Name: rlc circuit transfer functions
Uploaded: 2024-09-26T13:17:18.000Z
Duration: 1 min 21 s
Channel: JoVE

電気ネットワークは、まず、抵抗器、コンデンサ、インダクタの3つの受動線形部品で構成される等価回路で表されます。
これらの部品は回路に組み合わされ、入力と出力が決定され、伝達関数はキルヒホッフの法則を使用して求められます。
RLC回路では、コンデンサの両端の電圧を入力電圧に関連付ける伝達関数は、キルヒホッフの電圧則を使用して導き出すことができます。
これにより、初期条件がゼロであると仮定したネットワークの積分微分方程式が得られます。
変数は最初に電流から充電に変更され、次にコンデンサの電圧-充電関係が適用されます。
この方程式のラプラス変換を取り、それを単純化すると、この回路の伝達関数が得られます。
インピーダンスは抵抗と同様の伝達関数ですが、コンデンサやインダクタに適用できます。
伝達関数は、キルヒホッフの現在の法則を使用して、節点解析を使用して取得することもできます。
システム内の電流は、コンデンサを流れる電流と、直列抵抗とインダクタを循環する電流で構成されます。
単純化した後、伝達関数についても同じ結果が得られます。

電気システム

Electrical Systems

電気工学では、抵抗器 (R)、コンデンサ (C)、インダクタ (L)) などの受動線形要素で構成されるネットワークの解析が基本です。これらの要素は回路に編成され、入力と出力の関係を伝達関数を使用して解析できます。コンデンサの両端の電圧を入力電圧に関連付ける RLC 回路の伝達関数は、キルヒホッフの法則を用いて導出できます。

伝達関数を導出するには、要素が直列に接続された RLC 回路を考えます。キルヒホッフの電圧法則 (KVL) は、閉回路内のすべての電圧の和がゼロであることを示しています。 RLC 回路の場合、これは初期条件がゼロであると仮定すると、次の積分微分方程式に変換されます。

<img alt="Equation1" src="/files/ftp_upload/16096/16096_PT%20EQUATION_1.svg" style="height: 32px;">

ここで、V(t) は入力電圧、VR(t)=i(t)R は抵抗器の電圧、VC(t) =q(t)/C はコンデンサの電圧、q(t) はコンデンサの電荷、VL(t) = L(di(t)/dt) はインダクタの電圧です。

電流i(t) は電荷 q(t) と次の式で関連しています i(t)=dq(t)/dt。これらの関係を KVL 方程式に代入すると、次のようになります。


<img alt="Equation2" src="/files/ftp_upload/16096/16096_PT%20EQUATION_2.svg" style="height: 48px;">

方程式を電圧形式で書き、両辺のラプラス変換を取り、すべての初期条件がゼロであると仮定すると、次の方程式が得られます。


<img alt="Equation3" src="/files/ftp_upload/16096/16096_PT%20EQUATION_3.svg" style="height: 30px;">

項を並べ替えて伝達関数を解きます。


<img alt="Equation4" src="/files/ftp_upload/16096/16096_PT%20EQUATION_4.svg" style="height: 46px;">

この伝達関数は、周波数領域での入力電圧に対するコンデンサの電圧を表します。インピーダンスは抵抗に似ていますが、コンデンサとインダクタに適用でき、伝達関数を定義する上で重要な役割を果たします。さらに、キルヒホッフの電流法則 (KCL) を使用して、ノード解析によって伝達関数を導出することもできます。この法則は、ノードに入る電流の和が、ノードから出る電流の和と等しくなければならないことを示しています。 RLC 回路では、総電流はコンデンサを流れる電流と、抵抗器とインダクタの直列接続を流れる電流の和です。KCL を適用して簡略化することで、同じ伝達関数を導出できます。

KVL と KCL は回路解析の強力なツールであり、周波数領域での電気ネットワークの動的動作を簡潔に記述する伝達関数の導出を可能にします。これらの方法は、さまざまなアプリケーションでの複雑な回路の設計と解析に不可欠です。

電気ネットワークは、まず、抵抗器、コンデンサ、インダクタの3つの受動線形部品で構成される等価回路で表されます。
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Education

JoVE Core

Electrical Engineering

電気工学

Unit 1

ユニット1

Modeling in Time and Frequency Domain

時間領域および周波数領域におけるモデリング

KEY TERMS AND CONCEPTS

主要用語と概念

rlc circuit transfer functions

RLC circuits use resistors, capacitors, and inductors to show how electrical signals change over time. In circuit analysis, these passive linear components are arranged so the link between an input and an output can be described with a transfer function. For this topic, the output is the voltage across the capacitor, compared with the input voltage.

A series RLC circuit can be analyzed with Kirchhoff's Voltage Law, or KVL. KVL says that the sum of all voltages around a closed loop is zero. Using that rule, the circuit can be written as an integro-differential equation when the initial conditions are zero.

In the equation, the input voltage is V(t). The resistor voltage is VR(t) = i(t)R, where i(t) is the current. The capacitor voltage depends on the charge q(t), and the inductor voltage depends on the current change in the loop. Since current and charge are related, these expressions can be substituted into the KVL equation.

The same circuit can also be written in voltage form and converted with the Laplace transform. With all initial conditions set to zero, this step gives an equation in the frequency domain. Rearranging the terms produces the transfer function for the capacitor voltage response.

Impedance plays an important role in this result. It works like resistance, but it applies to capacitors and inductors as well. The transfer function shows how the capacitor voltage responds to the input voltage in the frequency domain.

Kirchhoff's Current Law, or KCL, can also be used to reach the same result through nodal analysis. KCL says that the sum of currents entering a node must equal the sum of currents leaving it. In the RLC circuit, the total current is the current through the capacitor plus the current through the series resistor and inductor, and simplifying this relationship gives the same transfer function.

RLC Circuit Transfer Functions

transfer function in control systems

Transfer Function from Input to Output

The transfer function describes how a linear time-invariant, or LTI, system maps an input to an output in the frequency domain. It gives a compact way to study system behavior without working only in the time domain. It also connects the differential equations of the system to a form that is easier to handle mathematically. For an nth-order LTI system, the behavior is written as a differential equation with constant coefficients. In that equation, r(t) is the input and c(t) is the output. The...

electrical systems

RLC circuits use resistors, capacitors, and inductors to show how electrical signals change over time. In circuit analysis, these passive linear components are arranged so the link between an input and an output can be described with a transfer function. For this topic, the output is the voltage across the capacitor, compared with the input voltage. A series RLC circuit can be analyzed with Kirchhoff's Voltage Law, or KVL. KVL says that the sum of all voltages around a closed loop is zero.

mechanical systems

Transfer Functions in Mechanical Motion

Mechanical motion can be modeled with springs, masses, and viscous dampers. In these models, springs behave like inductors and masses act like capacitors in an electrical network. A viscous damper works like a resistor because it removes energy from the system. A force acts on the mass in the direction of motion. It is balanced by the spring force, the damping force, and the force from the mass’s acceleration. Newton’s second law is used to describe this balance, and the sum of the forces on...

electro-mechanical systems

DC Motor Modeling in Electromechanical Systems

DC motor modeling is a key part of electromechanical systems. These systems combine electrical parts and mechanical parts to produce a useful output. A DC motor is one common example because it turns electrical energy into mechanical motion. It is used in many places, from simple fans to complex robotic mechanisms. Inside the DC motor, the armature is a rotating circuit placed in a magnetic field. When current flows through the armature, it experiences a force from the magnetic field. That...

linear approximation in frequency domain

Linearizing Nonlinear RL Circuits

Linearizing nonlinear RL circuits helps turn a hard circuit problem into a simpler linear one. Linear systems follow superposition, which means the total response equals the sum of the responses to separate inputs. They also follow homogeneity, which means scaling the input scales the output by the same amount. Nonlinear systems do not automatically have those properties. For small changes around an operating point, though, a nonlinear system can often be treated as linear. This is done with a...

state space representation

RLC Circuit State Equations

RLC circuit state equations show how a circuit changes over time using state variables. This time-domain method is useful for linear, time-invariant systems, but it also helps when frequency-domain analysis is not enough. It is especially useful for nonlinear systems, time-varying systems, and multiple-input multiple-output systems. The state-space approach turns an nth-order system into a set of simultaneous first-order differential equations called state equations. For a common second-order...

transfer function to state space

Phase-Variable State Equations from Transfer Functions

Phase-variable state equations show how a transfer function can be rewritten for computer simulation. In state-space form, the system is described with state variables, which are the output and its first several derivatives. This makes it easier to model a physical system on a digital computer. The starting point is an nth-order linear differential equation with constant coefficients. Systems like an RLC circuit can be written this way. To build the state equations, the transfer function is...

state space to transfer function

Finding a Transfer Function from State Space

Finding a transfer function from state-space form is a key step in system analysis. It changes a time-domain model into a frequency-domain form. That makes it easier to study and design control systems. The process starts with the state-space equations. These include the state equation and the output equation. In these equations, x(t) is the state vector, u(t) is the input vector, y(t) is the output vector, and A, B, C, and D are the matrices that define the system dynamics. Next, the Laplace...

linear approximation in time domain

Linearizing Nonlinear State-Space Models

Linearizing nonlinear state-space models helps describe systems that change with time or operating conditions. State-space representation is often effective for this kind of modeling. It is especially useful for systems such as a simple pendulum or a translational mechanical system with nonlinear springs. A simple pendulum can be modeled by summing torques around the pivot point. In this case, the mass is evenly distributed along the pendulum, and the center of mass is located at half its...

key terms and concepts

Learn how KVL, KCL, and the Laplace transform are used to find the transfer function of a series RLC circuit and track capacitor voltage in the frequency domain.

RLC Circuit Transfer Functions | 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 Circuit Transfer Functions

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