Name: forced response in a series rlc circuit
Uploaded: 2024-04-11T12:48:44.000Z
Duration: 1 min 12 s
Channel: JoVE

火花の発生に不可欠なバッテリーから高電圧を生成する自動車の点火システムを考えてみましょう。
このシステムは、単純な直列RLC回路としてモデル化でき、回路の完全な応答を解析できます。
ここでは、入力DC電圧が強制ステップ関数として機能し、強制関数の特性を反映した強制ステップ応答が得られます。
キルヒホッフの電圧則を回路に適用すると、2次微分方程式が得られます。
この方程式は、ソースフリーRLC回路の2次微分方程式に似ており、DCソースが方程式の形式を変更しないことを示しています。
この方程式の完全な解は、過渡応答と定常状態応答の組み合わせです。
過渡応答は時間とともに減少し、過減衰、臨界減衰、および減衰不足のシナリオにおけるソースフリー回路のソリューションに対応します。
定常状態の応答はコンデンサ電圧の最終値に対応し、これはソース電圧と同一です。
関連する定数は、回路の初期条件から推測できます。

利益相反は宣言されていません。

ソースを備えた直列 rlc 回路

Series RLC Circuit with Source

自動車の点火システムの動作を例に考えてみましょう。これは、バッテリーから高電圧を生成して火花を発生させる重要な部品です。 このシステムは単純な直列 RLC 回路として説明でき、その完全な応答を詳細に分析できます。

この場合、入力 DC 電圧は強制ステップ関数として機能し、入力の特性を反映する強制ステップ応答が得られます。 キルヒホッフの電圧則を回路に適用すると、2 階微分Figureが得られます。 注目すべきことに、このFigureは、ソースフリー RLC 回路を特徴付ける 2 次微分Figureに非常によく似ています。 この類似性は、DC 電源の存在によってFigureの基本的な形式が変化しないことを強調しています。

<img alt="Equation1" src="/files/ftp_upload/15100/15100_PT_Equation_1.svg" style="text-align: center; height: 34px;">

このFigureの完全な解には、過渡応答と定常状態応答の両方が含まれます。

<img alt="Equation2" src="/files/ftp_upload/15100/15100_PT_Equation_2.svg" style="text-align: center; height: 16px;">

時間の経過とともに減少する過渡応答は、ソースフリー回路のソリューションと一致しており、過減衰、臨界減衰、および過小減衰の動作を含むシナリオを包含します。 一方、定常状態の応答は、電源電圧に等しいコンデンサ電圧の最終値に対応します。 これらの応答に関係する定数は、回路の初期条件から決定できます。

ソースを備えた直列 RLC 回路

火花の発生に不可欠なバッテリーから高電圧を生成する自動車の点火システムを考えてみましょう。
このシステムは、...

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

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 DC source does not change the basic equation structure.

The full response of the circuit has two parts: transient response and steady-state response. The transient response fades with time and matches the source-free circuit solution. It can be overdamped, critically damped, or underdamped.

The steady-state response is the final capacitor voltage. That final value equals the source voltage. The constants in both parts of the solution come from the initial conditions of the circuit.

Forced Response in a Series RLC Circuit

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...

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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

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

Study how a DC source drives a series RLC circuit, creating transient and steady-state responses with overdamped, critically damped, or underdamped behavior.

Forced Response in a Series RLC Circuit | 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

Forced Response in a Series RLC Circuit

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