Distribution system and power quality
GL25
9_Fourier_Series_Application_Supporting_PPT.pdf
ECE 5750 Supporting document Part 9: Fourier series and applications
Instructor: Dr. Ha Le Department of Electrical and Computer Engineering
California State Polytechnic University, Pomona
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What will be presented? 1. Trigonometric Fourier Series 2. Symmetry Considerations 3. Exponential Fourier Series 4. Applications: Steady-state response of circuits Average power calculations with periodic functions The RMS value of periodic functions Spectrum analyzers Filters
(Alexander and Sadiku “Fundamentals of Electric Circuits”, Chapter 17)
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Why need Fourier series? Fourier series is a means for analyzing circuits
with periodic, nonsinusoidal excitations. Most of the functions of a circuit are periodic. They can be decomposed into infinite number of sine
and cosine functions that are harmonically related (harmonics).
A complete response of a forcing function = ∑ Partial response to each harmonics
Practical applications: Steady-state response of circuits Spectrum analyzers Filters Average power calculations etc.
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Trigonometric Fourier series (1) Definition: 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. The Fourier series of f(t) has the form
where 0=2/T is the fundamental frequency in rad/s, a0 is the DC component or the average value of f(t), an and bn are Fourier coefficients.
Harmonics have frequencies: 0 20 30 40 …
How to calculate a0, an, bn?
0 0 0 1
( ) ( cos sin )n n ndc
ac
f t a a n t b n t
Sufficient conditions on f(t) to yield a convergent Fourier series:
1. f(t) is single-valued everywhere.
2. f(t) has a finite number of discontinuities in any one period. The discontinuities must be of finite size.
3. f(t) has a finite number of maxima and minima in any one period.
4. The integral
Fourier series (2)
Dirichlet conditions
0
0 0( ) for any
t T
t f t dt t
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Example: Convergence of Fourier series
Fourier series (3)
Dirichlet conditions
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0sin 0 0
dttnw T
0cos 0 0
dttnw T
(b)
0 0 0 00 0 1
sin cos sin sin 0 2
T T kw t nw tdt k n w t k n w t dt
nkif nkifT
dttwnktwnk
dttnwtkw
T
T
,0 ,2
coscos 2 1
sinsin
0 00
0 00
nkif nkifT
dttwnktwnk
dttnwtkw
T
T
,0 ,2
coscos 2 1
coscos
0 00
0 00
Trigonometric Fourier series (4)
Useful trigonometric integrals
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(a)
(c)
(d) (e)
T
n nn
TT dttnwbtnwadtadttf
0 1
000 00 sincos
Based on (a) (b)
0sincos 0
1 00
T
n nn dttnwbtnwa
T
dttf T
a 00
1
0a is called the DC component or the average value of tf
Trigonometric Fourier series (5) Evaluation of Fourier coefficient a0
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Integrating both sides of the Fourier series definition of f(t)
Based on (b)
T
n n
T
n n
TT
dttkwtnwb
dttkwtnwadttkwatdtkwtf
0 1
00
0 1
000 000 0
cossin
coscoscoscos
0 00 1
cos cos 2
T
n n n
T a nw t kw tdt a
0cossin 0
1 00
T
n n dttkwtnwb
0cos 0 00
dttkwa T
Based on (c)
Based on (e)
When k=n
Trigonometric Fourier series (6) Evaluation of Fourier coefficient an
00 2
cos T
na f t nw tdtT
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Based on (a)
T
n n
T
n n
TT
dttkwtnwb
dttkwtnwadttkwatdtkwtf
0 1
00
0 1
000 000 0
sinsin
sincossinsin
0 00 1
sin sin 2
T
n n n
T b nw t kw tdt b
0sincos 0
1 00
T
n n dttkwtnwa
0sin 0 00
dttkwa T
Based on (c)
Based on (d)
When k=n
Trigonometric Fourier series (7) Evaluation of Fourier coefficient bn
00 2
sin T
nb f t nw tdtT
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Summary of trigonometric Fourier series (8) FS and evaluation of coefficients a0, an, bn, sine and
cosine form
Alternative form of f(t): amplitude-phase form
T
on dttntfT a
0 )cos()(
2
T
on dttntfT b
0 )sin()(
2
ac
n nn
dc
tnAatf
1
00 )cos(()(
2 2 1 n , tan
n n n n
n
b A a b
a
T
dttf T
a 00
1
0 0 0 1
( ) ( cos sin )n n ndc
ac
f t a a n t b n t
0 0 0cos sin cosn n n na nw t b nw t A nw t
2 2 n n nA a b
Phase spectrum
Trig. Fourier series: Frequency spectrum (9)
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Harmonic amplitude
1tan nn n
b a
The frequency spectrum of a signal consists of the plots of the amplitudes and phases of the harmonics versus frequency.
Example 1 (Prob. 17.3): Calculate the Fourier coefficients a0, an, and bn of the following waveform. Plot the amplitude and phase spectra.
Examples: Trig. Fourier series (1)
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Solution:
Solution (cont.):
Examples: Trig. Fourier series (2)
0 00 0
00
2 1 ( ) cos( )
2 ( ) sin( )
T T
n
T
n
a f t n t dt a f t dt T T
b f t n t dt T
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Solution (cont.):
Examples: Trig. Fourier series (3)
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0 00 0
00
2 1 ( ) cos( )
2 ( ) sin( )
T T
n
T
n
a f t n t dt a f t dt T T
b f t n t dt T
Solution (end):
Examples: Trig. Fourier series (4)
2 2 n n nA a b
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1tan nn n
b a
Amplitude and phase spectrum
Example 2: Determine the Fourier series of the pulse waveform shown in the following figure.
Examples: Trig. Fourier series (5)
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Solution for Example 2: Matlab plot of the Fourier series of the pulse waveform, n=500
Examples: Trig. Fourier series (4)
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Three types of symmetry: even, odd, half-wave 1. Even Symmetry: a function f(t) if its plot is symmetrical about
the vertical axis (even function)
In this case,
Symmetry considerations (1)
)()( tftf
0
)cos()( 4
)( 2
2/
0 0
2/
00
n
T
n
T
b
dttntf T
a
dttf T
a
Examples of even periodic function 19
Note: FS is a cosine series (cosine function is even)
2. Odd Symmetry : a function f(t) if its plot is anti-symmetrical about the vertical axis (odd function)
In this case,
)()( tftf
0
/ 2
00
0, 0 4
( ) sin( )
n
T
n
a a
b f t n t dt T
Examples of odd periodic function 20
Symmetry considerations (2)
Note: FS is a sine series (sine function is odd)
3. Half-wave Symmetry: a function f(t)
Symmetry considerations (3)
0
/ 2
00
/ 2
00
0 4
( ) cos( ) , for n=odd
0 , for n=even
4 ( ) sin( ) , for n=odd
0 , for n=even
T
n
T
n
a
f t n t dt a T
f t n t dt b T
Typical examples of half-wave odd periodic functions 21
( ) 2 T
f t f t
Symmetry considerations (4) Summary of Fourier coefficients
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Example 17.3: Find the Fourier series expansion of f(t) given below.
Examples: Symmetry (1)
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Solution:
Determine the fundamental frequency and specify the type of symmetry present in the following functions.
Examples: Symmetry (2)
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Fourier series of selected waveforms (1)
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Fourier series of selected waveforms (2)
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Application to circuit analysis Steps for applying Fourier series (FS)
to find the steady-state response of a circuit forced by a nonsinusoidal periodic excitation (function)
1. Express the excitation signal as a Fourier series.
2. Transform the circuit from the time domain to the frequency domain.
3. Find the response of the DC and AC components in the Fourier series.
4. Add the individual DC and AC responses using the superposition principle Circuit response.
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Example 1: Find the steady-state response v0(t) of the following RC circuit, given that the input vs(t) consists of first 3 terms of the Fourier series of a square wave (n = 0, 1, 3, n=0 is the DC term)
Examples: Circuit analysis using FS (1)
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Example 2: Find i(t) in the following circuit given that
Examples: Circuit analysis using FS (2)
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Average power:
Voltage and current as FS (amplitude-phase form):
The average power is
Average Power and RMS Values (1)
dc n 0 dc n 0 1 1
( ) V V cos( ) and ( ) I I cos( )n n n n
v t n t i t m t
dc dc n n 1
1 P V I V I cos( )
2 n nn
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Total average power is the sum of the average powers resulting from each harmonically related voltage and current.
After substituting v(t) and i(t) and evaluating, we obtain:
RMS value: The RMS value, or the effective value, of a periodic function is given by
Average Power and RMS Values (2)
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► RMS value of a periodic function is the square root of the sum that involves the amplitudes of all the harmonic components in its FS.
After substituting f(t) and evaluating:
2 2 2 0
1
1 ( )
2rms n nn F a a b
ac
n nn
dc
tnAatf
1
00 )cos(()(
Example (Prob. 17.48): For the following circuit
Average Power and RMS Values (3)
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Solution:
(a) find v(t), and (b) calculate the average power dissipated in the resistor.
Average Power and RMS Values (4)
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Solution (end):
Average Power and RMS Values (5)
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Example 17.9: Find an estimate for the RMS value of the voltage
Solution:
Exponential Fourier series (1)
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0 0 0 1
( ) ( cos sin )n n ndc
ac
f t a a n t b n t
A compact way of expressing the Fourier series
Complex or exponential Fourier series representation of f(t)
The exponential Fourier series of a periodic function f(t) describes the spectrum of f(t) in terms of the amplitude and phase angle of AC components at positive and negative harmonic frequencies.
Exponential Fourier series (2)
0 0 0 1
( ) ( cos sin )n n ndc
ac
f t a a n t b n t
2 2 1 n , tan
n n n n
n
b A a b
a
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Exponential Fourier series versus Trigonometric Fourier series
2 2 2 0
1
1 ( )
2rms n nn F a a b
ac
n nn
dc
tnAatf
1
00 )cos(()(
0 0 0 1 T
c a f t dt T
Exponential Fourier series (3) Example 17.10: Find the exponential Fourier series expansion of the
periodic function f(t)=et, 0<t<2 with f(t+2)=f(t)
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Solution:
Exponential Fourier series (4) Solution (end): Plotting the complex frequency spectrum of f(t)
38(a) Complex amplitude spectrum (b) Complex phase spectrum
Practice: Exp. Fourier series
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Determine the coefficients c0 and cn and the exponential Fourier series of f(t) in the figure below.
Exponential Fourier series (5)
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Symmetry that impacts Cn
0
)cos()( 4
)( 2
2/
0 0
2/
00
n
T
n
T
b
dttntf T
a
dttf T
a
)()( tftf Even f(t)
Cn = Cn
Odd f(t) )()( tftf
0
/ 2
00
0, 0 4
( ) sin( )
n
T
n
a a
b f t n t dt T
Cn = - Cn
Exponential Fourier series (6) Example: Find the exponential Fourier series expansion of the square wave v(t) below.
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Solution:
Verify ?
A is amplitude and T is period of v(t)
Exp. Fourier series of some waveforms (1)
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Exp. Fourier series of some waveforms (2)
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Reference: Table 15.5-1, p.759, Introduction to Electric Circuits, Richard C.Dorf, James A. Svoboda, 9th Edition, Wiley, 2013.
The Fourier series provides amplitudes and phases of the harmonics versus frequency, showing which frequencies are playing an important role in the shape of the output and which ones are not.
Spectrum analyzer: is an instrument that displays the amplitude of the components of a signal versus frequency.
can be used to conduct noise and spurious signal analysis, phase checks, electromagnetic interference and filter examinations, vibration measurements etc.
Application: Spectrum analyzer
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Average power A spectrum analyzer indicates that a signal is made up of three components only: 640 kHz at 2 V, 644 kHz at 1 V, 636 kHz at 1 V. If the signal is applied across a 10-Ω resistor, what is the average power absorbed by the resistor?
Example: Spectrum analyzer
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Solution
dc dc n n 1
1 P V I V I cos( )
2 n nn
After analyzing the signal of a voltage v(t) and a current i(t), a spectrum analyzer indicates the following result:
For v(t), this signal is made up of three AC components: 120 rad/s and phase 450 at 7 V, 240 rad/s at 5 V, and 300 rad/s at 3 V.
For i(t), this signal is made up of a DC and three AC components: The DC component is 3.6741 A; the three AC components are 120 rad/s at 6A, 240 rad/s at 4A, 300 rad/s at 2A.
1) Write v(t) and i(t) as Fourier series using the amplitude-phase form.
2) Calculate the RMS value of v(t) and i(t) 3) Determine the average power absorbed by the
resistor if v(t) is applied across a 5-Ω resistor.
Practice: Spectrum analyzer
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Band-limited periodic function is the one whose amplitude spectrum contains only a finite number of coefficients An or cn In this case, the Fourier series is
Sampling theorem
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Sampling theorem: A band-limited periodic function whose Fourier series contains N harmonics is uniquely specified by its values at 2N+1 instants in one period.
Filters are an important components of electronics and communications system. This filtering process cannot be accomplished without the Fourier series expansion of the input signal. Filter design: To select the fundamental component (or any desired harmonics) of the input signal and reject other harmonics.
Example:
Application: filters (1)
(a) Input and output spectra of a low-pass filter (b) The low-pass filter passes only the dc component when c << 0
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► : ωc large, a large number of harmonics can be passed. ► : ωc small, a large number of the ac components are blocked, only DC passed.
Application: filters (2)
(a) Input and output spectra of a band-pass filter
(b) The band-pass filter passes only the fundamental component when << 0
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► Highly-selective filter:
► Passed band of frequencies
where B is its bandwidth (B = ω2 - ω1). The filter passes only the fundamental component ω0 (n=1)
Application: filters (3)
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Example 17.14: If the sawtooth waveform in Fig. 17.45(a) is applied to an ideal low-pass filter with the transfer function shown in Fig. 17.45(b), determine the filter output.
Solution:
Only the DC and fundamental components are passed. Filter output:
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References 1. C. K. Alexander and M. N. O. Sadiku, “Fundamentals of Electric
Circuits”, 4th Ed. (2009), 5th Ed. (2013), 6th Ed. (2017). McGraw Hill.
2. H. W. Jackson, D. Temple, B. Kelly, Introduction to Electric Circuits, 9th Ed. (2012), Oxford University Press.
3. H. Saadat, Power System Analysis, 3rd and previous editions, McGraw-Hill.
4. R. C. Dorf, J. A. Svoboda, Introduction to Electric Circuits, 9th Ed. (2014) and previous editions, Wiley.
5. J.D. Irvin, R.M. Nelms, Basic Engineering Circuit Analysis, 10th Ed. (2010) and previous editions, Wiley.
6. J. W. Nilsson, S. A. Riedel, Electric Circuits, 10th Ed. (2014) and previous editions, Prentice Hall.
7. Other sources.
