ColorGrayscale Jeffrey Schiano 2014. All rights reserved. Rec 4. EE 350Continuous-Time Linear SystemsRecitation 41 Jeffrey Schiano 2014. All rights reserved. Rec 4. Recitation 4 Topics• Solved Problems– ODE representation of circuits– Solution of ODEs– Complex Numbers• MATLAB Programming– Numerical solution of ODEs2 Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 1• The passive filter shown below has input f(t) and output y(t)1. Is the circuit a low-pass or high-pass filter? Why?2. Does C1effect the output y(t)? Why?3. Derive an ODE representation of the network3Lf(t)1CCy(t)R Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 1 Solution4 Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 1 Solution5 Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 2• The passive circuit shown below has input f(t) and output y(t)1. What are the DC and high-frequency gains?2. Derive an ODE representation of the network6f(t)1C2Cy(t)1R2R Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 2 Solution7 Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 2 Solution8 Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 2 Solution9 Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 2 Solution10 Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 3• The active circuit shown below has input f(t) and output y(t). 1. What are the DC and high-frequency gains?2. Derive an ODE representation of the network assuming an ideal operational amplifier111Cy(t)1R2C2Rf(t) Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 3 Solution12 Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 3 Solution13 Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 3 Solution14 Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 3 Solution15 Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 4• A LTI system with input f(t) and output y(t) has the ODE representation• Consider the following input and initial conditions1. Determine the total response y(t) using the classical solution method2. Determine the zero-input and zero state responses3. Do the zero-input and zero-state responses correspond to the homogeneous and particular solutions from part 1?16324()yyy ft 3( ) 1 + e for t 0, (0) 4, (0) 8tft y y Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 4 Solution17 Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 4 Solution18 Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 4 Solution19 Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 4 Solution20 Jeffrey Schiano 2014. All rights reserved. Rec 4. Is yzs(0) always zero?• When a system is strictly proper, that is when the degree, n, of the Q(D) is greater then the degree, m, of P(D), the zero-state response at time t = 0 must be zero• For proper (m=n) or improper (m>n) systems, the zero-state response may be nonzero at time t = 021 Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 5• The passive circuit shown below has input f(t) and output y(t). 1. Determine the ODE representation2. Is the system strictly proper, proper, or improper?3. Determine and sketch the zero-state unit-step response when both resistors are 10 k and C is 1 µF22f(t)Cy(t)2R1R Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 5 Solution23 Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 5 Solution24 Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 5 Solution25 Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 5 Solution26 Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 6• Using Euler’s identity show that27*() ()22 11.cos( )22.sin( )23. cos( )224. cos( ) sin( ) cos Tan,,,,,and are real, is complexjjjjjt jt teeeejcceecetcBAtBtAB tAAB t c Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 6 Solution28 Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 6 Solution29 Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 6 Solution30 Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 6 Solution31 Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 6 Solution32 Jeffrey Schiano 2014. All rights reserved. Rec 4. Using MATLAB for ODE Analysis• The MATLAB command roots provides a tool for determining the roots of the characteristic equation• Represent Q(D) = D3+ 4D2+ D – 3 in MATLAB as• Determine the roots of the characteristic equation 33>>Q=[1,4,1,-3];>>roots(Q) Jeffrey Schiano 2014. All rights reserved. Rec 4. Using MATLAB for ODE Analysis• The MATLAB command step determines the zero-state unit-step response• Consider the ODE• The following commands generates the zero-state unit-step response 344 100 200 ( )yy y ft >>Q=[1,4,100]; P = 200;>>step(P,Q) Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 7• The passive circuit shown below has input f(t) and output y(t). 1. Determine the ODE representation2. Let L = 1 H, and C = 4 µF. Write a m-file that determines the roots of the characteristic equation and plots the zero-state unit-step responses for R = 50 , 1k, and 2.2k in a single figure for 0 < t < 80 ms using 1000 points35f(t)CRLy(t) Jeffrey Schiano 2014. All rights reserved. Rec 4. Problem 736 Jeffrey Schiano 2014. All rights reserved. Rec 4. EE 350Continuous-Time Linear SystemsRecitation 41 Jeffrey Schiano 2014. All rights reserved. Rec 4. Recitation 4 Topics• Solved Problems– ODE representation of circuits– Solution of ODEs– Complex Numbers• MATLAB Programming– Numerical solution of ODEs2 Jeffrey Schiano 2014. All rights
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