Name: rlc damping and resonant frequency
Uploaded: 2024-04-11T12:48:44.000Z
Duration: 1 min 17 s
Channel: JoVE

2つの既約エネルギー貯蔵要素を含む電気回路は、2次回路と呼ばれます。
例としては、RLC回路のほか、デュアルコンデンサとインダクタを備えたRC回路とRL回路があります。
これらの回路は、回路の入力信号と出力信号を相関させる2次微分方程式によって特徴付けられます。
入力信号は通常、電圧源または電流源から発信されますが、出力は通常、コンデンサの両端の電圧および/またはインダクタを流れる電流です。
コンデンサとインダクタに蓄えられた初期エネルギーが回路を駆動するRLC回路を考えてみましょう。
キルヒホッフの電圧法則を回路に適用し、時間微分を取ると、2次微分方程式が得られます。
この方程式の係数は、抵抗、静電容量、およびインダクタンスで表され、減衰係数と共振周波数として知られています。
減衰係数は、抵抗による減衰を示し、システム内のエネルギー散逸率を決定します。
共振周波数は、インダクタとコンデンサの間でエネルギーが交換される際の回路の固有発振周波数を表します。

利益相反は宣言されていません。

2 次回路

Second-Order Circuits

電気回路に 2 つの基本的なエネルギー蓄積要素を統合すると、RLC 回路とデュアル コンデンサまたはインダクタを備えた回路 (RC および RL 回路) を含む 2 次回路が生成されます。 2 次回路は、入力信号と出力信号を結び付ける 2 次微分Figureによって識別されます。

入力信号は通常、電圧源または電流源から発生し、大半の出力の場合では、コンデンサの両端の電圧および/またはインダクタを流れる電流を表します。 たとえば、RLC 回路では、コンデンサとインダクタに蓄えられた初期エネルギーによって回路が開始されます。 キルヒホッフの電圧則を適用して時間微分を実行すると、2 階微分Figureが得られます。

<img alt="Equation1" src="/files/ftp_upload/15097/15097_PT_Equation_1.svg" style="text-align: center; height: 36px;">

 抵抗、静電容量、インダクタンスによって決定されるその係数は、減衰係数と共振周波数として現れます。

<img alt="Equation2" src="/files/ftp_upload/15097/15097_PT_Equation_2.svg" style="text-align: center; height: 34px;">
<img alt="Equation3" src="/files/ftp_upload/15097/15097_PT_Equation_3.svg" style="text-align: center; height: 36px;">

減衰係数はこれらの回路において重要な役割を果たし、主に抵抗によって引き起こされる減衰の程度を示します。 これはシステム内でエネルギーが消散するペースに直接影響し、エネルギー損失の速度を効果的に制御します。 一方、共振周波数は、回路の固有発振周波数を表す重要な特性です。 これは、回路内のインダクタとコンデンサの間でエネルギーが交換される速度を測定し、特定の周波数で発振する回路の自然な傾向を示します。

減衰係数は、抵抗によってシステム内でエネルギーがどれだけ速く失われるかを決定します。 同時に、共振周波数は、インダクタとコンデンサの間でエネルギーがシフトするときの回路固有の発振速度を強調しています。 これら 2 つの要素は、2 次回路の動作を理解して分析する際に重要です。

2つの既約エネルギー貯蔵要素を含む電気回路は、2次回路と呼ばれます。
いくつかの例には、RLC回路や...

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

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 current through the inductor, or both. In an RLC circuit, energy already stored in the capacitor and inductor starts the circuit response.

To analyze an RLC circuit, Kirchhoff's voltage law is applied and then differentiated with respect to time. This step produces a second-order differential equation. The equation's coefficients depend on resistance, capacitance, and inductance.

Those coefficients show up as the damping coefficient and the resonant frequency. The damping coefficient describes how much the resistor slows the circuit down. It also shows how quickly energy is lost from the system.

The resonant frequency is the circuit's natural oscillation frequency. It shows how fast energy moves back and forth between the inductor and capacitor. Together, the damping coefficient and resonant frequency are key to understanding how second-order circuits behave.

RLC Damping and Resonant Frequency

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

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

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

Learn how second-order circuits use capacitors, inductors, and resistors. Explore damping coefficient, resonant frequency, and RLC circuit response.

RLC Damping and Resonant Frequency | 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

RLC Damping and Resonant Frequency

Description

Transcript

Tags