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# 1.

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Department of Electrical Engineering

Linear and Non Linear Systems and Their Effects

Definition of a Linear System

Characteristic of a Linear System

Non Linear Systems

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Department of Electrical Engineering
2. LINEARITY & NON-LINEARITY & THEIR EFFECTS

2.1 Linearity

(i) Definition:

A system is said to be linear if it obeys the principles of superposition. That is, the
response of a linear system to two or more inputs is equal to the sum of the
responses of the system to each input acting separately in the absence of all other
inputs
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Department of Electrical Engineering
2. LINEARITY & NON-LINEARITY & THEIR EFFECTS

2.1 Linearity

(ii) 3 basic characteristics:

(i) Scaling

(ii) Superposition

(iii) Time-Invariance
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Department of Electrical Engineering
2.2 Linear Operations

If

where e = angular frequency = 2tf , then output will also have angular frequency
e.

Frequency remains unchanged in response to the allowable linear operations of:

Subtraction
Differentiation
Integration

For a linear system, the frequency components at the input are preserved at the
output normally with scaling and delay.

x t t ( ) sin( ) = e
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Department of Electrical Engineering
2.3 Non-Linearity

Power series model:

where a,b,c,d,etc. are coefficients which specify the nature of the non-linearity,
e.g.

For non-linear 1, number of coefficients in series < non-linear 2.
y t a b x t c x t d x t ( ) [ ( )] [ ( )] [ ( )] = + + + +
2 3

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Department of Electrical Engineering
2.4 Single Sinusoidal Input to Non-Linear System

(i) (linear)

Fundamental

(ii) Simplest non-linearity (2nd order)

DC 2nd Harmonic

(iii) (3rd order)

Fundamental 3rd Harmonic

(iv) (4th order)

DC 2nd Harmonic 4th Harmonic

y x
k t
=
= cos ( ) e
| |
y x
k t
k
t
=
=
= +
2
2 2
2
2
1 2
cos ( )
cos( )
e
e
| |
y x
k t
k
t t
=
=
= +
3
3 3
3
4
3 3
cos ( )
cos ( ) cos( )
e
e e
| |
y x
k t
k
t t
=
=
= + +
4
4 4
4
8
3 4 2 4
cos ( )
cos( ) cos( )
e
e e
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Department of Electrical Engineering
2.5 Multi-Term Non-Linear Characteristic

e.g. Series:
with input:

Corresponding output:

Rearranging terms:

y a b x c x d x e x = + + + +
2 3 4
x t = cos( ) e
y a
b t
c c
t
d
t
d
t
e e
t
e
t
= +
+
+ +
+ +
+ +
cos( )
cos ( )
cos( ) cos( )
cos( ) cos( )
e
e
e e
e e
2 2
2
3
4 4
3
3
8 2
2
8
4
y a
c e
DC
b
d
t Fundamentals
c e
t nd Harmonics
d
t rd Harmonic
e
t th Harmonic
= + +

(
+
+

(
+
+

(
+
+
2
3
8
3
4
2 2
2 2
4
3 3
8
4 4
cos( )
cos( )
cos( )
cos( )
e
e
e
e
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Department of Electrical Engineering
2.6 Multiple Sinusoidal Input to Non-Linear System

(i) If

and

y x nd order =
2
2 ( )
x t t = + cos( ) cos( ) e e
1 2
| |
| | | |
y t t
t t t t
t t
t t
DC
t nd Harmonic
t nd Harmonic
t Difference Frequency Term
t Sum Frequency Term
= +
= + +
= + + + +
+ +
= +
+
+
+
+
cos ( ) cos ( )
cos ( ) cos ( ) cos ( ) cos ( )
cos ( ) cos ( )
cos ( [ ] ) cos ( [ ] )
cos ( )
cos ( )
cos ( [ ] )
cos ( [ ] )
e e
e e e e
e e
e e e e
e
e
e e
e e
1 2
2
2
1
2
2 1 2
1 2
1 2 1 2
1
2
1 2
1 2
2
1
2
1 2
1
2
1 2
1
1
2
2 2
1
2
2 2
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Department of Electrical Engineering
2.6 Multiple Sinusoidal Input to Non-Linear System (contd.)
(ii) If

and

Similarly for y = x
4
, x
5
, etc.
The sum & difference terms are known as Intermodulation Products (IPs).
y x rd Order =
3
3 ( )
x t t = + cos( ) cos( ) e e
1 2
| |
| | | |
| | | |
y t t
t t t t t t
t t t t
t t t t
t t t t
t
= +
= + + + +
+
= + + + +
+ + +
= +
cos ( ) cos ( )
cos ( ) cos ( ) cos ( ) cos ( ) cos ( ) cos ( )
cos ( ) cos ( ) cos ( ) cos ( )
cos ( ) cos ( ) cos ( ) cos ( )
cos ( ) cos ( ) cos ( ) cos ( )
cos ( ) cos (
e e
e e e e e e
e e e e
e e e e
e e e e
e e
1 2
3
3
1 1
2
2
2
1 2
3
2
2
2
1
2
2 1
1 1 2 2
1 2 2 1
1
2
2
1
4
3 3
1
4
3 3
3
2
1 2
3
2
1 2
9
4
9
4
| |
| |
2
1 2
2 1 2 1
1 2 1 2
1
4
3
1
4
3 3
3
4
2 2
3
4
2 2
3
t Input Frequencies
t t rd Harmonics
t t
t t
rd Order Sum Difference Terms
)
cos ( ) cos ( )
cos ( [ ] ) cos ( [ ] )
cos ( [ ] ) cos ( [ ] )
&
+
+ +
+ + +
+ +

e e
e e e e
e e e e
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Department of Electrical Engineering
2.7 Non-Linear Operations

In general, operations occurring within non-linear system models are:

multiplication
division
raising to the n
th
power ( n > 2 )
taking the n
th
root ( n > 2 )

(a) These operations have the effect of increasing the number of frequency
components present at the output in comparison with the number at the input.

(b) The additional components are harmonics and IPs of the input frequencies.

(c) As the order of the non-linearity increases, so the number of output frequency
components increases.

(d) An n
th
order non-linearity can give rise to harmonics and IPs up to and
including n
th
order.

(e) The additional components will produce signal distortion.
1.11
Department of Electrical Engineering
2.8 Distortion in Linear Systems
Linear systems are not distortion-free, e.g.

Output components will combine to give a waveform which is different from the
input.
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