Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part14

Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part14

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Chapter 4 Circuit Analysis with Laplace Transforms 4 2 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications Figure 4.3. Capacitive circuit in time domain and complex frequency domain Note: In the complex frequency domain, the terms and are referred to as complex inductive impedance , and complex capacitive impedance respectively. Likewise, the terms and and are called complex capacitive admittance and complex inductive admittance respectively. Example 4.1 Use the Laplace transform method and apply Kirchoff’s Current Law (KCL) to find the voltage across the capacitor for the circuit of Figure 4.4, given that . Figure 4.4. Circuit for Example 4.1 Solution: We apply KCL at node as shown in Figure 4.5. Figure 4.5. Application of KCL for the circuit of Example 4.1 Then, or + Time Domain + C v C t () i C t i C t C dv C dt -------- = v C t 1 C --- i C t d t = + Complex Frequency Domain + + V C s v C 0 s ---------------- I C s 1 sC ------ I C s sCV C s Cv C 0 = V C s I C s ------------ v C 0 s + = sL 1 sC 1 sL v C t v C 0 6 V = R C + + V 1 F 1 Ω 12u 0 t v S v C t A R C + + V 1 F 1 Ω 12u 0 t v S v C t A i R i C i R i C + 0 =
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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 4 3 Copyright © Orchard Publications Circuit Transformation from Time to Complex Frequency (4.1) The Laplace transform of (4.1) is By partial fraction expansion, Therefore, Example 4.2 Use the Laplace transform method and apply Kirchoff’s Voltage Law (KVL) to find the voltage across the capacitor for the circuit of Figure 4.6, given that . Figure 4.6. Circuit for Example 4.2 Solution: This is the same circuit as in Example 4.1. We apply KVL for the loop shown in Figure 4.7. v C t () 12u 0 t 1 ------------------------------------- 1 dv C dt -------- + 0 = C v C t + 12u 0 t = sV C s v C 0 V C s + 12 s ----- = s1 + V C s 12 s 6 + = V C s 6s 12 + ss 1 + ------------------ = V C s + + r 1 s --- r 2 + ---------------- + == r 1 + + ----------------- s0 = 12 r 2 + s = 6 V C s 12 s 6 + ----------- = 12 6e t 12 t u 0 t v C t v C t v C 0 6 V = C + + V 1 F 1 Ω 0 t v S v C t R
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Chapter 4 Circuit Analysis with Laplace Transforms 4 4 Signals and Systems with MATLAB Computing and Simulink
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Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part14

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