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ELEC1704 Further Engineering Mathematics Problem sheet Week 23 1. Draw the s-domain equivalent of the circuit below (Fig 1), and hence derive expressions for the Laplace Trans- forms of the voltage and current across the inductor Iy. Let R =5Q, Ly = 1H, [, = 1.25H and V; = 18V. Use inverse transforms to obtain the voltage and current in the time domain. R [ ] Ly Lo Figure 1 2. Draw the s-domain equivalent of the circuit below (Fig 2), and hence derive an expression for the Laplace 3. transform of the output voltage. For each of the cases: (@) L=0.25H, C=0.01F, R =2Q and Ey = 6V (b) L=0.8H, C =0.01F, R = 10Q and Ey = 6V determine the system poles and plot these on the complex s-plane; hence deduce the form of the temporal response. Use inverse Laplace transforms to obtain the output voltage as a function of time, and compare with your deduction. [ R SL Vour(t) & f/-i\\"\\ ' \\' v Figure 2 (a) An RC series circuit is driven by a d.c. voltage source Vj volts, which is switched on at time t = 0. An initial charge go is stored by the capacitor at t = 0. Write down the differential equation for the circuit and hence obtain the Laplace transform of the capacitor charge. Use the Final Value Theorem to determine the steady state charge. By taking inverse Laplace Transforms, obtain the temporal response of the capacitor charge, and verify that it converges to the expected final value. Sketch the response. (b S' Consider the same circuit, with stored charge g at time t = 0 as before, but in this case the voltage supply is not switched on until + = T seconds. Find the transform of the charge, the steady state charge, and the temporal response as above. Verify that the charge converges to the expected value and sketch the response. \f