예제 Nyquist 선도 그리기 2

나이퀴스트 선도 그리기

• $G(s)H(s)= \dfrac{K}{s(s+2)(s+10)}$

1. $GH(s)$에 $s = j\omega$ 를 대입 :

$$GH(j\omega) = \dfrac{K}{j\omega(j\omega+2)(j\omega+10)}$$

2. $\omega \to 0^+$ (A점에 해당)

$$GH(j0) = \dfrac{K}{j0(j0+2)(j0+10)}=\infty \angle(-90^\circ)$$

4. $\omega \to + \infty$ (B점에 해당)

$$GH(j\infty) = \dfrac{K}{j\infty(j\infty+2)(j\infty+10)}=0 \angle (-270^\circ)$$

5. $s = R^{j \theta} \vert_{\begin{cases} R\to \infty \\ \theta \to 90^\circ \to 0 \to -90^\circ \end{cases}}$ (C점에 해당)

$$GH(R^{j\theta}) = \dfrac{K}{R^{j\theta}(R^{j\theta}+2)(R^{j\theta}+10)}=0 \angle (-3 \theta)= 0\angle(-270^\circ \to 0^\circ \to 270^\circ )$$

6. $\omega \to -\infty$ (D점에 해당)

$$GH(-j\infty) = \dfrac{K}{-j\infty(-j\infty+2)(-j\infty+10)}=0 \angle (-90^\circ)$$

7. $\omega \to 0^-$ (E점에 해당)

$$GH(j0^{-}) = \dfrac{K}{j0^-(j0^-+2)(j0^-+10)} = \infty \angle(90^\circ)$$

8. $s = r^{j \theta} \vert_{\begin{cases} r\to 0 \\ \theta \to -90^\circ \to 0 \to 90^\circ \end{cases}}$ (F점에 해당)

$$GH(r^{j\theta}) = \dfrac{K}{r^{j\theta}(r^{j\theta}+2)(r^{j\theta}+10)}=\infty \angle (-\theta)= \infty \angle(90^\circ \to 0^\circ \to -90^\circ )$$

3. 실수축과의 교차점 :

$$\begin{aligned} GH(j\omega) &= \dfrac{K}{j\omega(j\omega+2)(j\omega+10)}\\ &=\dfrac{K}{-12\omega^2+j\omega(20-\omega^2)} \cdot \dfrac{-12\omega^2-j\omega(20-\omega^2)}{-12\omega^2-j\omega(20-\omega^2)} \\ &= \dfrac{-12K\omega^2-jK\omega(20-\omega^2)}{144\omega^4+\omega^2} \end{aligned}$$

$$-K\omega(20-\omega^2)=0 \Rightarrow \omega=\sqrt{20}$$

$$GH(j\sqrt{20})=\dfrac{-12K\sqrt{20}^2}{144\sqrt{20}^4+\sqrt{20}^2}= \dfrac{-240K}{57620}$$

$K=1$

$k=250$