异步电机磁链环PI参数设计

前置阅读

PI参数设计
异步电机数学模型

异步电机磁链环传递函数

$$\begin{array}{l}G(s)=(Kp+\frac{Ki}{s})\frac{L_m}{s{\tau }_{\mathrm{r}}+1}\\ G(j2\pi f_{\mathrm{c}\mathrm{u}\mathrm{t}})=(Kp+\frac{Ki}{j2\pi f_{\mathrm{c}\mathrm{u}\mathrm{t}}})\frac{L_m}{j2\pi f_{\mathrm{c}\mathrm{u}\mathrm{t}}{\tau }_{\mathrm{r}}+1}\end{array}$$

根据相位裕度需求计算截止频率

$$\begin{array}{l}\angle G(j\omega )=-8.88-(arc\tan (2\pi f_{\mathrm{c}\mathrm{u}\mathrm{t}}{\tau }_{\mathrm{r}}))\\ PM{}_{Set}=180+(-8.88-(arc\tan (2\pi f_{\mathrm{c}\mathrm{u}\mathrm{t}}{\tau }_{\mathrm{r}})))\\ f_{\mathrm{c}\mathrm{u}\mathrm{t}}=\frac{\tan (171.12-PM{}_{Set})}{2\pi {\tau }_{\mathrm{r}}}\end{array}$$

PI参数计算
$$\begin{array}{l}|G(j2\pi f_{\mathrm{c}\mathrm{u}\mathrm{t}})|=|(Kp+\frac{Ki}{j2\pi f_{\mathrm{c}\mathrm{u}\mathrm{t}}})\frac{L_m}{j2\pi f_{\mathrm{c}\mathrm{u}\mathrm{t}}{\tau }_{\mathrm{r}}+1}|=|(Kp+\frac{Ki}{j2\pi f_{\mathrm{c}\mathrm{u}\mathrm{t}}})\frac{L_m(j2\pi f_{\mathrm{c}\mathrm{u}\mathrm{t}}{\tau }_{\mathrm{r}}-1)}{(j2\pi f_{\mathrm{c}\mathrm{u}\mathrm{t}}{\tau }_{\mathrm{r}}+1)(j2\pi f_{\mathrm{c}\mathrm{u}\mathrm{t}}{\tau }_{\mathrm{r}}-1)}|\\ =\frac{L_m}{{(2\pi f_{\mathrm{c}\mathrm{u}\mathrm{t}}{\tau }_{\mathrm{r}})}^2-1}|j(Kp2\pi f_{\mathrm{c}\mathrm{u}\mathrm{t}}{\tau }_{\mathrm{r}}+\frac{Ki}{2\pi f_{\mathrm{c}\mathrm{u}\mathrm{t}}})+(Ki{\tau }_{\mathrm{r}}-Kp)|\\ =\frac{L_{m}Kp}{{(2\pi f_{\mathrm{c}\mathrm{u}\mathrm{t}}{\tau }_{\mathrm{r}})}^2-1}|j(2\pi f_{\mathrm{c}\mathrm{u}\mathrm{t}}{\tau }_{\mathrm{r}}+\frac{1}{6.4})+(2\pi \frac{f_{\mathrm{c}\mathrm{u}\mathrm{t}}}{6.4}{\tau }_{\mathrm{r}}-1)|\\ =\frac{L_{m}Kp}{{(2\pi f_{\mathrm{c}\mathrm{u}\mathrm{t}}{\tau }_{\mathrm{r}})}^2-1}\sqrt{{(2\pi f_{\mathrm{c}\mathrm{u}\mathrm{t}}{\tau }_{\mathrm{r}}+\frac{1}{6.4})}^2+{(2\pi \frac{f_{\mathrm{c}\mathrm{u}\mathrm{t}}}{6.4}{\tau }_{\mathrm{r}}-1)}^2}=1\\ Kp=\frac{{(2\pi f_{\mathrm{c}\mathrm{u}\mathrm{t}}{\tau }_{\mathrm{r}})}^2-1}{\sqrt{{(2\pi f_{\mathrm{c}\mathrm{u}\mathrm{t}}{\tau }_{\mathrm{r}}+\frac{1}{6.4})}^2+{(2\pi \frac{f_{\mathrm{c}\mathrm{u}\mathrm{t}}}{6.4}{\tau }_{\mathrm{r}}-1)}^2}}\\ Ki=2\pi \frac{f_{\mathrm{c}\mathrm{u}\mathrm{t}}}{6.4}Kp\end{array}$$

小结

• 根据相位裕度需求, 能够求出磁链环的截止频率
• 然后,只需要代入电机转子时间常数与截止频率, 则可求出PI参数