, LAST name: First name: Student ID: University of Toronto Faculty of Applied Science and Engineering ECE212H1F - Circuit Analysis Final Examination December 16, 2017 9:30am - noon Guidelines: Exam type: A Examiners: K. Phang, Z. Tate, and W. Wong Please write your answer in the space provided for each question. Show your work and use the back side of sheets as needed. This exam is closed-note. You may use a non-programmable calculator. The last two pages are the equation sheets previously provided to you via Portal. I Problem I Score 1 /4 2 /10 3 /9 4 /7 5 /14 6 /1 7 /3 8 /4 9 /6 10 /2 Total /60 I
Problem 1 Page 2 of 25 Problem 1 ( 4 points) 10 mh 100 µf a) (3 points) For the circuit shown above, find the transfer function G(s) = Vou((~), where Vout(s) = ½s {Vout(t)} and ½(s) = {v 5 (t)}. G(s) = b) (1 point) Determine all the poles of the transfer function G(s). List all poles of G ( s) in this box
Problem 2 Page 3 of 25 Problem 2 (10 points) - Vs(t) + L C H s Vo s = 104 - (.) Vs ( ) ( s + 10) ( s + 1000) The filter circuit shown above has component values R, L, and C such that its transfer function is equal to the given H(s), where.vout(s) =.C { Vout(~)} and ½(s) =.c {vs(t)}. a) ( 4 points) On the graphs provided below, sketch the Bode plot ( straight-line approximations of magnitude and phase) for H ( s). At the,top of the next page, th~re is a sample Bode plot ( corresponding to a different transfer function) that illustrates how the magnitude and phase plots should be labeled.. IHI [db] ; i. l! l...,.... -~...,.-i... l... I! l l I,! i i-! I i i i I I
Problem 2 Page 4 of 25 IHI [db] Sample Bode plot (based on a different transfer function) I..H -40 db -90 t ' i '' '_L_ - On the magnitude plot, label the slopes of all lines and provide (on the vertical axis) the magnitude for at least one frequency; On the phase plot, label the value of the phase for any constant asymptotes; and On both plots, clearly mark the scale on the horizontal and vertical axes and label all significant frequencies.
Problem 2 Page 5 of 25 b) ( 4 points) This is a commonly used filter circuit in de-de converters, which use switches ( operating at a frequency w 5 ) to reduce or increase a de supply voltage. If v 5 (t) = ~ + cos(w 5 t) V, find an approximate expression for the output waveform Vout(t) using the Bode plots from part (a), if i) W 5 = 100 rad/s Vout(t) = ii) W 5 = 10000 rad/s Vout(t) =
Problem 2 Page 6 of 25 c) (2 points) If the goal is to provide the load (represented by the resistor R) with a constant de voltage (i.e., minimize the peak-to-peak fluctuations of the output voltage), which switching frequency is a better choice based on your results from part (b)? Briefly explain. Circle one: W 5 =l00 rad/s W 5 =l0000 rad/s
Problem 3 Page 7 of 25 Problem 3 (9 points) A series of tests are run on a mutually-coupled set of windings with terminals A-B and C-D to determine the parameters of the corresponding circuit model (LAB, Lev, and M): a) (4 points) (Open-circuit test) With terminals C-D open-circuited, a voltage source VAB(t) = 2y'2cos(10t) Vis connected to the A-B winding, as shown in the figure below. Under this test condition, an ammeter connected to the voltage source reads 1 Arms, and a voltmeter connected to winding C-D reads 3 V rms. Using this test, determine LAB (the self-inductance of winding A-B) and M ( the mutual inductance between the windings) in henries. ias(t) A ----~ icn(t) (rms) +. LAB Len vcn(t) V (rms) Open-circuit test setup showing ammeter ~ (,;;;_,l ~ and voltmeter I (n~~l I connections. LAB=------------ M= ---------------------
Problem 3 Page 8 of 25 b) (4 points) (Short-circuit test) With terminals C-D short-circuited, a voltage source va8 (t) = 2v'2cos(10t) Vis connected to the A-B winding, as shown in the figure below. Under this test condition, an ammeter connected to the voltage source reads 10 A rms. Using this test, determine Len (the self-inductance of winding C-D) in henries. iab(t) A )![,. icd(t) VAs(t) ~ (rms) LA~ 8 Short-circuit test setup showing ammeter ~ (,;~,J ~ connection. Len=
Problem 3 Page 9 of 25 c) (1 point) The measurements collected during the open-circuit test in part (a) from the voltmeter and ammeter do not provide sufficient information to verify the dot markings on the device. Describe briefly how a 2-channel oscilloscope could be used to verify the dot markings during the open-circuit test.
Problem 4 Page 10 of 25 Problem 4 (7 points) Load 1 30 kva p.f. = 0.6 lagging Load 2 20 kva p.f. = 0.8 leading A 120-V rms, 60-Hz source v 8 (t) supplies two loads connected in parallel, as shown in the figure above. a) (4 points) Find the power factor of the parallel combination of the two loads. p.f
Problem 4 Page 11 of 25 b) ( 3 points) Calculate the value of the capacitance connected in parallel that will raise the power factor to unity. c= ----------------
Problem 5 Page 12 of 25 Problem 5 {14 points) t=0 t=0 + 40 mh vc 40 mf Assume the above circuit is in steady-state before time t = 0. Both switches move at t = 0. a) (2 points) Find il(0+) and vc(0+)- il(o+) = vc(o+) =
Problem 5 Page 13 of 25 b) (2 points) Draw the s-domain representation of the circuit fort ~ 0. You can ignore portions of the circuit that are disconnected from any sources. Draw the s-domain representation in this box
Problem 5 Page 14 of 25 c) (3 points) Use superposition to find vr(t) fort 2: 0. In this part, turn off the the e- 50 tu(t) source and only consider the response due to initial conditions. Determine VR(s) =.C{vR(t)}. VR(s) =
Problem 5 Page 15 of 25 d) (3 points) In this part, turn off all initial conditions and find the response VR(s) due only to the c 50 tu(t) source. VR(s) =
Problem 5 Page 16 of 25 e) (3 points) Write out the full solution for VR(s) (due to initial conditions and the source) and perform partial fraction expansion on it. VR(s) = f) (1 point) Find vr(t) fort~ 0. VR(t) =
Problem 6 Page 17 of 25 Problem 6 ( 1 points) For the following circuit, determine the filter type (low-pass, band-pass, or high-pass) with the input defined as Vin and the output defined as Vout Note: you do not need to derive the transfer function to solve this problem. + + Vout Circle the filter type: Low-pass Band-pass High-pass
Problem 7 Page 18 of 25 Problem 7 (3. points) + Vout(t) 30 a) (1 point) Find the time constant of the response Vout(t). T= b) (1 point) Find Vout(t) fort~ 0 if il(o+) = 1 A. Vout(t) =
Problem 7 Page 19 of 25 c) (1 point) Given il(o+) = 1 A, find the time, t 33 %, required for the output Vout to drop to one-third of its initial value (i.e., V 0 ut(taa%) = ½vout(O+)). t33% = -----------
Problem 8 Page 20 of 25 Problem 8 ( 4 points) 100 kn 1 nf 50 kn In the above op amp circuit, v 5 = 4 cos (10 4 t) V. Find the average power PL delivered to the 50-kn resistor. Note: 1 nf = 1 x 10-9 F. ~=-----------
Problem 9 Page 21 of 25 Problem 9 ( 6 points) 40 0.1 H + 1 mf. a) (2 points) If V 8 (t) = u(t) in the circuit above, determine the final value of v 0 (t). Verify your answer using the Final-Value Theorem for full marks. lim v 0 (t) = t-+oo
Problem 9 Page 22 of 25 b) (2 points) If vs(t) = u(t), what does the output waveform v 0 (t) look like? (Circle one and explain briefly) i) Va(t) ii) Va(t) -----------t _... t iii) Va( t) iv) Va(t) \ ~ t c) (2 points) Assuming Vs is a steady-state ac source, what is the minimum impedance "seen" by Vs (labeled Zeq ori the circuit diagram)? At what frequency does this minimum impedance occur? Zeq,min = -------------------- Wmin = --------------------
Problem 10 Page 23 of 25 Problem 10 (2 points) Source N What is the ideal turns ratio, N~, of a step-down transformer required to maximize the power delivered to a 40 load from a source with an internal (Thevenin) resistance of 900 O?
Page 24 of 25 ECE212 Equation Sheet Wye-Delta Transformations Bode Plots of quadratic poles a 20 2 3 4 5 6 7 891 2 3 4 5 6 7 891 'iii' ~ (U ".a c OD OS l:: 10 0-10 C b -20-30 0.2 0.5 1.0 2.0 4.0 10 w,. (rad/s)(log scale) Magnetically-Coupled Networks 1 2 3 4 5 6 7 891 2 3 4 5 6 7 89 1 L di2 + M di1 Vz = 2 dt dt ell (U ~,j: :.c "' (U "' OS..c. Cl. 0-40 -80-120 <l> = -tan-1 2tw,. 1-(w1') 2-160 -200 0.2 0.5 1.0 w,- (rad/s)(log scale) 2.0 4.0 10 Second-order Equations General second-order differential equation: d 2 x + 2 dx + 2 - a- wx= dt2 dt o Characteristic Equation: s 2 + 2( w 0 s + w5 = 0 s 2 + 2as + w5 = 0 Roots: Possible natural responses: Overdamped: Critically damped: Underdamped: f(t). x(t) = K1es1t + K2es2t x(t) = K 1 e-at + K 2 te-at x(t) = e-at (K 1 cos wdt + K2sinwdt) wd = Jw5 - s Bandpass Transfer Function: Hsp(s) = K sz+z(wos+ w~ 3dB (half-power) bandwidth: BW_ = whi - a2 Ww = 2( w 0, where w5 = WHJWw
Table of Laplace Transforms Page 25 of 25 f(t) F(s) f(t) F(s) 1 s u(t) - cos kt s 1 t - sz t2 2! - s3 tn n! -- sn+l eat 1 -- s-a sin kt cosh kt sinh kt eat cos kt eat sin kt s2 + k2 k s2 + k2 s s2 - k2 k s2 - k2 s-a (s - a) 2 + k 2 k (s - a) 2 + k 2 Table of Properties of the Laplace Transform f(t) F(s) f(t) F(s) af(t) + pg(t) af(s) + PG(s) eat f (t) F(s - a) r(n)(t) snf(s) - sn-lf(o) -... - rcn-1)(0) f (t - a)u(t - a) e-asf(s) ltf('r)dr ltf(r)g(t - r)dr tnf(t) F(s) -- s F(s)G(s) (-1rFCn)(s) eat f (t) F(s - a) f (t - a)u(t - a) e-asf(s) f(t)u(t - a) e-as L{f (t + a)} f(t) = f(t + T) 1 JT 1 - e-st o e-st f (t) dt f(t)u(t - a) f(t) = f(t + T) limf(t) t---¼0 t---¼oo limf(t) e-as L{f (t + a)} 1 JT 1 - e-st o e-st f (t) dt Jim sf(s) 5---¼00 Jim sf(s) S---¼0 Time-domain and s-domain Representations of Circuit Elements + v(r) ~ ~ ~ C + V(s) v(o) - s 1 sc V(s) 1 - sc i(r) l(s) l(s) L l i(o) Li(O) sl i(o) s