Fourier Series • Fourier Transform • Laplace Transform • Applications of Laplace Transform • Z ...

77 Pages·2011·1.11 MB·English

Basics in Systems and Circuits Theory
BSC Modul 4: Advanced Circuit Analysis
• Fourier Series
• Fourier Transform
• Laplace Transform
• Applications of Laplace Transform
• Z-Transform
Michael E.Auer 01.11.2011 BSC04
Basics in Systems and Circuits Theory
BSC Modul 4: Advanced Circuit Analysis
• Fourier Series
• Fourier Transform
• Laplace Transform
• Applications of Laplace Transform
• Z-Transform
Michael E.Auer 01.11.2011 BSC04
Basics in Systems and Circuits Theory
Trigonometric Fourier Series (1)
• The Fourier series of a periodic function f(t) is a
representation that resolves f(t) into a dc component and
an ac component comprising an infinite series of harmonic
sinusoids.
• Given a periodic function f(t) = f(t+nT) where n is an
integer and T is the period of the function.
∞
∑
f (t) = a + (a cos nωt + b sin nωt)
0 0 0 n 0
n=1
dc
ac

where ω =2π/T is called the fundamental frequency in
0
radians per second.
Michael E.Auer 01.11.2011 BSC04
Basics in Systems and Circuits Theory
Trigonometric Fourier Series (2)
• and a and b are as follow
n n

2 2
T T
a = ∫ f (t)cos(nωt)dt b = ∫ f (t)sin(nωt)dt
n o n o
T 0 T 0
• in alternative form of f(t)
∞
∑
f (t) = a + (c cos(nωt +φ )
0 n 0 n
n=1
dc
ac

b
where c = a2 + b2 , φ = − tan−1( n ) (Inverse tangent or arctangent)
n n n n
a
n

Michael E.Auer 01.11.2011 BSC04
Basics in Systems and Circuits Theory
Fourier Series Example

Determine the Fourier series of the
waveform shown right. Obtain the
amplitude and phase spectra.
1, 0 < t <1
f (t) =  and f (t) = f (t + 2)
0, 1< t < 2
 2/nπ,n = odd
a = 2 ∫T f (t)cos(nωt)dt = 0 and An =  0, n = even
n 0
T 0
 −90°,n = odd
2 T  2/nπ,n = odd φ = 
bn = T ∫0 f (t)sin(nω0t)dt = 0, n = even n  0, n = even ab)) APhmapslei tsupdeec t raunmd
∞
1 2 1
∑
f (t) = + sin(nπt), n = 2k −1
2 π n
k=1
Truncating the series at N=11
Michael E.Auer 01.11.2011 BSC04
Basics in Systems and Circuits Theory
Symmetry Considerations (1)
Three types of symmetry
1. Even Symmetry : a function f(t) if its plot is
symmetrical about the vertical axis.
f (t) = f (−t)
In this case,
2 T/2
a = ∫ f (t)dt
0
T 0
4 T/2
a = ∫ f (t)cos(nωt)dt
n 0
T 0
b = 0
n
Typical examples of even periodic function
Michael E.Auer 01.11.2011 BSC04
Basics in Systems and Circuits Theory
Symmetry Considerations (2)
2. Odd Symmetry : a function f(t) if its plot is
anti-symmetrical about the vertical axis.

f (−t) = − f (t)

In this case,
a = 0
0
4
T /2
b = ∫ f (t)sin(nωt)dt
n 0
T 0
Typical examples of odd periodic function
Michael E.Auer 01.11.2011 BSC04
Basics in Systems and Circuits Theory
Symmetry Considerations (3)
3. Half-wave Symmetry : a function f(t) if
a = 0
0
 4 T/2
 ∫ f (t)cos(nωt)dt , for n odd
T a =  0
T 0
f (t − ) = − f (t) n
 0 , for an even
2

 4 T/2
 ∫ f (t)sin(nωt)dt , for n odd
b =  0
T 0
n
 0 , for an even
Typical examples of half-wave odd periodic functions
Michael E.Auer 01.11.2011 BSC04
Basics in Systems and Circuits Theory
Symmetry Considerations (4)
Example 1
Find the Fourier series expansion of f(t) given
below.

2 ∞ 1  nπ  nπ 
∑
Ans: f (t) = 1− cos sin t 
π n  2   2 
n=1
Michael E.Auer 01.11.2011 BSC04
Basics in Systems and Circuits Theory
Symmetry Considerations (5)
Example 2
Determine the Fourier series for the half-wave
cosine function as shown below.

∞
1 4 1
∑
Ans: f (t) = − cos nt, n = 2k −1
2 π2 n2
k=1
Michael E.Auer 01.11.2011 BSC04