easily if the differences between the ordinates of the curves \(e_z(I)\) and \(u(I)\) (Figs. 2 and 3), divided by \(L(I)\), are plotted for the corresponding values of the current on the plane \(I_t'-I\). Such a construction is carried out in Fig. 4 for one quadrant.

It is obvious that the condition \(I_t'=0\), representing condition (3), is necessary and sufficient for finding equilibrium states. Therefore, on the basis of (15) and Fig. 4, the equilibrium state can be found in the phase plane as the point of intersection of the integral curve with the \(I\)-axis (point \(A_1\) in Fig. 4).

Further, it follows from (15) and from Fig. 4 that \(I_t'\) is a single-valued function of the current; therefore periodic motions and closed phase trajectories are impossible in the system under consideration.

Since the representative point moves along the phase trajectory in the first quadrant from left to right (which corresponds to an increase of \(I\) for positive \(I_t'\)) and in the fourth quadrant from right to left (which ...

corresponds to a decrease of \(I\) for negative \(I'_t\), then in Fig. 4 the representative point tends to point \(A_1\), both in the region of the first and in the region of the fourth quadrants, as is shown by the arrows, which is also a sign of the stability of the equilibrium state under consideration.

Differentiating (15) with respect to the current, we obtain:

\[ I''_{t,I}= \frac{[e'_z(I)-u'(I)]L(I)-L'(I)[e_z(I)-u(I)]}{[L(I)]^2}. \tag{16} \]

The last equality shows that \(I''_{t,I}\) vanishes at the extremal values of \(I'_t\). Therefore from (16) there follows the condition

\[ [e'_z(I)-u'(I)]L(I)-L'(I)[e_z(I)-u(I)]=0, \]

or

\[ \frac{e_z(I)-u(I)}{L(I)}= \frac{e'_z(I)-u'(I)}{L'(I)}, \tag{17} \]

which means that the phase trajectory assumes extremal values at those values of the current \(I_1\) (or \(I_2\)) for which the ratio

Fig. 4.                Fig. 5.

of the difference of the ordinates of the curves \(e_z(I)\) and \(u(I)\) to the corresponding ordinate \(L(I)\) is equal to the ratio of the difference of the derivatives of the same curves to the corresponding derivative of the curve \(L(I)\) (Figs. 3 and 5).

Bearing in mind the character of the change of \(L(I)\) with increasing \(I\), we conclude that \(L'(I)<0\) (Fig. 5); hence it follows that, in order to satisfy condition (17), it must always be the case that \(|e'_z(I)|<|u'(I)|\) when \(|e_z(I)|>|u(I)|\), which is possible only for \(|I_1|>|I_{K_1}|\) in the case of a maximum of the integral curve, and for \(|I_2|<|I_{K_2}|\) in the case of a minimum of the integral curve; here the points \(K_1\) and \(K_2\) have the peculiarity that at them the tangents to \(e_z(I)\) are parallel to the tangents to \(u(I)\) for identical currents, i.e., for these points the condition \(e'_z(I)-u'(I)=0\) is fulfilled.

In the special case when \(L(I)=\mathrm{const}\), from (15) we obtain:

\[ I''_{t,I}= \frac{e'_z(I)-u'(I)}{L(I)}, \tag{18} \]

whence it follows that in this case the integral curve assumes extremal values at such \(I_{K_1}\) and \(I_{K_2}\) for which \(e'_z(I)-u'(I)=0\), which, obviously, corresponds to the points \(K_1\) and \(K_2\) in Figs. 3 and 5.

If we take into account that \(I''_{t,I}=\dfrac{d^2 I}{dt\,dI}\) and that, when \(I''_{t,I}=0\), \(I'_t=f(I)\) assumes extremal values, then it is not difficult to understand that the points 1 and 2 and the corresponding points \(I_1\) and \(I_2\), or the points \(K_1\) and \(K_2\), and the corres-

the corresponding currents \(I_{k_1}\) and \(I_{k_2}\) in Fig. 4, for \(L(I)=\mathrm{const}\), determine the inflection points in the transient-process curve (Fig. 6).

From consideration of (15) and (16), and also from Fig. 4, it follows, moreover, that for equilibrium states for which \(I'_t=0\), \(I''_{t,t}\ne 0\) and \(|I'''_{t,t}|<\infty\), since at equilibrium points \(e'_z(I)\ne u'(I)\) and are finite. Thus the integral curve in the phase plane is not perpendicular to the \(I\)-axis, since at the points corresponding to equilibrium states (point \(A_1\) in Fig. 4) it has a bounded derivative.

Further, from (16) it follows that \(I''_{t,t}\) does not acquire indeterminate values for any values of \(I\). Hence it follows that the phase trajectory under consideration has no singularities (singular points are absent). The latter means that in this case the equilibrium states are not singular states of the system. The latter also follows from the fact that the equation for the second derivative with respect to time from (15) and the second equation of system (14) have the common factor \(\dfrac{dI}{dt}\), which vanishes in equilibrium states.

Fig. 6.

5. Influence of \(R\) on the behavior of the system. In Figs. 2, 3, and 4, the case is represented in which, when the current changes within the limits \(0<I<I_{\mathrm{ust}1}\), the curve \(e_z(I)\) lies above the curve \(u(I)\). Let us determine how the behavior of the system will change if we vary the value \(R\), which is a parameter in (1). This can be achieved by connecting an additional variable resistance into the power circuit of the system of Fig. 1,a. Taking \(R\) to be variable, we must now regard (15) as a function of two variables—the current and the resistance, i.e.:

\[ I'_t=f(I,R)=\frac{e_z(I)-RI}{L(I)}. \tag{19} \]

Let us investigate (19) for equilibrium states, for which it is necessary to set:

\[ e_z(I_{\mathrm{ust}})-RI_{\mathrm{ust}}=0. \tag{20} \]

It follows from (20) that, when \(R\) changes, the equilibrium states will correspond to different steady currents. At the same time, depending on the behavior of the functions \(e_z(I)\) and \(u(I)\), when \(R\) is increased several points of intersection of the curves \(e_z(I)\) and \(u(I)\) may appear. In Fig. 7 various characteristic cases are represented, corresponding to the curves \(u(I)\) given for the resistance values \(R_1, R_2, R_3, R_4\), and \(R_5\).

As we see,

for \(R=R_1\) there is one equilibrium point \(A_{11}\) at \(I=I_{\mathrm{ust}11}\);

for \(R=R_2\) there are two equilibrium points \(A_{12}\) and \(A_{22}\), respectively, at \(I_{\mathrm{ust}12}\) and \(I_{\mathrm{ust}22}\);

for \(R=R_3\)—three points \(A_{13}\), \(A_{23}\), and \(A_{33}\), corresponding to the currents \(I_{\mathrm{ust}13}\), \(I_{\mathrm{ust}23}\), and \(I_{\mathrm{ust}33}\);

for \(R=R_4\)—two points \(A_{14}\) and \(A_{24}\) at the currents \(I_{\mathrm{ust}14}\) and \(I_{\mathrm{ust}24}\);

for \(R=R_5\)—one point \(A_{15}\) at the current \(I_{\mathrm{ust}15}\).

Fig. 7.

Fig. 8.

Fig. 9.

The established values of the currents corresponding to the listed equilibrium points of the system can be determined if (20), for a given value of \(R\), is solved each time with respect to \(I_{\mathrm{st}}\). Proceeding from this and from Fig. 7, one can easily construct the dependence of \(I_{\mathrm{st}}\) on \(R\), shown in Fig. 8. It follows from it that, when the resistance is varied, at certain values of it the system acquires or loses certain equilibrium states. As is known, the values of the parameter at which the number of stable equilibrium states of the system changes are called “bifurcation” values. Such “bifurcation” values of \(R\), in accordance with Fig. 8, will be the values \(R_2\) and \(R_4\). Applying the method set forth above, it is easy to construct phase trajectories for different values of \(R\). These phase trajectories for \(R_1, R_2, R_3, R_4\), and \(R_5\) are shown in Fig. 9; from it it follows that, in the presence of three equilibrium states, the stability of these states alternates; thus, for example, the point \(A_{13}\) corresponds to a stable state, the point \(A_{23}\) to an unstable one, and the point \(A_{33}\) again to a stable equilibrium state. The points \(A_{22}\) and \(A_{14}\), corresponding to semistable equilibrium states, which occur when the phase trajectory touches the \(I\)-axis, should be especially noted. Returning again to Fig. 8, we can say that the branches \(A_{11}—A_{14}\) and \(A_{22}—A_{15}\) of the bifurcation diagram determine the regions of stable states, while the branch \(A_{14}—A_{22}\) determines the region of unstable equilibrium states of the generator.

Fig. 10.

Let us consider separately the curves \(e_z(I)\) and \(u(I)\) for the case of three equilibrium states (Fig. 10). As was already indicated above, the points \(A_1\) and \(A_3\) are points of stable equilibrium states, while the point \(A_2\) is an unstable equilibrium state. If we assume that the point \(A_1\) corresponds to the nominal operating condition of the generator, then the point \(A_3\), in comparison with the point \(A_1\), is located very close to the origin; since it determines a stable equilibrium state, when a load is connected to the generator the latter will excite not to the nominal voltage, but only to a small value of emf \(e_{23}\), corresponding to the point \(A_3\). In order to excite the generator to the nominal voltage corresponding to the point \(A_1\), it is necessary to apply to the excitation system an “external impulse” of such magnitude that the generator would be excited to an emf value \(e_z>e_{22}\), corresponding to the point \(A_2\) of unstable equilibrium. In view of what has been said, and neglecting the excitation of the generator to the emf \(e_{23}\), the case under consideration may be called a case of hard self-excitation. In contrast to it, the case considered earlier (Figs. 2, 3, and 4) is a case of soft self-excitation of the generator.

References

Andronov A. A., Khaikin S. E., Theory of Oscillations, ONTI NKTP USSR, 1937.