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電路學 - [第五章] 一階RC/RL電路

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電路學 - [第五章] 一階RC/RL電路

1. 1. RC/RL Department of Electronic Engineering National Taipei University of Technology
2. 2. • • • RC RL • Department of Electronic Engineering, NTUT2/32
3. 3. i + − C + − v dv i C dt = L vL −+ i L di v L dt = ( ) ( ) ( )0 0 1 t t i t v t dt i t L = +∫( ) ( ) ( )0 0 1 t t v t i t dt v t C = +∫ Department of Electronic Engineering, NTUT3/32
4. 4. RC RL RCv(t) + − ic iR + − v t( ) L R − + vR i t( ) KCL 0c Ri i+ = ( ) ( ) ( ) ( ) 1 0 0 dv t v t dv t C v t dt R dt RC + = ⇒ + = ( ) ( ) ( ) ( )0 0 di t di t R L Ri t i t dt dt L + = ⇒ + = KVL ( ) ( ) ( ) dy t ay t f t dt + = ( ) ( ) ( ) dy t ay t f t dt + = • RC RL Department of Electronic Engineering, NTUT4/32
5. 5. RC RL ( ) Vs C + − v vR + − it = 0 R t =0 iL + −vL RVs vs(t) C + − v vR + − iR iL + −vL Rvs(t) ( ) 0f t ≠ Department of Electronic Engineering, NTUT5/32
6. 6. • RC RL y v(t) i(t) a f(t) f(t) • c y(0) t c e-at eat ( ) ( ) ( ) dy t ay t f t dt + = ( ) ( ) ( )at at at dy t e e ay t e f t dt + = ( ) ( ) ( ) ( ) ( ) ( )at at at at at dy t dy td e y t e e ay t e ay t e f t dt dt dt     = + = + =     ( ) ( ) ( ) ( ) at at at at d e y t dt e f t dt e y t e f t dt c dt    = ⇒ = +∫ ∫ ∫ ( ) ( )at at at y t e e f t dt ce− − = +∫ Department of Electronic Engineering, NTUT6/32
7. 7. • (Complete response) ( ) ( )at at at y t e e f t dt ce− − = +∫ ( ) ( )at at at y t e e f t dt ce− − = +∫ yp (Particular solution) (Forced response) (Steady-state response) (Homogeneous solution) (Natural response) t (Transient response) t A Be τ − = + f ny y= + ( )at at fy A e e f t dt− = = ∫t at ny Be ceτ − − = = 1 a τ = Department of Electronic Engineering, NTUT7/32
8. 8. • f(t) (Natural response) f (t) = 0 f (t) = b ( ) ( ) ( ) ( ) ( )at at at dy t ay t f t y t e e f t dt ce dt − − + = ⇒ = +∫ ( ) ( )0at at y t ce y e− − = = ( ) atb y t ce a − = + ( )0 b y c a = + ( )0 b c y a = − ( ) ( )0 atb b y t y e a a −  = + −   (y (0) = 0) f (t) = b ( ) atb y t ce a − = + ( )0 0y = b c a = − ( ) atb b y t e a a − = − Department of Electronic Engineering, NTUT8/32
9. 9. 1 • 5-1 (a) f (t) = 0 y(0) = 10 ( ) (b) f (t) = 5 y(0) = 10 ( ) (c) f (t) = 5 y(0) = 0 ( = ) (a) (b) (c) ( ) ( ) ( )5 dy t y t f t dt + = ( ) ( )at at at y t e e f t dt ce− − = +∫ ( ) 5t y t ce− = ( )0 10y c= = ( ) 5 10 t y t e− ∴ = ( ) 5 5 5 5 5 1t t t t y t e e dt ce ce− − − = × × + = +∫ ( )0 10 1 9y c c= = + ⇒ = ( ) 5 1 9 t y t e− ∴ = + ( ) 5 5 5 5 5 1t t t t y t e e dt ce ce− − − = × × + = +∫ ( )0 0 1 1y c c= = + ⇒ = − ( ) 5 1 t y t e− ∴ = − Department of Electronic Engineering, NTUT9/32
10. 10. (I) • • RC (a) (a) RC 4 Ω 6 Ω 2 Ω 7 Ω C = 1 F 150 V + −v t =05 Ω + − vc(t) (b) t = 0 t = 0 t = 0 t = 0 (b) vc ( )0− 150 V + −v + − 4 Ω 6 Ω 2 Ω 7 Ω 5 Ω Department of Electronic Engineering, NTUT10/32
11. 11. (II) Rth =10 Ω 1 F + − vc t( ) (c) t > 0 ( )4 2 / /6 7 10thR = + + = Ω   ( ) 10 0 150 100V 10 5 cv −   = × =  +  t 0 vc 100V vc (c) ( ) ( )0 0 100Vc cv v+ − = = Department of Electronic Engineering, NTUT11/32
12. 12. RC (I) • t > 0 KCL ( ) t > 0 ( ) 00cv V− = 0c Ri i+ = ( ) ( ) 0c cdv t v t C dt R + = ( ) ( ) 1 0C C dv t v t dt RC + = ( ) t RC cv t ce − = ( ) 00cv V− = ( ) 00cv c V− = = ( ) ( ) 00 0c cv v v+ − = = ( ) 0 t RC cv t V e − = ( ) ( ) 0 t c RC c dv t V i t C e dt R − = = − ( ) ( ) 0 t c RC R v t V i t e R R − = = RCvc(t) + − ic iR Department of Electronic Engineering, NTUT12/32
13. 13. RC (II) • RC (a) K t R C RC ( , ) • RC (Time constant) v(t) i(t) (b) K 0 t f t( ) (a) (b) ( ) t f t Ke τ − = t → ∞ ( ) ( ) 0 dy t ay t dt + = RCτ = R v i= C q v= ( ) 0f t → τ 1 a τ τ τ V0 vc(t) 0 t 0.368V0 1 Kτ = 2 2Kτ = 3 3Kτ = K 2K 3K 1τ 2τ 3τ Department of Electronic Engineering, NTUT13/32
14. 14. 2 • (a) t = 0 t = 0 t 0 vc(t) , v(t) ic(t) (c) t ＞ 0 τ = RC (a) RC 4 Ω 6 Ω 2 Ω 7 Ω C = 1 F 150 V + +v t =05 Ω + − vc(t) Rth =10 Ω 1 F + − vc t( ) (c) t > 0 ( ) ( ) ( )0 0 100 Vc cv v+ − = = ( ) ( )0.110 100 100 V t t cv t e e − − = = ( ) ( ) ( ) ( )0.1 0.17 7 100 70 V 7 6// 2 4 10 t t cv t v t e e− − = = × = + +   ( ) ( ) ( ) ( )0.1 0.1 100 0.1 10 Ac t t c dv t i t C e e dt − − = = × − = − Department of Electronic Engineering, NTUT14/32
15. 15. 3 ( ) • v(0) = 4 (V) t > 0 i KCL t > 0 + − 1 8 F 3 Ω2i (V) + − v i 6 Ω 2 1 0 6 8 3 v i dv v dt − + + = 1 8 dv dv i C dt dt = = 1 14 0 6 0 6 8 3 dv v dv v dvdt v dt dt − + + = ⇒ + = ( ) ( ) ( )6 0 4 V t t v t v e eτ − − = = ( ) ( ) ( )6 61 4 6 3 A 8 t tdv i t C e e dt − − = = × × − = − Department of Electronic Engineering, NTUT15/32
16. 16. RC (I) • v(0–)=V0 t = 0 • t > 0 v(t) (b) ( ) ( ) RC K ( V0–Vs) ( RC ) ( ) ( )0 t RC s sv t V V V e − = + − sV ( )0 t RC sV V e − − (a) Vs 0 t v(t) ( )0 t RC sV V e − −( )0 sV V− (b)0 t V0 Vs v(t) ( )0 t RC s sV V V e − + − Vs C + − v vR + − it = 0 R Department of Electronic Engineering, NTUT16/32
17. 17. RC (II) • RC 1. RC 2. ( ) 3. ( ) τ t e τ − × Department of Electronic Engineering, NTUT17/32
18. 18. 4 • t = 0 v(0 ) = V0 t 0 v(t) v(t) t = 0+ KCL t 0 I + − vC t =0a b R iR iC dv v dv v I C I dt R dt RC C + = ⇒ + = ( ) t t t t RC RC RC RC I v t e e dt ce RI ce C − − − − = + = +∫ ( ) ( ) 00 0v v v+ − = = ( ) 00v RI c V+ = + = 0c V RI= − ( ) ( )0 , 0 t RC v t RI V RI e t − = + − > Department of Electronic Engineering, NTUT18/32
19. 19. 5 • (a) v(0 ) = 15 (V) t > 0 v(t) 24 Ω 1 3F + − v40V 8 Ωi a b (a) (b) 40V a b Rth = 6 Ω 8 Ω 24 Ω A. ( (b)) 30(V) ab Voc ( ) 24 40 30 V 24 8 × = + B. (c) 8 24 6 8 24 thR × = = Ω + 1 6 2 sec 3 thR Cτ = × = × = ( ) 0.52 15 30 15 t t e e − − − = − C. t > 0 ( ) ( )0.5 30 15 Vt v t e− = − 1 3F30V a b (c) 6 Ω Department of Electronic Engineering, NTUT19/32
20. 20. RC • RC v(0) = 0 t > 0 • RC Voc = ( ) Rth = ( ) ( )1 t tRC s s sv t V V e V e τ − − = − = − ( ) ( )1 tht R C ocv t V e− = − a t e − Vs C + − v vR + − it =0 R Department of Electronic Engineering, NTUT20/32
21. 21. 6 • v(0) = 0 t 0 v(t) i(t) (b) (a) + − 36 V 12 Ω + − v t( ) 6 Ω 1 8 F i t( ) + −Voc 24= V a b + − v t( ) Rth = 4 Ω (b) 1 8 F ( ) 12 36 24 V 12 6 ocV = × = + 6 / /12 4thR = = Ω 1 1 4 sec 8 2 thR Cτ = = × = ( ) ( )( )2 24 1 Vt v t e− = − ( ) ( ) ( )2 36 6=2+4e At i t v t − = −   Department of Electronic Engineering, NTUT21/32
22. 22. • I0 ( ) • RL RL 0di dt = 0Lv Ldi dt= = Department of Electronic Engineering, NTUT22/32
23. 23. RL • RL i (0 ) = I0 t > 0 • RL τ L/R τ ( L ) i(0+) = i(0 ) = I0 t > 0 i(0 ) = I0 + − v t( ) L R − + vR i t( ) ( ) ( ) ( ) ( )0 0 di t di t R L Ri t i t dt dt L + = ⇒ + = ( ) R t L i t ce − = ( ) 00i c I− = = ( ) 0 R t L i t I e − = ( ) 0 R t L L di v t L I Re dt − = = − ( ) 0 R t L Rv t iR I Re − = = ( ) t f t Ke τ − = Department of Electronic Engineering, NTUT23/32
24. 24. 7 • 1 ( 1 ) t = 0 2 t > 0 v(t) i(t) 2 Ω t =0 + − 30V 2 + − v t( ) 3 Ωa b i t( ) 2 Ω1H 3 Ω1 i 1H 15 8 Ω a b 15 8 45 4 t ab di v e dt − = = − ( ) 15 15 8 8 2 45 2 9 V 3 2 4 5 2 t t abv t t e e − − = × = − × = − + (3) 1 6A 2 a-b ( )0 30 30 6 A 2 3 5 i I= = = = + ( ) ( ) 3 2 3 15 3 2 3 8 thR × + = = Ω + + ( ) 15 8 6 t i t e − = 8 sec 15th L R τ = = (1) (2) t > 0 Department of Electronic Engineering, NTUT24/32
25. 25. 8 ( ) • L/R i(0) = 2 A v 6A a b 2 Ω 3 2H 4 Ω i + −v 3 4 2 2 6 ab di v v i i dt   = + = × −    3 2 di di v L dt dt = = 3 3 24 2 2 6 di di dti i dt     + = × −      6 0 di i dt + = 6 2 At i e− = 1 6 τ = L R τ = Department of Electronic Engineering, NTUT25/32
26. 26. RL (I) • RL • i(0 ) = I0 t = 0 KVL t > 0 t = 0+ t > 0 t =0 iL + −vL RVs s s di di R V L Ri V i dt dt L L + = ⇒ + = ( ) R R R R t t t t s sL L L L V V i t e e dt ce ce L R − − − = + = +∫ ( ) ( ) 00 0i i I+ − = = ( ) 0 00 s sV V i c I c I R R + = + = ⇒ = − ( ) 0 R t s s L V V i t I e R R −  = + −    s s V I R = ( ) ( )0 R t L s si t I I I e − = + − Department of Electronic Engineering, NTUT26/32
27. 27. RL (II) • i(t) Vs / R • • RL RL – τ ( ) ( )0 R t L s si t I I I e − = + − 0 R t s L V I e R −  −    t e τ − 0 R t s L V I e R −  −    Vs sV R sV R Department of Electronic Engineering, NTUT27/32
28. 28. 9 • i(0 ) = 2 A t > 0 i(t) A. t > 0 a-b ( ) B. a-b + − 3 Ω 6 Ω 3 Ω2 H b a it =0 6 Ω 36V + − 6V 2 Ω a 2 H b i t( ) 3 3 36 6(V) 6 3 3 3 oc abV v   = = × =  + +  ( )6 / /6 3 / /3 2thR = + = Ω   2 1 2th L R τ = = = ( ) ( ) ( )3 2 3 3 At t i t e e− − = + − = − ( ) ( ) 36 6 3 A 6 6 / /3 3 6 × = + + ( )6 / /6 3 / /3 2+ = Ω   ( ) ( ) ( )3 2 3 3 At t i t e e− − = + − = − Department of Electronic Engineering, NTUT28/32
29. 29. • K K = 1 u(t) (Unit step function) us(t) (a) t = 0 0 1 • t – t0 t R (b) u(t – t0) t0 t0 u t t K t ( ) = < = > 0 0 0 t t0 R t t0 , , u t t0( )− = < = > 0 1 0 t (a) (b) ( )u t 1 0 t0 t ( )0u t t− Department of Electronic Engineering, NTUT29/32
30. 30. 10 • i(t) KCL B. t ＞ 0 i t i i Aef n t ( ) = + = + − 3 2di dt i+ =2 6 A. t ＜ 0 i(0) = 0 A = 0 i(t) = 0 di dt i+ =2 0 i t Ae t ( ) = −2 C. t i(t) = 0 , t ＜ 0 = 3 – 3e-2t , t ＞ 0 i t e u tt ( ) ( ) ( )( )= − − 3 1 2 A i(0) = 0 = 3 + A = 0 , A = -3 i t e t ( ) ( )( )= − − 3 1 2 A + − 6u t( ) V 2 Ω 1H i Department of Electronic Engineering, NTUT30/32
31. 31. 11 • i(t) A. u(t) (b) B. –u(t – 2) (c) i2(t) i2 t( ) = – 0.5 [1 – e-2(t – 2)] , t ＞ 2 i1 t( ) = 0.5 (1 – e-2t)(A) , 0 ＜ t ＜ 2i t( ) = i1 t( ) += i2 t( ) = – 0.5 [1 – e-2(t – 2)] (A) , t ＞ 20.5 (1 – e-2t) i1(0) = 0 A = 0 i1(t) = 0 t 0 di dt i+ =2 0 i t Ae t( ) = −2 t 0 i1 t i1 i1 Aef n t ( ) = + = + − 0.5 2 di1 dt i1+ =2 0 i1(0) = 0.5 + A = 0 A = - 0.5⇒ 0.5 (1 – e-2t) (A) , t ＞ 0i1 t( ) = + −[u(t)−u(t−2)] V 2 Ω i 1H + − v t( ) u(t) i1(t) –u(t – 2) i2(t) i(t) = i1(t) + i2(t) + − 2 Ω 1H (c) i2(t) −u(t−2) V (a) + − u t( ) 2 Ω 1H (b) i1(t) Department of Electronic Engineering, NTUT31/32
32. 32. • RC RL yf yn • yf ( ) ( ) yf yf yn y(0) B yn = Be –at ( ) ( ) ( ) dy t ay t f t dt + = 1 a τ ( ) t at y t A Be A Beτ − − = + = + ( ) ( )0 at f f f ny t y y y e y y−  = + − = +  ( ) ( )0 0 fB y A y y= − = − Department of Electronic Engineering, NTUT32/32