Student ID:____________

2

Ph.D. Qualifying Exam

February 6, 2014

1:00-5:00pm

Electrical Engineering

Part II

Instructions:

This is a closed-book/closed-notes exam module, four hours in duration. Please answer all problems in two areas you have previously selected.

Write your name and student ID below.

Full Name: ____________________________________________

Student ID: ____________________________________________

Please write the last four digits of your student ID number on EVERY PAGE, but

DO NOT write yo ur name on any other pages of the exam.

Blank paper is provided if you need additional space to show your work. PLEASE

INDICATE THE PROBLEM NUMBER ON EACH ADDITIONAL PAGE YOU USE.

Good luck!

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2

Final grade for this section: __________ out of 40 points

Signal Processing

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3

Question 1, ElecEng 410

(a) Given the following frequency spectrum, determine theminimum sampling frequency Fs.

X(F)

-1000 1000 F

Draw the frequency spectrum after sampling.

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4

Given the following frequency spectrum determine the minimum sampling frequency Fs.

X(F)

-1200 -1000 -800 800 1000 1200 F

Draw the frequency spectrum after sampling.

Question 1, ElecEng 410 (Cont'd)

Student ID:____________

5

1. (b) Design an M=7 Blackman window FIR linear-phase digitalfilter with the desired frequency response:

1,ω( )

0,6

d

forH

for

π ω ≤= π

6

< ω ≤ π

Show the calculated values for the desired impulse response n( )dh

Note:

sin (ω )(n2(n) (nT )

( 1)2

h M( −1)2

c

d an

cd

M

h x

and

∞

=−∞

−1)−= M −)(nπ −

ω=

π

∑

Show the calculated values for Blackman window Blackmanw ( )n

Note: πn2 0.08cos −M 1

M −1B 4πnw (n) 0.42= 0.5cos−

+

Show the final values for the FIR filter coefficients h n( )

All values calculated and shown by hand or hand calculator. Show all work. No credit for results only.

Question 1, ElecEng 410 (Cont'd)

Student ID:____________

6

Question 2, ElecEng 410

2. (a) Given the following sequence

x(n) = [1 1 1 1 0 0 0 0]

Compute by hand the DFT showing all terms. No credit given without all work shown explicitly. Compute and sketch the Magnitude of the DFT. Compute and sketch the Phase of the DFT. Note:

1 1j− π2 kn N/

0 0X(k)

N Nkn

Nn n

− π2j / NN

x n( ) e x n( ) W

where=W e

− −

= =

= =∑ ∑

Student ID:____________

7

2. (b) Compute the 8 point radix-2 decimation in frequency FFTfor the given sequence. Show all work. No credit for just theanswer.

x(n) = [1 1 1 1 0 0 0 0]

Show the results of each stage of the FFT calculation. What is the main advantage of using the FFT over DFT ?

Question 2, ElecEng 410 (Cont'd)

Student ID:____________

8

Question 3, ElecEng 420

Let X be a random variable with density function

pX(x) =

A(1 − |x|), x ∈ [−1,+1]0, otherwise

(1)

where A > 0 and

Let random variable Y be related to X by

Y = g(X) =

−X, x < 02X, x ≥ 0

(2)

a.) Find the value of A and sketch pX(x).

b.) Find E[Y ].

c.) Find the PDF of Y , PY (y).

d.) Find and sketch the pdf of Y , pY (y).

Student ID:____________

9

Question 4, ElecEng 420

Student ID:____________

10

Final grade for this section: __________ out of 40 points

Power

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11

Question 1 Two balanced 3-phase, Y-connected loads are connected in parallel, one drawing 21 kW (3-phase) at 0.75 power factor lagging and the other drawing 12 kVA (3-phase) at 0.65 power factor leading, are supplied by a balanced, 3-phase, 480-volt (line-line) source. Determine:

1.1 The magnitude of the source line current;

1.2 The amount of capacitive VARs (3-phase) needed to drive the overall power factor to 1.0.

Student ID:____________

12

Question 2

A single-phase 50-kVA, 2400/240-volt, 60-Hz distribution transformer is used as a step-down transformer at the load end of a 2400-volt feeder whose series impedance is (1+j2.5) ohms. The equivalent series impedance of the transformer is (1+j2.0) ohms referred to the high-voltage (primary) side. The transformer is delivering rated load at 0.8 power factor leading and at rated secondary voltage. Neglecting the transformer exciting current, determine:

2.1 The voltage at the transformer primary terminals;

2.2 The voltage at the sending end of the feeder.

Student ID:____________

13

Question 3A three-phase PWM inverter has a 400V DC bus voltage and an output frequency that varies

from 25 to 200Hz. The RMS of the output line voltage is 196V and the carrier frequency is 20kHz. The load is a Y-connection with a series 20Ω resistance and 30mH inductance in each phase. Determine:

a) the range of the RMS value of the fundamental frequency component of load current asfrequency is varied

b) The effect of varying frequency on the THD of the load current and line to line voltage. Docalculations if required.

Bipolar switching

Unipolar Switching

Student ID:____________

14

Student ID:____________

15

Question 4

Explain in details using analytical method and stator and rotor magnetomotive forces (mmf) why a single-phase induction motor does not have starting torque.

Student ID:____________

16

Final grade for this section: __________ out of 40 points

Controls

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17

• Attach extra sheets as needed to provide space for your answers.

• Clearly identify each of your answers with the corresponding problem number and part, such

as 1.A, 2.B.ii, etc.

• Show your work, to show how you arrived at each answer.

• Several questions marked (Essay Question) are included. Answer these with one or

more complete sentences and complete and logical thoughts. Responses with incomplete

sentences and incompletely described thoughts will be heavily discounted.

Student ID:____________

18

Question 1, ElecEng 474, 574

y(t)r(t) +

-

e(t) u’(t)

+

+

d(t)

+ + v(t)

kp s + kis

73 s+20

u(t)Gp(s)=Gc(s)=

Figure 1: Block diagram for velocity servo. Signalr (t) is the reference input,d (t) is thedisturbance input andv(t) is the noise input.

A) Figure 1 presents a control loop block diagram with three inputs. Considering a control

system application to car cruise control, a typical system in a car that maintains steady car

speed while driving,

i ) What is represented byGp (s) ?

ii ) What is represented byGc (s) ?

iii ) What is the physical source of signalr (t) ?

iv ) What is the physical source of signalr (t) ?

v ) What is the physical source of signalr (t) ?

vi ) What signal(s) in the system are being regulated by the system ?

B) What is the system type ?

C) Using the valueskp =8, ki = 100

i ) Determine the closed-loop transfer functions

i) Try (s) =Y (s)R(s)

, ii) Try (s) =ER((ss)), iii) Tvy (s) =

Y (s)V (s)

ii ) Determine whether or not the closed-loop system is stable.

iii ) Determine the expected settling time and overshoot of the system.

Student ID:____________

19

Question 2, ElecEng 474, 574

y(t)r(t) +

-

e(t) u’(t)

+

+

d(t)

+ + v(t)

u(t)Kc Gc(s) Gp(s)

Figure 2: Block diagram for series compensation.

a)−12 −10 −8 −6 −4 −2 0 2 4−6

−4

−2

0

2

4

6

Pole−Zero Map

Real Axis (seconds−1)

Imag

inar

y A

xis

(sec

onds

−1 )

d)−12 −10 −8 −6 −4 −2 0 2 4−6

−4

−2

0

2

4

6

Pole−Zero Map

Real Axis (seconds−1)

Imag

inar

y A

xis

(sec

onds

−1 )

b)−12 −10 −8 −6 −4 −2 0 2 4−6

−4

−2

0

2

4

6

Pole−Zero Map

Real Axis (seconds−1)

Imag

inar

y A

xis

(sec

onds

−1 )

e)−12 −10 −8 −6 −4 −2 0 2 4−6

−4

−2

0

2

4

6

Pole−Zero Map

Real Axis (seconds−1)

Imag

inar

y A

xis

(sec

onds

−1 )

Figure 3: Root locus diagrams. A double circle indicates two co-located zeros, a triple ’x’indicates three co-located poles.

Student ID:____________

20

Question 2, ElecEng 474,574 (Cont’d.)

Figure 3 (continued): Root locus diagrams. A double circle indicates two co-located zeros, a triple

A) Figure 2 illustrates a basic block diagram for series compensation, with transfer functions

Gc (s) and Gp (s), and gain parameterKc. Figure 3, parts a) ... f) show pole-zero

constellations for a closed-loop system such as shown in figure 2.

For each of pole-zero constellations a) ... f) sketch the root-locus diagrams for each pole-zero

constellation.

B) (Essay Question) Interpretation of a root-locus plot. For each of your root locus plots above,

answer these two questions with one or more sentences.

i ) Describe system stability as gain parameterKc goes from 0 to a very large value

ii ) Describe the system speed as gain parameterKc goes from 0 to a very large value

(for this question, speed = the reciprocal of settling time)

Example: For the system of figure g) (not shown), the system is stable for small values of

gain, then becomes unstable, and then is stable for all large values of gain. OR g) The system

is stable for all values of gain.

For the system of figure g) (not shown) the system becomes faster asKc increases, until a

maximum speed is reached.

– Use extra sheets to answer this question –

c)−12 −10 −8 −6 −4 −2 0 2 4−6

−4

−2

0

2

4

6

Pole−Zero Map

Real Axis (seconds−1)

Imag

inar

y A

xis

(sec

onds

−1 )

f)−12 −10 −8 −6 −4 −2 0 2 4−6

−4

−2

0

2

4

6

Pole−Zero Map

Real Axis (seconds−1)

Imag

inar

y A

xis

(sec

onds

−1 )

c)−12 −10 −8 −6 −4 −2 0 2 4−6

−4

−2

0

2

4

6

Pole−Zero Map

Real Axis (seconds−1)

Imag

inar

y A

xis

(sec

onds

−1 )

f)−12 −10 −8 −6 −4 −2 0 2 4−6

−4

−2

0

2

4

6

Pole−Zero Map

Real Axis (seconds−1)

Imag

inar

y A

xis

(sec

onds

−1 )

’x’ indicates three co-located poles.

Student ID:____________

21

Question 3, ElecEng 474, 574

For the system of problem 1, figure 1, with kp = 8, k i = 100

A) Sketch the bode plot for the system

B) Determine the gain margin, phase margin and cross-over frequency

C) Describe how you would determine the stability of the closed-loop system.

Bode Plot, 3 Cycle

Mag

nitu

de, d

BP

hase

[deg

]

Frequency

Figure 4: Space for a Bode plot.

Student ID:____________

22

y(k)r(k) +

-

e(k) u’(k)

+

+

d(k)

z - 1

u(k)Gp(z)=Gc(z)=

4 z - 3 0.044z - 0.86

Figure 5: Discrete-time closed-loop system.

A) (Essay Question) In at least several sentences, describe how discrete-time control is

implemented and why it is important.

B) (Essay Question) In one or several sentences, describe an advantage of discrete-time control.

C) (Essay Question) In one or several sentences, describe a disadvantage of discrete-time

control.

D) True or False: With discrete-time control it is possible exactly reproduce the control action

of a continuous-time controller (True / False, circle one)

E) (Essay Question) Describe how you would determine these quantities for the discrete-time

system if figure 5

i ) Stability

ii ) Damping ratio

iii ) Settling time

F) (Essay Question) What pieces of equipment are required for a discrete-time controller that

are not required for a continuous-time controller.

G) (Essay Question) What mathematical tools are used to analyze a discrete-time controller,

and how are these different from the tools used to analyze a continuous-time controller.

H) The controller,Gc (z) produces control signalu(k) from error signale(k). Based on the

information in figure 5, describe howu(k) is produced.

Question 4, ElecEng 474, 574

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