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 with source

When a DC source is abruptly applied to an RC (Resistor-Capacitor) circuit, the voltage can be represented as a unit step function. The voltage across the capacitor, known as the step response, characterizes how the circuit reacts to this sudden change in input.
Due to the inherent properties of a capacitor, its voltage cannot change instantaneously. This means that immediately after the switch is closed, the capacitor&#39;s voltage remains the same as it was just before the switch was closed.
By applying Kirchhoff&#39;s current law at the moment the switch is closed (t=0), rearranging the terms, and rewriting the equation for times t&#62;0, a first-order differential equation is obtained. This equation describes how the current through the circuit changes with time following the abrupt application of the DC source.
This differential equation is solved by integrating it, applying the limits, and taking the exponential on both sides. This yields the step response of the capacitor for times t&#62;0. Combining this step response with the initial voltage across the capacitor (for t&#60;0) gives the complete response of the RC circuit.
As time progresses, the voltage across the capacitor increases exponentially and approaches the applied source voltage. This process represents the charging of the capacitor.
If the capacitor is initially uncharged, the complete response of the circuit gets modified accordingly. From this modified response, the current through the capacitor is calculated. This current is observed to decrease exponentially with time, representing the gradual charging of the capacitor until it reaches the source voltage.
In conclusion, understanding the step response of an RC circuit provides valuable insights into how these circuits respond to sudden changes in input voltage. This knowledge is essential for designing and analyzing electronic circuits, particularly in applications such as signal processing, where capacitors are used extensively to filter or shape signals.

RC Circuit with Source

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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JoVE Core

Electrical Engineering

Unit 1

First and Second-Order Circuits

KEY TERMS AND CONCEPTS

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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