E.1 Root-matching Technique Applied to a First Order Lag Function

This example illustrates the use of the root-matching technique to develop a difference equation as described in section 5.3. The first order lag function in the s-domain is expressed as:

The corresponding z-domain transfer function with pole matched (as z = e ^{s ? t}) is

Applying the final value theorem to the s-domain transfer function

Lin _{s ?0} [ s H( s)/ s} = G

Applying the final value theorem to the z-domain transfer function

Therefore

Rearranging gives:

Hence

I( z) = kV( z) + e ^{-? t / ?}z ^-1 I( z)

or in the time domain the exponential form of difference equation becomes:

(E.1)

E.2 Root-matching Technique Applied to a First Order Differential Pole Function

This example illustrates the use of the root-matching technique to develop a difference equation as described in section 5.3, for a first order differential pole function. The s-domain expression for the first order differential pole function is:

The z-domain transfer function with pole and zero matched (using z = e ^{s ? t}) is

Applying the final value...