Problem 7.4: This problem examines the relation between transient responses of linear systems.The network shown below is first driven by a step at t = 0, then driven by a ramp at t = 0, and finallydriven by a stepped ramp at t = 0. In the first two cases, the capacitor has zero initial voltage.(A) Find the capacitor voltage v(t) in response to the step shown below. Assume that v(0) = 0.(B) Find the capacitor voltage v(t) in response to the ramp shown below. Again assume thatv(0) = 0. Hint: since the step input can be constructed from the ramp input according tovStep(t) = vRamp(t), their respective ZSR responses are related in a similar manner.(C) Finally, find the capacitor voltage v(t) in response to the stepped-ramp shown below assumingthat the capacitor has the known initial voltage v(0) = vo. Hint: think superposition.R2Li(t)VR1R2Cv(t)IR1+-Network (A) Network (B)+-v(0) = voi(0) = io1α---ddt-----VoV(t) = RampV(t) = Steptt00V0αtV(t) = Stepped Rampt0V01 αt+()VoV(t)RCv(t)+-+-Part (A): v(0) = 0Part (B): v(0) = 0Part (C): v(0) = vo(B) Next, at t = T, vIN turns the MOSFET off. Determine both iR(t) and vDS(t) for t ≥ 0. Hint: iR(t) iscontinuous at t = T.(C) Sketch and clearly label graphs of both iR(t) and vDS(t) for t ≥ 0 assuming that T ≈ 5LR/RR andRX = RR.(D) The relay control circuit would be less expensive without the external resistor, which may be‘‘removed’’ from the circuit by considering the limit RX→∞. Why might such a cost reductionbe unwise?Problem 7.3: This problem illustrates the superposition of a zero-input response (ZIR) and azero-state response (ZSR) as a means of determining the total response of a network.(A) Solve the differential equation τ + x = S for t ≥ 0 given x(0). This is equivalent to finding thestep response of a general linear first-order time-invariant system having a nonzero initialcondition.(B) Use the result from Part (A) to show that the step response of a linear time-invariant first-ordersystem takes the form x(t) = x(0)e-t/τ + x(∞)(1 - e-t/τ). Explain why the two terms in this responseare the ZIR and ZSR of the system, respectively.(C) For each network shown below, find the network state at t = ∞ and the network time constant;note that I and V are constants. Hint: see Exercise 7.3. Next, use the results of Part (B) to find thenetwork state for t ≥ 0. You should consider whether you find the superposition of a ZIR andZSR to be a simple and intuitive method of determining the response of a linear system.VSvIN+-RxLRRelayRRiR+-vDSdxdt------Problem 7.1: At t = 0-, the networks shown below have zero initial state. That is, the capacitorvoltage v(t) and the inductor current i(t) are both zero at t = 0-. At t = 0, the voltage source producesan impulse of area Λ, and the current source produces an impulse of area Q.(A) Derive the differential equation which relates v(t) to V(t) and i(t) to I(t). Hint: consider usingThevenin or Norton equivalent networks to simplify the work.(B) Find the capacitor voltage v(t) and the inductor current i(t) at both t = 0+ and t = ∞. One way tofind the states at t = 0+ is to integrate the corresponding differential equations from t = 0-to t=0+ under the assumption that each state remains finite during that time; you shouldjustify this assumption. Then, substitute the initial conditions at t = 0- into the results todetermine the states at t = 0+. Try to determine the states at t = ∞ through physical, rather thanmathematical, reasoning.(C) Next, find the time constant by which each state goes from its initial value at t = 0+ to its finalvalue at t = ∞. Hint: see Exercise 7.3.(D) Using the previous results, and without necessarily solving the differential equations directly,construct v(t) and i(t) for t ≥ 0.(E) Verify that the solutions to Part (D) are correct by substituting them into the differentialequation found in Part (A).Problem 7.2: In the circuit shown below, a MOSFET and an external resistor having resistanceRX are used to control the current iRin the winding of a relay. Here, the relay is modeled as a seriesinductor and resistor having inductance LR and resistance RR, respectively. The MOSFET may bemodeled as an ideal switch.(A) At t = 0, vIN turns the MOSFET on so that vDS = 0. Determine iR(t) for t ≥ 0 given thatiR(t = 0) = 0.+-R1R2LCR2R1v(t)+-I(t)V(t)i(t)ttQΛI(t)V(t)00Massachusetts Institute of TechnologyDepartment of Electrical Engineering and Computer Science6.002 - Electronic CircuitsSpring 2000Homework #7Issued 3/15/2000 - Due 3/29/2000Exercise 7.1: Consider an amplifier with an input-output relation that takes the formvOUT=VA(vIN/VB)3, where VA and VB are voltage constants. Determine its output bias voltage VOUTand its small-signal gain vout/vin for a given input bias voltage VIN.Exercise 7.2: Find the capacitance of the all-capacitor network, and the inductance of the all-inductor network, shown below.Exercise 7.3: Each network shown below has a non-zero initial state at t = 0, as indicated. Findthe network state for t ≥ 0. Hint: what equivalent resistance is in parallel with each capacitor orinductor, and what decay time results from this combination?C1C2C3L1L2L3R1R2C+-v(t)R1R1R2+-v(t)R2CR1i(t)R2Li(t)Lv(0) = Vv(0) = Vi(0) = Ii(0) =
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