Fourier (1)

1. Fourier (1) Hany Ferdinando Dept. of Electrical Eng. Petra Christian University

2. Overview • History • Signal representation • Fourier in electrical engineering • Fourier Series • Application in electric circuit analysis Fourier (1) - Hany Ferdinando

3. History Jean Baptiste Joseph Fourier was born on March 21, 1768, in France He presented his paper about heat diffusion in 1807. He found series of harmonically related sinusoids to be useful in representing the temperature distribution through body But… no one accept his theory…! Fourier (1) - Hany Ferdinando

4. Signal Representation • A periodic signal can be represented as sum of sinusoidal waves • The sinusoidal wave has fundamental frequency and all waves are multiplication of that fundamental freq. Fourier (1) - Hany Ferdinando

5. Signal Representation Fundamental frequency = 50Hz, n = 1:10 Fourier (1) - Hany Ferdinando

6. Signal Representation Signal in frequency domain Fourier (1) - Hany Ferdinando

7. Signal Representation Fundamental frequency = 50Hz, n = 1:10, amplitude is half of the previous signal Fourier (1) - Hany Ferdinando

8. Signal Representation Signal in frequency domain, see the amplitude!! Fourier (1) - Hany Ferdinando

9. Fourier in Electrical Engineering • Power system: • To analyze the high freq. in the power line due to non linear load  power factor and power quality issue Fourier (1) - Hany Ferdinando

10. Fourier in Electrical Engineering • Electronics • Image and signal pre processing • Convolution • Electric circuit analysis for non DC signal Fourier (1) - Hany Ferdinando

11. Fourier in Electrical Engineering • Telematics: • Image and signal pre processing • Modulation analysis • Sampling analysis Fourier (1) - Hany Ferdinando

12. Fourier Analysis • Fourier Series • Exponential • Sinusoid • Fourier Transform • Discrete-time Fourier Transform • Discrete Fourier Transform (DFT) • Fast Fourier Transform (FFT) Fourier (1) - Hany Ferdinando

13. Fourier Series Exponential with Fourier (1) - Hany Ferdinando

14. Fourier Series Sinusoids with Fourier (1) - Hany Ferdinando

15. 1 1 2 3 4 5 6 7 8 1 1 2 3 4 5 6 7 8 Exercise: • Get ao, an and bn for square wave below: • Get ao, an and bn for sawtooth wave below: Fourier (1) - Hany Ferdinando

16. Electric Circuit Analysis • For electric circuit with non DC source, we can use Fourier analysis to solve the circuit • The source should be periodic, then we will use Fourier series • Usually, the sinusoid formula is used Fourier (1) - Hany Ferdinando

17. Electric Circuit Analysis • Find the ao, an and bn of all sources in the circuit • Find w of the signal (it is expressed in term of ‘n’) and transform all components into phasor representation • Get the equation and solve it (the solutions are in term of ‘n’) Fourier (1) - Hany Ferdinando

18. Example... Calculate the voltage drop at the inductor!! 5 1 2 3 4 5 6 7 8 Fourier (1) - Hany Ferdinando

19. Next… • The Fourier Series is discussed and students have to exercise how to use it. • For the next class, students have to read Fourier Transform: • Signals and Systems by A. V. Oppeneim ch 4 and 5, or • Signals and Linear Systems by Robert A. Gabel ch 5, or • Sinyal & Sistem (terj) ch 4 and 5 Fourier (1) - Hany Ferdinando