134 TIME-DEPENDENT CIRCUIT ANALYSIS WithiL(t)=iC(t)=i(t),vL(t)=LdiL(t)/dt, andvC(t)=C^1 ∫t −∞iC(τ)dτ, one obtains the following ...
3.2 TRANSIENTS IN CIRCUITS 135 xtr(t)=Aest (3.2.30) Substituting Equation (3.2.30) in Equation (3.2.28), one gets s^2 Aest+asAes ...
136 TIME-DEPENDENT CIRCUIT ANALYSIS Case 3: Complex Conjugate Roots For ( 4 b/a^2 ) >1, as seen from Equation (3.2.33), the r ...
3.2 TRANSIENTS IN CIRCUITS 137 The steady-state responsevC,ss(t)due to a dc source is determined by replacing inductors with sho ...
138 TIME-DEPENDENT CIRCUIT ANALYSIS The complete response isvC(t)= 1. 0 +(A 1 +A 2 t)e−t. Evaluating bothvCand dvC/dt att= 0 +, ...
3.2 TRANSIENTS IN CIRCUITS 139 steady-state value while it gradually approaches the steady-state value. Practical systems are ge ...
140 TIME-DEPENDENT CIRCUIT ANALYSIS FromiL( 0 +)=0 andvC( 0 +)=8 V, it follows thatA 1 +A 2 =0 andB 1 +B 2 + 4 =8. Simultaneous ...
3.2 TRANSIENTS IN CIRCUITS 141 f(t) t A 0 T Figure 3.2.6Rectangular pulse. The student is encouraged to justify this statement b ...
142 TIME-DEPENDENT CIRCUIT ANALYSIS The form of the natural response due to an impulse can be found by the methods presented in ...
3.3 LAPLACE TRANSFORM 143 f(t)=0 fort<0 and allf(t) exist fort≥0. Also note that in Table 3.3.1, functions 8 through 20 can b ...
144 TIME-DEPENDENT CIRCUIT ANALYSIS becomes an inherent part of the final total solution. For cases with zero initial condition, ...
3.3 LAPLACE TRANSFORM 145 The roots of the equation N(s)= 0 (3.3.7) are said to be thezerosofF(s); and the roots of the equation ...
146 TIME-DEPENDENT CIRCUIT ANALYSIS F 1 (s)= N 2 (s) (s+a+jb)(s+a−jb)D 2 (s) (3.3.14) or F 1 (s)= K 1 s+(a+jb) + K 2 s+(a−jb) + ...
3.3 LAPLACE TRANSFORM 147 Multiple Poles Let us consider thatF 1 (s) has all simple poles except, say, ats=p 1 which has a multi ...
148 TIME-DEPENDENT CIRCUIT ANALYSIS With the aid of theorems concerning Laplace transforms and the table of transforms, lin- ear ...
3.3 LAPLACE TRANSFORM 149 v(t) v(t) = Ri(t) i(t)^12 R Resistance +− V(s) V(s) = RI(s) I(s)^1 R Time − Domain Network Transformed ...
150 TIME-DEPENDENT CIRCUIT ANALYSIS (a) 2 F v(t) 4 Ω 1 2 10 V 2 Ω 1 H t = 0 S + − + − Figure E3.3.1 I(s) V(s) (b) 10 s 2 s 2 Ω s ...
3.3 LAPLACE TRANSFORM 151 EXAMPLE 3.3.2 Obtainv(t) in the circuit of Figure E3.3.2(a) by using the Laplace transform method. + − ...
152 TIME-DEPENDENT CIRCUIT ANALYSIS Substitution ofI(s) and solving forVBgives VB= 10 ( 3 s+ 1 )(s+ 1 ) 3 s ( s^2 + 3 s+ 2 ) VA= ...
3.3 LAPLACE TRANSFORM 153 Transfer functions V 2 (s); V 2 (s); I 2 (s); I 2 (s) V 1 (s) I 1 (s) V 1 (s) I 1 (s) + − + − V 1 (s) ...
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