Bode plots and frequency response

Overview

Only two things can happen to a sine wave passing through a linear, time-invariant system: it's magnitude can be changed; and the signal can be delayed. The delay and percentage change in the magnitude are a function of frequency. A transfer function, H(f), is a complex-valued function of frequency that specifies the magnitude and phase shift of a particular system for all frequencies. The change in amplitude is often called the gain, and the delay is usually thought of in terms of a phase shift of the sine wave.

One way to visualize a transfer function is to make two plots. The first plot shows the gain verses frequency on a set of log-log axes. The second plot shows the phase shift versus log frequency.

First-order R-C low-pass filter.

For example, consider a simple RC low-pass filter with the transfer function is given by

$ H(f) = \frac{V_{out}}{V_{in}} = \frac{1}{1 + j2\pi f RC} $

In this case the cutoff frequency is given by $ f_C = 1 / (2\pi RC) $.

Bode plot of low-pass filter with cutoff frequency $ f_C=1 $.

More examples

Bode plot of high-pass transfer function with cutoff frequency $ f_{HP}=1 $.

High-pass

$ H(f) = \frac{j f / f_{\rm HP}}{1 + j f / f_{\rm HP}} $

Second order low-pass transfer function. Note that the scale of the plots are double that of the 1st-order filter.

Second-order low-pass

$ H(f) = \left(\frac{1}{1 + j f / f_{\rm LP}}\right)^2 $

Frequency response of band-pass transfer function with $ f_{HP}=0.1 $ and $ f_{LP}=10 $.

Band pass

$ H(f) = \frac{j f / f_{\rm HP}}{1 + j f / f_{\rm HP}} \times \frac{1}{1 + j f / f_{\rm LP}} $