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

DC電源がRL回路から突然切断されると、ソースフリーになります。
インダクタの初期電流がi0であると仮定すると、インダクタに蓄積された初期エネルギーを決定できます。
ループの周囲にキルヒホッフの電圧則を適用し、インダクタと抵抗の両端の電圧を代入すると、1次微分方程式が得られます。
項を並べ替え、積分し、制限を適用すると、対数方程式が得られます。
両側で指数を取ることにより、回路の自然な応答の最終的な表現が決定されます。
電流対時間グラフは、初期電流の指数関数的な減少を示しています。
電流応答は、インダクタンスと抵抗の比である時定数で表すことができます。
電流式は、抵抗器の両端で消費される電圧と電力を決定するために使用されます。
時間の経過に伴う消費電力の統合により、抵抗によって吸収されるエネルギーの式が得られます。
時間が無限大に近づくと、抵抗によって吸収されたエネルギーはインダクタに蓄積された初期エネルギーに近づき、初期エネルギーが抵抗内で徐々に消費されることを意味します。

利益相反は宣言されていません。

ソースなしの rl 回路

DC 電源が RL (抵抗インダクター) 回路から突然切り離されると、回路の電源がなくなります。 インダクタの初期電流が I0 であると仮定する事で、インダクタに蓄えられる初期エネルギーを求めることができます。

キルヒホッフの電圧則を回路のループに適用し、インダクタと抵抗の両端の電圧を代入すると、一次微分Figureが得られます。 対数Figureは、このFigureの項を並べ替え、積分し、制限を適用することで得られます。 このFigureの両側の指数を取ると、回路の自然な応答の最終的な式が得られます。

電流を時間に対してプロットすると、初期電流の指数関数的な減少が観察されます。 この動作は時定数の観点から表現できます。RL 回路の場合、時定数はインダクタンスと抵抗の比です。 この時定数は、回路が入力信号の変化に応答する速度を表します。

電流式を使用すると、抵抗の両端の電圧と抵抗で消費される電力を計算できます。 消費電力は本質的に、エネルギーが熱の形で失われる割合です。

抵抗器によって吸収される総エネルギーを求めるには、消費電力を時間とともに積分します。 時間が無限大に近づくと、抵抗器によって吸収されるエネルギーは、インダクタに蓄えられる初期エネルギーに近づきます。 これは、インダクタに蓄えられた初期エネルギーが、インダクタのエネルギーが枯渇するまで抵抗器内で徐々に消散することを意味します。

結論として、DC 電源が除去されたときの RL 回路の動作を理解すると、これらの回路の過渡応答について貴重な洞察が得られます。 この知識は、信号のフィルタリングや整形にインダクタが広く使用されるパワー エレクトロニクスや通信システムなどのアプリケーションで回路を設計および解析するための基礎となります。

ソースなしの RL 回路

DC電源がRL回路から突然切断されると、ソースフリーになります。
インダクタに初期電流i0があると仮定すると、初期エネルギー...

Education

JoVE Core

Electrical Engineering

電気工学

Unit 1

ユニット1

First and Second-Order Circuits

一次および二次回路

KEY TERMS AND CONCEPTS

主要な用語と概念

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 電気工学 は、電気回路の基本概念と回路解析テクニックを網羅したアニメーションビデオシリーズです。オペアンプと回路変換および 1 次・ 2 次回路の概念を網羅しています。実験を撮影した実写動画では、設計例や現代社会での応用が含まれており、複雑な概念も分かりやすく学習いただけます。

electrical engineering

RL Circuit Decay After Source Removal

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