EE 350 Problem Set 3 Cover Sheet Fall 2014Last Name (Print):First Name (Print):ID number (Last 4 digits):Sectio n:Submission deadl ines:• Turn in the written solutions by 4:0 0 pm on Tuesday September 23 in the homework slot outside 121 EE East.Problem Weight Score11 2012 2013 2014 2015 20Total 100The solution submitted for grading represents my own analysis of the problem, and not that of another student.Signature:Neatly print the name(s) of the students you coll ab orated with on this assignment.Exam I is scheduled for Thursday, September 25 from 8:15 pm to 10:15 pm in room 10 Sparks for all sections. Theexam covers material problem sets 1 through 3, and recitations 1 through 3. The exam is closed-book, but you maybring one 8 1/2 by 11 inch note sheet, Cal culators are not allowed as graphical/scientific calculators are capabl e ofgraphing functions and solving ODEs.Reading assignment:• Lathi Chapter 2, Sections 2.1, 2.2, and 2.5• Priemer Chapter 6 sections 6.1 though 6.4Problem 11: (20 points)A LTI system with input f(t) and output y(t) may be represented as an ODE in the standard formdnydtn+ an−1dn−1ydtn−1+ · · · + a0y = bmdmfdtm+ bm−1dm−1fdtm−1+ · · · + b0f,where the coefficients aiand biare constant. Observe tha t the coefficient multiplying the highest derivative of y withrespect to time is unity. By introducing the deriva tive operator D ≡ d/dt and the polynomialsQ(D) = Dn+ an−1Dn−1+ · · · + a1D + a0P (D) = bmDm+ bm−1Dn−1+ · · · + b1D + b0,the ODE representation has the compact formQ(D) y(t) = P (D) f(t). (1)1. (10 points) The passive network in Figure 1 implements a l owpass filter. For a DC input, the output y isidentical to the input f as the inductors appear as short circuits. For a sinusoidal input, as the frequencyincreases the impedance of the inductors increases and as a result, the output amplitude decreases. Derive anexpression for the ODE tha t relates the output voltage y(t) to the input f(t). Express your answer using theform in equation 1 by providing expressions for the polynomials Q(D) and P (D) in terms of the param etersL1, R1, L2, and R2.Figure 1: Passive RL lowpass filter.2. (10 points) In most cases sensor noise is both unavoidable and undesirable. As an example, measurements ofthe bioelectric potential produced by the heart using an electrocardiogram (ECG) are typically corrupted by60 Hz signals originating from nearby devices. In this situation, where the noise si gnal is a sinusoid at a knownfrequency, it is possible to use a filter circuit to attenuate the noise component. For this problem, the sensoroutput i s corrupted by a 60 Hz sinusoidal signal . If the signal of interest has frequency components eithermuch lower or much higher than 60 Hz, we can pass the sensor output through a lowpass or high pass filter,respectively, to attenuate the 60 Hz noi se component. Suppose, however, that o ur sensor signal has frequencycomponents of interest that are both above and below 60 Hz. In this case, we can pass the output of the sensorthrough a notch filter that only attenuates frequency components around 60 Hz. Figure 2 shows an active RCnetwork, with input voltage f(t) and output voltage y(t), that implements a no tch filter. Assume that theoperational amplifiers are i deal, that is, they implement ideal voltage followers. Derive an expression for theODE that relates the output voltage y(t) to the input f(t). Express your answer using the form in equation 1by providing expressions for the polynom ial s Q(D) and P (D) in terms of the parameters R and C.Figure 2: Active RC notch filter.Problem 12: (20 points)Consider a LTI system with input f(t) and output y(t) that can be represented by the strictly-proper first-orderdifferential equationdydt+1τy(t) =Kτf(t), (2)where K and τ are constant parameters.1. (2 points) State the characteristic equation and find its characteristic root λ. A characteristic root λ maybe realor complex valued. If the real part of λ i s strictly negative, then the corresponding natural (or characteristic)mode eλtexponentially decays to zero as t increases. In this case the time constant associated with thenatural mode is defined as τ = −1/Re(λ). The time constant τ is a measure of how quickly a natural moderelaxes towards zero. The smaller τ is, the faster the mode decays.2. (3 points) Determine the zero-state unit-step response y(t) for t ≥ 0 in terms of the parameters τ and K.3. (2 points) What is the physical significance of the parameter K when τ > 0?4. (3 poi nts) For a unit-step input, the time required for the response to increase from 10% to 90% of its finalvalue is defined as the rise-time. Show tha t the rise-time for the first-order system considered in this problemistr= τ ln 9.This expression holds for any strictly-proper first-order system with a characteristic root that is strictly negative.5. (3 points) For a unit-step input, the time tsrequired for the zero-state response to reach and stay within 1%of its final value is called the settling time. Show that f or the first-order system the settling time is given byts= τ ln 100.This expression holds for any strictly-proper first-order system with a characteristic root that is strictly negative.6. (7 points) Show that the active RC circuit in Figure 3 can be represented in the form of equation 2, and specifythe parameters K and τ in terms of R1, R2, and C. Determine the rise-time, settling time, and DC gain ofthe network in Figure 3 , and express your results in terms of the parameters R1, R2, and C.Figure 3: Active RC circuit.Problem 13: (20 points)This problem illustrates the relationship between the natural response (homogeneous solution), the forced response(particular solution), the zero-state response, and the zero-input response of a LTI system. A certain linear time-invariant system with input f(t) and output y(t) is represented by the second-order differential equationd2ydt2+ 6dydt+ 9 y(t) = 18 f(t).Suppose that the initial conditions are y(0) = 1, ˙y(0) = −2 and that the input to the system isf(t) = (1 + 3e−2t)u(t).1. (3 points) State the characteristic equation and find its roots λ1and λ2.2. (5 points) Determine the total response y(t) = yh(t) + yp(t) by finding the homogeneous solution (naturalresponse) yh(t) and the particular solutio n (forced response) yp(t).3. (5 points) Determine the zero-input response, yzi(t) for t ≥ 0 by sol ving the appropriate ODE and using theappropriate initial
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