ECE 212H1F Circuit Analysis October 20, :15-19: Reza Iravani 02 Reza Iravani 03 Ali Nabavi-Niaki. (Non-programmable Calculators Allowed)

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1 Please Print Clearly Last Name: First Name: Student Number: Your Tutorial Section (CIRCLE ONE): 01 Thu 10:00 12:00 HA Thu 10:00 12:00 GB Thu 15:00 17:00 GB Thu 15:00 17:00 SF Fri 13:00 15:00 GB Fri 13:00 15:00 GB304 (YOU LOSE FOUR POINTS FOR INCORRECT TUTORIAL SECTION INFORMATION) Your Lecture Section (CIRCLE ONE): 01 Reza Iravani 02 Reza Iravani 03 Ali Nabavi-Niaki (YOU LOSE ONE POINT FOR INCORRECT LECTURE SECTION INFORMATION) Mid-term Answer All Questions (Non-programmable Calculators Allowed) 70 minutes [40 Marks] Instructions: Do not unstaple this exam book. Answer each question neatly and concisely. Write the final answer in the box where provided. The exam has 3 questions and 11 pages, including the cover page. Solutions steps and final answers must not be in pencil. Q1 /18 Q2 /8 Q3 /14 Total /40 University of Toronto 1 of 11

2 Question 1 a) Find the expressions for v o1(t) and v o2(t) in terms of v s1(t) and v s2(t) in the circuit in Figure Q1.a. (The op amps are ideal). [8 points] y i 1 (t) v s1 (t) x i 2 (t) v s2 (t) Figure Q1.a From the ideal op amp c/cs: The voltage of node x is v s2(t) and the voltage of node y is v s1(t). Using KVL: KCL at node x: Substitute in (1): Using KVL: KCL at node y: Where University of Toronto 2 of 11

3 Substitute in (2): University of Toronto 3 of 11

4 b) For the circuit in Figure Q1.b, (The op amp is ideal): x Figure Q1.b. 1) When the voltage sources V 3 and V 4 are off (disconnected), find V o in terms of V 1 and V 2. [3 points] When V 3 and V 4 are off, it becomes an inverting summer circuit: V o (V 3 and V 4 are off) = 2) When the voltage sources V 1 and V 2 are off (disconnected), find V o in terms of V 3 and V 4. [3 points] KCL at node x: V p =V n =V x (ideal op amp) and reducing it to a non-inverting circuit: Substitute in (1): V o (V 1 and V 2 are off) = University of Toronto 4 of 11

5 3) When all voltage sources are on (connected), find V o (Hint: opamp is linear and ideal). [2 points] Using superposition theorem: V o = 4) Draw a functional block diagram representing the circuit. [2 points] V 1-5 V 2-2 V o V 3 2 V 4 3 University of Toronto 5 of 11

6 Question 2 For the network given in Figure Q2: I Figure Q2 1) Calculate the Thévenin equivalent voltage V th of the left side of terminals A and B. [4 points] V th = V o.c. = V AB it is the open circuit voltage difference between A and B The dependent voltage source : and current I: Applying KVL to get : V th = University of Toronto 6 of 11

7 2) Calculate the Thévenin equivalent resistance R th of the left side of terminals A and B. [4 points] with a mix of independent and dependent sources = = Hence, terminals A and B is shorted. Using node-voltage analysis: KCL at node x: University of Toronto 7 of 11

8 Question 3 a) The switch in Figure Q3.a has been closed for a long time and it opens at t = 0. i(0) 1) Find v o (0). [2 points] Figure Q3.a In the given RC network:, 2) Find the expression of v o (t) for t 0. [3 points] v o (0) = = University of Toronto 8 of 11

9 Circuit time constant, Using 1 st order circuit equation: For v o (t) = = 3) Plot v o (t) (from part b above) for t 0. [2 points] v o (t) t University of Toronto 9 of 11

10 b) The switches, in Figure Q3.b, have been in their positions for a long time before instantaneously changing their positions at t = 0 as indicated. v c (t) Figure Q3.b 1) Determine the circuit response, i.e., i l (t) is over-, under- or critically damped. [3 points] Using the 2 nd order equation [a parallel RLC circuit - natural response] page 11: Laplace Transform of this equation: and can be calculated as follows: Using source transformation for the 240 V voltage source: and substitute in (1) Complex conjugate roots, hence the circuit response is underdamped. 2) Find the expression of i l (t) for t 0. [4 points] The circuit initial conditions: From part 1) we have complex conjugate roots, the current University of Toronto 10 of 11

11 Using the first initial condition Substitute in this equation with the second initial condition: t 0 University of Toronto 11 of 11

ECE 212H1F Circuit Analysis October 30, :10-19: Reza Iravani 02 Reza Iravani 03 Piero Triverio. (Non-programmable Calculators Allowed)

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