Chapter 16. Fourier Series Analysis


 Maximillian Wade
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1 Chapter 6 Fourier Series alysis 6 Itroductio May electrical waveforms are period but ot siusoidal For aalysis purposes, such waveform ca be represeted i series form based o the origial work of Jea Baptise Joseph Fourier he applicatio of Fourierseries method icludes sigal geerators, power supplies, ad commuicatio circuits Fourier series decomposes osiusoidal waveform ito series of siusoidal compoets of various frequecies With this property, frequecydomai represetatio or spectrum for periodic waveform is developed he spectral cocept ties the relatioship betwee timedomai ad frequecydomai properties of waveform I this chapter, we shall the various methods to geerate Fourier series ad the applicatio of Fourier series i ac steadystate circuit aalysis 6 Fourier Series he period waveform of fuctio f(t) is repetitio over time such that f(tm) f(t) m,, 3, (6) where is the period Whe m, m becomes, which is the smallest ad it is termed as fudametal period heoretically equatio (6) is true for value of t rages from  to But i practice, the waveform lasts oly for a fiite amout time he assumptio ca be true if the period is small as compared with duratio of repeatig waveform he et area uder a periodic waveform f(t) over ay period is idepedet of where the period begis hus, the itegratio of the f(t) over at ay begi poit is equal t + t t + f ( t)dt f (t)dt (6) t
2 6 rigoometric Fourier Series Fourier series state that almost ay periodic waveform f(t) with fudametal frequecy ω ca be expaded as a ifiite series i the form f(t) a + ( a cos ωt + b si ωt) (63) Equatio (63) is called the trigoometric Fourier series ad the costat C, a, ad b are depedet o f(t) ll the oscillatory compoets are iteger multiple of fudametal agular frequecy ω or harmoics Fourier series ca also be expressed i expoetial form, i which we will deal with later By icludig a ifiite umber of harmoics, Fourier series ca represet ay wellbehaved period fuctio his wellbehaved periodic fuctio is defied by Dirichlet s coditio, which states the fuctio must be siglevalued, must have a fiite umber of maxima, miima, ad discotiuities per period ad the itegral f (t) dt must be fiite Put i aother word Whe Dirichlet s coditio hold, the ifiite series summatio coverges to the value of f(t) wherever the waveform is cotiuous he ifiite series has orthogoal property meaig that that the itegral over oe period of the product of ay two differet terms vaishes hus, cos( ωt) dt si( ωt) dt, cos( ωt) si( mωt) dt, cos( ωt) cos( mωt) dt for m, si( ωt) si( mωt) dt for m However, for m, cos dt ( ωt) dt si ( ωt) Referece to equatio (63), itegratio of the f(t) over a period shall be f (t)dt a dt + ( a cos ωtdt + b si ωtdt) (64) Equatio (64) is equal to f (t)dt a hus, the costat a is a Note that a is also the average value of fuctio f(t) f (t)dt (65)
3 he coefficiet a is determied by multiplyig equatio (63) with cosmωt ad itegratig the equatio for a period his equatio used is ( m t) cos ω Kowig that f (t)dt C dt + ( a cos mωt cos ωtdt + b cos mωt si ωtdt) cos m ωt cos ωtdt cos( m ωt) f (t)dt (66) cos m ωt si ωtdt ad cos ωt cos ωtdt m for all m, the for m hus, equatio (66) shall be a he coefficiet a m shall follow equatio (67) a cos( m ωt) f (t)dt cos( ωt) f (t)dt for m (67) he coefficiet b is determied by multiplyig equatio (63) with simωt ad itegratig the equatio for a period he equatio used is Kowig that si( ωt) f (t)dt si m ωt si ωtdt m a dt + ( a simωt cos ωtdt + b si ωt si mωtdt) si m ωt si ωtdt ad si ωt cos ωtdt (68) m for all m, the for m hus, equatio (66) shall be b he coefficiet b shall follow equatio (67) b si( m ωt) f (t)dt si( m ωt) f (t)dt si( ωt) f (t)dt for m (69) lterative form of equatio (63) is the amplitudephase form, which is f(t) a + ( cos ωt + φ ) (6) Kowig that cos(α+β) cosαcosβsiαsiβ, equatio (6) shall become
4 f(t) a + ( cos φ cos ωt) ( si φ si ωt) (6) Equatig the coefficiet of equatio (63) ad (6), it gives rise to a cosφ ad b si φ his shall also mea that a b ad φ b ta a + he relatioship betwee amplitude ad phase ca also be expressed i phasor form, which is φ a jb Based o the above discussio, a fuctio f(t) cost Bsit + B B cos t + ta (6) he plot of amplitude of harmoic versus ω is called amplitude spectrum of f(t) ad the plot of phase φ versus ω is called phase spectrum of f(t) Both the amplitude ad phase spectra form the frequecy spectrum of f(t) Example 6 rectified half sie wave is defied over oe period f(t) siωt for < t < / ad f(t) for / < t < as show i Fig 6 Fid the Fourier series of this wave form Solutio he dc voltage shall be a cosie coefficiet a Figure 6: halfwave rectifier / / si ωtdt + si ωt cos ωtdt / dt siα cos αdα ω cos ω he, after lettig α ωt
5 he coefficiet a cos( ) cos( + ) + for Kowig that cos( ± )  whe is eve ad cos( ± ) whe is odd hus, a ( ) α α α si si s for, 4, 6, ad a for 3, 5, 7, a is foud to be he sie coefficiet b is si αsi αdα / for ad b for, 3, 4, 5 Kowig the coefficiet values, the rectified halfwave Fourier series shall be 3 f(t) + si ωt cos ωt cos 4ωt cos6ωt + si ωt cos (4 ) ω Fourier series is show i Fig 6 t 5 35 he plot of the rectified halfwave based o the Figure 6: he plot of f(t) + si ωt cos ωt cos 4ωt cos6ωt Expoetial Fourier Series other way of expressig Fourier series is i expoetial form It is doe by applyig Euler s rule to equatio (63) he equatio shall be f(t) a + jωt ωt (a b )e + (a + jb ) e j j (63) Lettig a c e jt ad summig over both positive ad egative values of, the compact expressio shall be  6 
6 f(t) c jωt e (64) Equatio (64) has a advatage that the trigoometric expasio because c represets all series coefficiets hus, expoetial series is preferred over the trigoometric series for aalytical ivestigatio Like the trigoometric series, the expoetial series are orthogoal i the j ωt jmωt j ωt jmωt sese that e e dt for m ad e e dt for m Multiply equatio (64) by e j mωt yields m j ωt f (t)e dt c e j ωt jmωt e dt (65) Kowig all terms with m varish ad remaiig term with m reduces to c m, thus, jmωt f (t)e dt c (66) m Hece the coefficiet c shall be c j ωt f (t)e dt (67) Equatio (67) holds for all values of icludig Whe, the equatio reduces to equatio (65) For, isert e j ωt cosωt  jsiωt to equatio (67), it becomes c f (t) cos ωtdt f (t) j si ωtdt (68) * Exchagig the sig of, it gives rise to c  c Comparig equatio (68) with equatio (67) ad (69), its cocludes that a Re[c ] ad b  Im[c ] for  6 
7 Based o the above aalysis, a ew set of coefficiets shall be defied, which are c a, c a jb, ad * a + jb c c Example 6 Determie the complex Fourier series for the waveform show i Fig 63 Solutio he coefficiet c f (t) he coefficiet c hus, the coefficiet for c is Let, c Figure 63: he square wave / 4 / 4 / dt + dt dt / / 4 / 4 / 4 / 4 / j ωt jωt ωt e dt + e dt e j dt / / 4 / 4 jωt / 4 jωt / 4 jωt / e e e + jω jω jω / / 4 / 4 j /  j j /  j / j e + e + e e e + e jω j /  j j j / e + e e + e jω 4 j si( / ) j si( ) j si( / ) si( ) for eve c si( / ) for odd j / [ ] [ ] [ ] [ ] his implies that c  is also equal to Let 3, c 3 his implies that c 3 is also equal to
8 Let 5, c 5 his implies that c 5 is also equal to 5 5 Let 7, c 7 his implies that c 7 is also equal to 7 Expasio of fuctio f(t) accordig to equatio (63) 7 yields f(t) 7 c j7ωt j 5ωt e + e 7 5 j 3ωt jωt j 3ωt jωt e + e e + e + e cos ωt cos3ωt + cos5ωt cos7ωt ( ) / 4 ( ) cos ωt odd jωt e for 7 to j5ωt 7ωt e 7 + j + he plot of the square wave fuctio based o Fourier series is show i Fig 64 Figure 64: he plot of square wave f(t) 4 odd ( ( ) ) / cos ωt for, 3, 5, 7 64 Symmetry Cosideratios he aalysis doe so far poited out that the Fourier series mostly cosists of either sie terms or cosie terms Oe may ask if there a method that ca be used i advace to avoid tedious mathematical process Such method does exist based o recogizig the existece of symmetry, which are eve symmetry, odd symmetry, ad halfwave symmetry
9 64 Eve Symmetry fuctio f(t) is eve if its plot is symmetrical about the vertical axis; that is f(t) f(t) (69) Examples of the eve fuctio are t, t 4, ad cos t Figure 65 shows a example of periodic eve fuctio Figure 65: eve periodic fuctio mai property of a eve fuctio f(t) is that f t)dt coefficiet of Fourier series for eve fuctio shall be a 4 / / f (t)dt a f (t) cos( ωt) dt / / / ( f (t)dt b (6) 64 Odd Symmetry he fuctio f(t) is said to be odd if the plot is atisymmetrical about the vertical axis; that is f(t) f(t) (6) he mai characteristic of a odd fuctio is Fourier series are / / f (t)dt he coefficiets of the
10 a a 4 / f (t)dt b f (t) si( ωt) dt (6) 643 HalfWave Symmetry fuctio is halfwave (odd) symmetric if f t  f (t) (63) his shall mea that each halfcycle is the mirror image of the ext halfcycle example of halfwave symmetrical fuctio is show i Fig 66 Figure 66: Halfwave symmetrical fuctio he coefficiet of the Fourier series shall be a a b / 4 f (t) cos 4 ( ωt) dt for odd for eve ( ω ) f (t) si t dt for odd (64) / for eve 65 Circuit pplicatios of Fourier Series I practice, may circuits are drive by osiusoidal periodic fuctio o obtai the steady state respose of the circuit to a osiusoidal periodic fuctio
11 requires the applicatio of Fourier series, ac phasor aalysis, ad superpositio priciple he procedure usually ivolves three steps, which are Express the excitatio as Fourier series rasform the circuit from time domai to the frequecy domai Fid the respose of dc ad ac compoets i the Fourier series dd the idividual dc ad ac respose usig the superpositio priciple Let s use two examples to illustrate the procedures Example 63 he circuit show i Fig 67 has a osiusoidal v s (t) source that has Fourier k series v s (t) + si( t) for k  Fid the voltage v o (t) at iductor ad the correspodig amplitude spectrum Solutio Figure 67: ac circuit jω L R + jω L he output voltage v o (t) is v (t) v (t) From the iput v s (t), that ω, j 5 + j o v o (t) shall be v t) v (t) he dc compoet shall be zero after substitutig ω ito v o (t) o ( s he phasor of sie compoet of the ac portio is 9 s hus, the output v o (t) shall be v o (t) ta ( / 5) ta ( ) /
12 I timedomai the output voltage shall be v o (t) 4 cos t ta ( / 5) [ ] odd [ ] he overall output v o (t) cos t ta ( / 5) he first four harmoics of the output voltage are v o (t) 5 cos(549 ) + cos(3754 ) + 3 cos(5896 ) + cos(783 ) + V he amplitude spectrum of the output voltage is show i Fig 68 V Figure 68: mplitude spectrum of the output voltage Example 64 Fid the respose i o (t) i the circuit show i Fig 69 if the iput voltage v(t) has ( ) + the Fourier series expasio v(t) + (cos t si t) Solutio Figure 69: ac seriesparallel circuit
13 Rewrite the iput voltage fuctio v(t) ( ) ( ) cos t si t + + covert the iput voltage fuctio to amplitudephase form usig equatio (6) ( ) + hus, + B ( ) + ( ) + B ad ad B he phase φ  ta ta ( ) he iput voltage v(t) shall the be equal to v(t) + (cos t + ta ) he phasor form is + ( ) ta + From the equatio ω rad/s ad ω rad/s he total impedace of the circuit Z 4 + jω 4 4+ he curret flows i the circuit shall be I he output curret shall be i o (t) 4 ( ) ta + jω jω jω 8 Settig ω, the dc curret shall be 4 + jω he ac compoet shall be ( ) + ta 4 ta ( ) + I timedomai format, ac compoet is + hus, the curret i o (t) shall be i o (t) + 4 ( ) jω jω 8 ( ) ta ( ) ta 4 + jx4 ( ) + cos t cos t 66 verage Power of Period Fuctios jω 8 + jω + jω 8 + jω 4 + jω ( ) ta j4 s show i equatio (6) for the amplitudephase Fourier series of a periodic fuctio, the voltage ad curret fuctios at the termial of the etwork are v(t) V dc + V cos( ωt ) θ v (65)
14 i(t) I dc + I cos( ωt ) θ i (66) I Chapter C Power alysis, we leart that the average power is P avg P(t)dt (67) hus, the average power expressed i amplitudephase Fourier series shall be P avg V I dt dc dc + V I cos( t ) cos( t ) dt ω θ v ω θi (68) Equatio (68) which is the average power, shall fially become P avg V dc I dc + V I cos( θv θi ) (69) Example 65 Determie the average power supplied to the circuit show i Fig 6 if i(t) + cos(t + ) + 6 cos(3t + 35 ) Solutio he impedace of the circuit Z is Z Hece the voltage v(t) is v(t) ZxI Figure 6: ac parallel circuit jω I + jω For dc compoet, I ad ω, v(t) V For I cos(t+ ) ad ω, v(t) 774 ) ta + jω I x + 4ω ta ( ) ( ω) cos(t
15 For I 6 cos(3t + 35 ) ad ω 3, v(t) cos(3t 544 ) x x9 ta ( x3) 544 he overall voltage v(t) + 5 cos(t 774 ) + cos(3t 544 ) he average power supplied to the circuit follows equatio P avg V dc I dc + ()(6)cos [ 544 ( 35 )] V I cos( θv θi ) () + (5)()cos[ 774 ( )] W he supply voltage is v(t) + 5 cos(t 774 ) + cos(3t 544 ) he supply curret is i(t) + cos(t + ) + 6 cos(3t + 35 ) he supply power is () + (5)()cos[ 774 ( )] + ()(6)cos [ 544 ( 35 )] W his average power P avg is same as the supply power sice capacitor absorbed o power  7 
16 Exercises 6 Fid the Fourier series of the waveform show i the figure Plot the amplitude ad phase 6 Fid the Fourier series of the waveform show i the figure Plot the amplitude ad phase (74) 63 Determie the fudametal frequecy ad specify the type of symmetry preset i the fuctios i the figures (78)  7 
17 64 Determie the Fourier series represetatio of the fuctio show i the figure (75) 65 Fid i(t) i the circuit give that i s (t) + cos3 t 66 I the circuit, the Fourier series expasio of v s (t) is v s (t) si( t) (733) 67 If the periodic voltage show i the figure is applied to the circuit, fid i o (t) (739)
18 68 he sigal show i the figure is applied to the circuit Fid output voltage v o (t) Bibliography Bruce Carlso, Circuits, Brooks/Cole James W Nilsso ad Susa Riedel, Electric Circuits, 6 th Editio, Pretice Hall 3 Charles K lexader ad Matthew NO Sadiku, Electric Circuits, d editio, McGrawHill
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