Lecture 1차지연요소

보드선도 일차지연요소

$$G(s) = \dfrac{1}{s \tau + 1}$$

$$\begin{align*} \vert G (j \omega ) \vert_{dB} &= 20 \log_{10} \vert \sqrt{1^2+ (\omega \tau)^2} \vert^{-1} \end{align*}$$

$$\begin{align*} \angle G (j \omega ) &= 0^\circ - \angle (1 + j \omega \tau) \end{align*}$$

이득

$$\omega \ll \dfrac1\tau \quad \Rightarrow \quad \vert \dfrac{1}{j \omega \tau + 1} \vert_{dB} = -20 \log_{10} \vert j \omega \tau + 1 \vert \approx -20 \log_{10} 1 = 0$$

$$\omega = \dfrac1\tau \quad \Rightarrow \quad \vert \dfrac{1}{j \omega \tau + 1} \vert_{dB} = -20 \log_{10} \vert j + 1 \vert = -20 \log_{10} \sqrt2 = -10 \log_{10} 2 \approx -3dB$$

$$\omega \gg \dfrac1\tau \quad \Rightarrow \quad \vert \dfrac{1}{j \omega \tau + 1} \vert_{dB} = -20 \log_{10} \vert \omega \tau +1\vert \approx -20 \log_{10} \vert \omega \tau \vert = -20 \log_{10} \vert \omega \vert -20 \log_{10} \vert \tau \vert$$

위상각

$$\omega \ll \dfrac1\tau \quad \Rightarrow \quad \angle (\dfrac{1}{j \omega \tau + 1}) \approx -\angle 1 = 0^\circ$$

$$\omega = \dfrac1\tau \quad \Rightarrow \quad \angle (\dfrac{1}{j \omega \tau + 1}) = -\angle (j+1) = -45^\circ$$

$$\omega \gg \dfrac1\tau \quad \Rightarrow \quad \angle (\dfrac{1}{j \omega \tau + 1}) \approx -\angle (j \omega \tau ) = -90^\circ$$

$\tau$ = 1 일때의 보드 선도