ENERGY CHARACTERISTICS OF AN ASYNCHRONOUS GENERATOR OPERATING IN PULSE MODE WITH AN ACTIVE LOAD

A. V. LOOS, Yu. A. ROMANOV, V. F. SERGEEV, G. A. SIPAILOV

Studies of the operation of an asynchronous generator in pulse mode with active and inductive loads, carried out on an experimental installation and on an analog computer using the complete system of differential equations, showed that it is advisable to analyze the operation of the machine without taking into account the exciting capacitance according to an equivalent circuit consisting of series-connected sinusoidal emf \(E_m \sin \omega t\), active resistance \(R_a\), and generator inductive reactance \(x'_a\) [1, 2].

The expression for the current in the general case can be found as

\[ i_{\text{н}} = \frac{E_m}{\sqrt{(R_a+R_{\text{н}})^2+(x'_a+x_{\text{н}})^2}} \left[ \sin(\omega t-\varphi)+\sin\varphi \cdot e^{-\frac{t}{T'_a}} \right], \tag{1} \]

where

\[ \varphi=\operatorname{arctg}\frac{x'_a+x_{\text{н}}}{R_a+R_{\text{н}}}, \qquad T'_a=\frac{x'_a+x_{\text{н}}}{\omega [R_a+R_{\text{н}}]} . \]

One of the most important tasks in studying the operation of an asynchronous surge generator into a load is to determine the matched resistance when the load is active. Analysis of equation (1) and experimental investigations showed that the expression for the current can be approximated by the following function:

\[ i_{\text{н}} = \frac{E_m}{\sqrt{(x'_a)^2+(R_a+R_{\text{н}})^2}} \sin 0.835\,\omega t . \tag{2} \]

The instantaneous power in the load \(R_{\text{н}}\) is:

\[ P_{\text{н}}=i_{\text{н}}^2 R_{\text{н}} = \frac{E_m^2 R_{\text{н}}}{(x'_a)^2+(R_a+R_{\text{н}})^2} \sin^2 0.835\,\omega t . \tag{3} \]

We determine the optimum load resistance from the condition of transferring maximum power to it:

\[ \frac{\partial P_{\text{н}}}{\partial R_{\text{н}}}=0 . \tag{4} \]

The solution of equation (4) with respect to \(R_{\text{н}}\) has the form:

\[ R_{\text{н}}=\sqrt{R_a^2+(x'_a)^2}=z_a . \tag{5} \]

Consequently, when the load resistance is equal to the total resistance of the generator, the surge power has the maximum value:

\[ P_{\text{sur}}=\frac{E_m^2\cdot \kappa_E^2}{2z'_a}, \tag{6} \]

where \(K_E\) is the damping coefficient of the e.m.f.

The rated power of a single-phase machine is

\[ P_1=U_1\cdot I_1=0.5\,U_{m1}\cdot I_{m1}. \tag{7} \]

The ratio of the surge and rated powers of the machine is

\[ \kappa_p=\frac{P_{\text{sur}}}{P_1} =\frac{E_m^2\cdot \kappa_E^2}{2z'_a}\cdot \frac{1}{0.5\,U_{m1}\cdot I_{m1}}. \tag{8} \]

Taking \(U_{m1}=E_m\), which is entirely permissible, we obtain:

\[ \kappa_p=\kappa_E^2\cdot \kappa_{\text{sur}}, \tag{9} \]

where \(\kappa_{\text{sur}}=\dfrac{1}{z_a}\) is the surge-current multiple.

Thus, the relation between the rated and surge power is established by the expression

\[ P_{\text{sur}}=\kappa_p\cdot P_1=\frac{\kappa_E^2}{z_a}\,P_1. \tag{10} \]

Let us compare the energy characteristics of an asynchronous generator operating into active and inductive loads.

In the case of a sudden short circuit at the stator terminals, the surge current, taking into account its aperiodic component, can be expressed as

\[ I_{\text{sur}}=\frac{2E_m\cdot \kappa_E}{x'_a}. \tag{11} \]

When a matched load is connected and the capacitance is sufficiently small, the maximum surge of the load current can be determined from the relation

\[ I_{\text{l}}=\frac{E_m\cdot \kappa_E}{x'_a}, \tag{12} \]

in which case the load energy is calculated from the expression

\[ W_L=0.5L_{\text{l}}\cdot I_{\text{l}}^2 =\frac{0.5E_m^2\cdot \kappa_E^2}{\omega x'_a}. \tag{13} \]

The energy transmitted to the active load during the current pulse is equal to:

\[ W_R=\int_{0}^{\frac{\pi+\varphi}{\omega}} u\,I\cdot i\cdot dt =\frac{E_m^2\cdot R_{\text{l}}\cdot 0.6\pi} {\left[(x'_a)^2+(R_{\text{l}}+R_a)^2\right]\cdot \omega}, \tag{14} \]

From the relations obtained, we determine the energy multiple transmitted by the generator to the active and inductive loads:

\[ \kappa_{\text{l}}=\frac{W_R}{W_L} =\frac{R_{\text{l}}\cdot x'_a}{(x'_a)^2+(R_{\text{l}}+R_a)^2} \cdot \frac{1.2\pi}{\kappa_E^2\cdot \kappa_{\text{sur}}^2}. \tag{15} \]

As analysis of expression (15) shows, the possible values of \(\kappa_{\text{l}}\) lie within the range \((2 \div 2.5)\). Thus, with an active load, utilization of the generator is twice as high as with an inductive load.

The theoretical studies carried out are in good agreement with the experimental results. The tested asynchronous generator had the following parameters in relative units:

\[ x_a = 1;\qquad x'_a = 0.047;\qquad R_a = 0.026;\qquad x_{\sigma s} = 0.035;\qquad T_r = 56.5\ \text{rad}. \]

Figure 1 shows the dependences of the energy in active and inductive loads for various values of its resistance. From a comparison of curves 1 and 2, corresponding to the active (1) and inductive (2) loads, it is evident that in the active load the energy is twice as large as in the inductive load.

Fig. 1

a)                Fig. 2                b)

Figure 2 shows oscillograms of the generator voltage and current in active (a) and inductive (b) loads.

Thus, the experimental data are in agreement with the theoretical conclusions. The expressions obtained for the current, power, and energy of an asynchronous generator make it possible to determine the rated capacity of the machine for producing the required energy in the load.

REFERENCES

1. Yu. A. Romanov, G. A. Sipailov. Analysis of the operation of an asynchronous impact generator using an analog computer. Izvestiya TPI, vol. 212, 1970.

2. Yu. A. Romanov, G. A. Sipailov, A. V. Loos, V. F. Sergeev. Pulse-mode operation of an asynchronous generator with an active load. Izvestiya TPI, the present collection.