Influence of Inductance on the Shape of Self-Oscillations in an Electromechanical System

L. I. Gandzha

(Presented at the scientific seminar of the Electromechanical Faculty)

In [1], self-oscillations were considered in an electromechanical system consisting of a separately excited generator feeding a separately excited motor (Fig. 1); in this treatment, the inductance of the power circuit was assumed to be zero, as a result of which the indicated system was described by a first-order nonlinear differential equation. It was shown that, under such an idealization, “discontinuous”

Fig. 1.

oscillations of the current and continuous oscillations of the speed are possible in the system. In general terms, the mechanism of these oscillations reduces to the following: when the current decreases from \(+I_{\max}\) to \(+I_{\mathrm{cr}}\), the generator is demagnetized along the segment \(ak\) of the hysteresis loop (Fig. 2a), while the motor accelerates along the segment \(ak\) of its mechanical characteristic (Fig. 2b), which is a segment of stable operation of the drive. When the system reaches point \(k\), it enters a regime in which stable operation changes to unstable operation; at the same time, the current changes “by a jump” from the value \(+I_{\mathrm{cr}}\) to \(-I_{\max}\) at the constant speed \(n_{\mathrm{cr}}\), and the motor finds itself in a braking regime on the stable segment \(a_1A_1\) of its mechanical characteristic, and then reverses and accelerates in the opposite direction along the segment \(A_1k_1\) of the mechanical characteristic. Point \(k_1\) corresponds to a change of stability, as a result of which the current, at constant speed, changes “by a jump” from \(-I_{\mathrm{cr}}\) to \(+I_{\max}\), and so on.

These “discontinuous” oscillations of the current and continuous oscillations of the speed are shown in Fig. 3. Taking into account the inductance of the system of Fig. 1 leads to the impossibility of the occurrence of “discontinuous” current oscillations, since the current can no longer change “by a jump.” The oscillogram corresponding to this case is shown in Fig. 4; the inductance of the system, correspond-

Fig. 2.

Fig. 3.

Fig. 4.

...corresponding to this oscillogram was determined only by the inductances of the machine elements. Let us consider in more detail the influence of inductance on the character of self-oscillations in the system under consideration.

Fig. 5.

In Fig. 5a and b are shown the hysteresis loop of the generator and the corresponding mechanical characteristic of the motor of the system in Fig. 1 for the case of “discontinuous” oscillations, bounded by the limiting current values \(+I_{\max}\) and \(-I_{\max}\). Let us now introduce into the power circuit of the system under consideration an additional, sufficiently large inductance and, by changing the resistance of the rheostat (Fig. 1), establish the former value \(R\) of the total

the circuit resistance so that the straight line \(I \cdot R\) (Fig. 5a) retains the former angle of inclination; this makes it possible to trace the influence of \(L\) on the form of the self-oscillations of the system, all other conditions being unchanged.

The system under consideration is described by the following equations:

\[ e_z=\psi(I)=R\cdot I+e_L+e_d, \tag{1} \]

\[ M-M_{st}=\frac{GD^2}{375}\frac{dn}{dt}, \tag{2} \]

where

\(e_z,\ e_L,\ e_d\) — the generator emf, the counter-emf of self-induction, and the counter-emf of the motor;

\(M,\ M_{st}\) — the torques of the motor and the static torque applied to the motor shaft, which will henceforth be assumed constant.

\(GD^2\) — the flywheel moment of the motor.

Let us now consider, from the qualitative standpoint, the self-oscillations arising in the system from the moment the motor is connected to the generator. After switching on, the motor will remain motionless until the current reaches the value \(+I_{tr}\), at which the motor starts moving (as is known [3], \(I_{tr}>I_{st}\)). In this case \(n=0\) and \(e_d=0\), and (1) takes the form

\[ e_z=R\cdot I+e_z. \tag{3} \]

As the current increases, \(e_z\) will also begin to increase along the initial magnetization curve \(6a\); in this case \(e_L\) will be directed opposite to the current and \(e_z\), and will be represented as the difference between the ordinates of curve \(6a\) and the straight line \(R\cdot I\). When the current reaches \(I_{tr}\), the motor will begin to accelerate and equation (1) will come into effect; simultaneously with the increase in motor speed, the current continues to increase up to a new maximum value \(+I'_{\max}<+I_{\max}\), since its magnitude is determined by \(e_L\), still directed opposite to \(I\) and \(e_z\) as the current increases up to \(+I'_{\max}\). From Fig. 5a, \(e_d\) is determined as the difference between the ordinates of curve 1–2, which represents the dependence

\[ e_z-e_L=R\cdot I+e_d \tag{4} \]

and the straight line \(R\cdot I\). When the current reaches the value \(+I'_{\max}\), \(e_L\) becomes equal to zero, since at \(I=I'_{\max}\),

\[ \left(\frac{dI}{dt}\right)_{\max}=0. \]

Thus, from (4) we obtain

\[ e_z=R\cdot I+e_d, \tag{5} \]

which corresponds to the position of point 2 on the magnetization curve \(6a\). Beginning from point 2, the motor current begins to decrease; in this case \(e_L\) will be directed in the same direction as \(I\) and \(e_z\), so that we obtain the dependence

\[ e_z+e_L=R\cdot I+e_d, \tag{6} \]

which determines the curve \(2k_2 3\), lying above the curve \(2-5\), along which the demagnetization of the generator will occur in the new regime. As the current decreases from \(+I'_{\max}\), the motor speed will increase in accordance with the equation of motion of the motor

\[ M-M_{st}=\frac{GD^2}{375}\frac{dn}{dt} \tag{7} \]

until it reaches the maximum value \(n_{\max}\), at which

\[ e_0=e_{0\max}=c_e\cdot n_{\max}\quad \text{and}\quad \left(\frac{dn}{dt}\right)_{\max}=0; \]

in this case \(M=M_{st}=c_M\cdot I_{st}\). In Figs. 5a and 6 the maximum speed corresponds to point \(k_2\) on the curve \(2-k_2 3\), which is determined by drawing a tangent to the curve parallel to the straight line \(R\cdot I\). It is not difficult to see that point \(k_2\) of the dynamic characteristic of the system, analogously to point \(k\) of the static characteristic, is the point at which the stability of the system is lost; indeed, since for it

\[ n=n_{\max}\quad \text{and}\quad \left(\frac{dn}{dt}\right)_{\max}=0, \]

the system at point \(k_2\) is in a state of instantaneous steady motion; this means that the system must lie on the static characteristic. Thus, if point \(k_2\) (Fig. 5b) is not situated on the static mechanical characteristic, the motor will begin to brake, tending to enter the section \(k b\) of the static mechanical characteristic. If, however, point \(k_2\) coincides with the section \(k b\), then the system will be in an unstable regime and will tend toward the stable motion corresponding to the section \(a_1 k_1\) of the static characteristic. Thus, in both cases the motor speed will begin to decrease with decreasing motor current. The speed values in this case may be found from the expression

\[ e_0=c_e\cdot n, \tag{8} \]

the left-hand side of which, in accordance with (6), is found as the difference between the ordinates of the curve \(k_2 3\) and the straight line \(R\cdot I\). When the current becomes equal to zero, from (6) we shall have:

\[ e_2+e_L=e_0. \tag{9} \]

With a further decrease of the current, when it begins to assume negative values, from (6) we obtain

\[ e_2+e_L=-R\cdot I+e_0, \tag{10} \]

whence it follows that \(e_2\) and \(e_L\) are directed against the current and that \(e_0\) becomes the source of current. Since the motor \(n\) and \(e_0\) decrease, an increase of the current into the region of negative values will be possible only up to a certain maximum value \(-I''_{\max}\). Thus,

\[ \left|I''_{\max}\right|<\left|I'_{\max}\right|. \tag{11} \]

Indeed, the source of current during its variation in the region of positive values was \(e_z\); in this case an increase of the current up to \(+I'_{\max}\) corresponded to an increase of \(e_z\). The source of current during its variation in the region of negative values is \(e_0\), and an increase of the current up to \(-I''_{\max}\) corresponds not to an increase, but to a decrease of \(e_0\); hence it follows that the absolute values of the maximum currents in the positive and negative regions of values must satisfy inequality (11). With a further decrease of the current from \(-I''_{\max}\) to zero, the speed will continue to decrease, since the negative current will produce a braking torque of the motor; at \(I=0\) the speed will also become equal to 0, and the curve \(3—4—0\), described by equation (10), will arrive at the origin. This will conclude the first period of oscillation of the system with large inductance. During this period the generator, which had an emf from the residual magnetism \(e_0\), will undergo the process of magnetization along the curve \(0—2\) and demagnetization along the curve \(2—5\).

in accordance with the direction indicated by the arrow (point 5 corresponds to \(-I''_{\max}\)). When the current, in the process of decreasing from \(-I''_{\max}\), becomes equal to zero, the generator will have an e.m.f. due to residual magnetism \(e'_0\), under the influence of which the generator will be magnetized along the curve \(5—2\) in the second period of oscillations.

In [1] it was shown that self-oscillations accompanied by reversal of the motor are possible in the system (Fig. 1) in the case where the negative current, in the process of changing in absolute value, can exceed the value of the current \(I_{B1}\), corresponding to the point \(B_1\) of unstable equilibrium of the system. If, however, the system possesses such considerable inductance that the absolute values of the currents \(-I_{B1}\) and \(-I''_{\max}\) satisfy the inequality

\[ |I''_{\max}| < |I_{B1}|, \]

then oscillations arise in this system that are not accompanied by reversal of the motor. Thus, the inductance significantly affects the form of the self-oscillations of the system of Fig. 1 and may cause the occurrence of discontinuous oscillations of the speed. Since the inductance also limits the peak values of the currents, the remagnetization of the generator will occur along a narrowed hysteresis loop. At the same time, equations (1, 3, 4, 5, 6, 9, 10) show that the generator of the system of Fig. 1 may be replaced by a certain equivalent generator whose electromotive force is equal to

\[ e_{\mathrm{eq}} = e_2 \pm e_L. \]

Since \(e_L = L \dfrac{dI}{dt}\), where both \(L\) and \(\dfrac{dI}{dt}\) are functions of the current, \(e_L\) may be represented by a certain function of the current \(e_L = \psi(I)\).

Then

\[ e_{\mathrm{eq}} = \varphi(I) \pm \psi(I) = F(I) \]

and the equations indicated above (1, 3, 4, 5, 6, 9, 10) may be replaced by a single equation of the general form

\[ e_{\mathrm{eq}} = F(I) = R \cdot I + e_d . \tag{12} \]

The function \(F(I)\) is the magnetization curve of the equivalent generator, which in Fig. 5 is represented by the curve \(0—1—2—k_2—3—4—0\). From Fig. 5a and from (12) it follows that the maximum value of the back e.m.f. \(e_{0\max}\) may be determined by drawing a tangent to the curve \(0—1—2—k_2—3—4—0\), parallel to the straight line \(R \cdot I\). This tangent also determines the point \(k_2\), analogous to the point \(k\) for the system with \(L = 0\), allowing discontinuous current oscillations. From all that has been said it follows that the point \(k_2\) is located closer to the ordinate axis than the point \(k\). Thus, the inductance of the circuit of the system of Fig. 1 contributes to increasing its stability at small loads. If it is assumed that the first oscillation arising in the system after switching on the motor corresponds to positive values of the current, then the absolute values of the maximum currents in the odd half-periods will be greater than the absolute values of the maximum currents in the even half-periods of the current curve. In accordance with what has been said, in the odd half-periods of the speed curve the latter will change, passing through a maximum value; in the even half-periods the same speed curve will be equal to zero under the condition \(|I''_{\max}| < |I_{\mathrm{tr}}|\). The oscillogram of the oscillations described is shown in Fig. 6; this oscillogram was taken for the same system as the oscillogram of Fig. 4, but in the series circuit of the armatures of the machines

a considerable inductance was introduced in the form of a coil with an iron core. The remaining experimental conditions were left unchanged, including \(R\).

Fig. 6.

REFERENCES

1. Gandzha L. I. Physics of oscillations in the system “generator with series—motor with independent excitation.” Izvestiya of the Tomsk Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov, vol. 76, 1953.

2. Gandzha L. I. and Potechin Yu. I. “On transient processes and oscillations in the system ‘generator with series—motor with independent excitation.’” Izvestiya of the Tomsk Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov, vol. 76, 1953.

3. Popov V. K. Fundamentals of Electric Drive. Gosenergoizdat, 1945.