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

Considera un sistema di accensione per automobili che produce alta tensione dalla batteria essenziale per generare una scintilla.
Questo sistema può essere modellato come un semplice circuito RLC in serie e la risposta completa del circuito può essere analizzata.
In questo caso, la tensione CC di ingresso funge da funzione di passo forzato che si traduce in una risposta a passo forzato che rispecchia le caratteristiche della funzione di forzatura.
Applicando la legge della tensione di Kirchhoff al circuito si ottiene un'equazione differenziale del secondo ordine.
Questa equazione assomiglia all'equazione differenziale del secondo ordine di un circuito RLC privo di sorgente, dimostrando che la sorgente CC non altera la forma delle equazioni.
La soluzione completa di questa equazione è una combinazione di risposte transitorie e stazionarie.
La risposta ai transienti, che diminuisce nel tempo, corrisponde alla soluzione per circuiti privi di sorgente in scenari sovrasmorzati, criticamente smorzati e sottosmorzati.
La risposta in stato stazionario corrisponde al valore finale della tensione del condensatore, che è identica alla tensione della sorgente.
Le costanti in gioco possono essere dedotte dalle condizioni iniziali del circuito.

Nessun conflitto di interessi dichiarato.

circuito rlc in serie con una sorgente

Series RLC Circuit with Source

Consideriamo il funzionamento del sistema di accensione di un'automobile, un componente cruciale responsabile della generazione di una scintilla prodotta dall’alta tensione della batteria. Questo sistema può essere descritto come un semplice circuito RLC in serie, consentendo un'analisi approfondita della sua risposta completa.

In questo contesto, la tensione CC in ingresso funge da funzione di gradino forzato, risultando in una risposta al gradino forzata, che rispecchia le caratteristiche dell'ingresso. Applicando la legge sulla tensione di Kirchhoff al circuito, si ottiene un'equazione differenziale del secondo ordine. Sorprendentemente, questa equazione assomiglia fortemente all'equazione differenziale di secondo ordine, che caratterizza un circuito RLC privo di sorgente. Questa somiglianza sottolinea che la presenza della sorgente continua non altera la forma fondamentale delle equazioni.

<img alt="Equation1" src="/files/ftp_upload/15100/15100_PT_Equation_1.svg" style="text-align: center; height: 34px;">

La soluzione completa a questa equazione comprende sia risposte transitorie che stazionarie.

<img alt="Equation2" src="/files/ftp_upload/15100/15100_PT_Equation_2.svg" style="text-align: center; height: 16px;">

La risposta transitoria, che diminuisce nel tempo, si allinea con la soluzione per circuiti privi di sorgente e comprende scenari che coinvolgono comportamenti sovrasmorzati, criticamente smorzati e sottosmorzati. D'altra parte, la risposta in regime stazionario corrisponde al valore finale della tensione del condensatore, che è uguale alla tensione della sorgente. Le costanti coinvolte in queste risposte possono essere determinate dalle condizioni iniziali del circuito.

Circuito RLC in Serie con una Sorgente

Considera un sistema di accensione per automobili che produce alta tensione dalla batteria essenziale per generare una scintilla.
Questo sistema può essere modellato ...

Education

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JoVE Core

Electrical Engineering

Ingegneria elettrica

Unit 1

Unità 1

First and Second-Order Circuits

Circuiti del primo e del secondo ordine

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

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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JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

Forced Response in a Series RLC Circuit

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