강의노트 2차지연요소

2차 지연 요소

$$G(s) = \frac{\omega^2}{s^2 + 2a \omega_n s + \omega^2_n } = \frac{1}{ (\frac{s}{\omega_n})^2 + (2a\frac{s}{\omega_n}) + 1}$$

$$G(j\omega) = \dfrac{1}{1-(\dfrac{\omega}{\omega_n})^2+j2a(\dfrac{\omega}{\omega_n})}$$

$$\vert G(j\omega) \vert_{dB}= 20 \log _{10} \vert \dfrac{1}{\sqrt{(1-(\dfrac{\omega}{\omega_n})^2)^2 +4a^2(\dfrac{\omega}{\omega_n})^2}} \vert \\ = - 20 \log _{10} \sqrt{(1-(\dfrac{\omega}{\omega_n})^2)^2 +4a^2(\dfrac{\omega}{\omega_n})^2}$$

이득

$$\omega \ll \omega_n \quad \Rightarrow \quad \vert G(j\omega) \vert _{dB} \approx -20 \log _{10} \sqrt1 = 0$$

$$\omega = \omega_n \quad \Rightarrow \quad \vert G(j\omega) \vert _{dB} = -20 \log _{10} \sqrt{4a^2} = -20 \log _{10}(2a) = -20 \log _{10} 2 - 20 \log _{10} a = -6 -20 \log _{10} a$$

$$\omega \gg \omega_n \quad \Rightarrow \quad \vert G(j\omega) \vert _{dB} \approx -20 \log _{10} \sqrt{(\frac{\omega}{\omega_n})^4} = -40 \log _{10} \omega +40 \log _{10} (\omega_n) \approx -40 \log _{10} \omega$$

위상

$$\omega \ll \omega_n \quad \Rightarrow \quad \angle G(j\omega) = - \tan ^{-1} (\dfrac{2 a (\dfrac{\omega}{\omega_n})}{1-(\dfrac{\omega}{\omega_n})^2}) = 0^\circ$$

$$\omega = \omega_n \quad \Rightarrow \quad \angle G(j\omega) =- \tan ^{-1} (\dfrac{2 a (\dfrac{\omega}{\omega_n})}{1-(\dfrac{\omega}{\omega_n})^2}) = -90^\circ$$

$$\omega \gg \omega_n \quad \Rightarrow \quad \angle G(j\omega) = - \tan ^{-1} (\dfrac{2 a (\dfrac{\omega}{\omega_n})}{1-(\dfrac{\omega}{\omega_n})^2}) = -180^\circ$$

• $\omega_n = 1$ 그리고 $a$는 0.1에서 0.9까지 0.1식 증가한 시스템의 보드선도이다.