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ECE 20200 Final Exam

December 15, 2012 Professor Clark Professor Furgason 8:30am MWF 9:30am MWF Section 0001 Section 0002

Name: __________________________________________ (Please print clearly)

PUID: _ _ _ _ _ _ _ _ _ _ Seat Number: _ _

INSTRUCTIONS

This is a closed book, closed notes exam. No communication devices can be on. The test consists of 20 multiple choice problems. Carefully mark your MC answers on the scantron form BEFORE THE END OF

THE EXAM PERIOD. When the exam ends, all writing is to stop. You will turn in both the scantron form and the test booklet. All students are expected to abide by the customary ethical standards of the

university, i.e., your answers must reflect only your own knowledge and reasoning ability.

Cheating will result in a zero on the exam and possibly an F in the course. Communicating with any of your classmates, in any language, by any means, for

any reason, at any time between the official start of the exam and the official end of the exam is grounds for immediate ejection from the exam site and loss of all credit for this test.

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1. For a frequency of 2 rad/sec, what is the phase of V at 2 sec?

sin exp2 2

V t t j t dt

(1) 0 (2) 45 (3) 90 (4) 180 (5) 45 (6) 90 (7) 60 (8) None of these

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2. What is the partial fraction expansion of

2

22

1sG ss

?

(1) 2

2 3 11 1

G ss s

(2) 2

3 11 1

G ss s

(3) 2

1 2 21 1

G ss s

(4) 2

2 1 31 1

G ss s

(5) 2

3 21 1

G ss s

(6) 2

3 4 11 1

G ss s

(7) 2

4 3 11 1

G ss s

(8) None of these

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3. What is inverse Laplace of 2

2

3 2 4s sF ss

?

(1) 3 2 4f t t u t

(2) 3 2 4f t t u t

(3) 4 5f t t t

(4) 12 2f t t u t

(5) 4 3 2f t u t t t

(6) 4 3 2f t u t tu t t

(7) 3 2 4f t t t u t (8) None of these

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4. What is the resonance frequency of the circuit shown below (in radians per second)? The parameters are: L = 8 H, C = ¼ F, and R = 2 .

(1) 12

(2) 14

(3) 12

(4) 23

(5) 32

(6) 23

(7) 32

(8) None of these

R L L R C C

Zin

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5. What is the steady state kinetic energy (hint: 212 Li , x yMi i ) stored this circuit (in units

of Joules)? Parameters: L1 = L2 = L3 = L4 = 2 H, R1 = 1 , R2 = R3 = 2 , Ma = Mb = 1 H, 3

1 1 2 tV t e V, and 42 2 6 cos(20 )tV t e t V.

(1) 1 (2) 2 (3) 3 (4) 4 (5) 5 (6) 6 (7) 7 (8) None of these

Ma

L1 L2 R2 L4 L3

V2(t) Mb

R3

V1(t)

R1

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6. What is the KVL equation for current loop i2 expressed in the s-domain notation?

(1) 1 3 2 3 1 2 2 2 1 0b aV sL i sM i R i sL i sM i (2) 1 3 2 3 1 2 2 2 1 0b aV sL i sM i R i sL i sM i (3) 1 3 2 3 1 2 2 2 1 0b aV sL i sM i R i sL i sM i (4) 1 3 2 3 1 2 2 2 1 0b aV sL i sM i R i sL i sM i (5) 1 3 2 3 1 2 2 2 1 0b aV sL i sM i R i sL i sM i (6) 1 3 2 3 1 2 2 2 1 0b aV sL i sM i R i sL i sM i (7) 1 3 2 3 1 2 2 2 1 0b aV sL i sM i R i sL i sM i (8) None of these

Ma

L1 L2 R2 L4 L3

V2 Mb

R3

V1

R1

i2 i3 i1

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7. Given the transmitter system below, determine the turn ratio a that is required for the antenna such that maximum power is transmitted. Parameters are: Rantenna = 2 , Zth = 200 .

(1) 1 (2) 2 (3) 4 (4) 5 (5) 10 (6) 20 (7) 100 (8) None of these

1:a

Rantenna

Signal source Transmitter

Zth

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8. For a particular circuit, its zeros and poles are plotted on the complex plane below. Determine which transfer function and stability condition best represents the plot.

(1) 210

3 4 3 4 2sH s

s j s j s

, stable

(2) 2

2172 6 25

sH ss s s

, unstable

(3) 2

282 6 25

sH ss s s

, stable

(4)

4 23 4 3 4 2

sH s

s j s j s

, unstable

(5)

6 23 4 3 4 2

sH s

s j s j s

, stable

(6) 2

2102 6 25

sH ss s s

, unstable

(7) 2

2102 6 25

sH ss s s

, stable

(8) None of these

X

O X

X

1

2

3

4

-4

-3

-2

-1 2 3 1 5 4 -2 -3 -1 -5 -4

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9. If 2

12 103 4

sF ss s

, then determine f in the time domain.

(1) 2/3 (2) 3 (3) 5/2 (4) 4 (5) 12 (6) 10/3 (7) 10 (8) None of these

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10. In the lab you attempt to determine the mutual inductance of an arbitrary transformer. You connect a current source, ammeter, and volt meter to the transformer as shown. From your experimental results plotted below, what is the mutual inductance M of this transformer?

(1) 1 (2) 4 (3) 1/4 (4) 2 (5) 1/2 (6) 3/4 (7) 4/3 (8) None of these

Current source

Ammeter

i1

Voltm

eter

V +

-

i [A]

t [sec] t [sec]

V [V] 12 8

-4

4

-8

3 2

-1

1

-2

2 4 8 6 10 2 4 8 6 10

-12 -3

M

i2

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11. A certain circuit has a Transfer Function, 2

out

in

V ( ) 3 2( )I ( ) (3 1)(3 2)

s s sH ss s s

.

The Impulse Response, h(t), of that circuit, in ohms, is :

(1) 2

3 31( ) 2 8 ( )9

t tt e e u t

(2) 2

3 31( ) 2 8 ( )3

t tt e e u t

(3) 2

3 31( ) 6 8 ( )9

t tt e e u t

(4) 2

3 31 6 8 ( )3

t te e u t

(5) 2

3 31 2 8 ( )9

t te e u t

(6) 2

3 31 3 ( ) 6 8 ( )9

t tt e e u t

(7) 2

3 31 ( ) 2 8 ( )3

t tt e e u t

(8) None of these

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12. The input impedance of the elements shown below is (1) 2 2 s

(2) 2

22

2s

s

(3) 2

222

( 1)s

s

(4) 2

22 6 2

1s s

s

(5) 2

24 3 1

1s s

s

(6) 2

24 2 2

1s s

s

(7) 2

24 2 1

2 1s s

s s

(8) None of these

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13. Solve the circuit shown below to obtain the s-domain capacitor voltage, VC1(s), which is valid for the time interval 0 ≤ t < 2. The initial conditions are switch open, VC1(0‾) = 4V, and VC2(0‾) = 2V.

(1) 10

2s s

(2)

2 201

ss s

(3) 2 10

2 1s

s s

(4)

4 102

ss s

(5) 4 20

2 1s

s s

(6)

8 202

ss s

(7) 8 20

2 1s

s s

(8) None of these

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14. Determine the Zero State Response part of the complete response, VC(s), of the circuit shown. The initial condition is VC(0‾) = 2V.

(1) 2 ( )te u t

(2) 8 ( )te u t (3) 22 ( )te u t (4) 4 ( )te u t (5) 5 ( ) 4 ( )tu t e u t (6) 10 ( ) 4 ( )tu t e u t (7) 10 ( ) 10 ( )tu t e u t (8) None of these

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15. The transfer function, out

in

V( )I

H s , for the circuit below is given by:

(1) 26 4

2 3 1s

s s

(2) 26 4

3 2s

s s

(3) 2 26 4

( 1)s

s

(4) 2

22 6 42 3 1

s ss s

(5) 2

22 6 22 3 1

s ss s

(6) 2

22 6 4

3 2s s

s s

(7) 2

22 4 2

3 2s s

s s

(8) None of these

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16. Suppose you have the 2nd order circuit shown below. You want to realize a 2nd order low pass Butterworth Filter with this circuit. The transfer function of this circuit is

H(s) = ( )( )

=

First with this circuit, you want to realize a 2nd order 3dB Normalized Low Pass Butterworth Filter. Where the transfer function of a 2nd order 3dB Normalized Low Pass Butterworth filter is given as

H (s) = √

Now suppose in your final design, you want the value of R to be √2 × 100 and value of 3dB down frequency to be 10k rad/ sec. What would be your final values of capacitor (in F) and Inductor (in H) ? (1) C = 10μF, L =10mH

(2) C = √2 μF, L = 10mH

(3) C = √2 μF, L = √2 mH

(4) C = 100 μF , L = 10 mH

(5) C = 100mF, L = 1mH

(6) C = 1 μF, L = 1mH

(7) C = 1 μF, L = 10mH

(8) None of these

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17. For the first order Butterworth filter shown below determine the frequency of the –3

dB point, the point where the gain falls to 70.7% of its maximum frequency value. The -3 dB frequency in radians/sec. is:

(1) 1/3 (2) 1/2 (3) 1 (4) 1.5 (5) 2 (6) 2.5 (7) 3 (8) None of these

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18. The Q of the bandpass filter shown below is: (1) 1/2 (2) 1 (3) 2 (4) 4 (5) 8 (6) 10 (7) 20 (8) None of these

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19. The resonant frequency, in rad/sec., of the circuit shown below is: (1) 10 (2) 1 (3) 2 (4) 3 (5) 4 (6) 10

(7) 110

(8) None of these

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20. Determine the h-parameter hie for the circuit shown below. (In the answers below || means parallel connection of components.) Vbe = hie Ib + hre Vce

Ic = hfe Ib + hoe Vce (1) x dr r r

(2) 1xr r

sC

(3) 1 1rsC sC

(4) 1 1xr r

sC sC

(5) 1 1d

mr

g sC

(6) 1 1x dr r r

sC sC

(7) 11 1m

x dgr r r

sC sC

(8) None of these

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Laplace Transforms and Properties (This page and next page)

Item Number f(t) L[f(t)] = F(s)

1 Kt K

2 Ku(t) or KKs

3 r(t) 1

s2

4 tnu(t) n!

sn1

5 e–atu(t) 1

s a

6 te–atu(t) 1

(s a)2

7 tne–atu(t) n!

(s a)n1

8 sin(t)u(t)

s2 2

9 cos(t)u(t) s

s2 2

10 e–atsin(t)u(t)

(s a)2 2

11 e–atcos(t)u(t) (s a)

(s a)2 2

12 tsin(t)u(t) 2s

(s2 2)2

13 tcos(t)u(t) s2 2

(s2 2)2

14 sin(t + )u(t) ssin() cos()

s2 2

15 cos(t + )u(t) scos() sin()

s2 2

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16 e–at

[sin(t) – tcost)]u(t) 23

[(s a )2 2 ]2

17 te–atsin(t)u(t) 2s a

[(s a)2 2 ]2

18 eat C1 cos(t) C2 C1a

sin(t)

u(t) C1s C2

s a 2 2

Property Transform Pair

Linearity L [a1f1(t) + a2f2(t)] = a1F1(s) + a2F2(s)

Time Shift L [f(t – T)u(t – T)] = e–sTF(s), T > 0

Multiplication by t L [tf(t)u(t)] = – dds

F(s)

Multiplication by tn L[tn f (t)] (1)n

dnF(s)dsn

Frequency Shift L [e–atf(t)] = F(s + a)

Time Differentiation Lddt

f (t)

= sF(s) – f(0–)

Second-Order Differentiation

L

d2 f (t)dt2

s2F(s) sf (0 ) f (1)(0 )

nth-Order Differentiation

Ld n f (t)

dtn

snF(s) sn1 f (0 ) sn2 f (1)(0 )

f (n1)(0 )

Time Integration

(i) L

t f (q)dq

F(s)s

0

f (q)dq

s

(ii) L

0t f (q)dq

F(s)s

Time/Frequency Scaling L [f(at)] = 1a

Fsa