直列 RLC 回路の応答の種類

A source-free series RLC circuit is represented by a second-order differential equation.

Its complete solution is a linear combination of two distinct solutions, each corresponding to the two roots expressed in terms of the damping factor and resonant frequency.

If the damping factor exceeds the resonant frequency, the roots are real and negative, leading to an overdamped response that decays over time.

When the damping factor equals the resonant frequency, the roots are equal.

In this scenario, the second-order differential equation reduces to a first-order with an exponential solution.

The natural response, a sum of a negative exponential and a negative exponential multiplied by a linear term, peaks at its time constant and then decays to zero, indicating critical damping.

If the damping factor is less than the resonant frequency, the complex roots can be expressed in terms of the damped natural frequency.

Euler's formula simplifies the complete response to functions of sine and cosine terms.

So, the natural response is underdamped and oscillatory with a time period proportional to the damped natural frequency.