2. •Sinusoidal functions occupy a unique position in engineering.
•They are easy to generate and the steady state response of a linear
system to dc and sinusoidal excitations can be found easily by using the
impedance concept.
•In practice input signals are of more complex nature being non
sinusoidal periodic and non periodic waveforms.
•The French mathematician, J.B.J. Fourier (1758-1830) showed that
arbitrary periodic functions could be represented by an infinite series of
sinusoids of harmonically related frequencies known as Fourier series.
•Non periodic waveforms can be represented by Fourier transform.
Fourier Series
3. •A signal f(t) is said to be periodic of period T if f(t)=f(t+T) for all t.
•Several signal waveforms that fulfill this requirement are shown in Fig.
below
t
•Fourier series can be represented either in the form of infinite
trigonometric series or infinite exponential series.
•Fourier series consists of dc terms as well as ac terms of all harmonics.
Fourier Series
4. Any periodic signal f(t) (satisfying certain conditions) can be expressed
as a summation of sine and cosine functions of frequencies 0,ω,2ω,3ω
…..kω, ω where refers to the fundamental frequency and time period.
Thus,
where the coefficients an and bn are given by
Trigonmetric Fourier Series
0
1
0 1 2 3
1 2 3
( ) [ cos( ) sin( )]
cos cos 2 cos3 ... cos ..
sin sin 2 sin 3 ..... sin ....
n n
n
n
n
f t a a n t b n t
a a t a t a t a n t
b t b t b t b n t
ω ω
ω ω ω ω
ω ω ω ω
∞
=
= + +
= + + + +
+ + + +
∑
0
2
( )cos
T
na f t nwt dt
T
= ∫ 0
2
( )sin
T
nb f t nwt dt
T
= ∫0
0
1
( )
T
a f t dt
T
= ∫
5. Combining the sine and cosine terms, a more compact representation of
the series, either in the sine or cosine form, may be obtained as follows:
Let us consider the nth harmonic term
where, the amplitude and phase, of the nth harmonic are given by
•Writing C0=a0
Trigonmetric Fourier Series
( )
( ) ( )
( ) [ ]
( )
2 2
2 2 2 2
2 2
( ) cos sin
.cos .sin
cos .cos sin .sin
cos
n n n
n n
n n
n n n n
n n n n
n n
f t a n t b n t
a b
a b n t n t
a b a b
a b n t n t
C n t
ω ω
ω ω
θ ω θ ω
ω θ
= +
= + +
+ +
= + +
= −
( )2 2 -1
; n=tann n n n nC a b b aθ+=
0
1
( ) cos( )n n
n
f t C C n tω θ
∞
=
= + −∑
6. In a similar way, the series may be represented in the sine form as
where,
The plot of Cn as a function of n or nω is known as the amplitude
spectrum and the plot of θn as a function of n or nω is known as the
phase spectrum.
0
1
( ) sin( )n n
n
f t C C n tω φ
∞
=
= + +∑
( )1
tann n nb aφ −
=
7. Find the Fourier series for the saw-tooth wave shown in Fig.
The analytical form of the function f(t) is given by
The coefficients are evaluated as below:
1
( ) , 0 2
2
f t t tω ω π
π
= ≤ ≤
2 2
0
0 0
1 1
( ) ( ) ( ) ( ) 0.5
2 2 2
t
a f t d t d t
π π ω
ω ω ω
π π π
= = =∫ ∫
2
2
2 20
0
1 1 cos ( )
( )cos ( ) ( ) sin ( ) 0
2 2
n
t t n t
a n t d t n t
n n
π
π ω ω ω
ω ω ω
π π π
= = + =
∫
2
2
2 20
0
1 1 sin ( ) 1
( )sin ( ) ( ) cos ( )
2 2
n
t t n t
b n t d t n t
n n n
π
π ω ω ω
ω ω ω
π π π π
= = − + = −
∫
8. So, the Fourier series can be written as:
Here, C0=a0=0.5 and as an=0 hence Cn=bn=1/nπ. So, the amplitude
spectrum is as shown in Fig.
1
1
( ) 0.5 sin
n
f t n t
n
ω ω
π
∞
=
= − ∑
9. Certain kinds of symmetries found in non sinusoidal periodic signal
waveforms leads to simpler Fourier coefficient calculations i.e. absence
of a0,an or bn coefficients from the series.
There are four types of waveform symmetries, which are discussed as
below.
•Even Symmetry
•Odd Symmetry
•Half Wave Symmetry
•Quarter Wave Symmetry
Wave Form Symmetry
10. Even Symmetry
A function f(x) is said to be even, if ,
Examples of even functions are x2
, cos x, cosh x, tanx2
, 1+2x2
+4x4
, etc.
It is to be noted that the even nature is preserved on addition of a
constant.
The sum of even functions remains even.
The waveforms of some even functions are shown in Fig.
It is found that these functions exhibit mirror symmetry about the
vertical axis, the negative half being the mirror image of the positive
( ) ( )f x f x= −
11. In Fourier series expansion of the even signal waveforms, the bn-terms
are absent i.e. only a0 and an terms are present.
2
0
4
( )cos
T
na f t nwt dt
T
= ∫
0nb =
2
0
0
2
( )
T
a f t dt
T
= ∫
12. Odd Symmetry
A function f(x) is said to be odd, if ,
Examples of odd functions are x, x3
, sin x, x cos 10x. etc.
The sum of odd functions remains odd.
The waveforms of some even functions are shown in Fig.
The addition of a constant removes the odd nature of the function.
( ) ( )f x f x= − −
13. The waveform symmetry associated with odd functions has the effect of
making the Fourier expression contain only the sine terms. Thus,
2
0
4
( )sin
T
nb f t nwt dt
T
= ∫
0na =
0 0a =
14. Half Wave Symmetry
A function f(x) is said to be half wave symmetric, if ,
The waveforms of some half wave symmetric functions are shown in
Fig.
In general, Fourier series expansion of these waveforms contains both an
and bn coefficients, i.e. odd harmonics sine and cosine terms are present
unless the function is even or odd as well.
( ) ( 2)f x f x T= − ±
15. Quarter Wave Symmetry
Waveforms showing half wave symmetry with either odd or even
symmetry are said to have quarter wave symmetry. Waveforms as shown
in Fig. a,b of slide, have both even as well as half wave symmetries,
similarly, waveforms as shown in Fig. 11.5-a,b and c , have both odd as
well as half wave symmetries. Hence, these waveforms have quarter
wave symmetry.
16. Find the Fourier series expansion of the periodic rectangular waveform
shown in Fig.
The function is even. Hence, only cosine terms will be present, i.e. bn=0.
The function f(t) is defined as
0 2 4
( ) 1 4 4
0 4 2
for T t T
f t for T t T
for T t T
− ≤ ≤ −
= − ≤ ≤
≤ ≤2 2
0
2 2
1 1 1
( ) (1)
2
T T
T T
a f t dt dt
T T− −
∴ = = =∫ ∫
2 4
2 4
2 2 2
( )cos cos
T T
n
T T
a f t n tdt n tdt
T T n
ω ω
π− −
= = =∫ ∫
17. Application of Fourier Series in Network Analysis
Consider a case, when a non sinusoidal voltage is applied to a linear
network. The Fourier series expansion of this waveform represents linear
combination of various harmonic voltages, which give rise to
corresponding harmonic currents in the circuit.
Each term of the Fourier series of voltage is considered to represent an
individual voltage source.
The equivalent impedance of the network at each harmonic frequency nω
is used to compute the current at that harmonic.
The sum of these individual responses provides the total response of the
circuit as per the principle of superposition.
18. Effective Value or RMS value
If,
by definition, the effective value or rms value can be written as
where c0 is the dc component and c1, c2, c3, … are the amplitudes of the
harmonics.
0
1
( ) [ cos( ) sin( )]n n
n
f t a a n t b n tω ω
∞
=
= + +∑
[ ]
[ ]
2 2 2 2 2 2 2
2 2 2 2
1 22
0
0
1
1 2
0 1 2 3 1 2 3
1 2
0 1 2 3
1
[ cos( ) sin( )]
1
........ .....
2
1
......
2
T
rms n n
n
F a a n t b n t dt
T
a a a a b b b
c c c c
ω ω
∞
=
= + +
= + + + + + + + +
= + + + +
∑∫
19. In general, if the voltage and current are given by
0( ) sin( )n nv t V V n tω φ= + +∑ 0( ) sin( )n ni t I I n tω θ= + +∑
Then, the effective values of the voltage and current are given as
( )2 2
1 2
1 22 2 2 2 2 2
0 1 2 3 0 1 2 3
1
...... ......
2
rms rms rms rmsV V V V V V V V V
= + + + = + + +
( )2 2
1 2
1 22 2 2 2 2 2
0 1 2 3 0 1 2 3
1
...... ......
2
rms rms rms rmsI I I I I I I I I
= + + + = + + +
20. Exponential Series:
• A compact way of expressing the Fourier series
•The sine and cosine forms of the trigonmetric series are represented
in exponential form using Euler’s identity.
Fourier Series
21. Exponential Series:
• A new coefficient cn is defined
• Now, f(t) becomes
• This is the complex or Exponential form of f(t)
Fourier Series
22. Exponential Series:
• This form is more compact than the sine-cosine form.
• The coefficient can be found as:
• The plots of the magnitude and phase of cn versus now are called
the complex amplitude spectrum and phase spectrum of f(t)
respectively.
Fourier Series
23. • Consider a periodic pulse train as shown. We have to find its phase
and amplitude spectra.
• The period of the pulse train T= 10
ω0= 2π/10=π/5
Fourier Series
24. • cn is the product of 2 and a function of the form sin x/x, which is
known as sinc function. Thus,
Some properties of the sinc function are:
For an integral multiple of π, the value of sinc is zero
• The sinc function always shows even symmetry
Fourier Series
26. • Fourier series enables us to represent a periodic function as a sum
of sinusoids and to obtain the frequency spectrum from the series.
• Fourier transform allows us to extend the concept of a frequency
spectrum to non periodic functions.
• The transform assumes that a non periodic function is a periodic
function with an infinite period.
• Thus the Fourier transform is an integral representation of a non
periodic function that is analogous to a Fourier series representation
of a periodic function.
Fourier Transform
27. • The Fourier transform is an integral transform like Laplace
transform, which transforms the function in time domain to
frequency domain.
•Fourier transform is useful in communication systems and digital
signal processing, in situations where Laplace transform does not
apply.
• While Laplace transform can only handle circuits with inputs for
t>0 with initial conditions, Fourier transform can handle circuits
with inputs for t<0 as well as those for t>0.
Fourier Transform
28. • Consider a non periodic function p(t) as shown below. Also
consider a periodic function f(t), whose shape for one period is same
as p(t).
•
• If we let period T→∞ , only a single pulse of width τ remains as all
other adjacent pulses move to ∞. The function f(t) is no longer
periodic.
Fourier Transform
31. Fourier Transform
The Inverse Fourier transform
0 0( ). ( ) ( )f t t t dt f tδ
∞
−∞
− =∫
Shifting Property of the impulse function:
When a function is integrated with impulse function, we obtain
the value of the function at the point where the impulse occurs
39. Filters
A filter is a frequency selective electrical network that allows signal of
desired band of frequencies to pass freely whilst attenuates the signals at
other frequencies.
Ideally, a filter should produce no attenuation in the desired band, called
the transmission band or pass band.
It should provide infinite attenuation at all other frequencies, called
attenuation band or stop band.
The frequencies that separate the pass band and stop band are called cut-
off frequencies(fc).
40. Classification of Filters
Filters can be classified on the basis of the range of the pass band and
stop band frequencies as:
1. Low Pass Filters (LPF)
2. High Pass Filters (HPF)
3. Band Pass Filters (BPF)
4. Band Stop or Band Elimination Filters (BSF)
41. Classification of Filters
- Passive Filters, which are made of L and C components
Filters also can be classified on the basis of components they are made of
as:
-Active Filters are made of components such as operational amplifiers
42. Active Filters
In the design of passive filters, inductor is an integral part. Inductor
creates some problems and because of other advantages of using active
elements, passive filters are seldom used in practical. Some of the
limitations of passive filters are:
-The use of inductor as a network element is not desirable especially
at low frequencies(less than 1 kHz) as at these frequencies practical
inductors of reasonable Q tend to become bulky, and expensive. The
dissipative losses start increasing when an inductor is minimized.
-It is necessary to cascade many sections to form a composite filter.
A composite filter has a prototype filter, two terminating half
sections(one at each end) and an m-derived section. Due to
presence of so many components, it becomes bulky.
43. Active Filters
While cascading different sections of filters a buffer or
isolation amplifier is required to prevent loading of the
circuit.
There is a need for an external amplifier to the required gain.
44. -The active filter provides gain and frequency adjustment
flexibility.
-Due to high input resistance and low output resistance of the
Op-Amp, they do not cause loading of the source or load.
-Reduction in size and weight and increased equipment density.
Reduction in power consumption.
-More economical than passive filters due to the use of Op-
Amps and absence of inductors.
45. References
1.C.K. Alexander and M.N.O. Sadiku, “Fundamentals of Electric
Circuits,” 3rd Edition, Tata McGraw Hill, 2008.
2.M. E. Valkenburg, “Network Analysis,” 3rd Ed., Pearson Prentice
Hall, 2006.
3.W. H. Hayt, J. E. Kemmerly and S. M. Durbin, “Engineering Circuit
Analysis,” 6th Edition, Tata McGraw Hill, 2007.
