Transfer Function

Unit 6 - Mechatronics PEARSON BTEC HND

Electrical & Electronic
Engineering
Randeera Liyanage
Introduction to Transfer Function
Introduction to Transfer Function

System Representation
• Expressed by establishing a variable function
that algebraically relates a system’s output to
its input.

• System representation separates input,

system and output.

• Allows to algebraically combine

mathematical representations of subsystems
to yield a total system representation.
Introduction to Transfer Function

• The relationship of a function between the input and output of

a system, is known as the transfer function.

• The relationship between the Laplace transform of the output

and the Laplace transform of the input.
Introduction to Transfer Function

“The transfer function of a system is defined as the ratio

of Laplace transform output to the Laplace transform
input with all initial conditions assumed to be zero.“
Introduction to Transfer Function

Why are Initial Conditions Assumed to be Zero?

• To satisfy the properties of linearity.

• System should be linear.

• When initial conditions are zero, corresponding outputs are

also zero.
Transfer Function Equation

𝑳𝒂𝒑𝒍𝒂𝒄𝒆 𝒕𝒓𝒂𝒏𝒔𝒇𝒓𝒐𝒎 𝒐𝒖𝒕𝒑𝒖𝒕 𝒀(𝒔)

𝑻𝒓𝒂𝒏𝒔𝒇𝒆𝒓 𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏 𝑮 𝒔 =
𝑳𝒂𝒑𝒍𝒂𝒄𝒆 𝒕𝒓𝒂𝒏𝒔𝒇𝒓𝒐𝒎 𝒊𝒏𝒑𝒖𝒕 𝑿(𝒔)

𝑿(𝒔)
𝑮 𝒔 =
𝒀(𝒔)
 Transfer Function of a Feedback System

𝒀(𝒔) 𝑮(𝒔)
=
𝑿(𝒔) 𝟏 + 𝑮 𝒔 𝑯(𝒔)

*Assuming all initial conditions are zero

 Transfer Function of a Feedback System

• If there is no H(s) block in the feedback, but the system still

consists of a feedback loop, it is considered as a unity
feedback. (unity = 1).

• Transfer function of a system with unity feedback is,

𝒀(𝒔) 𝑮(𝒔)
=
𝑿(𝒔) 𝟏 + 𝑮 𝒔

*Assuming all initial conditions are zero

QUESTIONS?
Transfer Function – First Order Systems
Transfer Function – First Order Systems

𝑑𝑦
+ 𝑘1 𝑦 𝑡 = 𝑘2 𝑥 𝑡
𝑑𝑡

𝑌 𝑠 𝑘2
=
𝑋 𝑠 𝑠 + 𝑘1

*For first order system transfer functions, the denominator should be in the form of s+1.
Transfer Function – First Order Systems

Standard First Order Transfer Function Equation is,

𝒀(𝒔) 𝒌𝒔𝒔
𝑮 𝒔 = =
𝑿(𝒔) 𝝉𝒔 + 𝟏

• 𝑘𝑠𝑠 - Gain of the system when there are steady-state conditions (Steady
State Gain)

• 𝜏 - Time which is taken for the output of a system to reach 63% (0.63) of
the final output value. (Time constant)
First Order Systems with Step Response

1
• Step Input : 𝑋 𝑠 =
𝑠

𝑘𝑠𝑠
𝑌 𝑠 =
𝑠 𝜏𝑠 + 1

Standard first order transfer function with a step response is,

𝟏/𝝉
𝒀 𝒔 = 𝒌𝒔𝒔
𝟏
𝒔 𝒔+
𝝉
First Order Systems with Step Response

First order systems response to a

unit step.

𝑇𝑟 - Rise time

𝑇𝑠 - Setting time
QUESTIONS?
Transfer Function – Second Order Systems
Transfer Function – Second Order Systems

𝑑 2 𝑦(𝑡) 𝑑𝑦(𝑡)
2
+ 𝑘1 + 𝑘2 𝑦(𝑡) = 𝑘3 𝑥(𝑡)
𝑑𝑥 𝑑𝑥

𝑌 𝑠 𝑘3
= 2
𝑋 𝑠 𝑠 + 𝑘1 𝑠 + 𝑘2

*For second order transfer functions to be in standard form, coefficient in-front of

𝑠 2 should always be 1.
Transfer Function – Second Order Systems

Standard Second Order Transfer Function Equation is,

𝒀(𝒔) 𝒌𝒔𝒔 𝝎𝟐𝒏 *Coefficient in-front of

= 𝟐
𝑿(𝒔) 𝒔 + 𝟐𝜻𝝎𝒏 𝒔 + 𝝎𝟐𝒏 𝑠 2 should always be 1.

• 𝑘𝑠𝑠 - Steady state gain

• 𝜔𝑛 - Natural frequency

• 𝜁 – Damping ratio
Second Order Systems with Step Response

1 𝑘𝑠𝑠 𝜔𝑛2
𝑌 𝑠 =𝑋 𝑠 𝐺 𝑠 = × 2
𝑠 𝑠 + 2𝜁𝜔𝑛 𝑠 + 𝜔𝑛2

Standard first order transfer function with a step response is,

𝒌𝒔𝒔 𝝎𝟐𝒏
𝒀 𝒔 =
𝒔 𝒔𝟐 + 𝟐𝜻𝝎𝒏 𝒔 + 𝝎𝟐𝒏

𝒌𝒔𝒔 𝝎𝟐𝒏
𝒀 𝒔 =
𝒔 𝒔 + 𝒑𝟏 (𝒔 + 𝒑𝟐 )
Damping Ratio

• Damping ratio of a second order transfer function provides information

about the system behavior for a step input.

• Damping ratio depends on its values and has a relation to the location of
system poles.
Damping Ratio : Types

Undamped
𝜻=𝟎
Damping Ratio : Types

Underdamped
𝟎<𝜻<𝟏
Damping Ratio : Types

Critically Damped
𝜻=𝟏
Damping Ratio : Types

Over Damped
𝜻>𝟏
Damping Ratio : Types

Over Damped
𝜻>𝟏
Step Response to Varying Damping Ratios
Damping Ratio

Generally, in controls engineering the underdamped second-

order system is considered as a common model for physical
problems that displays unique and natural behavior.
Second Order Systems with Step Response

First order systems response to a

unit step.

𝑇𝑟 - Rise time

𝑇𝑠 - Setting time

𝑇𝑝 - Time to first peak

Second Order System Specifications

• 𝑻𝒑 - Time to First Peak

Time taken to reach the peak value

𝝅
𝑻𝒑 =
𝝎𝒏 𝟏 − 𝜻𝟐
Second Order System Specifications

• 𝑻𝒔 - Settling Time

Time taken to reach within ±2% of the steady state value

𝟒
𝑻𝒔 =
𝜻𝝎𝒏
Second Order System Specifications

• 𝑻𝒓 - Rise Time

Time required for the response to rise from 10% (0.1) to 90% (0.9) of the
final steady state value.

It cannot be directly computed using an equation since a precise

analytical relationship between rise time and damping ratio cannot be
found.
 Second Order System Specifications

• %𝑶𝑺 – Percentage Overshoot

Percentage difference between the maximum peak value 𝑐max and the final

peak value 𝑐𝑓𝑖𝑛𝑎𝑙 .

𝑐𝑚𝑎𝑥 − 𝑐𝑓𝑖𝑛𝑎𝑙
%𝑂𝑆 = × 100
𝑐𝑓𝑖𝑛𝑎𝑙

Percentage overshoot is a function only of the damping ratio. Hence,

− 𝜻𝝅 𝟏−𝜻𝟐
%𝑶𝑺 = 𝒆 × 𝟏𝟎𝟎
QUESTIONS?
Transfer Functions in the Industry
 Transfer Functions in the Industry

• The theories of transfer functions, more specifically the transient response of

systems is widely used in the industry for control application.

• These theories are used whenever a specific parameter needs to be controlled

at a specific level.

• Common Examples
• Motor Control

• Temperature Control

• Pressure Control
Transfer Functions in the Industry

Transient response (natural second order underdamped Industrial temperature

response) of an industrial motor controller. controller.
Examples
RC Circuit – Transfer Function

𝟏
𝑮 𝒔 =
𝑹𝑪𝒔 + 𝟏

FIRST ORDER TRANSFER FUNCTION.

Example – Mathematical Model of a Spring Mass
Damper System

𝑋(𝑠) 1
𝐺 𝑠 = =
𝐹(𝑠) 𝑚𝑠 2 + 𝑏𝑠 + 𝑘

𝟏
𝑮 𝒔 = 𝒎
𝟐 𝒃 𝒌
𝒔 + 𝒔+
𝒎 𝒎

SECOND ORDER TRANSFER FUNCTION.

 Example 1

• Determine the natural frequency, damping ratio and the steady state value for
a system containing the following transfer function.

1
𝐺 𝑠 = 2
𝑠 + 8𝑠 + 16

Hence, determine the nature of the damping of this system using the damping
ratio and system poles.
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