Freq Resp and expπ - Part D

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.

In Part B we plotted gain and phase vs angular frequency (ω = 2 π * frequency).

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

In Part D we show how we model the system.

Here in Part E we show how we use these to find eπ.

See also Original Paper at Control 2012.

From Part D we know the equation for the phase and the phase asymptotes.

The middle asymptote is centred at the corner frequency ωc.

We will assume it goes from ωc/r to ωc*r, as shown below.

To find the frequency range of the middle asymptote, we need the gradient of phase.

It must be noted that phase is plotted vs log(ωT).

From the graph, the phase changes from 0 to -π/2 from log(ωc/r) to log(ωc*r), so :

But we also find this by differentiating phase with respect to log(ωT) and evaluating at ωc

Equating these two calculations of the gradient :

Hence we have a use for eπ.