1. CIRCUITS and SYSTEMS – part II Prof. dr hab. Stanisław Osowski Electrical Engineering (B.Sc.) Projekt współfinansowany przez Unię Europejską w ramach Europejskiego Funduszu Społecznego. Publikacja dystrybuowana jest bezpłatnie

2. Lecture 8 Circuits at nonsinusoidal excitation

3. Fourier series -Dirichlet conditions f(t) periodic of period T f(t) absolutely integrable, i.e. finite number of minimum and maximum points At non-continuous points we have

4. Trigonometric form of Fourier series General form Sinusoidal form k – multiplicity of harmonic component F k – magnitude of kth harmonic component – phase of kth harmonic component – angular frequency of of kth harmonic component – DC component T – period of nonsinusodal periodic signal

5. Fourier coefficients and

6. Example Decompose the given f(t) into Fourier series

7. Fourier coefficients Trigonometric series coefficients Trigonometric form of the time function

8. Simplified series At T 1 =1/4T we have Magnitude characteristicsPhase characteristics

9. Fourier approximation of the pulse

10. Exponential form of Fourier series Definition of trigonometric functions Exponential form Basic relations

11. Final exponential form where

12. Example Find the exponential Fourier series of f(t) After applying definition of sinus and cosine functions we get

13. Example (cont.) Magnitude spectrum Phase spectrum Frequency spectra of the signal

14. Parseval theorem The mean value over the period of the product of two periodic functions can be expressed in the form where f k and g k represent the exponential form coefficients of kth harmonics of f(t) and g(t).

15. The rms value of nonsinusoidal signal Given the voltage and current signals Their rms values are expressed in the form Total harmonic distortion (TDH)

16. Powers at nonsinuoidal signals The general expressions for real, reactive and apparent power The distortion power D

17. Analysis of circuits at nonsinusoidal excitations Voltage excitation

18. Analysis of circuits at nonsinusoidal excitations (cont.) Current excitation

19. Example Consider the circuit given below. Assume : R1=1Ω, R2=2 Ω, L1=1H, L2=2H, C1=1/4F, C2=1/2F, ω=1. Calculate the rms values of the currents and powers of the source.,,,,

20. DC component

21. First harmonic component Reactances Currents and power of the source

22. Second harmonic component Reactances and equivalent impedance Currents and power of the source

23. Total responses rms values of currents Powers calculation