Chapter 15: Fourier Series

Transcription

1 Chapter 5: Fourier Series Ex Ex Ex. 5.- f(t) K is a Fourier Series. he coefficiets are a K; a b for. f(t) AcosZ t is a Fourier Series. a A ad all other coefficiets are zero. Set origi at t, so have a odd fuctio; the a for,,... 8 ote: Also, f(t) is half wave symmetric, the b for eve b f(t) si f t dt si f t dt + si f t dt so f(t) si t; odd ad rad s where f si f t dt rad s (cos f), 3, 5,... f Ex. 5.-, a, a for all odd fuctio quarter wave symmetric b eve t < t < 8 6 b f(t) si t dt where f(t) 6 t < 6 hus b si 3 so f(t) si si (t) odd 3 Ex a) f(t) is either eve or odd. f(t) will cotai both sie ad cosie terms b) wave symmetry o eve harmoics c) average value of f(t) a 65

2 Ex Odd fuctio, C jt jt j t C f(t)e dt e dt e dt j j j j e + + e e (e ) j j odd j eve so f(t) e e e... e j jt j3t j5t jt j3t j5t e e Ex Ex Eve fuctio: a, a for eve, rad s t a cos t dt C j e t dt! odd so f (t) cos t+ cos 3t+ cos 5t C % & K ' K j j e t j / j / e e j ( ) ( ) odd eve, " $ # 66

3 Ex MathCad Spreadsheet Settig up the idex: :,..3 Settig the costraits: A : K : : he various values of delta: he coefficiets: Similarily, ow plottig, usig Cc Cc d : ad d : 3 8 Ad si (x ) d : where x: 8 also φ : arg (Cc ) x Ad si (x ) d : with x: φ : arg (Cc) 8 x : Ex v t si s() 3. si t V si si t 3 3 V 3 si si t V 5 si si t 5 5 RC s, rad s 67

4 use also V V s + (Rc ) V at.6 & φ ta (6) 86 V s +( 57 ) V3 9.3 & φ ta (.) 88.8 V s + (.) (or 3.7% of fudametal) So use oly ad 3 terms v 3. (.6) si (t 86 ) 3.(.3) si (t 89 ). si (t86 ).75 si (t89 ) Ex Ex 5.- at f() t e u() t ( ) a+ j t + jt at jt e F( ) f ( t) e dt e e dt a+ j a+ j ( ) ( ) jt { f at } f at e dt ( ) τ Let τ at t a jτ a τ j( a) τ { f ( at) } f ( τ) e d f ( τ) e dτ F a a a a Ex 5.- Ex jt f () t ( δ ( ) A) e dt ( δ ( ) A) dt A jt e dt e { δ( ) } δ( ) ake the Fourier rasform of both sides to get: jt ( e ) δ ( ) ( ) ( ) ( ) jt jt e + e A jt jt A { Acost} A ( e ) + ( e ) δ + δ + Aδ + Aδ + jt ( ) ( ) ( ) 68

5 Ex 5.- a.) V ( ) V ( ) i i + j b.) W W i out ta d 3 J ta d 6.3 J Wout 6.3 η % %.5% W 3 i Ex 5.3- () at () at ( ) f t te + f t te f t te at F () s ad F () s s+ a s+ a + ( ) ( ) he F F s + F s + + ( ) ( ) () s j s j ( s+ a) ( s+ a) s j s j ja ( a+ j) ( a j) ( a + ) 69

6 Problems Sectio 5.3: he Fourier Series P5.3- f(t) t for t a t dt 3 a t cost dt 6 b t si t dt ft + cos 6 t si 3 t P5.3- Eve fuctio:., 5 rad s % A cos t t. f(t) &K. t <.3 'K A cos t.3 t. Choose period. t. 3 for itegral a. Acos t A. a. Acost cos t dt. A so a 5A cos t dt a 5A cos t cos t dt 5A [ cos 5 (+)t+cos 5 ()t ] dt A cos ( /) ad b 7

7 P a cos t dt + cos t dt si t si t (si ) + (si ) si ( + ) si odd eve + b si t dt + si t dt cos t cos t (cos ( ) ) cos 3 is odd,6,,!,8,,! P5.3- f(t) A(t ) t P A A( ) ( ) + ( ( )) Sice A si t A cos t A cos t f (t)f t cos t a A e d ( ) + A A( ) + cos (t ) ( ) + A A( ) + ( ) A A cos( t) cos t ( ) ( ) his is the Fourier Series of a eve fuctio (Why?) deed, f (t) is eve. a A( t )cos t dt > b A( t ) si t dt A f(t ) A + A si t e 7

8 P5.3-6 MathCad Spreadsheet dex of summatio, : : 5 :,.. Defie parameters: : : Defie icremet of time. Set up idex to ru over to of the sigal. : dt: i :,.. t i : dt i Eter the formulas for the Fourier Series: a : t dt a : t cos( t) dt b : t si ( t) dt a a b : ( si( ) + cos( ) ) Eter the Fourier Series : f(i) : a + b si( t ) i Plot the periodic sigal : Sectio 5-: Symmetry of the Fuctio f(t) P5.- Choose t, average a b f t si t dt a sice have odd fuctio 6 b t si t t si t d! " b +! $# b b 3 3 t cos t f() t t < t < 7

9 P5.- Odd fuctio, half-wave symmetry, from able 5- a a b for eve for eve 9 a cos t dt odd a for all b 9si t dt cos " t 6 35,,,...! $ # ft 6 si t ; odd P5.-3 8s, eve fuctio b ( ) a average 8 a () 8 3 f t cos t dt cos t dt cos t dt 3 si si a.7, a.955, a.66 P5.-, a a b A A cost dt A cost cos t dt ( ) ( ) ( ) ( ) A si t si + t + + A si( ) si( + ) + + A A A + si ( ) cos( ) ( ) ( ) si ( ) ( ) due to symmetry ( ) ( ) ( ) 73

10 P5.-5 a because the average value is zero ext P5.-6 a because the fuctio is odd b for eve due to wave symmetry b t si t dt 6 Refer to able ake A si cos % & K ' K 8 8 for,5,9,... for 3,7,,... ad a average value of f t Also so ft + si t + si t + si 3t + si 5t rad s Sectio 5.5: Expoetial Form of the Fourier Series P5.5- MathCad Spredsheet: dex of summatio, : : 5 :,.. m :,.. Defie parameters :A : : : : Defie icremet of time. Set up idex to ru over two periods of the sigal. : dt : i :,.. t i: dt i Eter the formulas for the Fourier Series: cos si C i A cos m m si m Cm i A m Eter the Fourier Series: 6 i6 m i6 m fi: C exp j t + C exp j m t Plot the periodic sigal: 7

11 P5.5- MathCad Spreadsheet: dex of summatio, : : 5 :,.. m,.. Defie parameters : A : : 8 : 785. Defie icremet of time. Set up idex to ru over two periods of the sigal. : dt: i:,.. ti : dt i Eter the formulas for the Fourier Series: C : exp j t dt+ exp j t dt+ exp j t dt C m : exp j m t dt+ exp j m t dt+ exp j m t dt Eter the Fourier Series: 6 i6 m i6 m fi: C expj t + C expj m t Plot the periodic sigal: P5.5-3 ; C average value jt C f6 t e dt ! + jt jt 5 5 j j j j j j e jt dt e dt + 5. e dt. 5 e 5 e 3 e 5 e e e j " $ # 75

12 P5.5- MathCad Spreadsheet: dex of summatio: : 5 :,.. m :,.. Defie parameters: A : 6 : Fudametal frequecy: : 57. Defie icremet of time. Set up idex to ru over two periods of the sigal. : dt: i:,.. t i: dt i Eter the formulas for the Fourier Series: Aj C : m 6 6 Eter the Fourier Series: Aj C m: m 6 i 6 i 6 m fi: 6+ C expj t+ + C exp jm t+ Plot the periodic sigal: <he Fourier coefficiets foud usig able 5.5- he 6 shifts the plot vertically, while the (t+) shifts the plot horizotally. P5.5-5 C C ( t) exp( j t) dt ( exp j j ( ) + ) ( ) cos jsi ( ) ( ) j ( ) j f iteger, the C ( ) j 76

13 P5.5-6 MathCad Spreadsheet: Creatig the sigal: x:,... y:,... f(x): exp x. y g(y) : 5 exp. dex of summatio, : : 5 :,.. m:,.. Defie parameters: A: : Fudametal frequecy: : : 3. Defie icremet of time. Set up idex to ru over two periods of the sigal. : dt: i:,.. t i : dt i Eter the formulas for the Fourier Series: C : exp C m : exp t. t. t exp ( j t) dt+ 5 exp. exp( j t) dt t exp( j m t) dt + 5 exp exp( j m t) dt. Eter the Fourier Series: Plot the periodic sigal: f(i): C exp j t + C exp j m t + 5. * Solvig yields C for odd C, eve i6 m i6 m 5 j65+ j6 * 77

14 Sectio 5-6: he Fourier Spectrum P5.6- Average value a half wave symmetry A A a t cos t dt ( cos( ) ) f(t) ad A A b t si t dt cos 67 C a b θ ta 59. A A A A 88. b a P5.6- MathCad Spreadsheet: Determiig the Fourier Coefficiets: Settig the idex : 5 :,.. m :,.. Parameters: : wo : j : Fidig the coefficiets of the expoetial Fourier Series. Split the fuctio ito four regios. 6 C : t 3 exp jwot)dt C : ( si t exp( j wo t)dt C3 : 6 3 t exp( j t)dt C : si t exp( jwot)dt + 6 C : C + C + C3 + C 3 78

15 Verify that these coefficiets are ideed correct by usig them to plot the fuctio: Defie icremet of time. Set up idex to ru over two periods of the sigal. dt : i:,.. t i : dt i Eter the Fourier Series: f(i): C cos wo t + arg C Plot the periodic sigal: 7 i 6 Here are the coefficiets: C.6.7i.388i i.77.79i i.9.77i i.5.i i P5.6-3 From P5.5-3 C

16 P5.6- MathCad Spreadsheet: dex. of summatio: : :,.. m:,.. idexes pos, m idexes eg, Defie parameters: A: : as the summatios should be ru from to + Defie icremet of time. Set up idex to ru over two periods of the sigal. : Eter the formulas for the Fourier Series:! dt: i:,.. t i : dt i 6 6 " $! 6 6 C : t exp j t dt # C m: t exp j m t dt Eter the Fourier Series: " $ # f(i): C exp j t + C exp j m t + ow to get the Fourier Spectrum: 6 6. i m i 5 m 8 : φ : arg(c ) C φ

17 P5.6-5 MathCad Spreadsheet dex. of summatio: :5 :,,.. m:,.. idexes pos, m idexes eg, Defie parameters: A : : as the summatios should be ru from to + Defie icremet of time. Set up idex to ru over two periods of the sigal. : dt: i :,.. t i: dt i Eter the formulas for the Fourier Series: C: exp C : m exp f(i): C exp j t + C exp j m t + 5 Eter the Fourier Series: t. t t exp j t dt+ exp exp j t dt 5. t exp j m t dt+ exp exp j m t dt 5. i6 m i6. m 8 ow to get the Fourier SP ectrum: : φ: arg C 6 Cotiued C φ i i i i i i i i i i i i i i

18 Sectio 5.8: Circuits ad Fourier Series P5.8- Refer to able 5.-. ake A 5, ad a 5. he v(t) s o 5+ Let k k si k t k v(t) s 5+ sit 5+ sit+ si6t+ sit odd o 67 he trasfer fuctio of the circuit is ( ) ( ) V s6 6 6 () so 5 5 i(t) + + j + j odd + + e 6 ta 6 9 si tta P5.8- V(s) V(s) s V(s) V(s) s We require Z p (s) where z p (s) Z (s) + Z (s) s p s LC R L R C s R L LC RLLC RL + jr C L ad Z (s) R + jl. After some algebra s R 8 + LC R LC or L C R + R L L µ H 8

19 P5.8-3 Rather tha fid the Fourier Series of v(t) directly, cosider the sigal v(t) show above. hese two sigals are related by v(t) v (t) 6 sice v(t) is delayed by ms ad shifted dow by 6 V. For example, at t ms v (ms) 3 V v( ms) V he Fourier series of v (t) is obtaied as follows ms radias rad/ms ms average value of v (t) a because v (t) is a odd fuctio. b 63t6si tdt 3 3 si t dt t si tdt " cos t 3 3 si t tcos t! $ # cos 6 si 6cos 67 so v (t) si t he v(t) 6+ si t6 6+ si t where t is i ms. Equivaletly v(t) si t where t is i s 83

20 so he ad ext, the trasfer fuctio of the circuit is R R (s) L s Cs Ls R R s L s LC ( ) () R j L LC + Fially, v t 5 j 9 R L j 8 + j j j + 3 si t + 9 ta + e 9 j 9 ta P5.8- Rather tha fid the Fourier Series of v(t) directly, cosider the sigal v(t) ˆ show below. ( t ) hese two sigals are related by v(t) vˆ Let's calculate the Fourier Series of v(t), ˆ takig advatage of its symmetry. 6ms O rad rad ms 6ms 3 3. a o average value of v(t) ˆ V 6 b because v(t) ˆ is a eve fuctio ( t ) a 33 cos tdt 6 3 8

21 a cos t dt t cos t dt 3 3 si 3 cos t tsi t si cos si cos So 8 ˆv( t) + cos cos t v() t vˆ ( t) + cos cos t where t is i ms. Equivaletly v(t) + 8 t cos cos 3 where t is i secods. ext we calculate the trasfer fuctio of the circuit: ( ) R + j CR j CR R +jr jr C + C + jc 6 6 j j + j ta + ta he output voltage is v6 t At t ms v t cos cos 9 ta ta cos cos 9 ta ta

22 P 5.9- Let () at at g t e u() t e u( t). otice that f () t g() t ext ( ) lim. a ( a+ j) t ( aj) t at jt at jt e e G e e dt e e dt + F j limg lim a a a + j Fially ( ) ( ) ( a j) ( a j) j ( a j) ( a j) + a + P 5.9- ( ) ( ) ( a+ j ) t at jt at jt Ae A A F Ae u t e dt Ae e dt a+ j a+ j a+ j ( ) ( ) P First otice that A A he, from lie 6 of able 5.-: { f() t } Sa Sa { } () d A Also, from lie 7 of able 5.-: { f() t } f() t j { f t } j Sa dt si A A his ca be writte as: { f() t } j si j 86

23 P 5.9- First otice that: ( ) { } ( ) jt j t δ δ e d e j t e j5t j5t δ. ext, cos 5t 5 e + 5 e. herefore { } ( ) j5t j5t herefore { } { } { } cos 5t 5 e + 5 e δ ( 5) + δ ( + 5). P5.9-5 jt e F e dt e e j j j j j jt j j ( ) ( ) (( cos si ) ( cos si ) ) P j + ( cos cos ) ( si si ) B j t ( ) ( ) ( ) j B B A j t A e A e ( ) F t e dt jt jb B B j B jb jb A Be e B + j P jt jt e e F e dt e dt e e e e j j j j jt jt j j j j ( ) ( ) ( ) si si ( ) P5.- s () sigum() i t t s H ( ) ( ) 8 j j + j s ( ) ( ) 8 ( ) H( ) s ( ) + j j j + j t () sigum() () i t t e u t 87

24 P5.- () cos 3 A s ( ) δ( 3) + δ( + 3) ( ) ( ) s ( ) + j δ ( 3) + δ ( + 3) ( ) i t t s H + j ( 3) + δ ( + 3) j3t j3t δ jt e e i() t e d 5 j + + j3 + j3 ( ) ( ) e + e cos j t j t ( t ) P 5.-3 () cos t ( ) δ ( + ) + δ ( ) v t V Y ( ) + j ( + ) + δ ( ) δ ( ) Y( ) V( ) + j ( + ) + δ ( ) jt jt δ jt e e i() t e d 5 j + + j + j ( 5) ( 5) ( ) j t j t 5 e e + 5 cos t5 A 88

25 P5.- t () ( ) + () v t eu t u t ( j ) t t t jt t jt e { ( )} ( ) eu t eu t e dt ee dt j j + j { u() t} δ ( ) V ( ) + δ( ) + j j j + j, H ( ) j j j + j V o 3 δ ( ) j j j 3+ j j j 3+ j ( ) δ( ) ( ) δ ( ) δ jt e d 3+ j 3+ j 6 3t t vo () t e u() t + eu( t) + sigum() t P5.-5 5t 5 vs () t 5e u() t V V( ) 5 + j ( ) ( ) 5t 5t ( ()) ( ) W 5e u t dt 5e dt.5 J H s jc RC R+ + j jc RC C µ F. ry R k Ω. he V o 5 + j 5 + j Wo d d 5 J + j 5 + j

26 P 5.-6 H ( ) + j 8 8 Vs ( ) { 8u( t) 8u( t ) } 8δ( ) + 8δ( ) + e j j 8 ( ) ( j) sice j Vs e δ( ) e δ( ) j 8 j Vo ( ) ( e ) e + j j j + j j + j ext use V o + δ j j ( ) δ ( ) ( ) 8 δ ( ) δ ( ) 8 δ ( ) δ ( ) t () () () ( ) to write e j + j j + j ( t) ( ( )) j j δ ( ) 8 + δ ( ) e j + j j + j v t 8u t 8e u t 8u t 8e u t o 8( e t ( ) ) () 8( t u t e ) u( t) V j j 9

27 PSpice Problems SP 5- i pulse ( ) R.tra four.595 v().probe.ed FOURER COMPOES OF RASE RESPOSE V () DC COMPOE.88355E- HARMOC FREQUECY FOURER ORMALZED PHASE ORMALZED O (HZ) COMPOE COMPOE (DEG) PHASE (DEG).59E-.E+.E+ -.78E-.E+ 3.83E-.E+ 5.3E- -.8E+ -.8E E E E- -3.3E- -.53E E- 5.3E-.56E- -.8E+ -.8E E-.6E-.8E E- -.9E E E-.676E- -.8E+ -.8E+ 7.E+.88E-.E E- -6.5E- 8.73E+.56E-.63E- -.8E+ -.8E+ 9.3E+.5E-.6E- -9.6E- -8.5E- SP 5- i pulse ( ) R.tra..four V().probe.ed FOURER COMPOES OF RASE RESPOSE V() DC COMPOE E- HARMOC FREQUECY FOURER ORMALZED PHASE ORMALZED O (HZ) COMPOE COMPOE (DEG) PHASE (DEG).E+ 3.8E-.E+ -.77E+.E+.E+.59E-.996E- -.7E+.895E+ 3 3.E+.59E- 3.37E- -.73E E+.E+ 7.95E-.9E- -.68E E+ 5 5.E+ 6.36E-.988E E+.56E+ 6 6.E+ 5.57E-.65E- -.67E+.E+ 7 7.E+.9E-.E E+.73E+ 8 8.E+ 3.9E-.3E E+.7E+ 9 9.E+ 3.6E-.88E- -.5E+.33E+ 9

28 SP 5-3 Vi pulse (.5m.5m m) R.tra u m.four k v().probe.ed FOURER COMPOES OF RASE RESPOSE V() DC COMPOE E- HARMOC FREQUECY FOURER ORMALZED PHASE ORMALZED O (HZ) COMPOE COMPOE (DEG) PHASE (DEG).E E+.E E+.E+.E+3.6E- 3.E E+.796E+ 3 3.E+3.698E E- 8.89E+.793E+.E+3.6E- 3.E-3-9.E+ -.8E+ 5 5.E+3.9E+.E- -9.8E+ -.E+ 6 6.E+3.6E- 3.E E+.78E+ 7 7.E+3 7.7E+.8E- 8.78E+.778E+ 8 8.E+3.6E- 3.E E+ -.5E+ 9 9.E E+.E- -9.3E+ -.88E+ SP 5- Vi pulse (5 5 5) R.tra. 5.four. v().probe.ed FOURER COMPOES OF RASE RESPOSE V() DC COMPOE 8.96E+ HARMOC FREQUECY FOURER ORMALZED PHASE ORMALZED O (HZ) COMPOE COMPOE (DEG) PHASE(DEG).E- 7.9E+.E+.53E+.E+.E- 6.3E+ 8.7E-.66E+ 3.58E+ 3 6.E-.6E+ 5.73E- -.6E+ -.89E+ 8.E-.935E+.69E- -.89E+ -.5E+ 5.E+ 8.E-.78E E+ -.89E+ 6.E+.8E+.593E-.7E+ -3.6E+ 7.E+.7E+.97E-.57E+ 3.68E+ 8.6E+.537E+.7E E+ -.93E+ 9.8E+ 8.95E.7E- -.35E E+ 9

29 SP 5-5 Vi pulse ( ) R.tra..four v().probe.ed FOURER COMPOES OF RASE RESPOSE V() DC COMPOE.9937E- HARMOC FREQUECY FOURER ORMALZED PHASE ORMALZED O (HZ) COMPOE COMPOE (DEG) PHASE (DEG).E+ 6.36E-.E E+.E+.E+ 3.8E-.996E-.679E+.83E+ 3 3.E+.7E- 3.36E- -.73E+.68E+.E+.585E-.9E E+.87E+ 5 5.E+.6E-.987E E E+ 6 6.E+.5E-.65E-.7E+.97E+ 7 7.E+ 8.97E-.E E+.9E+ 8 8.E+ 7.87E-.8E-.88E+.965E+ 9 9.E+ 6.96E-.87E E+.883E+ Verificatio Problems VP 5- f(t) + cos t a, a ad all other coefficiets are zero. he computer pritout is correct. VP 5- able 5. - shows that the average value of a full wave rectified siewave is A () where A is the amplitude of the siewave. this case a 55. Ufortuately the report says, "half - wave rectified." he report is ot correct. 93

30 Desig Problems DP 5- For siusoidal aalysis, shift horizotal axis to average, which is 6V. Have odd fuctio ad half -wave symmetry a, / eed third harmoic : b b 3 so v. si 6t. cos (6t 9 ) f(t) si t dt / si 6t dt / cos6t. 6 V. ( assume si iput ad output for ease ), Z / 6 trasfer fuctio H(j ) 6 j/6c V VH (. ) H θ.36 choose so H 3. requires C F so H j3 third harmoic of v.36 si (6t+6.9 ) V c j for third harmoic 6c DP 5- Refer to able 5.-. So here v (t) 36 6 s v(t) A A s cos (377t) or v (t)v + v (t) ad v (t)v + v (t) ripple. dc output max v (t). v or v (t). v but v v whe dc the L becomes a short V o R R+j L V s so s o o o o o so s o cos( t) 9

31 R V R+j L V + j377l V, but V s s s So V + j377l 6 3 but V v ad v v 36. o o s 6 the +(377) L Solvig for L yields L >.5mH 6 () 3 DP 5-3 V o V to dc rasfer Fuctio First harmoic: s Zp V Z +Z V where Z R s p L p + jrc R So V + jrc / LC V R s j L+ (j ) +(j ) + + jrc RC LC ad V Vo Vs /LC or + RC + 8 with R75 kω LC ad choosig L. mh yields C. F 95