107

Appendix Past Exams

The following are actual exams for a fourth-year course of 3 credit hours

in an Electrical Engineering Program

at a major university

There are 3 class tests and 1 final exam each semester. Class tests are of duration 1:30 hrs

Final exams are of duration 2:00 hrs

108

109

EEEEEE 330088,, TTeesstt ##11,, OOcctt.. 55 ,, 22001100 Roll:_____________Name:_____________ Show all your work. Write answer in designated spaces. Pencil preferred for your convenience. (4) Question: Find Fourier transform, V(ω) = ____________________ (4) Question: Find inverse Fourier transform v(t) = _______________ (4) Question: Find Laplace transform V(s) = _________________ (4) Question: Derive L[cos ωt] = ______________ (7) Question: Given e(t) = u(t) or a step function, Use Laplace to find i(t) = ____________ (b) Use Laplace to find vL(t) = _____________ (4) Question: For a ramp function, f(t) = t u(t), derive Laplace transform = ________

110

(7) Question: In an RLC circuit (as in class), L = 1, R = 2, C = 0.2 (a) When input is a step function, find i(t) = _______________ (b) When input is a delta function δ(t), find i(t) = _________________ (6) Question: In RLC series circuit, when L = 1, R = 6, C = 1/9, for delta function input, find i(t) = _______________ (5) Question: Express v(t) = 12 sin 8t + 5sin 8t in the form P sin (Q+R), v(t) = __________________

111

EEEEEE 330088,, TTeesstt ##22,, OOcctt.. 3311 ,, 22001100 Roll:_____________Name:_____________ Show all your work. Write answer in designated spaces. Pencil preferred for your convenience. (5) (Q) Question: Draw the characteristics for an op-amp, labeling the (a) linear (b) non-linear region (c) What does it mean by the statement, "the output impedance of an op-amp is zero"

(5) Question: Given feedback is positive, derive )s(R)s(C =

(4) Question: Find )s(R)s(C =

(4) Question: Find )s(R)s(C =

112

(4) Question: Find )s(F

X(s) =

(5) Question: Derive )s(E)s(V =

(6) Question: Write state equations for x = A = B = C = (6) Use loop current method to find

)s(E)s(I =

(6) Question: Derive

)s(F)s(X1 = -----------------

(5) Question: Give a delta function input to a pendulum of length l, and mass m, at rest, to find the period of oscillation T = (as done in class).

113

EEEEEE 330088,, TTeesstt ##33,, NNoovv.. 2233 ,, 22001100 Roll:_____________Name:_____________ Show all your work. Write answer in designated spaces. Pencil preferred for your convenience.

(5) Question: Given system with G(s) = 2s

1+

find output when input is

(a) δ(t), output c(t)= __________, (b) u(t), output c(t) = __________

(4) Question: Given G(s) = 1s

42 +

, H = 2s

1+

, redesign the system so as to

have unity feedback. (4) Question: Complete block diagram for function

)s(R)s(C =

5s2s3s2

23 +++

(4) Question: Complete block diagram for function

)s(R)s(C =

5s2s3s7s3s4

23

2

+++++

114

(4) Question: x' = Ax + Bu and y = Cx + Du, Eliminate x and find an expression in terms of A, B, C, D, u Y(s) =

(5) Question: Given )s(R)s(C =

5s2s3s2

23 +++, find A = and B =

(as done in class) (8) Question: Given x = and i = y = Cx + Du Find A = B = Find C = D = (6) Question: (4) Question: Complete Routh's criterion table to find the number of unstable poles N (a) 2s5 + s4 + 3s3 + s2 + 6s + 2 (N = ____) (b) 2s5 + s4 + 3s3 + 4s2 + 7s + 3 (N = _____)

115

(7) Question: Given state equation, x = Ax + Bu and y = Cx + Du. Find eigenvalues = ___________ A = B = C = Find [sI - A]-1 = (5) Question: For above system, write controllability and observability matrices. Is system controllable? yes / no (circle one) Is system observable? yes/ no (circle one)

116

EEEEEE 330088,, FFiinnaall EExxaamm,, DDeecc.. 77?? ,, 22001100 Roll:_____________Name:_____________ Show all your work. Write answer in designated spaces. Pencil preferred for your convenience. Expand: MIMO ________________________________________ ___ Q. The Butterfly Effect refers directly to ___ Q. Chaos is seldom present when there is ______ in a system ___ Q. Such systems are very complex, and require much training. (a) slight non-linearity (b) neural network(c) sensitivity to initial conditions ___ Q. A PID controller generally adds ___ poles and ___ zeroes to a system (a) 1, 2 (b) 1, 1 (c) 0, 2 (d) 2, 1 (e) 2, 2 ___ Q. In a unity feedback system, addition of a zero to G(s) pulls the root locus ___ Q. In a unity feedback system, addition of a pole to G(s) pulls the root locus (a) to the right (b) to the left (c) upwards (d) downwards ___ Q. The root locus is (a) always symmetrical about the x axis (b) always symmetrical around the y axis (c) sometimes symmetrical about the x axis (d) sometimes symmetrical around the y axis (6) Given displacement (distance) of a body x(t). Write and draw the velocity v(t) = ______________ Write and draw the acceleration a(t) = ________________

117

(6) Given e(t) = u(t) + 2u(t-3). Draw e(t). Write and draw de/dt = ____________________ Draw ∫e(t)dt (4) Question: Given f(t) = 5δ(t) + u(t) + 2u(t-1) + 4u(t-3), write L[f(t)] =

(4) Question: Given, F(s) = 1 + e-s + s

e s−

Find f(t) = ______________

(4) Given input e(t) = e-2t u(t) to a system G(s) = 3s2s

++ ,

find output v(t) = ____________

(5) Given input e(t) = e-2t u(t), to a system G(s) = 3s

1+

, find output

v(t) = _____________ (6) Question: Find

)s(T)s(1θ =

(should not show θ2) (6) Question: Complete Routh's criterion table to find the number of unstable poles N

118

(a) 2s5 + s4 + 5s3 + 2s2 + 3s + 1 (N = _) (b) 2s5 + s4 + 4s3 + 3s2 + s + 1 (N = _) (8) Question: Given x = and i = y = Cx + Du Find A = B = Find C = D =

(4) Question: In a unity feedback system, G(s) = )3s)(1s(

)2s(K++

+ .

Draw root locus, clearly showing the poles and zeroes.

(6) Question: In a unity feedback system, given G(s) = )3s)(2s(s

K++

Clearly draw root locus, labeling all points. Use Routh's criterion to find K = __________, above which system becomes unstable. At this marginal stability, what is the value of s = __________

(6) Question: For unity feedback system, given G(s) = )5s)(1s(

K+−

, find

K = ___________ above which system becomes stable. Clearly draw root locus, showing all points. When system is critically damped, what is K = _________

119

(5) Question: For unity feedback system, given G(s) = s12s9s2

K23 ++

,

using dK/ds = 0, Determine the possible breakaway points in the system, s1 = __, s2 =__

(4) Question: In a PID controller, K1 + s

K2 + s, the zeroes are at - 3, and -

5. Find values of K1 = __ and K2 = __ (4) Question: Complete block diagram for function

)s(R)s(C =

7s2s4s3s5s2

23

2

+++++

(6) Question: Find

)s(E)s(V =

120

XXXXXX FFCCKK,, TTeesstt ##11,, MMaarrcchh 11 ,, 22001111 Roll:_____________Name:_____________ Show all your work. Write answer in designated spaces. Pencil preferred for your convenience. (5) Question: Express f(t) = ______________________in terms of u(t) and ramp function, and then find L[f(t)] = ____________________ Question: Differentiate the above function, f’(t) = ________________ Draw: Find L[f’(t)] = _______________ (6) Question: In a circuit,

)s(E)s(V =

4ss+

Given input e(t) = δ(t), Find and

draw v(t) = ___________ (5) Question: Given mechanical system, with K = 1, M = 1, B = 2, and f(t) = δ(t), Find and draw x(t) = _______________

121

(5) Question: Find inverse Laplace transform of F(s) = 2s3s

3s2 ++

+ , f(t) =

_____________ (5) Question. Derive L[sin ωt] = ----------------- (5) Question: Find L[t sin(ωt)] =

(5) Question: Given, )s(E)s(I =

3s4ss

2 ++, find step response, i(t) =

________

(5) Question: Given, )s(E)s(I =

13s4ss

2 ++, find response to e(t) = δ(t), i(t) =

(5) Question: Given, )s(E)s(I =

3s4ss

2 ++, and e(t) = ramp function,

find i(t) = ________________

(5) Question: Find )s(E)s(VC = -------------------------- in terms of L, C, R

Also, )(E)(VC

ωω = -----------------------------, in terms of L, C, R

122

EEEEEE 330088,, TTeesstt ##22,, AApprriill 55 ,, 22001111 Roll:_____________Name:_____________ Show all your work. Write answer in designated spaces. Pencil preferred for your convenience. ___ Q. A circuit with three loops, 2 current sources and one inductor is a ____ order circuit (a) zeroth (b) first (c)second (d) third (e) fourth (f) fifth ___ Q. Characteristic equation: s2 + 4s - 5: Roots are s1 = _____________, s2 = ________________ ___ Q. Characteristic equation: s2 - 4s + 4 Roots are s1 = _____________, s2 = ________________ ___ Q. Characteristic equation: s2 + 4s + 13 Roots are s1 = _____________, s2 = ________________ ___ Q. Characteristic equation: s2 - 4s + 13 Roots are s1 = _____________, s2 = ________________ ___ Q. Characteristic equation: s2 + 4s + 5 Roots are s1 = _____________, s2 = ________________ (a) underdamped (b) overdamped (c) critically damped (d) unstable with oscillations (e) unstable without oscillations (4) Question: Find IL(s)/E(s) =

(5) Question: Write state equation for [vC, iL] A = B = (4) Question: For above, If the output is iR, find C = D =

123

(6) Question: Use loop-current method to find )s(E)s(I2 = ------------------------

--

(6) Question: Use node-voltage to write state equations, with [vC, iL]. C = D = (4) Question: If output is iC, find C = D = (5) Question: In an e(t) – R – C series circuit, R = C = 1. Given e(t) = δ(t), Find and draw vR(t) = ___________

(5) Question: In a unity feedback circuit, with G = 8s

12 +

. Given r(t) =

δ(t), find and draw c(t) = __________ Roots are ___________This system is stable / unstable / marginally stable (circle one) (5) Question; Given G(s) =

2s1+

, and H(s) = s1 , find

)s(R)s(C = ---------------

LC

e(t)+

-vc

R+

_

iL iC

124

The system is (a) overdamped (b) underdamped (c) critically damped (d) unstable with oscillations (e) unstable without osc (circle one)

125

EEEEEE 330088,, TTeesstt ##22,,((IIII)) AApprriill 1100 ,, 22001111 Roll:_____________Name:_____________ Show all your work. Write answer in designated spaces. Pencil preferred for your convenience.

(5) Question: Find )s(E)s(V =

(5) Question: Find )s(F)s(X =

(5) Question:

)s(T)s(1θ

-

+R

C

e(t)v(t)

L

M

KM

x(t)

f(t)

B

126

EEEEEE 330088,, TTeesstt ##33,,((IIII)) AApprriill 1199 ,, 22001111 Roll:_____________Name:_____________ Show all your work. Write answer in designated spaces. Pencil preferred for your convenience ___ Q. Upon adding a pole to G(S) a unity feedback system, it will become ___ Q. Upon adding a zero (to the numerator) to G(s) in a unity feedback system, it will become (a) more stable (b) less stable (c) remain the same (6) Question: Complete Routh's criterion table to find the number of unstable poles N (a) 2s5 + s4 + 5s3 + 2s2 + 6s + 3 (N = _) (b) s5 + s4 + 4s3 + 3s2 + 2s + 1 (N =___ )

(4) Question: In a unity feedback system, G(s) = )4s)(1s(

)3s(K++

+ .

Draw root locus, clearly showing the poles and zeroes.

(4) Question: In a unity feedback system, G(s) = )3s(s)2s(K

++ .

Draw root locus, clearly showing the poles and zeroes. (4) Question: In a unity feedback system, G(s) =

)3s)(2s()1s(K++

+ .

Draw root locus, clearly showing the poles and zeroes. Find the breakaway points σ = _________, ________ (5) Question: Calculate C(s)/R(s) =

127

EEEEEE 330088,, FFiinnaall EExxaamm,, MMaayy 1100,, 22001111 Roll:_____________Name:_____________ Show all your work. Write answer in designated spaces. Pencil preferred for your convenience ___ Q. Characteristic equation: s2 + 3s - 4: Roots are s1 = _____________, s2 = ________________ ___ Q. Characteristic equation: s2 - 6s + 9 Roots are s1 = _____________, s2 = ________________ ___ Q. Characteristic equation: s2 + 6s + 10 Roots are s1 = _____________, s2 = ________________ ___ Q. Characteristic equation: s2 + 6s + 13 = 0 Roots are s1 = _____________, s2 = ________________ (5) Question: Express f(t) = _________________ in terms of u(t) and ramp function, and then find L[f(t)] = _____ (4) Question: Differentiate the above function, f’(t) = ________________ Draw: Find L[f’(t)] = _______________ (5) Question: Draw and label underdamped and overdamped cases for i(t), vL(t), and vC(t)

128

(5) Question: Draw underdamped and overdamped cases, i(t) = u(t), step function. (4) For state equation for above circuit, with ⎥

⎦

⎤⎢⎣

⎡

L

C

iv , for iC, find [C] =

[D] = (6) Question Write state equation with ⎥

⎦

⎤⎢⎣

⎡

L

C

iv , [A] = [B] =

(4) For state equation for above circuit, with ⎥

⎦

⎤⎢⎣

⎡

L

C

iv , for iC, find [C] =

[D] = (6) Question: Find

)s(E)s(V = --------------

(5) Question: Find L[t cos(ωt)] =

129

(5) Question: In an e(t) – R – L series circuit, R = L = 1. Given e(t) = δ(t), Find and draw vL(t) = ___________ (5) Question: Express v(t) = 5cos 3t + 12sin 3t in the form P sin (Q+R), v(t) = __________________ (6) Question: Complete Routh's criterion table to find the number of unstable poles N (a) s5 + s4 + 5s3 + 2s2 + 3s + 1 (N = ____) (b) s5 + s4 + 4s3 + 3s2 + s + 1 (N = _____)

(6) Question: Use loop current method to find )s(E)s(V = ----------------

(6) Question: In a unity feedback system, G(s) = )4s)(3s(s

K++

.

Draw root locus, clearly showing the poles. Find the breakaway point σ = _________ Find K = ________, and s=_____________, when system just becomes unstable

130

(4) Question: In a unity feedback system, G(s) = )1s(s

)3s)(2s(K+

++ .

Draw root locus, clearly showing the poles and zeroes. Find the breakaway points σ = _________, ________ (5) Question: Complete block diagram for function

)s(R)s(C =

5ss3s4s6s2

23

2

+++++

(5) Question: In a unity feedback system, G(s) = )3s(s)4s(K

++ .

Draw root locus, clearly showing the poles and zeroes. Find the breakaway points σ = _________, ________

(5) Question: In a unity feedback system, G(s) = )1s(s)2s(K

−+ .

Draw root locus, clearly showing the poles and zeroes. Find K = __________, and s = ____________when system is just stable.

(4) Question: In a PID controller, K1 + s3 + K2s, the zeroes are at - 3, and -

4. Find values of K1 = __ and K2 = __

(6) Question: Find )s(T)s(θ = -----------------

Given delta function input torque, θ(t)= __________________