CIRCUITS and SYSTEMS part II Prof dr hab
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
Lecture 8 Circuits at nonsinusoidal excitation
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
Trigonometric form of Fourier series General form Sinusoidal form k – multiplicity of harmonic component Fk – 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
Fourier coefficients and
Example Decompose the given f(t) into Fourier series
Fourier coefficients Trigonometric series coefficients Trigonometric form of the time function
Simplified series At T 1=1/4 T we have Magnitude characteristics Phase characteristics
Fourier approximation of the pulse
Exponential form of Fourier series • Definition of trigonometric functions • Exponential form • Basic relations
Final exponential form where
Example Find the exponential Fourier series of f(t) After applying definition of sinus and cosine functions we get
Example (cont. ) Frequency spectra of the signal Magnitude spectrum Phase spectrum
Parseval theorem The mean value over the period of the product of two periodic functions can be expressed in the form where fk and gk represent the exponential form coefficients of kth harmonics of f(t) and g(t).
The rms value of nonsinusoidal signal Given the voltage and current signals Their rms values are expressed in the form Total harmonic distortion (TDH)
Powers at nonsinuoidal signals The general expressions for real, reactive and apparent power The distortion power D
Analysis of circuits at nonsinusoidal excitations Voltage excitation
Analysis of circuits at nonsinusoidal excitations (cont. ) Current excitation
Example Consider the circuit given below. Assume : R 1=1Ω, R 2=2 Ω, L 1=1 H, L 2=2 H, C 1=1/4 F, C 2=1/2 F, ω=1. , , Calculate the rms values of the currents and powers of the source.
DC component
First harmonic component Reactances Currents and power of the source
Second harmonic component Reactances and equivalent Currents and power impedance of the source
Total responses rms values of currents Powers calculation
- Slides: 23