The frequency range required in frequency response is often so wide that it is inconvenient to use a linear scale for the frequency axis. For these reason, it has become standard practice to plot the transfer function on a pair of semilogarithmic plots: The magnitude in decibels is plotted against the logarithm of the frequency; on a separate plot, the phase in degrees is plotted against the logarithm of the frequency.
Bode plots are semilog plots of the magnitude (in decibels) and phase (in degrees) of a transfer function versus frequency.
Since transfer function is a complex number, \(\displaystyle \mathbf{H}(\omega) = H(\omega)\angle\phi(\omega) = \textrm{Re}\left[\mathbf{H}(\omega)\right] + j\,\textrm{Im}\left[\mathbf{H}(\omega)\right]\).
In a Bode magnitude plot, the gain \(\displaystyle H_{\textrm{dB}}\) is
\(\boxed{H(\omega)_{\textrm{dB}} = 20\log_{10}\left|\mathbf{H}(\omega)\right| = 20\log_{10}\left[H(\omega)\right]}\)
where \(\displaystyle H(\omega) = \sqrt{\left(\textrm{Re}\left[\mathbf{H}(\omega)\right]\right)^2 + \left(\textrm{Im}\left[\mathbf{H}(\omega)\right]\right)^2}\)
In a Bode phase plot, the phase \(\phi\) is
\(\boxed{\phi(\omega) = \textrm{Arg}\left[\mathbf{H}(\omega)\right] = \tan^{-1}\left(\frac{\textrm{Im}\left[\mathbf{H}(\omega)\right]}{\textrm{Re}\left[\mathbf{H}(\omega)\right]}\right)}\)
A transfer function \(\mathbf{H}(\omega)\) may be written in terms of factors that have real and imaginary parts.
\(\boxed{\mathbf{H}(\omega) = K\frac{\displaystyle \left(j\omega\right)^{\pm 1} \left(1 + \frac{j\omega}{z}\right) \left[1 + \frac{j2\zeta_{1}\omega}{\omega_n} + \left(\frac{j\omega}{\omega_n}\right)^2\right]\cdots}{\displaystyle \left(1 + \frac{j\omega}{p}\right) \left[1 + \frac{j2\zeta_{2}\omega}{\omega_k} + \left(\frac{j\omega}{\omega_k}\right)^2\right]\cdots}}\)
where \(z\), \(p\), \(\omega_n\), and \(\omega_k\) are the corner frequency (or break frequency), and \(\zeta\) is the damping factor.
The Bode plot can be constructed by plotting each factors of transfer function \(\mathbf{H}(\omega)\) separately and adding them graphically. The factors can be considered one at a time and then combined additively because of the logarithms involved.
Straight-line Plots of the Factors
For positive integer \(N\).
| Factor | Magnitude | Phase |
|---|---|---|
| \(\displaystyle K\) |  |  |
| \(\displaystyle (j\omega)^N\) |  |  |
| \(\displaystyle \frac{1}{(j\omega)^N}\) |  |  |
| \(\displaystyle \left(1 + \frac{j\omega}{z}\right)^N\) |  |  |
| \(\displaystyle \frac{1}{\displaystyle \left(1 + \frac{j\omega}{p}\right)^N}\) |  |  |
| \(\displaystyle \left[1 + \frac{2j\omega\zeta}{\omega_n} + \left(\frac{j\omega}{\omega_n}\right)^2\right]^N\) |  |  |
| \(\displaystyle \frac{1}{\displaystyle \left[1 + \frac{2j\omega\zeta}{\omega_k} + \left(\frac{j\omega}{\omega_k}\right)^2\right]^N}\) |  |  |
A decade is an interval between two frequencies with a ratio of 10. Thus, \(20\;\textrm{dB}/\textrm{decade}\) means that the magnitude changes \(20\;\textrm{dB}\) whenever the frequency changes tenfold or one decade.
If \(K\) is negative, the magnitude remains \(20\log_{10}K\) but the phase is \(\pm 180^\circ\).