To analyze the given RLC series circuit and derive the conditions for underdamped, overdamped, and critically damped vibrations, let's start by rewriting the equation you've provided and deriving the corresponding second-order differential equation.

1. Formulation of the Differential Equation

The given equation for the RLC circuit is:

$RI + L \frac{dI}{dt} + \frac{q}{C} = V(t)$

Using the relationship $I = \frac{dq}{dt}$, we can rewrite $I$:

$R \frac{dq}{dt} + L \frac{d^2q}{dt^2} + \frac{q}{C} = V(t)$

For the case where $V(t) = 0$ (no voltage source), the equation becomes:

$L \frac{d^2q}{dt^2} + R \frac{dq}{dt} + \frac{q}{C} = 0$

2. Characteristic Equation

Assuming a solution of the form $q(t) = e^{\lambda t}$, we can substitute this into the differential equation:

$L \lambda^2 e^{\lambda t} + R \lambda e^{\lambda t} + \frac{1}{C} e^{\lambda t} = 0$

Factoring out $e^{\lambda t}$ (which is never zero) gives us the characteristic equation:

$L \lambda^2 + R \lambda + \frac{1}{C} = 0$

3. Solutions of the Characteristic Equation

The characteristic equation is a quadratic equation in the form $a\lambda^2 + b\lambda + c = 0$. The solutions for $\lambda$ can be found using the quadratic formula:

$\lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Here, $a = L$, $b = R$, and $c = \frac{1}{C}$:

$\lambda = \frac{-R \pm \sqrt{R^2 - 4\frac{L}{C}}}{2L}$

4. Conditions for Damping

1. Underdamped: The system is underdamped if the discriminant $D = R^2 - 4\frac{L}{C} < 0$. This leads to complex roots, which result in oscillatory behavior.

2. Critically damped: The system is critically damped if $D = R^2 - 4\frac{L}{C} = 0$. This results in a repeated real root, leading to the fastest return to equilibrium without oscillation.

3. Overdamped: The system is overdamped if $D = R^2 - 4\frac{L}{C} > 0$. This results in two distinct real roots and the system returns to equilibrium slowly without oscillation.

5. Harmonic Oscillation

To prove that the system exhibits harmonic oscillation when $R = 0$:

The differential equation reduces to:

$L \frac{d^2q}{dt^2} + \frac{q}{C} = 0$

This is a second-order linear differential equation with constant coefficients, which can be rewritten as:

$\frac{d^2q}{dt^2} + \frac{1}{LC} q = 0$

The characteristic equation is:

$\lambda^2 + \frac{1}{LC} = 0$

This has solutions:

$\lambda = \pm j\sqrt{\frac{1}{LC}}$

The solution for $q(t)$ will be:

$q(t) = A \cos\left(\sqrt{\frac{1}{LC}} t + \phi\right)$

where $A$ is the amplitude and $\phi$ is the phase constant. The angular frequency $\omega$ of the harmonic oscillation is given by:

$\omega = \sqrt{\frac{1}{LC}}$

Summary

• Underdamped: $R^2 - 4\frac{L}{C} < 0$
• Critically damped: $R^2 - 4\frac{L}{C} = 0$
• Overdamped: $R^2 - 4\frac{L}{C} > 0$
• Harmonic Oscillation (if $R=0$): $\omega = \sqrt{\frac{1}{LC}}$

If you have any questions or need further details, feel free to ask! Here are some related questions you might consider:

1. How does the value of resistance $R$ affect the transient response in an RLC circuit?
2. What practical applications utilize the concepts of damping in electrical circuits?
3. How would you analyze an RLC circuit with a constant voltage source?
4. What is the impact of varying capacitance $C$ and inductance $L$ on the system's natural frequency?
5. Can you describe a physical system that exhibits similar behavior to the RLC circuit in terms of damping?

Tip: When analyzing circuits, always pay attention to the signs and units of parameters to avoid mistakes in calculations!