CHAPTER 5 FOURIER SERIES SAIDATUL SHEMA BINTI SAAD

CHAPTER 5: FOURIER SERIES SAIDATUL SHEMA BINTI SAAD 016 -6651449 JABATAN TEKNOLOGI KEJURUTERAAN ELEKTRIK , FAKULTI TEKNOLOGI KEJURUTERAAN , ROOM NO. 7, ARAS 2 BLOK S 4 (S 4 -L 2 -90), KAMPUS UNICITI ALAM , UNIVERSITI MALAYSIA PERLIS. PLT 201 – ELECTRIC CIRCUIT THEORY II

What to learn in this chapter? Trigonometric Fourier Series � Fourier Series Concept � Fourier Coefficients � Alternative Trigonometric (Amplitude-Phase ) Symmetry Considerations � Even, Odd & Half-Wave Circuit Applications Average Power & RMS Value Exponential Fourier Series PLT 201 – ELECTRIC CIRCUIT

5. 1 TRIGONOMETRIC FOURIER SERIES Fourier Series Concept FOURIER SERIES of a periodic function of f(t) : A representation that resolves f(t) into a DC and AC component comprising an infinite series of harmonic sinusoids. DC AC Fourier coefficients PLT 201 – ELECTRIC CIRCUIT Harmonic frequency

5. 1 TRIGONOMETRIC FOURIER SERIES Fourier Series Concept ω0 = 2π/T is the fundamental frequency (rad/second) sin nω0 t or cos nω0 t = nth harmonic of f(t) an and bn (for n ≠ 0) are the Fourier Coefficients which is amplitude of the sinusoids in AC component. a 0 = coefficient in DC component or average value of f(t). PLT 201 – ELECTRIC CIRCUIT

5. 1 TRIGONOMETRIC FOURIER SERIES Fourier Series Concept Condition of convergent Fourier series (Dirichlet conditions): f(t) is single-valued f(t) has a finite number of finite discontinuities in any one period f(t) has a finite number of maxima and minima in any one period The integral PLT 201 – ELECTRIC CIRCUIT

Trigonometric Integrals (for any integers m & n) PLT 201 – ELECTRIC CIRCUIT

5. 1 TRIGONOMETRIC FOURIER SERIES Fourier Coefficients DC AC PLT 201 – ELECTRIC CIRCUIT

5. 1 TRIGONOMETRIC FOURIER SERIES Alternative Trigonometric (Amplitude-Phase) Basic Fourier form alternative trigonometric form PLT 201 – ELECTRIC CIRCUIT

5. 1 TRIGONOMETRIC FOURIER SERIES Alternative Trigonometric (Amplitude-Phase) Amplitude and Phase Spectrum: 1) AMPLITUDE 2) PHASE PLT 201 – ELECTRIC CIRCUIT

Trigonometric Integrals to Evaluate Fourier coefficients: PLT 201 – ELECTRIC CIRCUIT

Values of cosine, and exponential functions for integral multiples of π Function Value 1 0 0 1 PLT 201 – ELECTRIC CIRCUIT

EXAMPLE 1 Determine the Fourier series of the waveform shown in Fig 5. 1. Obtain the amplitude and phase spectra. Figure 5. 1 PLT 201 – ELECTRIC CIRCUIT

EXERCISE 1 Determine the Fourier series of the sawtooth waveform in Figure 5. 2 below: Figure 5. 2 Answer: PLT 201 – ELECTRIC CIRCUIT

5. 2 SYMMETRY CONSIDERATIONS Even symmetry Odd symmetry Half-wave (odd) symmetry Only these are covered in the note, others please refer to the ref. book Half wave (even) symmetry Quarter wave symmetry PLT 201 – ELECTRIC CIRCUIT

5. 2 SYMMETRY CONSIDERATIONS Even Symmetry PLT 201 – ELECTRIC CIRCUIT

5. 2 SYMMETRY CONSIDERATIONS Even Symmetry The coefficients for even function become; PLT 201 – ELECTRIC CIRCUIT

5. 2 SYMMETRY CONSIDERATIONS Odd Symmetry PLT 201 – ELECTRIC CIRCUIT

5. 2 SYMMETRY CONSIDERATIONS Odd Symmetry The coefficients for odd function become; PLT 201 – ELECTRIC CIRCUIT

5. 2 SYMMETRY CONSIDERATIONS Half-Wave (Odd) Symmetry PLT 201 – ELECTRIC CIRCUIT

5. 2 SYMMETRY CONSIDERATIONS Half-Wave (Odd) Symmetry The coefficients for half-wave odd property function become; PLT 201 – ELECTRIC CIRCUIT

5. 3 FOURIER SERIES IN CIRCUIT APPLICATIONS Steps for applying Fourier series: Express the excitation as a Fourier Series Transform the circuit from the time domain to the frequency domain Find the response of the dc and ac components in the Fourier series Add the individual dc and ac responses using the superposition principle. PLT 201 – ELECTRIC CIRCUIT

5. 3 FOURIER SERIES IN CIRCUIT APPLICATIONS Fourier Series Expression with DC component AC component PLT 201 – ELECTRIC CIRCUIT

5. 3 FOURIER SERIES IN CIRCUIT APPLICATIONS Circuit Transformation PLT 201 – ELECTRIC CIRCUIT

5. 3 FOURIER SERIES IN CIRCUIT APPLICATIONS DC and AC Response 1. DC component: Frequency domain: set n=0 or ω=0 Time domain: inductor = short circuit capacitor = open circuit 2. AC component: apply phasor techniques PLT 201 – ELECTRIC CIRCUIT

5. 3 FOURIER SERIES IN CIRCUIT APPLICATIONS Add Response (Using Superposition) PLT 201 – ELECTRIC CIRCUIT

EXAMPLE 2 Find the response v 0(t) of the circuit below when the voltage source vs(t) is given by PLT 201 – ELECTRIC CIRCUIT

5. 4 AVERAGE POWER & RMS VALUE Average Power By using Fourier, we know that for periodic function: and for average power: PLT 201 – ELECTRIC CIRCUIT

5. 4 AVERAGE POWER & RMS VALUE Average Power Therefore average power defined in Fourier series form is: ∆ with is the angle of Vn while is the angle of In PLT 201 – ELECTRIC CIRCUIT

5. 4 AVERAGE POWER & RMS VALUE RMS Value RMS value of periodic function (either V or I) is: For trigonometri c form PLT 201 – ELECTRIC CIRCUIT

EXAMPLE 3 Get the average power for the circuit shown in figure below if: PLT 201 – ELECTRIC CIRCUIT

5. 5 EXPONENTIAL FOURIER SERIES Cos and sin function can be expressed in exponential form as: PLT 201 – ELECTRIC CIRCUIT

5. 5 EXPONENTIAL FOURIER SERIES Remind of basic Fourier function: By using sinusoidal form, Fourier series in exponential form is written as: PLT 201 – ELECTRIC CIRCUIT

5. 5 EXPONENTIAL FOURIER SERIES Simplified exponential form: Or it can be written in the form of: with PLT 201 – ELECTRIC CIRCUIT

5. 5 EXPONENTIAL FOURIER SERIES The plots of the magnitude and phase of cn versus nω0 are called complex amplitude spectrum and complex phase spectrum 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. PLT 201 – ELECTRIC CIRCUIT

5. 5 EXPONENTIAL FOURIER SERIES Coefficients of the three forms of Fourier series (sine-cosine form, amplitude-phase form & exponential form) related by: Or , If only an > 0 PLT 201 – ELECTRIC CIRCUIT

5. 5 EXPONENTIAL FOURIER SERIES rms value for exponential form: with therefore rms value for the exponential form can also be stated as: PLT 201 – ELECTRIC CIRCUIT