Chapter 4 Fourier Series 1 TOPIC Fourier series
- Slides: 71
Chapter 4: Fourier Series 1
TOPIC: Fourier series definition Fourier coefficients The effect of symmetry on Fourier series coefficients Alternative trigonometric form of Fourier series Example of Fourier series analysis for RL and RC circuit Average power calculation of periodic function rms value of periodic function Exponential form of Fourier series Amplitude and phase spectrum 2
FOURIER SERIES 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. 3
FOURIER SERIES Periodic function 4
trigonometric form of Fourier series AC DC Fourier coefficients Harmonic frequency 5
Condition of convergent a Fourier series (Dirichlet conditions): 1. F(t) is single-valued 2. F(t) has a finite number of finite discontinuities in any one period 3. F(t) has a finite number of maxima and minima in any one period 4. The intergral 6
Fourier coefficients Integral relationship to get Fourier coefficients 7
av coefficient 8
an coefficient 9
bn coefficient 10
Example 1 Obtain the Fourier series for the waveform below (given ωo=π): 11
Solution: Fourier series: 12
Waveform equation: 13
av coefficient 14
an coefficient Note: w 0= π 15
bn coefficient 16
Fit in the coefficients into Fourier series equation: 17
By using n=integer…. 18
THE EFFECT OF SYMMETRY ON FOURIER COEFFICIENTS Even symmetry Odd symmetry Half-wave symmetry Quarter-wave symmetry 19
Even Symmetry A function is define as even if 20
Even function example 21
Even function property: 22
Fourier coefficients 23
Odd Symmetry A function is define as odd if 24
Odd function example 25
Odd function property: 26
Fourier coefficients 27
Half-wave symmetry half-wave function: 28
half-wave function 29
Fourier coefficients for half wave function: 30
Quarter-wave symmetry A periodic function that has half-wave symmetry and, in addition, symmetry about the mid-point of the positive and negative half-cycles. 31
Example of quarter-wave symmetry function 32
Even quarter-wave symmetry 33
Odd quarter-wave symmetry 34
ALTERNATIVE TRIGONOMETRIC FORM OF THE FOURIER SERIES Fourier series • Alternative form 35
Trigonometric identity • Fourier series 36
Fourier coefficients 37
Example 2 Find the Fourier series expansion of the function below 38
Solution This is an even function, bn = 0 W 0 = 2π/T, Thus, W 0 = 2π/2π = 1 Integration by parts (see next slide) 39
Integration by parts (revision) 40
Example 3 Obtain the trigonometric Fourier series for the waveform shown below:
Solution Integration by parts 42
Example 4 Determine the Fourier series expansion of the function below:
Solution: The function is half wave symmetry
Fourier coefficients for half wave function:
An coefficient:
Bn coefficient:
Fourier series:
Steps for applying Fourier series: Express the excitation as a Fourier Series Find the response of each term in Fourier Series Add the individual response using the superposition principle 49
Periodic voltage source: 50
Step 1: Fourier expansion 51
Step 2: find response DC component: set n=0 or ω=0 Time domain: inductor = short circuit capacitor = open circuit 52
Steady state response (DC+AC) 53
Step 3: superposition principle 54
example: 55
Question: If Obtain the response of vo(t) for the circuit using ωn=nωo. 56
Solution: Using voltage divider: Note: L= 2 H R= 5Ω 57
DC component (n=0 @ ωn=0) • nth harmonic 58
Response of vo: Change V 0 into polar form and perform summation at the denominator; Vs 59
In time domain: 60
Example of symmetry effect on Fourier coefficients (past year): A square voltage waveform, vi (t) ( as in Fig (b)) Is applied to a circuit as in Fig. (a). If Vm = 60π V and the period is T = 2π s, a) Obtain the Fourier Series for vi (t). b) Obtain the first three nonzero term for vo (t). 61
Figure (a) Figure (b) 62
Solution (a): Response is the Odd Quarter-wave symmetry… 63
Equation of vi (t) for 0<t< T/4: Harmonic frequency: 64
bn coefficient: 65
Fourier series for vi(t): 66
Solution (b): Voltage vi for first three harmonic: 67
Circuit transfer function: 68
Transfer function for first three harmonic: 69
Voltage vo for first three harmonic: 70
First three nonzero term: 71
- Painted paragraph strategy
- Narrow down topic
- Parseval's identity for fourier transform
- Serie de fourier del seno
- Formula series de fourier
- Serie de fourier
- Frequency domain to time domain
- Fourier transform equations
- Half range fourier sine series formula
- Periodic function fourier transform
- Odd quarter wave symmetry
- Use of fourier series
- Impulse train fourier transform
- Rectified sine wave fourier series
- Fourier transform formula
- Fourier transform of multiplication of two signals
- Bn formula in fourier series
- Polar fourier series
- Orthogonality fourier series
- Fourier series multiplication property
- Fourier series representation of periodic function
- Fourier series half range
- Properties of fourier transform
- Fourier transform
- Fourier
- Fourier's theorem
- Half range fourier series is defined in
- Discrete time fourier series
- Discrete time fourier series
- Discrete time fourier series
- Fourier series circuit analysis
- Fourier series of even and odd functions
- Delta function fourier transform
- Series complejas de fourier
- Matlab fourier series coefficients
- Series fourier
- Orthogonal series expansion
- Series de fourier
- Parseval's theorem in signals and systems
- Fourier series
- Series de fourier
- Series de fourier
- Orthogonal functions in fourier series
- Ejercicios resueltos de series de fourier
- Fourier series coefficients formula
- Wolfram fourier series
- Define fourier series of a function
- Synthesis equation fourier series
- Unit impluse
- Fourier series formula
- Fourier series
- Polar fft
- Fourier series
- Fourier series of periodic function
- Three phase full wave controlled rectifier with rl load
- Fourier series and orthogonal functions
- Maclaurin series vs taylor series
- Balmer series lyman series
- Maclaurin series vs taylor series
- Deret maclaurin
- Ibm p series models
- Shunt feedback amplifier analysis
- Series aiding and series opposing
- Sum of infinite series
- Chapter 5 selecting a topic and a purpose
- Specific purpose statements
- Chapter 12 sequences and series answers
- Chapter 23 series and parallel circuits
- Geometric sequence
- Chapter 1 sequences and series
- Sequences and series math 20-1
- What is a topic sentence