DEPARTMENT OF ELECTRICAL & COMPUTER ENGINEERING OLD DOMINION UNIVERSITY

Ph.D. DIAGNOSTIC EXAMINATION Spring 2008

ODU HONOR PLEDGE

I pledge to support the Honor system of Old Dominion University. I will refrain from any form of academic dishonesty or deception, such as cheating or plagiarism. I am aware that as a member of the academic community, it is my responsibility to turn in all suspected violators of the Honor Code. I will report to a hearing if summoned. Student Signature: ___________________________________________________ Student Name (BLOCK CAPITALS): ___________________________________ UIN Number: _________________________________________________ Please turn in this examination paper, with the pledge above signed with your answer books. 1. This examination contains 24 problems from the following six areas:

A. MATH A1 A2 A3 A4 B. CIRCUITS & ELECTRONICS B1 B2 B3 C. DSP/CONTROLS/COMMUNICATION C1 C2 C3 C4 D. EMAG/QUANTUM ELEC./LASERS D1 D2 D3 D4 OPCIAL COMMUNICATIONS E. SOLID STATE/PHYS. ELEC./ E1 E2 E3 E4 PLASMA ELECTRONICS F. COMPUTER SYSTEMS F1 F2 F3 F4 F5

2. You must answer Eight questions (no more than three from MATH group). 3. Answer in the blue books provided. Use a separate book for each question. Put

the title and number of question on front of each book (ex. MATH A-1) 4. Return all the 24 problems. 5. You will be graded on EIGHT questions only. 6. The examination is “closed-book”; only writing material and a scientific

calculator are allowed. No formula sheet is allowed. Formulas are included where needed. No material shall be shared without prior permission of the proctor(s).

7. You have four hours to complete this examination.

SECTION A1 – MATH

Complex Variables and Differential Equations

SECTION A2 – MATH

Vector Calculus Evaluate the path integral: ∫ dt [t1/2 ax + 2t ay – 8t3 az ] along a straight line from the point (0,0,0) to the point (1, 2, -8).

SECTION A3 – MATH

Linear Algebra Let njiijaA ≤≤= ,1][ be an nn × matrix with real elements, and let )(det A denote the determinant of A .

1. Write the expressions of )det(2A , )(det A− , and )det( 2A in terms of )(det A . 2. What is the minor ijM and the cofactor ijA of matrix A ? 3. Write the expression of )(det A in terms of cofactors ijA . 4. Use Cramer’s rule to solve the system of linear equations:

032014

=+=++=−+

zxzyxzyx

SECTION A4 – MATH

Probability Two random variables have the joint probability density function given by:

f(X,Y) = A e–(3x+4y) for x > 0 , y > 0 ; and f(X,Y) = 0 for x < 0 , y < 0 . Find the expected value of XY.

SECTION B1 – CIRCUITS AND ELECTRONICS

Laplace Application to Circuit Analysis There is no energy stored in the circuit shown at the time the current source turns on. Given that 100 ( )gi u t A=

a. Draw the Laplace Transformed equivalent circuit in s-domain. b. Find the s-domain expression for the current Io(s) c. Use the s-domain expression derived in (b) and the initial and final value

theorems to find io(0+) and io(∞). d. Find the time domain expression for io(t).

25 Ω

5 Ω20 mF

+−

ig

20 ix

ix

25 H

io

SECTION B2 – CIRCUITS AND ELECTRONICS

Sinusoidal Steady State Response

Find the average power dissipated in the 20 Ω resistor in the following circuit. The sinusoidal source current is 15 cos(10,000 )gi t A= . Use phasor analysis.

ig ix+−10 ix

2.5 µF

1 mH

20 Ω

SECTION B3 – CIRCUITS AND ELECTRONICS Electronic Circuit The NMOS transistors in the circuit shown below have Vt = 1V, μnCox = 120 μA/V2, λ=0, and L1 = L2 = 1μm. Find the required values of gate width for each of Q1and Q2, and the values of R, to obtain the voltage and current values indicated.

SECTION C1 – DSP/CONTROLS/COMMUNICATIONS DIGITAL SIGNAL PROCESSING

Q. A causal LTI system is characterized by the following difference equation:

y[n] -127 y[n-1] +

121 y[n-2] = x[n]

(a). Determine the system function H(z) for the system (2 points) (b). Determine the ROC (1 point). (c). Determine the impulse response h[n] for the LTI system (3 points). (d). If x[n] = (1/2)nu[n], what is y[n] (4 points)?

SECTION C2 – DSP/CONTROLS/COMMUNICATIONS CONTROLS

CONTROL SYSTEMS The block diagram in Figure 1 is the simplified roll control system for a test aircraft.

Figure 1. Block diagram of an aircraft roll control system.

a) For what amplifier gain values is the closed-loop system stable? b) Sketch the root locus. There is no need to determine the break-away or break-in

points, if any. c) Suppose that the last control engineer set the amplifier gain to be K=2.5. For this

value of the amplifier, the Bode plots of the loop gain are given in Figures 2-3. a. What is the transfer function of the closed loop system? Where are the

closed-loop poles and zeros? Hint: One of the closed-loop poles is at s=-5.87.

b. Determine approximate values for the settling time and percent overshoot. c. What is the steady-state error to step changes in the desired roll angle? d. Determine approximately the gain and phase margins. e. Qualitatively explain if this is a good design. If you were a passenger in an

aircraft with this roll control system, what would you notice? d) Determine a new amplifier gain to give a 60° phase margin. Explain your

procedure and compare this design to the one with K=2.5.

-100

-80

-60

-40

-20

0

20

40

60

Mag

nitu

de (d

B)

10-2 10-1 100 101 102-270

-225

-180

-135

-120

-90

Phas

e (d

eg)

Bode Diagram

Frequency (rad/sec)

Figure 2. Bode plots of the loop gain when K=2.5.

-5

-4

-3

-2

-1

0

1

2

3

4

5

Mag

nitu

de (d

B)

100 101-185

-180

-175

-170

Phas

e (d

eg)

Bode Diagram

Frequency (rad/sec)

Figure 3. Zoomed-in Bode plots of the loop gain when K=2.5.

REVIEW For a prototype second order open-loop transfer function G(s) = ωn2/(s2 + 2ςωns + ωn2) the following unit step response relations are useful: • percent overshoot = 100 exp(-ςπ / sqrt(1 - ς2) ) • 2% settling time ≈4 / (ςωn) Suppose that the loop gain of the closed-loop system can be written as ( )KG s with

( ) 1

1

( - ),

( - )

m

ii

G n

jj

s zG s K

s p

=

=

=Π

Π

where K is the gain of the controller that needs to be determined,

G(s) represents the loop gain when K=1, and the loop gain has m zeros at zi and n poles at pj. The magnitude condition of root locus states that

n

jj=1

m

ii=1

|s- |p = , whenever a closed-loop pole

|s-z |

Π

ΠGK s

K

.

Laplace’s TheoremsLet F(s) be the Laplace transform of f(t) .

Initial Value Theorem Now, if F(s) be a strictly proper rational transfer

function (degree denominator > degree numerator), then

Final Value Theorem If all the poles of sF(s) have negative real parts, then

( ) ( )0 lim .s

f sF s+→∞

=

( ) ( )0

lim lim .t s

f t sF s→∞ →

=

SECTION C3 – DSP/CONTROLS/COMMUNICATIONS

Communications Consider the simple modulation system shown below. The multiplier, together with the squarewave input signal p(t) having amplitude zero or one, represents a sampler. In this system, the information signal is f(t) = cos(t). The squareware p(t) has radian frequency 8. The bandpass filter has center (radian) frequency 24, bandwidth 4 (radians per second), delay time zero, and gain 1. Answer the following questions concerning this

system. 1. Determine and accurately sketch the signal x(t). 2. Determine and accurately sketch the Fourier transform of the signal y(t). 3. Determine and accurately sketch the signal y(t). 4. Identify the modulation form of the signal y(t). 5. Describe a system which will reconstruct f(t) from y(t).

Multiplier

Ideal Bandpass Filter

f(t)

p(t) p(t)

t

x(t) y(t)

SECTION C4 – DSP/CONTROLS/COMMUNICATIONS

Communication Signals

SECTION D1 – EMAG/QUANTUM ELECTRONICS/ LASERS/OPTICAL COMMUNICATIONS Electromagnetics

1. An electromagnetic wave whose frequency is 100 MHz, traveling in air along the z axis, encounters a wall at z = 0. The amplitude of the incident wave's E field is 1Volt/meter. The wall is in the x-y plane. The wall can be taken to have a reflection coefficient of 0.6R = − . (a) calculate the distances, on the negative z axis, at which a detector would read maximum values (b) What is the amplitude of the total magnetic field intensity, in Amps/m, H at z =- 2 meter?

SECTION D2 – EMAG/QUANTUM ELECTRONICS/ LASERS/OPTICAL COMMUNICATIONS Quantum Electronics and Lasers

A gas whose molecular energy level diagram is given above is to be used to make a laser. The laser transition is expected to be between levels 2 and 1. It is known that there are three paths by which spontaneous emission can occur in this molecule: from levels 3 to 2, A32; from levels 2 to 1, A21; and, from levels 3 to 1, A31. (a) Write the three relevant equations that would describe how the populations of the three levels changes. Assume that the total number of molecules, NT, is fixed. (b) Use these equations to find the densities of levels 3 and 2 as a function of time, assuming the pump is steady. (c) Assuming that, eventually at time " "t →∞ , a steady state equilibrium is achieved what is the population difference N2- N1? (d) Discuss the conditions needed to produce lasing between levels 2 and 1.

N1, E1

N3, E3

N2, E2

Pump R3

SECTION D3 – EMAG/QUANTUM ELECTRONICS/ LASERS/OPTICAL COMMUNICATIONS

Discuss, carefully, each of the following concepts: (a)Radiation Pressure (b) intrinsic impedance of a material (c) characteristic impedance of a transmission line

SECTION D4 – EMAG/QUANTUM ELECTRONICS/ LASERS/OPTICAL COMMUNICATIONS

The condition for single mode operation of a cylindrical fiber is V ≤ 2.4. Find the maximum core radius for single mode operation at λ = 1320 nm of a step-index fiber with n1 = 1.48 and n2 = 1.478. What is the angle of acceptance of this fiber?

V: V-parameter = 2πa(NA)/λ NA: Numerical Aperture λ: Wavelength

SECTION E1-Solid State Electronics

Solid State Electronics

a) Draw the I-V characteristics of a p-n junction diode.

b) Briefly discuss the principle of a device operating in each quadrant.

SECTION E2 – PHYSICAL ELECTRONICS

Physical Electronics

a) For a free electron with a velocity of 107 cm/s, what is its de Broglie wavelength?

b) What is the voltage required to produce the above velocity for the free electron?

SECTION E3-Plasma Electronics Plasma Electronics (10 points) A diffusion-controlled discharge in Ar is modeled as an infinite slab separated by length L. The discharge is left on for a long time; that is, we assume the plasma density profile is cosine between the two boundaries of the slab. If L = 2 cm, the plasma density at the center of the slab is 1 x 1010 cm-3, and the ambipolar diffusion coefficient in Ar is 5 x 105

cm2.s-1, find out how many Ar+ per unit area per unit time (flux) are crossing at 4Lx =

assuming x = 0 is at the center of the slab.

SECTION E4-Plasma Electronics Plasma Electronics

Derive the expressions of the mean velocity and most probable velocity for the Maxwellian distribution.

Hint: use dVkT

mVExpVkTmNdNv

−

=

224 222

3

π

Average velocity: VdVdNN v∫

∞

0

1

Most probable: 0=vdNdVd

SECTION F1 – COMPUTER SYSTEMS Microprocessors

Write a subroutine (in assembly language) to sort an array of N 16-bit numbers into ascending

order in a M68000 microprocessor based system. On entry to the subroutine, address register A0

contains the first address (i.e., lowest address) of the array and data register D0 contains the size

(i.e., number of 16-bit numbers) of the array.

Assume the following instructions are available. These instructions with different addressing

modes or any other suitable M68000 instructions may be used.

MOVE.L R1, R2 : Move a longword (32 bits) – register direct addressing

MOVE.W (R1), (R2) : Move a word (16 bits) – register indirect addressing

MOVE.B #n, R3 : Move a byte (8 bits) – Data to R3

CLR.B R : Clear register R (8 bits)

CMP.B ea1, ea2 : Compare byte (8 bits) – flags are set

CMP.W ea1, ea2 : Compare word (16 bits) – flags are set

BEQ label : Branch to label if equal – relative addressing

BNE label : Branch to label if not equal – relative addressing

BCC label : Branch to label if carry clear – relative addressing

ROL.B #n, R : Rotate R left n times and move MSB into carry

AND.B #%data, R : AND byte (8 bits) – AND immediate data to R

OR.B #%data, R : OR byte (8 bits) – OR immediate data to R

ADDQ.B #n, R : Quick Add a byte (8 bits) R ← R – n

SUB.W ea1, ea2 : Subtract a word (16 bits) ea2 ← ea2 – ea1

SUBQ.B #n, R : Quick Subtract a byte (8 bits) R ← R + n

ea : Effective address

R : One of the address registers A0, A1,.., A7 or One of the data registers D0, D1,.., D7

SECTION F2– COMPUTER SYSTEMS

SECTION F3 – COMPUTER SYSTEMS Computer Architecture Instruction Design Take the instruction sequence provided below and determine the operation of the sequence. Then propose a new instruction to accomplish the same thing. Provide abstract RTL for the new instruction.

shr r2, r2, 1 shl r0, r1, 31 shr r1, r1, 1 or r2, r2, r0

Note: r0 is used as a temporary register in this sequence and should not be part of your new instruction. Your new instruction should be of the form new_instr rn, rm and should not affect any other general purpose registers than rn and rm.

Deliverables:

• Description of instruction sequence. • Abstract RTL for new instruction.

SECTION F4 – COMPUTER SYSTEMS

1

SECTION F5 – COMPUTER SYSTEMS Data Structures Exam Questions A) Define the following:

a. Inheritance

b. Operator Overloading

c. Constructor

d. Static Data Member

e. Dynamic Memory Allocation B) Describe the following code segments. Indicate A)overall functionality, B)inputs, C)outputs, and D)object-oriented programming/software engineering concepts used. Comment every line in the code containing a “//” at the end. Code Segment #1 // strng3.h -- String class definition #include using namespace std; class String { private: char * str; // pointer to string int len; // length of string public: String(const char * s); // ? String(); // default constructor

2

String(const String & st); ~String(); // ? int length() const { return len; } // overloaded operators String & operator=(const String & st); // Assignment operator String & operator=(const char * s); // Assignment operator #2 // friend functions friend bool operator>(const String &st1, const String &st2); friend bool operator

3

// pe11-2.cpp #include using namespace std; #include "strng3.h" int main() { String s1(" and I am a C++ student."); String s2 = "Please enter your name: "; String s3; cout s3; // overloaded >> operator s2 = "My name is " + s3; // overloaded =, + operators cout

4

A) Functionality B) Inputs C) Outputs D) Programming / Software Engineering concepts

ODU HONOR PLEDGEStudent Signature: ___________________________________________________Student Name (BLOCK CAPITALS): ___________________________________UIN Number: _________________________________________________Sinusoidal Steady State ResponseREVIEWSECTION F4 – COMPUTER SYSTEMS