The Effect of Changes in Rotational Speed on the Current of a Shock Generator

G. A. Sipailov, A. V. Loos

(Submitted by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)

When a shock generator is short-circuited, or when it is switched on to a load whose resistance is comparable with the resistance of the generator in a supertransient regime, very large torques arise as a result of the surge of the stator current; these torques have a large multiple in relation to the torque of the driving motor. Therefore, for the duration of the pulse the driving motor is disconnected from the network, and the torques that arise are covered only by the kinetic energy of the rotating masses of the rotor. As the stator current increases from zero to its maximum value, the mechanical energy accumulated in the rotating masses of the rotor is transformed into the electromagnetic energy of the load and of the leakage fields of the generator. The magnitude of the converted kinetic energy of the rotor is [1]:

\[ W=\frac{1}{2}I(\omega^{2}-\omega_{2}^{2})=T-\frac{1}{2}I\omega_{2}^{2} \tag{1} \]

where \(\omega\) is the angular velocity of the generator rotor at no load;

\(\omega_{2}\) is the angular velocity of the generator rotor at the maximum value of the stator current;

\[ I=\sum mr^{2} \]

is the moment of inertia of the rotor;

\[ T=\frac{1}{2}I\omega^{2}=\frac{\pi^{3}}{16}\gamma lD_{p}^{4}\left(\frac{f}{p}\right)^{2} \]

is the total reserve of kinetic energy of the rotor of the shock generator.

From expression (1) one can find the angular velocity of the rotor:

\[ \omega_{2}=\omega\sqrt{1-\frac{W}{T}}, \tag{2} \]

where

\[ W\sim\frac{E_{m}^{2}}{x_{\mathrm{ud}}}\int_{0}^{t}\sin\omega t \left( e^{-\frac{t}{T_{a}}} - \frac{1+e^{-\frac{t}{T_{d}^{\prime\prime}}}}{2} \right) \cos\omega t\,dt \]

is the amount of kinetic energy of the rotor converted, by the time \(t\), into electromagnetic energy.

As the stator current decreases from its maximum value to zero, the reverse conversion of electromagnetic energy into

kinetic energy, which causes an increase in the rotational speed of the rotor. However, this entire process takes place with the dissipation of energy in the active resistances of the load and of the generator windings; therefore, after the generator is disconnected at the moment when the current passes through zero, the rotor rotational speed will be less than the initial one.

The changes in rotational speed that occur during operation of a shock generator affect the generator parameters, e.g., etc. However, the amplitude of the stator current for \(r \ll x_{\text{sh}}\) is practically independent of rotational speed. The shock current is determined by the expression:

\[ I_{\text{sh}}=\frac{2E_m}{x_{\text{sh}}}k_z, \tag{3} \]

where the numerator and denominator are directly proportional to the rotational speed \(\omega\); therefore \(I_{\text{sh}}=\mathrm{const}\), independently of the rotational speed. A decrease in rotational speed leads to a decrease in the maximum power

\[ P_{\max}=I_{\text{sh}}E_m k_z, \]

since \(E_m=\omega\), while \(I_{\text{sh}}=\mathrm{const}\); however, it does not affect the maximum value of the electromagnetic energy:

\[ W_{\max}=\frac{P_{\max}}{\omega}=\frac{I_{\text{sh}}E_{\max}k_z}{\omega}. \tag{4} \]

Thus, from general physical considerations it has been established that a change in rotational speed does not affect the magnitude of the energy delivered to the load; only the rate of its delivery and the shape of the current curve change.

For a more accurate investigation of the influence of speed variation on the current of a shock generator, it is necessary to solve jointly the voltage-balance equations of the machine circuits and the equation of motion of the rotor. These investigations were carried out on an analog computer by the method set forth in [2]. In the modeling, the following parameters of the shock generator were adopted in relative units: \(x_a=1\); \(x_{s1}=0.05\); \(x_{Dd}=x_{Dq}=1.03\); \(x_B=1.1\); \(r_c=0.005\); \(r_B=0.02\); \(r_{Dd}=r_{Dq}=0.01\); \(T_1=6\) sec.

Fig. 1. Oscillogram of the solution on an analog computer.

Figure 1 shows an oscillogram of the solution for the given parameter values. It is convenient to compare the results of the solution obtained for different parameter values by the maximum-

... value of the current \(i_d\), since the phase current, determined by expression (5),

\[ i = i_d \cos \theta + i_q \sin \theta , \tag{5} \]

at the instant when the rotor has turned by \(180^\circ\) from the initial short-circuit position, as is evident from (5) and from the oscillogram (Fig. 1), is determined only by the current \(i_d\).

The increase in the stator current corresponds to a decrease in the rotational speed of the rotor, which, as was clarified above, is caused by the conversion of the kinetic energy of the rotor into the electromagnetic energy of the fields of the generator windings and of the load. With further rotation of the rotor, the magnitude of the slip decreases, which is caused by the reverse conversion of the electromagnetic energy of the fields into the kinetic energy of the rotating rotor. From the smallest value of the slip at \(\Theta = 2\pi\), one can find that part of the losses during the pulse which corresponds to the decrease in the kinetic energy of the rotor.

If the stator winding is not opened when the current passes through zero, then the process considered—the conversion of kinetic energy into electromagnetic energy—will continue, with gradual braking of the rotor. The manner in which the speed changes is shown in Fig. 1.

We carried out investigations of the influence of the rotor inertia constant \(T_I\) on the stator current for the following values: 2 sec, 6 sec, 10 sec, \(\infty\).

Fig. 2. Stator current for various values of \(T_I\).

Figure 2 presents, for comparison, the stator-current curves for the following cases:

a) the stator current without taking into account the change in rotational speed, which corresponds to the value \(T_I = \infty\);

b) the stator current taking into account the change in rotational speed for a rotor inertia constant \(T_I = 6\) sec;

c) the stator current taking into account the change in rotational speed for a rotor inertia constant \(T_I = 2\) sec.

The stator current is determined from expression (5) using the direct-axis \(i_d\) and quadrature-axis \(i_q\) components of the stator current. Comparison of these curves clearly makes it possible to estimate the influence of the change in speed on the character of the stator current. The magnitude of the rotor inertia constant \(T_I\) does not exert...

has no effect on the maximum value of the current, but changes only the shape of the current curve.

Therefore, in all cases in which it is necessary to know accurately the shape of the current curve, the duration of the rising part, and the negative half-wave of the current—for example, in selecting and calculating arc-free ion-mechanical devices—it is necessary to take into account the change in rotational speed. However, in most practical problems connected with determining the maximum energy transferred to the load, calculating various operating circuits of a shock generator, etc., the problem can be substantially simplified by not considering the change in rotational speed. Such an assumption does not disturb the qualitative picture of the process under consideration.

REFERENCES

1. M. P. Kostenko. Electrical Machines. Special Part, GEI, 1949.

2. A. V. Loos, G. A. Sipailov. Mathematical modeling of a shock generator with allowance for saturation and changes in rotational speed. Izvestiya TPI, vol. 160, 1966.