Transfer Function Vs Sinusoid Freqency Response

Relationship Between Transfer Function And Frequency Response By Yong-Nien Rao In transfer function theory, it is often addressed that the transfer function with s = jw could be derived from the frequency response by experiments. This article is to prove it mathematically. Apply a sinusoid wave sin ( wt ) to the input of a circuit h( t ) , the sinusoidal steady state output of h( t ) must be Asin ( wt + θ ) if this circuit is stable. Let the circuit’s transfer function is H ( s ) , we want to prove that H ( s ) | s = jw = A

∠ H ( s ) |s = jw = θ

The time domain equation of this sinusoidal-input sinusoidal-output circuit could be describe as1 sin( wt ) ∗ h(t ) = A sin( wt + θ ) …(eq.1) Apply the Laplace transform to the (eq.1), we get w H ( s ) = L{ A sin( wt + θ )} 2 s + w2 = A • L{ sin( wt ) cos(θ ) + sin(θ ) cos( wt )} w s   = A cos(θ ) 2 + sin(θ ) 2 2 s + w s + w 2    e iθ + e − iθ w e iθ − e − iθ s  = A + 2 2 2 2 s + w 2j s + w 2  

…(eq.2)

Then  e iθ + e − iθ e iθ − e − iθ s  H ( s ) = A + 2 2j w  

…(eq.3)

Let s = jw in (eq.3), we get  e jθ + e − jθ H ( s ) | s = jw = A + 2   e jθ + e − jθ e jθ − e − jθ = A + 2 2 

e jθ − e − jθ jw  2j w    2e jθ  = A   2    

…(eq.4)

= Ae jθ

And from (eq.4), it is trivial that

1

Leon O. Chua et al, “Linear And Nonlinear Circuits”, p620~p623

© Copyright 2009 by Yong-Nien Rao

Page 1 of 2

H ( s ) | s = jw = A

∠ H ( s ) |s = jw = θ Q.E.D.

© Copyright 2009 by Yong-Nien Rao

Page 2 of 2