Hw10_sols - ECE 210 Spring 2005 Homework 10 Solutions From...

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ECE 210, Spring 2005 Homework 10 Solutions From Rizzoni, Problems: 5.59, 5.63, 5.68. In addition, for the circuit in 5.63, find the voltage across the resistor for t>0 for the case where the 10 V DC source is replaced by an AC source with voltage 10*cos(w*t) where w is the natural radian frequency of the circuit.
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For the supplementary problem where the 10 V source is replaced by an AC source with voltage 10cos( ω t ) , we do KVL around the circuit, then write the equation in differential form, 10cos( ω t ) = v R + v L + v C = iR + L di dt + 1 C idt ω = 1/ LC = 10 5 rad / s d 2 i dt 2 + R L di dt + ω 2 i = ω L 10sin ω t The roots of the homogeneous part of this equation are, for the given values, s = R 2 L ± 1 2 R L 2 4 ω 2 ≈ − R 2 L ± j ω The homogenous solution (or natural response) is then i H ( t ) = e tR /2 L A sin ω t + B cos ω t ( ) For the inhomogeneous solution (forced response), we can do phasor analysis. Note that there are other ways to approach this as well. So, i I ( t ) = Re{ ˆ i ( t )} = Re{ i 0 e j ω t } ω 2 + j ω R
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