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 are the relevant maths.

See also Original Paper at Control 2012.

The system is modelled by a complex number involving i = √-1 and ω. T is its time constant.

The gain is the modulus of this, the phase is the argument, as shown below.

We consider these when ω is much less or much greater than 1/T (the corner frequency).

The low frequency gain asymptote is when ωT << 1, so √(1 + (ωT)2) ~ 1.

The high frequency asymptote, is when ωT >> 1, so √(1 + (ωT)2) ~ ωT.

The first and last phase asymptotes are where the phase is 0 and -π/2 rads.

To find the frequency range of the other, we need the gradient of phase noting it is plotted vs log(ωT).

Let the asymptote be from 1/r to 1*r, ie over a range of r2, which we show is eπ.