Chapter 1 Electric Circuit Variables

Transcription

1 Chaper 1 Elecric Circui Variables Exercises Exercise Find he charge ha has enered an elemen by ime when i = A, 0. Assume q() = 0 for < Answer: q () = 2 C 3 () 2 i = 8 4 A q () = idτ + q(0) = (8τ 4 τ) dτ 0 τ 2τ 2 C 0 + = = Exercise The oal charge ha has enered a circui elemen is q() = 4 sin 3 C when 0 and q() = 0 when < 0. Deermine he curren in his circui elemen for > 0. Answer: i () = d 4sin3= 12cos3 A d dq d i() = = 4sin3 = 12cos3 A d d Exercise Which of he hree currens, i 1 = 45 μa, i 2 = 0.03 ma, and i 3 = A, is larges? Answer: i 3 is larges. i 1 = 45 μa = A < i 2 = 0.03 ma = A = A < i 3 = A 1-1

2 Exercise Figure E shows four circui elemens idenified by he leers A, B, C, and D. (a) Which of he devices supply 12 W? (b) Which of he devices absorb 12 W? (c) Wha is he value of he power received by device B? (d) Wha is he value of he power delivered by device B? (e) Wha is he value of he power delivered by device D? Answers: (a) B and C, (b) A and D, (c) 12 W, (c) 12 W, (e) 12 W + 4 V 2 V V 3 V + 6 A 2 A 4 A 3 A (A) (B) (C) Figure E (D) (a) B and C. The elemen volage and curren do no adhere o he passive convenion in B and C so he produc of he elemen volage and curren is he power supplied by hese elemens. (b) A and D. The elemen volage and curren adhere o he passive convenion in A and D so he produc of he elemen volage and curren is he power delivered o, or absorbed by hese elemens. (c) 12 W. The elemen volage and curren do no adhere o he passive convenion in B, so he produc of he elemen volage and curren is he power received by his elemen: (2 V)(6 A) = 12 W. The power supplied by he elemen is he negaive of he power delivered o he elemen, 12 W. (d) 12 W (e) 12 W. The elemen volage and curren adhere o he passive convenion in D, so he produc of he elemen volage and curren is he power received by his elemen: (3 V)(4 A) = 12 W. The power supplied by he elemen is he negaive of he power received o he elemen, 12 W. 1-2

3 Problems Secion 1-2 Elecric Circuis and Curren Flow P1.2.1 The oal charge ha has enered a circui elemen is q() = 1.25(1 e 5 ) when 0 and q() = 0 when < 0. Deermine he curren in his circui elemen for 0. Answer: i() = 6.25e 5 A d i e e d 5 5 = = 6.25 A () ( ) P The curren in a circui elemen is i() = 4(1 e 5 ) A when 0 and i() = 0 when < 0. Deermine he oal charge ha has enered a circui elemen for 0. Hin: q(0) = i( τ) = 0dτ = 0 Answer: q() = e C for q i d q e d d e d e () ( ) ( 0) 4( 1 τ τ = τ τ + = ) τ + 0= 4 τ 4 τ = 4 + C P The curren in a circui elemen is i() = 4 sin 3 A when 0 and i() = 0 when < 0. Deermine he oal charge ha has enered a circui elemen for 0. Hin: q(0) = i( τ) dτ = 0 d τ = q( ) = i( τ) dτ + q( 0) = 4sin 5 d 0 cos 3 cos 3 C 0 τ τ + = τ =

4 0 < 2 2 2< < 4 P The curren in a circui elemen is i () = where he unis of curren are A 1 4 < < 8 0 8< and he unis of ime are s. Deermine he oal charge ha has enered a circui elemen for 0. Answer: 0 < < < 4 q ( ) = where he unis of charge are C. 8 4 < < 8 0 8< () ( ) 0 0 C for 2 so q(2) = 0. q = i τ dτ = dτ = () ( τ) τ ( 2) 2 τ 2τ C for 2 4. In paricular, q(4) = 4 C. q = i d + q = d = = 2 2 () ( τ) τ ( ) τ τ 4 8 C for 4 8. In paricular, q(8) = 0 C. q = i d + q 4 = 1d + 4= + 4= 4 4 () ( ) ( ) q = i τ dτ + q 8 = 0dτ + 0= C for 8. P The oal charge q(), in coulombs, ha eners he erminal of an elemen is 0 < 0 q () = ( 2) 3 + e > 2 Find he curren i() and skech is waveform for 0. 0 < 0 dq() i() = i() = 2 0< < 2 d 2( 2) 2 e > 2 1-4

5 P An elecroplaing bah, as shown in Figure P 1.2-6, is used o plae silver uniformly ono objecs such as kichen ware and plaes. A curren of 600 A flows for 20 minues, and each coulomb ranspors mg of silver. Wha is he weigh of silver deposied in grams? Objec o be plaed i Bah i Silver bar Figure P i C = 450 A = 450 s C s mg s min C 5 Silver deposied = min = mg= g 1-5

6 P1.2-7 Find he charge q() and skech is waveform when he curren enering a erminal of ab elemen is as shown in Figure P Assume ha q() = 0 for < 0. 1 when 0< 2 i () = 1 when 2 Figure P1.2-7 and since q(0) =0. q () = i( τ ) dτ + q(0) = i( τ) dτ 0 0 When 0 < 2, we have When 2, we have q = 1 dτ = C q= i( τ) dτ = 1 dτ + ( τ 1) dτ τ 0 2 τ = τ + = + 2 C The skech of q() is shown o he righ:. 1-6

7 Secion 1-3 Sysems of Unis P A consan curren of 3.2 μa flows hrough an elemen. Wha is he charge ha has passed hrough he elemen in he firs millisecond? Answer: 3.2 nc Δ q = iδ = A 1 10 s = As = C = nc ( )( ) P A charge of 45 nc passes hrough a circui elemen during a paricular inerval of ime ha is 5 ms in duraion. Deermine he average curren in his circui elemen during ha inerval of ime. Answer: i = 9 μa 9 Δ q i = = = Δ = 9 μa P Ten billion elecrons per second pass hrough a paricular circui elemen. Wha is he average curren in ha circui elemen? Answer: i = na Soluion elecron 19 C 9 elecron 19 C i = 10 billion = s elecron s elecron elecron C = s elecron 9 C = = na s 1-7

8 P1.3-4 The charge flowing in a wire is ploed in Figure P Skech he corresponding curren. Figure P1.3-4 P = = 7.5 ma when 0 < < 2μs d i() = q() = he slope of he q versus plo = 6 = 5 10 = 5 ma when 4μs< < 7μs d oherwise 1-8

9 P1.3-5 The curren in a circui elemen is ploed in Figure P Skech he corresponding charge flowing hrough he elemen for > 0. Figure P1.3-5 < < 450μA dτ when 0 80 ms () ( τ) τ ( )( ) ( ) 0 μ τ 80 ms ( )( ) ( )( ) 6 ( ) when 0 < < 80 ms 6 6 = ( ) + ( ) when 80 ms < < 140 ms q = i d = A d when 80 ms < < 140 ms dτ 0 C when 140 ms < While 0 < < 80 ms q() increases linearly from 0 o 36 μc and while 80 < < 140 ms q() decreases linearly from 36 o 0 μc. Here s he skech: 140 ms 1-9

10 P1.3-6 The curren in a circui elemen is ploed in Figure P Deermine he oal charge ha flows hrough he circui elemen beween 300 and 1200 μs. Figure P μ s () ( ) q = i τ dτ = "area under he curve beween 300 μs and 1000 μs" 300 μs pc = + = + = 2 () ( ) ( )( ) ( ) q 1-10

11 Secion 1-5 Power and Energy P1.5-1 Figure P1.5-1 shows four circui elemens idenified by he leers A, B, C, and D. (a) Which of he devices supply 30 mw? (b) Which of he devices absorb 0.03 W? (c) Wha is he value of he power received by device B? (d) Wha is he value of he power delivered by device B? (e) Wha is he value of he power delivered by device C? Figure P1.5-1 (a) A and D. The elemen volage and curren do no adhere o he passive convenion in Figures P1.5- A and D so he produc of he elemen volage and curren is he power supplied by hese elemens. (b) B and C. The elemen volage and curren adhere o he passive convenion in Figures P1.5-1 B and C so he produc of he elemen volage and curren is he power delivered o, or absorbed by hese elemens. (c) 30 mw. The elemen volage and curren adhere o he passive convenion in Figure P1.5-1B, so he produc of he elemen volage and curren is he power received by his elemen: (5 V)(6 ma) = 30 mw. The power supplied by he elemen is he negaive of he power received o he elemen, 30 W. (d) 30 mw (e) 30 mw. The elemen volage and curren adhere o he passive convenion in Figure P1.5-1C, so he produc of he elemen volage and curren is he power received by his elemen: (5 V)(6 ma) = 30 mw. The power supplied by he elemen is he negaive of he power received o he elemen, 30 W. 1-11

12 P An elecric range has a consan curren of 10 A enering he posiive volage erminal wih a volage of 110 V. The range is operaed for wo hours. (a) Find he charge in coulombs ha passes hrough he range. (b) Find he power absorbed by he range. (c) If elecric energy coss 12 cens per kilowa-hour, deermine he cos of operaing he range for wo hours. 4 a.) q = id iδ ( )( )( ) b.) P= vi= ( 110 V)( 10 A) = 1100 W c.) = = 10 A 2 hrs 3600s/hr = C 0.12$ Cos = 1.1kW 2 hrs = $ kwhr P A walker s cassee ape player uses four AA baeries in series o provide 6 V o he player circui. The four alkaline baery cells sore a oal of 200 wa-seconds of energy. If he cassee player is drawing a consan 10 ma from he baery pack, how long will he cassee operae a normal power? P = ( 6V)( 10 ma ) = 0.06 W Δw 200 W s Δ = = = P 0.06 W s 1-12

13 P The curren hrough and volage across an elemen vary wih ime as shown in Figure P Skech he power delivered o he elemen for > 0. Wha is he oal energy delivered o he elemen beween = 0 and = 25 s? The elemen volage and curren adhere o he passive convenion. v (vols) 30 i (amp) (s) (a) Figure P (s) (b) 30 for 0 10 s: v = 30 V and i = = 2 A P = 30(2 ) = 60 W for s: v() = + b v( 10) = 30 V b= 80 V 5 v ( ) = and i ( ) = 2 A P= = W ( )( ) ( )( ) 2 30 for s: v= 5 V and i( ) = + b A 10 i(25) = 0 b = 75 i( ) = A P = = W ( ) ( ) Energy = Pd = 60d d d = = J

14 P An auomobile baery is charged wih a consan curren of 2 A for five hours. The erminal volage of he baery is v = V for > 0, where is in hours. (a) Find he energy delivered o he baery during he five hours. (b) If elecric energy coss 15 cens/kwh, find he cos of charging he baery for five hours. Answer: (b) 1.84 cens a.) Assuming no more energy is delivered o he baery afer 5 hours (baery is fully charged). 5( 3600) 5( 3600) 0.5τ w= Pd = vidτ = d 0 + τ = + τ = J = 441 kj b.) 1 hr 15 Cos = 441kJ = s kwhr 1-14

15 P Find he power, p(), supplied by he elemen shown in Figure P when v() = 4 sin 3 V and i() = (1/12) sin 3 A. Evaluae p() a =0.5 s and = 1 s. Observe ha he power supplied by his elemen has a posiive value a some imes and a negaive value a oher imes. 1 Hin: (sin a)(sin b) = (cos( a b) cos( a + b) ) 2 Answer: p() = (1/6)cos(6) W, p(0.5) = W, p(1) = W p() v()() i ( 4cos3) = = sin 3 = ( sin 0 + sin 6) = sin W 1 p ( 0.5) = sin 3 = W 1 p () 1 = sin 6 = W Here is a MATLAB program o plo p(): clear 0=0; % iniial ime f=2; % final ime d=0.02; % ime incremen =0:d:f; % ime v=4*cos(3*); i=(1/12)*sin(3*); for k=1:lengh() p(k)=v(k)*i(k); end % device volage % device curren % power plo(,p) xlabel('ime, s'); ylabel('power, W') 1-15

16 P Find he power, p(), supplied by he elemen shown in Figure P when v() = 8 sin 3 V and i() = 2 sin 3 A. 1 Hin: (sin a)(sin b) = (cos( a b) cos( a + b) ) 2 Answer: p() = 8 8cos 6 W () ()() ( )( ) ( ) p = v i = 8sin 3 2sin 3 = 8 cos0 cos6 = 8 8cos 6 W Here is a MATLAB program o plo p(): clear i + v Figure P =0; f=2; d=0.02; =0:d:f; v=8*sin(3*); i=2*sin(3*); for k=1:lengh() p(k)=v(k)*i(k); end % iniial ime % final ime % ime incremen % ime % device volage % device curren % power plo(,p) xlabel('ime, s'); ylabel('power, W') 1-16

17 P Find he power, p(), supplied by he elemen shown in Figure P The elemen volage is represened as v() = 4(1 e 2 )V when 0 and v() = 0 when < 0. The elemen curren is represened as i() = 2e 2 A when 0 and i() = 0 when < 0. Answer: p() = 8(1 e 2 )e 2 W () = ()() = ( ) = ( ) p v i e e e e W i + v Figure P Here is a MATLAB program o plo p(): clear 0=0; f=2; d=0.02; =0:d:f; v=4*(1-exp(-2*)); i=2*exp(-2*); for k=1:lengh() p(k)=v(k)*i(k); end % iniial ime % final ime % ime incremen % ime % device volage % device curren % power plo(,p) xlabel('ime, s'); ylabel('power, W') 1-17

18 P The baery of a flashligh develops 3 V, and he curren hrough he bulb is 200 ma. Wha power is absorbed by he bulb? Find he energy absorbed by he bulb in a five-minue period. The power is P = V I= =0.045 W. Nex, he energy is w= PΔ = =13.5 J. 1-18

19 P Medical researchers sudying hyperension ofen use a echnique called 2D gel elecrophoresis o analyze he proein conen of a issue sample. An image of a ypical gel is shown in Figure a. The procedure for preparing he gel uses he elecric circui illusraed in Figure b. The sample consiss of a gel and a filer paper conaining ionized proeins. A volage source causes a large, consan volage, 500 V, across he sample. The large, consan volage moves he ionized proeins from he filer paper o he gel. The curren in he sample is given by i() = e a ma where is he ime elapsed since he beginning of he procedure and he value of he consan a is 1 a = 0.85 hr Deermine he energy supplied by he volage source when he gel preparaion procedure lass 3 hours. Devon Svoboda, Queen s Universiy (a) i () V 500 V (b) sample Figure (a) An image of a gel and (b) he elecric circui used o preparaion a gel. T T () () ()() energy = w = p d = v i d = ( 2 30e ) ( ) 0 0 = d+ e d 15 = = = 19.3 J d 2.55 ( 3 0) ( e 1) 1-19

20 Secion 1.7 How Can We Check? P Conservaion of energy requires ha he sum of he power absorbed by all of he elemens in a circui be zero. Figure P shows a circui. All of he elemen volages and currens are specified. Are hese volage and currens correc? Jusify your answer. + 2 V 5 V 2 A V 3 V 5 A 2 A 3 A 5 A Figure P V Hin: Calculae he power absorbed by each elemen. Add up all of hese powers. If he sum is zero, conservaion of energy is saisfied and he volages and currens are probably correc. If he sum is no zero, he elemen volages and currens canno be correc. Noice ha he elemen volage and curren of each branch adhere o he passive convenion. The sum of he powers absorbed by each branch are: (-2 V)(2 A)+(5 V)(2 A)+(3 V)(3 A)+(4 V)(-5 A)+(1 V)(5 A) = -4 W + 10 W + 9 W -20 W + 5 W = 0 W The elemen volages and currens saisfy conservaion of energy and may be correc. 1-20

21 P Conservaion of energy requires ha he sum of he power absorbed by all of he elemens in a circui be zero. Figure P shows a circui. All of he elemen volages and currens are specified. Are hese volage and currens correc? Jusify your answer. Hin: Calculae he power absorbed by each elemen. Add up all of hese powers. If he sum is zero, conservaion of energy is saisfied and he volages and currens are probably correc. If he sum is no zero, he elemen volages and currens canno be correc. + 4 V 3 A A 3 A 3 V 3 V 3 V 3 V 3 A 2 A V + Figure P A Noice ha he elemen volage and curren of some branches do no adhere o he passive convenion. The sum of he powers absorbed by each branch are: -(3 V)(3 A)+(3 V)(2 A)+ (3 V)(2 A)+(4 V)(3 A)+(-3 V)(-3 A)+(4 V)(-3 A) = -9 W + 6 W + 6 W + 12 W + 9 W -12 W 0 W The elemen volages and currens do no saisfy conservaion of energy and canno be correc. 1-21

22 3V + P The elemen currens and volages shown in Figure P are correc wih one excepion: he reference direcion of exacly one of he elemen currens is reversed. Deermine which reference direcion has been reversed. a + 5V 3A 1V + b 2V + 4A 2A 7A 6V 2A 8V + + c 5A d Figure P Le s abulae he power received by each elemen. We ll idenify each elemen by is nodes. So nodes Power received, W 3 3 = 9 a c ( )( ) a b ( 1)( 4) = 4 b c ( 2)( 2) = 4 a d ( 5)( 7) = 35 b d ( 6)( 2) = 12 c d ( 8)( 5) = 40 Toal power received = ( ) + 40= 24 0 Changing he curren reference direcion for a paricular elemen will change he oal power by wice he power of he paricular elemen. Since he elemen conneced beween nodes b and d receives -12 W, changing he reference direcion of is curren will increase he oal power received by 24 W, as required. Afer making ha change Toal power received = ( ) + ( ) = 0 We conclude ha i is he reference direcion of he elemen conneced beween nodes and b ha has been reversed. 1-22

23 Design Problems DP 1-1 A paricular circui elemen is available in hree grades. Grade A guaranees ha he elemen can safely absorb 1/2 W coninuously. Similarly, Grade B guaranees ha 1/4 W can be absorbed safely and Grade C guaranees ha 1/8 W can be absorbed safely. As a rule, elemens ha can safely absorb more power are also more expensive and bulkier. The volage across an elemen is expeced o be abou 20 V and he curren in he elemen is expeced o be abou 8 ma. Boh esimaes are accurae o wihin 25 percen. The volage and curren reference adhere o he passive convenion. Specify he grade of his elemen. Safey is he mos imporan consideraion, bu don specify an elemen ha is more expensive han necessary. The volage may be as large as 20(1.25) = 25 V and he curren may be as large as (0.008)(1.25) = 0.01 A. The elemen needs o be able o absorb (25 V)(0.01 A) = 0.25 W coninuously. A Grade B elemen is adequae, bu wihou margin for error. Specify a Grade B device if you rus he esimaes of he maximum volage and curren and a Grade A device oherwise. 1-23

24 DP 1-2 The volage across a circui elemen is v() = 20 (1 e 8 )V when 0 and v() = 0 when < 0. The curren in his elemen is i() = 30e 8 ma when 0 and i() = 0 when < 0. The elemen curren and volage adhere o he passive convenion. Specify he power ha his device mus be able o absorb safely. Hin: Use MATLAB, or a similar program, o plo he power. () = ( ) = ( ) p e e e e W Here is a MATLAB program o plo p(): Here is he plo: clear 0=0; f=1; d=0.02; =0:d:f; % iniial ime % final ime % ime incremen % ime v=20*(1-exp(-8*)); % device volage i=.030*exp(-8*); % device curren for k=1:lengh() p(k)=v(k)*i(k); end % power plo(,p) xlabel('ime, s'); ylabel('power, W') The circui elemen mus be able o absorb 0.15 W. 1-24