Problem Weight Score Total 100

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1 EE 350 Exam # 1 25 September 2014 Last Name (Print): First Name (Print): ID number (Last 4 digits): Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO Problem Weight Score Total You have 2 hours to complete this exam. INSTRUCTIONS 2. This is a closed book exam. You may use one 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 solution on 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 a grade of ZERO will be assigned. 6. The quality of your analysis and evaluation is as important as your answers. Your reasoning must be precise and clear; your complete English sentences should convey what you are doing. To receive credit, you must show your work. 7. Any student caught cheating on an exam will receive a grade of zero for the exam. Additional sanctions, including assigning an XF grade, will be pursued following university guidelines. 1

2 Problem 1: (25 Points) 1. (6 points) Consider the signal f(t) = ( 1 e (t + T)/T ) [u(t + T) u(t)] + ( 1 e 1) e t/t [u(t) u(t T)]. (a) (4 points) Sketch f(t) in Figure 1. Label the values of f(0) and f(t) on the y axes in terms of the number e (do not provide a numeric value). Figure 1: Sketch of f(t). (b) (2 points) State whetherf(t) is a causal or noncausal signal. Justify your answer using a short sentence. 2

3 2. (5 points) Consider the signal g(t) = 4e (t + 3T)/T u(t + 3T). Determine whether g(t) is an energy signal, a power signal, or neither. If g(t) is either an energy or a power signal, determine the corresponding metric E g or P g, respectively. 3

4 3. (14 points) A system with input f(t) and output y(t) has the zero-state response y(t) = t f 2 (τ)e (t τ) dτ. (a) (7 points) Is the system zero-state linear or nonlinear? To receive partial credit justify your answer. (b) (7 points) Is the system time invariant or time-varying? To receive partial credit justify your answer. 4

5 Problem 2: (25 points) 1. (8 points) A linear time-invariant system with input f(t) and output y(t) is represented by the ODE ÿ 2ẏ + 10y(t) = f(t) (a) (6 points) Determine the roots of the characteristic equation and sketch these roots in the λ-plane provided in Figure 2. To obtain full credit, appropriately label the real and imaginary axes to specify the location of the characteristics roots. Figure 2: Sketch of the characteristic roots in the λ-plane. (b) (2 points) Based on the location of the characteristic roots, specify if the system is asymptotically stable, marginally stable, or unstable. Justify your answer using a short sentence. 5

6 2. (8 points) Figure 3 show the roots of the characteristic equation for a certain fourth-order LTI system. Figure 3: Location of the characteristic roots in the λ-plane. (a) (3 points) State the form of the homogeneous solution y h (t). (b) (2 points) Which characteristic root dominates the transient response characteristics of the homogeneous solution? Justify you answer using a short sentence. (c) (3 points) Suppose that you are determining the zero-state response of the system for the input f(t) = 1 + e 2t, t 0 Specify the form of the particular solution y p (t) for t 0. 6

7 3. (9 points) A LTI system with input f(t) and output y(t) is represented by the ODE d 5 y dt 5 + y 7d4 dt 4 + y 18d3 dt 3 + y 23d2 dt dy dt + 6y(t) = f 8d2 dt 2 3f(t) Complete the m-file in Figure 4 so that it accomplishes the following tasks. (a) (3 points) Determines and display the roots of the characteristic equation. (b) (6 points) Determines and plots the zero-state unit-step response for the input f(t) = e 2t u(t) over the time range 0 t 100 using a time vector of one thousand points. Appropriately label the x and y axes of the figure. EE 350 Fall 2014 Exam 1 Problem 2 Part 3 clc; close all; clear Q = ; specify Q(D) P = ; specify P(D) Determine and display roots of the characteristic equation display( roots of Q are ) disp( ) Generate the time vector t = ; Generate the input f = ; Determine the zero-state response y = ; Plot the response y(t) plot y versus t label the x-axis label the y-axis Figure 4: MTALB m-file. 7

8 Problem 3: (25 points) 1. (12 points) The circuit in Figure 5 is driven by an independent voltage source with a constant strength V s. Prior to the switch closing at time t = 0, the current and voltages within the circuit have reached steady-state values. Figure 5: The switch in the passive RL circuit closes at time 0. (a) (8 points) Determine expressions for the current i(0 + ) and voltage y(0 + ) in terms of the circuit parameters. (b) (4 points) Determine expressions for the current i( ) and voltage y( ) in terms of the circuit parameters. 8

9 2. (13 points) A LTI system with input f(t) and output y(t) has the ODE representation The system is driven by the input and the initial value y(0) = 2. dy + 2 y(t) = 4 f(t). dt f(t) = 1 + e 2t, t 0, (a) (3 points) Determine the the zero-input response y zi (t) for t 0. (b) (9 points) Determine the the zero-state response y zs (t) for t 0. 9

10 Problem 4: (25 points) 1. (12 points) The circuit in Figure 6 has input voltage f(t) and output voltage y(t). Figure 6: Passive RLC filter circuit with input f(t) and output y(t). (a) (2 points) Determine the DC gain and high frequency gain of the network. Justify your answers using short sentences. (b) (10 points) Determine a second-order ODE representation of the form ÿ + a 1 ẏ + a 0 y = b o f for the circuit. Specify the parameters b 0, a 1 and a 0 in terms of R, L, and C. Obtain the ODE representation by applying nodal analysis at nodes A and B, which are indicated in Figure 6. 10

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12 2. (13 points) Consider another second-order system with input f(t), output y(t), and ODE representation ÿ + 4ẏ + 8y = 24f(t). (a) (2 points) Determine the natural frequency and dimensionless damping ratio for the system. (b) (1 point) Based on your result in part 1, will the zero-state response be overdamped, critically damped, or underdamped? (c) (10 points) Given that that y(0) = 1, ẏ(0) = 8, and f(t) = u(t), determine the total response y(t). To receive full credit, your solution must not contain any complex-valued terms. 12

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14 EE 350 Exam # 1 25 September 2014 Last Name (Print): First Name (Print): ID number (Last 4 digits): Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO Problem Weight Score Total You have 2 hours to complete this exam. INSTRUCTIONS 2. This is a closed book exam. You may use one 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 solution on 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 a grade of ZERO will be assigned. 6. The quality of your analysis and evaluation is as important as your answers. Your reasoning must be precise and clear; your complete English sentences should convey what you are doing. To receive credit, you must show your work. 7. Any student caught cheating on an exam will receive a grade of zero for the exam. Additional sanctions, including assigning an XF grade, will be pursued following university guidelines. 1

15 Problem 1: (25 Points) 1. (6 points) Consider the signal f(t) = ( 1 e (t + T)/T ) [u(t + T) u(t)] + ( 1 e 1) e t/t [u(t) u(t T)]. (a) (4 points) Sketch f(t) in Figure 1. Label the values of f(0) and f(t) on the y axes in terms of the number e (do not provide a numeric value). Figure 1: Sketch of f(t). (b) (2 points) State whetherf(t) is a causal or noncausal signal. Justify your answer using a short sentence. 2

16 2. (5 points) Consider the signal g(t) = 4e (t + 3T)/T u(t + 3T). Determine whether g(t) is an energy signal, a power signal, or neither. If g(t) is either an energy or a power signal, determine the corresponding metric E g or P g, respectively. 3

17 3. (14 points) A system with input f(t) and output y(t) has the zero-state response y(t) = t f 2 (τ)e (t τ) dτ. (a) (7 points) Is the system zero-state linear or nonlinear? To receive partial credit justify your answer. (b) (7 points) Is the system time invariant or time-varying? To receive partial credit justify your answer. 4

18 Problem 2: (25 points) 1. (8 points) A linear time-invariant system with input f(t) and output y(t) is represented by the ODE ÿ 2ẏ + 10y(t) = f(t) (a) (6 points) Determine the roots of the characteristic equation and sketch these roots in the λ-plane provided in Figure 2. To obtain full credit, appropriately label the real and imaginary axes to specify the location of the characteristics roots. Figure 2: Sketch of the characteristic roots in the λ-plane. (b) (2 points) Based on the location of the characteristic roots, specify if the system is asymptotically stable, marginally stable, or unstable. Justify your answer using a short sentence. 5

19 2. (8 points) Figure 3 show the roots of the characteristic equation for a certain fourth-order LTI system. Figure 3: Location of the characteristic roots in the λ-plane. (a) (3 points) State the form of the homogeneous solution y h (t). (b) (2 points) Which characteristic root dominates the transient response characteristics of the homogeneous solution? Justify you answer using a short sentence. (c) (3 points) Suppose that you are determining the zero-state response of the system for the input f(t) = 1 + e 2t, t 0 Specify the form of the particular solution y p (t) for t 0. 6

20 3. (9 points) A LTI system with input f(t) and output y(t) is represented by the ODE d 5 y dt 5 + y 7d4 dt 4 + y 18d3 dt 3 + y 23d2 dt dy dt + 6y(t) = f 8d2 dt 2 3f(t) Complete the m-file in Figure 4 so that it accomplishes the following tasks. (a) (3 points) Determines and display the roots of the characteristic equation. (b) (6 points) Determines and plots the zero-state unit-step response for the input f(t) = e 2t u(t) over the time range 0 t 100 using a time vector of one thousand points. Appropriately label the x and y axes of the figure. EE 350 Fall 2014 Exam 1 Problem 2 Part 3 clc; close all; clear Q = ; specify Q(D) P = ; specify P(D) Determine and display roots of the characteristic equation display( roots of Q are ) disp( ) Generate the time vector t = ; Generate the input f = ; Determine the zero-state response y = ; Plot the response y(t) plot y versus t label the x-axis label the y-axis Figure 4: MTALB m-file. 7

21 Problem 3: (25 points) 1. (12 points) The circuit in Figure 5 is driven by an independent voltage source with a constant strength V s. Prior to the switch closing at time t = 0, the current and voltages within the circuit have reached steady-state values. Figure 5: The switch in the passive RL circuit closes at time 0. (a) (8 points) Determine expressions for the current i(0 + ) and voltage y(0 + ) in terms of the circuit parameters. (b) (4 points) Determine expressions for the current i( ) and voltage y( ) in terms of the circuit parameters. 8

22 2. (13 points) A LTI system with input f(t) and output y(t) has the ODE representation The system is driven by the input and the initial value y(0) = 2. dy + 2 y(t) = 4 f(t). dt f(t) = 1 + e 2t, t 0, (a) (3 points) Determine the the zero-input response y zi (t) for t 0. (b) (9 points) Determine the the zero-state response y zs (t) for t 0. 9

23 Problem 4: (25 points) 1. (12 points) The circuit in Figure 6 has input voltage f(t) and output voltage y(t). Figure 6: Passive RLC filter circuit with input f(t) and output y(t). (a) (2 points) Determine the DC gain and high frequency gain of the network. Justify your answers using short sentences. (b) (10 points) Determine a second-order ODE representation of the form ÿ + a 1 ẏ + a 0 y = b o f for the circuit. Specify the parameters b 0, a 1 and a 0 in terms of R, L, and C. Obtain the ODE representation by applying nodal analysis at nodes A and B, which are indicated in Figure 6. 10

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25 2. (13 points) Consider another second-order system with input f(t), output y(t), and ODE representation ÿ + 4ẏ + 8y = 24f(t). (a) (2 points) Determine the natural frequency and dimensionless damping ratio for the system. (b) (1 point) Based on your result in part 1, will the zero-state response be overdamped, critically damped, or underdamped? (c) (10 points) Given that that y(0) = 1, ẏ(0) = 8, and f(t) = u(t), determine the total response y(t). To receive full credit, your solution must not contain any complex-valued terms. 12

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