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Kirchhoff's Voltage Law:

Kirchhoff's voltage law can be stated with the following equation:

$$ v_R + v_L + v_C = v(t) $$

Where:

$v_R$, $v_L$, $v_C$ are the voltages across the resistor R, inductor L, and capacitor C respectively.
$v(t)$ is the time-varying source voltage.

By substituting the constitutive equations, we get:

$$ RI(t) + L \frac{dI}{dt} + \frac{1}{C} \int_{-\infty}^{t} I(\tau) , d\tau = v(t) $$

Differential Equation in the Case of a Constant Source Voltage:

When the source voltage is constant, deriving the above equation and dividing by L leads to the following second-order differential equation:

$$ \frac{d^2I(t)}{dt^2} + \frac{R}{L} \frac{dI(t)}{dt} + \frac{1}{LC} I(t) = 0 $$

This equation can be written in a more commonly used form:

$$ \frac{d^2I(t)}{dt^2} + 2\alpha \frac{dI(t)}{dt} + \omega_0^2 I(t) = 0 $$

Where:

$\alpha$, termed as the 'damping factor', measures the rate at which the transient response of the circuit decays after removal of external input.
$\omega_0$ is the angular resonance frequency.

The coefficients $\alpha$ and $\omega_0$ are given by:

$$ \alpha = \frac{R}{2L}, \quad \omega_0 = \frac{1}{\sqrt{LC}} $$

Damping Coefficient:

The damping coefficient $\zeta$ is another commonly used parameter, defined as the ratio of $\alpha$ to $\omega_0$:

$$ \zeta = \frac{\alpha}{\omega_0} $$

The damping coefficient can also be derived from R, L, and C as follows:

$$ \zeta = \frac{R}{2} \sqrt{\frac{C}{L}} $$

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