lab_1_f08

lab_1_f08 - EE 350 CONTINUOUS-TIME LINEAR SYSTEMS ANALYSIS...

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EE 350 CONTINUOUS-TIME LINEAR SYSTEMS ANALYSIS FALL 2008 Laboratory #1: Time Response Characteristics The objectives of the Frst laboratory are to: Review the representation of passive circuits using ODE models. Utilize MATLAB to Fnd the roots of the characteristic equation, numerically solve ODEs, and determine time-response characteristics. Estimate component values using time response characteristics acquired with MATLAB and the dSPACE data acquisition system. Results from Laboratory #1 are required to complete Problem Set 4. 1 ODE Representation of Passive Circuits The passive circuits in ±igure 1 produce an output voltage y ( t ) in response to a driving voltage f ( t ), and may be represented by an ODE of the form Q ( D ) y ( t )= P ( D ) f ( t ) ( D n + a n - 1 D n - 1 + ··· + a 1 D + a 0 ) y ( t ( b m D m + b m - 1 D m - 1 + + b 1 D + b 0 ) f ( t ) where D k d k dt k is the differential operator, and a i and b i are constant coefficients. The parameter n is the highest derivative of y ( t ) and represents the order of the system. ±or a circuit, the number of independent energy storage elements determines the system order. The adjective independent speciFes that capacitors (inductors) connected in series or parallel count as a single energy storage element.The laboratory instructor will review determination of the polynomials Q ( D ) and P ( D ) using circuit analysis techniques from EE 210 and will help you complete Table 1 for the circuits shown in ±igure 1. Circuit Q ( D ) P ( D ) Frst-order RC Frst-order RL second-order RLC Table 1: Polynomials P and Q representing the passive circuits in ±igure 1.
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Figure 1: Passive circuits considered in Laboratory #1: (A) RC circuit, (B) RL circuit, and (C) RLC circuit.
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2 Simulation and Analysis using a MATLAB M-fle This section introduces several MATLAB commands that are useful for studying ODEs and their solutions. The MATLAB command roots determines the roots of the characteristic equation Q ( λ )= λ n + a n - 1 λ n - 1 + ··· + a 1 λ + a o =0 . For example, if Q ( D D 3 +4 D 2 + D - 3 , enter >> Q =[1 , 4 , 1 , - 3]; roots( Q ) and MATLAB will return the roots of Q ( λ ) = 0. Note that the polynomial Q ( D ) is entered as a row vector in MATLAB, where the ±rst element is the coefficient associated with D 3 . The command step determines the unit-step response of a system. For example, suppose we want to determine the unit-step response of a second-order system described by the ODE d 2 y dt 2 dy dt + 100 y ( t ) = 200 f ( t ) . First represent the ODE as Q ( D ) y ( t P ( D ) f ( t ) , where Q ( D D 2 D + 100 P ( D )=2 0 0 . Represent these polynomials in MATLAB as row vectors >> Q =[ 1 , 4 , 100]; >> P = [200]; The MATLAB command >> step( P,Q ) generates a plot of the unit-step response over a time interval determined by MATLAB. To generate the unit-step response at 200 points over a speci±ed time interval, say 0 t 5, use the commands >> Q , 4 , 100]; >> P = [200]; >> t = linspace(0 , 5 , 200); >> step( P,Q,t ) You can also save the response to a vector y using >> Q , 4 , 100]; >> P = [200]; >> t = linspace(0 , 5 , 200); >> y = step( ); MATLAB provides a command, lsim , for determining the zero-state response of a LTI system to an arbitrary input. As an example, the zero-state response of the system ˙ y + y ( t )=6 f ( t
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This note was uploaded on 10/13/2008 for the course EE 350 taught by Professor Schiano,jeffreyldas,arnab during the Fall '07 term at Penn State.

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lab_1_f08 - EE 350 CONTINUOUS-TIME LINEAR SYSTEMS ANALYSIS...

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