Fourier Series 1 Chapter 4:

TOPIC: 2 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

FOURIER SERIES DEFINITION 3 Fourier Series 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.

FOURIER SERIES Periodic function 4

5 trigonometric form of Fourier series Fourier coefficients Harmonic frequency DC AC

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

a v coefficient 8

a n coefficient 9

b n coefficient 10

Example 1 11 Obtain the Fourier series for the waveform below (given ω o = π ):

Solution: Fourier series: 12

Waveform equation: 13

a v coefficient 14

a n coefficient 15 Note: w 0 = π

b n coefficient 16

17 Fit in the coefficients into Fourier series equation:

By using n=integer…. 18

THE EFFECT OF SYMMETRY ON FOURIER COEFFICIENTS 19 Even symmetry Odd symmetry Half-wave symmetry Quarter-wave symmetry

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 31 A periodic function that has half-wave symmetry and, in addition, symmetry about the mid-point of the positive and negative half- cycles.

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 35 Alternative form

Trigonometric identity 36 Fourier series

Fourier coefficients 37

Example 2 Find the Fourier series expansion of the function below 38

Solution This is an even function, bn = 0 39 W 0 = 2π/T, Thus, W 0 = 2π/2π = 1 Integration by parts (see next slide)

Integration by parts (revision) 40

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

Solution 42 Integration by parts

Example 4 Determine the Fourier series expansion of the function below:

Solution: The function is half wave symmetry

Fourier coefficients for half wave function:

A n coefficient:

B n coefficient:

Fourier series:

Steps for applying Fourier series: 49 Express the excitation as a Fourier Series Find the response of each term in Fourier Series Add the individual response using the superposition principle

Periodic voltage source: 50

Step 1: Fourier expansion 51

Step 2: find response 52 DC component: set n=0 or ω =0 Time domain:inductor = short circuit capacitor = open circuit

Steady state response (DC+AC) 53

Step 3: superposition principle 54

example: 55

Question: 56 If Obtain the response of v o (t) for the circuit using ω n =n ω o.

Solution: Using voltage divider: 57 Note: L= 2H R= 5Ω

DC component ω n =0) 58 nth harmonic

Response of v o : 59 Change V 0 into polar form and perform summation at the denominator; VsVs

In time domain: 60

Example of symmetry effect on Fourier coefficients (past year): 61 A square voltage waveform, v i (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 v i (t). b) Obtain the first three nonzero term for v o (t).

62 Figure (a) Figure (b)

Solution (a): Response is the Odd Quarter-wave symmetry… 63

Equation of v i (t) for 0<t< T/4: 64 Harmonic frequency:

b n coefficient: 65

Fourier series for v i (t): 66

Solution (b): 67 Voltage v i for first three harmonic:

68 Circuit transfer function:

Transfer function for first three harmonic: 69

Voltage v o for first three harmonic: 70

First three nonzero term: 71