Given the Transfer function characterizing a dynamic system, its response is well categorized by considering the steady state response to sinusoidal inputs across a range of frequencies. For this, we will consider the gain and the phase shift across frequencies.
$$ |G|{dB}=20\log{10}|G(j\omega)| $$
$$ \phi=\tan^{-1}\frac{\Im (G(j\omega))}{\Re(G(j\omega))} $$
Plotting gain in decibels and phase shift versus log scaled frequency renders Bode plots. There are three major classes of factors in transfer functions which recognizably change the form of these Bode plots.
| Factor | $K(j\omega)^n$ | $(j\omega\tau+1)$ | $((\frac{j\omega}{\omega_n})^2+2\zeta(\frac{j\omega}{\omega_n})+1)^{\pm 1}$ |
|---|---|---|---|
| Transition | Constant | Transition at $\omega_b=\frac 1\tau$ | Transition at $\omega_b=\omega_n$ |
| Gain | Slope of $20n~(\frac{dB}{dec})$ | Slope of $0\rarr20~(\frac{dB}{dec})$ | Slope of $0\rarr40~(\frac{dB}{dec})$ |
| Gain spike at transition of $ | G | =\frac 1{2\zeta}$ | |
| Phase | Phase of $90n~(\degree)$ | Phase from $0\rarr90~(\degree)$ | Phase from $0\rarr180~(\degree)$ |
| Reversal | Holds with $n<0$ | Reflection over $G_{dB}=0$, $\phi=0~(\degree)$ | Reflection over $G_{dB}=0$, $\phi=0~(\degree)$ |