Frequency Control Law for an Induction Motor During Current Formation
Abstract Generated abstract
This paper examines frequency control of an induction motor when a semiconductor frequency converter forms a sinusoidal stator current despite a nonsinusoidal voltage. Starting from the conventional voltage control law associated with M. P. Kostenko, the authors derive a torque expression for current formation using the motor equivalent parameters and reduce it to a form analogous to the standard mechanical characteristic. The analysis indicates that, under current control, the parameters and initial stiffness of the mechanical characteristic do not depend on stator frequency, but are determined by rotor frequency and relative current. To keep the static overload capacity constant, the slip parameter must remain constant, which implies a relative current proportional to the square root of the load parameter.
Full Text
FREQUENCY CONTROL LAW FOR AN INDUCTION MOTOR DURING CURRENT FORMATION
V. I. IVANCHURA, B. P. SOUSTIN
(Presented by the scientific-technical seminar of the Research Institute of AEM)
In frequency control of an induction motor (IM), the law of frequency and voltage control—the law of M. P. Kostenko [1]—has been studied quite fully:
\[ \gamma_u=a\sqrt{\mu}. \]
The stator voltage is assumed to be sinusoidal
\[ \dot U=U_m e^{j2\pi f_1 t}. \]
In this case the motor torque has the expression
\[ M=\gamma_u^2 M_{ku}\frac{2(1+q\beta_{ku})\beta\beta_{ku}}{\beta^2+\beta_{ku}^2+2q\beta\beta_{ku}}. \tag{1} \]
Here \(\beta=\dfrac{f_2}{f_{1N}}\) is the absolute slip parameter; \(q\) is a coefficient determined from the motor parameters.
The parameters of the mechanical characteristic \(M_{ku}\), \(q\), and \(\beta_{ku}\) depend on frequency. The stiffness of the mechanical characteristic \(K_{M0}=\dfrac{\partial M}{\partial \beta}\) at \(\beta=0\) also depends on the stator frequency.
Usually, in frequency control, a semiconductor frequency converter (FC) is used as the source, at whose output the voltage has a nonsinusoidal form. Therefore the expression for the torque does not reflect the actual picture.
The developed current-formation systems based on semiconductor FCs make it possible to obtain a sinusoidal current with nonsinusoidal voltage [2]
\[ \dot i=I_m e^{j2\pi f_1 t}. \]
In this case the current is formed by feedback and does not depend on the stator parameters; then the IM torque can be expressed as
\[ M=\frac{1}{\sqrt{2}}m_1 w_1 k_{ob1}p\Phi_m\cos\psi_2. \tag{2} \]
Here \(w_1, k_{\mathrm{ob}1}\) are the number of turns and the winding factor of the stator winding, and \(m, p\) are the number of phases and pole pairs of the stator.
\[ \cos \psi_2 = \frac{r_2'}{\beta \sqrt{(x_2')^2+\left(\dfrac{r_2'}{\beta}\right)^2}}; \qquad I_2' = \frac{I_1 Z_{qu}}{\sqrt{(x_2')^2+\left(\dfrac{r_2'}{\beta}\right)^2}}, \]
\[ Z_{qu} = \frac{\left(jx_2' + \dfrac{1}{\beta} r_2'\right)(r_0+jx_0)} {\left(\dfrac{r_2'}{\beta}+r_0\right)+j(x_0+x_2')}; \qquad \Phi_m = \frac{I_1 Z_{qu}}{4.44\, W_1 k_{\mathrm{ob}1} f_{1N}} . \]
The expression for the torque can be brought to the form
\[ M=\gamma_i^2 M_{ki}\, \frac{2(1+q_i \beta_{ki})\beta \beta_{ki}} {\beta^2+\beta_{ki}^2+2q_i\beta\beta_{ki}} . \tag{3} \]
\[ M_{ki}= \frac{pm}{2\pi f_{1N}} \frac{I_1^2 x_0^2}{2(x_0+x_2')}; \qquad \gamma_i=\frac{I_1}{I_{1N}}; \qquad \beta_{ki}=\pm \frac{r_2'}{\sqrt{r_0^2+(x_0+x_2')^2}}, \]
\[ q_i=\frac{r_0}{r_2'} . \]
From expression (3) it is evident that the parameters of the mechanical characteristic and the slope of the mechanical characteristic at \(\beta=0\), \(K_{m0}=\dfrac{dM}{d\beta}\), do not depend on the stator frequency, but are completely determined by the frequency in the rotor and the relative current \(\gamma_i\).
In order to keep the static overload capacity coefficient constant, \(\lambda=\dfrac{M_k}{M}\), it is necessary to maintain the equality \(\beta=\mathrm{const}\). Then \(\gamma_i=\sqrt{\mu}\). When this condition is observed in open-loop systems, the law of M. P. Kostenko is fulfilled; in closed-loop systems, it is necessary that \(\beta=\mathrm{const}\).
REFERENCES
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A. A. Bulgakov. Frequency Control of Induction Motors. Moscow, “Nauka,” 1966.
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I. F. Kalinin, B. P. Soustin. Closed-loop system for forming the current curve of an inverter. In: Devices of Converter Engineering, issue 2, Kiev, “Naukova Dumka,” 1969.