Freq Resp and expπ - Part B

Three interesting numbers in Maths are π, the ratio of a circle's circumference to its diameter, Euler's number e, approx 2.71818, and i, √-1. It is well known that eiπ = -1. Here we discuss a use of eπ in frequency response, which uses i itself!

In frequency response, we input a sinusoid into a system (here first order linear), and consider its output as we change the frequency f of the input. The output is a sinusoid of the same frequency, but its amplitude changes, as does the delay or phase shift between input and output. Strictly we examine changes in gain (ratio of output and input amplitudes) and phase, with angular frequency (2πf).

In Part A we explained the basic concepts.

Here in Part B we plot gain and phase vs angular frequency (ω = 2 π * frequency).

In Part C we look at asymptotic approximations to the graphs - and get eπ.

In Part D we show how we model the system.

In Part E we show how we use these to find eπ.

See also Original Paper at Control 2012.

As gain and phase vary with ω, so we plot graphs.

We scale ω and Gain logarithmically as they vary over a wide range, and Phase linearly - it varies from 0 to -90O or 0 to -π/2 rads.

Vary ω by moving the slider and so see the values on the gain and phase graphs at each ω. You will see the phase almost reaches 0 at the lowest freq and -π/2 at the highest.