5.9 : Types of Responses of Series RLC Circuits

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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.

A second-order differential equation characterizes a source-free series RLC circuit, marking its distinct mathematical representation. The complete solution of this equation is a blend of two unique solutions, each linked to the circuit's roots expressed in terms of the damping factor and resonant frequency.

Exponential decay equation: It = A1e^(s1t) + A2e^(s2t); science formula illustration.

When the damping factor surpasses the resonant frequency, both roots are real and negative, leading to an overdamped response. In this scenario, the circuit's reaction gradually decays over time.

When the damping factor matches the resonant frequency, the second-order differential equation simplifies to a first-order equation with an exponential solution. The natural response follows a pattern of peaking at its time constant and then decaying to zero, signifying critical damping.

$Time-dependent intensity decay formula \(I_t=A_1te^{-\alpha t}+A_2e^{-\alpha t}\).$

For situations where the damping factor is less than the resonant frequency, complex roots emerge, characterized by the damped natural frequency. Euler's formula simplifies the complete response to sine and cosine functions, resulting in an underdamped and oscillatory natural response with a time period proportional to the damped natural frequency.

$Decay curve equation \(I_t = e^{-\alpha t}(B_1 \cos \omega_d t + B_2 \sin \omega_d t)\).$

These different response behaviors illustrate the significance of source-free RLC circuits in circuit analysis, offering intriguing insights into electrical circuit behavior and applications.

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second order differential equation source free RLC circuit damping factor resonant frequency overdamped response critical damping underdamped response damped natural frequency circuit behavior mathematical representation

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5.9 : Types of Responses of Series RLC Circuits

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