Explanation Given data: Refer to Figure 8.1 in the textbook for the given circuit. The underdamped response of the circuit is given as follows: v ( t ) = ( A 1 + A 2 ) e − α t cos ( ω d t ) + j ( A 1 − A 2 ) e − α t sin ( ω d t ) (1) The initial value of the inductor current is I 0 . The initial value of the capacitor voltage is V 0 . Formula used: Write the expression of voltage response v ( t ) for underdamped response of a parallel RLC circuit as follows: v ( t ) = B 1 e − α t cos ( ω d t ) + B 2 e − α t sin ( ω d t ) , t ≥ 0 (2) Here, B 1 , B 2 are arbitrary constants, α is the neper frequency. and ω d is the damping frequency of response. Refer to Equation (8.16) in the textbook and write the expression to obtain the value of B 1 . v ( 0 + ) = B 1 (3) Refer to Equation (8.17) in the textbook and write the expression to obtain the value of B 2 . d v ( 0 + ) d t = − α B 1 + ω d B 2 (4) Calculation: Compare Equation (1) with (2) and write the expression for B 1 as follows: B 1 = A 1 + A 2 Substitute ( A 1 + A 2 ) for B 1 in Equation (3) as follows: v ( 0 + ) = A 1 + A 2 It is clear that, the value of v ( 0 + ) is always a real number. Therefore, consider the value of v ( 0 + ) as K 1 . Substitute K 1 for v ( 0 + ) as follows: K 1 = A 1 + A 2 A 1 = K 1 − A 2 (5) Compare Equation (1) with (2) and write the expression for B 2 as follows: B 2 = j ( A 1 − A 2 ) Substitute ( A 1 + A 2 ) for B 1 and j ( A 1 − A 2 ) for B 2 in Equation (4) as follows: d v ( 0 + ) d t = − α ( A 1 + A 2 ) + ω d j ( A 1 − A 2 ) It is clear that, the value of d v ( 0 + ) d t is always a real number. Therefore, consider the value of d v ( 0 + ) d t as K 2

Electric Circuits Plus Mastering Engineering with Pearson eText 2.0 - Access Card Package (11th Edition) (What's New in Engineering)

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ISBN: 9780134814117

Author: NILSSON, James W., Riedel, Susan

Publisher: PEARSON

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Chapter 8, Problem 22P

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Prove that the arbitrary constant A2 is the complex conjugate of A1 in the given underdamped voltage response.

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Chapter 8, Problem 21P

Chapter 8, Problem 23P

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Electric Circuits Plus Mastering Engineering with Pearson eText 2.0 - Access Card Package (11th Edition) (What's New in Engineering)

Ch. 8.1 - The resistance and inductance of the circuit in...Ch. 8.2 - Use the integral relationship between iL and v to...Ch. 8.2 - Prob. 3AP Ch. 8.2 - Prob. 4AP Ch. 8.2 - Prob. 5AP Ch. 8.3 - Prob. 6AP Ch. 8.4 - Prob. 7AP Ch. 8.4 - Prob. 8AP Ch. 8.4 - Repeat Assessment Problems 8.7 and 8.8 if the 80 Ω...Ch. 8 - The resistance, inductance, and capacitance in a...

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