Massachusetts Institute of TechnologyDepartment of Electrical Engineering and Computer Science6.002 – Electronic CircuitsFall 2005Homework 9Issued 11/8/2005 - Due 11/18/2005Helpful readings for this homework: Chapter 12Exercise 9-1: Using one 3-nF capacitor and two resistors, construct a network that has thefollowing zero-state response (ZSR) to a 1-V step input. Provide a diagram of the network, andspecify the values of the two resistors.Massachusetts Institute of TechnologyDepartment of Electrical Engineering and Computer Science6.002 - Electronic CircuitsFall 2004Homework #9Handout F04-046Issued 11/04/2004 - Due 11/12/2004Helpful Readings for this Homework: Chapter 12Exercise 9-1: Using one 3-nF capacitor and two resistors, construct a network that has the following zero-state response to a 1-V step input. Provide a diagram of the network, and specify the values of the two resistors.1Vt1Vt23--VNetwork23--V13--Vet 20µs⁄–+v2(t)+-v2(t)v1(t)v1(t)+-Exercise 9-2: Exercise 12.4, Chapter 12 (p. 985)Exercise 9-3: Consider a linear time-invariant system. Suppose its ZSR to a unit step applied at t 0= is A 1 et τ⁄––(). What would be its ZSR to the input SMt+ , applied at t 0= , where S and M are constants?Problem 9.1: Problem 12.6, Chapter 12 (p. 991)Problem 9.2: In the network shown below, the inductor and capacitor have zero states prior to t 0= . At t 0= , a step in voltage from 0 to V0 is applied by the voltage source as shown.a) Find vC, vL, i , and didt----- at t 0+= .b) Argue that i 0= at t ∞= so that it() has no constant component.c) Find a second-order differential equation which describes the behavior of it() for t 0≥ .Exercise 9-2: Exercise 12.4, Chapter 12, page 695.Exercise 9-3: Consider a linear time-invariant system. Suppose its ZSR to a unit step appliedat t = 0 is A(1 − e−t/τ). What would be its ZSR to the input S + M t, applied at t = 0, where Sand M are constants?Problem 9.1: Problem 12.2, Chapter 12, page 697.Problem 9.2: Problem 12.6, Chapter 12, page 698.Problem 9.3: In the network shown below, the inductor and capacitor have zero states prior tot = 0. At t = 0, a step in voltage from 0 to Vois applied by the voltage source as shown.a) Find vc, vl, i, anddidtat t = 0+.b) Argue that at t = ∞, i = 0 so that i(t) has no constant component.d) Following Part b, the current it() takes the form it() Ieαt–ωt φ+()sin= . Find I , ω , φ , and α . Hint: first find ω and α from the differential equation, and then find I and φ from the initial conditions; alternatively, solve this problem by any method you wish.e) Suppose that the input is a voltage impulse with area Λ0 (in Volt-seconds), where Λ0τV0= , the volt-age V0 is the amplitude of the voltage step shown below, and τ is a given time constant. Find the response of the network shown below to the impulse. Hint: Before solving this problem directly, con-sider the relation between step and impulse responses.Save a copy of your answers to this problem. They will be useful during the pre-lab exercises for Lab #3.vR+-i(t)vLC+-LR+-V0tv(t)v(t)vC+-Problem 9.3: Problem 12.7, Chapter 12 (p. 991). Use the values R1 = 4.75kΩ, Rn = 25Ω, L1 = 10µm, W1 = 1µm, Cp = 1nF, Lp = 4mH for the underdamped case, and use R1 = 750Ω, Lp = 6.25mH (with the rest the same) for the over-damped case.Explicitly set up the characteristic equation for the circuit, and then use intuitive analysis (i.e., without finding the full solutions to differential equations) to find the form of the responses (See section 12.7). You don’t have to solve for the maximum amplitude in either case, but important values such as frequencies, initil and final values, time constants, decay envelopes, etc., should be indicated numerically when applicable. Hint: use circuit analysis to set up a differen-tial equation, but don’t solve it. Use the constants from the differential equation, along with your initial conditions to determine the parameters necessary for your plots. Don’t forget about initial slopes when you determine your initial conditions!Comparing your results with Problem 10.8 from Chapter 10 is optional, but highly recommended.c) Find a second-order differential equation which describes the behavior of i(t) for t ≥ 0.d) Following part (b), the current i(t) takes the form i(t) = Ie−αtsin(ωt + φ). Find I, ω, φ,and α.e) Suppose that the input is a voltage impulse with area Λoin Volt-seconds where Λo= τ Vo,the voltage Vois the amplitude of the voltage step shown below, and τ is a given time constant.Find the response of the network to the impulse.Save a copy of your answers to this problem. They will be useful during the pre-lab exercisesfor Lab 3.Problem 9.4: Problem 12.7, Chapter 12, page 699 with the following parts.a) Explicitly set up the characteristic equation for the circuit.For parts (b) and (c) use the values Rn= 25Ω, L1= 10µm, W1= 1µm, and Cp= 1nF . Useintuitive analysis to find the form of the responses. Solve for the frequency, initial and final values,time constants, and decay envelope s. You don’t have to s olve for the maximum amplitude.b) Sketch vPfor the underdamped case. Use R1= 4.75kΩ and Lp= 4mH.c) Sketch vPfor the overdamped case. Use R1= 750Ω and LP= 6.25mH.d) Compare the results from parts (b) and (c) with that for the inductor acting alone as shownin Figure 10.107 on page
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