On Oscillations in the Exciter of the “Generator–Motor” System

L. I. Gandzha and S. S. Kropanin

(Presented to the scientific seminar of the Electromechanical Faculty)

In electric-drive practice, the “generator–motor” system is widely used with various arrangements for supplying the generator field winding, including supply from a separately excited exciter (Fig. 1).

It is known that, with such a supply circuit for the generator field winding, undamped oscillations are possible in the generator field current $i_{vg}$, the exciter field current $i_{vv}$, the armature current $i_{av}$, the electromotive force $e_v$, and the voltage $U_v$ of the exciter.

Fig. 1.

These oscillations, in a linear interpretation, were studied by M. V. Shuleikin [1, 2], who gave an elementary explanation of the cause of their occurrence. In [3] it is shown that these oscillations arise due to the energy accumulated in the magnetic field of excitation of the generator at the current $i_{vg\,\max}$, corresponding to the maximum load current of the exciter according to its external characteristic (Fig. 2), if this energy is sufficiently large to change the direction of the field current and, in this way, change the polarity of the voltage at the brushes of the exciter. The consequence of this is a change in the direction of the generator field current $i_{vg}$, a change in the polarity of the voltage $U_g$ at its brushes, and, ultimately, a change in the sign of the speed $n$ of motor $D$.

Subsequently these processes repeat, which leads to periodic oscillations of the motor speed, accompanied by reversal.

It is obvious that these oscillations can be used for the construction of a reversing drive whose reversal can be carried out without the use of the corresponding apparatus. Without entering into a detailed study of the question of the criterion for the occurrence of oscillations, we shall consider their character on the basis of the fact that these oscillations are present in the given nonlinear exciter system (Fig. 1).

Fig. 2.

Oscillations for \(L_{vv}=0,\ L_{yav}=0,\ L_{vg}\ne 0\)

Since the energy of the magnetic field of the generator

\[ \frac{L_{vg}\, i_{vg\max}}{2} \]

must be sufficiently large in order to produce oscillations in the system, it is natural, in a first approximation, to assume that the inductance of the generator field-winding circuit is sufficiently large in comparison with the inductances of the exciter field-winding circuit and of the exciter armature-winding circuit. Admitting this idealization, we assume (Fig. 3)

\[ L_{vg}\ne 0,\quad L_{vv}=0,\quad L_{yav}=0, \]

where:

\(L_{vg}\) is the coefficient of self-induction of the generator field-winding circuit,
\(L_{vv}\) is the same for the exciter field winding, and
\(L_{yav}\) is the same for the exciter armature winding.

In what follows, the constancy of \(L_{vg}\), \(L_{vv}\), and \(L_{yav}\) is assumed everywhere.

Fig. 3.

The system (Fig. 3) consists of two circuits, whose coupling is effected by means of the exciter armature circuit. Thus it is a system with two degrees of freedom if the currents \(i_{vv}\) and \(i_{vg}\) are taken as the independent variables.

Bearing in mind that the exciter e.m.f. \(e_v\) is a nonlinear function of the excitation current \(i_{vv}\), which is represented by the magnetization curve,

denoting this function by \(e_v=\varphi(i_{vv})\) and proceeding from the idealization adopted above, we describe the system (Fig. 3) by the following equations:

\[ \varphi(i_{vv})=r_{yav}i_{yav}+r_{vv}i_{vv}, \tag{1} \]

\[ \varphi(i_{vv})=r_{yav}i_{yav}+r_{v2}i_{v2}+L_{v2}\frac{di_{v2}}{dt}, \tag{2} \]

\[ i_{yav}=i_{vv}+i_{v2}. \tag{3} \]

Substituting (3) into (1) and (2), we obtain:

\[ \varphi(i_{vv})=(r_{yav}+r_{vv})i_{vv}+r_{yav}i_{v2}, \tag{4} \]

\[ \varphi(i_{vv})=r_{yav}i_{vv}+(r_{yav}+r_{v2})i_{v2}+L_{v2}\frac{di_{v2}}{dt}, \tag{5} \]

whence from (4) we have:

\[ i_{vv}=\frac{\varphi(i_{vv})-r_{yav}i_{v2}}{r_{yav}+r_{vv}}, \tag{6} \]

\[ i_{v2}=\frac{\varphi(i_{vv})-(r_{yav}+r_{vv})i_{vv}}{r_{yav}}. \tag{7} \]

Substituting (6) into (5) and solving the latter with respect to \(\dfrac{di_{v2}}{dt}\), we finally obtain:

\[ \frac{di_{v2}}{dt}= \frac{r_{vv}\varphi(i_{vv})-(r_{vv}r_{yav}+r_{v2}r_{yav}+r_{vv}r_{v2})i_{v2}} {(r_{vv}+r_{yav})L_{v2}}. \tag{8} \]

Substituting (7) into (5) and solving the latter with respect to \(\dfrac{di_{vv}}{dt}\), we obtain:

\[ \frac{di_{vv}}{dt}= \frac{r_{v2}\varphi(i_{vv})-(r_{vv}r_{yav}+r_{v2}r_{yav}+r_{vv}r_{v2})i_{vv}} {\left[\varphi'(i_{vv})-(r_{vv}+r_{yav})\right]L_{v2}}. \tag{9} \]

Differentiating (3) with respect to time, we have:

\[ \frac{di_{yav}}{dt}=\frac{di_{vv}}{dt}+\frac{di_{v2}}{dt}, \tag{10} \]

and, substituting (8) and (9) into (10) and omitting the intermediate calculations, we finally obtain:

\[ \left. \begin{aligned} \frac{di_{yav}}{dt} &=\frac{A-B}{(r_{vv}+r_{yav})\left[\varphi'(i_{vv})-(r_{vv}+r_{yav})\right]L_{v2}},\\ \text{where:}\qquad A&=\left[\varphi'(i_{vv})-(r_{vv}+r_{yav})\right] \left[r_{vv}\varphi(i_{vv})-(r_{vv}r_{yav}+r_{v2}r_{yav}+r_{v2}r_{vv})i_{v2}\right],\\ B&=(r_{vv}+r_{yav}) \left[r_{v2}\varphi(i_{vv})-(r_{vv}r_{yav}+r_{v2}r_{yav}+r_{v2}r_{vv})i_{vv}\right]. \end{aligned} \right\} \tag{11} \]

The value of derivative (11) can easily be expressed through \(i_{yav}\); indeed, from (1)

\[ i_{vv}=\frac{\varphi(i_{vv})-r_{yav}i_{yav}}{r_{vv}}, \tag{12} \]

and, differentiating (12) with respect to time and solving everything with respect to \(\dfrac{di_{vv}}{dt}\), we have

\[ \frac{di_{vv}}{dt}= \frac{r_{yav}}{\varphi'(i_{vv})-r_{vv}}\, \frac{di_{yav}}{dt}. \tag{13} \]

Determining further \(i_{v2}\) from (3) and substituting its value into (2), we obtain:

\[ \varphi(i_{v v})=r_{l v}i_{l v}+r_{v2}(i_{l v}+i_{v v})+L_{v2}\frac{d(i_{l v}-i_{v v})}{dt}. \tag{14} \]

Finally, substituting (12) and (13) into (14) and solving everything with respect to \(\dfrac{di_{l v}}{dt}\), we obtain finally:

\[ \frac{di_{l v}}{dt} = \frac{\left[(r_{v v}+r_{v2})\varphi(i_{v v})-(r_{v v}r_{l v}+r_{v2}r_{l v}+r_{v v}r_{v2})i_{l v}\right]\left[\varphi'(i_{v v})-r_{v v}\right]} {r_{v v}L_{v2}\left[\varphi'(i_{v v})-(r_{v v}+r_{l v})\right]}. \tag{15} \]

Analysis of equation (8) shows that, for finite values of the current, the derivative \(\dfrac{di_{v2}}{dt}\) cannot assume values equal to infinity, since the denominator in (8) does not vanish. Hence it follows that, in the process of oscillations, the current in the generator field winding cannot change instantaneously, “by a jump,” i.e., the current oscillations can only be continuous, and for it there can be no “discontinuous” oscillations. Physically, the impossibility of changing \(i_{v2}\) “by a jump” follows from the fact that we have taken into account the inductance of the generator field-winding circuit, putting \(L_{v2}\ne 0\); consequently, at every instant of time the magnetic field of the generator stores energy equal to

\[ A=\frac{L_{v2}i_{v2}^{2}}{2}, \]

and the latter cannot change instantaneously in time, since otherwise

\[ \frac{dA}{dt}=\text{power}=\infty, \]

i.e., the system would have to develop infinitely great power.

Analyzing (9) and (15), we arrive at the conclusion that both \(\dfrac{di_{v v}}{dt}\) and \(\dfrac{di_{l v}}{dt}\) become infinite under the condition

\[ \varphi'(i_{v v})-(r_{v v}+r_{l v})=0; \tag{16} \]

therefore, when the system satisfies condition (16), both \(i_{v v}\) and \(i_{l v}\) can change “by a jump”; thus, continuous oscillations are impossible for the currents \(i_{v v}\) and \(i_{l v}\), and only “discontinuous” oscillations are possible. We obtained the idea of discontinuous oscillations for \(i_{v v}\) and \(i_{l v}\) as a result of the idealization of the system admitted above, by putting \(L_{v v}=0\) and \(L_{l v}=0\) and thus considering a twice-degenerate system. By admitting such an idealization, we introduce a certain inaccuracy in solving the problem; however, the idea of “discontinuous” oscillations helps to a considerable extent in forming a correct conception of the complex mechanism of nonlinear oscillations in the given system and in correctly describing it from the qualitative side.

Condition (16) for the “jump” of \(i_{vv}\) and \(i_{yav}\) has a definite geometrical meaning, following from consideration of the magnetization curve and the characteristic of the exciter shunt (Fig. 4)\(^1\).

In Fig. 4 the shunt characteristic, whose equation is

\[ U_v = r_{vv} i_{vv}, \]

is inclined to the abscissa axis at an angle equal to

\[ \alpha = \operatorname{arc\,tg} r_{vv}; \]

in the same axes draw the straight line \(oa\), inclined to the abscissa axis at an angle

\[ \beta = \operatorname{arc\,tg}(r_{vv} + r_{yav}). \]

Fig. 4.

If now a tangent is drawn to the curve \(\varphi(i_{vv})\), parallel to the straight line \(oa\), then at the point \(K\) of tangency the following condition will be satisfied:

\[ \gamma = \beta. \tag{a} \]

Since

\[ \gamma = \operatorname{arc\,tg}\left[\varphi'(i_{vv})\right], \]

\[ \beta = \operatorname{arc\,tg}(r_{vv} + r_{yav}), \]

\(^1\) It should be noted that these characteristics are “static”: \(e_v\) will follow the law shown in Fig. 4 in the case of slow changes of \(i_{vv}\). In a real system, with rapid changes of \(i_{vv}\), the reaction of eddy currents will manifest itself, as a result of which the “dynamic” magnetization characteristic will differ from the “static” one. The analysis of oscillations using the static characteristics of the exciter is therefore also an idealization of the system and does not contradict the original assumption about considering a system with two degrees of freedom; if we took into account the reaction of eddy currents, we would arrive at a system with three degrees of freedom, the analysis of which would prove considerably more complicated and would differ substantially from the analysis of the given system with two degrees of freedom.

then

\[ \varphi'(i_{vv}) = (r_{vv}+r_{av}) \]

or

\[ \varphi'(i_{vv})-(r_{vv}+r_{av})=0, \]

which coincides with (16).

Thus, “jumps” of the currents \(i_{vv}\) and \(i_{av}\) occur when the system reaches the state corresponding to point \(K\).

Let us draw the tangent to the curve \(\varphi(i_{vv})\) at the point \(K'\), parallel to the shunt characteristic, and inscribe the corresponding characteristic triangles for the points \(K\) and \(K'\). It is not difficult to see that the dimensions of characteristic triangle 1, corresponding to the point \(K'\), are greater than the dimensions of triangle 2, corresponding to the point \(K\). From Fig. 4 it is also clear that triangle 1 corresponds to the maximum armature current \(i_{av\max}=i_{avk'}\), determined from the external characteristic of the exciter (Fig. 2); then the current \(i_{av}\), corresponding to triangle 2, will be less than \(i_{av\max}\). In other words, if the system approaches the point \(K\) from the side of currents greater than the current \(i_{vv}\), in the direction shown by the arrow, then “jumps” of \(i_{vv}\) and \(i_{av}\) occur at a certain point \(K\) of the external characteristic (Fig. 2) and after \(i_{v2}\) has reached its maximum value. These “jumps” occur in such a way that, at the instant of the “jump,” on the one hand, equality (3) is preserved, and on the other—the condition is satisfied

\[ i_{v2}=\text{const}, \tag{6} \]

since \(i_{v2}\) cannot change by a “jump.”

If the values of the currents before the “jump” are denoted by \(i_{vv1}\) and \(i_{av1}\), and after the “jump” by \(i_{vv2}\) and \(i_{av2}\), then, by virtue of (6), we shall have

\[ i_{v2k}=i_{av1}-i_{vv1}, \]

\[ i_{v2k}=i_{av2}-i_{vv2}, \]

whence the condition for the “jump” is written in the form:

\[ i_{av2}-i_{vv2}=i_{av1}-i_{vv1}. \tag{17} \]

Proceeding from (1), and also from the equation

\[ U_v=e_v-r_{av}\cdot i_{av}, \]

which is valid under the idealization we have admitted \((L_{av}=0)\), we conclude that both \(e_v\) and \(U_v\) will also change by a “jump” simultaneously with \(i_{vv}\) and \(i_{av}\).

The “jump” occurring at the point \(K\) (Fig. 4) evidently predetermines the presence of the values \(i_{vv}>i_{vv1}\) and \(i_{vv}<i_{vv1}\). This means that, when approaching the point \(K\), the system changes its state so that it tends to the value \(i_{vv1}\); \(i_{av}\) also tends to the value \(i_{av1}\), corresponding to the point \(K\), but in doing so passes through its maximum value \(i_{av\max}\), corresponding to the point \(K'\), as was already said above in considering Figs. 2 and 4, and also in [3]. This leads to the following conclusions:

1) The function \(i_{av}=f(t)\) is nonmonotonic over the course of one half-period, since it assumes an extremal value.

2) The system approaches the point \(K\), i.e., the “jump,” in such a way that \(|i_{vv}|\) tends to \(|i_{vv1}|\) from the region of larger absolute values, which is shown by the arrow in Fig. 4.

3) After the jump \(|i_{vv}|\) assumes the value \(|i_{vv2}|\), and since before the next “jump,” in the process of oscillations, the current \(i_{vv}\) changes so that it tends from the region of larger absolute values to the value \(|i_{vv1}|\), we have:

\[ |i_{vv2}|>|i_{vv1}|. \tag{18} \]

4) Since, under a nonmonotonic variation of \(i_{vv}=f(t)\), the function \(i_{vv}=\psi(t)\) changes monotonically, the value \(|i_{vv2}|\) after the “jump” is the greatest absolute value of this function over the course of a half-period, while \(|i_{vv1}|\) is the smallest.

Further, since during oscillations the exciter is subjected to periodic reversals of magnetization, its static magnetization characteristic will appear in the form of a hysteresis loop (Fig. 5).

Applying what was said above to Fig. 5, we obtain the following picture of “discontinuous” oscillations.

Fig. 5.

1. The current \(i_{vv}\), changing from a value satisfying the condition

\[ +i_{vv1}<+i_{vv}<+i_{vv2}, \]

tends to the value \(+i_{vv1}\), and the system tends toward the state corresponding to point \(K\).

1. At point \(K\) there occurs a “jump” of \(i_{vv}\) from the value \(+i_{vv1}\) to the value \(-i_{vv2}\), provided the condition

\[ |i_{vv2}|>|i_{vv1}| \]

is satisfied.

1. The current \(i_{vv}\) tends from values satisfying the condition

\[ -i_{vv2}<-i_{vv}<-i_{vv1}, \]

to the value \(-i_{vv1}\), corresponding to point \(K_1\).

1. At point \(K_1\) there occurs the next “jump” of \(i_{vv}\), from the value \(-i_{vv1}\) to the value \(+i_{vv2}\), and the condition

\[ |i_{vv2}|>|i_{vv1}| \]

is satisfied.

Thereafter these processes repeat periodically.

1. Since \(e_v\) and \(U_v\) also change by “jumps” simultaneously with \(i_{vv}\) and in accordance with the magnetization curve and with the load characteristic,

the portions of the hysteresis loop shown in Fig. 5 by the dashed line will be inoperative for the exciter.

The closed curve \(AKA_1K_1\) gives an idea of the limiting cycle of self-oscillations in the plane \(i_{vv}, e_v\). As for the values \(|i_{vv1}|\), \(|i_{vv2}|\), \(|i_{vzk}|\), \(|i_{yv1}|\), and \(|i_{yv2}|\), they are found from the following considerations.

1. The value \(i_{vv1}\) is found as the abscissa of the point \(K\), whose position is easily determined by the corresponding graphical constructions mentioned earlier (Figs. 4 and 5).

2. From the no-load and short-circuit tests the parameters of the characteristic triangle can be determined, the sides of which are proportional to \(i_{yv}\).

3. Carrying out the construction of the characteristic triangle for the point \(K\) and determining the value of the emf equal to \(\varphi(i_{vv1})\) (Fig. 5), one can compute from (1) the value \(i_{yv1}\):

\[ i_{yv1}=\frac{\varphi(i_{vv1})-r_{vv}i_{vv1}}{r_{yv}}. \]

1. The found values \(i_{vv1}\) and \(i_{yv1}\) make it possible to compute from (3) the value \(i_{vzk}\):

\[ i_{vzk}=i_{yv1}-i_{vv1}. \]

1. Since during the “jump” \(i_{vzk}=\mathrm{const}\), for determining \(i_{vv2}\) one may use (7) in the form:

\[ i_{vzk}\cdot r_{yv} =\varphi(i_{vv1})-(r_{yv}+r_{vv})i_{vv1} =\varphi(i_{vv2})-(r_{yv}+r_{vv})\cdot i_{vv2} =\mathrm{const}. \tag{19} \]

Since the left-hand side of (19) is known, \(i_{vv2}\) is determined by inscribing the segment \(i_{vzk}\cdot r_{yv}\) between the curve \(\varphi(i_{vv})\) and the straight line \(i_{vv}(r_{yv}+r_{vv})\). In order to take account of the armature reaction, one should proceed in accordance with Fig. 6, shifting the characteristic triangle into the position in which the ordinate of its vertex \(B\) becomes equal to the length of the segment

\[ EM=EN+NM, \]

where \(EN\) is the left-hand side of (19), laid off on the corresponding scale, and \(NM\) expresses, on the same scale, the quantity \((r_{yv}+r_{vv})i_{vv2}\). Then the vertex \(D\) of the triangle, or the point \(M\), gives the value of the current \(i_{vv2}\).

1. Finally, the value of the current \(i_{yv2}\) can be found from (3) as

\[ i_{yv2}=i_{vzk}+i_{vv2}. \]

Taking into account all that has been said above, the changes of the quantities under consideration as functions of time may be qualitatively represented by Fig. 7.

Let us further represent these oscillations in the plane \(i_{yv}, U_v\). Since the curve \(U_v=f_1(i_{yv})\) is an external characteristic, we shall first construct it in all quadrants, using the characteristic triangle and the hysteresis loop (Fig. 5). Bearing in mind that the vertex \(D\) of the triangle must always lie on the shunt characteristic, and the vertex \(B\) on that branch of the hysteresis loop along which the process of remagnetizing the machine takes place, it is not difficult to see that the external characteristic \(U_v=f_1(i_{yv})\) will have the form shown in Fig. 8.

Plotting on it the points corresponding to the values of \(i_{yv}\) before the “jump” and after it, we conclude that the branches of the external characteristic shown by the dashed line are inoperative, while the oscillations and “jumps” have the definite direction indicated by the arrows. The closed curve \(AKA_1K_1\) gives an idea of the limiting cycle of self-oscillations in the plane \(i_{yv}, U_v\).

Fig. 6.

Visible labels in the diagram: \(e_b\), \(i_{vb}\), \(O\), \(K\), \(B\), \(C\), \(D\), \(M\), \(N\), \(a\), \(\alpha\), \(\beta\), \(i_{vb1}\), \(i_{vb2}\), \(i_{vb}(z_{ve}+z_{ov})\), \(i_{vk}z_{ov}\), “shunt characteristic.”

Fig. 7.

Visible labels in the diagram: \(i_{vb}\), \(i_{vz}\), \(i_{vb\,\max}\), \(i_{vz}\), \(i_{vb1}\), \(i_{vb2}\), \(i_{vk}\), \(t\).

Fig. 8.

Visible labels in the diagram: \(+U_b\), \(-i_{gz}\), \(-i_{g1}\), \(-i_{g2}\), \(O\), \(-i_{g2}^{0}\), \(+i_{g2}\), \(+i_{g1}\), \(+i_g\), \(U_b=f(i_{gb})\), \(A\), \(K\), \(K_1\), \(A_1\), \(i_g\).

Oscillations for \(L_{yav}\ne 0,\ L_{vv}\ne 0,\ L_{v2}\ne 0\)

In this case the calculation scheme remains the same (Fig. 3), but equations (1) and (2), taking account of \(L_{yav}\) and \(L_{vv}\), are written differently:

\[ \varphi(i_{vv})=r_{yav}i_{yav}+r_{vv}i_{vv}+L_{yav}\frac{di_{yav}}{dt}+L_{vv}\frac{di_{vv}}{dt}, \tag{20} \]

\[ \varphi(i_{vv})=r_{yav}i_{yav}+r_{v2}i_{v2}+L_{yav}\frac{di_{yav}}{dt}+L_{v2}\frac{di_{v2}}{dt}. \tag{21} \]

Substituting (3) into (20) and (21), we obtain, respectively:

\[ \varphi(i_{vv})=(r_{vv}+r_{yav})i_{vv}+r_{yav}i_{v2}+(L_{vv}+L_{yav})\frac{di_{vv}}{dt}+L_{yav}\frac{di_{v2}}{dt}, \tag{22} \]

\[ \varphi(i_{vv})=(r_{v2}+r_{yav})i_{v2}+r_{yav}i_{vv}+(L_{v2}+L_{yav})\frac{di_{v2}}{dt}+L_{yav}\frac{di_{vv}}{dt}. \tag{23} \]

From (22) we have:

\[ \frac{di_{vv}}{dt} = \frac{\varphi(i_{vv})-(r_{vv}+r_{yav})i_{vv}-r_{yav}i_{v2}-L_{yav}\dfrac{di_{v2}}{dt}} {L_{yav}+L_{vv}}, \tag{24} \]

and, substituting (24) into (23), we shall have:

\[ \frac{di_{v2}}{dt}= \]

\[ = \frac{ L_{vv}\varphi(i_{vv})+(r_{vv}L_{yav}-r_{yav}L_{vv})i_{vv} -\left[r_{v2}(L_{vv}+L_{yav})+r_{yav}L_{vv}\right]i_{v2} } {L_{vv}L_{yav}+L_{v2}L_{yav}+L_{vv}L_{v2}}. \tag{25} \]

In exactly the same way, from (22) we have:

\[ \frac{di_{v2}}{dt} = \frac{ \varphi(i_{vv})-(r_{vv}+r_{yav})i_{vv}-r_{yav}i_{v2} -(L_{vv}+L_{yav})\dfrac{di_{vv}}{dt} } {L_{yav}}, \tag{26} \]

and, substituting (26) into (23), we have:

\[ \frac{di_{vv}}{dt} = \frac{ L_{v2}\varphi(i_{vv})+(r_{v2}L_{yav}-r_{yav}L_{v2})i_{v2} -\left[r_{vv}(L_{v2}+L_{yav})+r_{yav}L_{v2}\right]i_{vv} } {L_{vv}L_{yav}+L_{v2}L_{yav}+L_{vv}L_{v2}}. \tag{27} \]

It follows from (25) and (27) that, when all the inductances of the system are taken into account (Fig. 3), and for finite values of the currents \(i_{vv}\) and \(i_{v2}\), these currents cannot change by a “jump,” since \(\dfrac{di_{v2}}{dt}\) and \(\dfrac{di_{vv}}{dt}\) cannot take values equal to infinity, because the denominators in (25) and (27) cannot become equal to zero.

Since \(i_{vv}\) and \(i_{v2}\) cannot change by “jumps,” then, proceeding from (3), we conclude that \(i_{yav}\) also cannot change by a “jump”; it is not difficult to show that for \(e_v\) and \(U_v\) such changes likewise cannot occur. Thus, for all the quantities under consideration, only continuous periodic solutions are possible. A qualitative analysis in this case presents great difficulties and lies beyond the scope of the present work.

In conclusion, we shall point to Fig. 9, which reproduces the oscillogram of these oscillations for a real system. Comparing Fig. 9 with Fig. 7, we see that the “discontinuous” interpretation describes well, from the qualitative standpoint, the oscillations occurring in the real system.

Fig. 9.

As a result of processing the oscillogram (Fig. 9), the “dynamic external characteristic” of the exciter during the oscillation process was obtained; it is presented in Fig. 10.

Fig. 10.

Comparing it with the closed curve \(AKA_1K_1\) (Fig. 8), we are once again convinced of the advisability of the “discontinuous” interpretation in describing oscillations in the system considered, which has two degrees of freedom.

References

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

2. Gorshkov, V. A. Works of Academician M. V. Shuleikin on electrical machines. Elektrichestvo, No. 5, 1952.

3. Ganzha, 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, Tomsk, 1952.

4. Andronov, A. A. and Khaikin, S. E. Theory of Oscillations, Part One, ONTI, 1937.