FOURIER SERIES CHAPTER 5 1 TOPIC Fourier series

FOURIER SERIES CHAPTER 5 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

8

9

av coefficient 10

an coefficient 11

bn coefficient 12

THE EFFECT OF SYMMETRY ON FOURIER COEFFICIENTS • • Even symmetry Odd symmetry Half-wave symmetry Quarter-wave symmetry 13

Even Symmetry • A function is define as even if 14

Even function example 15

Even function property: 16

Fourier coefficients 17

Odd Symmetry • A function is define as odd if 18

Odd function example 19

Odd function property: 20

Fourier coefficients 21

Half-wave symmetry • half-wave function: 22

half-wave function 23

Fourier coefficients for half wave function: 24

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. 25

Example of quarter-wave symmetry function 26

Even quarter-wave symmetry 27

Odd quarter-wave symmetry 28

ALTERNATIVE TRIGONOMETRIC FORM OF THE FOURIER SERIES • Fourier series • Alternative form 29

• Trigonometric identity • Fourier series 30

Fourier coefficients 31

Example 1 • Obtain the Fourier series for the waveform below (given ωo=π): 32

Solution: • Fourier series: 33

Waveform equation: 34

av coefficient 35

an coefficient 36

bn coefficient 37

Fit in the coefficients into Fourier series equation: 38

By using n=integer…. 39

Example 2 Find the Fourier series expansion of the function below 40

Solution This is an even function, bn = 0 41

Example 3 • Obtain the trigonometric Fourier series for the waveform shown below:

Solution: • Function is an odd function

Fourier coefficients of odd function:

Bn coefficient:

Fourier series:

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 53

Periodic voltage source: 54

Step 1: Fourier expansion 55

Step 2: find response • DC component: set n=0 atau ω=0 • Time domain: inductor = short circuit capacitor = open circuit 56

Steady state response (DC+AC) 57

Step 3: superposition principle 58

example: 59

Question: • If Obtain the response of vo(t) for the circuit using ωn=nωo. 60

Solution: • Using voltage divider: 61

• DC component (n=0 @ ωn=0) • nth harmonic 62

Response of vo: 63

In time domain: 64

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). 65

Rajah (a) Rajah (b) 66

Solution (a): • Response is the Odd Quarter-wave symmetry… 67

Equation of vi (t) for 0<t< T/4: Harmonic frequency: 68

bn coefficient: 69

Fourier series for vi(t): 70

Solution (b): • Voltage vi for first three harmonic: 71

Circuit transfer function: 72

Transfer function for first three harmonic: 73

Voltage vo for first three harmonic: 74

First three nonzero term: 75