PROCEEDINGS
OF THE TOMSK ORDER OF THE OCTOBER REVOLUTION
AND ORDER OF THE RED BANNER OF LABOR
POLYTECHNIC INSTITUTE named after S. M. KIROV

Volume 284

1974

SOME FEATURES OF MODELING THE PULSED OPERATING MODE OF A THREE-PHASE INDUCTION GENERATOR

G. A. SIPAILOV, Yu. A. ROMANOV, M. A. SUTORMIN, Yu. I. PEREZHIROV

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

Studies of the transient processes of an induction impact generator have been reflected in a number of articles [1, 2]. A drawback of the operation of a single-phase induction generator (IG) in the pulsed mode is the fact that, with a large duty cycle of the operating pulses, significant no-load power losses occur due to the electrical asymmetry of the generator stator windings. These power losses, caused by a counter-rotating field in a single-phase induction machine, increase as the saturation of the magnetic circuit increases. Therefore it appears advisable to carry out self-excitation of a pulsed induction generator on the three phases of the stator.

Fig. 1. Schematic circuit diagram of the operation of a three-phase pulsed induction generator

An induction generator may be used as a pulsed power source according to the circuit of Fig. 1.

At the first stage, the rotor of the induction machine is brought into rotation at nominal speed. At the second stage, self-excitation of the machine is carried out, for which switch Л is closed. Then, as the generator voltage increases, switch 1Л is closed, connecting the load to the generator for the duration of the positive half-wave of the current.

To construct the generator model, it is necessary to combine the system of equations of a generalized two-phase machine with the equations of the external circuits of the three-phase machine (Fig. 1).

In the case when the induction machine operates in generator mode with a single-phase load, differential equations can be applied both in the axes \(d, q, o\) and in \(\alpha, \beta, o\).

Analysis showed that in the latter case the number of equations is reduced and the block diagram for solving the generator equations is simplified. To analyze the operation of the circuit in Fig. 1, the equations of the IG should be supplemented by the equilibrium equations for the voltages and currents of the external-circuit loops:

\[ \left. \begin{aligned} u_{\text{н}} &= u_a - u_b,\\ i_a - C_a \frac{d u_a}{dt} - i_{\text{н}} &= 0,\\ i_b - C_b \frac{d u_b}{dt} + i_{\text{н}} &= 0,\\ i_c - C_c \frac{d u_c}{dt}, \end{aligned} \right\} \tag{1} \]

where

\(i_a, i_b, i_c\) and \(i_{\text{н}}\) are the generator and load currents;

\(u_a, u_b, u_c\) and \(u_{\text{н}}\) are the voltages across the excitation capacitors and the load.

The load current when active resistance is connected is determined from the expression

\[ i_{\text{н}} = \frac{u_{\text{н}}}{R_{\text{н}}}, \]

and when inductance is connected,

\[ i_{\text{н}} = \frac{1}{L_{\text{н}}} \int u_{\text{н}}\, dt. \]

With equal phase parameters of the three-phase IG and equal values of the excitation capacitances in the phases, the rms values of the generator phase output voltage are equal to the rms values of the corresponding voltages across the capacitors for any values of the load resistance. We shall prove the latter proposition using the classical method of nodal voltages. Let us show that the neutral points of the generator and of the excitation capacitors have equal potentials \(U_0 = U_1\).

\[ \left. \begin{aligned} \dot U_1\left(2\frac{1}{Z_c}+\frac{1}{Z_c+Z_r}\right) -\dot U_2\frac{1}{Z_c} -\dot U_3\frac{1}{Z_c} &=\dot E_c\frac{1}{(Z_r+Z_c)},\\ -\dot U_1\frac{1}{Z_c} +\dot U_2\left(\frac{1}{Z_n}+\frac{1}{Z_c}+\frac{1}{Z_r}\right) -\dot U_3\frac{1}{Z_n} &=\dot E_b\frac{1}{Z_r},\\ -\dot U_1\frac{1}{Z_c} -\dot U_2\frac{1}{Z_n} +\dot U_3\left(\frac{1}{Z_r}+\frac{1}{Z_c}+\frac{1}{Z_n}\right) &=\dot E_a\frac{1}{Z_r}, \end{aligned} \right\} \tag{2} \]

where

\[ \dot U_1,\dot U_2,\dot U_3 \]

are the potentials of nodes 1, 2, and 3 of the circuit in Fig. 1;

\[ Z_n,\ Z_r,\ Z_c \]

are the total impedances of the load, the generator phases, and the exciting capacitance;

\[ \dot E_a,\dot E_b,\dot E_c \]

are the complexes of the generator phase e.m.f.

To prove the equality \(U_0=U_1\), it is necessary that \(U_1\) [[unclear: superscript expression]] be equal to zero.

Forming the determinant of system (2),

\[ \Delta_1= \left| \begin{array}{ccc} \dot E_c\dfrac{1}{(Z_r+Z_c)} & \left(-\dfrac{1}{Z_c}\right) & \left(-\dfrac{1}{Z_c}\right) \\[1.2em] \dot E_b\dfrac{1}{Z_r} & \dfrac{1}{Z_n}+\dfrac{1}{Z_c}+\dfrac{1}{Z_r} & -\dfrac{1}{Z_n} \\[1.2em] \dot E_a\dfrac{1}{Z_r} & \left(-\dfrac{1}{Z_n}\right) & \left(\dfrac{1}{Z_r}+\dfrac{1}{Z_c}+\dfrac{1}{Z_n}\right) \end{array} \right| \]

and expanding it, after the simplest transformations we find \(\Delta_1=0\).

It follows from this that the rms values of the voltages and e.m.f. are equal:

\[ \dot E_a=\dot U_a;\qquad E_b=\dot U_b;\qquad E_c=\dot U_c. \]

However, the analysis presented is valid for the steady-state operating mode of the generator, when the e.m.f. moduli and the parameters of the three generator phases are equal. In the transient process this equality may be violated.

a                  b

Fig. 2. Oscillograms of the measurement of the voltage \(U_{01}\) during operation of a three-phase IG under a pulsed load \((i_n)\): a) active load; b) short circuit; c) inductive load

In this connection, on the experimental setup, studies were carried out of the change in the voltage between the neutral points of the generator and the exciting capacitances \(U_{01}\) during the transient process under

operation of the generator under load. Figure 2 shows oscillograms of the variation of \(U_{01}\). It should be noted that the voltage \(U_{01}\) at no load proved not to be equal to zero and amounted to \(0.5\%\) of the rated phase voltage of the generator. During the load-current pulse, the value of the voltage \(U_{01}\) changes, and this change occurs in the direction of decreasing voltage.

Therefore, with full justification it may be assumed that the instantaneous values of the voltages and e.m.f.’s are equal:

\[ e_a = u_a,\qquad e_b = u_b,\qquad e_c = u_c. \]

The last conclusion is very important.

In this case, the number of required equations is reduced and the structural diagram of the IG model is simplified.

c

The complete system of differential equations describing the transient process of a three-phase IG operating in pulsed mode into a single-phase load is reduced to a form convenient for modeling on an analog computer [3]:

the flux linkages are determined from the expressions

\[ \left. \begin{aligned} p\psi_{\alpha} &= -u_{\alpha} - R_{\alpha} i_{\alpha};\\ p\psi_{\beta} &= -u_{\beta} - R_{\beta} i_{\beta};\\ p\psi_{\alpha r} &= \omega' \psi_{\beta r} - R_r i_{\alpha r};\\ p\psi_{\beta r} &= -\omega' \psi_{\alpha r} - R_r i_{\beta r}, \end{aligned} \right\} \tag{3} \]

the flux linkages in the air gap along the axes

\[ \left. \begin{aligned} \psi_{\delta\alpha} &= \psi_{\alpha} - x_{s\sigma} i_{\alpha},\\ \psi_{\delta\beta} &= \psi_{\beta} - x_{s\sigma} i_{\beta}, \end{aligned} \right\} \tag{4} \]

the equations of the machine currents

\[ \left. \begin{aligned} i_{\alpha} &= \frac{\psi_{\delta\alpha}}{x_m} - i_{\alpha r};\\ i_{\beta} &= \frac{\psi_{\delta\beta}}{x_m} - i_{\beta r};\\ i_{\alpha r} &= \frac{1}{x_{\sigma r}}\left(\psi_{\alpha r} - \psi_{\delta\alpha}\right);\\ i_{\beta r} &= \frac{1}{x_{\sigma r}}\left(\psi_{\beta r} - \psi_{\delta\beta}\right). \end{aligned} \right\} \tag{5} \]

From the known values of the stator-current components, from (5) the line currents of the generator phases are determined:

\[ \left. \begin{aligned} i_a &= i_{\alpha};\\ i_b &= -\frac{1}{2}i_{\alpha} + \frac{\sqrt{3}}{2}i_{\beta};\\ i_c &= -\frac{1}{2}i_{\alpha} - \frac{\sqrt{3}}{2}i_{\beta}. \end{aligned} \right\} \tag{6} \]

The zero-sequence component is excluded from the equations, since we consider the case where \(i_a+i_b+i_c=0\), which corresponds to the operation of a three-phase machine with an arbitrary type of asymmetry and an isolated neutral.

From (1), the instantaneous values of the phase voltages across the capacitors are determined in relative units:

\[ \begin{aligned} p u_a &= x_{ca}(i_a-i_{\mathrm{n}});\\ p u_b &= x_{cb}(i_b+i_{\mathrm{n}});\\ p u_c &= x_{cc}i_c . \end{aligned} \tag{7} \]

Fig. 3. Structural diagram for solving the system of equations of a three-phase IG when operating into a single-phase load

Load voltage and current

\[ u_{\mathrm{n}}=u_a-u_b, \]

for an active load resistance

\[ i_{\mathrm{n}}=\frac{u_{\mathrm{n}}}{R_{\mathrm{n}}}, \]

for an inductive resistance

\[ p i_{\mathrm{n}}=\frac{u_{\mathrm{n}}}{x_{\mathrm{n}}}. \]

The stator-voltage components \(u_\alpha\) and \(u_\beta\) are determined from the known values of the voltages across the capacitors:

\[ \left. \begin{aligned} u_\alpha &= \frac{2}{3}\left(u_a-\frac{u_b+u_c}{2}\right),\\ u_\beta &= \frac{1}{\sqrt{3}}(u_b-u_c). \end{aligned} \right\} \tag{8} \]

Summing integrators, solving elements, and elements implementing the algebraic operations required for solving the last equations are shown in Fig. 3. The calculations performed showed stable operation of the model.

Figure 4 presents an oscillogram of the load current obtained from the solution of differential equations (3)—(8) for the pulsed operating mode of the generator with an active load resistance.

Fig. 4. Oscillogram of the load current of a three-phase induction generator for three values of the load resistance:
1 — \(R_{\mathrm{n}} = 0.2\); 2 — \(R_{\mathrm{n}} = 0.05\); 3 — \(R_{\mathrm{n}} = 0.01\)

Thus, the developed model makes it possible to obtain a quantitative assessment of the influence of various generator parameters on the energy characteristics, to investigate the armature reaction, and also the influence of asymmetry of the exciting capacitances on generator operation.

REFERENCES

1. Yu. A. Romanov, G. A. Sipailov, A. V. Loos. Results of studies of the operation of an asynchronous shock generator on an inductive load. Izvestiya TPI, vol. 212, 1971.

2. Yu. A. Romanov, G. A. Sipailov, A. V. Loos. Operation of an asynchronous generator in pulsed mode on an inductive load. Elektrotekhnicheskaya apparatura, vol. 2, 1971.

3. L. M. Gruzov. Methods of Mathematical Investigation of Electrical Machines. Moscow, Gosenergoizdat, 1953.