Stability Study of an Inverter Voltage Control System
Abstract Generated abstract
This paper examines the stability of a closed-loop inverter output-voltage control system using pulse-width regulation under the assumed equivalence of continuous and pulse systems. It formulates transfer functions for the rectifier and resonant AC output filter, reduces the filter model to two second-order sections, and applies logarithmic frequency characteristics to a 500 Hz, 150 W static converter example. The analysis indicates instability for the considered parameter set, then uses the root-locus method to determine ranges of harmonic coefficient, gain, and filter parameters that can preserve stability. The paper notes that such width-method stabilization systems may have infinitely many critical frequencies, each associated with critical filter and gain values.
Full Text
STUDY OF THE STABILITY OF AN INVERTER VOLTAGE CONTROL SYSTEM
S. I. Volkov, B. P. Soustin
(Presented by the Scientific and Technical Seminar of the NII AEM)
When stabilizing the output voltage of an inverter, it is necessary to solve the problem of the stability and quality of closed-loop systems. It is assumed that the condition for equivalence of continuous and pulse systems, i.e., that the width of the frequency spectrum of the external disturbance is always less than the difference between the switching frequency and the cutoff frequency (V. A. Kotelnikov’s theorem), is fulfilled.
The block diagram of control of the output voltage of an inverter by the width method is shown in Fig. 1, where \(K_{\mathrm{u}}, K_{\mathrm{m}}, K_{\mathrm{i}}, K_{\mathrm{io}}\) are the gain coefficients of the amplifier, the modulator (the converter of a continuous signal into a pulse of variable width), the inverter, and the measuring element.
Fig. 1
The transfer functions of the rectifier and of the output resonant AC filter are
\[ W_{\mathrm{в}}(p)=\frac{K_{\mathrm{в}}}{1+T_{\mathrm{в}}p}, \]
\[ W_{\phi}(p)=\frac{b_{1}p^{2}}{a_{1}p^{4}+a_{2}p^{3}+a_{3}p^{2}+a_{4}p+a_{5}}. \]
Here \(K_{\mathrm{в}}\) and \(T_{\mathrm{в}}\) are the gain coefficient and time constant of the filter. It is assumed that transmission of the first harmonic occurs without attenuation. Therefore
\[ a_1=C_1L_1C_2L_2,\qquad a_2=\frac{1}{R_{\mathrm{n}}}\left[C_1L_1(L_2+L_{\mathrm{n}})+L_2L_{\mathrm{n}}(C_1+C_2)\right], \]
\[ a_3=C_1L_1+C_2L_2+C_1L_2;\quad a_4=\frac{1}{R_{\mathrm{n}}}(L_2+L_{\mathrm{n}});\quad a_5=1;\quad b_1=C_1L_2. \]
For known roots of the characteristic equation, the transfer function of the filter has the form
\[ W_{\phi}(p)= \frac{b_1p^2} {a_1(\omega_1^2+\alpha_1^2)(\omega_2^2+\alpha_2^2) \left(\dfrac{1}{\omega_1^2+\alpha_1^2}p^2+\dfrac{2\alpha_1}{\omega_1^2+\alpha_1^2}p+1\right)} \cdot \left( \frac{1}{\omega_2^2+\alpha_2^2}p^2+ \frac{2\alpha_2}{\omega_2^2+\alpha_2^2}p+1 \right). \]
Fig. 2
The transfer function of the filter is reduced to two series-connected sections:
\[ W_{\phi}(p)= \frac{K_{\phi}p^2} {(T_1^2p^2+2\xi_1T_1p+1)(T_2^2p^2+2\xi_2T_2p+1)}, \]
where
\[ K_{\phi}=\frac{b_1}{a_1(\omega_1^2+\alpha_1^2)(\omega_2^2+\alpha_2^2)},\qquad T_{1,2}=\frac{1}{\sqrt{\omega_{1,2}^2+\alpha_{1,2}^2}},\qquad \xi_{1,2}=\frac{\alpha_{1,2}}{\sqrt{\omega_{1,2}^2+\alpha_{1,2}^2}}. \]
For the study, a static converter with an output frequency of 500 Hz and a power of 150 W was taken, with the parameters given in the table.
Table
| $\alpha_1$ | $\alpha_2$ | $\omega_1$ | $\omega_2$ | $T_1$ | $T_2$ | $K_{\phi}$ | $K$ |
|---|---|---|---|---|---|---|---|
| $-2880$ | $-636$ | $\pm 6560$ | $\pm 524$ | $1.39\cdot 10^{-4}$ | $1.24\cdot 10^{-3}$ | $1.31$ | $54.4$ |
In the analysis, the method of logarithmic characteristics was used (Fig. 2). Curve 1 shows that the system is unstable. To obtain the stability condition for the system, the root-locus method [1] was applied; it makes it possible to determine the range of variation of the harmonic coefficient while maintaining stability and the selected gain coefficient by changing the filter parameters.
It should be noted that, for a stabilization system using the width method, there is always an infinite number of critical frequency values, each of which corresponds to one critical value of the filter parameters and the gain coefficient.
REFERENCES
G. A. Bendrikov, K. F. Teodorchik. Root Loci of Linear Automatic Systems. “Nauka,” 1964.