APPLICATION OF AN ANALOG-DIGITAL COMPUTER TO THE STUDY OF SELF-OSCILLATIONS OF A SYNCHRONOUS RELUCTANCE MOTOR BY THE FREQUENCY METHOD

E. V. KONONENKO, A. S. GITMAN, A. L. KISLITSYN, A. F. FINK

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

The use of automatic digital computing machines (ADCMs) opens up great possibilities for modern technology. It is enough to say that the best models of ADCMs are now capable of carrying out tens of thousands of operations per second. The use of computing machines makes it possible to choose optimal variants for solving engineering problems and thereby yields an enormous economic effect.

Many problems whose solution until recently seemed exceptionally difficult because of the large volume of computations can now, with the advent of electronic computing machines, be solved in a short time, and their results used for practical purposes.

In the present work an ADCM is used to investigate the self-oscillations of a synchronous reluctance motor.

The problem of this investigation reduces to determining the influence of the parameters of the supply network and of the synchronous-reluctance motor on self-excited oscillations and to establishing the limits of stable operation with the use of an ADCM.

In considering this problem, equations obtained by the frequency method [1] are taken as the basis.

The equation of motion of the rotor of a 3-phase synchronous-reluctance motor may be represented in the form

\[ \left[-h^{2}T_j\omega^{2}+M_{so}+M_{sd}+jh\left(M_{do}+M_{dr_1}+M_{dr_2}\right)\right]\cdot \Delta\Theta=0, \tag{1} \]

where \(h\) is the frequency of oscillations,

\(T_j\) is the inertia constant of the rotating masses of the synchronous-reluctance motor,

\(\omega\) is the frequency of the supply network,

\[ M_{so}=U^{2}\left(\frac{1}{x_q}-\frac{1}{x_d}\right)\cdot \cos 2\Theta \tag{2} \]

is the principal synchronizing torque caused by the difference of the resistances along the \(d\) and \(q\) axes,

where \(x_d=x_{a.d}+x_l+x\) is the inductive resistance of the stator winding of the synchronous-reluctance motor along the longitudinal axis \(d\), taking into account the inductive resistance of the external network,

\(x_q=x_{aq}+x_1+x\) — inductive resistance of the stator winding along the transverse axis \(q\), taking into account the inductive resistance of the external network;

\(\Theta\) — the angle formed by the transverse axis \(q\) and the voltage vector;

\[ M_{sd}=U^2\cdot\cos^2\Theta\left(\frac{1}{x_q''}-\frac{1}{x_q}\right)\cdot \frac{h^2T_q''{}^2}{1+h^2T_q''{}^2} + U^2\cdot\sin^2\Theta\left(\frac{1}{x_d''}-\frac{1}{x_d}\right)\cdot \frac{h^2T_d''{}^2}{1+h^2T_d''{}^2} \tag{3} \]

— the additional synchronizing torque due to the presence of a winding on the rotor along the direct and transverse axes,

where \(T_q''\) is the time constant of the transverse damping circuit,

\(T_d''\) is the time constant of the direct damping circuit,

\(x_d''\) and \(x_q''\) are the subtransient reactances along the direct and transverse axes.

\[ M_{d0}=U^2\cdot\cos^2\Theta\left(\frac{1}{x_q''}-\frac{1}{x_q}\right)\cdot \frac{T_q''}{1+h^2T_q''{}^2} + U^2\cdot\sin^2\Theta\left(\frac{1}{x_d''}-\frac{1}{x_d}\right)\cdot \frac{T_d''}{1+h^2T_d''{}^2} \tag{4} \]

— the main asynchronous torque due to the presence of a winding on the rotor;

\[ \begin{aligned} M_{dr_1}={}& -R\Bigg\{ \left[\left(\frac{x_{dh}}{z_{dh}^{2}}-\frac{1}{x_q}\right)\cdot U\cdot\sin\Theta\right]^2 + \left[\left(\frac{x_{qh}}{z_{qh}^{2}}-\frac{1}{x_q}\right)\cdot U\cdot\cos\Theta\right]^2 \\ &\quad -U^2\cdot\frac{r_{dh}}{z_{dh}^{2}}\cdot\sin^2\Theta -U^2\cdot\frac{r_{qh}}{z_{qh}^{2}}\cdot\cos^2\Theta \Bigg\} -\frac{2R}{h}\left[ -\frac{U^2\cdot\cos\Theta\cdot\sin\Theta}{x_d}\cdot \frac{r_{dh}}{z_{dh}^{2}} + \frac{U^2\cdot\sin\Theta\cdot\cos\Theta}{x_q}\cdot \frac{r_{dh}}{z_{qh}^{2}} \right] \end{aligned} \tag{5} \]

— the additional asynchronous torque depending on the total resistance of the motor and of the external network,

Fig. 1. Equivalent circuit of substitution of the rotor winding of a synchronous-reluctance motor along the direct axis.

where \(r_{dh}, x_{dh}, z_{dh}\) are the resultant active, inductive, and total resistances computed from the equivalent substitution circuit (Fig. 1) for the direct axis \(d\) of the synchronous-reluctance motor;

\(r_{qh}, x_{qh}, z_{qh}\) are the resultant active, inductive, and total resistances, calculated from the equivalent circuit of substitution (Fig. 2) for the transverse axis of the synchronous reluctance motor,

Fig. 2. Equivalent circuit of substitution of the rotor winding of a synchronous reluctance motor along the transverse axis.

\[ M_{dr_z}=-\frac{R^2}{h}U^2\left\{ -\cos^2\Theta\cdot\frac{A'x_{dh}-B'r_{dh}}{x_d\cdot x_q} +\cos^2\Theta\frac{A'x_{qh}+B'r_{qh}}{z_{qh}^2} +\right. \]
\[ \left. +\sin^2\Theta\left( \frac{A'x_{dh}+B'r_{dh}}{z_{dh}^2} +\frac{B'r_{qh}-A'x_{qh}}{x_d\cdot x_q} \right)\right\} \tag{6} \]

— the additional asynchronous torque, depending on \(R^2\), where

\[ A'=\frac{r_{dh}\cdot x_{qh}}{z_{dh}^2\cdot z_{qh}^2} +\frac{r_{qh}\cdot x_{dh}}{z_{dh}^2\cdot z_{qh}^2}, \]

\[ B'=\frac{x_{dh}\cdot x_{qh}}{z_{dh}^2\cdot z_{qh}^2} -\frac{r_{dh}\cdot r_{qh}}{z_{dh}^2\cdot z_{qh}^2}. \]

Let us denote the real part of equation (1)—the total synchronizing torque—by \(X(h)\), and the imaginary part—the total asynchronous torque—by \(Y(h)\); then

\[ M(jh)=X_h(h)+jY(h). \tag{7} \]

Plotting \(Y(h)\) on the ordinate axis and \(X(h)\) on the abscissa axis, one can construct the hodograph of the torque and frequency characteristic if \(h\) varies within the limits from 0 to \(\infty\).

According to the frequency criterion [1], the system will be stable if the total asynchronous torque at resonance of the forced oscillations is positive, i.e., when \(X(h_0)=0\),

\[ Y(h_0)>0 \tag{8} \]

To determine the boundary of stable operation, we take the total synchronizing torque to be zero and determine the resonant frequency \(h_0\).

The stability analysis reduces to determining the boundary of stable operation as a function of the parameters \(r\) and \(x\) of the supply network (Fig. 3).

Substituting the values \(\Theta\), \(\omega\), \(x\), \(h_0\) into the imaginary part \(jY(h_0)\) of equation (1), and taking \(Y(h_0)\) equal to zero, we determine the critical resist-

…solution, for which the system is on the stability boundary. In solving this problem it becomes necessary to carry out a large, complex, and labor-intensive computational procedure. In order to speed up the computations, the “Minsk-1” digital electronic computer was used. Since scaling the formulas proved extremely difficult, the solution was carried out with floating-point numbers. To solve equations (1)—(8), a block diagram was compiled, shown in Fig. 4. As is seen from this diagram, the change in the inductive resistance of the external network \(x\) occurs in the inner loop with respect to the change in the angle \(\Theta\), and the change in the angle \(\Theta\) in the inner loop with respect to the change in the external-network frequency \(\omega\). The parameters \(x\), \(\Theta\), \(\omega\) were varied with constant increments.

To solve the problem, the following data were entered into the ADCC memory:

\[ \begin{aligned} \omega_{\mathrm{n}}&\text{ — initial value of }\omega,\\ \Theta_{\mathrm{n}}&\text{ — initial value of }\Theta,\\ x_{\mathrm{n}}&\text{ — initial value of }x,\\ x_l&\text{ — leakage reactance,}\\ \Delta\omega&\text{ — increment of }\omega,\\ \Delta\Theta&\text{ — increment of }\Theta,\\ \Delta x&\text{ — increment of }x\text{ and the parameters} \end{aligned} \]

\[ x''_q,\; x_{aq},\; x_{rq},\; x''_d,\; x_{ad},\; x_{rd},\; T_j,\; r_{rd},\; r_{rq}. \]

The first half of the solution of the problem is connected with finding the resonant frequency \(h_0\), with the interval of variation of \(h_0\) of interest lying in the region of positive numbers.

The quantity \(h_0\) is determined from the equation

\[ - h_0^2 T_j \omega^2 + M_{so} + M_{sd} = 0. \tag{9} \]

Since all the quantities entering equations (1)—(6) are expressed in relative units, \(\omega\) and \(U\) are represented by one and the same quantities. Therefore all terms of equation (9) have a common factor that can be taken out, \((\omega^2)\) or \((U^2)\). Then, expressing \(T_j\) through \(T'_j\omega^2\), \(M_{so}\) through \(M'_{so}U^2\), and \(M_{sd}\) through \(M'_{sd}U^2\), equation (9) can be transformed to the form

\[ - h^2 T'_j + M'_{so} + M'_{sd} = 0. \tag{10} \]

Substituting into equation (10) the expression for \(M'_{sd}\), we obtain

\[ - h^2 T'_j + M'_{so} + \cos^2\Theta \left( \frac{1}{x''_q} - \frac{1}{x_{aq}+x_l+x} \right) \cdot \frac{h^2 T_q''{}^2}{1+h^2 T_q''{}^2} + \tag{11} \]

\[ + \sin^2\Theta \left( \frac{1}{x''_d} - \frac{1}{x_{ad}+x_l+x} \right) \cdot \frac{h^2 T_d''{}^2}{1+h^2 T_d''{}^2} =0. \]

Equation (11) is cubic with respect to \(h^2\), or, correspondingly, of the sixth degree with respect to \(h\). Investigations have shown that the cubic equation with respect to \(h^2\), in the region bounded by the conditions of the problem, has only one positive root, the determination of which is the aim of solving equation (3). Solving the cubic equation by the known Cardano formulas is inadvisable on an electronic digital computer. From this point of view, iterative methods are very advantageous. Ite-

Fig. 3. Equivalent substitution circuit of the system: synchronous-reluctance motor—supply network.

Fig. 4. Flowchart for solving the equations on an ADC.

Flowchart labels:

• Conversion of initial data
• Multiplication of the parameters by \(\omega\)
• \(\omega_n+\Delta\omega\)
• \(\theta_n+\Delta\theta\)
• Computation of trigonometric functions
• \(x_n+\Delta x\)
• Computation of \(M_{d0}, M_{dr1}, M_{dr2}\)
• Computation of \(r\)
• Printing of results
• Has the change in \(x\) ended?
• no
• yes
• Has the change in \(\theta\) ended?
• no
• yes
• Has the change in \(\omega\) ended?
• no
• yes
• Stop

Side labels:

• Determination of the resonant-frequency \(h_0\)
• Determination of \(r_{d1}\) and \(r_{d2}\)

iterative methods presuppose representing the equation \(F(x)=0\) in the form \(x=f(x)\), and the desired root is found as the limit of the sequence

\[ x_{i+1}=f(x_i),\qquad i=(0,1,2,3\ldots) \]

Iterative methods usually require a preliminary study of the equations because of the possibility of divergence of the iterative processes. In 1962, [2] published Wegstein’s iterative method, which is free from this drawback. An undoubted advantage of the indicated method is also that it is fairly easy to program. Taking this into account, one may consider Wegstein’s method to be among the best iterative methods used for solving equations on electronic digital computers. The solution of equation (11) was separated into a separate subroutine.

Finding the quantities \(z_{dh}\) and \(z_{qh}\) is associated with the transformation of expressions that include complex numbers. Since they differ only in their parameters, it proved expedient to create a subroutine for finding \(z_{dh}\) and \(z_{qh}\).

After each calculated variant, the following results were printed:

\(N\) — variant number,

\(r, x\) — critical parameters of the external network,

\(h_0\) — resonant frequency,

\(\omega\) — frequency of the supply network, as well as the components of the synchronizing and asynchronous torques

\[ M_{s0},\ M_{d0},\ M_{dr_1},\ M_{dr_2}. \]

Printing was performed at a time interval of 4 seconds. After the input of the initial data, the machine calculated 1000 variants, which required about 70 minutes of machine time. For comparison, it may be noted that carrying out a similar amount of computation manually would require about 1.5–2 years of work by one calculator.

The results of the studies obtained on the “Minsk-1” analog-digital computer made it possible, in a comparatively short interval of time, to determine the influence of the parameters of synchronous reluctance motors and of the external network, as well as of the load magnitude, on the region of stable operation; they are presented in [1].

REFERENCES

1. E. V. Kononenko, A. L. Kistitsyn, A. F. Fink. Study of self-oscillations of a synchronous-reluctance motor when operating from a controlled generator. Izvestiya TPI, vol. 132, 1965.

2. J. H. Lance. Numerical Methods for High-Speed Computing Machines. Moscow, IL, 1962.