EE 350 Problem Set 3 Cover Sheet Fall 2016 Last Name (Print): First Name (Print): ID number (Last 4 digits): Section: Submission deadlines: Turn in the written solutions by 4:00 pm on Tuesday September 20 in the homework slot outside 121 EE East. Problem Weight 10 20 11 20 12 20 13 20 14 20 Total 100 Score The 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 collaborated with on this assignment.
Exam I is scheduled for Thursday, September 22 from 8:15 pm to 10:15 pm. The location will be announced during lecture and on the EE 350 web site. The exam covers material in problem sets 1 through 3, and recitations 1 through 5. The exam is closed-book, but you may bring one 8 1/2 by 11 inch note sheet, Calculators are not allowed as graphical/scientific calculators are capable of graphing functions and solving ODEs. Reading assignment: Lathi Chapter 2, Sections 2.1, 2.2, and 2.5 Problem 10: (20 points) A LTI system with input f(t) and output y(t) may be represented as an ODE in the standard form d n y dt + a d n 1 y n n 1 dt + + a d m f 0y = b n 1 m dt + b d m 1 f m m 1 dt + + b 0f, m 1 where the coefficients a i and b i are constant. Observe that the coefficient multiplying the highest derivative of y with respect to time is unity. By introducing the derivative operator D d/dt and the polynomials the ODE representation has the compact form Q(D) = D n + a n 1 D n 1 + + a 1 D + a 0 P(D) = b m D m + b m 1 D n 1 + + b 1 D + b 0, Q(D) y(t) = P(D) f(t). (1) 1. (10 points) The passive network in Figure 1 implements a lowpass filter. For a DC input, the output y is identical to the input f as the inductors appear as short circuits. For a sinusoidal input, as the frequency increases the impedance of the inductors increases and as a result, the output amplitude decreases. Derive an expression for the ODE that relates the output voltage y(t) to the input f(t). Express your answer using the form in equation 1 by providing expressions for the polynomials Q(D) and P(D) in terms of the parameters L 1, R 1, L 2, and R 2. Figure 1: Passive RL lowpass filter. 2. (10 points) In most cases sensor noise is both unavoidable and undesirable. As an example, measurements of the bioelectric potential produced by the heart using an electrocardiogram (ECG) are typically corrupted by 60 Hz signals originating from nearby devices. In this situation, where the noise signal is a sinusoid at a known frequency, it is possible to use a filter circuit to attenuate the noise component. For this problem, the sensor output is corrupted by a 60 Hz sinusoidal signal. If the signal of interest has frequency components either much 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 noise component. Suppose, however, that our sensor signal has frequency components of interest that are both above and below 60 Hz. In this case, we can pass the output of the sensor through a notch filter that only attenuates frequency components around 60 Hz. Figure?? shows an active RC network, with input voltage f(t) and output voltage y(t), that implements a notch filter. Assume that the operational amplifiers are ideal, that is, they implement ideal voltage followers. Derive an expression for the ODE that relates the output voltage y(t) to the input f(t). Express your answer using the form in equation 1 by providing expressions for the polynomials Q(D) and P(D) in terms of the parameters R and C.
Figure 2: Active RC notch filter. Problem 11: (20 points) Consider a LTI system with input f(t) and output y(t) that can be represented by the strictly-proper first-order differential equation dy dt + 1 τ y(t) = K f(t), (2) τ 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 real or complex valued. If the real part of λ is strictly negative, then the corresponding natural (or characteristic) mode e λt exponentially decays to zero as t increases. In this case the time constant associated with the natural mode is defined as τ = 1/Re(λ). The time constant τ is a measure of how quickly a natural mode relaxes 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 points) For a unit-step input, the time required for the response to increase from 10% to 90% of its final value is defined as the rise-time. Show that the rise-time for the first-order system considered in this problem is t r = 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 t s required for the zero-state response to reach and stay within 1% of its final value is called the settling time. Show that for the first-order system the settling time is given by t s = 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 specify the parameters K and τ in terms of R 1, R 2, and C. Determine the rise-time, settling time, and DC gain of the network in Figure 3, and express your results in terms of the parameters R 1, R 2, and C. Figure 3: Active RC circuit. Problem 12: (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 timeinvariant system with input f(t) and output y(t) is represented by the second-order differential equation d 2 y dt 2 + 6 dy + 9 y(t) = 18 f(t). dt Suppose that the initial conditions are y(0) = 1, ẏ(0) = 2 and that the input to the system is f(t) = (1 + 3e 2t )u(t). 1. (3 points) State the characteristic equation and find its roots λ 1 and λ 2. 2. (5 points) Determine the total response y(t) = y h (t) + y p (t) by finding the homogeneous solution (natural response) y h (t) and the particular solution (forced response) y p (t). 3. (5 points) Determine the zero-input response, y zi (t) for t 0 by solving the appropriate ODE and using the appropriate initial conditions. 4. (5 points) Determine the zero-state response, y zs (t) for t 0 by solving the appropriate ODE and using the appropriate initial conditions. 5. (2 points) It must be true that y h (t) + y p (t) = y zi (t) + y zs (t) because y(t) = y h (t) + y p (t) and y(t) = y zi (t) + y zs (t). In one or two sentences, explain why y zs (t) differs from y p (t), and why y zi (t) differs from y h (t).
Problem 13: (20 points) A certain linear time-invariant system with input f(t) and output y(t) is represented by the second-order differential equation d 2 y dt 2 + α dy + 400 y(t) = 800 f(t), dt where α is a real valued parameter. 1. (4 points) Determine the numeric value of the natural frequency and express the dimensionless damping ratio in terms of the parameter α. 2. (3 points) Determine the range of values α for which the system admits a DC gain and determine the numeric value of the DC gain. 3. (3 points) Determine the range of values of α for which the zero-state unit-step response is underdamped, overdamped, and critically damped. 4. (5 points) Determine the zero-input response y zi (t) for α = 32, y(0) = 10, and ẏ(0) = 0. 5. (5 points) Determine the zero-state unit-step response for α = 40. Problem 14: (20 points) Consider a set of three systems with input f(t) and output y(t) that are represented by the following ODEs for t 0: System 1: System 2: System 3: d 3 y dt + y 3 2d2 dt + 2dy + y(t) = f(t) 2 dt d 3 y dt + y 3 2d2 dt 5dy 6y(t) = f(t) 2 dt d 3 y dt + d2 y 3 dt + dy + y(t) = f(t) 2 dt 1. (3 points) Use MATLAB to obtain the characteristic roots for each system, and sketch these roots in the λ-plane. 2. (3 points) Based on the location of the characteristic roots, state whether the system is asymptotically stable, marginally stable, or unstable. 3. (4 points) For the asymptotically stable and unstable systems, justify your answer in part 2 by stating the form of the homogeneous solution for these systems for t 0. If the system has complex-valued characteristic roots, express the homogeneous solution in terms of real-valued sinusoidal terms, rather than complex-valued exponentials. 4. (10 points) (a) (2 points) Consider the zero-state response y 1 (t) of system 3 to the unit-step input f 1 (t) = u(t). Using your results from part 1, state the form of the homogeneous and particular solutions in terms of undetermined coefficients. As time approaches infinity, will the response y 1 (t) remain bounded? (b) (2 points) Consider the zero-state response y 2 (t) of system 3 to the sinusoidal input f 2 (t) = cos(t)u(t). Using your results from part 1, state the form of the homogeneous and particular solutions in terms of undetermined coefficients. As time approaches infinity, will the response y 2 (t) remain bounded? (c) (6 points) Verify your qualitative results in parts (a) and (b) by using MATLAB to numerically calculate the zero-state response. As an example, consider a system with input f(t) and output y(t) that has the ODE representation ÿ + 30ẏ + 200y = 8f + 400f. This system can be represented in MATLAB by the row vectors >> Q = [1, 30, 200];P = [8, 400]; Suppose we desire to find the zero-state response of the system to two separate inputs f 1 (t) = 4u(t)
and f 2 (t) = 4u(t) + 2e t u(t) over the time range 0 t 2 using a time vector of two hundred points. Construct the time vector using the command linspace >> t = linspace(0, 2, 200); while the input functions are generated as and >> f1 = 4 ones(size(t)); >> f2 = 4 + 2 exp( t); Find the zero-state responses with the command lsim as and To receive credit: >> y1 = lsim(p, Q, f1, t); >> y2 = lsim(p, Q, f2, t); Using this example as a guide, write an m-file that calculates and plots the zero-state response of system 3 system to the inputs f 1 (t) = u(t) f 2 (t) = cos(t) over the time range 0 t 100 using a time vector of one thousand points. Turn in a copy of your m-file that generates the results for parts 1 and 4. Include your name and section number in the m-file. Plot the zero-state responses for part 4 in two separate subplots on a single page, with the response to f 1 (t) in the upper subplot and the response to f 2 (t) in the lower subplot. Use the MATLAB command gtext to place your name and section number within the figure. Make sure that you appropriately label the x and y axes; no credit is given for MATLAB plots whose axes are unlabeled. Add an appropriate title to the figure, for example, Problem Set 3 Problem 14, Parts 1 and 4.