FOURIER SERIES CHAPTER 5

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

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.

FOURIER SERIES Periodic function

trigonometric form of Fourier series Fourier coefficients Harmonic frequency DC AC

Condition of convergent a Fourier series (Dirichlet conditions): 1.F(t) adalah 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

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

Fourier coefficients Integral relationship to get Fourier coefficients

a v coefficient

a n coefficient

b n coefficient

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

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

Even Symmetry A function is define as even if

Even function example

Fourier coefficients

Odd Symmetry A function is define as odd if

Odd function example

Odd function characteristic

Fourier coefficients

Half-wave symmetry half-wave function:

half-wave function

Fourier coefficients for half wave function:

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.

Example of quarter-wave symmetry function

Even quarter-wave symmetry

Odd quarter-wave symmetry

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

ALTERNATIVE TRIGONOMETRIC FORM OF THE FOURIER SERIES Fourier series Alternative form

Trigonometric identity Fourier series

Fourier coefficients

Example 1 Obtain the Fourier series for the waveform below:

Solution: Fourier series:

Waveform equation:

a v coefficient

a n coefficient

b n coefficient

Fit in the coefficients into Fourier series equation:

By using n=integer….

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

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

Periodic voltage source:

Step 1: Fourier expansion

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

Steady state response (DC+AC) (c) (d)

Step 3: superposition principle

example:

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

Solution: Using voltage divider:

DC component ω=0) nth harmonic

Response of v o :

In time domain:

Example of symmetry effect on Fourier coefficients (past year): Satu voltan berkala segiempat, v i (t) ( Rajah (b)) digunakan ke atas litar seperti yang ditunjukkan pada Rajah (a). Jika Vm = 60π V dan tempoh, T = 2π s, a)Dapatkan persamaan Siri Fourier untuk v i (t). b)Dapatkan tiga sebutan pertama bukan sifar bagi Siri Fourier untuk v o (t).

Rajah (a) Rajah (b)

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

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

b n coefficient:

Fourier series for v i (t):

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

Circuit transfer function:

Transfer function for first three harmonic:

Voltage v o for first three harmonic:

First three nonzero term: