Explanation Given: The value of inductance is L = 1 2 h , resistance R = 20 Ω , capacitance C = 0.001 f and voltage in phase is E ( t ) = 100 sin 60 t + 200 cos 40 t V. Calculation: The general equation of the LRC -circuit is L d 2 q d t 2 + R d q d t + 1 C q = E ( t ) . From the given values, the general equation can written as, 1 2 d 2 q d t 2 + 20 d q d t + 1 , 000 q = 100 sin 60 t + 200 cos 40 t (1) Multiply the above equation by 2. d 2 q d t 2 + 40 d q d t + 2 , 000 q = 200 sin 60 t + 400 cos 40 t The corresponding homogeneous equation is, d 2 q d t 2 + 40 d q d t + 2 , 000 q = 0 . Now, the auxiliary equation is m 2 + 40 m + 2 , 000 = 0 . Solve the auxiliary equation as follows. m = − 40 ± ( 40 ) − 4 ( 2 , 000 ) 2 ( 1 ) = − 40 ± 80 i 2 = − 20 ± 40 i Therefore, the complementary function of the above value is q e = e − 20 t [ c 1 cos 40 t + c 2 sin 40 t ] . Let, the value of steady state charge is q p . Then, q p ( t ) = A cos 60 t + B sin 60 t + C cos 40 t + D sin 40 t (2) Differentiate both sides with respect to t in equation (2). q ′ p = 60 A sin 60 t + 60 B cos 60 t − 40 C sin 40 t + 40 D cos 40 t (3) Again differentiate both sides with respect to t . q ″ p = − 3 , 600 A cos 60 t − 3 , 600 B sin 60 t − 1 , 600 C cos 40 t − 1 , 600 D sin 40 t Now, substitute the values of q ″ p , q ′ p , and q p in the equation d 2 q d t 2 + 40 d q d t + 2 , 000 q = 200 sin 60 t + 400 cos 40 t . [ − 3 , 600 A cos 60 t − 3 , 600 B sin 60 t − 1 , 600 C cos 40 t − 1 , 600 D sin 40 t ] + 40 [ 60 A sin 60 t + 60 B cos 60 t − 40 C sin 40 t + 40 D cos 40 t ] + 2 , 000 [ A cos 60 t + B sin 60 t + C cos 40 t + D sin 40 t ] = 200 sin 60 t + 400 cos 40 t Simplify and obtain as, − 1 , 600 ( A cos 60 t + B sin 60 t ) + 400 ( C cos 40 t + D sin 40 t ) − 2 , 400 A sin 60 t + 2 , 400 B cos 60 t − 1 , 600 C sin 40 t = 200 sin 60 t + 400 cos 40 t Now, combine the coefficient of similar terms. ( − 1 , 600 A + 2 , 400 B ) cos 60 t + ( − 1 , 600 B − 2 , 400 A ) sin 60 t + ( 400 C + 1 , 600 D ) cos 40 t + ( 400 D − 1 , 600 C ) sin 40 t = 200 sin 60 t + 400 cos 40 t Compare the coefficients of the above equation

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

11th Edition

ISBN: 9781305965720

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 5.1, Problem 56E

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The value of steady state current in an LRC-series circuit.

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Chapter 5.1, Problem 55E

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The value of steady state current in an LRC -series circuit.

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