Name: forced response in a series rlc circuit
Uploaded: 2024-04-11T12:48:44.000Z
Duration: 1 min 12 s
Channel: JoVE

Consider an automobile ignition system that produces high voltage from the battery essential for generating a spark.
This system can be modeled as a simple series RLC circuit and the complete response of the circuit can be analyzed.
Here, the input DC voltage serves as a forcing step function resulting in a forced step response that mirrors the characteristics of the forcing function.
Applying Kirchhoff&#39;s voltage law to the circuit yields a second-order differential equation.
This equation resembles the second-order differential equation of a source-free RLC circuit, showing that the DC source does not alter the form of the equations.
The complete solution to this equation is a combination of transient and steady-state responses.
The transient response, which diminishes over time, corresponds to the solution for source-free circuits in overdamped, critically damped, and underdamped scenarios.
The steady-state response corresponds to the final value of the capacitor voltage, which is identical to the source voltage.
The constants involved can be deduced from the initial conditions of the circuit.

series rlc circuit with source

Consider the operation of an automobile ignition system, a crucial component responsible for generating a spark by producing high voltage from the battery. This system can be described as a simple series RLC circuit, allowing for an in-depth analysis of its complete response.
In this context, the input DC voltage serves as a forcing step function, resulting in a forced step response that mirrors the characteristics of the input. Applying Kirchhoff's voltage law to the circuit yields a second-order differential equation. Remarkably, this equation strongly resembles the second-order differential equation characterizing a source-free RLC circuit. This similarity underscores that the presence of the DC source does not alter the fundamental form of the equations.
<img alt="RLC circuit differential equation, &#931;d&#178;v/dt&#178; + Rdv/Ldt + v/LC = Vs/LC, formula illustration." src="/files/ftp_upload/15100/15100_PT_Equation_1.svg" style="text-align: center; height: 34px;" data-altupdatedat="1761848533000" data-alt="Equation1" />
The complete solution to this equation comprises both transient and steady-state responses.
<img alt="Velocity equation \(v(t)=v_t(t)+v_{ss}(t)\) in dynamic system analysis." src="/files/ftp_upload/15100/15100_PT_Equation_2.svg" style="text-align: center; height: 16px;" data-altupdatedat="1761848533000" data-alt="Equation2" />
The transient response, which diminishes over time, aligns with the solution for source-free circuits and encompasses scenarios involving overdamped, critically damped, and underdamped behaviors. On the other hand, the steady-state response corresponds to the final value of the capacitor voltage, which equals the source voltage. The constants involved in these responses can be determined from the initial conditions of the circuit.

Series RLC Circuit with Source

Consider an automobile ignition system that produces high voltage from the battery essential for generating a spark.
This system can be modeled as a ...

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JoVE Core

Electrical Engineering

Unit 1

First and Second-Order Circuits

KEY TERMS AND CONCEPTS

forced response in a series rlc circuit

A series RLC circuit with a DC source can model an automobile ignition system. The circuit shows how a battery can produce the high voltage needed for a spark. In this case, the input DC voltage acts like a forcing step function.

Applying Kirchhoff's voltage law leads to a second-order differential equation. A second-order differential equation is an equation with a term for the second derivative. The form of this equation is very similar to the one used for a source-free RLC circuit, so the DC source does not change the basic equation structure.

The full response of the circuit has two parts: transient response and steady-state response. The transient response fades with time and matches the source-free circuit solution. It can be overdamped, critically damped, or underdamped.

The steady-state response is the final capacitor voltage. That final value equals the source voltage. The constants in both parts of the solution come from the initial conditions of the circuit.

Forced Response in a Series RLC Circuit

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electrical engineering

Forced Response in a Series RLC Circuit

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