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

DC 소스가 RC 회로에 갑자기 적용되면 해당 전압은 단위 단계 기능으로 표시됩니다.
커패시터 양단의 전압은 스텝 응답입니다.
커패시터 양단의 전압은 순간적으로 변할 수 없기 때문에 스위칭 직후의 값은 스위칭 직전의 값과 동일하게 유지됩니다.
0과 같은 시간에 Kirchhoff의 현재 법칙을 적용하고, 항을 재배열하고, 0보다 큰 시간에 대해 방정식을 다시 작성하면 1차 미분 방정식을 얻을 수 있습니다.
방정식을 적분하고, 한계를 적용하고, 양쪽에서 지수를 취하면 0보다 큰 시간에 대한 커패시터의 계단 응답이 생성됩니다.
이를 커패시터의 초기 전압과 결합하면 0보다 작은 시간 동안 회로 응답으로 RC 회로의 완전한 응답을 얻을 수 있습니다.
시간이 지남에 따라 커패시터의 전압은 기하급수적으로 증가하고 소스의 전압에 접근합니다.
커패시터가 초기에 충전되지 않은 경우 전체 응답이 그에 따라 수정됩니다. 이를 통해 커패시터를 통과하는 전류가 결정되고 시간이 지남에 따라 기하급수적으로 감소하는 것으로 관찰됩니다.

소스가 포함된 rc 회로

RC(ResI_stor-Capacitor) 회로에 DC 전원이 갑자기 인가되면 전압은 단위 계단 함수로 표현될 수 있습니다. 계단 응답으로 알려진 축전기 양단의 전압은 입력의 급격한 변화에 회로가 어떻게 반응하는지를 나타냅니다.

축전기의 고유한 특성으로 인해 전압이 순간적으로 변경될 수 없습니다. 이는 스위치가 닫힌 직후 축전기의 전압이 스위치가 닫히기 직전과 동일하게 유지된다는 것을 의미합니다.

스위치가 닫히는 순간(t=0)에 키르히호프의 전류법칙을 적용하고, 항을 재배열하고, t>0 시간에 대한 방정식을 다시 쓰면 1차 미분방정식이 얻어진다. 이 방정식은 DC 소스의 갑작스러운 적용에 따라 회로를 통과하는 전류가 시간에 따라 어떻게 변하는지를 설명합니다.

이 미분 방정식은 이를 적분하고, 극한을 적용하고, 양쪽에 지수를 취하여 해결됩니다. 이는 시간 t>0에 대한 축전기의 계단 응답을 산출합니다. 이 단계 응답을 축전기 양단의 초기 전압(t<0의 경우)과 결합하면 RC 회로의 완전한 응답이 제공됩니다.

시간이 지남에 따라 축전기 양단의 전압은 기하급수적으로 증가하고 적용된 소스 전압에 가까워집니다. 이 과정은 축전기의 충전을 나타냅니다.

축전기가 처음에 충전되지 않은 경우 회로의 전체 응답이 이에 따라 수정됩니다. 이 수정된 응답으로부터 축전기를 통과하는 전류가 계산됩니다. 이 전류는 시간이 지남에 따라 기하급수적으로 감소하는 것으로 관찰되며, 이는 소스 전압에 도달할 때까지 축전기가 점진적으로 충전되는 것을 나타냅니다.

결론적으로 RC 회로의 계단 응답을 이해하면 이러한 회로가 입력 전압의 급격한 변화에 어떻게 반응하는지에 대한 귀중한 통찰력을 얻을 수 있습니다. 이러한 지식은 특히 신호를 필터링하거나 형성하기 위해 축전기가 광범위하게 사용되는 신호 처리와 같은 응용 분야에서 전자 회로를 설계하고 분석하는 데 필수적입니다.

소스가 포함된 RC 회로

DC 소스가 RC 회로에 갑자기 적용되면 해당 전압은 단위 단계 기능으로 표시됩니다.
커패시터 양단의 전압은 단계입니다 ...

Education

교육

JoVE Core

Electrical Engineering

전기공학

Unit 1

제1단원

First and Second-Order Circuits

1차 및 2차 회로

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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

first-order circuits

First-Order Circuits in Flashing Lamps

First-order circuits use a resistor and one energy storage element, either a capacitor or an inductor. They are common in electronic systems and are described by a first-order differential equation that links the input and output signals. One important example is the RC circuit, which uses a resistor and a capacitor. In a relaxation oscillator such as a neon lamp circuit, the capacitor starts charging when voltage is applied. At first, the lamp behaves like an open circuit. When the capacitor...

rc circuit without source

RC Capacitor Discharge After Power Loss

An RC circuit discharges after the DC source is removed. If the capacitor was fully charged first, its initial voltage, V0, provides the energy that drives the source-free circuit. Kirchhoff’s current law is applied at the top node of the circuit. The current terms for the resistor and capacitor are then substituted, which leads to a first-order differential equation. Rearranging, integrating, and taking the exponential gives the natural response of the circuit. The integration constant in...

rc circuit with source

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...

rl circuit without source

RL Circuit Decay After Source Removal

An RL circuit begins to decay after the DC source is removed. At that moment, the circuit becomes source-free. If the inductor has an initial current, I0, it also stores initial energy. That stored energy drives the natural response of the circuit. Kirchhoff’s voltage law is applied around the loop. The inductor voltage and resistor voltage are substituted into the equation. This gives a first-order differential equation for the current. Rearranging the terms produces a logarithmic form, which...

rl circuit with source

RL Step Response Under a DC Source

An RL circuit connected to a DC source has a step response that changes over time. The response has two parts: the transient response and the steady-state response. The transient response is the short-lived reaction that happens right after the source is applied. During the transient phase, the current changes exponentially. If the circuit starts with current in the inductor, that current decays toward zero as time goes on. At this stage, the inductor acts like a short circuit, so the source...

design example: frog muscle response

RL Circuit and Frog Leg Current Response

An RL circuit is used to model the frog leg’s response to electrical stimulation. In this setup, the student studies how current changes in a resistor-inductor circuit can trigger muscle contraction. The circuit provides a way to control and measure the electrical impulses involved in the experiment. When the switch is closed, the frog’s leg shows a brief contraction. At steady state, the inductor acts like a short circuit. That lets current bypass the resistor and produces a mild, short...

second-order circuits

RLC Damping and Resonant Frequency

Second-order circuits use two energy storage elements, such as a capacitor and an inductor, to shape how current and voltage change over time. These circuits include RLC circuits, along with RC and RL circuits that have two capacitors or two inductors. Their behavior is described by second-order differential equations that connect the input signal to the output signal. The input usually comes from a voltage source or a current source. The output is often the voltage across the capacitor, the...

series rlc circuit without source

Natural Response of a Source-Free RLC Circuit

A source-free RLC circuit is a series circuit with a resistor, inductor, and capacitor, but no external energy source. It begins with the energy already stored in the capacitor and inductor. That stored energy drives the circuit at first. The circuit is described by a second-order differential equation. This equation shows how the resistor, inductor, and capacitor interact over time. The resistor removes energy from the circuit, so the solution has an exponential form. When the exponential...

types of responses of series rlc circuits

types-of-responses-of-series-rlc-circuits

Damping in Series RLC Circuit Responses

A source-free series RLC circuit shows three response patterns based on damping. The circuit is described by a second-order differential equation. Its complete solution is made from two distinct solutions. Those solutions depend on the roots, which are written in terms of the damping factor and the resonant frequency. When the damping factor is greater than the resonant frequency, both roots are real and negative. This gives an overdamped response. The circuit output decays slowly over time. <br...

series rlc circuit with source

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...

parallel rlc circuits

Parallel RLC Step Response in Surge Protectors

Parallel RLC circuits help explain how a surge protector works in a street lamp. The circuit model uses a DC input source that creates a step response when the switch is turned on. This setup shows how the lamp’s components can be protected from sudden voltage spikes. When the switch closes, Kirchhoff’s current law is applied to the parallel RLC circuit. That leads to a second-order differential equation. The solution has two parts: a transient response and a steady-state response. The...

second-order op amp circuits

Second-Order Op Amp Low-Pass Filters

Second-order op-amp low-pass filters remove unwanted high-frequency noise from audio signals. They are used in audio systems to refine sound quality. These filters are often built as voltage followers and have two nodes with storage elements. The circuit analysis follows the same basic method used for second-order RLC circuits. In practice, bulky inductors are usually avoided because of their size and weight. For that reason, the focus here is on RC second-order op-amp circuits, which are...

design example: underdamped parallel rlc circuit

design-example-underdamped-parallel-rlc-circuit

Oscillator Design with a Parallel RLC Circuit

A parallel RLC circuit can be designed to work as an oscillator with underdamped behavior. This example targets a damped natural frequency of 4 kHz and a damping factor of 4 radians per second. A fixed resistance of 200 Ω is used as the starting point for the design. The first step is to find the capacitance. The damping factor is the reciprocal of twice the product of resistance and capacitance. Using that relationship, the needed capacitance value can be calculated from the given resistance...

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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