ANALYSIS OF THE OPERATION OF AN ASYNCHRONOUS IMPACT GENERATOR USING AN ANALOG COMPUTER

Yu. A. Romanov, G. A. Sipailov

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

The results of studies of the transient processes of an impact generator (IG) have been reflected in a number of works [1, 2]. From the published works it follows that the authors’ efforts were directed mainly toward the study of the transient processes of an IG created on the basis of synchronous machines. In view of the prospects for using asynchronous machines as storage devices and sources of large energies, the study of the transient processes of an asynchronous IG is of interest.

The difficulty of analytical calculations of electromechanical transient processes in electrical machines is due to the need to solve a system of nonlinear differential equations. The use of computing technology for solving nonlinear problems has become the principal means of theoretical investigation of transient modes in electrical machines.

The aim of the present work is to obtain the equations of a single-phase asynchronous machine in generator mode with capacitor excitation, taking into account saturation along the path of the main flux of the machine, in a form convenient for solution on analog computing machines (ACM).

In contrast to work [3], in which the operation of a synchronous IG is analyzed by classical methods using the Park–Gorev equations, in the present work the analysis is carried out under the assumption that a single-phase asynchronous machine is a special case of a generalized two-phase machine in the coordinate axes \(\alpha\) and \(\beta\) [4].

In modeling, the following assumptions were adopted for the generalized two-phase machine:

1) the stator has two phases \(\alpha, \beta\), displaced by 90 electrical degrees;

2) the rotor has a short-circuited winding, which is reduced to a system of two circuits \(d, q\) with perpendicular axes;

3) the electromagnetic torque \(M_{\psi}\) is taken as positive in the mode of operation of the machine as a generator;

4) the inductive and active resistances of the windings are constant;

5) iron losses are absent.

Equations of a Single-Phase Asynchronous Machine Taking into Account Saturation Along the Path of the Main Magnetic Flux

An impact asynchronous generator can be used as an energy storage device under pulse supply according to the circuit of Fig. 1. The load

Fig. 1. Operating circuit of an asynchronous impact generator with an inductive load

is connected to the stator winding through switches 1B and 2B. Excitation of the generator is provided by the capacitance \(C\).

For analysis of the operation of the circuit (Fig. 1), the fundamental equations of a two-phase asynchronous machine in the \(\alpha\) and \(\beta\) axes should be used. We take the \(\beta\) axis to lead the \(\alpha\) axis in the direction of rotor rotation. Then the equations of the asynchronous machine are written in the following form [5]:

\[ \begin{aligned} p\Psi_\alpha + i_\alpha R_\alpha &= -u_\alpha, & \Psi_\alpha &= x_d i_\alpha + x_{ad} i_{\alpha r}, \\ p\Psi_\beta + i_\beta R_\beta &= -u_\beta, & \Psi_\beta &= x_d i_\beta + x_{ad} i_{\beta r}, \\ p\Psi_{\alpha r} + \Psi_{\beta r}\omega_r + i_{\alpha r}R_r &= 0, & \Psi_{\alpha r} &= x_r i_{\alpha r} + x_{ad} i_\alpha, \\ p\Psi_{\beta r} - \Psi_{\alpha r}\omega_r + i_{\beta r}R_r &= 0, & \Psi_{\beta r} &= x_r i_{\beta r} + x_{ad} i_\beta, \\ M_\psi &= i_\alpha \Psi_\beta - i_\beta \Psi_\alpha, \\ p\omega_r &= \frac{1}{H_j}(M_m - M_\psi), \end{aligned} \tag{1} \]

where \(\omega_r\) is the angular frequency of rotor rotation.

Here all quantities are expressed in per-unit values; the notation corresponds to that adopted in [5, 6].

It is known that transformation of the differential equations of a two-phase machine in phase coordinates by the two-reaction method to the stator axes frees the latter from harmonic functions under the condition of complete symmetry of the rotor parameters \((x_{dr}=x_{qr}=x_r;\ R_{dr}=R_{qr}=R_r)\). At the same time, no special requirements are imposed on the stator parameters \((x_{\alpha s}; x_{\beta s}; R_{\alpha s}; R_{\beta s})\),

where

\(x_{dr}; x_{qr}; R_{dr}; R_{qr}\) are the inductive and active resistances of the rotor circuits in phase coordinates;

\(x_{\alpha s}; x_{\beta s}; R_{\alpha s}; R_{\beta s}\) are the leakage inductive resistances and active resistances of the stator circuits.

Using the property of this method, we exclude the second equation from system (1) and, taking \(i_\beta=0\), obtain the complete system of differential equations of a single-phase asynchronous machine:

\[ \begin{aligned} p\Psi_\alpha + i_\alpha R_\alpha &= -u_\alpha, & \Psi_\alpha &= x_{\sigma s} i_\alpha + \Psi_{\delta\alpha}, \\ p\Psi_{\alpha r} + \Psi_{\beta r}\omega_r + i_{\alpha r}R_r &= 0, & \Psi_\beta &= \Psi_{\delta\beta}, \\ p\Psi_{\beta r} - \Psi_{\alpha r}\omega_r + i_{\beta r}R_r &= 0, & \Psi_{\alpha r} &= x_{\sigma r}\cdot i_{\alpha r} + \Psi_{\delta\alpha}, \end{aligned} \tag{2} \]

\[ p\omega_r = \frac{1}{H_j}\left(M_m - i_\alpha \Psi_{\delta\beta}\right), \tag{2} \]

\[ \Psi_{\beta r} = x_{\sigma r}\cdot i_{\beta r} + \Psi_{\sigma\beta}, \]

where

\[ \Psi_{\delta\alpha} = x_{ad}(i_\alpha + i_{\alpha r}) - \Delta\Psi_{\delta\alpha}, \]

\[ \Psi_{\delta\beta} = x_{ad}i_{\beta r} - \Delta\Psi_{\delta\beta} \]

are the flux linkages in the air gap along the \(\alpha\) and \(\beta\) axes of the machine, taking into account saturation along the path of the main flux; \(x_{ad}\) is the stator reaction inductive reactance of the unsaturated machine. Saturation is taken into account by introducing the additional \(\Delta\Psi_{\delta\alpha}\), \(\Delta\Psi_{\delta\beta}\) into the expression for the flux linkages. Dividing the values \(\Delta\Psi_{\delta\alpha}\), \(\Delta\Psi_{\delta\beta}\) by \(x_{ad}\), we obtain the values of the additional current \(\Delta i\) required to compensate for the influence of saturation.

Mathematical Modeling of the Transient Processes of an Asynchronous UG

To analyze the operation of the circuit in Fig. 1, the above equations of the single-phase machine should be supplemented by the equilibrium equations for the voltages and currents of the generator external-circuit loops:

\[ u_\alpha = u_c = \frac{1}{C}\int i_c\,d\tau, \]

\[ i_H = \frac{1}{L_H}\int u_c\,d\tau, \tag{3} \]

\[ i_c = i_\alpha - i_H, \]

where \(i_\alpha\), \(i_c\), \(i_H\) are the currents of the generator, capacitance, and load.

The complete system of equations (2)—(3) should be represented in a form convenient for investigating transient processes with the use of computing machines:

\[ p\Psi_\alpha = -i_\alpha R_a - u_c, \]

\[ p\Psi_{\alpha r} = -i_{\alpha r}R_r - \omega_r\Psi_{\beta r}, \]

\[ p\Psi_{\beta r} = -i_{\beta r}R_r + \omega_r\Psi_{\alpha r}, \]

\[ pu_c = x_c i_c, \]

\[ pi_H = \frac{1}{x_H}\cdot u_c, \]

\[ p\omega_r = \frac{1}{H_j}\left(M_m - i_\alpha\Psi_{\delta\beta}\right), \]

\[ i_c = i_\alpha - i_H, \tag{4} \]

\[ i_\alpha = \frac{1}{x_{ad}}\cdot \Psi_{\delta\alpha} + \frac{1}{x_{ad}}\cdot \Delta\Psi_{\delta\alpha} - i_{\alpha r}, \]

\[ i_{\alpha r} = \frac{1}{x_{\sigma r}}\left(\Psi_{\alpha r} - \Psi_{\delta\alpha}\right), \]

\[ i_{\beta r} = \frac{1}{x_{ad}}\cdot \Psi_{\delta\beta} + \frac{1}{x_{ad}}\cdot \Psi_{\sigma\beta}, \]

\[ \Psi_{\delta\alpha}=\Psi_{\alpha}-x_{\sigma s} i_{\alpha}, \]

\[ \Psi_{\delta\beta}=\Psi_{\beta r}-x_{\sigma r}\cdot i_{\beta r}, \]

\[ \Psi_{\delta}=\sqrt{\Psi_{\delta\alpha}^{2}+\Psi_{\delta\beta}^{2}}, \]

where

\[ \frac{1}{x_{ad}}\cdot\Psi_{\delta\alpha}=\Delta i_{\alpha},\qquad \frac{1}{x_{ad}}\cdot\Delta\Psi_{\delta\beta}=\Delta i_{\beta}. \]

The equations are written in relative units. The nominal phase values and the synchronous speed are taken as the base quantities.

The summing integrators, variable-multiplication blocks, and solving elements that implement the operations required for solving equation (4) are shown in Fig. 2a. The expressions for the additional currents

Fig. 2a. Modeling of a single-phase asynchronous generator. Structural diagram of the model.

\(\Delta i_{\alpha}\) and \(\Delta i_{\beta}\) are implemented in the structural diagram as follows. The flux linkages \(\Psi_{\delta\alpha}\) and \(\Psi_{\delta\beta}\) are taken from the outputs of the corresponding summers and, by means of functional converter 15N, the total flux linkage in the air gap is obtained. By means of the second functional converter, 25N, the dependence of the additional conductance of the magnetizing branch, caused by saturation, is formed. Figure 2b shows the obtaining of the required dependence. Curve 3 gives the additional current arising due to saturation. Curve 4 reflects the dependence of the additional conductance \(\Delta i/\Psi_{\delta}\) on the air-gap flux linkage \(\Psi_{\delta}\). Multiplying the components of the flux linkages along the axes \((\Psi_{\delta\alpha}\) and \(\Psi_{\delta\beta})\) by the quantity \(\Delta i/\Psi_{\delta}\), we obtain the values of the additional currents \((\Delta i_{\alpha}\) and \(\Delta i_{\beta})\), which we feed to the inputs of the summers.

Fig. 2b. Determination of the characteristic of the 2BN functional converter. 1 — no-load characteristic; 2 — magnetization curve of the unsaturated machine; 3 — additional magnetizing current caused by saturation; 4 — functional dependence of the 2BN converter.

In general, the computing elements shown in Fig. 2a constitute a circuit for solving the system of equations (4). By changing the coefficients at the outputs of the corresponding amplifiers, one can investigate the operation of the impact generator for different parameter values.

The circuit shown in Fig. 2a makes it possible to investigate: the operation of a single-phase asynchronous machine in the self-excitation mode; the influence of the machine parameters on the rate of excitation; the short-circuit mode of an asynchronous generator; the influence of the capacitance and of the parameters of the stator and rotor circuits on the energy delivered to the load, etc.

Figure 3 presents the results of calculating the differential equations (4) for the self-excitation mode of a single-phase asynchronous generator with the following parameters, in relative units:

Fig. 3. Comparison of experimental and calculated data for the degree of variation of the generator voltage during self-excitation

\[ x_c = 0.525;\quad R_s = 0.13;\quad x_{ad} = 1;\quad R_r = 0.023; \]

\[ x_{\sigma s} = 0.253;\quad x_{\sigma r} = 0.107;\quad \left(U_B = 750\ \mathrm{V};\ I_B = 54.4\ \mathrm{A};\ \omega_s = 628\ \frac{1}{\mathrm{sec}}\right) \]

The calculations were carried out on an MN-14 analog computer. For comparison, the results of an experimental study of the self-excitation regime on a prototype asynchronous impact generator with the parameters for which the calculations were performed are also presented there. A comparison of the experimental results with the calculation confirms the sufficient accuracy of the solutions of equations (4).

REFERENCES

1. V. V. Kaplan, V. M. Nashatyr. On the question of the application of electrical-machine energy storage devices for physical research. Elektrichestvo, 1966, No. 4.

2. A. V. Loos, G. A. Sipailov. Mathematical modeling of the electromagnetic processes of an impact generator. Izvestiya TPI, vol. 152, 1965.

3. A. V. Loos, G. A. Sipailov. Mathematical modeling of an impact generator taking account of saturation and changes in rotational speed. Izvestiya TPI, vol. 160, 1966.

4. I. M. Postnikov. Generalized theory and transient processes of electrical machines. Tekhnika, 1966.

5. L. M. Gruzov. Methods of mathematical investigation of electrical machines. Gosenergoizdat, 1953.

6. A. I. Vazhnov. Fundamentals of the theory of transient processes of a synchronous machine. Gosenergoizdat, 1960.