Proceedings of the Tomsk Polytechnic Institute named after S. M. Kirov,
holder of the Order of the October Revolution and the Order of the Red Banner of Labor

Vol. 284
1974

ANALYTICAL SOLUTION OF THE EQUATIONS OF TRANSIENT PROCESSES IN A SYNCHRONOUS PULSE GENERATOR

V. P. LITVINOV, A. V. LOOS, E. I. SOBKO

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

Reducing the windings of a single-phase synchronous impact generator to a single coordinate system and the appropriate choice of the rotational speed of this system do not eliminate the periodic coefficients in the system of differential equations describing electromagnetic transients [1]. However, in [2] it is shown that a system of linear differential equations in which all coefficients are periodic with one and the same period can be reduced to a system of equations with constant coefficients. Criteria for the reducibility of such systems of equations are given in [3]. A broad range of questions connected with the analytical study of asymmetric operating modes of AC machines by means of differential equations with periodic coefficients is solved in [4]. In that work, on the basis of integral equations, a method is substantiated for reducing equations with periodic coefficients for asymmetric modes to the same number of equations with constant coefficients. The error of such a reduction does not exceed the order of the active resistances of the machine circuits.

The equations of transient processes of a single-phase synchronous impact generator can be written in the following form [1]:

\[ 0=(r_c+r_H)i_c+\frac{d\Psi_c}{dt}, \]

\[ U_f=r_f i_f+\frac{d\Psi_f}{dt}, \]

\[ 0=r_{Dd}i_{Dd}+\frac{d\Psi_{D\alpha}}{dt}, \]

\[ 0=r_{Dq}i_{Dq}+\frac{d\Psi_{Dq}}{dt}. \tag{1} \]

The flux linkages, depending on the mutual position of the windings, are equal to:

\[ \Psi_c=i_c(x_c+x_H)+(i_f+i_{Dd})x_a\cos\gamma+i_{Dq}x_a\sin\gamma, \]

\[ \begin{aligned} \Psi_f&=i_c x_a \cos\gamma+i_f x_f+i_{Dd}x_a,\\ \Psi_{Dd}&=i_c x_a \cos\gamma+i_f x_a+i_{Dd}x_{Dd},\\ \Psi_{Dq}&=i_c x_a \sin\gamma+i_{Dq}x_{Dq}. \end{aligned} \tag{2} \]

The angle between the axis of the stator winding and the direct axis of the rotor is determined by the expression

\[ \gamma=\int_0^t \omega\,dt. \tag{3} \]

At constant rotational speed, \(\gamma=\omega t\).

In equations (1–3) the following notation is adopted:

\(i_c,\ i_f,\ i_{Dd},\ i_{Dq}\) are the currents, respectively, in the stator winding, the excitation winding, and the damper windings along the direct and quadrature axes;

\((r_c+r_n),\ r_f,\ r_{Dd},\ r_{Dq}\) are the active resistances, respectively, of the stator winding with load, the excitation winding, and the damper windings along the direct and quadrature axes;

\((x_c+x_n),\ x_f,\ x_{Dd},\ x_{Dq}\) are the inductive reactances, respectively, of the stator winding with load, the excitation winding, and the damper windings along the direct and quadrature axes;

\(x_a\) is the inductive reactance of mutual induction when the winding axes coincide;

\(U_f\) is the voltage of the excitation winding.

The equations are written in the generally accepted system of relative units [1, 4].

Equations (1), (2) are a characteristic example of the general case of asymmetry of a machine; they cannot be reduced to equations with constant coefficients, and their solution becomes complicated, which makes it unacceptable for practical use.

Let us find an approximate solution of the system of equations (1), (2), using the method given in [4] for reducing them to equations with constant coefficients.

We shall seek the solution of the equations not with respect to currents, but to flux linkages. This is due to the fact that the flux linkages of the windings vary in time much less than the currents. The effect of active resistances on their variation in this case is also weaker.

The currents, expressed through the flux linkages, are determined from the solution of the algebraic equations (2):

\[ \begin{aligned} i_c&=f_{cc}\Psi_c+f_{cf}\Psi_f+f_{cd}\Psi_{Dd}+f_{cq}\Psi_{Dq},\\ i_f&=f_{fc}\Psi_c+f_{ff}\Psi_f+f_{fd}\Psi_{Dd}+f_{fq}\Psi_{Dq},\\ i_{Dd}&=f_{dc}\Psi_c+f_{df}\Psi_f+f_{dd}\Psi_{Dd}+f_{dq}\Psi_{Dq},\\ i_{Dq}&=f_{qc}\Psi_c+f_{qf}\Psi_f+f_{qd}\Psi_{Dd}+f_{qq}\Psi_{Dq}, \end{aligned} \tag{4} \]

where

\[ f_{cc}=\frac{1}{x_2},\quad f_{ff}=\frac{1}{x_f\sigma_{fDd}}+ \frac{x_a^2\sigma_{Dd}^2\cos^2\gamma}{x_f^2\sigma_{fDd}x_2}, \]

\[ f_{cf}=f_{fc}=-\frac{x_a\sigma_{Dd}\cos\gamma}{x_f\sigma_{fDd}x_2},\quad f_{qq}=\frac{1}{x_{Dq}}+\frac{x_a^2\sin^2\gamma}{x_{Dq}^2x_2}, \tag{5} \]

\[ f_{dd}=\frac{1}{x_{Dd}\sigma_{fDd}}+ \frac{x_a^2\sigma_f^2\cos^2\gamma}{x_{Dd}^2\sigma_{fDd}x_2},\quad f_{cd}=f_{dc}=-\frac{x_a\sigma_f\cos\gamma}{x_{Dd}\sigma_{fDd}x_2}, \]

\[ f_{cq}=f_{qc}=-\frac{x_a\sin\gamma}{x_{Dq}x_2},\quad f_{fq}=f_{qf}=\frac{x_a^2\sigma_{Dd}\cos\gamma\sin\gamma}{x_f\sigma_{fDd}x_2x_{Dq}}, \]

\[ f_{fd}=f_{df}=\frac{x_a^2\sigma_f\sigma_{Dd}\cos^2\gamma}{x_fx_{Dd}\sigma_{fDd}x_2} -\frac{x_a}{x_fx_{Dd}\sigma_{fDd}}, \]

\[ f_{dq}=f_{qd}=\frac{x_a^2\sigma_f\cos\gamma\sin\gamma}{x_{Dq}x_{Dd}\sigma_{fDd}x_2} \]
— periodic functions relating the currents and flux linkages.

In formulas (5)

\[ \sigma_f=1-\frac{x_a}{x_f},\quad \sigma_{Dd}=1-\frac{x_a}{x_{Dd}},\quad \sigma_{fDd}=1-\frac{x_a^2}{x_fx_{Dd}} \]

are the leakage coefficients, respectively, of the field winding, of the damper winding along the \(d\) axis, and the common coefficient for the field and damper windings; \(x_2\) is the negative-sequence reactance.

For a pulse generator, which, as a rule, has a powerful damper system along the \(d\) and \(q\) axes and is a nonsalient-pole machine, the assumption \(x_d''=x_q''\) is justified [1]. It follows from this that the leakage inductive reactance of the field winding is considerably greater than that of the damper winding along the \(d\) axis; moreover, it is well shielded by the damper winding, and therefore the transients in it during one current pulse are insignificant and may be neglected [1, 4].

Putting in (1), (2) \(\Psi_f=i_{f0}x_f=\mathrm{const}\), \(r_f=0\), and \(i_f=i_{f0}\), where \(i_{f0}\) is the initial field current, we obtain the system of equations:

\[ 0=r_ci_c+\frac{d\Psi_c}{dt}, \]

\[ 0=r_{Dd}i_{Dd}+\frac{d\Psi_{Dd}}{dt}, \tag{6} \]

\[ 0=r_{Dq}i_{Dq}+\frac{d\Psi_{Dq}}{dt}, \]

\[ \Psi_c=i_cx_c+(i_{f0}+i_{Dd})x_a\cos\gamma+i_{Dq}x_a\sin\gamma, \]

\[ \Psi_{Dd}=i_cx_a\cos\gamma+i_{f0}x_a+i_{Dd}x_{Dd}, \tag{7} \]

\[ \Psi_{Dq}=i_cx_a\sin\gamma+i_{Dq}x_{Dq}. \]

In special electromagnetic pulse sources with shaping of the magnetic flux by means of armature reaction, in order to achieve maximum values of the flux in the gap, the damper winding along the \(q\) axis is opened at the initial stage of the pulse [1].

In this case the currents are determined by formulas (4), where it is necessary to put \(x_{Dq}=\infty\), and

\[ x_2=\frac{x_d''+x_c}{2}+\frac{x_d''-x_c}{2}\cos2\gamma. \tag{8} \]

The flux linkages can be found from the solution of the system of equations (6), (7) for \(i_{Dq}=0\). Taking this condition into account, the system of equations (6), (7) in the \(\alpha,\beta,0\) axes is written in the form:

\[ \frac{d\Psi_c}{dt}+i_cr_c=0, \]

\[ \frac{d\Psi_{D\alpha}}{dt}+i_{D\alpha}r_{Dd}+\omega\Psi_{D\beta}=0, \tag{9} \]

\[ \frac{d\Psi_{q3}}{dt}+i_{D\beta}r_{Dd}-\omega\Psi_{D\alpha}=0, \]

where

\[ i_c=\frac{1}{x_2}\Psi_c-\frac{x_a}{x_{Da}x_2}\Psi_{D\alpha}, \]

\[ i_{D\alpha}=\frac{x_c}{x_{Da}x_2}\Psi_{D\alpha}-\frac{x_a\cos^2\gamma}{x_{Dd}x_2}\Psi_c, \tag{10} \]

\[ i_{D\beta}=\frac{1}{x_{Dd}}\Psi_{D\beta}+\frac{x_a^2\sin 2\gamma}{2x_{Da}^2x_2}\Psi_{D\alpha} -\frac{x_a\sin 2\gamma}{2x_{Dd}x_2}\Psi_c. \]

As is seen from (9), (10), in this case it is not possible to eliminate the periodic coefficients by a coordinate transformation. However, equations (9), (10) can be transformed using the approximate reduction method. To this end, the functions relating the currents and flux linkages in equations (10) are expanded in a Fourier series, and subsequently, when solving equations (9), (10), only the constant components of the expansions are taken into account. By neglecting the periodic functions of the series obtained from expanding the functions in (10), we introduce an error of the \(r\)-th order of smallness in comparison with unity, which does not exceed the error caused by the assumptions made in formulating the original equations, and is quite acceptable.

After the indicated transformations, equations (10) take the form:

\[ i_c=\frac{1}{\sqrt{x_d''x_c}}\Psi_c-\frac{x_a}{x_{Dd}\sqrt{x_d''x_c}}\Psi_{D\alpha}, \]

\[ i_{D\alpha}=\frac{1}{\sqrt{x_{3d}''x_{Dd}}}\Psi_{D\alpha} -\frac{D}{\sqrt{x_{3d}''x_{Dd}}}\Psi_c, \tag{11} \]

\[ i_{D\beta}=\frac{1}{x_{Dd}}\Psi_{D\beta}. \]

The ratios

\[ \frac{x_a}{x_{Dd}} \]

and

\[ D=\frac{x_a}{2x_c}\left(1+\frac{x_{Dd}-x_{3d}''}{x_{3d}''+x_{Dd}+2\sqrt{x_{3d}''x_{Dd}}}\right) \]

for small leakage reactances of the circuits, characteristic of electromechanical pulse sources, tend to unity; therefore, in order to simplify the subsequent calculations, we take them equal to this limit.

Substitution of (11) into (9), taking into account the adopted assumptions, gives the system of differential equations:

\[ \frac{d\Psi_c}{dt} +\frac{r_c}{\sqrt{x_d''x_c}}\Psi_c -\frac{r_c}{\sqrt{x_d''x_c}}\Psi_{D\alpha}=0, \]

\[ -\frac{d\Psi_{D\alpha}}{dt} +\frac{r_{Dd}}{\sqrt{x_{3d}''x_{Dd}}}\Psi_{D\alpha} -\frac{r_{Dd}}{\sqrt{x_{3d}''x_{Dd}}}\Psi_c +\omega\Psi_{D\beta}=0, \tag{12} \]

\[ \frac{d\Psi_{D\beta}}{dt} +\frac{r_{Dd}}{x_{Dd}}\Psi_{D\beta} -\omega\Psi_{D\alpha}=0. \]

\[ T_a=\frac{\sqrt{x_d''x_c}}{r_c},\qquad T_{\mathrm{ed}}=\frac{\sqrt{x_{Dd}x_{\mathrm{ed}}''}}{r_{Dd}},\qquad T_{Dd}=\frac{x_{Dd}}{r_{Dd}}. \]

From the solution of the system of differential equations (12), we determine the flux linkages:

\[ \Psi_c= \frac{i_{f0}x_a}{ \dfrac{1}{T_a^2}+\omega^2-\dfrac{1}{T_aT_{\mathrm{ed}}''}} \left\{ \left(\omega^2-\frac{1}{T_aT_{\mathrm{ed}}''}\right)e^{-\frac{t}{T_a}} + \frac{e^{-\frac{t}{2T_{\mathrm{ed}}''}}}{\omega} \cdot \left[ \frac{\omega}{T_a^2}\cos\gamma+ \left( \frac{\omega^2}{T_a}-\frac{1}{2T_a^2T_{\mathrm{ed}}''} \right)\sin\gamma \right] \right\}, \tag{13} \]

\[ \Psi_{Da}= \frac{i_{f1}x_a}{ \dfrac{1}{T_a^2}+\omega^2-\dfrac{1}{T_aT_{\mathrm{ed}}''}} \left\{ \frac{e^{-\frac{t}{2T_{\mathrm{ed}}''}}}{\omega} \left[ \left(\omega^2+\frac{1}{T_a^2}\right)\omega\cos\gamma+ \left( \frac{1}{T_{\mathrm{ed}}''{}^2}+\omega^2-\frac{1}{T_a^2} \right) \frac{1}{2T_{\mathrm{ed}}''}\sin\gamma \right] - \frac{1}{T_aT_{\mathrm{ed}}''}e^{-\frac{t}{T_a}} \right\}, \tag{14} \]

\[ \Psi_{D\beta}= \frac{i_{f0}x_a\omega}{ \dfrac{1}{T_a^2}+\omega^2-\dfrac{1}{T_aT_{\mathrm{ed}}''}} \left\{ \frac{1}{T_{\mathrm{ed}}''}e^{-\frac{t}{T_a}} + \frac{e^{-\frac{t}{2T_{\mathrm{ed}}''}}}{\omega} \cdot \left[ \left(\omega^2+\frac{1}{T_a^2}\right)\sin\gamma - \frac{\omega}{T_{\mathrm{ed}}''}\cos\gamma \right] \right\}. \tag{15} \]

By means of inverse transformations from (14) and (15), one can determine the law of variation of the flux linkage of the equivalent damper circuit along the \(d\)-axis in physical coordinates:

\[ \Psi_{Dd}= \frac{i_{f0}x_a}{ \dfrac{1}{T_a^2}+\omega^2-\dfrac{1}{T_aT_{\mathrm{ed}}''}} \left[ \frac{1}{T_{\mathrm{ed}}''} \left( \omega\sin\gamma-\frac{1}{T_a}\cos\gamma \right)e^{-\frac{t}{T_a}} + \frac{e^{-\frac{t}{2T_{\mathrm{ed}}''}}}{\omega} \left(\omega^2+\frac{1}{T_a^2}\right) \left( \omega-\frac{1}{4T_{\mathrm{ed}}''}\sin2\gamma \right) \right]. \tag{16} \]

It is necessary to note that, in solving the system of equations (12), the finding of the roots of the characteristic equation determining the components “linked” with the rotor is carried out for ideal conduc-

...of the stator winding, and, conversely, to find the root that determines the components “associated” with the stator, it is assumed that \(r_{Dd}=0\).

Such determination of the roots is admissible if the rotor speed is high and the active resistances of the windings are significantly smaller than the total inductive resistances at the nominal frequency, which is the case in an impulse generator [1]. In addition, a number of quantities of the second order of smallness, which complicate the calculation and have little effect on accuracy, were neglected.

Having calculated, by formulas (13) and (16), the flux linkages \(\Psi_c,\ \Psi_{Dd}\) from relations (4), one can calculate the currents \(i_c,\ i_f,\ i_{Dd}\) in the circuits of the impulse generator.

As noted earlier, in order to attain maximum values of the flux in the gap, at the initial stage of the pulse the damper winding along the \(q\)-axis is opened. When the maximum flux linkage with the damper winding along the \(q\)-axis is reached, it is closed.

The flux linkage of the open damper winding along the transverse axis is \(\Psi_{Dq}=i_c x_a \sin\gamma\). When it is closed, the processes occurring in the source are described by the systems of equations (1) and (2).

The solution (6) and (7) at this stage of generator operation will be carried out with the initial conditions \(\Psi_{c0},\ \Psi_{Dd0},\ \Psi_{Dq0}\), which are the final results of the solutions of the system of equations (12).

Equations (6) and (7), transformed to the axes \(\alpha,\ \beta,\ 0\) and written in terms of flux linkages, have the form:

\[ \frac{d\Psi_c}{dt}+\frac{1}{T_a}\Psi_c-\frac{1}{T_a}\Psi_{D\alpha}=0, \]

\[ \frac{d\Psi_{D\alpha}}{dt} +\frac{1}{T''_{ed}}\Psi_{D\alpha} -\frac{1}{T''_{ed}}\Psi_c +\omega\Psi_{D\beta}=0, \tag{17} \]

\[ \frac{d\Psi_{D\beta}}{dt} +\frac{1}{T_{Dq}}\Psi_{D\beta} -\omega\Psi_{D\alpha}=0, \]

where

\[ T_a=\frac{x''_d}{r_c},\qquad T''_{ed}=\frac{x''_{ed}}{r_{Dd}},\qquad T_{Dq}=\frac{x_{Dq}}{r_{Dq}}. \]

In the transformation, the damping system was assumed to be symmetrical, and

\[ \frac{x_a}{x_{Dq}}\approx 1. \]

As is seen from (17), under the adopted assumptions the machine becomes symmetrical from the rotor side and has no periodic coefficients in the axes \(\alpha,\ \beta,\ 0\).

From the solution of the system of differential equations (17) we determine the flux linkages:

\[ \Psi_c = \Psi_{c0}e^{-\frac{t}{T_a}} + \frac{\Psi_{D\alpha 0}\cdot \frac{1}{T_a}} {\frac{1}{T_a^2}+\omega^2-\frac{1}{T_aT''_{ed}}} \left\{ -\frac{1}{T_a}e^{-\frac{t}{T_a}} + \frac{e^{-\frac{t}{2T''_{ed}}}}{\omega} \left[ \frac{\omega}{T_a}\cos\gamma + \left( \omega^2-\frac{1}{2T_aT''_{ed}} \right)\sin\gamma \right] \right\} - \frac{\Psi_{D\beta 0}} {\frac{1}{T_a^2}+\omega^2-\frac{1}{T_aT''_{ed}}}. \]

\[ -\frac{\omega}{T_a}\left\{e^{-\frac{t}{T_a}}+\frac{e^{-\frac{t}{2T''_{эd}}}}{\omega}\left[-\omega\cos\gamma+\left(\frac{1}{T_a}-\frac{1}{2T''_{эd}}\right)\sin\gamma\right]\right\}. \tag{18} \]

\[ \Psi_{D\alpha}= \frac{\Psi_{c0}}{\dfrac{1}{T_a^2}+\omega^2-\dfrac{1}{T_aT''_{эd}}}\cdot \frac{1}{T''_{эd}} \left\{ -\frac{1}{T_a}e^{-\frac{t}{T_a}} +\frac{e^{-\frac{t}{2T''_{эd}}}}{\omega} \left[ \frac{\omega}{T_a}\cos\gamma+ \right.\right. \]

\[ \left.\left. +\left(\omega^2-\frac{1}{2T_aT''_{эd}}\right)\sin\gamma \right]\right\} +\Psi_{D\alpha0}e^{-\frac{t}{2T''_{эd}}} \left(\cos\gamma-\frac{1}{\omega 2T''_{эd}}\sin\gamma\right) - \]

\[ -\Psi_{D\beta0}e^{-\frac{t}{2T''_{эd}}}\sin\gamma, \tag{19} \]

\[ \Psi_{D\beta}= \frac{\Psi_{c0}\omega}{\dfrac{1}{T_a^2}+\omega^2-\dfrac{1}{T_aT''_{эd}}}\cdot \frac{1}{T''_{эd}} \left\{ e^{-\frac{t}{T_a}} +\frac{e^{-\frac{t}{2T''_{эd}}}}{\omega} \left[ -\omega\cos\gamma+ \right.\right. \]

\[ \left.\left. +\left(\frac{1}{T_a}-\frac{1}{2T''_{эd}}\right)\sin\gamma \right]\right\} +\Psi_{D\alpha0}e^{-\frac{t}{2T''_{эd}}}\sin\gamma+ \]

\[ +\frac{\Psi_{D\beta0}}{\dfrac{1}{T_a^2}+\omega^2-\dfrac{1}{T_aT''_{эd}}} \left\{ -\frac{1}{T_aT''_{эd}}e^{-\frac{t}{T_a}} +\frac{e^{-\frac{t}{2T''_{эd}}}}{\omega}\cdot \right. \]

\[ \left. \cdot\left[ \omega\left(\omega^2+\frac{1}{T_a^2}\right)\cos\gamma +\frac{1}{2T''_{эd}}\left(\omega^2-\frac{1}{T_a^2}\right)\sin\gamma \right]\right\}, \tag{20} \]

where

\[ \Psi_{D\alpha0}=\Psi_{Dd0}\cos\gamma_0+\Psi_{Dq0}\sin\gamma_0, \]

\[ \Psi_{D\beta0}=\Psi_{Dd0}\sin\gamma_0-\Psi_{Dq0}\cos\gamma_0, \tag{21} \]

\(\gamma_0\) is the angle at which the damper winding along the \(q\) axis is closed.

By inverse transformations from (19) and (20), one can determine the law of variation of the flux linkages \(\Psi_{Dd}\) and \(\Psi_{Dq}\) of the equivalent damper circuits in physical coordinates.

In formulas (21), the quantity \(\Psi_{Dq0}\) is taken with allowance for saturation of the magnetic circuit along the \(q\) axis.

Investigation of the pulse source under consideration on an analog model showed that the current amplitude depends strongly on the magnitude of the flux \(\Psi_{Dq0}\).

With an unsaturated magnetic circuit, the value \(\Psi_{Dq0}\) reaches 4–5 times the initial flux, which cannot be achieved in real machines. With saturation of the magnetic circuit, the value of the flux-

linkage flux \(\Psi_{Dq0}\), as studies have shown, can reach \(1.5 \div 2\) times the initial flux. To determine \(\Psi_{Dq0}\) with allowance for saturation, we shall use the approximation of the no-load characteristic [5]:

\[ E_0 = 0.99 \sqrt[3]{i}. \tag{22} \]

Since, in relative units, \(\Psi_\delta = E_0\), (22) can be written in the form

\[ \Psi_\delta = 0.99 \sqrt[3]{i_\delta}, \]

where

\[ i_\delta = \sqrt{(i_{f0}+i_{Dd}+i_c \cos \gamma)^2 + (i_c \sin \gamma)^2} \]

is the resultant current in the air gap of the generator with the damper winding along the \(q\)-axis open.

This makes it possible to determine approximately the value of the forced flux in the air gap of the machine, as well as the value of the flux along the \(q\)-axis, which is equal to \(\Psi_{Dq0}=\Psi_\delta \cos \varphi\), where \(\varphi\) is the angle between the vector \(\Psi_\delta\) and the \(q\)-axis. As studies show, the angle \(\varphi\) is very small; then \(\cos \varphi \approx 1\), \(\Psi_{Dq0} \approx \Psi_\delta\).

In Fig. 1, for comparison, the results are shown of calculating the current of the pulse source, obtained on the MN-14 analog machine from the complete system of equations (1), (2), and from the analytical expressions obtained, for the machine parameters in relative units

\[ x_c = 1.02; \]

\[ x_{Dd}=1.02;\quad x_{Dq}=1.02;\quad x_f=1.2; \]

\[ r_c = r_{Dd}=r_{Dq}=r_f=0.005. \]

As can be seen from Fig. 1, the maximum discrepancy in the calculations is no more than \(7\%\).

Fig. 1. Current in the stator winding of a synchronous pulse generator:
— solution on an analog computer;
\(+\) analytical calculation

The greatest inaccuracy is admitted in calculating the currents at the instant when winding \(D_q\) is switched in. This is explained by the fact that in the first stage, when winding \(D_q\) is open, saturation of the magnetic circuit was not taken into account, but it was taken into account in determining the initial conditions in the second stage.

Thus, an analytical solution of the equations of a synchronous pulse generator has been obtained, and the possibility has been shown of using the method of reducing equations with periodic coefficients to the same number of equations with constant coefficients in the study of complex asymmetrical operating conditions.

LITERATURE

1. E. I. Sobko. Studies of the operation of a shock generator with an active load characteristic. Author’s abstract of dissertation, Tomsk, 1972.

2. A. M. Lyapunov. Collected Works, Vol. II. Publishing House of the Academy of Sciences of the USSR, 1956.

1. N. P. Erugin. Reducible Systems. Proceedings of the V. A. Steklov Mathematical Institute. Issue 13. Moscow, 1946.

2. I. I. Treshchev. Methods for the Investigation of Alternating-Current Machines. Moscow, “Energiya,” 1969.

3. V. I. Arkhangel’skii. Analytical expression for the magnetization curve of electrical machines. “Elektrichestvo,” 1967, No. 3.