Engineering Circuit Analysis CH8 Fourier Circuit Analysis 8.1 Fourier Series 8.2 Use of Symmetry

Ch8 Fourier Circuit Analysis 8.1 Fourier Series Most of the functions of a circuit are periodic functions They can be decomposed into infinite number of sine and cosine functions that are harmonically related. A complete responds of a forcing function = Partial response to each harmonics.

Ch8 Fourier Circuit Analysis 8.1 Fourier Series Harmonies: Give a cosine function : fundamental frequency ( is the fundamental wave form) Harmonics have frequencies Freq. of the 1st harmonics (=fund. freq) Freq. of the 3rd harmonics Freq. of the 4th harmonics Freq. of the 2nd harmonics Amplitude of the nth harmonics (amplitude of the fundamental wave form) Freq. of the nth harmonics

Ch8 Fourier Circuit Analysis 8.1 Fourier Series Example Fundamental: v1 = 2cosw0t v3a = cos3w0t v3b = 1.5cos3w0t v3c = sin3w0t

Ch8 Fourier Circuit Analysis 8.1 Fourier Series - Fourier series of a periodic function Given a periodic function can be represented by the infinite series as

Ch8 Fourier Circuit Analysis 8.1 Fourier Series Example 12.1 Given a periodic function It is knowing It can be seen , we can evaluate

Ch8 Fourier Circuit Analysis 8.1 Fourier Series Review of some trigonometry integral observations (a) (c) (d) (b) (e)

Ch8 Fourier Circuit Analysis 8.1 Fourier Series Evaluations of Based on (a) (b) ( is also called the DC component of )

Ch8 Fourier Circuit Analysis 8.1 Fourier Series Based on (b) Based on (c) Based on (e) When k=n

Ch8 Fourier Circuit Analysis 8.1 Fourier Series Based on (a) Based on (c) Based on (d) When k=n

Ch8 Fourier Circuit Analysis 8.1 Fourier Series Harmonic amplitude Phase spectrum

Ch8 Fourier Circuit Analysis 8.2 Use of Symmetry - Depending on the symmetry (odd or even), the Fourier series can be further simplified. Even Symmetry Observation: rotate the function curve along axis, the curve will overlap with the curve on the other half of . Example : Odd Symmetry Observation: rotate the function curve along the axis, then along the axis, the curve will overlap with the curve on the other half .

Ch8 Fourier Circuit Analysis 8.2 Use of Symmetry Symmetry Algebra odd func. =odd func. × even func. Example: (b) even func. =odd func. × odd func. (c) even func. =even func. × even func.

Ch8 Fourier Circuit Analysis 8.2 Use of Symmetry (d) even func. =const. +∑ even func. (No odd func.) Example: (e) odd func. =∑odd func. odd func. odd func.

Ch8 Fourier Circuit Analysis 8.2 Use of Symmetry Apply the symmetry algebra to analyze the Fourier series. If is an even function If is an odd function

Ch8 Fourier Circuit Analysis 8.2 Use of Symmetry Half-wave symmetry f(t) = -f(t - ) or f(t) = -f(t + )

Ch8 Fourier Circuit Analysis 8.2 Use of Symmetry Fourier series: