(b) y(t) is not periodic although sin t and 4 cos 2πt are independently periodic.

Transcription

1 Chapter 7, Sluti. (a) his is peridic with ω which leads t /ω. (b) y(t) is t peridic althugh si t ad cs t are idepedetly peridic. (c) Sice si A cs B.5[si(A B) si(a B)], g(t) si t cs t.5[si 7t si( t)].5 si t.5 si7t which is harmic r peridic with the fudametal frequecy ω r /ω. (d) h(t) cs t.5( cs t). Sice the sum f a peridic fucti ad a cstat is als peridic, h(t) is peridic. ω r /ω. (e) (f) (g) he frequecy rati.6..5 makes z(t) peridic. ω. r /ω. p(t) is t peridic. g(t) is t peridic. Chapter 7, Sluti. (a) he frequecy rati is 6/5.. he highest cmm factr is. ω / r. (b) ω r /ω. (c) f (t) si 6 t (/)( cs t) ω r /ω /() /6. (d) f (t) e jt cs t jsi t. ω r /ω..

2 Chapter 7, Sluti., ω / / g(t) 5, < t <, < t <, < t < a (/) g(t)dt.5[ 5dt ] dt.75 a (/) g(t) cs(ω t) dt (/)[ 5 cs( t)dt cs( t)dt ].5[ 5 si t si t ] ( /())5 si(/) a (5/())( ) ()/, dd, eve b (/) g(t) si(ω t) dt (/)[ 5 si( t)dt si( t)dt ] x5.5[ cs t x cs t ] (5/())[ cs cs(/)] Chapter 7, Sluti. f(t) 5t, < t <,, ω / a (/) f (t)dt (/) ( 5t)dt.5[t (5t / )] 5 a (/) f (t) cs(ω t) dt (/) ( 5t) cs(t)dt ( ) cs(t)dt (5t) cs(t)dt 5 cs t 5t si t [ 5/( )](cs )

3 b (/) ( 5t) si(t)dt ( ) si(t)dt (5t) si(t)dt 5 si t 5t cs t [/()](cs ) /() Hece f(t) 5 si(t) Chapter 7, Sluti 5., ω / a z(t)dt [x x ].5 a z(t)cs ωdt cs tdt cs tdt si..t si t b hus, z(t).5 z(t)cs ωdt 6 dd si t si tdt si tdt cs t cs t 6, dd, eve

4 Chapter 7, Sluti 6., ω a y(t)dt (x x) 6 Sice thisis a dd fucti,a. b y(t) si(ω t)dt si(t)dt si(t)dt cs(t) cs(t) (cs() ) (cs() cs()) ( cs()) ( cs()) ( cs()),, eve dd y(t) dd si(t) Chapter 7, Sluti 7., ω /, a 6 a f (t)cs ωdt [ cs t / 6dt ( )cs t / 6dt] 6 si t / 6 si t / 6 b f (t)si ωdt [ si t / 6dt ( )si t / 6dt] 6 [ si / si / si 5 / ]

5 cs t / 6 cs t / 6 [ cs5 / cs / si / ] f (t) ( a cs t / 6 b si t / ) 6 where a ad b are defied abve. Chapter 7, Sluti 8. f (t) ( t), -< t <,, ω / a f (t)dt (t )dt t t t a f (t)cs dt (t )cs tdt cs t si t si t ω b t f (t)si ωdt (t )si tdt si t cst cst cs f (t) ( ) cs t Chapter 7, Sluti 9. f(t) is a eve fucti, b. 8, ω / / a f ( t) dt cs / ( )si / 8 t dt t.8

6 [ ]dt t t dt t t dt t f a / ) / ( cs ) / ( cs 5 ] / cs / cs [ 8 )cs ( ω Fr, / si 5 ] / [cs 5 t dt t dt t a Fr >, ) ( si ) ( ) ( si ) ( ) ( si ) ( ) ( si ) ( t a si si 6.66, / si si a a hus,, 6.6,,.8, b b b a a a a Chapter 7, Sluti. ω /, ω t j t j t j t j t j j e j e dt )e ( dt e dt h(t)e c [ ], eve, dd, 6j 6] [6cs j e e e j c j j j hus, t j dd e j6 f (t)

7 Chapter 7, Sluti., ω / / c jω jt / jt / y(t)e t dt (t )e dt ()e dt c jt e / / ( jt / ) j jt e / j jt e j / j / j / e (j / ) e e j j j / j But e j / cs / jsi / jsi /, e j / cs / jsi / jsi / c [ j(j / )si / si / ] y(t) j / [ j(j / )si / si / ] e t Chapter 7, Sluti. A vltage surce has a peridic wavefrm defied ver its perid as v(t) t( - t) V, fr all < t < Fid the Furier series fr this vltage. v(t) t t, < t <,, ω / a (/) f (t)dt (t t ) dt ( t ( / ) t / )

8 a (t t )cs(t)dt t cs(t) si(t) [ t cs(t) si(t) t si(t ] ) ( ) cs() b (t t )si(t)dt (t t )si(t) dt (si(t) t cs(t)) (t si(t) cs(t) t cs(t)) Hece, f(t) cs(t) Chapter 7, Sluti., ω a (/) h(t)dt [ si t dt si(t ) dt] [ cs t cs(t ) ] a (/) h(t) cs(ωt) dt [/()] si t cs(t)dt si(t ) cs(t)dt Sice si A cs B.5[si(A B) si(a B)] si t cs t.5[si(( )t) si(( ))t] si(t ) si t cs cst si si t si(t )cs(t) si(t)cs(t)

9 a [si([ ]t) si([ ]t)]dt [si([ ]t) si([ ]t)]dt 5 cs([ ]t) cs([ ]t) cs([ ]t) cs([ ]t) a 5 cs([ ] ) cs([ ] ) But, [/()] [/(-)] /( ) cs([ ]) cs([]) cs cs si si cs a (5/)[(6/( )) (6 cs()/( ))] [/(( ))]( cs ) [ 6/(( ))], eve, dd b (/) h(t) si ω t dt [/()][ si t si t dt ( si t) si t dt But, si A si B.5[cs(A B) cs(ab)] si t si t.5[cs([ ]t) cs([]t)] b (5/){[(si([ ]t)/( )) (si([]t)/ ( )] [(si([-]t)/(-)) (si([]t)/( )] } 5 si([ ] ) si([ ] ) 6 hus, h(t) cs(kt) (k ) k

10 Chapter 7, Sluti. Sice cs(a B) cs A cs B si A si B. f(t) cs( / )cs(t) si( / )si(t ) Chapter 7, Sluti 5. (a) Dcs ωt Esi ωt A cs(ωt - θ) where A D E, θ ta - (E/D) 6 A 6 ( ), θ ta - (( )/( )) f(t) 6 ( ) 6 cs t ta (b) Dcs ωt Esi ωt A si(ωt θ) where A D E, θ ta - (D/E) f(t) 6 ( ) 6 si t ta Chapter 7, Sluti 6. If v (t) is shifted by alg the vertical axis, we btai v * (t) shw belw, i.e. v * (t) v (t). v * (t) t

11 Cmparig v * (t) with v (t) shws that v * (t) v ((t t )/) where (t t )/ at t - r t Hece v * (t) v ((t )/) But v * (t) v (t) v (t) v ((t)/) v (t) - v ((t)/) 8 t t t - cs cs cs v (t) 8 t cs 9 t cs 5 5t cs 5 v (t) 8 t si 9 t si 5 5t si Chapter 7, Sluti 7. We replace t by t i each case ad see if the fucti remais uchaged. (a) t, either dd r eve. (b) t, eve (c) cs (-t) si (-t) - cs t si t, dd (d) si (-t) (-si t) si t, eve (e) e t, either dd r eve.

12 Chapter 7, Sluti 8. (a) leads t ω / f (-t) -f (t), shwig that f (t) is dd ad half-wave symmetric. (b) leads t ω / f (t) f (-t), shwig that f (t) is eve. (c) leads t ω / f (t) is eve ad half-wave symmetric. Chapter 7, Sluti 9. his is a half-wave eve symmetric fucti. a b, ω / / / a t cs(ωt) dt [/() ]( cs ) 8/( ), dd, eve f (t) 8 dd t cs Chapter 7, Sluti. his is a eve fucti. b, 6, ω /6 / a / f (t)dt (t )dt 6 dt

13 (t t) ( ) / a f (t) cs(t / )dt (/6)[ ( t ) cs(t / )dt cs(t / )dt ] t cs t t si t si 6 t si 6 [/( )][cs(/) cs(/)] hus f(t) At t, t cs cs cs f() (/ )[(cs(/) cs(/))cs(/) (/)(cs(/) cs(/))cs(/) (/9)(cs() cs())cs() -----].( ) f().756 Chapter 7, Sluti. his is a eve fucti. b,, ω / /. f(t) t, < t <, < t < t a ( t)dt t.5 a / f (t) cs(ω t)dt ( t t) cs dt

14 [8/( )][ cs(/)] f(t) 8 t cs cs Chapter 7, Sluti. Calculate the Furier cefficiets fr the fucti i Fig f(t) t Figure 7.6 Fr Prb. 7. his is a eve fucti, therefre b. I additi, ad ω /. a f (t)dt tdt t a f (t) cs( ω t)dt t cs(t / ) dt t cs(t / ) si(t / ) 6 8 a (cs( / ) ) si( / ) Chapter 7, Sluti. f(t) is a dd fucti. f(t) t, < t < a a,, ω /

15 b / f (t) si(ω t)dt t si(t) dt [ si(t) t cs(t ] ) [/()]cs() () /() f(t) ( ) si(t) Chapter 7, Sluti. (a) his is a dd fucti. a a,, ω / / b f (t) si( ωt) dt f(t) t/, < t < b ( t / ) si(t) dt t cs(t) si(t) cs(t ) [/()][ cs()] [/()][ () ] a, b [/()][ ()] /.8 (b) ω ω r a, b [/()][ cs()] /(5) hus the magitude is A a b /(5).666 ad the phase is φ ta (b /a ) 9

16 (c) f(t) [ cs()] si(t) f(/) [ cs()] si( / ) Fr, f (/)( ) 6/ Fr, f Fr, f [/()][ cs()]si(/) 6/() Fr, f Fr 5, f 5 6/(5), ---- hus, f(/) 6/ 6/() 6/(5) 6/(7) (6/)[ / /5 / ] f(/).8 which is withi 8% f the exact value f.5. (d) Frm part (c) f(/).5 (6/)[ / /5 / ] (/)(/6) [ / /5 / ] r / / /5 / Chapter 7, Sluti 5. his is a dd fucti sice f(t) f(t). a a,, ω /. b / f (t) si(ω t)dt t si(t / ) dt

17 9 t si t t cs 9 si cs f(t) si t cs si Chapter 7, Sluti 6., ω / / a f (t)dt dt dt dt a f (t) cs( t) dt ω a cs(t / )dt cs(t / )dt cs(t / )dt t si si si t si t si b f (t) si(ωt) dt t si dt t si dt t si dt t cs t cs t cs

18 [ cs() ] Hece f(t) [(si( / ) si( / )) cs(t / ) (cs() ) si(t / ) ] Chapter 7, Sluti 7. (a) dd symmetry. (b) a a,, ω / / f(t) t, < t <, < t < b t t t t t si dt si cs si cs () ()/ /( ), dd () / /(), eve a, b ()/(9 ).5 (c) b /, b /, b /(9 ), b /(), b 5 (5 ) F rms a ( a b ) F rms.5σb [/( )][(6/ ) (6/(8 )) (/) (6/(65 ))] (/9.79)( ) F rms Cmpare this with the exact value f F rms t dt / 6.8

19 Chapter 7, Sluti 8. his is half-wave symmetric sice f(t /) f(t). a,, ω / a / f (t) cs(ω t)dt ( t) cs(t) dt si(t) t cs(t) si(t) [/( )][ cs()] 8/( ), dd, eve b ( t) si(t)dt cs(t) t si(t) cs(t) /(), dd f(t) 8 k cs(t) si(t ), k Chapter 7, Sluti 9. his fucti is half-wave symmetric., ω /, f(t) t, < t < Fr dd, a ( t) cs(t)dt cs(t) t si(t) [ ] /( ) b ( t) si(t)dt [ si(t) t cs(t) ] /

20 hus, f(t) cs(t) si(t), k k Chapter 7, Sluti. / jω / / t c f (t)e dt f (t) cs ω ω / tdt j f (t)si / tdt () / (a) he secd term the right had side vaishes if f(t) is eve. Hece c / f (t)cs ωtdt (b) he first term the right had side f () vaishes if f(t) is dd. Hece, c / j f (t)si ωtdt Chapter 7, Sluti. If h(t) f ( αt), ' / α ω ' ' / α αω a ' ' ' h(t) cs ω ' tdt ' ' f ( αt) cs ω ' tdt Let α t λ,, d t dλ / α, α' α a ' f ( λ)cs ωλdλ / α a Similarly, b ' b

21 Chapter 7, Sluti. Whe i s (DC cmpet) i /( ) / Fr, ω, I s / I [/( jω )]I s I s /( j6) hus, ta() ta (6 / ) i(t) cs( ta ()) Chapter 7, Sluti. Fr the DC case, the iductr acts like a shrt, V. Fr the AC case, we btai the fllwig: V V s V j jv 5 j.5 V V s V Vs j.5 5 A Θ j.5 5 j(.5 5) A (.5 5) ; Θ ta.5 5

22 v (t) A si(t Θ Chapter 7, Sluti. Fr ay, V [/ ] (/), ω. H becmes jω L j ad.5 F becmes /(jω C) j/ Ω j ) V V j/ V V {j(/)/[ j j(/)]}v {j/[ j( )]}[(/ ) (/)] (( / ) / ) ( ) ta (( [( / ) ( / ) ta ) / ) (( ) / )] v (t) Chapter 7, Sluti 5. cs t ta If v s i the circuit f Fig. 7.7 is the same as fucti f (t) i Fig. 7.57(b), determie the dc cmpet ad the first three zer harmics f v (t). Ω H v S F Ω v Figure 7.7 Fr Prb. 7.5

23 f (t) t Figure 7.57(b) Fr Prb. 7.5 he sigal is eve, hece, b. I additi,, ω /. v s (t) fr all < t < fr all < t <.5 a. 5 dt dt a cs(t / )dt.5 cs(t / )dt 6. 5 si( t / ) si( t / ) si( / ) v s (t) si( / ) cs(t / ) Nw csider this circuit, Ω j/ v S -j/() Ω v Let Z [-j/()]()/( j/()) -j/( - j) herefre, v Zv s /(Z j/). Simplifyig, we get v j9v s j( 8)

24 Fr the dc case, ad v s ¾ V ad v v s / /8 V. We ca w slve fr v (t) v (t) 8 A t cs Θ vlts where A 6 6 si( / ) 6 ad Θ 9 ta where we ca further simplify A t this, A 9si( / ) 8 Chapter 7, Sluti 6. v s (t) A cs(t θ ) dd where θ ta [(/())/(/())] ta ().5 A 9 si 9 si ω ad H becmes jω L j Let Z j j/( j) If V is the vltage at the -referece de r acrss the -H iductr. V ZV s /( Z) [j/( j)]v s /{ [j/( j)]} jv s /( j) But V s A θ V j A θ /( j)

25 I V /j [ A θ ]/ 6 ta 9 si 6.5 ta Sice si(/) () ()/ fr dd, si (/) I.5 6 ta i (t) cs(t.5 ta ) 6 dd Chapter 7, Sluti 7. Frm Example 5., v s (t) 5 si(t), k k Fr the DC cmpet, the capacitr acts like a pe circuit. Fr the th harmic, V 5 V s [/()] mf becmes /(jω C) j/(xx ) j/() v j Vs j j j 9 ta v (t) si(t 9 ta ) 5

26 Chapter 7, Sluti 8. v s (t) si t, k k V jω Vs, ω jω Fr dc, ω, V.5, V s Fr th harmic, Vs 9 V 9 9 ta ta v (t) k cs(t ta ), k Chapter 7, Sluti 9. Cmparig v s (t) with f(t) i Figure 5., v s is shifted by.5 ad the magitude is 5 times that f f(t). Hece v s (t) 5 si(t), k k, ω //, ω ω Fr the DC cmpet, i 5/( ) / Fr the kth harmic, V s (/()) mh becmes jω L jx. j. 5 mf becmes /(jω C) j/() I Ω Ω I V S j/() j. Z

27 Let Z j/() ( j.) j ( j.) j j. j( j. j j. j8 j(. ) Z i Z 8 j( ) j(. ) I Vs Z i [8 j( ) j( )] I j j I ( j.) ji j(. ) j [8 j( )] hus 9 ta (8) {( ( i (t) k ) /(8)} ) I si(t θ ), k where θ 9 ta 8 I (8) ( )

28 Chapter 7, Sluti., ω / t a v(t)dt ( t)dt t / a v(t) cs(t)dt ( t) cs(t)dt si(t) t cs( t) si( t) ( cs ),, eve dd ( ) b v(t) si(t)dt ( t) si(t)dt cs(t) si(t) t cs(t) v s (t) A cs(t ϕ ) ( ) 6 where φ ta, A ( ) Fr the DC cmpet, v s /. As shw i Figure (a), the capacitr acts like a pe circuit. Ω V x V x V.5V i V x Ω V (a)

29 Ω V x V x V V S (/)F Ω V (b) Applyig KVL t the circuit i Figure (a) gives.5 V x i () But.5 i V x r V x i () Addig () ad (),.5 6i r i.5 V i.75 Fr the th harmic, we csider the circuit i Figure (b). ω, V s A φ, /(jω C) j/() At the superde, (V s V x )/ [/(j)]v x V / V s [ j/]v x V / () But V x V x V r V V x Substitutig this it (), V s [ j/]v x V x [ j/]v x (/)[ j/]v (/)[8 j]v V V s /(8 j) 6 A φ ta ( / 8) 6 V [ta ( / 8) ta ( ( ) /())] 6 ( )

30 hus v (t) V cs(t θ ) 6 where V ) 6 ( θ ta (/8) ta (( )/()) Chapter 7, Sluti. Fr the full wave rectifier,, ω /, ω ω Hece t v i (t) cs( ) Fr the DC cmpet, V i / he iductr acts like a shrt-circuit, while the capacitr acts like a pe circuit. V V i / Fr the th harmic, V i [ /(( ))] H becmes jω L j. F becmes /(jω C) j5/ Z ( j5/) j/( j) V [Z/(Z j)]v i jv i /( j(8 )) j j(8 ) ( )

31 {9 ta ( ) (.5)} 6 (8 ) Hece v (t) where A ( ) 6 A cs(t θ ) 9 θ 9 ta (.5) Chapter 7, Sluti. v s 5 si t, k - k Vs R jωc( V ) V j Vs, ωrc ω ω Fr (dc cmpet), V. Fr the th harmic, Hece, 9 V 9 RC 9 x xx v 5 (t) cs t, k - k 5 Alteratively, we tice that this is a itegratr s that v 5 (t) vsdt cs t, k - RC k

32 Chapter 7, Sluti. (a) V rms a (a b ) ( ).9 V (b) I rms 6 ( ) 6.78 A (c) P V dc I dc V I cs(θ Φ ) x6.5[xcs(5 - ) xcs(-5 6 )] W Chapter 7, Sluti. (a) p vi [ 6cs 5 cs 5 ] W (b) he pwer spectrum is shw belw. p ω Chapter 7, Sluti 5. ω jω L jxx j /(jω C) j/(xx 6 ) j5/

33 Z R jω L /(jω C) j j5/ I V/Z Fr, V, Z j j5 j I /( j) Fr, V 5, Z j j.5 j8.5 I 5/( j8.5).8.6 Fr, V 5, Z j6 j5/ j. I 5/( j.).5. I rms A p V DC I DC V cs( θ φ I ).5[x.987cs(7.89 ) 5x.8cs(.6 ) 5x.5cs(. )].5[ ] 57.5 watts Chapter 7, Sluti 6. (a) his is a eve fucti I rms f (t)dt / f (t)dt f(t) t,, < t < < t <, ω / / I rms ( t) dt (t t t / )

34 ( /) / r I rms.865 A (b) Frm Prblem 6., a [8/( )][ cs(/)], a.5 a 8/, a /, a 8/(9 ), a, a 5 9/(5 ), a 6 /(9 ) I rms a A I rms.86 A. 666 Chapter 7, Sluti 7. Let I I DC I I Fr the DC cmpet I DC [5/(5 )]() A I j8 I s 5 Ω Ω Fr AC, ω jωl jx8x j8 I 5I s /(5 j8) Fr I s.5 6 I 6 /(5 j8) r I / Fr I s.5 I.5 /(5 j8) r I.5/ p (I DC I / I /) ( [/(x89)] [6.5/(x89)])x p.88 watts

35 Chapter 7, Sluti 8. (a) Fr the DC cmpet, i(t) ma. he capacitr acts like a pe circuit s that v Ri(t) x xx Fr the AC cmpet, ω,, /(jω C) j/(xx 6 ) ( j/) kω Z ( j/) ( j/)/( j/) j/( j) V ZI [ j/( j)]i Fr, Fr, V [ j/( j)] mv V [ j/( j)] mv v(t).cs(t 8. ).58cs(t 5.96 ) V (b) p V DC I DC V cs( θ φ I ) x.5xx.cs(5 8. ).5xx.58cs( ) 8. mw Chapter 7, Sluti 9. (a) Z rms z (t)dt dt dt (5). 5 (b) Z rms Z rms.58 6 a (a b ) Z rms (c ) %errr x

36 Chapter 7, Sluti 5. jω t c f (t)e dt, ω t te j dt Usig itegrati by parts, u t ad du dt dv e jt dt which leads t v [/(j)]e jt t jt jt c e e dt j j j j jt [ e e ] ( j) e jt [j/()]cs() [/( )](e j e j ) hus c j( ) j si() j( ) f(t) jωt c e ( ) j e jt Chapter 7, Sluti 5., ω / ( t jt ) jt jωt jt e f (t)e dt t e dt ( j) c c j ( j) ( j) f (t) ( j)e jt

37 Chapter 7, Sluti 5. jω t c f (t)e dt, ω t te j dt Usig itegrati by parts, u t ad du dt dv e jt dt which leads t v [/(j)]e jt t jt jt c e e dt j j j j jt [ e e ] ( j) e jt [j/()]cs() [/( )](e j e j ) hus c j( ) j si() j( ) f(t) jωt c e ( ) j e jt Chapter 7, Sluti 5. ω / c e t e jωt dt e ( jω )t ( j)t ( j) e [ e ] j dt j [/(j)][ e (cs() jsi())] ( e )/( j).6/( j f(t).6e j jt

38 Chapter 7, Sluti 5., ω / / c f (t)e jωt dt e jt / dt jt / e dt e jt / dt j j / j j / j j [ e e e e e ] j j / j [ e e ] f(t) c e jωt Chapter 7, Sluti 55., ω / c i(t)e jωt dt But i(t) si(t),, < t < < t < c jt si(t)e dt j (e jt e jt )e jt dt j jt( ) e j( ) jt( ) e j( ) e j ( ) e j ( )

39 ( ) j( ) j( ) j( ) j( [ e e e e ] ) But e j cs() jsi() e j hus j j j j c [ e e e e ] ( ) i(t) j e ( e ) jt j e ( ) Chapter 7, Sluti 56. c a, ω c (a jb )/ ( j)/[( )] f(t) ( j) ( ) jt e Chapter 7, Sluti 57. a (6/ ) c c.5(a jb ) a / /( ) f(t) j5t e Chapter 7, Sluti 58. c (a jb )/, ( ) cs(), ω / c [(cs() )/( )] j cs()/() hus f(t) cs() cs() j jt e

40 Chapter 7, Sluti 59. Fr f(t),, ω /. a DC cmpet (x )/.5 Fr h(t),, ω /. a (x x)/.5 hus by replacig ω with ω ad multiplyig the magitude by five, we btai j( ) t j5e h(t) ( ) Chapter 7, Sluti 6. Frm Prblem 6.7, a a, b [/()][ cs()], c c (a jb )/ [j/()][ cs() ],. Chapter 7, Sluti 6. (a) ω. f(t) a A cs(ω t φ ) 6 cs(t 5 ) cs(t 5 ) cs(t 5 ).5cs(t ) 6 cs(t)cs(5 ) si(t)si(5 ) cs(t)cs(5 ) si(t)si(5 ) cs(t)cs(5 ) si(t)si(5 ).5cs(t)cs( ).5si(t)si( ) 6.57cs(t).7si(t).65cs(t).7si(t).96cs(t).si(t).7cs(t).7si(t)

41 (b) f rms a A f rms 6.5[ (.5) ] 6.65 f rms 6.88 Chapter 7, Sluti 6. (a) ω /.s (b) f (t) a A cs(ωt φ ) cs(t 9 ) 5.cs(t 9 )... f (t) si t 5.si t.7si 6t.8si 8t... Chapter 7, Sluti 6. his is a eve fucti., ω /, b. f(t),, < t < < t <.5 a / f (t)dt dt.5 dt (/)[ ] / a / f (t)cs(ω t)dt t si cs(t / )dt 6 t si.5.5 cs(t / ) dt [ /()]si(/)

42 f (t) t si cs a /., ω /, a [/()]si(t/) A a b si A.55, A.757, A, A.75, A 5. he amplitude spectra are shw belw.. A Chapter 7, Sluti 6. he amplitude ad phase spectra are shw belw.

43 A ω φ 6 ω -8 Chapter 7, Sluti 65. a /( ), b /(), ω 9 A a b.,,, 5, 7, 9, etc. A

44 φ ta (b /a ) ta {[ /()][ /]} ta ( x.7) φ ω 5. A φ ω Chapter 7, Sluti 66. he schematic is shw belw. he wavefrm is iputted usig the attributes f VPULSE. I the rasiet dialg bx, we eter Prit Step.5, Fial ime, Ceter Frequecy.5, Output Vars V() ad click eable Furier. After simulati, the utput plt is shw belw. he utput file icludes the fllwig Furier cmpets.

45 DC COMPONEN 5.995E FOURIER COMPONENS OF RANSIEN RESPONSE V() HARMONIC FREQUENCY FOURIER NORMALIZED PHASE NORMALIZED NO (HZ) COMPONEN COMPONEN (DEG) PHASE (DEG) 5.E-.8E.E.78E.E.E.59E 5.E-.56E.78E.5E.6E.8E- 5.7E.56E.E 7.978E-.56E- 7.9E 5.7E 5.5E 6.9E-.8E- 8.9E 7.9E 6.E 5.6E-.676E-.69E 8.9E 7.5E.58E-.E-.8E.69E 8.E.E-.6E-.6E.8E 9.5E.58E-.6E-.6E.6E OAL HARMONIC DISORION 7.66E PERCEN Chapter 7, Sluti 67. he Schematic is shw belw. I the rasiet dialg bx, we type Prit step.s, Fial time 6s, Ceter frequecy.667, Output vars v(), ad click Eable Furier. After simulati, the utput file icludes the fllwig Furier cmpets,

46 FOURIER COMPONENS OF RANSIEN RESPONSE V() DC COMPONEN.96E HARMONIC FREQUENCY FOURIER NORMALIZED PHASE NORMALIZED NO (HZ) COMPONEN COMPONEN (DEG) PHASE (DEG).667E-.E.E E.E.E E-.75E- -8.9E 6.67E- 5.E- 5.E-.E- 9.E.8E 6.668E-.E-.75E- 9.E.8E 5 8.5E- 9.76E-.996E E.E- 6.E 7.8E-6.76E-6-9.E -.58E- 7.67E.968E-.E E.7E- 8.E.6E- 6.6E-5-8.7E.78E 9.5E 6.E-.68E- 9.E.8E OAL HARMON IC DISORION.865E PERCEN

47 Chapter 7, Sluti 68. he schematic is shw belw. We set the fial time 6s ad the ceter frequecy /.5. Whe the schematic is saved ad ru, we btai the Furier series frm the utput file as shw belw. FOURIER COMPONENS OF RANSIEN RESPONSE V() DC COMPONEN.99E HARMONIC FREQUENCY FOURIER NORMALIZED PHASE NORMALIZED NO (DEG) (HZ) COMPONEN COMPONEN (DEG) PHASE 5.E-.7E.E 9.E-.E.E 6.67E- 5.E- -.78E.79E.5E.6E-.E-.7E.8E.E.85E-.5E- -.76E -.77E 5.5E.59E-.E-.5E.6E 6.E.5E-.669E- -.76E -.755E 7.5E.8E-.E- 6.E 5.E 8.E.596E-.5E- -.78E -.77E 9.5E.9E-.5E- 8.E 7.E Chapter 7, Sluti 69.

48 he schematic is shw belw. I the rasiet dialg bx, set Prit Step.5 s, Fial ime, Ceter Frequecy.5, Output Vars V() ad click eable Furier. After simulati, we btai V() as shw belw. We als btai a utput file which icludes the fllwig Furier cmpets. FOURIER COMPONENS OF RANSIEN RESPONSE V() DC COMPONEN 5.85E- HARMONIC FREQUENCY FOURIER NORMALIZED PHASE NORMALIZED NO (HZ) COMPONEN COMPONEN (DEG) PHASE (DEG) 5.E-.56E-.E -9.9E.E.E.977E- 7.E E.8E.5E.5E-.7E E -.76E.E.969E- 7.E- -8.E 6.757E 5.5E.68E-.6E- -9.E -.7E 6.E.955E- 7.85E- -8.E 9.659E 7.5E 8.55E-.E E -.9E 8.E.95E- 7.8E E.5E 9.5E 5.58E-.96E- -9.7E -6.97E OAL HARMONIC DISORION.85E PERCEN

49 Chapter 7, Sluti 7. he schematic is shw belw. I the rasiet dialg bx, we set Prit Step. s, Fial Step s, Ceter Frequecy.5, Output Vars V() ad V(), ad click eable Furier. After simulati, we cmpare the utput ad utput wavefrms as shw. he utput icludes the fllwig Furier cmpets.

50 FOURIER COMPONENS OF RANSIEN RESPONSE V() DC COMPONEN E- HARMONIC FREQUENCY FOURIER NORMALIZED PHASE NORMALIZED NO (HZ) COMPONEN COMPONEN (DEG) PHASE (DEG) 5.E-.7E.E.E.E.E.758E-.5E- -.9E -.98E.5E.E-.97E E -.99E.E.7E-.66E E -6.87E 5.5E 8.58E- 7.98E E E 6.E 6.9E- 5.78E E E 7.5E.7E-.E- -6.5E -7.5E 8.E.7E-.69E- -7.E -8.6E 9.5E.997E-.8E E -8.9E OAL HARMONIC DISORION.5895E PERCEN

51 Chapter 7, Sluti 7. he schematic is shw belw. We set Prit Step.5, Fial ime s, Ceter Frequecy.5, Output Vars I(), ad click eable Furier i the rasiet dialg bx. After simulati, the utput wavefrm is as shw. he utput file icludes the fllwig Furier cmpets.

52 FOURIER COMPONENS OF RANSIEN RESPONSE I(L_L) DC COMPONEN E- HARMONIC FREQUENCY FOURIER NORMALIZED PHASE NORMALIZED NO (HZ) COMPONEN COMPONEN (DEG) PHASE (DEG) 5.E-.87E-.E -6.79E.E.E.89E- 8.68E- 8.7E 7.566E.5E.78E-.E E -.E.E 9.58E-5.9E- -.8E 6.565E 5.5E.7E-.6E E -.76E 6.E 6.66E-5.78E- -7.8E 6.8E 7.5E 5.97E-.596E E -.8E 8.E 6.59E-5.69E- -.88E.9E 9.5E.E- 9.E- -.E -5.87E OAL HARMONIC DISORION.8E PERCEN Chapter 7, Sluti 7. 5, ω / /5 f(t) is a dd fucti. a a / b f (t)si(ω t)dt si(. 5 t) dt 8x5 cs(.t) [ cs(.)] f(t) [ cs(.)]si(.t) Chapter 7, Sluti 7. p V DC R V R.5[( )/] mw

53 Chapter 7, Sluti 7. (a) A A A a b, φ ta (b /a ) 6 8, φ ta (6/8) , φ ta (/) 6.87 i(t) { cs(t 6.87 ) 5cs(t 6.87 )} A (b) p I DCR.5 I R [.5( 5 )] 57 W Chapter 7, Sluti 75. he lwpass filter is shw belw. R v s C v - - v s Aτ A τ si cs ωt V jωc V R jω C s jω V RC s, ω ω / Fr, (dc cmpet), V Aτ V s ()

54 Fr the th harmic, A τ V si 9 ω R C ta ω RC Whe, A τ V si () R C Frm () ad (), Aτ A 5x si R C R C.9 τ.9x R C C 5 R x.9x x.59 mf Chapter 7, Sluti 76. v s (t) is the same as f(t) i Figure 6. except that the magitude is multiplied by. Hece v (t) 5 si(t), k k, ω /, ω ω jω L j; Z R R/( R) V ZV s /(Z j) [R/(R j( R))]V s V R ta {( / 5R)( R)} V R ( R) s V s [/()] he surce curret I s is ( R) Vs Vs I s Z j R R j( R) j R

55 ( R) ta {( / )( R)} R ( R) p s V DC I DC Vs Is cs(θ φ ) Fr the DC case, L acts like a shrt-circuit. I s 5 R R 5( R), V s 5 V R p s 5( R) R ( R)cs ta R ( R) 5 ( R) ( R) cs ta ( R) 5 R 6 ( R) V p s DC V R R 5 R R R R ( R) R ( R) We wat p (7/)p s.7p s. Due t the cmplexity f the terms, we csider ly the DC cmpet as a apprximati. I fact the DC cmpet has the latgest share f the pwer fr bth iput ad utput sigals ( R) x R R 7 7R which leads t R /7.86 Ω

56 Chapter 7, Sluti 77. (a) Fr the first tw AC terms, the frequecy rati is 6/.5 s that the highest cmm factr is. Hece ω. /ω / (b) he average value is the DC cmpet (c) V rms a (a b ) V rms ( ) ( 8 6 ).5 V rms. V Chapter 7, Sluti 78. V DC DC (a) p R V R V R V,rms ( /5) ( /5) ( /5) W (b) 5% icrease (5/) DC p DC W which leads t VDC R 5 R V DC.5 V R V Chapter 7, Sluti 79. Frm able 7., it is evidet that a, b A/[( )], A. A Frtra prgram t calculate b is shw belw. he result is als shw.

57 C FOR PROBLEM 7.79 DIMENSION B() A PIE. C.*A/PIE DO N, B(N) C/(.*FLOA(N).) PRIN *, N, B(N) CONINUE SOP END b Chapter 7, Sluti 8. Frm Prblem 7.55, c [ e j ]/[( )] his is calculated usig the Frtra prgram shw belw. he results are als shw. C FOR PROBLEM 7.8 COMPLEX X, C(:) PIE.597 A.*PIE DO N, IF(N.EQ.) GO O X CMPLX(, PIE*FLOA(N)) C(N) (. CEXP( X))/(A*( FLOA(N*N))) PRIN *, N, C(N) CONINUE SOP END

58 c.88 j.6 j.x j x j x j 9.5x j Chapter 7, Sluti 8. (a) A A A f(t) cs(ω t) he ttal average pwer is p avg F rms R F rms sice R hm. (b) P avg F rms f Frm the Furier series abve (t)dt c A/, c A/[( )].5A

59 ω c c % pwer A/ A /( ) 8.% ω A/() 8A /(9 ) 8.% ω A/(5) A /(5 ).7% 6ω A/(5) 8A /(5 ).% 8ω A/(6) 8A /(969 ).% (c) 8.% (d).7% Chapter 7, Sluti 8. P VDC V R R Assumig V is a amplitude-phase frm f Furier series. But A C, c a A C Hece, P c c R R Alteratively, where V rms a P V rms R A c c c ( ) x P 89./ 7. W