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Chapter 4 Circuit Analysis with Laplace Transforms 4 34 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications From (1) (4) From (2) and with (4) (5) By substitution of (4) and (5) into (3) we obtain and thus By substitution of the given values and after simplification we obtain We use the MATLAB script below to plot this function. w=1:10:10000; s=j.*w; Gs=7.83.*10.^7./(s.^2+1.77.*10.^4.*s+5.87.*10.^7);. .. semilogx(w,abs(Gs)); xlabel('Radian Frequency w'); ylabel('|Vout/Vin|');. .. title('Magnitude Vout/Vin vs. Radian Frequency'); grid V 1 s () 1R 4 R 3 R 4 + R 3 R 4 ----------------------------------------V out s R 3 R 3 R 4 + -----------------------V s == V 2 s R 2 1 R 2 ----- C 1 s +   V 1 s 2 C 1 s + V 1 s V 2 s R 3 2 C 1 s + R 3 R 4 + ------------------------------------V s = 1 R 1 1 R 2 C 2 s ++ R 3 2 C 1 s + R 3 R 4 + s V in s R 1 --------------- 1 R 2 R 3 R 3 R 4 + s C 2 sV s = 1 R 1 1 R 2 C 2 s R 3 2 C 1 s + R 3 R 4 + ------------------------------------ 1 R 2 R 3 R 3 R 4 + ----------------------- C 2 s V s 1 R 1 -----V s ( ) = Gs V s V s ----------------- 1 R 1 1 R 1 1 R 2 C 2 s R 3 2 C 1 s + R 3 R 4 + 1 R 2 R 3 R 3 R 4 + C 2 s ---------------------------------------------------------------------------------------------------------------------------------------------- 7.83 10 7 × s 2 1.77 10 4 s × 5.87 10 7 × ---------------------------------------------------------------------- =

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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 4 35 Copyright © Orchard Publications Solutions to End of Chapter Exercises The plot above indicates that this circuit is a second order low pass filter. 10 0 10 1 10 2 10 3 10 4 0.4 0.6 0.8 1 1.2 1.4 Radian Frequency w |Vout/Vin| Magnitude Vout/Vin vs. Radian Frequency
Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 5 1 Copyright © Orchard Publications Chapter 5 State Variables and State Equations his chapter is an introduction to state variables and state equations as they apply in circuit analysis. The state transition matrix is defined, and the state space to transfer function equivalence is presented. Several examples are presented to illustrate their application. 5.1 Expressing Differential Equations in State Equation Form As we know, when we apply Kirchoff’s Current Law (KCL) or Kirchoff’s Voltage Law (KVL) in networks that contain energy storing devices, we obtain integro differential equations. Also, when a network contains just one such device (capacitor or inductor), it is said to be a first order circuit . If it contains two such devices, it is said to be second order circuit, and so on. Thus, a first

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