Transfer Functions

The Idea

The idea here is that we take the LaPlace Transforms and Z Transforms w/ zero initial conditions and to build a direct relationship between the input and an output in a sort of way that looks like $y=mx$, where the $m$ term is our “gain”.

Given as: $P\left(s\right)$ in Continuous Time, or $G\left[z\right]$ in Discrete Time (See Difference Equations)

• Rational if $P\left(s\right)=\frac{N\left(s\right)}{D\left(s\right)}$ where N and D are polynomials (where the same applies in discrete time)
• Real if in addition the coefficients of N, D are all real numbers
• Proper if in addition, the degree of D ≥ degree of N
• Strictly proper if it is proper and the degree of D > degree of N

Has Poles and Zeroes.

Important

A real, rational transfer function $P\left(s\right)$ is stable if all the poles of P(s) lie in the open left half plane ${\mathbb{C}}^{-}$

Why Use Transfer Functions??

• Differential equations are difficult to handle
• We can dramatically simplify the input-output relationship by using these transfer functions
• The physical meaning of the transfer function is that it is the LaPlace transform of the impulse response

The general form…

• The poles are the roots of $P\left(s\right)$ and the zeroes are the roots of $Q\left(s\right)$
• The equation $P\left(s\right)=0$ is called the characteristic equation of $G\left(s\right)$
• The order of the denominator cannot be larger than the highest order of the denominator
• With block diagrams, the TF lets us do operations on dynamic systems in a transform domain
• Only defined for LTI

Frequency Response

• A response of a system to a sinusoidal input is also sinusoidal
• The gain $M\left(\omega \right)$ and phase delay $\varphi \left(\omega \right)$ can be obtained by the frequency response given by $G\left(j\omega \right)$

You use Transfer Functions in Block Diagrams