Name: rl circuit decay after source removal
Uploaded: 2024-04-11T12:48:43.000Z
Duration: 1 min 14 s
Channel: JoVE

DC 소스가 RL 회로에서 갑자기 분리되면 소스가 없어집니다.
인덕터가 초기 전류 i0를 갖는다고 가정하면 인덕터에 저장된 초기 에너지를 결정할 수 있습니다.
루프 주위에 Kirchhoff의 전압 법칙을 적용하고 인덕터와 저항기의 전압을 대체하면 1차 차동 방정식이 생성됩니다.
항을 재정렬하고, 적분하고, 한계를 적용하면 로그 방정식이 생성됩니다.
양쪽에서 지수를 취함으로써 회로의 자연 응답의 최종 표현이 결정됩니다.
전류 대 시간 그래프는 초기 전류의 기하급수적인 감소를 보여줍니다.
전류 응답은 인덕턴스와 저항의 비율인 시간 상수로 표현할 수 있습니다.
전류 표현은 저항기에서 발산되는 전압과 전력을 결정하는 데 사용됩니다.
시간이 지남에 따라 소산되는 전력의 통합은 저항기가 흡수하는 에너지에 대한 표현을 제공합니다.
시간이 무한대에 가까워짐에 따라 저항기에 의해 흡수된 에너지는 인덕터에 저장된 초기 에너지에 접근하며, 이는 초기 에너지가 저항기에서 점차적으로 소산됨을 의미합니다.

소스가 없는 rl 회로

RL(저항-유도자) 회로에서 DC 소스의 연결이 갑자기 끊어지면 회로에 소스가 없게 됩니다. 유도자에 I0 로 표시된 초기 전류가 있다고 가정하면 유도자에 저장된 초기 에너지가 결정될 수 있습니다.

회로 루프 주위에 키르히호프의 전압 법칙을 적용하고 유도자와 저항기의 전압을 대체하면 1차 미분 방정식이 생성됩니다. 이 방정식의 항을 재배열하고 이를 적분한 후 극한을 적용하여 로그 방정식을 얻습니다. 이 방정식의 양쪽에 지수를 취하면 회로의 자연스러운 응답이 최종적으로 표현됩니다.

전류를 시간에 대해 그래프로 표시하면 초기 전류가 기하급수적으로 감소하는 것이 관찰됩니다. 이 동작은 RL 회로의 경우 인덕턴스 대 저항의 비율인 시상수로 표현될 수 있습니다. 이 시정수는 회로가 입력 신호의 변화에 반응하는 속도를 나타냅니다.

전류 식을 사용하면 저항기 양단의 전압과 저항기에서 소비되는 전력을 계산할 수 있습니다. 소비되는 전력은 본질적으로 열의 형태로 에너지가 손실되는 속도입니다.

저항기에 의해 흡수된 총 에너지를 찾기 위해 소비된 전력을 시간에 따라 적분합니다. 시간이 무한대에 가까워질수록 저항기에 의해 흡수된 에너지는 유도자에 저장된 초기 에너지에 가까워집니다. 이는 유도자에 저장된 초기 에너지가 유도자의 에너지가 고갈될 때까지 저항기에서 점차적으로 소산된다는 것을 의미합니다.

결론적으로, DC 소스가 제거될 때 RL 회로의 동작을 이해하면 이러한 회로의 과도 응답에 대한 귀중한 통찰력을 얻을 수 있습니다. 이 지식은 신호를 필터링하거나 형성하기 위해 유도자가 광범위하게 사용되는 전력 전자 장치 및 통신 시스템과 같은 응용 분야의 회로를 설계하고 분석하는 데 기본입니다.

소스가 없는 RL 회로

DC 소스가 RL 회로에서 갑자기 분리되면 소스가 없어집니다.
인덕터가 초기 전류 i0를 갖는다고 가정하면 초기 에너지 ...

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

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 can then be integrated with limits. Taking the exponential of both sides gives the final current expression.

The current decreases exponentially with time. This drop can be described with the time constant of an RL circuit. The time constant is the ratio of inductance to resistance. It shows how quickly the circuit responds to a change in the input signal.

The current expression also lets us find the voltage across the resistor. It also lets us calculate the power dissipated in the resistor. Power is the rate at which energy is lost as heat. To find the total energy absorbed by the resistor, the power is integrated over time.

As time approaches infinity, the energy absorbed by the resistor approaches the initial energy stored in the inductor. This means the inductor’s energy is gradually transferred to the resistor until it is depleted. Understanding this transient response helps with circuit analysis. It is also useful in power electronics and communication systems, where inductors are used to filter or shape signals.

RL Circuit Decay After Source Removal

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

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

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

See how an RL circuit responds after the DC source is removed, with exponential current decay, time constant, resistor power, and energy loss as heat.

RL Circuit Decay After Source Removal | 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

RL Circuit Decay After Source Removal

Description

Transcript

Tags