TUTORIAL SOLUTIONS. F.1 KCL, KVL, Power and Energy Q.1. i All units in VAΩ,,

Transcription

1 F TUTOIAL SOLUTIONS F. KCL, KVL, Power and Energy Q All uns n VAΩ,,

2 Appendx F Tuoral Soluons Applyng KCL o he doed surface: + + Q. All uns n V, A, Ω Nework A Nework B Applyng KCL o he doed surface: + A regardless of he alue of. For Ω, For KΩ, V KV

3 Appendx F Tuoral Soluons Q. All uns n VAΩ,, 8 8+ V; ; olage across Ω ressor V

4 Appendx F Tuoral Soluons 6 Q. (a) Boh meers ge pose readngs A B + AM + VM Snce he arrows for and are n oppose drecons Power consumed Also, AM wll ge a pose readng f s pose, whle VM wll ge a pose readngs f s pose. Snce boh and are pose n hs case, s pose and power s consumed. (b) Boh meers ge negae readngs Boh and are negae, s pose and power s consumed. (c) One meer ges a pose readng and he oher ges a negae readng and hae oppose sgns, s negae and power s suppled by he dece. (d) One meer or boh meer ge zero readngs Power neher s consumed nor suppled by he dece.

5 Appendx F Tuoral Soluons 7 Q. Curren n crcu All uns n VAΩ,, Applyng KVL: + + Power consumed/suppled If he olage and curren arrows are n oppose drecons, Thus: or Power consumed ( )( ) olage curren Power consumed by Ω ressor ( ) 6W Power consumed by Ω ressor ( ) W Power consumed by V source ( ) W Power consumed by V source ( ) W Power suppled by V source W

6 Appendx F Tuoral Soluons 8 Q.6 All uns n VAΩ,, The A curren source s supplyng a power of ( )( ) W.

7 Appendx F Tuoral Soluons 9 Q.7 Effcency Elecrcal power suppled ( )( ) W. h.p. 76W h.p. 86W Mechancal power delered ( )( ) 86 Effcency Torque 9. % Moor speed ( re mn) π 6 rad re smn 6 π rad s Mechancal power delered 86 Torque moor speed π 6 78.Nm Energy los Power los 86 W Energy los per mn ( )( 6) 8J Q.8 Generaor oupu power ( )( ) W Generaor npu power 9. Generaor shaf speed. Torque 8. Nm..W re mn π 6 rad re smn. rad s

8 Appendx F Tuoral Soluons Q.9 Volage, curren and power gans for sysem Volage gan g log( ) db db Curren gan g 8 Power gan g gg 8 6 log( 6) db 76dB p elaonshp beween hese gans g g g p ( g p ) db ( g db) + ( g db) [ ] ( )( 6) log ( )( 6) db + [ ( ) + ( )] 6 76 log log db db db gp g g f load ressance equals amplfer's npu ressance Audo amplfer Mos loudspeakers hae ressances n he order of a few Ω. Howeer, n order no o load he CD player or oher audo npu equpmen, he npu ressance of he amplfer wll hae o be large and s usually greaer han many kω.

9 Appendx F Tuoral Soluons F. KCL, KVL and Groundng Q. Currens 8 8 All uns n VAΩ,, Value for Applyng KVL o he loop wh he sources and :

10 Appendx F Tuoral Soluons Q. (a) (open crcu or no load suaon) All uns n VAΩ,, 8 8 (b) 8Ω A 8 8 8

11 Appendx F Tuoral Soluons V (c) Ω + A 6V

12 Appendx F Tuoral Soluons (d) (shor crcu).a I may be slghly faser o dere wo general formulas for and and hen subsue he alues for.

13 Appendx F Tuoral Soluons Q. Equalence of crcus The wo crcus are equalen because he connecons (opology), elemens and currens beween he arous nodes are dencal: E D All uns n VAΩ,, A X C Ground B C B X A E D Ground

14 Appendx F Tuoral Soluons 6 Curren and olage s s X x Ground Applyng KCL o node X and hen KVL: s + ( ) 8 x s When, s A and x V When, s 7A and x V

15 Appendx F Tuoral Soluons 7 KCL for ground node Snce here may be oher componens conneced o ground, he applcaon of KCL mus nclude all he oher connecons no shown n he orgnal dagram. The mplcaon s ha all hese oher componens mus be delerng a combned curren of s o ground: Oher componens X s s Ground All hese are acually conneced ogeher Q. (a) Pon C grounded and Pon B conneced o Pon D A B C All uns n VAΩ,, D + VM Applyng KCL o node B: Applyng KVL o loop wh olage source, A, B and C: + A

16 Appendx F Tuoral Soluons 8 Applyng KVL o loop wh B, C and VM : 6 V (b) No connecon for Pon C and Pon B conneced o Pon D C A B All uns n VAΩ,, D + VM Applyng KCL o node C and hen o node B: A + A Applyng KVL o loop wh olage source, A, B and VM : + V (c) Pon B grounded and Pon C conneced o Pon D A All uns n VAΩ,, B C D + VM

17 Appendx F Tuoral Soluons 9 Applyng KVL o loop wh olage source, A and B: A Applyng KVL o loop wh B, C, D and VM : + V Q. Orgnal crcu All uns n VAΩ,, 6 X Applyng KCL o he doed surface: Thus: All uns n VAΩ,, 6 6 X Poenal of node X wr ground V

18 Appendx F Tuoral Soluons When he crcu s no grounded All uns n VAΩ,, 6 X 6 Undeermned The poenal of node X wr ground canno be deermned. In pracce, s alue wll depend on facors such as he exsence of sac charges and oher elecrcal and magnec effecs. When Pon X s grounded X 6 All uns n VAΩ,, The poenal of node X wr ground s now. Q.6 Frs crcu All uns n VAΩ,, Applyng KVL:

19 Appendx F Tuoral Soluons + Second crcu Applyng KCL: + + Thrd crcu + +

20 Appendx F Tuoral Soluons + + From KVL: Equalence The hree crcus are dfferen n crcu opology and he componens used. Howeer, hey hae he same olage-curren relaonshp and are elecrcally equalen from a olage-curren pon of ew. I s mpossble for an exernal crcu conneced o he oupus of hese crcus o ell whch crcu has acually been used: Exernal crcu Crcu, or Only sees - relaonshp Impossble o ell wheher crcu, or has been used

21 Appendx F Tuoral Soluons F. DC Crcu Analyss I Q. Source curren 6 All uns n VAΩ,, Value for + 6 Source curren A All uns n VAΩ,, 6

22 Appendx F Tuoral Soluons Q. Orgnal crcu A All uns n VAΩ,, B A B A B A + ( ) + ( ) B

23 Appendx F Tuoral Soluons 6 + ( ) Equalen ressance Ω When ouer wo ressors are shor-crcued A B A B Equalen ressance. 666Ω

24 Appendx F Tuoral Soluons 6 Q. Mesh analyss 7 All uns n VAΩ,, 6 A B C A B C 8 Applyng KVL for he hree loops shown: 8 ( ) 6+ ( ) + ( ) ( ) + ( ) + 6 ( ) 7

25 Appendx F Tuoral Soluons 7 Smplfyng: In marx form: 8 Solng (acually no requred n hs queson): + () 8 + ( + ) ( ) ( ) ( 78) + ( 7) 6( 78) + ( 7) 6( 99 ) + ( + 6) 6( 99) ( ) 9 ( 78) + 99( 7) ( 78) + 99( 7) ( 99 ) + 99( + 6) ( ) + 99( 6) ( ) Volages of nodes A, B and C wh respec o ground: A B C A B C A 8 6

26 Appendx F Tuoral Soluons 8 + B + C Nodal analyss + C A B A 6 + C B A B C D + 8 A B B E C 8 A C D E D E Applyng KCL o nodes A, B, C, D and E: D + 8 A 6 B A C A B A B C B E C A C + B C D + 8 A D + E C E + In marx form: A B C + 7 ` 8 D E

27 Appendx F Tuoral Soluons 9 Q. All uns n VAΩ,, 8 A B B A A B 8 A B Applyng KCL: A B A A A B A A A B A B B B A B + + B + B A + B A + B Elmnang A and B : ( ) ( ) ( ) ( ) A B + A + B B B 6 7 A B + A + B A A A B The equalen ressance whou he Ω ressor s herefore 7 8. Ω and he equalen ressance wh he Ω seres ressor s

28 Appendx F Tuoral Soluons Equalen ressance. 8Ω Q. Equalen ressance n Snce he ressors are n parallel, he equalen ressance s + + n Shor crcu curren n n n sc n Applyng KCL: sc + + n n Noron's equalen crcu sc + + n

29 Appendx F Tuoral Soluons sc + + n n Theenn's equalen crcu oc + + n oc sc n + + n + + n

30 Appendx F Tuoral Soluons Q.6 e-drawng orgnal crcu AM All uns n VAΩ,, 6 Equalen ressance whou Ω ressor 6 Equalen ressance ( ) Usng Noron's equalen crcu Ω + 6 AM sc 6 Snce AM reads A sc A Thus, when he swch s open

31 Appendx F Tuoral Soluons AM A Q.7 + +

32 Appendx F Tuoral Soluons F. DC Crcu Analyss II Q. Load curren due o Baery... All uns n VAΩ,, Curren from source A... + Load curren due o Baery... Curren from source A... +

33 Appendx F Tuoral Soluons Load curren due o Baery... Curren from source A... + Acual load curren A Q. Curren due o V olage source All uns n VAΩ,, 6 Curren from source 9. + ( 6 + ) A + 8

34 Appendx F Tuoral Soluons 6 Curren due o V olage source 6 Curren from source 8. + ( 6 + ) A + 8 Curren due o curren source ( ) A ( ) Acual curren A

35 Appendx F Tuoral Soluons 7 Q. Open crcu olage All uns n VAΩ,, V Equalen ressance. essance across ermnals. 667Ω + Theenn's equalen crcu and maxmum power.667 Maxmum power ransferable (wh a. 667Ω load) W. 667

36 Appendx F Tuoral Soluons 8 Q. All uns n VAΩ,, Combned curren of curren sources Equalen parallel ressance + 7 essor ha draws A 7 Ω essor ha absorbs he maxmum power Ω 7 Maxmum power ha can be ransferred 96 W

37 Appendx F Tuoral Soluons 9 Q. Theenn's equalen crcu All uns n V, A, Ω.6 d.6 d D D d Dece curren gen 6. A Applyng KVL: d d d and d can be found from solng hs (whch ges rse o he load lne) and he relaonshp f ( ) d gen by he characersc cure. Specfcally, when d, d 6.. Also, when d, d. d Dece characersc d f ( d ) All uns n V A Ω,, d. Load lne from crcu.6.6 d + d d The pon of nersecon ges d 6. A Source curren for power dsspaed n D o be 6. W.6 d All uns n VA,, Ω D d Power dsspaed n D d d 6.

38 Appendx F Tuoral Soluons 6 The dece olage and curren can be found from solng hs and he relaonshp f ( ) d gen by he characersc cure: d d. Dece characersc d f ( d ) Power requremen.6 d d All uns n V, A, Ω d The pon of nersecon ges d 8. V d 7. A From KVL: ( 7. ) + 8. A d d Q.6 Volage gan Applyng KCL o he second half of he crcu: ( ) Applyng KVL o he frs half of he crcu: + Elmnang : The olage gan s 8. The gan n olage magnudes s

39 Appendx F Tuoral Soluons 6 8 log db db..8 Equalen ressance To deermne he equalen ressance as seen from he oupu ermnals, all ndependen sources hae o be replaced by her nernal ressances and a olage source has o be appled o hese wo ermnals: All uns n VmA,,kΩ Transsor amplfer Equalen ressance 9 kω Noe ha n calculang hs ressance or n usng superposon, dependen sources mus no be replaced by her nernal ressances. Theenn's equalen crcu 9 All uns n VmA,,kΩ.8

40 Appendx F Tuoral Soluons 6 F. AC Crcu Analyss I Q. (a) AC waeform sn( ω ) π cos ω e jπ [( jω e )( e )] (b) cos ω e ( ) j [( jω e )( e )] Peak alue Frequency ω ω ωrad s Hz ωrad s Hz π π MS alue π Phase 9 Phasor e jπ/ 9 o e j o (c) AC waeform sn( + ) ( ) cos 6 e jπ j [( e )( e )] (d) ( ) cos e jπ j [( e )( e )] Peak alue Frequency rad s. 8Hz rad s Hz MS alue π Phase 6 π Phasor e jπ/ 6 o e jπ/ o

41 Appendx F Tuoral Soluons 6 AC waeform (e) π sn π π. cos + π jπ [( 6 j e )( e ) ] e. (f) ( + ). cos ( ) 77. cos + 7. e. j7. j [( 77e )( e )] Peak alue. Frequency rad s. 67Hz rad s. 8Hz MS alue.. 77 Phase Phasor π e jπ/ 6. o.77 e j.7.77 o Q. (a) (b) (c) (d) e j V jπ 6 jπ π e e ( e ) cos e jπ V ( j e )( j ) [ ] + V 6 e e cos + V π π π [ ] [( )( )]. e jπ A e (. jπ jπ )( ) e jπ. jπ jπ e e e e e. cos + π A jπ jπ jπ jπ jπ [ ] [( )( )].69 6 o A e ( 69. e )( e ) e e 69. e e. 69 cos cos π π A

42 Appendx F Tuoral Soluons 6 Q. From ( ) cos( ) V Frequency ω rad s Impedance of capacor j Ω jω. j. Impedance of nducor jω j Ω V j e I All uns n VAΩ,, V L j V I j V C I V ( ) cos( ) A π ji 6j L 6 cos + V V L ( ) V ( ) V V + V + 6 j C L VC + 6 j I I + j j +. j j Imag. I Arg [ I ] an ( ).8 eal I j

43 Appendx F Tuoral Soluons 6 Arg [ I] Arg[ 8 j]. +. an j ( ) I j. e. cos +. A Q. Impedances and phasors ( ) ( ) cos + V V e j V. Impedance of nducor j( )(. ) jω Impedance of capacor j( )(. ) I V All uns n VAΩ,, jω V e j o j V L j V C j an 6. 9 Toal mpedance + j j j + e e j V e I e j j j e j ( )( ) V I e e j66. 9 ( )( ) V ji e e e L j66. 9 V j9 j66. 9 j6. 9 ( )( ) V ji e e 8e C A V j9 j66. 9 j. V Ω

44 Appendx F Tuoral Soluons 66 Phasor dagram Imag All uns n VAΩ,, V V L V V + + V V L C eal V C V n phase wh I V L ( I lags V L by 9 o ) I VC ( I leads V by 9 o ) C Q. Componens The mpedances of seres L, C and LC crcus are ZL + j ω L + j π L Z j C j C + ω πc Z j L j C j LC + L ω π ω π C + j mus correspond o a seres L crcu wh componens:

45 Appendx F Tuoral Soluons 67 + jπ L + j Ω and L π. 9 H j mus correspond o a seres C crcu wh componens: j j C C Ω and π π( ). μ F Crcu admance Crcu mpedance Z ( j) ( j) j j j + j Crcu admance + + Z + j j + + ( ). j j j. Ω Power facor Power facor alue cos[ Arg( I) Arg( V) ] leadng, Arg( I) Arg( V) > leadng / laggng laggng, Arg( I) Arg( V) < I Arg( I) Arg( V) Arg Arg Arg( Z) V Z Arg( + j ) an Power facor alue cos(.6).89 leadng,.6 > leadng / laggng laggng,.6 <. 89 leadng. Imag Z Value of p.f. cos θ leadng p.f. as I lags V I θ θ V.6 eal

46 Appendx F Tuoral Soluons 68 F.6 AC Crcu Analyss II Q. Crcu dagram I jπl All uns n VAΩ,, Hz Equalen crcu for col Man equaons 9 Volage across Ω I 9 I ( π ) Volage across col I + j L + jl ( L) I ( π ) Supply I + + j L + + j L ( ) ( ) I + + L. Componen alues [( ) ( L) ] [ ( L) ] Ω ( L) 8. 7 ( L) L. 7H

47 Appendx F Tuoral Soluons 69 Power and power facor Power absorbed by col ( ) I 9.. W [ Arg( ) Arg( )] alue cos curren olage Power facor leadng, Arg leadng / laggng laggng, Arg curren Arg( curren) Arg( olage) Arg olage alue cos Arg Arg Impedance [ Arg( mpedance) ] Power facor leadng, Arg leadng / laggng laggng, Arg ( curren) Arg( olage) ( curren) Arg( olage) ( mpedance) ( mpedance) > < ( Impedance) < > ( mpedance ) ( + + jl) (. + + j. 7) Arg Arg Arg Arg( + ) an j Power facor alue cos (. 67). 79 leadng,. 67 < leadng / laggng laggng,. 67 > 79. laggng

48 Appendx F Tuoral Soluons 7 Q. Load curren I Z All uns n VAΩ,, V Hz Z Load acual power Load power facor. IZ A apparen power I Z V All uns n VAΩ,, I Z Laggng power facor lagsv I Z [ ( I ) ( ) Z V ] ( I ) Arg( V) <. laggng cos Arg Arg. load p.f. Arg Z Arg Arg ( I ) ( ) Z ( I ) < Z ± cos. ±. ( I ) Arg. Arg I I e [ Z ] e Z Z Z j I j. A Power facor mproemen I I Z Hz I C j C Z Load I ( j C)( ) C j68 C (. ) I I + I e + j68c + j 68C 8 66 Z C j.

49 Appendx F Tuoral Soluons 7 I C V I I Z 9. laggng cos [ Arg( ) Arg( )] 9. Arg( ). oerall p.f. I V I ± Arg( I) Arg( V) < Arg( I ) < Arg( I ). an - 68C an (. ) C F 7.6μ F uny [ ( ) ( )] oerall p.f. cos Arg I Arg V Arg( I) Arg( V) Arg( I ) - 68C an C. 8 F.8μ F leadng cos [ Arg( ) Arg( )] 8. Arg( ). 6 oerall p.f. I V I ± Arg( I) Arg( V) > Arg( I ) > Arg( I ). 6 an - 68C an (. 6) C F7.8μ F

50 Appendx F Tuoral Soluons 7 Q. Power a + jb I Z All uns n VAΩ,, V V Z Z + jx Elecrcal sysem V IZ ( a jb) ( jx) ( a+ ) + j( b+ X) ( ) V I + jx Z Z V [ ] [ ] e ( ) Power absorbed p e I V I + jx Maxmum power ransfer [ ] I e + jx I Z Z Z Z Z V ( a+ ) + j( b+ X) ( a+ ) + ( b+ X) V For maxmum p, he denomnaor should be as small as possble. As he numeraor does no depend on X and he smalles alue for ( b+ X) s, maxmum power wll be absorbed f X so ha p b V ( a+ ) Dfferenang: dp d V a V a ( + ) ( a+ ) ( a + ) Thus, maxmum p occurs when a

51 Appendx F Tuoral Soluons 7 and he maxmum power ransferable s p V ( + ) V V a W In general, maxmum power ransfer occurs when he load mpedance s equal o he conjugae of he Theenn's or Noron's mpedance. When hs occurs, he oal mpedance s purely resse and he curren and olage n he crcu are n phase: a+ jb V Maxmum power ransfer Z a j b ( a + jb) * Q. Noron's and Theenn's equalen crcu j All uns n VAΩ,, o o j Z I + j o j Z I j e + j j e j86.. e 8. j. j6. 8e

52 Appendx F Tuoral Soluons 7 Z I Z Z I Z Z ( ) Z j + j. e j8.. 6 j. 6. 6e j j + j j + j8. j I. 9 j. e. 7 j e j6 Maxmum power ransfer From he preous problem, hs occurs when Z I I Z Z Z * ( ) j 8. j8. Z Z. e. e [ ] Toal mpedance + j8. j8 Z Z Z + Z e.. +. e. cos ( 8. ) I IZ e. e 87. e [ ] cos ( 8. ) ( Z + Z ).cos( 8. ) j j j Thus, he maxmum power ransferable s e [ I ( IZ) ] I e[ Z ] j 87. e j8. e. e cos 8. ( ) [ ] cos ( 8. ) W cos 8. ( ) 87. ( ).cos 8.

53 Appendx F Tuoral Soluons 7 Q. Ω μh Col pf esonan frequency π 6 6 ( )( ) π 9. Q facor 6 ( )( ) 9. MHz Snce he Q facor s large, he crcu s bandpass n naure wh db cuoff frequences 9. ± MHz 9. Bandwdh 6. 9 khz Q.6 L Col C C pf resonan frequency π L khz L L For he lowes unable frequency o be 666 khz: ( ) π L( ) L. mh L 666 The hghes unable frequency s hen L.. MHz

54 Appendx F Tuoral Soluons 76 F.7 Perodc Sgnals Q. (a) Perod of olage and curren waeforms Snce he shores me needed for he waeforms o repea hemseles s 6s, he perod s 6s. (b) Aerage or mean alue of olage and curren waeforms The aerage alues of boh ( ) and ( ) are obously. (c) Tme when he dece s consumng power and when s supplyng power Snce he olage and curren arrows are n oppose drecons, he nsananeous power consumed by he dece s p ( ) ( ) ( ) Graphcally:

55 Appendx F Tuoral Soluons 77 () 6 ( ) p () () ( ) Consume power Supply power (d) Perod of nsananeous power cure From he aboe cure, he perod of ( ) p s s. (e) Aerage power consumed by dece In perod of s, he dece consumes W of power for s and supples W of power for s. Thus: Ne energy consumed n s ( )( ) ( )( ) J Aerage power consumed W

56 Appendx F Tuoral Soluons 78 Q. (a) Perod and mean alue of ( ) From he waeform self: Perod of ( ) T 6s Mean alue of ( ) V (b) MS alue () Mean alue Quadrac 6 (), < 9 Perod of ( ) T 6s Mean alue of ( ) mean alue of ( ) from o area under ( ) from o d 9 7 V MS alue of ( ) mean alue of ( ) (c) Aerage power rms ( ) Insananeous power consumed p ( ) ( ) mean alue of ( ) Aerage power consumed mean alue of [ mean alue of ( ) ] rms V

57 Appendx F Tuoral Soluons 79 Q. (a) ( ) cos( π + ) and ( ) cos( π + ) To an obserer wh a me orgn of : ( ) ( ) ( ) ( ) V e j cos π + and frequency V e j cos π + and frequency Phase dfference,v leadng To an obserer wh a me orgn of. : ( ) ( ) ( ) ( ) V e j 8 +. cos π + π + and frequency V e j +. cos π + π + and frequency Phase dfference 8,V leadng Snce he me orgn has changed, he olages obsered are me shfed and he phasors change n phase. Howeer, snce he frequences are he same, he relae phase dfference remans consan. (b) ( ) cos( π + ) and ( ) cos( π + ) To an obserer wh a me orgn of : ( ) ( ) V e j cos π + and frequency ( ) ( ) V e j cos π + and frequency To an obserer wh a me orgn of. : ( ) ( ) V e j 8 +. cos π + π + and frequency (. ) cos (. ) V e j + π + π + and frequency Snce he frequences are no he same, he phase dfference beween ( ) and ( ) s no consan and depends on he me orgn. In fac, s neher meanngful nor useful o compare he phases of ( ) and ( ), as hese sgnals hae dfferen frequences. Phasor analyss can only be used when all he sources are snusodal and hae he same frequences. In suaons where hs s no he case bu he sources are sll snusodal, superposon has o be used ogeher wh phasor analyss.

58 Appendx F Tuoral Soluons 8 Q. In general, for a sgnal ( ) wh perod T : ( ) a + a π π π π cos + b sn a b T T + cos + sn T T + T a d ( ) T Aerage alue of ( ) a, n n T ( ) nπ cos d T T Twce he aerage alue of n ( ) cos π T b, n n T ( ) nπ sn d T T Twce he aerage alue of n ( ) sn π T

59 Appendx F Tuoral Soluons 8 For he perodc waeform aboe: T s a mean of ( ) () a π mean of ( ) cos π π d 6 cos sn π π () cos(π /) () cos(π /) a π mean of ( ) cos () () cos(π /) cos(π /) a 6π mean of ( ) cos 6π π d 6 cos sn 6π π () cos(6π /) () cos(6π /) a 8π mean of ( ) cos () () cos(8π /) cos(π /)

60 Appendx F Tuoral Soluons 8 Mahemacally, for n > : a n nπ nπ nπ mean of () cos d d cos + cos 9 nπ nπ n n n + n π π sn sn sn sn 6 π π nπ 9 a, a, a, a, a, a6,,,,,,, π π π b n nπ nπ nπ mean of () sn d d sn + sn 9 nπ nπ n n n + n π π cos cos cos cos 6 π π nπ 9 b, b, b, b, b, b,,,,,,, 6 The Fourer seres represenaon for ( ) s hus: ( ) π 6π π + cos cos + cos π π π Q. Frequency response 9 All uns n VA,, Ω,H, F 9 () o() V Vo jπ f + Vo j π f + jπ f Frequency response H( f ) V + jπ f + j π f Magnude response ( ) Magnude response H f + jπ f + jπ f + π f + π f

61 Appendx F Tuoral Soluons 8 ( ) H + + ( ) H f π f f π H( f ) Lowpass. f The response s obously lowpass n naure. Ths can also be deduced from he low and hgh frequency characerscs of he capacors: Low frequency approxmaon 9 9 V Vo V V o V jπ f Hgh frequency approxmaon 9 9 V jπ f Vo V V V o

62 Appendx F Tuoral Soluons 8 Non-perodc excaon wh snusodal componens The oupu due o a general snusodal excaon can be found as follows: () cos( π θ) r f+ V re jθ ( ) ( ) ( ) ( ) [ ] ( ) o jarg H f j θ + Arg H( f ) V H f V H f e H f V V o { [ ]} () ( ) cos π θ Arg ( ) o r H f f+ + H f + jπ f + jπ f r cos π f + θ+ Arg + jπ f + jπ f + π f r cos an an + jπ f jθ re H f r e { [ ]} [ π f + θ+ ( π f ) ( π f )] π π s gen Thus, from superposon, he oupu due o () cos( ) + sn( ) by () cos( π ) () () sn( π ) () o o + jπ + jπ cos π + Arg + jπ + jπ + π cos an an + π [ π + ( π) ( π) ] ( π ). cos. + j π sn + j π [ π + ( π) ( π) ] ( π ). sn j π π + Arg + j π + π sn an an + π () cos( π ) + sn ( π ) (). cos ( π. ) +. sn ( π 99. ) Perodc square excaon () sn( π ) o o ( 6π) ( π) ( πn) sn sn sn n,,, { [ ]} n H f n n H f n () ( ) sn π + Arg ( ) n,,, n n,,, + π n n + π n { πn + ( πn) ( πn) } sn an an

63 Appendx F Tuoral Soluons 8 F.8 Transens I Q. Volages across nducor L () d () d () H () (ma) 6 Graden Graden Graden (ms) () L (V) (ms) Volages across capacor C () d C () () C d C ( ) d ( ) C d C d Cd ( ) C ( ) C C C C d [ ] [ ( ) ( ) ] C

64 Appendx F Tuoral Soluons 86 ( ) ( ) ( ) C C + C d [ nal olage a ] + area under ( ) from o C For hs problem: ( ) ( ) C 6 area under from o [ ] [ ] () (ma) 6 (ms) () C. (V) Area 6 Inegral ncreases lnearly Area 8 6 Inegral ncreases quadracally Area 6 Inegral ncreases lnearly.. (ms)

65 Appendx F Tuoral Soluons 87 Q. (a) Volages and currens for < Wh he source beng a dc one and akng he swches o be n he posons shown sarng from, all he olages and currens wll hae seled down o consan alues for praccally all < : () 6 d () d d () d () () () 6 () () (b) Volages and currens a (jus afer he swches are hrown) Snce he olages across capacors mus be connuous, ( ) and ( ) mus hae he same alues before he swches are hrown:

66 Appendx F Tuoral Soluons 88 () 6 () () () 6 6 () () (c) Volages and curren for () 6 d () d d () d () () From KCL and KVL: d ( ) ( ) d ( ) d d ( ) 6( ) ( ) Elmnang ( ) and ( ) : d ( ) ( ) ( ) ( ) ( ) ( ) d d ( ) d 6 d d d d 6 + d d

67 Appendx F Tuoral Soluons 89 Solng hs homogeneous equaon: ( ) ke, Applyng nal condon: ( ) k ( ) e, 6 6 Solng for ( ) and ( ) : d ( ) () e ( ) k + e d 6 ( ) k k ( ) + + e, ( ) ( ) ( ) 6 + e e e, (d) Volages and curren waeforms 6 () () ()

68 Appendx F Tuoral Soluons 9 Q. (a) Volages and curren and energy sored for < Wh he source beng a dc one and akng he swches o be n he posons shown sarng from, all he olages and currens wll hae seled down o consan alues for praccally all < : L d () d L () () L L ( ) L Energy sored n nducor (b) Curren a (jus afer he swches s opened) Snce he curren n he nducor mus be connuous, ( ) mus hae he same alue before he swch s opened: L ()

69 Appendx F Tuoral Soluons 9 (c) Curren for L d () d () L () () ( ) L d d d( ) + + ( ) + ( ) + ( ) d L + ( ) L ke, + ( ) k ( ) e L, (d) Energy los for Insananeous power los n ressors ( ) ( ) Toal energy los ( + ) + + ( + ) ( + ) e L d e + L + L e + L, L From he conseraon of energy, hs mus also be equal o he decrease n energy sored by he nducor: L ( ) L ( ) L Decrease n energy sored by nducor

70 Appendx F Tuoral Soluons 9 Q. Tme < () () Tme jus afer he frs swch s acaed Snce he curren n he nducor mus be connuous, he nducor mus be carryng he same 6A of curren jus afer he frs swch s acaed: () () 6 Tme and < afer acang s swch bu before closng nd swch d () d () () () d( ) d( ) ( ) + ( ) + ( ) ke, < d d, ( ) k ( ) e < ( ) ( ) 6e, <

71 Appendx F Tuoral Soluons 9 Tme jus afer closng second swch Snce he curren n he nducor mus be connuous, ( ) mus be he same as e e A jus before closng hs swch: () e (). () Tme afer closng second swch d () d () (). () d( ) d( ) ( ) +. ( ) + ( ) he 6, d d 6 ( ) ( ) he 6 e h e ( ) e e 6, ( ) ( ). ( ) 8e e 6, Volage and curren waeforms

72 Appendx F Tuoral Soluons 9 () 6 ()

73 Appendx F Tuoral Soluons 9 F.9 Transens II Q. Volages and currens for < Takng he swches o be n he posons shown sarng from, all he olages and currens wll hae seled down o consan alues for praccally all <. () d C C () d () L () C () C L d L L () d () ( ) ( ) C L () () () L ( ) C () C L

74 Appendx F Tuoral Soluons 96 Volages and currens jus afer closng swch a Snce he olage across a capacor and he curren hrough an nducor mus be connuous, he nal condons are () L C () C L Volages and currens for () d C C () d () L () L C C () L d L L () d ( ) ( ) ( ) ( ) C d C dc C C + + d d C C ( ) ( ) ( ) L d L dl ( ) d d L L + + L L The general soluons are ( ) ( ) + ( ) C ss r ( ) ( ) + ( ) L ss r The seady sae responses are ( ) k, ss d ( ) ( ) ss ss k + k ( ) ss, d C C C ( ) k, ss

75 Appendx F Tuoral Soluons 97 d ( ) ss d ( ) ( ) L k + ss k ss, L L The ransen responses are gen by d ( ) ( ) r r C ( ) + r he d C, d ( ) r ( ) ( ) d L r r he + L, Combnng and applyng nal condons: C ( ) ( ) + ( ) + h e, C ss r C ( ) h h ( ) + e, C C ( ) ( ) ( ) L ss + r + he L, ( ) h h ( ) L L e + L, Source curren: for ( ) ( ) C d C ( ) d C e e e C C C e + L L For hs o be me ndependen: C L L C L

76 Appendx F Tuoral Soluons 98 Q. Volages and currens for < before he swch s opened All uns n VA,, Ω, F () ( ) a jus afer he swch s opened Snce he olage across he capacor mus be connuous, ( ) mus be ( ) ( ) for afer he swch s opened d () d + () () d () d () d( ) ( ) ( ) + ( ) ( ) + d d + d The general soluon of hs s gen by ( ) ( ) + ( ) ss r ( ) ss s he seady sae response and can be found by ryng ( ) ss k,

77 Appendx F Tuoral Soluons 99 d ( ) ss + ( ) 7 ( ) ss + k k ss 7, d ( ) r s he ransen response and s equal o he general soluon of d ( ) r + ( ) ( ) he r r, d Combnng: ( ) ( ) + ( ) 7 + he, ss r ( ) 7 + h h ( ) 7+ e, Q. (a) Volage and curren before closng swch () C () C L (b) Volages and curren jus afer closng swch a Snce he olage across a capacor and he curren hrough an nducor mus be connuous, he nal condons are () C () C L (c) Goernng dfferenal equaon for curren for () d C () () C d C () C L L d () d

78 Appendx F Tuoral Soluons ( ) L d d d ( ) d( ) d ( ) C + ( ) + ( ) C + + d L d L d d ( ) d( ) ( ) + + d L d LC, wh nal condons ( ) ( ) ( ) ( ) L d d C d d L Subsung esonan frequency ω LC ω L L Q C he goernng dfferenal equaon s d ( ) ω d( ) ( ) + + ω, d Q d (d) Oerdamped suaon when Q < The roos of he polynomal are d ( ) ω d( ) ω ( ) z d Q d ( ) Q z + + ω + + ω z, z d d replaced by z ω ω ± ω Q Q ω ± Q ( Q ) When Q <, boh roos are real, negae and dsnc. Thus: z z ( ) ke + ke, z z ( Q ) ω < Q ( Q ) ω + < z Q Usng nal condons: ( ) k + k k k

79 Appendx F Tuoral Soluons d( ) d kω kz + kz k( z z) Q L Q L ( ) ( z z ) ( z z ) k e e Q L Q e e, ω Q L ω Q e z () Q L ω Q e z (e) Underdamped suaon when Q > When Q >, he wo roos form a complex par: z z ( j Q ) Q Q j ω ω + ω Q ( j Q ) Q Q j ω ω + ω z Q z z ( ) he + he, Usng nal condons: ( ) h + h h h d( ) d hz + hz h( z z) jhω L Q L j ( ) h ( e e ) Lω Q e ω z z Q j Q ω Q + j ω ω Q e ω Q j e Lω Q e Q e e j Q Q ω Lω Q jω ω Q sn ω,

80 Appendx F Tuoral Soluons () (e) Undamped suaon when When, he Q facor and curren s Q ω L ( ) ω ( ω) ω Q e Lω Q () Q sn sn, Lω

81 Appendx F Tuoral Soluons Q. Volages and curren for < before he swch s closed () sn (ω ) C L Volages and curren a jus afer he swch s closed As he olage across he capacor and he curren hrough he nducor mus be connuous: () sn (ω) C L Curren for afer he swch s closed () sn (ω ) C L d () L d C () d ( ) L d d + + C jω ( ) d ( ) sn( ω) e[ je ] ( ) ( ) L d d + + e ωej ω d d C wh nal condons ( ) ( ) [ ]

82 Appendx F Tuoral Soluons ( ) Inal olage across nducor L d d Seady sae curren The seady sae response ( ) ss s gen by jω ( ) [ ] e I e, ss L d e d ss [ I e jω] [ I e jω e ] ss d ss + + e ss d C jω jω [ I e ] e[ ωe ] e L d e jω de jω jω e e e[ ] d d C I ss j L j C I ss e jω e jω + + ω ω ω + + j ω L+ jω+ I I C ss ω ss j jωl+ + jc ss ( ) ( ) ( ) jz e jω jω e e e Z j Z sn ω Arg Arg jze Z jz [ ] + ω L ωc ω L C ω sn ω an,

83 Appendx F Tuoral Soluons F. Magnec Crcus Q. Toal flux.8 mwb.mm N Area mm Lengh mm elae permeably 6 elucance of ron core core 7 6 ( 6 π )( ). 66 A Wb. elucance of ar gap gap 7 6 ( π )( ) 99. ( 8. )( core gap ) MMF N A Wb 6 6 ( 8. )( ) 776A ( Amp.urns) + Q. Force of aracon MMF for ar gaps ( )( ) 8A ( ) elucance of ar gaps. gap 7 ( π )(. ) 8 Flux n crcu Φ Wb 77. A Wb

84 Appendx F Tuoral Soluons 6 To deermne he force, suppose he gap lengh s decreased by δx and he curren s changed by δ whle he flux remans he same. Snce here s no back emf nduced and no power s suppled by he elecrcal crcu: ( ) 8 δ x δ x Decrease n relucance δgap 7 ( π )(. ) π δgapφ δ x Decrease n energy sored 9. δ x π Work done by moemen of armaure ( aracon force) δ x From energy conseraon: 9. δx Aracon force δx 9. N Doublng of curren If he curren s doubled, he flux Φ wll be doubled. Snce he energy sored and he force are proporonal o Φ, hey wll ncrease by mes o (. 9) 8kN heorecally. Howeer, n a praccal dece, he ron may ge sauraed as he curren s ncreased. The flux may herefore no ncrease by a facor of and he acual force wll be smaller. Q. I : I : n I/n Ω V V.V.nV Ω For maxmum power ransfer, he source mus be seeng a load ressance of Ω: I Ω V V Ω V V

85 Appendx F Tuoral Soluons 7 I ( + ). A. nv nv n 9 n In I Q. I : n I n 66 Z n raed curren 66 I 8. laggng p.f. 78. A cos [ Arg( I )] 8. Arg( I ) ± 6. 9 Arg( I ) < Arg( I ) < Arg( I ) 6. 9 Z e In j 7 8e j6. 9 Ω Load mpedance referred o prmary load mpedance seen by source e I j e j6. 9 Ω Q. Hz () () () V N N () V Prmary wndng Flux Φ() Secondary wndng

86 Appendx F Tuoral Soluons 8 N N V N 66 V Snce he currens are snusodal a Hz, he flux ( ) Φ wll also be snusodal: Φ ( ) Φ ( + θ ) max cos π Thus, he prmary olage wll be ( ) ( ) N d Φ π NΦ max sn ( π + θ ) d The rms alue of hs s π NΦ max Φmax. Wb π ( ) Φmax Φmax Cross seconal area of core cm maxmum flux densy 6. Q.6 Effcences under dfferen operang condons Full load, uny pf full load, uny pf Apparen power delered ( VA ). 7 7 Acual power delered ( W ) 7 7 Copper loss ( W ) Iron loss ( ) W Toal loss ( W ) Power suppled ( W ) Effcency 98. % % 8

87 Appendx F Tuoral Soluons 9 Full load, 8. pf full load, 8. pf Apparen power delered ( VA ). 7 7 Acual power delered ( W ) Copper loss ( W ) Iron loss ( ) W Toal loss ( W ) Power suppled ( W ) Effcency 97. 6% % 7 All-day effcency Energy loss ( kw hr ) Energy delered ( kw hr) 6 hr no load 6 (. ). 6 hr full load, uny pf hr 6 (. +. 6) full load, uny pf ( ). 8 hr full load,. 8pf 8 hr full load, (. +. 6) ( ). pf 8. hr no load (. ) 7. Toal All-day effcency 97. %