UY1: L-R-C Series Circuit



LRC Circuit

The diagram above shows a L-R-C series circuit. Let’s analyze the circuit:

From Kirchhoff’s loop rule:

$$\begin{aligned} L\frac{di}{dt} + \frac{q}{C} &=-Ri \\ \frac{d^{2}q}{dt^{2}} + \frac{R}{L}\frac{dq}{dt} + \frac{1}{LC}q &= 0 \end{aligned}$$

The equation above resembles the equation for damped simple harmonic motion, whereby:

Overdamped:

$$\frac{R^{2}}{L^{2}} > \frac{4}{LC}$$

Critical damped:

$$\frac{R^{2}}{L^{2}} = \frac{4}{LC}$$

Underdamped:

$$\frac{R^{2}}{L^{2}} < \frac{4}{LC}$$

Solving further:

$$\begin{aligned} q &= Q e^{-\frac{R}{2L}t} \cos{\left( \omega^{\prime} t + \phi \right)} \\ \omega^{\prime} &= \sqrt{\frac{1}{LC}-\frac{R^{2}}{4 L^{2}}} \end{aligned}$$

In the L-R-C circuit, the resistance is the damping mechanism.

Next: R-L Circuit

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