Name: rc circuit charging after a dc step
Uploaded: 2024-04-11T12:48:43.000Z
Duration: 1 min 15 s
Channel: JoVE

Quando una sorgente CC viene applicata bruscamente a un circuito RC, la sua tensione è rappresentata come una funzione di passo unitario.
La tensione attraverso il condensatore è la risposta al gradino.
Poiché la tensione attraverso un condensatore non può cambiare istantaneamente, il suo valore immediatamente dopo la commutazione rimane lo stesso del valore immediatamente prima della commutazione.
Applicando l'attuale legge di Kirchhoff in un tempo uguale a zero, riorganizzando i termini e riscrivendo l'equazione per un tempo maggiore di zero, si ottiene un'equazione differenziale del primo ordine.
Integrando l'equazione, applicando i limiti e prendendo l'esponenziale su entrambi i lati si ottiene la risposta al gradino del condensatore per volte maggiori di zero.
Combinando questo con la tensione iniziale del condensatore come risposta del circuito per un tempo inferiore a zero, si ottiene la risposta completa del circuito RC.
Col tempo, la tensione del condensatore aumenta in modo esponenziale e si avvicina alla tensione della sorgente.
Se il condensatore è inizialmente scarico, la risposta completa viene modificata di conseguenza. Da questo, viene determinata la corrente attraverso il condensatore e si osserva che diminuisce esponenzialmente con il tempo.

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circuito rc con sorgente

Quando una sorgente CC viene applicata improvvisamente a un circuito RC (resistenza-condensatore), la tensione può essere rappresentata come una funzione a gradino unitario. La tensione ai capi del condensatore, nota come risposta al gradino, caratterizza il modo in cui il circuito reagisce a questo improvviso cambiamento di ingresso.

A causa delle proprietà intrinseche di un condensatore, la sua tensione non può cambiare istantaneamente. Ciò significa che immediatamente dopo la chiusura dell'interruttore, la tensione del condensatore rimane la stessa di prima della chiusura dell'interruttore.

Applicando la legge della corrente di Kirchhoff al momento della chiusura dell'interruttore (t=0), riordinando i termini e riscrivendo l'equazione per tempi t>0, si ottiene un'equazione differenziale del primo ordine. Questa equazione descrive come la corrente attraverso il circuito cambia nel tempo, in seguito all'applicazione improvvisa della sorgente CC.

Questa equazione differenziale si risolve integrandola, applicando i limiti e prendendo l'esponenziale su entrambi i membri. Ciò produce la risposta al gradino del condensatore per tempi t>0. Combinando questa risposta al gradino con la tensione iniziale ai capi del condensatore (per t<0), si ottiene la risposta completa del circuito RC.

Con il passare del tempo, la tensione ai capi del condensatore aumenta esponenzialmente e si avvicina alla tensione della sorgente applicata. Questo processo rappresenta la carica del condensatore.

Se il condensatore è inizialmente scarico, la risposta completa del circuito viene modificata di conseguenza. Da questa risposta modificata viene calcolata la corrente che attraversa il condensatore. Si osserva che questa corrente diminuisce esponenzialmente con il tempo, rappresentando la carica graduale del condensatore, fino a raggiungere la tensione della sorgente.

In conclusione, comprendere la risposta al gradino di un circuito RC fornisce preziose informazioni su come questi circuiti rispondono a cambiamenti improvvisi della tensione di ingresso. Questa conoscenza è essenziale per progettare e analizzare circuiti elettronici, in particolare in applicazioni come l'elaborazione dei segnali, dove i condensatori vengono ampiamente utilizzati per filtrare o modellare i segnali.

Circuito RC con Sorgente

Quando una sorgente CC viene applicata bruscamente a un circuito RC, la sua tensione è rappresentata come una funzione di passo unitario.
La tensione attraverso il condensatore è il passo ...

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Electrical Engineering

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First and Second-Order Circuits

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RC Circuit with Source

rc circuit charging after a dc step

An RC circuit with a DC source shows how a capacitor charges after a sudden voltage step. The applied source is treated as a unit step function, and the capacitor voltage is the step response of the circuit. This response describes how the circuit reacts right after the input changes.

A capacitor cannot change its voltage instantly. So, immediately after the switch closes at t=0, the capacitor voltage stays at its initial value. Kirchhoff’s current law is then applied at that moment and rewritten for times t&gt;0. This produces a first-order differential equation for the circuit current.

The differential equation is solved by integration. After applying the limits and taking the exponential form, the capacitor’s step response is found for t&gt;0. When this result is combined with the capacitor voltage for t&lt;0, it gives the complete response of the RC circuit. If the capacitor starts uncharged, the response is adjusted accordingly.

As time passes, the capacitor voltage rises exponentially and approaches the source voltage. This is the charging process of the capacitor. The current through the capacitor can also be calculated from the modified response. It decreases exponentially with time as the capacitor gradually charges until it reaches the source voltage.

Understanding the step response of an RC circuit helps explain how the circuit reacts to a sudden DC input. This is useful for analyzing and designing electronic circuits. It is especially important in signal processing, where capacitors are often used to filter or shape signals.

RC Circuit Charging After a DC Step

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key terms and concepts

Study how an RC circuit responds to a sudden DC step, including capacitor charging, the exponential voltage rise, and the decreasing current over time.

RC Circuit Charging After a DC Step | JoVE K12

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

RC Circuit Charging After a DC Step

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