Electromagnetic Processes in an Asynchronous Electric Drive with Nonsinusoidal Supply Voltage
Abstract Generated abstract
This paper develops a mathematical description of electromagnetic processes in an induction motor drive supplied by a nonsinusoidal, stepped voltage. By transforming the supply voltage to a reference frame rotating with the stator field, the authors decompose the machine equations into zero-frequency and higher harmonic components, especially the 6th and 12th harmonics, and formulate matrix linear equations that include rotor speed deviations in the impedance coefficients. The method enables calculation of harmonic torque pulsations and associated speed pulsations, with higher harmonics treated analogously. A numerical example for a DAT 250-8 motor at nominal frequency gives torque pulsations of 12.1 percent of nominal torque for the 6th harmonic and 1.78 percent for the 12th.
Full Text
ELECTROMAGNETIC PROCESSES IN AN ASYNCHRONOUS ELECTRIC DRIVE WITH NONSINUSOIDAL SUPPLY VOLTAGE
A. A. EFIMOV, V. I. PANTELEEV, B. P. SOUSTIN
(Presented by the scientific and technical seminar of the NII AEM)
If a stepped voltage is transformed to a reference frame rotating with the frequency of the stator field, then the voltage along the two axes is written as
\[ U_{\alpha s}^{k}=\frac{2U_1'}{\pi}\left(1+\frac{2}{35}\cos 6\omega_k t-\frac{2}{143}\cos 12\omega_k t+\ldots\right), \tag{1} \]
\[ U_{\beta s}^{k}=\frac{2U_1'}{\pi}\left(\frac{12}{35}\sin 6\omega_k t-\frac{24}{143}\sin 12\omega_k t+\ldots\right). \tag{2} \]
The axis \(\alpha\) and the magnetic axes of the stator and rotor phases coincide at the initial instant of time. The 5th and 7th harmonics are transformed into quantities of the 6th harmonic, and the 11th and 13th harmonics into quantities of the 12th, etc. [2]. Figure 1 presents the equivalent circuit of the system [3], which may be implemented—
Fig. 1
can be, with any desired degree of accuracy, replaced by adding current and voltage sources to it. The established values of voltage and current, the rotor speed, and also the inverter voltages and currents are determined by constant components (zero frequency) and by harmonics 6, 12, etc.
If the base electrical angular frequency \(\omega_b\) is introduced, and the relative quantities of frequency \(\alpha=\dfrac{\omega_k}{\omega_b}\) and absolute slip \(\beta=\alpha s\) are adopted, then the equations of the asynchronous machine in matrix form are written as
\[ \begin{bmatrix} R_s+\left(\dfrac{p}{\omega_b}\right)x_s & \alpha x_s & \left(\dfrac{p}{\omega_b}\right)x_m\\ -\alpha s & R_s+\left(\dfrac{p}{\omega_b}\right)x_s & -\alpha x_m\\ \left(\dfrac{p}{\omega_b}\right)x_m & \alpha s x_m & R_r+\left(\dfrac{p}{\omega_b}\right)x_r\\ -\alpha s x_m & \left(\dfrac{p}{\omega_b}\right)x_m & -\alpha s x_r \end{bmatrix} \begin{bmatrix} \alpha x_m\\ \left(\dfrac{p}{\omega_b}\right)x_m\\ \alpha s x_r\\ R_r+\left(\dfrac{p}{\omega_b}\right)x_r \end{bmatrix} \times \begin{bmatrix} i_{\alpha s}\\ i_{\beta s}\\ i_{\alpha r}\\ i_{\beta r} \end{bmatrix} = \begin{bmatrix} U_{\alpha s}\\ U_{\beta s}\\ U_{\alpha r}^{*}\\ U_{\beta r} \end{bmatrix}. \tag{3} \]
\[ M=x_m(i_{\alpha s}\cdot i_{\beta r}-i_{\beta s}\cdot i_{\alpha r}), \tag{4} \]
\[ M-M_c=\left(\frac{2H}{\omega_b}\right)p\cdot\omega_r. \tag{5} \]
These equations can be divided into three systems: for the quantities of zero frequency, and the 6th, 12th, etc. harmonics. For zero frequency,
\[ \begin{bmatrix} U_{10}\\ 0\\ 0\\ 0 \end{bmatrix} = \begin{bmatrix} R_s & \alpha x_s & 0 & \alpha x_m\\ -\alpha x_s & R_s & -\alpha x_m & 0\\ 0 & \alpha s_0 x_m & R_r & \alpha s_0 x_r\\ -\alpha s_0 x_m & 0 & -\alpha s_0 x_r & R_r \end{bmatrix} \times \begin{bmatrix} i_{\alpha s0}\\ i_{\beta s0}\\ i_{\alpha r0}\\ i_{\beta r0} \end{bmatrix}, \tag{6} \]
\[ M_0=x_m(i_{\alpha s0}\cdot i_{\beta r0}-i_{\beta s0}\cdot i_{\alpha r0}), \tag{7} \]
\[ S_0=\left(\frac{\omega_k-\omega_{r0}}{\omega_k}\right), \tag{8} \]
\[ M_0-M_{c0}=0. \tag{9} \]
For the quantities of the 6th harmonic the following equations hold:
\[ \begin{bmatrix} R_s+j6\alpha x_s & \alpha x_s & j6\alpha x_m & \alpha x_m\\ -\alpha x_s & R_s+j6\alpha x_s & -\alpha x_m & j6\alpha x_m\\ j(K_m\Psi_{\beta r0}\cdot i_{\beta r0}+6\alpha x_m) & \alpha S_0x_m-jK_m\Psi_{\beta r0}\cdot i_{\alpha r0} & R_r+j(6\alpha x_r-K_m\Psi_{\beta r0}\cdot i_{\beta s0}) & \alpha S_0x_r+jK_m\Psi_{\beta r0}\cdot i_{\alpha s0}\\ -\alpha S_0x_m-jK_m\Psi_{\alpha r0}\cdot i_{\alpha r0} & j(6\alpha x_m+K_m\Psi_{\alpha r0}\cdot i_{\alpha r0}) & -\alpha S_0x_r+jK_m\Psi_{\alpha r0}\cdot i_{\beta r0} & R_r+j(6\alpha x_r-K_m\Psi_{\alpha r0}\cdot i_{\alpha s0}) \end{bmatrix} \begin{bmatrix} \overline{i}_{\alpha s6}\\ \overline{i}_{\beta s6}\\ \overline{i}_{\alpha r6}\\ \overline{i}_{\beta r6} \end{bmatrix} = \begin{bmatrix} \dfrac{2}{35}U_{10}\\ -j\dfrac{12}{35}U_{10}\\ 0\\ 0 \end{bmatrix}. \tag{10} \]
\[ |\overline{M}_6|=[\operatorname{Re}(\overline{M}_6)^2+\operatorname{Im}(\overline{M}_6)^2]^{1/2}. \tag{11} \]
The relative speed deviations caused by the 6th harmonic are
\[ \left|\frac{\overline{\omega}_{r6}}{\omega_k}\right| = \frac{1}{12a^2\omega_6H}\cdot|\overline{M}_6|. \tag{12} \]
Expression (10) is a matrix linear equation that takes account of the speed deviation when calculating the pulsations of the torque of the sixth harmonic. The influence of the speed deviation is taken into account by the corresponding coefficients at the impedances in the third and fourth rows.
For the quantities of the 12th harmonic the derivation of the equations and their form are analogous, but
\[ \frac{p}{\omega_6}=j12a, \tag{13} \]
\[ \overline{M}_{12}=x_m\cdot \begin{bmatrix} i_{\beta s0} & -i_{\alpha r0} & -i_{\beta s0} & i_{\beta s0} \end{bmatrix} \cdot \overline{i}_{12}. \tag{14} \]
The relative speed deviations are
\[ \left|\frac{\overline{\omega}_{r12}}{\omega_k}\right| = \frac{1}{24a^2\omega_6H}\cdot|\overline{M}_{12}|. \tag{15} \]
In an analogous manner, the deviations of the speed from the 18th, 24th, etc. harmonics are calculated; however, in practice, even for motors with a very small moment of inertia it is sufficient to determine these two.
The pulsations of the torque of the DAT 250-8 motor, calculated on an electronic digital computer by the proposed method at the nominal frequency, amount to 12.1% of the nominal value for the 6th harmonic, and 1.78% for the 12th.
Conclusions
A mathematical description has been proposed for an asynchronous machine under nonsinusoidal voltage, making it possible to determine torque pulsations while taking into account changes in rotor speed. The expressions obtained are matrix linear equations. The effect of speed deviation is taken into account by the corresponding components of the coefficients at the impedances.
LITERATURE
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Lipo T. A., Krause P. C., Jordan Howard E. “Harmonic torque and speed pulsation in a rectifier—inverter induction motor drive.” “IEEE Trans. Power. Appar. and Syst.” No. 5, Part 1, 1969.