12. Bode Plots

\[\small \log \left(\frac{\text{AR}}{K_p}\right) = -\frac{1}{2}\log(1+\tau_p^2\omega^2) \tag*{(1)}\]

As \(\omega\rightarrow0\), then \(\tau_p\omega\rightarrow0\), and from Eqn.(1), \( \log \left(\frac{\text{AR}}{K_p}\right)\rightarrow0\) or \(\left(\dfrac{\text{AR}}{K_p}\right)\rightarrow1\). This is the low-frequency asymptote shown by a blue dashed line.

As \(\omega\rightarrow\infty\), then \(\tau_p\omega\rightarrow\infty\), and from Eqn.(1), \(\log \left(\frac{\text{AR}}{K_p}\right)\approx-\log(\tau_p\omega)\). This is the high-frequency asymptote shown by a red dashed line. It is a line with a slope of \(-1\) passing through the point \(\dfrac{\text{AR}}{K_p}=1\) for \(\tau_p\omega=1\).

The frequency \(\omega=1/\tau_p\) is known as the corner frequency. At the corner frequency, the deviation of the true value of AR/\(K_p\) from the asymptotes is maximum.

The plot of \(\phi\) vs. \(\tau_p\omega\) is constructed from the following characteristics:

As \(\omega\rightarrow 0\), then \(\phi\rightarrow0\).

As \(\omega\rightarrow\infty\), then \(\tan^{-1}(-\infty)=-90^\circ\).

At \(\omega=1/\tau_p\) (corner frequency), \(\phi=\tan^{-1}(-1)=-45^\circ\).