85-4 The driven RL circuit r at t<0 i(t=0 Lai for t>0 Ri+ d t In(v-Ri=t+k dt V-Ri R L i(0)=0→i(0)=0∴k Inv i V/R--forced-response R V/R V-Ri IIn(v-ri)-Inv=t R -complete-resnnn or e (t> R R ( L )u(t) (for all t) R -V/R natural- response
§5-4 The driven RL circuit (1 e )u(t) ( for all t) R V i t L R − = − complete response e R V t L R − − − − − (1 ) −V / R natural response e R V t L R − − − − − i t 0 V / R V / R− − forced − response at t 0 i(t) = 0 V Ri t k R L dt V Ri Ldi V dt di for t Ri L = − − = + − 0 + = ln( ) V R L i(0 ) = 0 → i(0) = 0 k = − ln − t L R e V V Ri V Ri V t R L − = − − [ln( − ) − ln ] = = − = (1− ) ( 0) − − e t R V e R V R V or i t L R t L R
The complete response-the natural response+forced response I=L+ i=Ae (Ri+L==0 R (t→> i=i +i=ae+ Ri(0)=0→ 0=4+ R R i=n(1-e)=r(1-eo (t=L/R R R V/R 0.632V/R t=T=L/R, →i(z)=0.632/R
The complete response=the natural response+forced response n f i = i + i (1 ) (1 e ) ( L/ R) R V e R V i t t L R = − = − = − − = ( + = 0) − dt di i Ae Ri L t L R n = (t → ) R V i f R V i i i Ae t L R = n + f = + − i t 0 V / R 0.632V / R i V R t L R ( ) 0.632 / / , → = = = R V A R V i(0) = 0 → 0 = A+ = −
Example: Find i(t) Solution 50uctNv 2 I=L+ Ae 3/1.5=2s R 5 H 75/15=50 ∴i=Ae2+50(t>0) 1.5 i(0+)=i(0)=50/2=25 (t) .25=A+50→A=-25 i=50-25e2=50-25e4(t>0) 75y i=25A(t<0 st or i=25+ 25(1-e u()a (for all f)
Example: Find i(t). 25 25(1 ) ( ) ( ) 0.5 or i e u t A for all t − t = + − Solution: n f i = i + i s R L i Ae eq t n = = = 3/1.5 = 2 − = 75 / 1.5 = 50 f i 50 ( 0) 2 = + − i Ae t t i = 25A (t 0) 50 25 50 25 ( 0) 2 0.5 = − = − − − i e e A t t t 25 50 25 (0 ) (0 ) 50 / 2 25 = + → = − = = = + − A A i i
V1 3H 50V 0 +-+ 40A 30A 6 10s 口I(L1)
Time 0s 2s 4s 6s 8s 10s I(L1) 20A 30A 40A 50A V1 I 2 6 3H 1 2 50V 0
i=50-25e057(t>0) ori=25e0+50(1-e05)(t>0) zero-input zero-state response response Zero input response--term due to the excitation of the network by a nonzero initial condition Zero state response--terms due to the excitation of the network by sources
25 50(1 ) ( 0) 50 25 ( 0) 0.5 0.5 0.5 = + − = − − − − or i e e t i e t t t t response zero − input response zero − state Zero input response--term due to the excitation of the network by a nonzero initial condition. Zero state response--terms due to the excitation of the network by sources