Faculty of Applied Science and Engineering ECE212H1F ...

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University of Toronto Faculty of Applied Science and Engineering

ECE212H1F - Circuit Analysis

Final Examination December 16, 2017

9:30am - noon

Exam type: A Examiners: K. Phang, Z. Tate, and W. Wong

Guidelines:

• 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 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) = £{v5 (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

Problem 2 (10 points) -

L

+ Vs(t) C

of 25

⇒ 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

IHI [dB]

Sample Bode plot (based on a different transfer function)

I..H

of 25

-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 of 25

b) ( 4 points) This is a commonly used filter circuit in de-de converters, which use switches ( operating at a frequency w5 ) to reduce or increase a de supply voltage. If v5 (t) = ~ + cos(w5 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) W5 = 10000 rad/s

Vout(t) = ___________ _

Problem 2 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: W5 =l00 rad/s W 5 =l0000 rad/s

Problem 3 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 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 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


Load 1 30 kVA

p.f. = 0.6 lagging

Load 2 20 kVA

p.f. = 0.8 leading

of 25

A 120-V rms, 60-Hz source v8 (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 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

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+) = _________ _

of 25

Problem 5 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 of 25

c) (3 points) Use superposition to find vR(t) fort 2: 0. In this part, turn off the the e-50tu(t) source and only consider the response due to initial conditions. Determine VR(s) = .C{vR(t)}.

VR(s) = ________ _

Problem 5 of 25

d) (3 points) In this part, turn off all initial conditions and find the response VR(s) due only to the c 50tu(t) source.

VR(s) = ________ _

Problem 5 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 of 25


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 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 of 25

c) (1 point) Given iL(O+) = 1 A, find the time, t33%, required for the output Vout to drop to one-third of its initial value (i.e., V0 ut(taa%) = ½vout(O+)).

t33% = -----------

Problem 8 of 25


100 kn

1 nF 50 kn

In the above op amp circuit, v5 = 4 cos (104t) V. Find the average power PL delivered to the 50-kn resistor. Note: 1 nF = 1 x 10-9 F.

~=-----------

Problem 9 of 25


40 0.1 H

+

1 mF.

a) (2 points) If V8 (t) = u(t) in the circuit above, determine the final value of v0 (t). Verify your answer using the Final-Value Theorem for full marks.

lim v0 (t) = __________ _ t-+oo

Problem 9 of 25

b) (2 points) If vs(t) = u(t), what does the output waveform v0 (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 of 25


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?

of 25

ECE212 Equation Sheet

Wye-Delta Transformations Bode Plots of quadratic poles

2 3 4 5 6 7 891 2 3 4 5 6 7 891 a

20

'iii' 10 ~ (U

0 " .a ·c OD -10 OS l::

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

-160

-200 0.2 0.5 1.0

<l> = -tan-1 2tw,. 1-(w1')2

2.0 4.0

w,- (rad/s)(Log scale)

Second-order Equations

General second-order differential equation: d

2x + 2 dx + 2 - a- wx= dt2 dt o

f(t).

Characteristic Equation: s 2 + 2( w0 s + w5 = 0 s2 + 2as + w5 = 0

Roots: Possible natural responses:

• Overdamped: • Critically damped: • Underdamped:

x(t) = K1es1t + K2es2t x(t) = K1e-at + K2te-at x(t) = e-at (K1 cos wdt + K2sinwdt)

wd = Jw5 - a2 s

Bandpass Transfer Function: Hsp(s) = K sZ+z(wos+ w~

3dB (half-power) bandwidth: BW_ = wHI - Ww = 2( w0 , where w5 = WHJWw

10

Table of Laplace Transforms of 25

f(t) F(s) f(t) F(s)

u(t) 1 -s 1

t -

s cos kt

s2 + k2

sin kt k

s2 + k2 sz s

t2 2! -s3

tn n! --sn+l

cosh kt s2 - k2

sinh kt k

s2 - k2 s-a

eat cos kt (s - a)2 + k 2

eat 1 --s-a

eat sin kt 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 F(s) --

s

ltf(r)g(t - r)dr F(s)G(s)

tnf(t) (-1rFCn)(s)

eat f (t) F(s - a)

f(t)u(t - a) e-as L{f (t + a)}

1 JT f(t) = f(t + T) 1 - e-sT o e-st f (t) dt

limf(t) Jim sF(s) t---¼0 5---¼00

limf(t) Jim sF(s) t---¼OO S---¼0

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

Time-domain and s-domain Representations of Circuit Elements

+

v(r)

~ ~ ~

C

i(r)

L

i(O) l

+

V(s)

1 sC

V(s)

v(O) - s

l(s) l(s)

Li(O)

1 -sC

sL i(O)

s

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