4. Consider a series LRC circuit, with an inductor L with no current initially flowing through it, a capacitor C which is initially charged to charge qo, and a resistor R that is small enough so underdamped. d) Rewrite your answer from part (c) so that it has the form: Аcos(or + ф) + B sin(wt + ф) — 0. Since this must hold for all times, the coefficients A and B must be zero. that the circuit is e) Set the coefficient of the sin term equal to zero and solve for t. This is the decay time for the R W- circuit. f) Set the coefficient of the cos term equal to zero, plug in your result for t, and solve for w. This is the natural frequency of the circuit. g) Plug in the initial conditions to finish solving for q(t). h) Show that, in the limit as the resistance of the circuit goes to zero, the solution reduces to the solution for an LC circuit. C a) At the instant the switch is closed, what . current flows through the circuit? i) What is the maximum value of resistance that allows underdamped behavior? b) At some time t after the switch is closed, the charge on the capacitor is q(t) and the current going through the circuit is i(t). Write the loop rule for this circuit. Rewrite this as a second order differential equation for q(t). j) What is the quality factor, Q (defined as 27 times the number of cycles needed for the energy stored in the circuit to decay be a factor of 1/e) of the circuit? с) Plug q(t) = ae cos(ot +@) into the loop rule. the in -t/t trial solution k) Show that when the resistance is such that the circuit is critically damped, the quality factor reduces to 0. ** (Since ex is never 0, we can cancel out the common exponential in all of the terms.)
4. Consider a series LRC circuit, with an inductor L with no current initially flowing through it, a capacitor C which is initially charged to charge qo, and a resistor R that is small enough so underdamped. d) Rewrite your answer from part (c) so that it has the form: Аcos(or + ф) + B sin(wt + ф) — 0. Since this must hold for all times, the coefficients A and B must be zero. that the circuit is e) Set the coefficient of the sin term equal to zero and solve for t. This is the decay time for the R W- circuit. f) Set the coefficient of the cos term equal to zero, plug in your result for t, and solve for w. This is the natural frequency of the circuit. g) Plug in the initial conditions to finish solving for q(t). h) Show that, in the limit as the resistance of the circuit goes to zero, the solution reduces to the solution for an LC circuit. C a) At the instant the switch is closed, what . current flows through the circuit? i) What is the maximum value of resistance that allows underdamped behavior? b) At some time t after the switch is closed, the charge on the capacitor is q(t) and the current going through the circuit is i(t). Write the loop rule for this circuit. Rewrite this as a second order differential equation for q(t). j) What is the quality factor, Q (defined as 27 times the number of cycles needed for the energy stored in the circuit to decay be a factor of 1/e) of the circuit? с) Plug q(t) = ae cos(ot +@) into the loop rule. the in -t/t trial solution k) Show that when the resistance is such that the circuit is critically damped, the quality factor reduces to 0. ** (Since ex is never 0, we can cancel out the common exponential in all of the terms.)
Physics for Scientists and Engineers, Technology Update (No access codes included)
9th Edition
ISBN:9781305116399
Author:Raymond A. Serway, John W. Jewett
Publisher:Raymond A. Serway, John W. Jewett
Chapter33: Alternating Current Circuits
Section: Chapter Questions
Problem 33.60AP
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