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Transfer Function

The document discusses transfer functions and important definitions related to linear time-invariant systems. It defines impulse response and transfer function, and how the transfer function is obtained from the impulse response or by taking the Laplace transform of the input-output relation. It also provides more details on properties and characteristics of transfer functions.

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Ananya Sinha
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38 views10 pages

Transfer Function

The document discusses transfer functions and important definitions related to linear time-invariant systems. It defines impulse response and transfer function, and how the transfer function is obtained from the impulse response or by taking the Laplace transform of the input-output relation. It also provides more details on properties and characteristics of transfer functions.

Uploaded by

Ananya Sinha
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Transfer

Function
Important Definitions
• Impulse Function
• Consider that a linear time invariant (LTI) system with input u(t) and output y(t).
• Impulse response is g(t), which is defined as the output when the input is a
unit-impulse function δ(t).
• Transfer Function
• The transfer function G(s) of an LTI single-input-single-output (SISO) system is
defined as the Laplace transform of the impulse response, with all the initial
conditions set to zero.

• The transfer function G(s) is defined as


•

•
Characterizing an LTI System
• Input-output relation of a linear time-invariant system is described by the following n th
order differential equation with constant real coefficients:

• ao, a1, ... , an-1 and bo, b1, ... , bm are real constants.

• Transfer function of the linear system - take the Laplace transform on both sides of the
equation and assume zero initial conditions. The result is

•
More about transfer function
● Defined only for LTI system, not for nonlinear systems.
● The transfer function between an input variable and an output variable of a system -
Laplace transform of the impulse response.
● Alternately, ratio of the Laplace transform of the output to the Laplace transform of the
input.
● All initial conditions of the system are set to zero.
● The transfer function is independent of the input of the system.
● The transfer function of a continuous-data system is expressed only as a function of the
complex variable s; not a function of the real variable, time, or any other variable that is
used as the independent variable.
● For discrete~data systems modeled by difference equations, the transfer function is a
function of z when the z-transfonn is used.
More about transfer function
• The transfer function is
• strictly proper if the order of the denominator polynomial is greater than that of the numerator
polynomial (i.e., n > m).
• proper if n = m
• improper if m > n.
• The characteristic equation of a linear system is defined as the equation obtained by
setting the denominator polynomial of the transfer function to zero. So, for

Characteristic equation is:

• Stability of linear, single-input, single-output systems is completely governed by the roots


of the characteristic equation
Laplace Transform Tables
Voltage Current relationships
Force - Velocity Impedance Relations
Analogy between Electrical and Mechanical Systems
Analogy between Electrical and Mechanical Systems

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