Bode plots for simple transfer functions

This lecture also appears in Control: an introduction.

It turns out that bode plots, both magnitude and phase, given their logarithmic scale (recall that the \(\omega\)-axes are also plotted logarithmically), are quite asymptotic to straight-lines for first- and second-order systems. Furthermore, higher-order system transfer functions can be re-written as the product of those of first-and second-order. For instance, $$\begin{align} H(s) &= \frac{\emp[1em] s + \emp[1em]}{s^3 + \emp[1em] s^2 + \emp[1em] s + \emp[1em]} \\ &= \emp[1em] \cdot \left( \emp[1em] s + 1 \right) \cdot \frac{1}{\emp[1em] s + 1} \cdot \frac{1}{s^2 + \emp[1em] s + \emp[1em]} \end{align}$$

Recall (from, for instance, phasor representation) that for products of complex numbers, phases \(\phi_i\) add and magnitudes \(M_i\) multiply. For instance, \[\begin{aligned} M_1 \angle{\phi_1} \cdot \frac{1}{M_2 \angle{\phi_2}} \cdot \frac{1}{M_3 \angle{\phi_3}} = \frac{M_1}{M_2 M_3} \angle{\left(\phi_1-\phi_2-\phi_3\right)}. \end{aligned}\] And if one takes the logarithm of the magnitudes, they add; for instance, \[\begin{aligned} \log\frac{M_1}{M_2 M_3} = \log M_1 - \log M_2 - \log M_3. \end{aligned}\] There is only one more link in the chain: first- and second-order Bode plots depend on a handful of parameters that can be found directly from transfer functions. There is no need to compute \(\abs{H(j\omega_0)}\) and \(\angle H(j\omega_0)\)!

In a manner similar to , we construct Bode plots for several simple transfer functions in this lecture. Once we have these simple “building blocks,” we will be able to construct sketches of higher-order systems by graphical addition because logarithmic magnitudes and phases combine by summation, as shown in .

Pole and zero at the origin

In , we have already demonstrated how to derive from the transfer function \(H(s) = s\), a zero at the origin, the frequency response function plotted in . Similarly, for \(H(s) = 1/s\), a pole at the origin, the frequency response function plotted in figure 13.2.

Real pole and real zero

The derivations for real poles and zeros are not included, but the resulting Bode plots are shown in figure 13.3.

Complex conjugate pole pairs and zero pairs

The derivations for complex conjugate pole pairs and zero pairs are not included, but the resulting Bode plots are shown in figure 13.4.