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 that the circuit is underdamped. d) Rewrite your answer from part (c) so that it has the form: Acos(ot + p) + Bsin(æt + q) = 0. Since this must hold for all times, the coefficients A and B must be zero. e) Set the coefficient of the sin term equal to zero and solve for t. This is the decay time for the R Wr 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 in the trial solution k) Show that when the resistance is such that the circuit is critically damped, the quality factor reduces to 0. ** q(t) = aet cos(ax +q) into the loop rule. (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 that the circuit is underdamped. d) Rewrite your answer from part (c) so that it has the form: Acos(ot + p) + Bsin(æt + q) = 0. Since this must hold for all times, the coefficients A and B must be zero. e) Set the coefficient of the sin term equal to zero and solve for t. This is the decay time for the R Wr 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 in the trial solution k) Show that when the resistance is such that the circuit is critically damped, the quality factor reduces to 0. ** q(t) = aet cos(ax +q) into the loop rule. (Since ex is never 0, we can cancel out the common exponential in all of the terms.)
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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Transcribed Image Text: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 that the circuit is
underdamped.
d) Rewrite your answer from part (c) so that it
has the form:
Acos(ot + q) + Bsin(wt + q) = 0. Since this
must hold for all times, the coefficients A and
B must be zero.
e) Set the coefficient of the sin term equal to zero
and solve for t. This is the decay time for the
R
Wr-
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.
L
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 2t
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
in
the
trial
solution
k) Show that when the resistance is such that
the circuit is critically damped, the quality
factor reduces to 0. **
q(t) = ae- cos(ot +9) into the loop rule.
(Since ex is never 0, we can cancel out the
common exponential in all of the terms.)
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