07_2_Ph.D_Diagnostic_Exam_Fall_2007

07_2_Ph.D_Diagnostic_Exam_Fall_2007 - DEPARTMENT OF...

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DEPARTMENT OF ELECTRICAL & COMPUTER ENGINEERING OLD DOMINION UNIVERSITY Ph.D. DIAGNOSTIC EXAMIlYATION Fall 2007 ODU HONOR PLEDGE Ipledge 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 ifsummoned. Student Signature: Student Name (BLOCK CAPITALS): Social Sec. Number: Please turn in this examination paper, with the pledge above signed with your answer books. 1. This examination contains 23 problems from the following six areas: A. MATH A1 A2 A3 A4 B. CIRCUITS & ELECTRONICS B1 B2 B3 D. EMAGIQUANTUM ELEC. & LASERS Dl D2 D3 E. SOLID STATEPHYS. ELEC./GAS.ELEC. El E2 E3 E4 F. COMPUTER SYSTEMS F1 F2 F3 F4 F5 2. You must answer Eight questions (no more than three fiom 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 23 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.
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SECTION A1 - MATH Complex Variables and Differential Equations Let C be the contour shown above. Assume f is a function which is analytic everywhere in the complex plane @ except at the points p = -1 and q = 1. Compute the contour integral fC f (z) dz. Ezplicitly justify your answer.
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SECTION A2 - MATH Vector Calculus Consider a vector function F(x,y) = xg a, + (x2 + y) a, . Find the curl of F.
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SECTION A3 - MATH Linear Algebra Let A = [u,],,,,~,, be an n x n matrix with complex elements. 1. Define the transposed matrix AT, the complex conjugate matrix A*, and the Hermitian matrix AH . 2. Write the equation that determines the eigenvalues and eigenvectors of matrix A . 3. Prove that if matrix = the quadratic form xH Ax is real for all complex vectors x . 4. Prove that if matrix A = all its eigenvalues are real.
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SECTION A4 - MATH Proba biliQ The probability density function of a random variable X has the form: f(X) = ~e-~" u(x), where K is a constant and u(x) is the unit step function. Find the value of K, and the probability that X > 1.
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SECTION Bl - CIRCUITS AND ELECTRONICS Sinusoidal Steady State Response The sinusoidal voltage source in the circuit shown is generating the voltage vg = 1.2 cos 100 t V. The variable capacitor is adjusted until the output voltage leads the input voltage by 120'. a. Find the value of C in microfarads. b. Write the steady state expression for the output voltage vo(t) when C has the value found in (a).
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07_2_Ph.D_Diagnostic_Exam_Fall_2007 - DEPARTMENT OF...

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