Consider the parallel RLC circuit below with a dc current source (Iac = 1 A). Assume...

 

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Consider the parallel RLC circuit below with a dc current source (Iac = 1 A). Assume that no initial energy is stored in the circuit at the instant when the switch, which is shorting the dc current source, is opened. The problem is to find the time-domain expression for v(t) when t > 0 and determine if the voltage rating threshold of the capacitor has been exceeded when the dc current source is first switched on. t = 0 Idcu(t) R L (1)a Paste Use the Laplace transform method to determine the complete response of the voltage v(t). To begin, the integrodifferential equation that governs the circuit response is the following: v(t) 1 dv(t) v(x) dx + C dt Iacu(t) Equation 1 • u(t) is the unit step function defined as t<0 u(t) = {", t > 0 Initial condition: The initial voltage across the circuit is v( 0 because the initial energy stored in the circuit is zero. The dc current value is Idc = 1 A. Use the specific values for R, L, C, and the capacitor's voltage rating threshold assigned to you. You can introduce those values right away in part (a) or stay with variables up until plotting in part (e), your choice. (a) Using Laplace transforms, transform the integrodifferential equation (Equation 1) into an algebraic equation in the s-domain in which V(s) is the unknown variable. (10 points) (b) Solve the algebraic equation for the unknown V (s). The final result will be a V (s) = a rational function. (10 points) (c) Find the partial fraction expansion of V (s). i. Find the roots of the polynomial in the denominator. (5 points) ii. Solve for the unknown coefficients for each partial fraction factor. (22 points) (d) Take the inverse Laplace transform of the partial fraction expansion of V(s) to find the time-domain expression for the voltage response: v(t) = L-1{V(s)}. (10 points) Consider the parallel RLC circuit below with a dc current source (Iac = 1 A). Assume that no initial energy is stored in the circuit at the instant when the switch, which is shorting the dc current source, is opened. The problem is to find the time-domain expression for v(t) when t > 0 and determine if the voltage rating threshold of the capacitor has been exceeded when the dc current source is first switched on. t = 0 Idcu(t) R L (1)a Paste Use the Laplace transform method to determine the complete response of the voltage v(t). To begin, the integrodifferential equation that governs the circuit response is the following: v(t) 1 dv(t) v(x) dx + C dt Iacu(t) Equation 1 • u(t) is the unit step function defined as t<0 u(t) = {", t > 0 Initial condition: The initial voltage across the circuit is v( 0 because the initial energy stored in the circuit is zero. The dc current value is Idc = 1 A. Use the specific values for R, L, C, and the capacitor's voltage rating threshold assigned to you. You can introduce those values right away in part (a) or stay with variables up until plotting in part (e), your choice. (a) Using Laplace transforms, transform the integrodifferential equation (Equation 1) into an algebraic equation in the s-domain in which V(s) is the unknown variable. (10 points) (b) Solve the algebraic equation for the unknown V (s). The final result will be a V (s) = a rational function. (10 points) (c) Find the partial fraction expansion of V (s). i. Find the roots of the polynomial in the denominator. (5 points) ii. Solve for the unknown coefficients for each partial fraction factor. (22 points) (d) Take the inverse Laplace transform of the partial fraction expansion of V(s) to find the time-domain expression for the voltage response: v(t) = L-1{V(s)}. (10 points)

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