EE310_examI_f11

EE 350 EXAM I 22 September 2011

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Problem Weight Score

1 25

2 25

3 25

4 25

Total 100

Test Form A

INSTRUCTIONS

1. You have 2 hours to complete this exam.

2. This is a closed book exam. You may use one 8.5” × 11” note sheet.

3. Calculators are not allowed.

4. Solve each part of the problem in the space following the question. If you need more space, continue your solutionon the reverse side labeling the page with the question number; for example, Problem 1.2 Continued. NO

credit will be given to solutions that do not meet this requirement.

5. DO NOT REMOVE ANY PAGES FROM THIS EXAM. Loose papers will not be accepted and agrade of ZERO will be assigned.

6. The quality of your analysis and evaluation is as important as your answers. Your reasoning must be preciseand clear; your complete English sentences should convey what you are doing. To receive credit, you must

show your work.

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Problem 1: (25 Points)

1. (12 points) Consider the signal f(t) shown in Figure 1.

Figure 1: Signal f(t)

(a) (2 points) Is the signal f(t) a causal or noncausal signal? To receive credit you must justify your answerin a single sentence.

(b) (5 points) Determine an expression for the signal f(t).

(c) (5 points) Determine whether f(t) is an energy signal, a power signal, or neither. If f(t) is either anenergy or a power signal, calculate the corresponding metric Ef or Pf , respectively.

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2. (7 points) A linear time-invariant system generates the zero-state responses shown in Figure 2, where thezero-state responses y1(t) and y2(t) correspond to the inputs f1(t) and f2(t), respectively.

Figure 2: Zero-state responses for a LTI system.

(a) (2 points) Is the system causal or noncausal? To receive credit you must justify your answer in a shortsentence.

(b) (5 points) The input f3(t) in Figure 3(A) is applied to the LTI system. Sketch the resulting zero-stateresponse y3(t) in Figure 3(B).

Figure 3: For the input f3(t) in (A), the LTI system generates the zero-state response y3(t) in (B).

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3. (6 points) Consider another system whose zero-state response y(t) to the input f(t) is

y(t) −1

2y(t − 2) = f(t).

Is this system zero-state linear or nonlinear? To receive partial credit you must justify your answer.

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Problem 2: (25 points)

1. (13 points) A linear time-invariant system with input f(t) and output y(t) is represented by the ODE

y + 8y + 20y(t) = 40f(t) + 20f

(a) (5 points) Determine the roots of the characteristic equation.

(b) (6 points) Determine the value of the dimensionless damping ratio and the natural frequency.

(c) (2 points) Is the system asymptotically stable, marginally stable, or unstable? To receive credit you mustjustify your answer in a short sentence.

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2. (12 points) An engineer is studying the characteristics of a linear time-invariant system represented as

Q(D)y(t) = P (D)f(t),

where f(t) and y(t) represent the input and output, respectively. The engineer represents the polynomialsP (D) and Q(D) as row vectors P and Q in MATLAB. Execution of the m-file

disp(’roots(Q): ’), roots(Q)

disp(’length(P): ’), length(P)

disp(’P(1)/Q(2):’), P(1)/Q(2)

results in the output

roots(Q):

ans =

-2

length(P):

ans =

1

P(1)/Q(2):

ans =

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Suppose that the system is driven by the input f(t) = 2u(t) and that y(0) = 0.

(a) (4 points) Determine the steady-state value of the response y(t).

(b) (4 points) At what time will the response y(t) reach and stay within 1% of the steady-state value of y(t)?

(c) (4 points) Determine the ODE representation of the system and place your answer in the standard form

dny

dtn+ an−1

dn−1y

dtn−1+ · · ·+ aoy = bm

dmf

dtm+ bm−1

dm−1f

dtm−1+ · · ·+ bof,

by providing numeric values for the coefficients ai and bi.

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Problem 3: (25 points)

1. (13 points) The circuit in Figure 4 takes an input voltage f(t) and generates an output current y(t).

Figure 4: RC circuit with input voltage f(t) and output current y(t).

(a) (8 points) Determine the ODE representation of the system and place your answer in the standard form

dny

dtn+ an−1

dn−1y

dtn−1+ · · ·+ aoy = bm

dmf

dtm+ bm−1

dm−1f

dtm−1+ · · ·+ bof,

by expressing the coefficients ai and bi in terms of the parameters R and C.

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(b) (5 points) Suppose that f(t) = u(t), R = 1kΩ, and C = 1µF . Determine the initial value, y(0+), andfinal value, y(∞), of the zero-state output current.

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2. (12 points) The response of a system represented by the ODE

dy

dt+ a y(t) = b f(t).

to the inputf(t) = e−3tu(t)

isy(t) = 3e−2t + 6e−3t

for t ≥ 0.

(a) (8 points) Determine the numeric value of the parameters a and b.

(b) (4 points) In terms of the parameters a and/or b (do not use your numeric answer from part a), specifythe rise-time of the zero-state unit-step response of the system.

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Problem 4: (25 points)

1. (13 points) The circuit in Figure 5 has input voltage f(t) and output voltage y(t).

Figure 5: RC filter circuit with input f(t) and output y(t).

(a) (2 points) What is the DC gain of the circuit? In order to receive credit you must justify your answer ina short sentence.

(b) (11 points) Determine a second-order ODE representation of the form

y + a1 y + a0 y = f + bof

for the circuit. Specify the parameters b0, a1 and a0 in terms of R, and C. Obtain the ODE representationby using nodal analysis at the three nodes, a, b, and c, shown in Figure 5.

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2. (12 points) Consider a second-order system, with input f(t) and output y(t), that has the ODE representation

y + 2y + y = f(t).

(a) (2 points) Determine if the zero-state unit-step response would be overdamped, critically damped, orunderdamped. Justify your answer by showing appropriate calculations, but do not attempt to calculatethe zero-state unit-step response.

(b) (5 points) Suppose that y(0) = 1 and y(0) = 0. Determine the zero-input response, yzi(t) of the system.

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(c) (5 points) Suppose that f(t) = 6e−2tu(t). Determine the zero-state response, yzs(t) of the system.

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