Question

4. Assuming that the switch has been open for a long time, find io(f) for >0. 800 Ω r0 30 V Find the roots of the characteristic equation: a) b) Write the general form of the solution (with unknown constants) io(t) c) What is the value at time infinity? d) What is the value at time zero? What is the value of the derivative at time zero? e) f) What is the final solution? io()

Answer 1

Apply the nodal analysis at node v_0 (t) - v_s/8 R + v_0 c d v_0 (t) dt + I_0 (t) = 0 c d v_0 (t)/d t + v_0 (t)/R + i_0 (t) = v_s/R substitute v_0 (t) = L di/dt in above equations c L d^2 i_0 (t)/dt^2 + 1/RC d_i_0 (t)/d r + i_0 (T)/LC = v_s/RLC D = d/dt D^2 = d^2/dt^2 (D^2 + 1/RC D + 1/LC) I (t) = v_s/RLC characteristics equation f(n) = 0 D^2 + 1/RC D + 1/LC = 0 D = -40, -160 roots of characteristics equation -40, -60 \r\n the enteral solution i_0(t) = c_1 e^-40 t = c_2 e^-160 t particular solution v_s/R L C/f(D)/D = 0 v_s/R L C/1/L C = 30/800 Total solution i_0 (t) = c_1 e^-40 t + c_2 e^-160 t + 3/8 t > 0 - (1) As r rightarrow infinity (steady state) inductor rightarrow short circuit capacitors rightarrow open circuit i_0 (infinity) = 30/800 = 3/80 = 0.375 A