EE201 Lecture 23 P 1 Undriven RLC Circuits RLC series circuits R iL t L C vc t iL as circuit variable d2 iL t dt2 R d iL t L dt 1 iL t 0 1 vc t 0 2 LC vc as circuit variable d2vc t dt2 R dvc t L dt 1 LC EE201 Lecture 23 P 2 RLC parallel circuits iL R L vc t C iL as circuit variable d2 iL t 1 d iL t dt2 RC dt 1 iL t 0 LC 3 vc as circuit variable d2 vc t dt2 1 d vc t RC dt 1 vc t LC 0 4 EE201 Lecture 23 P 3 General Form of Differential Equations for RLC Circuits d2 x t dt2 b d x t dt c x t 0 5 Trial solution for Eqn 5 x t K e st K d2 e st bK d e st dt2 dt K e st s2 bs c 0 c K e st 0 For non trivial solutions s2 bs c 0 This is the characteristic equation 6 EE201 Lecture 23 P 4 Solution for Eqn 6 b b2 4c s1 s2 2 s1 s2 are natural frequencies Three solution cases for Eqn 7 7 Case 1 Solution forms b2 4c 0 s1 s2 are real and distinct General form is x t K1 e s1t K2 e s2t 8 Find K1 and K2 using initial conditions If t0 0 x 0 K1 e s10 K2 e s20 K1 K2 9 dx 0 10 x 0 s1 K1 s2 K2 dt Solve Eqns 9 and 10 to obtain K1 K2 EE201 Lecture 23 P 5 Case 2 Solution forms b2 4c 0 s1 s2 are complex and distinct Complex number Z X jY where j 1 b s1 s2 j 2 4c b2 2 j d 11 where b 2 and d 4c b2 2 s1 j d s2 j d s1 s2 s1 and s2 are complex conjugates General form is x t K1 e s1t K2 e s2t Because s1 s2 es1t and e s2t are complex conjugates Therefore for x t to be real K2 K1 EE201 Lecture 23 P 6 The solution becomes x t e t A cos dt B sin dt 12 This is equivalent to x t Ke t cos dt K A2 B2 tan 1 B A A B are found from initial conditions x 0 e t A cos dt B sin dt t 0 x 0 A x 0 A dB Note that d n2 2 1 2 EE201 Lecture 23 P 7 Case 3 Solution forms b2 4c s1 s2 and are real The solution form is x t K1 K2t e s1t 13 where x 0 K1 x 0 s1 K1 K2 As before if to 0 solve simultaneous equations for x to and x to to find K1 and K2 EE201 Lecture 23 P 8 Undamped response x t Typical for LC circuits Underdamped response x t decays as exp t t Typical of case 2 LCR circuits where Re s1 0 EE201 Lecture 23 P 9 Critically damped x t exponential decay t Typical of case 3 LCR circuits where s1 s2 Over damped response x t Non oscillatory t Typical of case 1 LCR circuit where s1and s2 are negative EE201 Lecture 23 P 10 Example Find the value of the capacitance C if iL t 50 e 10t sin 10 3 t A for t 0 10 iL t L C vc t he amplitude of iL t decays exponentially with time but is oscillatory Therefore response is underdamped General solution for case 2 is iL t e t A cos dt B sin dt EE201 Lecture 23 Comparing both expressions for iL t A 0 d 10 3 B 50 Comparing equations 1 5 and 12 b 2 R 2L 0 5 R L 10 L 0 5 H But d n2 2 1 2 10 3 n2 1 LC 300 100 400 n 20 LC 1 400 C 1 400L 1 400 0 5 C 0 005 F P 11
View Full Document
Unlocking...