ANALYSIS OF ELECTROMAGNETIC PROCESSES IN A MODULATION SOURCE WHEN IT OPERATES WITH AN INDUCTION MOTOR

V. S. SAYAPIN, V. A. DENISOV, Sh. S. ROIZ

In recent years, sources of stable frequency have been developed in which frequency stabilization is carried out by means of amplitude modulation of the emf of a synchronous generator. In this connection, the operation of an induction motor when supplied from a modulation source is of particular interest.

In analyzing the electromagnetic processes, we represent the modulation-source—induction-motor system by the equivalent circuit of Fig. 1. Here the induction motor is represented by a T-shaped equivalent circuit without allowance for iron losses.

Fig. 1. Equivalent circuit of the modulation-source—induction-motor system: MS—modulation source; IM—induction motor

The following notation is adopted in the circuit:

$r_g$ — active resistance of the generator winding;

$x_g$ — leakage inductive reactance of the generator;

$e_g$ — emf of the generator;

$r_1$ — active resistance of the motor stator;

$x_1$ — leakage inductive reactance of the motor stator;

$x_\mu$ — inductive reactance of the magnetizing circuit;

$r_2'$ — referred active resistance of the motor rotor;

$x_2'$ — referred leakage inductive reactance of the motor rotor.

In the given equivalent circuit, the motor current can be found as the sum of the magnetizing current and the rotor current:

\[ i_{\mathrm{d}} = i_\mu + i'_{\mathrm{p}} . \tag{1} \]

The expressions for \(i_\mu\) and \(i'_{\mathrm{p}}\), taking into account commutation of the cycloconverter valves, are obtained using the method of [1].

Then the equation describing the transient process in the magnetizing branch at the instant of commutation has the form

\[ k(x_1+x_\mu)\frac{di_\mu}{d(\omega t)} -kx_\Gamma\frac{di^1_\Gamma}{d(\omega t)} +i_\mu r_1-i^1_\Gamma r_\Gamma = \]

\[ = E_m\left[1-\frac{a}{2}-\frac{a}{2} \cos\left(\frac{2m+1}{k}\pi\right)\right]\sin k\omega t, \tag{2} \]

where \(i^1_\Gamma\) is the part of the generator current going to create the main magnetic flux of the motor;

\[ E_m\left[1-\frac{a}{2}-\frac{a}{2} \cos\left(\frac{2m+1}{k}\pi\right)\right]\sin k\omega t \]

is the emf of the modulation source [2];

\(a\) is the depth of amplitude modulation of the generator emf;

\(m\) is the interval number \((m=0,1,2,\ldots,k-1)\);

\(k\) is the ratio of the carrier frequency to the modulating frequency.

After integrating (2) over the limits from \(t=0\) to \(t=\varepsilon\), where \(\varepsilon\) is the commutation angle \((\varepsilon \to 0)\), we obtain an equation that makes it possible to determine the motor magnetizing current after commutation:

\[ i_\mu(\varepsilon)=\zeta_\mu I_{0\mu}, \tag{3} \]

where

\[ \zeta_\mu=\frac{x_\mu+x_1-x_\Gamma}{x_\mu+x_1+x_\Gamma}; \]

\(I_{0\mu}\) is the value of the magnetizing current in the preceding interval.

For the operating interval one may write

\[ kx_{\Sigma\mu}\frac{di_\mu}{d(\omega t)}+r_{\Sigma\mu}i_\mu =E_m\left[1-\frac{a}{2}-\frac{a}{2} \cos\left(\frac{2m+1}{k}\pi\right)\right]\sin k\omega t, \tag{4} \]

where

\[ x_{\Sigma\mu}=x_\Gamma+x_1+x_\mu; \]

\[ r_{\Sigma\mu}=r_\Gamma+r_1. \]

The solution of equation (4), taking into account the initial conditions (3), is the expression

\[ i_\mu(\omega t)= \frac{ E_m\left[1-\frac{a}{2}-\frac{a}{2} \cos\left(\frac{2m+1}{k}\pi\right)\right] }{ \sqrt{r_{\Sigma\mu}^{\,2}+(kx_{\Sigma\mu})^2} } \left[\sin(k\omega t-\varphi_{\Sigma\mu})+\sin\varphi_{\Sigma\mu}\right. \]

\[ \left.\times \exp(-\operatorname{ctg}\varphi_{\Sigma\mu}\, k\omega t)\right] +\zeta_\mu I_{0\mu}\exp(-\operatorname{ctg}\varphi_{\Sigma\mu}\, k\omega t), \]

where

\[ \varphi_{\Sigma\mu}=\operatorname{arctg}\frac{kx_{\Sigma\mu}}{r_{\Sigma\mu}}, \qquad \operatorname{ctg}\varphi_{\Sigma\mu}=\frac{r_{\Sigma\mu}}{kx_{\Sigma\mu}}. \tag{5} \]

To calculate the magnetizing-current curve over all intervals, we use the method of matching.

Analogously, the equation for the rotor circuit at the instant of commutation is written as follows:

\[ i_p'(r_1+r_2)-i_\Gamma''r_\Gamma+k(x_1+x_2')\frac{di_p'}{d(\omega t)} -kx_\Gamma\frac{di_\Gamma''}{d(\omega t)}= \]
\[ =E_m\left[1-\frac{a}{2}-\frac{a}{2}\cos\left(\frac{2m+1}{k}\pi\right)\right]\sin k\omega t-e_{\mathrm{dv}}(\omega t), \tag{6} \]

where \(i_\Gamma''\) is the fraction of the generator current used to create the motor rotor current;

\(e_{\mathrm{dv}}\) is the e.m.f. of motor rotation;

\[ e_{\mathrm{dv}}(\omega t)=\frac{r_2'(1-s)}{s}\,i_2 \quad \text{or} \quad e_{\mathrm{dv}}=U_{12}k_\sigma(1-s), \]

where

\[ k_\sigma=1-\frac{x_2'}{x_\mu}, \]

\(s\) is the motor slip,

\[ U_{12}=x_\mu\frac{di_\mu}{d(\omega t)}. \tag{7} \]

Substituting (5) into (7), we determine the e.m.f.

\[ e_{\mathrm{dv}}(\omega t)= \frac{k^2x_\mu(1-s)k_\sigma}{\sqrt{r_{\mathrm{ts}\mu}^{\,2}+(kx_\mu)^2}}\, E_m\left[1-\frac{a}{2}-\frac{a}{2}\cos\left(\frac{2m+1}{k}\pi\right)\right]\times \]
\[ \times\left[\cos(k\omega t-\varphi_{\mathrm{z}\mu}) -\sin\varphi_{\mathrm{z}\mu}\operatorname{ctg}\varphi_{\mathrm{z}\mu} \exp(-\operatorname{ctg}\varphi_{\mathrm{z}\mu})\right]+ \]
\[ +k^2x_\mu\operatorname{ctg}\varphi_{\mathrm{z}\mu}\,\zeta_\mu k_\sigma(1-s) I_{0\mu}\exp(-\operatorname{ctg}\varphi_{\mathrm{z}\mu}k\omega t). \tag{8} \]

After substituting (8) into (6) and integrating over the limits from \(t=0\) to \(t=\varepsilon\), we obtain

\[ i_p'(\varepsilon)=\zeta_p I_{0p}', \tag{9} \]

where

\[ \zeta_p=\frac{x_1+x_2'-x_\Gamma}{x_1+x_2'+x_\Gamma}; \]

\(I_{0p}'\) is the value of the rotor current on the preceding interval.

On the operating interval, the referred rotor current is determined from the expression

\[ kx_{\mathrm{sr}}\frac{di_p'}{d(\omega t)}+r_{\mathrm{sr}}i_p' = E_m\left[1-\frac{a}{2}-\frac{a}{2}\cos\left(\frac{2m+1}{k}\pi\right)\right]\sin k\omega t-e_{\mathrm{dv}}(\omega t), \tag{10} \]

where

\[ x_{\mathrm{sr}}=x_\Gamma+x_1+x_2', \qquad r_{\mathrm{sr}}=r_\Gamma+r_1+r_2'. \]

The solution of equation (10), taking (8) and (9) into account, is written as

\[ i_p'(\omega t)= E_m\left[1-\frac{a}{2}-\frac{a}{2}\cos\left(\frac{2m+1}{k}\pi\right)\right] \frac{1}{\sqrt{r_{\mathrm{sr}}^{\,2}+(kx_{\mathrm{sr}})^2}}\times \]
\[ \times\left\{ \sin(k\omega t-\varphi_{\mathrm{z}\mu}) -\frac{k^2x_\mu(1-s)k_\sigma}{\sqrt{r_{\mathrm{z}\mu}^{\,2}+(kx_\mu)^2}} \left[\cos(k\omega t-\varphi_{\mathrm{z}\mu})-\right.\right. \]
\[ \left.\left. -\cos\varphi_{\mathrm{z}\mu}\exp(-\operatorname{ctg}\varphi_{\mathrm{z}\mu}k\omega t) \right]\right\} -kr_{\mathrm{z}\mu}\zeta_\mu(1-s)k_\sigma\times \]
\[ \times I_{0\mu}\exp(-\operatorname{ctg}\varphi_{\mathrm{z}\mu}k\omega t) +\left\{\zeta_p I_{0p}+ \frac{kx_\mu k_\sigma(1-s)}{\sqrt{r_{\mathrm{sr}}^{\,2}+(kx_{\mathrm{sr}})^2}}\times \]

Fig. 2. Motor current curves at different depths of amplitude modulation of the generator e.m.f.:
a) at \(a=0\), b) at \(a=0.6\), c) at \(a=1\).

\[ \times \left\{ E_m \left[ 1 - \frac{a}{2} - \frac{a}{2} \cos \left( \frac{2m+1}{k}\,\pi \right) \right] \sin \varphi_{\Sigma p} + k \operatorname{ctg} \varphi_{\Sigma \mu}\,\varphi_\mu I_{0\mu} \right\} \times \]
\[ \times \exp\left(-\operatorname{ctg}\varphi_{\Sigma p}\,k\omega t\right), \tag{11} \]

where

\[ \varphi_{\Sigma p} = \operatorname{arctg}\frac{kx_{\Sigma p}}{r_{\Sigma p}}, \qquad \operatorname{ctg}\varphi_{\Sigma p} = \frac{r_{\Sigma p}}{kx_{\Sigma p}}. \]

Solving (5), (11), and (1) jointly, it is possible to calculate, by intervals, the motor-current curve for different depths of amplitude modulation of the synchronous generator emf, taking into account the emf of rotation of the induction motor. Figure 2 shows the calculated curves for \(a=0\), \(a=0.6\), and \(a=1\). In all cases one can observe a distortion of the current waveform caused by the influence of the motor emf. It should be noted that the current waveform also depends substantially on the depth of amplitude modulation.

Conclusions

1. As a result of the analysis of the electromagnetic processes carried out, equations have been obtained that make it possible to construct the motor-current curve for different depths of modulation of the synchronous generator emf, taking into account commutation of the valves of the cycloconverter.

2. In a modulation source, the leakage inductive reactances of the induction motor have a substantial influence on the commutation processes.

References

1. Yu. Ya. Korobitsin, V. N. Kochnev, B. A. Lopato, G. G. Orduyants, I. M. Seryi, A. A. Yanko-Trinitskii. “A method for calculating electromagnetic processes in controlled valve converters.” Republican interdepartmental scientific and technical collection Theoretical Electrical Engineering, issue 12, Lvov, 1971.

2. V. A. Denisov, Sh. S. Roiz, N. F. Zhmaev. “Analysis of the output voltage of modulation autonomous sources of stable frequency.” Proceedings of TPI, vol. 284, Tomsk, TTU Publishing House, 1974.