“How can we plot the gain and phase shift for a transfer function?”
A transfer function will change the magnitude and phase of a sinusoid in some way. So wouldn’t it be logical if we could plot this out on a graph? Well, let’s think about how we could do this ourselves. First, since we have to plot two different outputs (gain and phase shift) let’s put make two separate graphs side by side. Then, let’s put the input (frequency) on the x axis and the output (gain or phase shift) on the y axis. Now, since our input variable will cover an extremely large range, let’s make it on a logarithmic scale. Specifically, let’s take a frequency as an input, plug it into the formula 20log10(omega)), and then graph. Since the units on the x axis are not normal numbers but rather ratios, let’s give them the unit decibels. This type of plot is called a bode plot and is used for analyzing control systems worldwide.