Question

Please if you could help me understand how to do this. Thank you!

RLC example A dc source is connected to a series RLC circuit by a switch that closes at t = 0 as shown in Figure 4.21. The initial conditions are i(0) 0 and vc(0) 0. Write the differential equation for vct). Solve for vcl) if R 300,200, and 100 2. 10 mH i(0)

Answer 1

Given that A DC source is connected to a series RLC circuit by a switch that closes at t = 0 as shown in figure The initial conditions are i(0) = 0 and v_c(0) = 0 if R = 30,200,100 l current through capacitor i(t) = c d v_c (t)/dt rightarrow (1) Apply KVL at sh i(t) -V_S + L di(t)/dt + R i(t) + V_c (t) = 0 d^2 v_c(t)/dt + R/L d v_c(t)/dt + 1/Lc v_c (t) = V_s/LC rightarrow (2) w_0 = 1/squareroot LC = 1/squareroot 10 m times 1 mu w_0 = 10,000 rad/sec For R = 300 alpha = R/2L = 300/20 m = 15000 rad/sec Damping ratio epsilon = alpha/w_0 = 15000/10,000 = 1.5 epsilon > 1, this is the over damped case The roots of characteristics equation are S_12 = -alpha plusminus squareroot alpha^2 - w^2_0 = -15000 plusminus squareroot (15000)^2 - (10,000)^2 = -15000 plusminus 11180.33 = -3819.66, -26180.33 The general solutions is V_c