Engineering LCR circuit, calculating R when frequency width is given

[Pushoam]

Homework Statement

Homework Equations

The Attempt at a Solution

LCR circuit, calculating R when frequency width is given

Applying Kirchoff Voltage Loop Law,

$V_A – V_B + V_B - V_C + V_C – V_A = 0$ ...(2)

$V_A – V_B$ is potential drop across inductor. Since current is flowing from A to B, I tend to take $V_A > V_B$ . This way I am taking inductor as a resistor. Inductor has a very low resistance. So, $V_A – V_B$ , taking this way is negligible.

Since the current is decreasing through inductor, there will be an EMF across it. This EMF will tend to increase the current according to Lenz's law. This EMF = - L $\frac { dI} {dt}$ = $V_A – V_B$ ...(3)

I doubt whether (3) is correct.

From (2) and (3),

- L $\frac { dI} {dt} + IR - \frac { q} C = 0$

Is this correct?

 

[NascentOxygen]

Do you know some equations involving Q, the Quality Factor of a resonant circuit?

 

[Pushoam]

Do you know some equations involving Q, the Quality Factor of a resonant circuit?

In mechanics, I knew.
Q = $\frac {\omega_0}{\Delta \omega}$
Here, $\Delta \omega = \gamma = \frac b m$ where b is the coefficient of damping
I want to have a similar eqn for LCR circuit and then find out what is b and m in this circuit.
But I am stuck in applying emf across inductor = $-L\frac {dI}{dT}$ in writing the second order linear differential eqn for the circuit.
I need help here.

 

[gneill]

Wikipedia has a decent coverage of the RLC circuit, including a relevant section on expressions for its Q

RLC Circuit