Chapter 16 Fourier Series Analysis

1 - 257 - Chapter 16 Fourier Series Analysis Introduction Many electrical waveforms are period but not sinusoidal. For Analysis purposes, such waveform can be represented in Series form based on the original work of Jean Baptise Joseph Fourier . The application of Fourier - Series method includes signal generators, power supplies, and communication circuits. Fourier Series decomposes non-sinusoidal waveform into Series of sinusoidal components of various frequencies. With this property, frequency-domain representation or spectrum for periodic waveform is developed. The spectral concept ties the relationship between time-domain and frequency-domain properties of waveform. In this Chapter , we shall the various methods to generate Fourier Series and the application of Fourier Series in ac steady-state circuit Analysis .

2 Fourier Series The period waveform of function f(t) is repetition over time such that f(t-mT) = f(t) m = 1, 2, 3, .. ( ) where T is the period. When m = 1, mT becomes T, which is the smallest T and it is termed as fundamental period. Theoretically equation ( ) is true for value of t ranges from - to . But in practice, the waveform lasts only for a finite amount time. The assumption can be true if the period T is small as compared with duration of repeating waveform. The net area under a periodic waveform f(t) over any period is independent of where the period begins. Thus, the integration of the f(t) over at any begin point is equal. ++=TttTtt2211dt)t(dt)t(ff ( ) - 258 - Trigonometric Fourier Series Fourier Series state that almost any periodic waveform f(t) with fundamental frequency can be expanded as an infinite Series in the form f(t) = a0 + = + 1n)tsinbtcosa(nnnn ( ) Equation ( ) is called the trigonometric Fourier Series and the constant C0, an, and bn are dependent on f(t).

3 All the oscillatory components are integer multiple of fundamental angular frequency or harmonics. Fourier Series can also be expressed in exponential form, in which we will deal with later. By including an infinite number of harmonics, Fourier Series can represent any well-behaved period function. This well-behaved periodic function is defined by Dirichlet s condition, which states the function must be single-valued, must have a finite number of maxima, minima, and discontinuities per period and the integral Tdt|)t(|f must be finite. Put in another word. When Dirichlet s condition hold, the infinite Series summation converges to the value of f(t) wherever the waveform is continuous. The infinite Series has orthogonal property meaning that that the integral over one period of the product of any two different terms vanishes.

4 Thus, ()()0dttsindttcosTT= = nn, ()()0dttsintcosT= mn, ()()0dttcostcosT= mn for n m, ()()0dttsintsinT= mn for n m. However, for n = m, ()()2 TdttsindttcosT2T2= = nn. Reference to equation ( ), integration of the f(t) over a period T shall be Tdt)t(f = T0dta + = + 1nTnT)dttsinbtdtcosa(nnn ( ) Equation ( ) is equal to Tdt)t(f = a0T. Thus, the constant a0 is a0 = T1 Tdt)t(f ( ) Note that a0 is also the average value of function f(t). - 259 - The coefficient an is determined by multiplying equation ( ) with cosm t and integrating the equation for a period T. This equation used is () Tdt)t(tcosfm = T0dtC + = + 1nTnT)dttsintcosbtdtcostcosa(nmnmn ( ) Knowing that = 1nTdttsintcosnm= 0 and 0dttcostcosT= nm for all n m, the = 1nTdttcostcosnm= 2T for n = m.

5 Thus, equation ( ) shall be Tdt)t(t)cos(fm= 2 Tan. The coefficient am shall follow equation ( ). an = = TTdt)t(t)cos(T2dt)t(t)cos(T2fnfm for n = m ( ) The coefficient bn is determined by multiplying equation ( ) with sinm t and integrating the equation for a period T. The equation used is Tdt)t(t)sin(fm = T0dta + = + 1nTT)dttsintsinbtdtcostsina(mnnmnn ( ) Knowing that = 1nTdttsintsinnm= 0 and 0dttcostsinT= nm for all n m, the = 1nTdttsintsinnm= 2T for n = m. Thus, equation ( ) shall be Tdt)t( t)sin(fm= 2 Tbn. The coefficient bn shall follow equation ( ). bn = = TTdt)t(t)sin(T2dt)t(t)sin(T2fnfm for n = m ( ) Alternative form of equation ( ) is the amplitude-phase form, which is f(t) = a0 + = + 1nn)tcosA(nn ( ) Knowing that cos( + ) = cos cos -sin sin , equation ( ) shall become - 260 - f(t) = a0 + = = 1n1n)tsinsinA()tcoscosA(nnnnnn ( ) Equating the coefficient of equation ( ) and ( ), it gives rise to an = nn cosA and bn = nn sinA.

6 This shall also mean that 22baAnnn+= and = nnnabtan1. The relationship between amplitude and phase can also be expressed in phasor form, which is nnnnjbaA = . Based on the above discussion, a function f(t) = A cosnt Bsinnt = ++ ABtantcosBA122n ( ) The plot of amplitude An of harmonic versus n is called amplitude spectrum of f(t) and the plot of phase n versus n is called phase spectrum of f(t). Both the amplitude and phase spectra form the frequency spectrum of f(t). Example A rectified half sine wave is defined over one period f(t) = Asin t for 0 < t < T/2 and f(t) = 0 for T/2 < t < T as shown in Fig. Find the Fourier Series of this wave form. Figure : A half-wave rectifier Solution The dc voltage shall be a0 = + T2/T2/T0dt0T1dttsinAT1= = A12 TcosTA.

7 The cosine coefficient an = = dcossinAtdtcostsinAT102/T0nn, after letting = t. - 261 - The coefficient an = + + 1)1cos(11)1cos(12 Annnn for n 1. Knowing that cos(n 1) = -1 when n is even and cos(n 1) = 1 when n is odd. Thus, an = )1(A22 n for n = 2, 4, 6,.. and an = 0 for n = 3, 5, 7,.. a1 is found to be ssinsinA0n= 0. The sine coefficient bn is 0dsinsinAn= A/2 for n = 1 and bn = 0 for n = 2, 3, 4, Knowing the coefficient values, the rectified half-wave Fourier Series shall be f(t) = ..t6cos35A2t4cos15A2t2cos3A2tsin2AA + = tsin2AA + t2cos)14(A212 =nnn. The plot of the rectified half-wave based on the Fourier Series is shown in Fig. Figure : The plot of f(t) = t6cos35A2t4cos15A2t2cos3A2tsin2AA + Exponential Fourier Series Another way of expressing Fourier Series is in exponential form.

8 It is done by applying Euler s rule to equation ( ). The equation shall be f(t) = a0 + = = ++ 1ntnnn1ntnnne)ba(21e)ba(21jjjj ( ) Letting a0 = c0ej0t and summing over both positive and negative values of n, the compact expression shall be - 262 - f(t) = = = nntnecjn ( ) Equation ( ) has an advantage that the trigonometric expansion because cn represents all Series coefficients. Thus, exponential Series is preferred over the trigonometric Series for analytical investigation. Like the trigonometric Series , the exponential Series are orthogonal in the sense that = T0tmtn0dteejj for n m and = T0tmtnTdteejj for n = m. Multiply equation ( ) by tme j yields T0tmdte)t(jf = = = nnT0tmtnndteecjj ( ) Knowing all terms with n m varnish and remaining term with n = m reduces to cmT, thus, Tcdte)t(T0tmmjf= ( ) Hence the coefficient cn shall be cn = T0tndte)t(T1jf ( ) Equation ( ) holds for all values of n including n = 0.

9 When n = 0, the equation reduces to equation ( ). For n 0, insert tne j = cosn t - jsinn t to equation ( ), it becomes cn = T0T0tdtsin)t(T1tdtcos)t(T1njfnf ( ) Exchanging the sign of n, it gives rise to c-n = *cn. Comparing equation ( ) with equation ( ) and ( ), its concludes that an = 2Re[cn] and bn = -2 Im[cn] for n 1. - 263 - Based on the above Analysis , a new set of coefficients shall be defined, which are c0 = a0, 2bacnnnj =, and 2bacc*nnnnj+== . Example Determine the complex Fourier Series for the waveform shown in Fig. Figure : The square wave Solution The coefficient c0 = =T0)t(T1f + 4/T2/T4/T4/T2/T4/TAdtAdtAdtT1= 0 The coefficient cn = + 4/T2/T2/T4/Ttn4/T4/TtntndtAedtAedtAeT1jj j = + 2/T4/Ttn4/T4/Ttn4/T2/TtneeeTAjnjnjnjjj =[]2/nn2/n2/nn2/neeeeeeTA + ++ jjj-jj-jjn = []2/nnn2/ne2eee2TA + + jjj-jjn = [])sin(2)2/sin(42A njnjnj = [])sin()2/sin(2A nnn Thus, the coefficient for cn is = odd for )2/sin(2 Aeven for 0cnnnnn Let n = 1, c1 = A2.

10 This implies that c-1 is also equal to A2. Let n = 3, c3 = 3A2. This implies that c-3 is also equal to 3A2. - 264 - Let n = 5, c5 = 5A2. This implies that c-5 is also equal to 5A2. Let n = 7, c7 = 7A2. This implies that c-7 is also equal to 7A2. Expansion of function f(t) according to equation ( ) = = nntnecjn for n = -7 to n = 7 yields f(t) = ..t7e7A2 jt5e5A2 +j t3e3A2 j+teA2 jt3e3A2 j+teA2 jt5e5A2 +jt7e7A2 j+.. = t5cos5A4t3cos3A4tcosA4 + t7cos7A4 +.. = tcos)1(A4odd12/)1( == nnnnn. The plot of the square wave function based on Fourier Series is shown in Fig. Figure : The plot of square wave f(t) = tcos)1(A4odd12/)1( == nnnnn for n =1, 3, 5, 7 Symmetry Considerations The Analysis done so far pointed out that the Fourier Series mostly consists of either sine terms or cosine terms.