Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part16

Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part16

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Chapter 4 Circuit Analysis with Laplace Transforms 4 18 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications (4.34) and for simplicity, let , and . By substitution into (4.34) we obtain (4.35) Next, we let the input be the unit step function , and as we know from Chapter 2, . Therefore, (4.36) To find , we perform partial fraction expansion, and for convenience, we use the MAT- LAB residue function as follows: num= 1; den=[1 3 1 0];[r p k]=residue(num,den) r = -0.1708 1.1708 -1.0000 p = -2.6180 -0.3820 0 k = [] Therefore, (4.37) The plot for is obtained with the following MATLAB script, and it is shown in Figure 4.23. t=0:0.01:10; ft= 1+1.171.*exp( 0.382.*t) 0.171.*exp( 2.618.*t); plot(t,ft); grid The same plot can be obtained using the Simulink model of Figure 4.24, where in the Function Block Parameters dialog box for the Transfer Fcn block we enter for the numerator, and for the denominator. After the simulation command is executed, the Scope block dis- plays the waveform of Figure 4.25. Gs () V out s V in s ------------------ = 1 R 1 1 R 1 1 R 2 1 R 3 sC 1 +++ sR 3 C 2 1 R 2 + [] -------------------------------------------------------------------------------------------------------------------------------- = R 1 R 2 R 3 1 === C 1 C 2 1 F == G s V s V s = 1 s 2 3s 1 ++ ------------------------- = u 0 t u 0 t 1s V s V s 1 s -- 1 s 2 1 1 s 3 2 s --------------------------- = v t 1 s 1 s 2 1 1 s 1.171 s0 . 3 8 2 + --------------------- 0.171 s2 . 6 1 8 + + =   1 1.171e 0.382t 0.171e 2.618t + v t = v t 1 3 1

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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 4 19 Copyright © Orchard Publications Using the Simulink Transfer Fcn Block Figure 4.23. Plot of for Example 4.8. Figure 4.24. Simulink model for Example 4.8 Figure 4.25. Waveform for the Simulink model of Figure 4.24 0 2 4 6 8 10 -1 -0.8 -0.6 -0.4 -0.2 0 v out t ()
Chapter 4 Circuit Analysis with Laplace Transforms 4 20 Signals and Systems with MATLAB Computing and Simulink

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Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part16

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