Physics of Oscillations in the Shunt-Excited Exciter—Generator Field Winding System

S. S. Kropyanin

(Presented by the scientific seminar of the Electromechanical Faculty)

In a system whose circuit is shown in Fig. 1, for a certain ratio of the parameters it is possible for undamped periodic oscillations to arise in the excitation current of the exciter \(i_{\mathrm{в}}\), the excitation current of the generator \(i_2\), and the armature current of the exciter \(i_{\mathrm{я}}\). The explanation of the physical nature of the undamped oscillations given in [1, 2, 3, 4] is not sufficiently complete. In [1, 2] the system was considered as linear; in [3] the explanation was given without sufficient substantiation. The works mentioned above do not reveal the physics of the processes taking place and do not answer a number of questions that arise in studying the operation of the system, especially when the principal nonlinearities are taken into account.

Fig. 1.

Let us consider the physical processes occurring in the system after the switch \(P\) is closed (Fig. 1). We shall carry out all reasoning with the following assumptions, justified in [4, 5, 6].

1. We take the rotational speed of the exciter armature to be constant.
2. We take the leakage coefficients of the poles of the exciter field winding \(\sigma_{\mathrm{в}}\) and of the generator \((\sigma_2)\) to be constant.

1. We do not take into account the reaction of the exciter armature.

2. We do not take hysteresis phenomena into account.

3. We neglect the mutual inductance between the armature and excitation windings.

4. We neglect the influence of eddy currents in the massive parts of the system’s electric machines.

5. We neglect the magnitude of the inductance of the armature winding \(L_{\mathrm{a}}\).

6. We take as substantially nonlinear the function of the exciter emf \(e_{\mathrm{v}}(i_{\mathrm{v}})\).

7. We take the inductances of the excitation windings of the exciter \(L_{\mathrm{v}}\) and of the generator \(L_{\mathrm{g}}\), and the resistance of the exciter armature circuit \(r_{\mathrm{a}}\), to be constant.

At the instant the switch \(P\) is closed, the exciter is excited (operates at no load), which is characterized by points \(a\) on the no-load characteristic (curve 1) and \(b\) on the external characteristic of the exciter (curve 3) in Fig. 2. The coordinates of point \(a\) are determined

Fig. 2.

by the intersection of the no-load characteristic and the characteristic \(i_{\mathrm{v}} r_{\mathrm{v}}\) (characteristic 2), where

\[ r_{\mathrm{v}}=\frac{e_{\mathrm{va}}}{i_{\mathrm{va}}}=\tan \gamma_1 \]

is the total resistance of the exciter excitation winding. The coordinates of point \(b\) are determined by the no-load emf of the exciter at armature current equal to zero. After the switch is closed, the exciter armature current \(i_{\mathrm{a}}\) will increase, while the excitation current \(i_{\mathrm{v}}\), the emf \(e_{\mathrm{v}}\), and the voltage \(U_{\mathrm{v}}\) will decrease. The direction of the change is shown in Fig. 2 by arrows. As is known, the current in the armature of the exciter is equal to

\[ i_{\mathrm{a}}=i_{\mathrm{v}}+i_{2}. \tag{1} \]

To simplify the reasoning, in (1) we shall neglect \(i_{\mathrm{v}}\), since the current \(i_{\mathrm{v}}\) amounts to 2–3 percent of the armature current \(i_{\mathrm{a}}\) [5]. Taking into account the assumptions we have adopted, let us construct in Fig. 2 the external characteristic of the exciter (curve 3) and the load characteristics \(i_{\mathrm{a}} r_{\mathrm{c}}-i_{2} r_{2}\) for various values of the resistances of the circuit of the generator excitation winding (curves 4, 5, 6), for resistances \(r_{21}>r_{22}>r_{23}\). Obviously, the tangent of the angle of inclination \((\beta)\) of the load characteristic \(i_{2} r_{2}\) to the abscissa axis is determined by the value of the load resistance \(r_{2}\) as:

\[ \tan \beta_{2}=-\frac{U_{\mathrm{v}0'2}}{i_{\mathrm{a}0'2}} =\frac{U_{\mathrm{v}0'2}}{i_{2 0'2}}=r_{2}. \tag{2} \]

If, for a given load resistance \(r_z\), the slope angle of the characteristic \(i_z r_z\) \((\beta)\) is such that the point of intersection of the external characteristic and the load characteristic lies on the segment \(\delta-\delta_k\) (curve 3), then the exciter will operate stably with the given load (for example, points \(\delta_1\) at \(r_{z1}\), or \(\delta_2\) at \(r_{z2}\) in Fig. 2). The static stability of a system operating on the segment \(\delta-\delta_k\) can be proved by the method used in drive theory [7]. Indeed, if tangents are drawn to the external characteristic and to the load characteristic \(i_z r_z\) from their point of intersection (for example, the equilibrium point \(\delta_1\) in Fig. 2) to the abscissa axis, then the condition of static stability may be written as

\[ \tan \beta_1 > \tan \alpha_1, \tag{3} \]

where \(\alpha_1\) is the angle between the tangent drawn to the external characteristic and the abscissa axis;

\(\beta_1\) is the angle between the load characteristic \(i_z r_z\) and the abscissa axis (Fig. 2).

As is evident from the figure, this condition will be satisfied on the segment \(\delta-\delta_k\) of the external characteristic. As an example, the oscillogram in Fig. 3a is given, showing the transient process from no-load operation to a new stable state under load at \(r_{z2}>r_{zk}\) \((\beta_k)\). When the load resistance is decreased, the point of intersection of the external characteristic and the load characteristic (\(\delta\)) lies on the segment \(\delta_k-a\). Point \(\delta_3\) will be a point of unstable equilibrium, as is easily seen from relation (3). Point \(\delta_k\) on the external characteristic of the exciter determines the largest “critical” value of the armature current \(i_{yak}\), and, if the magnitude of the excitation current of the exciter is neglected (because of its smallness), the largest value of the load current \(i_{zk}\). We shall now increase the resistance of the exciter field winding to the value \(r_{v2}>r_{v1}\) \((i_2>i_1)\). No-load operation of the exciter will be characterized by the points \(a'\) and \(\delta'\). Point \(\delta'\) will belong to the new external characteristic 7. After the switch is closed, at the same load resistance \(r_{z1}\) \((\beta_1)\), point \(\delta'_1\) will, as before, characterize the static stability of the system. When the load resistance is reduced to \(r_{z2}<r_{z1}\) \((\beta_2<\beta_1)\), point \(\delta'_2\) will already characterize an unstable state of the system, whereas at \(r_{v1}<r_{v2}\) this load \(r_{z2}\) characterized stable operation of the exciter.

\(r_z = 0.25\ \Omega\)
\(r_b = 1.0\ \Omega\)
\(r_g = 0.2\ \Omega\)

\(L_z = 0.11\ \Omega\cdot\mathrm{s}\)
\(L_g = 0.35\ \Omega\cdot\mathrm{s}\)

Fig. 3a.

It follows from what was said above that the state of the system is determined by the ratio of the resistances \(r_2\) and \(r_v\). The oscillograms shown in Figs. 3a and 3b confirm the correctness of the conclusions drawn.

Fig. 3b.

Let us describe by equations the equilibrium state of the system under load for the point \(a_x\):

\[ e_v(i_{vx})=i_{vx}(r_v+r_{ya})+i_{2x}r_{ya}, \tag{4} \]

\[ e_v(i_{vx})=i_{2x}(r_2+r_{ya})+i_{vx}r_{ya}. \tag{5} \]

Solving equations (4) and (5) simultaneously, we obtain:

\[ e_v(i_{vx})=i_{vx}\frac{r_v r_{ya}+r_2 r_{ya}+r_v r_2}{r_2} =i_{vx}R_e, \tag{6} \]

or

\[ \frac{e_v(i_{vx})}{i_{vx}}=R_e \tag{7} \]

where

\[ R_e=\frac{r_v r_{ya}+r_2 r_{ya}+r_v r_2}{r_2} \]

is the equivalent resistance of the system. Draw the characteristic \(i_vR_e\) from the origin to the point \(a_x\), located on the no-load characteristic (characteristic 8 in Fig. 2). Denote by \(\varphi\) the angle of inclination of this characteristic to the abscissa axis. The coordinates of the point \(a_x\) are determined by the exciter emf \(e_{vx}\) and the excitation current \(i_{vx}\) for a given load current \(i_{2x}\). After the constructions and analysis of equation (7), we see that the left-hand side of the equation is \(\tan\varphi\), and the right-hand side is the equivalent resistance of the system \(R_e\); then equation (7) can be rewritten as

\[ \tan\varphi=R_e. \tag{8} \]

As we proved earlier, the point \(b_k\) on the external characteristic determines the greatest value of the load current \(i_{2k}\), that is, the smallest load resistance \(r_{2k}\) at which stable operation of the system is still possible. By graphical constructions we determine the value of the exciter emf \(e_{vk}\) and the excitation current \(i_{vk}\) at the greatest possible value of the load current \(i_{2k}\) (point \(a_k\) in Fig. 2). From the origin draw the characteristic \(i_{vk}R_{ek}\) (9) to the point \(a_k\). Denote the angle formed by this ray and the abscissa axis by \(\varphi_k\). According to equation (8), we can write

\[ \tan\varphi_k=R_{ek}. \tag{9} \]

On the basis of the assumptions adopted above, it may be said that the angle \(\varphi_k\) is the largest of all possible values at which stable operation of the system under load is still possible. If, for the chosen resistances \(r_v, r_2, r_a\), \(R_e\) is greater than \(\tan \varphi_k\), then the system becomes statically unstable and in it there may arise—as will be shown below—undamped periodic oscillations. What has been said above may be written in the form of the condition

\[ R_e > \tan \varphi_k . \tag{10} \]

Condition (10) is necessary, but not unique.

Suppose that, for some ratio of the system parameters, condition (10) is satisfied. After the switch \(P\) is closed, as the load current increases, the emf of the exciter decreases. At a small excitation current, with accuracy sufficient for practice, the characteristic \(e_v(i_v)\) may be regarded as linear; then

\[ e_v(i_v) = K i_v , \tag{11} \]

where \(K\) is a parameter of the exciter, determined by the winding data, the rotational speed of the armature, and the resistance of the magnetic circuit (Fig. 2) [8].

The equations of motion in the system for the general case of transient conditions will have the form

\[ e_v(i_v) = i_a r_a + i_v r_v - L_v \frac{d i_v}{dt}, \tag{12} \]

\[ e_v(i_v) = i_a r_a + i_2 r_2 + L_2 \frac{d i_2}{dt}. \tag{13} \]

Substituting (11) into (12) and (13) and solving the resulting equations together with equation (1) with respect to the exciter excitation current, we shall have:

\[ L_v L_2 \frac{d^2 i_v}{dt^2} + \left[ L_2(r_a+r_v-K)+L_v(r_a+r_2) \right] \frac{d i_v}{dt} + \]

\[ \left[ (r_a+r_2)(r_a+r_v-K)+r_a(K-r_a) \right] i_v = 0 . \tag{14} \]

Dividing both sides of equation (14) by the expression \((r_a+r_2)[K-(r_a+r_v)]\), we obtain the equation of motion in the system

\[ T_v T_2 \frac{d^2 i_v}{dt^2} + \left(T_v-T_2\right)\frac{d i_v}{dt} + \left\{ \frac{r_a(K-r_a)} {(r_a+r_2)[K-(r_a+r_v)]} -1 \right\} i_v = 0 , \tag{15} \]

where

\[ T_v=\frac{L_v}{K-(r_a+r_v)} \]

is the electromagnetic time constant of the exciter excitation winding,

\[ T_2=\frac{L_2}{r_a+r_2} \]

is the electromagnetic time constant of the generator excitation winding.

Equation (15) is a linear differential equa-

... the equation of motion in the system. Equation (15) corresponds to the roots of the characteristic equation

\[ P_{1,2}=-\frac{T_{\mathrm{v}}-T_{2}}{2T_{\mathrm{v}}T_{2}} \pm \sqrt{ \left(\frac{T_{\mathrm{v}}-T_{2}}{2T_{\mathrm{v}}T_{2}}\right)^2 - \left\{ \frac{r_{\mathrm{e}}(K-r_{\mathrm{sh}})} {(r_{\mathrm{sh}}+r_{2})[K-(r_{\mathrm{sh}}+r_{\mathrm{v}})]T_{\mathrm{v}}T_{2}} -\frac{1}{T_{\mathrm{v}}T_{2}} \right\} }. \tag{16} \]

Depending on the form of the roots of the characteristic equation, the nature of the motion in the system will be determined. If it is assumed that the roots of the characteristic equation \(P_1\) and \(P_2\) are real and positive, then, as is known, the linear system is unstable [9]. For the case under consideration, the solution of the equation of motion takes the form

\[ i_{\mathrm{v}}=C_1 e^{P_1 t}+C_2 e^{P_2 t}, \tag{17} \]

where \(C_1\) and \(C_2\) are constants of integration determined by the initial conditions.

According to the experimental data (the parameters are indicated in Fig. 3b), we have:

\[ R_{\mathrm{e}}=5.33\ \Omega;\qquad \tan\varphi_{\mathrm{k}}=4.07\ \Omega;\qquad T_{\mathrm{v}}=0.1272\ \mathrm{sec}. \]

\[ T_2=0.2445\ \mathrm{sec};\qquad P_1=3.675\ \frac{1}{\mathrm{sec}};\qquad P_2=0.095\ \frac{1}{\mathrm{sec}}. \]

To determine the constants of integration \(C_1\) and \(C_2\), we impose the conditions:

\[ t=0;\qquad i_{\mathrm{v}}=i_{\mathrm{v\,init.}}=1\ \mathrm{A};\qquad \frac{d i_{\mathrm{v\,init.}}}{dt}=-7.32\ \frac{\mathrm{A}}{\mathrm{sec}}. \]

Differentiating equation (17) with respect to time \(t\) and imposing the initial conditions adopted by us, we determine the constants of integration:

\[ C_1= \frac{ P_2 i_{\mathrm{v\,init.}}-\dfrac{d i_{\mathrm{v\,init.}}}{dt} }{ P_2-P_1 }; \qquad C_2= \frac{ \dfrac{d i_{\mathrm{v\,init.}}}{dt}-P_1 i_{\mathrm{v\,init.}} }{ P_2-P_1 }; \tag{18} \]

\[ C_1=-2.07\ \mathrm{A};\qquad C_2=3.07\ \mathrm{A}. \]

Substituting the obtained results into equation (17), we shall have:

\[ i_{\mathrm{v}}=-2.07 e^{3.675t}+3.07 e^{0.095t}. \tag{19} \]

Denoting the first component of the excitation current by \(i'_{\mathrm{v}}\), the second component by \(i''_{\mathrm{v}}\), and assigning the time \(t\) from zero to \(0.15\ \mathrm{sec}\), we compute the coordinates of the points from which we construct the corresponding characteristics \(i'_{\mathrm{v}}(t)\), \(i''_{\mathrm{v}}(t)\), \(i_{\mathrm{v}}(t)\) in Fig. 4. As can be seen, the exciter excitation current \(i_{\mathrm{v}}\) changed its direction, which means the possibility of reversal of the exciter. If we consider the portion of the external characteristic near the short-circuit point (point \(v\), Fig. 2), then for equations (12) and (13) the signs of the derivatives \(\dfrac{d i_{\mathrm{v}}}{dt}\) and \(\dfrac{d i_2}{dt}\) become negative (Fig. 3b). Solving these equations jointly,

Taking into account the signs of the derivatives with respect to the exciter excitation current, we shall have

\[ i_{\mathrm{v}}= \frac{\left(L_{\mathrm{v}}\left|\frac{di_{\mathrm{v}}}{dt}\right|-i_2r_2\right)}{r_{\mathrm{v}}} - L_2\left|\frac{di_2}{dt}\right|. \tag{20} \]

Analysis of the numerator of equation (20) makes it possible to say that if, under transient conditions, the subtracted term \(\left|L_2\dfrac{di_2}{dt}\right|\) becomes, in absolute value, greater than the diminished term \(\left|\left(L_{\mathrm{v}}\dfrac{di_{\mathrm{v}}}{dt}+i_2r_2\right)\right|\), then the direction of the current in the excitation winding may change. Let us write what has been said in the form of the inequality

\[ \left|L_2\frac{di_2}{dt}\right| > \left|\left(L_{\mathrm{v}}\frac{di_{\mathrm{v}}}{dt}+i_2r_2\right)\right|. \tag{21} \]

The left-hand side of inequality (21) is the self-induction emf of the generator excitation winding; the right-hand side is the sum of the self-induction emf of the exciter excitation winding and the voltage drop across the active resistance of the generator excitation winding. Multiplying both sides of inequality (21) by the product \(i_2\,dt\), we obtain

\[ \left|L_2 i_2\,di_2\right| > \left| \left(L_{\mathrm{v}} i_{\mathrm{v}}\,di_{\mathrm{v}}\right)\frac{i_2}{i_{\mathrm{v}}} + i_2^2 r_2\,dt \right|. \tag{22} \]

Inequality (22) is an inequality for the change of energy in the circuit: generator excitation winding--exciter excitation winding. Analyzing inequality (22), we can say that the current in the exciter excitation winding can change its direction if the change in electromagnetic energy of the generator excitation winding is greater than the sum of the change in the energy of the exciter excitation winding and the energy lost in the active resistance of the generator excitation winding.

Fig. 4.

Since the direction of the current in the exciter excitation winding can change, reversal of the exciter magnetization is possible. Let us determine under what conditions reversal of the exciter magnetization is possible. In Fig. 5 a part of the no-load characteristic of the exciter is shown, taking account of hysteresis, which in the present case we cannot neglect. Also shown there is the characteristic \(i_{\mathrm{v}} r_{\mathrm{v}}\). As we established above, the excitation current can change direction. If the current in the reverse direction, in absolute value, becomes greater than some value of the current \(\left|i_{\mathrm{vs}}\right|\) (the self-excitation current), then reversal of the exciter magnetization will occur, be-

since the point \(a_c\) is a point of unstable equilibrium [8]. What was said above can be written in the form of the inequality

\[ |i_v|>|i_{vs}|. \tag{23} \]

Such a self-excitation regime, when a push of a certain finite magnitude is required in order for the system to self-excite, bears the name of the regime of hard self-excitation. After self-excitation in the reverse direction, the entire process will repeat; periodic undamped oscillations arise in the system (Fig. 5).

Fig. 5.

In Fig. 6 an oscillogram is given of the transient regime for the case when conditions (10, 11, 22) were satisfied, but condition

Fig. 6.

condition (23), that is, \(|i_{\mathrm{v}}|\) was less than \(|i_{\mathrm{vs}}|\). Despite the fact that the direction of the current changed, no reversal of magnetization of the exciter occurred. From the oscillogram data of Fig. 6 we have

\[ i_{\mathrm{v}} = 0.35\ \mathrm{A}; \qquad R_{\mathrm{e}} = 4.67\ \mathrm{ohm}. \]

From the data of Fig. 2 we have: \(\tan \varphi_{\mathrm{k}} = 3.88\ \mathrm{ohm}\), and from Fig. 5 \(i_{\mathrm{vs}} = 0.38\ \mathrm{A}\). From what has been said it follows that the necessary conditions for reversal of magnetization of the exciter are conditions (10, 22, 23).

Let us consider the phenomena occurring in the system from the energy point of view. After simple transformations in equations (1, 12, 13), we obtain the power-balance equation for the system

\[ e_{\mathrm{v}}(i_{\mathrm{v}}) i_{\mathrm{ya}} - \left(r_{\mathrm{v}} i_{\mathrm{v}}^{2} + r_{2} i_{2}^{2} + r_{\mathrm{ya}} i_{\mathrm{ya}}^{2}\right) - L_{\mathrm{v}} i_{\mathrm{v}} \frac{d i_{\mathrm{v}}}{d t} - L_{2} i_{2} \frac{d i_{2}}{d t} = 0 . \tag{24} \]

Each term of equation (24) represents, respectively: the first, the source power \((P_{\mathrm{i}})\); the second (the sum in parentheses), the power of losses in the active resistances of the system \((P_{\mathrm{p}})\); the third, the power of the magnetic field of the exciter field winding \((P_{\mathrm{m1}})\); and the fourth, the power of the magnetic field of the generator field winding \((P_{\mathrm{m2}})\). Using the oscillogram of the transient process (Fig. 6), one can construct the components of the power in coordinates whose axes are power and time. The graph \(P(t)\) is plotted in Fig. 7. It is evident that the area bounded by the corresponding curve \(P(t)\) and the abscissa axis represents the corresponding energy. As can be seen from the graph of Fig. 7, during the time

Fig. 7.

from \(t_{1}\) to \(t_{2}\), the energy of the source is expended on covering the losses in the system. During this same time, a certain amount of magnetic energy is stored in the generator and exciter field windings. During the time from \(t_{2}\) to \(t_{3}\), the exciter field winding gives back to the system the energy stored in it. During the time from \(t_{3}\) to \(t_{4}\), the generator field winding likewise becomes a source of energy for the system. During the time from \(t_{5}\) to \(t_{6}\), the generator field winding becomes the only source of energy in the system. The energy of this winding is expended on covering the losses in the active resistances of the system, is partially returned to the source, and creates a reserve of energy in the exciter field winding. Let us note that, when energy is delivered by the generator field winding, reversal of magnetization of the exciter occurs.

The equation of motion in the system (15) that we derived is valid only for the linear part of the magnetization characteristic of the exciter. In order to cover the entire process completely, it is necessary to take into account the whole magnetization characteristic. To derive the general equation of motion in the system, we approximate the nonlinear magnetization characteristic of the exciter \(e_{\mathrm{v}}(i_{\mathrm{v}})\), entering into equations (12) and (13), as

\[ e_{\mathrm{v}}(i_{\mathrm{v}})=A e_{\mathrm{vn}} \operatorname{arctg} B i_{\mathrm{v}}, \tag{25} \]

where \(e_{\mathrm{vn}}\) is the nominal value of the no-load emf of the exciter, equal in our case to 12 volts,

\(A\) and \(B\) are coefficients determined by the form of the experimental magnetization characteristic of the exciter and, in our case, equal to \(A=1.2\);

\[ B=0.386\,\frac{1}{a}. \]

The justification of the adopted approximation is given in [10].

Substituting the adopted approximation into (12) and (13) and solving the resulting equations jointly with equation (1) with respect to the excitation current of the exciter, we obtain the nonlinear differential equation of motion in the system

\[ \frac{d^{2} i_{\mathrm{v}}}{dt^{2}} + \left[ \frac{r_{\mathrm{v}}+r_{\mathrm{ya}}}{L_{\mathrm{v}}} + \frac{r_{2}+r_{\mathrm{ya}}}{L_{2}} - \frac{e_{\mathrm{vn}}AB}{(1+B^{2}i_{\mathrm{v}}^{2})L_{\mathrm{v}}} \right] \frac{d i_{\mathrm{v}}}{dt} + \frac{R_{\mathrm{e}}r_{2}i_{\mathrm{v}}-r_{2}A e_{\mathrm{vn}}\operatorname{arctg} B i_{\mathrm{v}}}{L_{\mathrm{v}}L_{2}} =0. \tag{26} \]

Solving this equation for the investigation of transient regimes in the system at various parameters presents considerable difficulties. For a physical analysis of the possible motions obtained, one should adopt a method that would make it possible, with sufficient simplicity and accuracy for practice, to investigate all possible phenomena in the system. The representation of the phenomena occurring on the phase plane gives an idea of the nature of the process without solving the nonlinear differential equations in finite form, reducing their solution to a simple geometric construction [9, 11].

In equation (26), let us denote the derivative \(\dfrac{d i_{\mathrm{v}}}{dt}\) by \(Y\), and \(\dfrac{dY}{d i_{\mathrm{v}}}\) by \(\Theta\), and rewrite it with respect to \(Y\):

\[ Y = - \frac{ r_{2}A e_{\mathrm{vn}}\operatorname{arctg} B i_{\mathrm{v}}-R_{\mathrm{e}}r_{2}i_{\mathrm{v}} }{ L_{\mathrm{v}}L_{2} \left[ \dfrac{r_{\mathrm{v}}+r_{\mathrm{ya}}}{L_{\mathrm{v}}} + \dfrac{r_{2}+r_{\mathrm{ya}}}{L_{2}} - \dfrac{e_{\mathrm{vn}}AB}{(1+B^{2}i_{\mathrm{v}}^{2})L_{\mathrm{v}}} + \Theta \right] }. \tag{27} \]

The equation obtained is the equation of the isoclines. Using the method set forth in [9, 11], on the phase plane (Fig. 8) we construct the isoclines for the case recorded in the oscillogram of Fig. 3b. From the constructed isoclines we construct the integral curve, which is a stable limit cycle. The existence of a limit cycle on the phase plane is the principal indication of the existence of self-oscillations [9].

The investigations carried out above make it possible to draw the following conclusions.

1. The physical picture of the phenomenon of overexcitation of the exciter is revealed from the linear (idealized) consideration of the system.

1. Remagnetization of the exciter occurs at the expense of the energy stored in the generator field winding.

2. When the phenomenon of hysteresis in the exciter magnetization curve is taken into account, a regime of hard self-excitation is revealed.

3. The actual picture of the motion in the system is explained quite fully on the phase plane, taking into account the principal nonlinearity \(e_{\mathrm{в}}(i_{\mathrm{в}})\).

4. The presence of a limit cycle on the phase plane characterizes the existence of self-oscillations.

Fig. 8.

REFERENCES

1. Shuleikin M. V. On electrical oscillations, Journal of the Russian Physico-Chemical Society, vol. XLII, physical section, issue I, 1910.

2. Govorkov V. A. The works of Academician M. V. Shuleikin on electrical machines, Elektrichestvo, 1952.

3. Gandzha L. I. Self-reversing electric-drive systems, Proceedings of the Tomsk Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov, vol. 72, 1952.

4. Ryudenberg R. Transient Processes in Electrical Systems. Translation from the first American edition edited by V. Yu. Lomonosov, IL, 1955.

5. Ermolin N. P. Transient Processes in Direct-Current Machines, GEI, 1951.

6. Basharin A. V. Graphical calculation of transient processes in cascade-excitation systems of electrical machines with feedbacks and nonlinearities, Vestnik elektropromyshlennosti, No. 5, 1956.

7. Andreev V. P. and Sabinin Yu. A. Fundamentals of Electric Drive, GEI, 1956.

8. Tolvinsky V. P. Direct-Current Electrical Machines, GEI, 1956.

9. Andronov A. A. and Khaikin S. E. Theory of Oscillations, ONTI, part 1, 1937.

10. Arkhangelsky B. I. Analytical expression for the magnetization curve of electrical machines, Elektrichestvo, No. 7, 1950.

11. Edited by Polivanov K. M. Physical Foundations of Electrical Engineering, GEI, 1950.