1 of 4 pages University of Rochester Department of Electrical & Computer Engineering ECE111 Final Examination 12-17-08 This is a closed-book examination, but you may use a hand-held calculator. Write your name on your exam booklet and then do all work in it. Clearly indicate your answers. Partial credit is more likely to be granted for work that is neat, well-organized, and easy to read. There are eight (8) problems, all weighted equally. Do not overstate or understate the precision of numerical answers. 1) A battery, Vdc, and an AC voltage source, vac(t) = Vocos(ωt), are connected to a linear LR circuit as shown below. Find an expression for the total output voltage vout(t). Explain your methods fully. 2) The voltage and current to a linear, passive AC load are v(t) = Vo cos (ωt) & i(t) = Io cos (ωt+ θ). The nominal values and uncertainties of these are: Vo = 100 (±5) Volts, Io = 1 (±0.03) Amps, θ = 30° (±5°). frequency: f = 100 (±2) Hz. Find the nominal, time-average power P delivered to the load and its ± uncertainty, both in Watts. vac(t)LVdcvout(t) Linear passive load v(t) i(t)2 of 4 pages 3) Find the Thevenin equivalent for the AC circuit shown below, assuming i(t) = Iocos(ωt) as the instantaneous value for the current source. What, if any, frequency restrictions might apply to this equivalent circuit. 4) A voltage source connected to an RLC circuit makes a step change from 2Vo to Vo at t = 0. (a) Replicate the table below in your exam booklet and then fill it in with expressions for the defined voltages and currents at the times given. (b) Explain concisely and clearly how these answers were obtained. (c) Without actually solving for it, plot voltage, vb(t) versus time t. t = 0- t = 0+ t  ∞ va(t) vb(t) iT(t) iR(t) iL (t) i(t)j100 !100 !-j100 !RR+va(t)vb(t)iT(t)iR(t)iL(t)++--vs(t)vs(t)t2VoVo3 of 4 pages 5) For the RLC circuit below, find an algebraic expression for the complex admittance Y(ω) = G(ω) + jB(ω). Explain the low frequency limiting behavior of this circuit that is exhibited by G(ω) and B(ω). 6) The integrated circuit-based analog computer below solves an integro-differential equation relating vin(t) and vout(t). Find this equation. You may employ the time-based vi characteristics for the passive components (R,L,C) directly or you may use phasors, but your answer must be a differential equation in the time domain. 7) At time t = 0, the dc voltage Vo is suddenly removed from the LR circuit below, that is, vs(t) = Vo[1-u(t)]. (a) Find expressions for the voltage vout(t) for both t < 0 and t > 0, making sure to define all quantities used in your answer. (b) Sketch this voltage carefully versus time t. r Y = ?vout(t)Rfvin(t)CLop ampRR+vout(t)+-vs(t)4 of 4 pages 8) A linear system has two independent inputs, X and Y, and a single output, Z. The table below gives the results of some tests performed with this system. Replicate the table in your exam booklet and then answer the following questions about this system. (a) Complete the empty table elements. (b) Find the algebraic expression relating X, Y, and Z. (c) If you had not been told that the system is linear, would the first four rows of test results provide proof of linearity? Explain how you arrived at your answer. row X Y Z 1 1 0 0.5 2 2 0 1.0 3 0 1 -1.0 4 0 2 -2.0 5 1 1 6 2 1 7 1