Write your four digit code here... NEW MEXICO STATE UNIVERSITY THE KLIPSCH SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING Ph.D. QUALIFYING EXAMINATION

Transcription

1 Write your four digit code here... NEW MEXICO STATE UNIVERSITY THE KLIPSCH SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING Ph.D. QUALIFYING EXAMINATION JANUARY 12, :00 AM 1:00PM CLOSED BOOK Exam Instructions: 1. Write the last four digits of your Banner ID number of the top of every page. 2. Work six (6) problems from the three (3) areas of specialization selected at the time of registration. Do not work more than two (2) problems in any one area. A. Circuits and Electronics B. Communications C. Computers D. Control Systems E. Digital Signal Processing F. Electric Energy Systems G. Electromagnetics H. Photonics 3. Check the boxes below indicating which six (6) problems you want graded. (You must work two (2) problems each from each of the three (3) areas you specified at the time of registration) A. Circuits and Electronics B. Communications... C. Computers... D. Control Systems... E. Digital Signal Processing F. Electric Energy Systems G. Electromagnetics... H. Photonics...

2 A (1) Circuits and Electronics Design a fully differential telescopic operational amplifier for a minimum gainbandwidth product GB min =50MHz, and minimum slew rate SR min =50V/µs and a load capacitance C L =20pF. Assume following 0.13µm CMOS technology parameters: kn n =340μA/V 2, Vth n =0.40V, kn p =50µA/V 2, Vth p =0.40V, λ n =λ p =0.25V -1, C ox =10fF/μm 2 C gd =1fF/μm and a capacitive load C L =10pF (kn=μc ox /2). Determine all transistor sizes including those in the biasing branch and in the common mode feddback network. Assume a single supply voltage V DD =1.2V The design requires to determine bias currents, W/L sizes for all transistors. Estimate the dominant and high frequency poles and the open loop DC gain. Calculate the static power dissipation of your design input common mode range and output swing Determine the output impedance and bandwidth of a unity gain buffer (Av=1) and b) a gain ten (Av=10v/v) inverting amplifiers implemented using your op-amp.

3 A (2) Circuits and Electronics Design a Miller (two stage) single-ended operational amplifier for a minimum gainbandwidth product GB min =100MHz, minimum slew rate SR min =20V/μs and a load capacitance C L =20pF. Assume following 0.18µm CMOS technology technology parameters: kn n =170μA/V 2, Vth n =0.45V, kn p =25uA/V 2, Vth p =0.45V, λ n =λ p =0.1V -1, C ox =4.2fF/μm 2 (kn=μc ox /2), C gd =1fF/μm. Show the scheme of the circuit. Determine all transistor sizes including those in the biasing branch. Assume a single supply voltage V DD =1.8V. Estimate the open loop DC gain and the high frequency pole. The design requires to determine bias currents, W/L size all transistors, and to determine the value of the compensation capacitance Cc and the compensation resistance Rc.

4 A (3) Circuits and Electronics Design a cascode amplifier with minimum gain bandwidth product GB=100MHz in 0.13µm CMOS technology with kn n =340μA/V 2, Vth n =0.40V, kn p =50µA/V 2, Vth p =0.40V, λ n =λ p =0.25V -1, C ox =10fF/μm 2, C gd =1fF/μm and a capacitive load C L =10pF (kn=μc ox /2), ν=0.4v 1/2. Use an NMOS input transistor. Determine biasing currents, cascode voltages (VCP, VCN) all transistor sizes including those in the biasing branch. Assume a single supply voltage V DD =1.2V, a load capacitance C L =10pF. Assume the source resistance is 20kΩ. Estimate the open loop DC gain, the bandwidth, the common mode input range, the peak to peak output swing the dominant and the high freqency poles. For your calculations of the high frequency poles take only into account the Cgs parasitic capacitances of the cascode transistors multiplied by a factor 2.

5 B (1) Communications The random variable pair (X, Y ) has a joint pdf f X,Y (x, y) = 2e x e 2y x > 0, y > 0 (i) Find the probability P [Y X 1]. (ii) If Z = X + Y, find the pdf of Z. (iii) Discuss if X and Y are independent, orthogonal and/or uncorrelated. You need to use the fact that the mean of an exponential RV with pdf (for x > 0), f X (x) = λ exp( λx), is 1/λ.

6 B (2) Communications (i) In each lot of 100 items, 2 items are tested, and the lot is rejected if either of the tested items is found defective. Find the probability of rejecting a lot with 5 defective items. (ii) Consider a random process X(t) = A cos ωt + B sin ωt, where A and B are iid Gaussian RVs, each with zero mean and a variance of 4. (a) Discuss if X(t) is wide sense and strict sense stationary. (b) Discuss if X(t) has a mean square derivative. (c) Suppose A and B are not iid, but each one is still zero-mean Gaussian with a variance of 4, and the correlation coefficient between them is 0.1. Assuming ω = 2π, find the probability that X(t) at t = 1/8 exceeds the value of 1.5.

7 B (3) Communications A random variable X has a cumulative distribution function (CDF) 0 x < 1 2(x + 1) 1 x < 0 F X (x) = x 0 x x 1 (a) Find the probabilities P [X 0.2] and P [ X 0.2]. (b) Find the variance of X. (c) If Y = ln(1 X), find the pdf of Y.

8 C (1) Computers The sign table of a 2^2 factorial design is provided below. Build the regression model and compute the variations explained by each factor. I A B AB y

9 C (2) Computers In a queuing system that can hold at most 4 customers, the arrival and service times are exponentially distributed. The arrival rate is 5 customers per sec. The serve rates are 8,7, 6, 5, customers per second when the system has 1, 2, 3, 4 customers, respectively. Find the average response time and the blocking probability of system.

10 C (3) Computers Given 10 uniformly distributed random numbers generated by MATLAB function rand( ) as follows: , , , , , , , , , Use those random numbers to generate 10 random numbers with the Pareto distribution with its pdf given as f(x)=3x -4 in the range [1, ].

11 C (4) Computers Amdahl s law is named after computer architect Gene Amdahl, and is used to find the maximum expected improvement to an overall system when only part of the system is improved. For this question you are asked to compare two alternatives by comparing the speedups. Floating-point square root (FPSQR) operations are often part of scientific workloads. For a particular workload, FPSQR is responsible for 20% of the execution time. The first design option is to make hardware enhancements that will speedup the FPSQR by a factor of 10. The second design option will speedup all floating-point (FP) operations by a factor of 1.6. All FP operations makeup 50% of the execution time for the workload. Show which option is better and by how much. 1

12 C (5) Computers Consider the MIPS code shown below. Loop: L.D F0,0(R1) ;F0=vector element ADD.D F4,F0,F2 ;add scalar from F2 S.D 0(R1),F4 ;store result DADDUI R1,R1,-8 ;decrement pointer BNEZ R1,Loop ;branch R1!=zero 1. Indicate the location and number of stalls due to data dependences and hazards. 2. Revised the code for this loop to minimize the number of stalls. Be sure to indicate the stalls in your new code. 2

13 C (6) Computers Consider a computer that uses an in-order execution. The cache miss penalty is 100 clock cycles, All instructions take 1 clock cycle (without memory stalls). They system has an average miss rate of 2% and there is an average of 1.5 memory references per instruction. The average number of cache misses per 100 instructions is 3. What is the impact on performance when the behavior of the cache in included? (looking for a comparison of the CPU time ) Calculate the impact using both the misses per instruction and the miss rate. 3

14 D (1) Control Systems Given the discrete-time system [ 0 1 x(t + 1) = 0 2 y (t) = [ 1 0 ] x (t) ] [ 0 x (t) + 1 ] u (t) Find an output-feedback controller such that the closed-loop system tracks a constant input, and all the closed-loop eigenvalues are at the origin.

15 D (2) Control Systems For the system illustrated Δ M where we have the continuous-time transfer function M(s) = s s 2 + s + 10 and where the uncertainty,, lies in the uncertainty set Determine if this system is robustly stable. = { : 1}

16 D (3) Control Systems Consider the real, controllable, continuous-time system ẋ = Ax + Bu. In this problem, there exists a similarity transformation, T, such that Ā = T 1 AT and such that Ā ) T = Ā Define also Prove any of the following: K = 1 2 BT T T T ) 1 a) the eigenvalues of A are purely imaginary. b) the matrix T 1 e At T is unitary. c) the matrix A + BK is Hurwitz. d) there exists Q > 0 for which AQ + QA T = 0. e) there is no matrix, Q, that will satisfy AQ + QA T + BB T = 0. f) there exists Q > 0 for which (A + BK)Q + Q(A + BK) T + BB T = 0.

17 E (1) Digital Signal Processing Given: a real, linear phase FIR filter. a) Two zeros are shown on the z-plane plot. Fill in the missing zeros. Assume that the filter has minimum possible order. What type of linear phase filter must this be? Explain what it means for a filter to be linear phase. b) Plot the poles and zeros of the minimum phase filter having the same magnitude response as the filter in part (a). If this is not possible, write "no solution." Also, explain what it means for a filter to be minimum phase.

18 E (1) Digital Signal Processing c) Plot the poles and zeros for every other real filter having the same magnitude response as the filter in part (a)? d) Write the transfer function of a filter that can be cascaded with (b) to create the filter of part (c). If this is not possible, write "no solution."

19 E (2) Digital Signal Processing Consider a digital filter whose transfer function has a single pole at z = 1/2, a single pole at z = 0, and two zeros at z = 1. The transfer function includes a scale factor so that the DC response is 0 db. (a) Sketch the pole-zero pattern of the filter. (b) Determine the transfer function H(z). factor. Be sure to include (and determine) the scale (c) Assume the filter is stable and determine the impulse response h[n]. (d) Sketch the magnitude response of the filter H(e jω ) vs. ω for 0 ω π.

20 E (3) Digital Signal Processing Continuous-time signal x(t) has spectrum X(jΩ) as shown below, where Ω M = 10, 000π. (a) What is the minimum sampling rate Ω s that x(t) can be sampled and not alias? (b) Assuming the minimum sampling rate Ω s determined in part (a), sketch the spectrum X(e jω ) of the discrete-time signal x[n]. Label your axes and important amplitudes and frequencies. (c) Assume that discrete-time signal x[n] has spectrum as in part (b). Sketch the specrum X down (e jω ) for the downsampled signal x down [n] for a downsampling factor M = 2. Label your axes and important amplitudes and frequencies.

21 E (3) Digital Signal Processing (d) What is the effective sampling rate Ω s,down for x down [n]? (e) Assume that discrete-time signal x[n] has spectrum as in part (b). Sketch the specrum X decimate (e jω ) for the decimated signal x decimate [n] for a downsampling factor M = 2. Label your axes and important amplitudes and frequencies.

22 F (1) Electric Energy Systems Consider the system shown in the figure below. Assume 100 MVA, 230 kv base in the transmission line. Transformer T 1 is rated 100 MVA, 15/230 kv. The load is drawing 0.9 pu real power at 0.9 power factor lagging. Sequence reactances are shown in the figure as per usual convention. X G1 (1) =j0.1 PU X G1 (2) =j0.1 PU X G1 (0) =j0.05 PU G T 1 X T1 = j0.1 PU 1 2 Transmission Line X TL (1) = j0.1 PU X TL (0) = j0.25 PU Load A line to ground fault on phase a takes place at bus 1. Without neglecting pre- fault conditions, 1) calculate the three- phase voltages (kv) at bus 1 after the fault. 2) Calculate the current contribution (Ampere) from phase a of the generator G after the fault.

23 F (2) Electric Energy Systems Find the per unit values of I B, I AB, and I a in the transformer diagram shown below. I a =? A =1 0 I A.3 30 PU c a LV C HV I AB =? B I B =? b =1 0 I C.0 50 PU

24 F (3) Electric Energy Systems G1 L G2 Load G1 100 MVA, 20kV, three phase, 2pole Synchronous Generator, Xd =0.1pu, H = 4 s G2 Infinite Bus, Voltage 1pu L Transmission line z = j 0.1 pu, y=0 on 100MVA, 20 kv base. Load 25 MVA, unity power factor In the system shown Generator G1 supplies the entire load with a terminal voltage of 1 pu. There is no power flow between Generator G1 and the Infinite Bus G2. The breaker to the load opens, dropping the entire load. Is the system first swing stable? Show all work.

25 G (1) Electromagnetics Shown below is the so-called dielectric-loaded parallel plate waveguide for microwave signal transmission. The structure is uniform in the z direction. It has two infinite and perfectly conducting plates located at y = 0 and y = a. Between the two conducting plates, we have a lossless dielectric slab with permittivity ε and permeability µ. Above the dielectric slab, we simply have air ( ε o and µ o ). As the structure is uniform in the z direction, the field solutions will not depend on z. Furthermore, we have a source free region. In one solution, the electric field E in the air region only has a z component and it is given by E z = A sin [ g ( a y)] e j β x, t y a, A - constant, g, β - real Inside the slab, the electric field also only has a z component sinusoidally with respect to y. E z, and it is found to vary a) Inside the slab, i.e., 0 y t, should E z be B cos ( hy) -j e β x or B sin ( hy) -j e β x Why? Explain it in 1-2 sentences. (B constant, h real, β as given in Equation above) b) Find an equation relating the constants A and B (the equation can contain A, B, g, h, a and t). c) Find an equation relating g, β, and k o, where ko is the propagation constant for air, i.e., k o = ω µ oε o.

26 G (1) Electromagnetics

27 G (2) Electromagnetics Consider a thin dipole or thin wire directed in the z direction and located at a distance h above an infinite perfectly conducting ground plane, as shown below. The incident field is created by an infinitesimally small circular loop of magnetic current (i.e., equivalent to an electric hertzian dipole) located just above the ground plane, i.e., at z = 0+, which has radius a and a constant current M. As far as calculating the scattered field is concerned, we can consider the thin wire as represented by an unknown line current I(z) along its axis and the typical thin wire approximation holds. Also, we can assume that the thin wire is in the far-field region of the infinitesimal magnetic current loop. a) Find the total E-field in the z direction at the line segment L as indicated in the figure. 1 b) Enforce a boundary condition at the line segment L and obtain an integral equation for the 1 unknown current I.

28 G (2) Electromagnetics

29 G (3) Electromagnetics Consider a thin dipole or thin wire directed in the z direction and located at a distance d above an infinite PEC ground plane, as shown below. The (fixed) incident field is created by an infinitesimal or Hertzian electric dipole I (z) ˆz I ( R ) located at the origin. As far as e calculating the scattered field is concerned, we can consider the thin wire as represented by an unknown line current I(z) along its axis and the typical thin wire approximation holds. Also, we can assume that the thin wire is in the far-field region of the Hertzian dipole. a) Find the total E at the line segment L as indicated in the figure. The contribution from the z 1 thin wire can be represented by some integral(s), but you need to give me all the details about the integral(s). b) Enforce a boundary condition at line segment L and obtain an integral equation for the 1 unknown current I.

30 G (3) Electromagnetics

31 H (1) Photonics a) Write an expression for the Fourier shift theorem. Assume two dimensions. b) Use the integral definition of the Fourier transform to prove the shift theorem.

32 H (2) Photonics a) Find an expression for the field distribution of the Fraunhofer diffraction pattern for the two rectangular apertures shown below. b) Find an expression for the Fraunhofer intensity (irradiance) distribution. c) Sketch the Fraunhofer intensity pattern along the horizontal (x) axis. Assume the field illuminating the apertures is a monochromatic unitamplitude, normally incident plane-wave. Assume the rectangles are the same size. 2a 2b 2c

33 H (3) Photonics Consider the arrangement below where a monochromatic point source illuminates a transparency at a distance a and the transmitted light continues another distance a to a lens of focal length f. a) Find the distance from the lens where the Fraunhofer pattern of the transparency is formed. b) Find the distance from the lens where an image is formed of the transparency. c) Find the transverse magnification of the image of the transparency. d) Find the focal length required such that the Fraunhofer pattern of the transparency is a distance a from the lens. e) Find the focal length such that the magnification of the transparency image is 1. a a Source Transparency Lens Focal length = f