Freq Resp and expπ - Part C

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 add asymptotic approximations to the graphs - and get eπ.

In Part D we show the maths.

See also Original Paper at Control 2012.

Here the gain and phase plots are shown to which straight line asymptotes are added.

The actual curves start on one asymptote then move away and then approach the next.

The gain has two asymptotes - one where gain 1, the other has gradient -1 (on loglog) curves. These meet where ωT = 1.

The phase has three, where: phase = 0, phase = -π/2, and joined by the tangent to the phase at ωT = 1.

This tangent goes from ωT = 0.21 to 4.81, and 4.81/0.21 = eπ.