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Consider the two differential equations: тx" + bx' + kx%3D F(t) dI + RI +1 = dt E(t) C where m, b, and k are constants for a spring-mass system and L, R, and C are constants for an electrical circuit consisting of an inductor, resistor, and capacitor in series. a) If we write the circuit equation in terms of q and its derivatives, we see that the circuit equation is analogous to forced motion of a spring. If the electromotive force E(t) is given by E(t) = E, cos(@t), find the steady-state solution for charge (so just a particular solution). Remember that if R>0, then your guess function could not possibly overlap with the complementary solution.

Consider the two differential equations: тx" + bx' + kx%3D F(t) dI + RI +1 = dt E(t) C where m, b, and k are constants for a spring-mass system and L, R, and C are constants for an electrical circuit consisting of an inductor, resistor, and capacitor in series. a) If we write the circuit equation in terms of q and its derivatives, we see that the circuit equation is analogous to forced motion of a spring. If the electromotive force E(t) is given by E(t) = E, cos(@t), find the steady-state solution for charge (so just a particular solution). Remember that if R>0, then your guess function could not possibly overlap with the complementary solution.

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
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Consider the two differential equations: \[ mx'' + bx' + kx = F(t) \] \[ L \frac{dI}{dt} + RI + \frac{q}{C} = E(t) \] where \( m, b, \) and \( k \) are constants for a spring-mass system and \( L, R, \) and \( C \) are constants for an electrical circuit consisting of an inductor, resistor, and capacitor in series. a) If we write the circuit equation in terms of \( q \) and its derivatives, we see that the circuit equation is analogous to forced motion of a spring. If the electromotive force \( E(t) \) is given by \[ E(t) = E_1 \cos(\omega t) \] find the steady-state solution for charge (so just a particular solution). Remember that if \( R > 0 \), then your guess function could not possibly overlap with the complementary solution.
Transcribed Image Text:Consider the two differential equations: \[ mx'' + bx' + kx = F(t) \] \[ L \frac{dI}{dt} + RI + \frac{q}{C} = E(t) \] where \( m, b, \) and \( k \) are constants for a spring-mass system and \( L, R, \) and \( C \) are constants for an electrical circuit consisting of an inductor, resistor, and capacitor in series. a) If we write the circuit equation in terms of \( q \) and its derivatives, we see that the circuit equation is analogous to forced motion of a spring. If the electromotive force \( E(t) \) is given by \[ E(t) = E_1 \cos(\omega t) \] find the steady-state solution for charge (so just a particular solution). Remember that if \( R > 0 \), then your guess function could not possibly overlap with the complementary solution.
b) Find the amplitude of your answer from (a). c) Consider the values of L, R, and C to be constants of the circuit, but allow \( E(t) \) to be varied by changing the period of the oscillations by changing \(\omega\). What value of \(\omega\) maximizes the amplitude from (b)? Give your answer in terms of L, R, and C. d) Calculate the limit as R approaches 0 from the right of your answer to (c). The result is the natural frequency of an LC circuit.
Transcribed Image Text:b) Find the amplitude of your answer from (a). c) Consider the values of L, R, and C to be constants of the circuit, but allow \( E(t) \) to be varied by changing the period of the oscillations by changing \(\omega\). What value of \(\omega\) maximizes the amplitude from (b)? Give your answer in terms of L, R, and C. d) Calculate the limit as R approaches 0 from the right of your answer to (c). The result is the natural frequency of an LC circuit.
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