ECE 212 H1F Pg 1 of 12 ECE 212 - Circuit Theory Final Examination December 5, 2008 1. Policy: closed book, calculators allowed. Show all work. 2. Work in the provided space. 3. The exam has 3 problems and 12 pages, including the cover page and extra sheet. Name: Student Number: Problem 1 [33 pts] Problem 2 [33 pts] Problem 3 [34 pts] Total [100 pts]:
Problem 1 [33 marks] In the circuit below, the switch has been closed for a long time. At t=0, the switch moves from closed to open position t = 0 R S R + C _ v C (t) i L (t) 12 V + _ L (a) [5 points] Derive a differential equation for i L (t) for t>0. (5 marks) (b)[2 points] Find the circuit characteristic equation for t>0. (2 marks)
(c)[5 points] For R = 8 Ω, R S = 2 Ω, C = 50 mf and L = 200 mh, determine the initial conditions for i L (t) and v C (t) at t = 0. (d) [4 points] For the given parameters in (c), determine the final conditions for i L (t) and v C (t). (e) [4 points] For the given parameters in (c), determine if the circuit is overdamped, underdamped, or critically damped.
(f) [8 points] Use your results from (e) to find the response of i L (t) for t>0. (g) [5 points] If we replace the 8 Ω resistor with a 4Ω resistor (R = 4Ω), determine if the circuit is overdamped, underdamped, or critically damped.
Problem 2 [33 points] The circuit of Fig.2 operates in the sinusoidal steady-state. (a) [2 points] Transfer the circuit into the phasor domain and, in the provided diagram clearly label values of all components. (b) [3 points] Draw the phasor domain equivalent circuit for the case when only the current source is active, i.e. the voltage source is turned off.
i) [6 points] Find the phasor Vx 1, corresponding to the sole contribution of the current source to Vx. ii) [5 points] Find the phasor Ix 1, corresponding to the sole contribution of the current source to the phasor current Ix.
(c) [3 points] Draw the phasor domain equivalent circuit for the case when only the voltage source is active i) [6 points] Find the phasor Vx 2, corresponding to the sole contribution of the voltage source to Vx.
ii) [5 points] Find the phasor Ix 2, corresponding to the sole contribution of the voltage source to the phasor current Ix. (d) [3 points] Find v x (t), and i x (t)
Problem 3 [34 Points] In the following, all OP-AMPs are ideal and no energy is stored initially in any of these circuits. 0.01 μf 200 kω 10 kω 5 kω v s (t) 100 kω v o (t) v s (t) 0.05 μf 5 kω 10 kω v o (t) v s (t) 0.02 μf v o (t) (i) (ii) (iii) Fig. 1 Useful Laplace Transform Pairs F f () ( s) t f () t F( s ) f ( t ) F( s ) f () t F( s ) u() t 1 s b sin bt u t 2 2 s + b δ () t 1 ( ) ( ) e at sin ( bt) u( t) b ( ) 2 2 s+ a + b e () u t at 1 s+ a te at () u t 1 ( s+ a) 2 s cos( bt) u ( t) 2 s + b e at cos( bt) u( t) s ( ) 2 2 s+ a + b (a) [3 Points] Transform the three circuits in Fig. 1 into the s-domain and label all values in the diagrams below. 5 kω v s (t) 0.02 μf v o (t) (i) (ii) (iii)
(b) [15 Points] For each of the three circuits in Fig. 1 derive expressions for the s- domain output voltage waveform in terms of the source voltage.
(c) [2 Points] Transform the circuit in Fig. 2 into the s-domain and label all values in the diagram below. 0.01 μf 2u(t) 100 kω 200 kω 10 kω 0.05 μf 5 kω 10 kω 5 kω 0.02 μf + v o (t) _ Fig. 2 (d) [5 Points] Derive an expression for the s-domain output voltage waveform in the circuit in Fig. 2.
(e) [4 Points] Determine the poles and zeroes of the waveform from part (d) and enter them in the pole-zero-diagram below. jω σ (f) [3 Points] Transform the waveform from part (d) into the time domain. (g) [2 Points] Determine the DC steady state response of the circuit in Fig. 2.
Extra sheet: