Symmetry Of Nyquist Plot

Symmetry of Nyquist Plot By Yong-Nien Rao For a LTI(Linear time invariant) circuit, the Nyquist plot of the transfer function is symmetric to the real axis. Now we’d like to prove it here. Let a LTI circuit with its transfer function H ( s ) . Thus H ( s ) is a rational function of real coefficient polynomials expressed as N ( s ) D( s ) , both N ( s ) and D( s ) are polynomials of real coefficients. For all the polynomials of real coefficients,

( )

∗

( )

∗

N s∗ = ( N ( s) ) D s ∗ = ( D( s ) )

Thus we could have

( )

H s∗ = ( H ( s) )

∗

…(eq.1)

Since Nyquist path NP is also symmetric to the real axis1. It means that if s is on the Nyquist path, so does s ∗ . That is s ∗ ∈ NP, ∀ s ∈ NP …(eq.2) For the Nyquist plot H ( s ) , ∀ s ∈ NP , From (eq.2), H s ∗ , ∀ s ∈ NP is on the Nyquist plot.

( )

And further then from (eq.1), we get ( H ( s ) ) ∗

, ∀ s ∈ NP is also on the Nyquist

plot. Since both H ( s )

, ∀ s ∈ NP and ( H ( s ) ) ∗

, ∀ s ∈ NP are on the Nyquist plot, the

Nyquist plot is symmetric to the real axis Q.E.D.

Joseph J. DiStefano, III et al, “Theory and Problems of Feedback and control systems” 2nd ed. P253~p256 1

© Copyright 2009 by Yong-Nien Rao

Page 1 of 1