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University of Toronto Faculty of Applied Science and Engineering

ECE212H1F - Circuit Analysis

Final Examination December 17, 2018

6pm - 8:30pm

Exam type: A Examiners: A. Prodic, 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 Quercus.

Problem Score

1 /15 2 /10- 3 /15 4 /7 5 /9 6 /18

Total /74

V

Problem 1 Page 2 of 20

Problem 1 (15 points)

The above circuit is being used to implement the following transfer function, with Vi, (s) = {v(t)} and V0t(s) =

2 +3000s+ 108

a) (2 points) Should v0 (t) be defined across the resistor R, the inductor L, or the capacitor C?

Circle one R L C

Problem 1 Page 3 of 20

(4 points) If the capacitor value is 5 1iF, find R and L to implement the above transfer function. Assume v0t (t) is taken across the element identified in part (a).

L=

(1 point) Does the transfer function correspond to a low-pass, band-pass, or high-pass filter?

Circle one Low-pass Band-pass High-pass

Problem 1 Page 4 of 20

d) (8 points) On the graphs provided below, sketch the Bode plot (straight-line approximations of magnitude and phase) for H(s) = , which is repeated here:

H(s) - V0 (s)

— V," ( S ) - 2 +3000s+ 108

At the top of the next page, there is a sample Bode plot (corresponding to a different transfer function) that illustrates how the magnitude and phase plots should be labeled.

II [dB]

11111111 I 11111111 I 11111111 I 11111111 I 11111111 I 11111111 IiIIIIII)L)

W

Problem 1 Page 5 of 20

Sample Bode plot (based on a different transfer function)

II [dB]

lrad/s lOrad/s 100rod/s c dB il I I 111111 I I I I 1111111 I I II III

4O dB

LII

1 rad/ 1 rad/ 100 rod/ 00 I I44._ IIIIIIIIIIJ

I I 1111:11

-900

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

Problem 2 (10 points)

1Q 200mH 10:1 '1011t(O

v(1) E111c2 vout (t)

Using an oscilloscope, it is observed that v0 (t) = 10cos (20t) V.

a) (3 points) Find i(t).

Page 6 of 20

Problem 2

1Q 200 m jim (1;) 0:1

Vi", (t)

EQ vout(t)

c

(Circuit repeated from previous page)

b) (3 points) Find v(t).

Page 7 of 20

Problem 2 Page 8 of 20

(2 points) Find the average and reactive power supplied by the voltage source v(t).

Average power =

Reactive power =

(2 points) A capacitor Cpf c is added in series with the voltage source (see diagram below) to reduce the reactive power supplied by the voltage source to zero. Find the capacitor value.

u n (t) 1 Q 200 mH

1 i0 (t)

C r —

Cp f c =

Problem 3 Page 9 of 20

Problem 3 (15 points)

i(t) 2H

120e_3t?j (t) 3H

1Q

a) (4 points) The initial condition for the circuit is i(0) = 1 A. Draw the s-domain circuit with all initial conditions modeled as voltage sources.

Draw the s-domain circuit in this box

Problem 3 Page 10 of 20

(4 points) Write down the mesh equation for the circuit. DO NOT SIMPLIFY OR SOLVE FOR I(s) YET! For full marks, you must clearly label each inductance term as being either "self' or "mutual" underneath the term.

Mesh equation

(2 points) Solve for I(s). DO NOT PERFORM THE INVERSE LAPLACE TRANSFORM YET!

I(s) =

Problem 3 Page ii of 20

d) (5 points) Do partial fraction expansion and apply the inverse Laplace transform to find i(t) for t> 0.

Problem 4

Problem 4 (7 points)

Load 1 50 kVA

P. f. = 0.6 lagging

Page 12 of 20

Load 2 30 kVA

i).f. = 0.8 leading

A 120-V iins, 60-Hz source 'v(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 13 of 20

b) (3 points) Calculate the value of the capacitance connected in parallel that will raise the power factor to unity.

c=

Problem 5 Page 14 of 20

Problem 5 (9 points)

vout

a) (3 points) Find 'A as a function of the input Vi,, and appropriate resistance value(s).

Problem 5 Page 15 of 20

b) (6 points) Find V0 = f(V, R1, R2, R3, R4 ).

VO?Jt

Problem 6 Page 16 of 20

Problem 6 (18 points)

1=0 3A (t)

iov() 5 A ( t 10 .? 20l 100 yF

a) (4 points) Assuming the switch was iii the oil position for a long time, find v(0).

Problem 6 Page 17 of 20

h) (8 points) Find the Thévenin equivalent at the capacitor terminals for I > 0.

Vth=

Rt/L =

Problem 6 Page 18 of 20

c) (6 points) Using the Thévenin equivalent at the capacitor terminals and time-domain analysis, find and plot v(t) for t > 0.

Use the following graph to plot v(t):

Equation Sheet Page 19 of 20

ECE2I2 Equation Sheet

Bode Plots of quadratic poles

1 2 3 4 5 6 7 891 2 3 4 5 6 7 891 I 1111111 1 11111111 ,

20 0 20Iog1oI[H 2(In) ± (/aiT)2 J_11

Wye-Delta Transformations

c

R,

/,

_ R1 R2

_ R1 - RaRb+RbRc+RaRc

Ra - Rb R1 +R 2 +R3

- 10

c 2! —10

6 —20

—30

Magnetically-Coupled Networks

v1 =L1 +M- v2 =L2 +M

M•

V1 L, D Ev2

02 0.5 to 2.0 40 10

air (rad/s))Log scale)

1 2 3 4 567891 2 3 4567891 I 111111

0- = 0.1 0.2 0.4

_::

- -tan-1

I0.8

—120-

—160

—200- I I I 0.2 0.5 1.0 20 4.0 10

)T )rad/s)(Log scale)

Second-order Equations

General second-order differential equation: dx --+

dx 2 2a—+w0x= f(t)

Characteristic Equation: s2 + 2<w0s + w = 0 s2+ 2as +w=0

Roots: s12 = —a ± Ja2 -

Possible natural responses: Overdamped: x(t) Kie'l' + K2e t

Critically damped: x(t) = Kie_rt + K2te 1

Underdamped: x(t) = e_at (K1 cos (odt + K2sinwdt)

- a2

Bandpass Transfer Function: HBP (s) = K 2 s 2 ±2w 0s+ w0

MB (half-power) bandwidth: BW = (()HI - ()LO = 2w0, where oi = (t)HI(J)LO

Sc

Equation Sheet Page 20 of 20

Table of Laplace Transforms

f(t) F(s)

u(t) 1- S

1 t -

s2

2! -

s3

t n!

sn+1

eat s — a

f(t) F(s)

cos kt S

s 2 +k 2 k

sin kt s2 + k 2

cosh kt S

s2 - J2

k sinh kt

s2 -

eat cos kt s — a

(s - a)2 + k 2

e t sin kt k

(s - a)2 + k 2

Table of Properties of the Laplace Transform

f(t) F(s) af(t) + f3g(t) aF(s) + 13G(s)

f'(t) sF(s) - s'f(0) - - f'(0)

ff (T)d-E

F (s)

f t f(T)9(t -

tf(t) (-1)'F()(s)

eat f(t) F(s - a)

f(t - a)u(t - a) e_F(s)

f(t)u(t - a) e sLff(t + a))

f(t) = f(t + T) Te_stf(t) dt

1 _ e_sTf 0

f(t) F(s) eat f(t) F(s — a)

f(t - a)u(t - a) e'F(s)

f(t)u(t - a) e'Lff(t + a))

f(t) = f(t + T) 1 _—sT fT e —sIf (t) dt

0

limf(t) lim sF(s) t-.o

Iirnf(t) lim sF(s)

Time-domain and s-domain Representations of Circuit Elements