Name: rc capacitor discharge after power loss
Uploaded: 2024-04-11T12:48:43.000Z
Duration: 1 min 16 s
Channel: JoVE

DC 소스가 RC 회로에서 갑자기 제거되면 회로는 소스가 없어집니다.
완전히 충전된 커패시터의 초기 전압이 V0라고 가정하면 회로를 자극하는 초기 에너지를 얻을 수 있습니다.
상단 노드에 Kirchhoff의 전류 법칙을 적용하고 구성 요소 전체에 전류 값을 대체하면 1차 미분 방정식이 제공됩니다.
항을 재정렬하고, 적분하고, 양쪽에서 지수를 취하면 회로의 자연스러운 응답이 생성되며, 여기서 적분 상수는 초기 전압과 같습니다.
전압 대 시간 그래프는 초기 전압이 시간에 따라 기하급수적으로 감소한다는 것을 보여줍니다.
시간 상수 tau는 커패시터가 초기 전압의 36.8%까지 방전되는 데 필요한 시간을 나타냅니다.
전압 응답 표현식에 tau의 값을 대입하여 저항기에서 소비되는 전류와 전력을 결정할 수 있습니다.
시간이 지남에 따라 소산되는 전력을 통합하면 저항기가 흡수하는 에너지를 제공합니다.
시간이 무한대에 가까워짐에 따라 이 에너지는 커패시터에 저장된 초기 에너지에 접근하며, 이는 커패시터의 초기 에너지가 저항기에서 점차 소멸됨을 의미합니다.

소스가 없는 rc 회로

RC Circuit without Source

DC 소스가 RC(저항기-축전기) 회로에서 갑자기 분리되면 회로는 소스가 없게 됩니다. 소스가 제거되기 전에 축전기가 완전히 충전되었다고 가정하면 V0로 표시되는 초기 전압은 회로를 자극하는 초기 에너지로 간주될 수 있습니다.

회로의 최상위 노드에 키르히호프의 전류 법칙을 적용하고 구성 요소 전체의 전류 값을 대체하면 1차 미분 방정식이 얻어집니다. 이 방정식의 항을 재배열하고 적분한 다음 양쪽에 지수를 취함으로써 회로의 자연스러운 응답이 결정됩니다. 이 방정식의 적분 상수는 초기 전압과 같습니다.

전압 대 시간 그래프는 초기 전압이 시간에 따라 기하급수적으로 감소함을 보여줍니다. 이는 축전기의 전하가 점진적으로 감소하여 이를 통과하는 전압이 감소함을 의미합니다.

그리스 문자 타우(τ)로 표시되는 회로의 시상수는 축전기가 초기 전압의 36.8%까지 방전하는 데 필요한 시간을 나타냅니다. 이 시상수는 축전기가 방전되는 속도와 결과적으로 회로가 변화에 반응하는 속도를 결정하는 데 중요한 역할을 합니다.

전압 응답 표현식에 타우 값을 대입하면 저항기를 통해 흐르는 전류와 저항기에서 소비되는 전력을 계산할 수 있습니다. 저항기에서 소비되는 전력은 열의 형태로 에너지가 손실되는 비율입니다.

시간이 지남에 따라 소산되는 전력을 통합하면 저항기에 의해 흡수되는 총 에너지가 제공됩니다. 시간이 무한대에 가까워질수록 이 에너지는 축전기에 저장된 초기 에너지에 가까워집니다. 이는 축전기의 초기 에너지가 저항기에서 점차적으로 소실되어 결국 축전기의 전하가 고갈됨을 의미합니다.

결론적으로, DC 소스가 제거될 때 RC 회로의 동작을 이해하면 이러한 회로의 과도 응답에 대한 귀중한 통찰력을 얻을 수 있습니다. 이러한 지식은 축전기의 급속 충전 및 방전이 기본 과정인 신호 처리, 전력 전자 장치, 통신 시스템과 같은 응용 분야의 회로를 설계하고 분석하는 데 필수적입니다.

소스가 없는 RC 회로

DC 소스가 RC 회로에서 갑자기 제거되면 회로는 소스가 없어집니다.
완전히 충전된 커패시터의 초기 전압이 ...

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 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 this expression equals the initial voltage.

The voltage-time graph shows an exponential drop in voltage. As the capacitor loses charge, the voltage across it also falls. The time constant, tau (τ), is the time needed for the capacitor to discharge to 36.8% of its initial voltage. It helps determine how fast the capacitor discharges and how quickly the circuit responds to change.

Using tau in the voltage response expression also allows the current through the resistor to be found. From that current, the power dissipated in the resistor can be calculated. Power is the rate at which energy is lost as heat. When that power is integrated over time, the total energy absorbed by the resistor is obtained.

As time approaches infinity, the energy absorbed by the resistor approaches the initial energy stored in the capacitor. In other words, the capacitor’s initial energy is gradually converted into heat in the resistor until the charge is depleted. Understanding this transient response is useful for analyzing RC circuits in signal processing, power electronics, and communication systems, where capacitor charging and discharging are key processes.

RC Capacitor Discharge After Power Loss

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

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

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...

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...

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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

Learn how a source-free RC circuit discharges after the DC source is removed, including exponential voltage decay, time constant, current, and energy loss.

RC Capacitor Discharge After Power Loss | 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 Capacitor Discharge After Power Loss

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