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

When a DC source is applied abruptly to an RC circuit, its voltage is represented as a unit step function.
The voltage across the capacitor is the step response.
Since the voltage across a capacitor cannot change instantaneously, its value immediately after switching remains the same as the value immediately before switching.
By applying Kirchhoff&#39;s current law at a time equal to zero, rearranging the terms, and rewriting the equation for a time greater than zero, a first-order differential equation is obtained.
Integrating the equation, applying the limits, and taking the exponential on both sides yields the step response of the capacitor for times greater than zero.
Combining this with the capacitor&#39;s initial voltage as the circuit response for a time less than zero gives the complete response of the RC circuit.
In time, the capacitor&#39;s voltage increases exponentially and approaches the source&#39;s voltage.
If the capacitor is initially uncharged, the complete response gets modified accordingly. From this, the current through the capacitor is determined and is observed to be exponentially decreasing with time.

rc-circuit met bron

Wanneer een DC-bron abrupt wordt toegepast op een RC-circuit (Resistor-Capacitor), kan de voltage worden weergegeven als een eenheidsstapfunctie. De voltage over de condensator, bekend als de staprespons, karakteriseert hoe het circuit reageert op deze plotselinge verandering in invoer. 

 Vanwege de inherente eigenschappen van een condensator kan de voltage ervan niet onmiddellijk veranderen. Dit betekent dat onmiddellijk nadat de schakelaar is gesloten, de voltage van de condensator hetzelfde blijft als vlak voordat de schakelaar werd gesloten. 

 Door de voltageswet van Kirchhoff toe te passen op het moment dat de schakelaar wordt gesloten (t=0), de termen te herschikken en de vergelijking te herschrijven voor tijden t>0, wordt een differentiaalvergelijking van de eerste orde verkregen. Deze vergelijking beschrijft hoe de stroom door het circuit in de loop van de tijd verandert na de abrupte toepassing van de gelijkstroombron. 

 Deze differentiaalvergelijking wordt opgelost door deze te integreren, de limieten toe te passen en de exponentiële waarde aan beide kanten te nemen. Dit levert de stapresponsie van de condensator op voor tijden t>0. Het combineren van deze stapresponsie met de initiële voltage over de condensator (voor t<0) geeft de volledige responsie van het RC-circuit. 

 Naarmate de tijd vordert, neemt de voltage over de condensator exponentieel toe en benadert deze de aangelegde bronvoltage. Dit proces vertegenwoordigt het opladen van de condensator. 

 Als de condensator aanvankelijk ongeladen is, wordt de volledige respons van het circuit overeenkomstig aangepast. Uit deze gewijzigde respons wordt de stroom door de condensator berekend. Er wordt waargenomen dat deze stroom exponentieel afneemt met de tijd, wat het geleidelijke opladen van de condensator vertegenwoordigt totdat deze de bronvoltage bereikt. 

 Concluderend biedt het begrijpen van de staprespons van een RC-circuit waardevolle inzichten in hoe deze circuits reageren op plotselinge veranderingen in de ingangsvoltage. Deze kennis is essentieel voor het ontwerpen en analyseren van elektronische circuits, vooral in toepassingen zoals signaalverwerking, waarbij condensatoren op grote schaal worden gebruikt om signalen te filteren of vorm te geven.

RC-circuit met bron

When a DC source is applied abruptly to an RC circuit, its voltage is represented as a unit step function.
The voltage across the capacitor is the step ...

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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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