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
0 +∑(a0 cos nω0t + bn sin nω0t)

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 T 2 T
a
n =
T ∫0 f (t) cos(nωot)dt bn = T ∫0 f (t)sin(nωot)dt
• in alternative form of f(t)
∞

f (t) = a + (c cos(nω t +φ )
0 ∑ n 0 n

dc n=1
ac

2 2 −1 bn
where cn = an + bn , φn = − tan ( ) (Inverse tangent or arctangent)
an

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
2 T An = 
an =
T ∫0 f (t) cos(nω0t)dt = 0 and  0, n = even
 −90°,n = odd
bn =
T2 ∫0T f (t)sin(nω0t)dt = 02, / n π , n = eovdedn φn =  0, n = even ab) APhmapslei tsupdeec t raunmd
∞
1 2 1
f (t) = + sin(nπt), n = 2k −1
∑
2 π k=1 n
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
a0 =
T ∫0 f (t)dt
4 T / 2
an =
T ∫0 f (t) cos(nω0t)dt
bn = 0
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(nω t)dt
n
T ∫0 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
a0 = 0
 4 T / 2
T an =  T ∫0 f (t) cos(nω0t)dt , for n odd
f (t − ) = − f (t) 
 0 , for an even
2

 4 T / 2
bn =  T ∫0 f (t)sin(nω0t)dt , for n odd

 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=1 n  2   2 
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) = −
2 ∑ 2 cos nt, n = 2k −1
2 π k=1 n
Michael E.Auer 01.11.2011 BSC04