Main Topics: The Past, H(s): Poles, zeros, s-plane, and stability; Decomposition of the complete response.

Transcription

1 EE202 HOMEWORK PROBLEMS SPRING 18 TO THE STUDENT: ALWAYS CHECK THE ERRATA on the web. Quote for your Parent' Partie: 1. Only with nodal analyi i the ret of the emeter a poibility. Ray DeCarlo 2. (The need for uniquene.) Anybody, who i any good, i different from anybody ele. Felex Frankfurter 3. Politic i the art of preventing people from participating in affair that properly concern them. Paul Venerey. 4. The true worth of your travel lie not in where you came to be at journey end, but in whom you came to be along the way. ---????? (And o, the true worth of a coure i not in the grade you receive at the end, but in the intellectual maturity you gained along the way.) 5. A ingle arrow i eaily broken, but not ten in a bundle. Japanee Proverb 6. Laugh at yourelf before anyone ele can. Ela Maxwell 7. If you are patient in one moment of anger, you will ecape a hundred day of orrow. Chinee Proverb 8. There are two kind of people thoe who do the work and thoe who take the credit. Try to be in the firt group; there i le competition there. Indira Gandhi LAWS OF COMPUTER PROGRAMMING: I. Any given program, when running, i obolete. II. If a program i ueful, it will have to be changed. III. If a program i uele, it will have to be documented. V. Any program will expand to fill available memory. DUE WEDNESDAY FEBRUARY 14, HW 7 Main Topic: The Pat, H(): Pole, zero, -plane, and tability; Decompoition of the complete repone. 25. The pole-zero plot of a tranfer function H() i given below. (a) If the dc gain i 20, find H(). (b) Compute the impule repone. (c) Compute the tep repone. (d) Compute the derivative of the tep repone howing all calculation and explaining any intereting tep. Check: Your anwer to (b) hould be the derivative of your anwer to (c), ince the delta function i the derivative of the tep function. (e) If the input i 10e at u(t), find the poitive number a uch that the repone doe not have a term of the form Ke at u(t). Find the zero-tate repone under thi condition. (f) If the input i 10e at u(t), find the poitive number a uch that the repone doe not have a term of the form Ke at u(t). Find the zero-tate repone under thi condition.

2 page 2 Spring The DeCarlo-Meyer Motor Company, maker of the famou Laplace GTX, wihe to implement a new cruie control controller. The controller accept the error between the driver deired vehicle peed and the actual vehicle peed, e(t) = v de (t) v(t), and then output a throttle command, T (t), baed upon thi error. The following diagram (given in the -domain) illutrate how the controller and the vehicle interact for peed regulation: de V () E( ) = V de ( ) V( ) Cruie T() + Control TF, - H() Throttle-to- Vehicle Speed TF, P() V() In the -world, V de () i the deired vehicle peed, V () i the actual vehicle peed, E() i the difference in peed, T () i the throttle command output from the controller, H() i the controller tranfer function, and P() i the throttle-to-vehicle peed tranfer function that decribe the relationhip between a throttle command and vehicle peed repone. For 202 we are only intereted in the H() -part of the diagram. The control engineer at DMMC Reearch have decided to ue a Proportional-Integral-Derivative (PID) type controller. The tranfer function of a PID controller ha the form: H() = T () E() = K 1 + K 2 + K 3 = K n() 3 d() n() Part 1: Compute H() = K 3, i.e., pecifically compute n() and d() whoe leading d() coefficient are 1.

3 page 3 Spring 18 Part 2: The tranfer function i to be implemented uing the following circuit. Compute the tranfer function, denoted H cir (), of the circuit under the condition that R 4 = 1 Ω, C 2 = 0.5 F, and C 1 = C 3 Part 3: The PID controller form wa choen becaue it can (i) drive E() to zero over time (thu maintaining the exact driver commanded peed), (ii) output a nonzero T () ignal when E() i zero (thu maintaining the throttle command to keep the vehicle at the deired peed once it i reached), and (iii) anticipate velocity error (the derivative or multiplication by term) and take early corrective action. The deired reult of the control action i that the driver feel that the vehicle i acceptably maintaining the commanded velocity. The generally out-of-control control engineer have decided it i bet to place the tranfer function zero at 4 and 5. Further, h( ) i deired to be equal to 0.1, which i what i termed a mall teady tate allowable error. For part 3, your job i to: (a) Compute numerical value for K 1, K 2, and K 3. (b) Determine the remaining value for the circuit element in the implementation circuit, a reult of cot cutting meaure o that company executive can get $500k bonue in an off hore account in the Cayman Iland tax free, incentive pay, jut like at univeritie. (Thee tatement are fully fictitiou being fallaciou fabrication of the author faulty imagination and thu any bai in reality i completely coincidental.) Part 4: An error input to the controller i e(t) = 2t 2 e 4t u(t) V, find T (t) = L 1 T () { } auming zero initial condition.

4 page 4 Spring (a) The circuit below ha tranfer function H() = V out V in and R 1 = 4 Ω and R 2 = 2 Ω.. It i known that L = 0.5 H (i) Compute the tranfer function. (ii) Compute the COMPLETE range of r m for which the circuit i BIBO table. (b) Let R = 2 Ω, C = 0.25 F. Compute H() = V out () V in () the circuit i BIBO table.. Determine the complete range of m for which 28. In the circuit hown below, R 1 = 10 Ω, R 2 = 40 Ω, C = 0.5 F, the witch S i open for t < 0 and cloe at t = 0. (a) Draw the equivalent circuit in the -domain uing the current ource model for the capacitor uing the literal v C (0 ). (b) For t 0, ue nodal analyi in the -domain to obtain the relationhip between V C (), V 1 (), I 2 () and the initial condition v C (0 ). Specifically, find an expreion for V C (). Check: V C () =?? ( +??) V 1 () +?? +?? ) ( ) I 2 () + v C (0 ( +??)

5 page 5 Spring 18 For the remainder of the problem aume v 1 (t) = 50u(t) V, i 2 (t) = 2e 0.5t u(t) u(t + 20)u( t) A. (c) Find v C (0 ) and then find the zero input repone, v C,zi (t) for t 0 by the Laplace tranform method. (d) Find the zero-tate repone, v C,z (t) for t 0 by the Laplace tranform method. Separate the zero-tate repone into one term due to v 1 and another term due to i 2. (e) Write down the complete repone. Identify the teady-tate repone and the tranient repone. (f) Utilize the anwer of previou part to write down the complete repone for t 0 for the following two cae without reolving the circuit equation: v C (0 ) v 1 (t) i 2 (t) Cae u(t) V 4e 0.5t u(t) A Cae u(t) V 4e 0.5t u(t) A Remark: the circuit i LINEAR in the zero-tate repone relative to each ource term which i why you were aked to eparate them in part (c). Further, the circuit i linear in the zero-input repone. You mut look up and figure out what linear mean if you do not already know. DUE WEEK 8, MONDAY FEBRUARY 19; HW 8 Main Topic: Sinuoidal Steady State Analyi and Frequency Repone 29. (a) The tranfer function of a complex circuit i given by H() = V out () V in () = 5 ( 3) ( + 3)( 2 + 4). The circuit i excited by an input v in (t) = 2co(3t)u(t) V. Compute the magnitude and phae of v out (t) in SSS. Plot the magnitude and phae of H( jω ) for 0 ω 25 rad/ in MATLAB or equivalent. Turn in your code and printout.

6 page 6 Spring 18 (b) For the circuit immediately below, C = F, L = H, and R = 4 Ω. Plot the magnitude and phae repone (in degree) for the tranfer function H() = V out () for 0 ω 200 rad/. V in () Conider the following code in MATLAB: w = 0:0.02:200; num = (Inert coefficient vector of numerator polynomial.) den = (Inert coefficient vector of denominator polynomial.) h = freq(num,den,w); plot(w,ab(h)) plot(w,angle(h)*180/pi) (i) The magnitude repone i approximately what type of filtering action. (ii) Compute v out (t) in inuoidal teady tate for the circuit when v in (t) = 10co(80t)u(t) V. (iii) Compute v out (t) in inuoidal teady tate for the circuit when v in (t) = 10co(70t)u(t) V. (iv) Are your magnitude value in (ii) and (iii) conitent with the magnitude plot of part (i)? ( + 80)( 80) (c) Suppoe H() = 125 ( + 40) ha input v in (t) = co(ωt) V. (i) Compute v out, (t) = K co(ωt +θ) for ω = 0,80,400,800 rad/ec. (ii) Now plot the frequency repone in MATLAB, both magnitude and phae in degree on a emilog cale from 1 rad/ec to 2000 rad/ec. (iii) Verify graphically that your numerical calculation are conitent with the magnitude frequency repone. (iv) Turn in the plot and your MATLAB code. 30. The tranfer function of a NORMALIZED Butterworth band pa (BP) characteritic i H NormBP ( ) = A BP filter pae frequencie in a range and reject frequencie outide that range, meaning that any ignal containing a bunch of frequencie loe any frequency content outide the o called pa band and keep thoe frequencie within the o called pa band. Such filter are ued for

7 page 7 Spring 18 example on high end peaker ytem to direct mid-range audio frequencie (ay 500 to 3000 Hz) to the mid-range peaker and reject low frequency and high frequency content of the audio. Such normalized filter need to be frequency caled o that the appropriate range of frequencie are paed and not rejected. In thi cae the frequency cale factor K f (in which in the tranfer function) i computed a the geometric mean of the low and high end pa band frequencie: (a) Compute f mean = 500*3000 and ω mean = 2*π * f mean. The actual unnormalized bandpa tranfer function, H BP () = H NormBP H BP ( ) = 5 ω mean ω mean 16 3 ω mean ω mean 2 K f K f i now given a: ω mean + 5 (b) Uing MATLAB or equivalent, plot the frequency repone for 0 f 10 4 Hz uing the emilogx plot command in MATLAB. You mut turn in your code and your plot. f = 0:2:10000; w = 2*pi*f; 2 num = [16/ω mean 0 0]; den = [ U R on your own]; h = freq(num,den,w); emilogx(w/(2*pi), ab(h)) emilogx(w/(2*pi),angle(h)*180/pi) (c) From your plot in part (b), compute approximately the magnitude and phae of the teady tate output voltage, v out, (t) = K co(ωt +θ) V (θ in degree) when the input i: (i) 10co(2π 250t), (ii) 10co(2π 750t), (iii) 10co(2π 2500t), (iv) 10co(2π 7500t). 31. (a) In the op am circuit below, R 1 = R 2 = 1 Ω, C 1 = 5 mf, C 2 = C 3 = 2 mf, and R 4 = 1 Ω. The magnitude frequency repone i of which (general) filter type? Jutify your anwer analytically. Avoid the tuff of bull.

8 page 8 Spring 18 (b) (Filtering and impdence) The circuit below i of what filter type? Jutify Your Anwer. (c) The tranfer function having a firt order numerator and denominator that bet meet the phae repone plot below i H() =?? Jutify your anwer by looking at critical point. Check in MATLAB or equivalent.

9 page 9 Spring Phae in degree frequency in rad/ (d) The tranfer function that bet matche the magnitude frequency repone below ha a third order numerator and a third order denominator. Contruct it. Then check it in MATLAB and modify your number a needed.

10 page 10 Spring Magnitude H(jw) w in rad/ 32. Thi problem emphaize a qualitative undertanding of pole and zero on frequency repone. Conider the upper part of the pole-zero plot below. Note complex pole and zero mut occur in conjugate pair. (a) If H( ) = 1, contruct H(). (b) In MATLAB or equivalent, (YOU MUST TURN IN YOUR MATLAB CODE AS WELL AS THE PLOT) (i) Plot the magnitude of the frequency repone for 0 ω 25 rad/. (ii) Then locate the half power point (in thi cae the gain magnitude i 1/qrt(2)) and the frequencie where they occur. (Power i proportional to gain-quared; (1/qrt(2))-quared = 0.5.) (iii) What type of filtering i taking place with thi tranfer function? (iv) What frequency range i rejected? Why? What frequency range i paed? Why? Hint: Half power i the generally accepted cutoff for paed and non-paed frequencie. You hould be able to read thee range from your frequency repone plot. (c) If an input ignal of the form v in (t) = 10in(10.1t)u(t) V were applied to thi tranfer function, APPROXIMATELY what would be the teady tate magnitude of the repone? Why? (d) If an input ignal of the form v in (t) = 4in(36t)u(t) V were applied to thi tranfer function, APPROXIMATELY what would be the teady tate magnitude of the repone? Why?

11 page 11 Spring 18 HW 9 DUE FRIDAY FEBRUARY (Scaling Problem) (a) Compute the tranfer function H1() = Figure (a) below and then chooe parameter value o that H1() = that C2 = 1 F. Vout () of the op amp circuit in Vin () 4 under the contraint ( + 2)( + 8)

12 page 12 Spring 18 (b) Compute the tranfer function H 2 () = V out () V in () parameter value o that H 2 () = ( + 8) (c) Compute H() = H 1 () H 2 () = ( Stage1) (Stage2). of the circuit of Figure (b) below and compute under the contraint that G = 1 mho. (d) The pole location of uch normalized tranfer function are not uually phyically meaningful, and require o called frequency caling. The reitor value and capacitor value are alo not phyically meaningful for an op amp circuit and thee require o called magnitude caling. Magnitude caling i achieved by modifying the value of the reitor and capacitor. Frequency caling i achieved by modifying the value of the capacitor. Suppoe the mallet non-zero pole location hould be at 2500 (for the entire tranfer function). The mallet reitor value hould be at 2.2 kω for tage 1 and for tage 2, C 1 final = 100 nf. (i) Determine the appropriate frequency cale factor K f to achieve thee new pole location. (ii) Determine K m1 for tage 1 and K m2 for tage 2. Becaue the op amp circuit are in cacaded tage, eparate magnitude cale factor are allowed. However, i common to both tage o only one frequency cale factor i allowed. (iii) Now compute all the final parameter value of your circuit labeling them a C i, final and R i, final for appropriate integer i beneath the word Figure (a) or Figure (b) a appropriate. (e) Doe magnitude caling change the voltage ratio? Figure (a) Figure (b)

13 page 13 Spring (a) (i) Uing the integral equation for convolution and the fact that r(t) = tu(t) OR uing Laplace tranform method, how that the following i true: y(t) = u(t) r(t) = 1 2 t2 u(t) (ii) Similar to part a-i, how that the following i true: y(t) = r(t) r(t) = 1 6 t3 u(t) (b) Conider y(t) (the firt figure below) and f (t) hown below on the right. (i) Expre y(t) and f ( t) a um of caled/hifted tep and ramp. (ii) Now compute g(t) = y(t)* f ( t) uing cla formula and part (a) above a needed. (c) Conider f (t) (hown in the figure below on the left) and h(t) alo hown below. (i) Expre f (t) and h(t) a um of caled/hifted tep and ramp. (ii) Now compute y(t) = h(t)* f (t) uing cla formula and part (a) above a needed. 35. Compute the impule repone, h(t), of the circuit below in which R 1 = 4 Ω, R 2 = 8 Ω, and

14 page 14 Spring 18 C = 0.5 F, and v C (t) i the output. Suppoe f 1 (t) = K 1 e at u( t), a > 0 and f 2 (t) = K 2 e bt u(t), b > 0. Compute each of the indicated convolution. Aume a b. When plotting, let K 1 = K 2 = 10. (a) y 1 (t) = h(t)*u( t) in term of a and b, and ketch for 4 < t < 4 when a = 1, b = 2 in MATLAB. (b) y 2 (t) = h(t)* f 1 (t) in term of a and b, and ketch for 4 < t < 4 when a = 1, b = 2 in MATLAB. (c) y 3 (t) = h(t)* f 2 (t) in term of a and b, and ketch for 4 < t < 4 when a = 1, b = 2 in MATLAB. 36. (a) In the circuit below, uppoe C = 0.25 F, R 1 = 1 Ω, R = 100 Ω, and K = 1 i a contant. (i) Find the tranfer function of the op amp circuit, i.e., find H() = V out () V in () = A p. (ii) Determine the impule repone, h(t). Thi will be ued in part (b). (b) Suppoe f 1 (t) = K 1 e at u(1 t), a > 0 and f 2 (t) = K 2 e bt u(t +1), b > 0. (i) Compute v out,1 (t) = h(t)*u( t). Then ketch for 4 < t < 4 uing MATLAB or equivalent. (ii) v out,2 (t) = h(t)* f 1 (t) when a = 1 and K 1 = 2. Then graph for 4 < t < 4 uing MATLAB or equivalent. (iii) v out,3 (t) = h(t)* f 2 (t) when b = 2 and K 2 = 2. Then graph for 4 < t < 4 uing MATLAB or equivalent.