Name: transfer function from input to output
Uploaded: 2024-09-26T13:17:18.000Z
Duration: 1 min 21 s
Channel: JoVE

伝達関数は、周波数領域の可能な各入力に対するシステムの出力を記述する数学的表現です。
一般的な n次、線形、時間不変微分方程式について考えます。この方程式は、1 つの変数が入力を表し、別の変数が出力を表すシステムを特徴付けます。
この方程式の両側にラプラス変換を適用すると、代数式が得られます。
すべての初期条件がゼロであると仮定すると、この方程式はさらに単純化されます。
出力のラプラス変換と入力のラプラス変換の比率は、伝達関数と呼ばれます。
伝達関数はブロック線図として表され、左側に入力、右側に出力、ブロック内にシステム伝達関数があります。
伝達関数の分母は、微分方程式の特性多項式と同じです。
1 階の微分方程式について考えます。この方程式の伝達関数は、初期条件がゼロであると仮定して、両側でラプラス変換を取ることによって計算されます。
単純化すると、結果は周波数領域の入力に対するシステムの応答を表す伝達関数になります。

制御システムにおける伝達関数

Transfer Function in Control Systems

伝達関数は、線形時間不変 (LTI) システムの分析と設計における基本的な概念です。伝達関数は、システムが周波数領域でさまざまな入力にどのように応答するかを簡潔に理解する方法を提供します。伝達関数は、システムの動作を記述する時間領域微分方程式と、操作と分析を容易にする周波数領域表現との間の橋渡しとして機能します。

伝達関数を導出するには、次の形式の一般的な n 次線形時間不変微分方程式を考えます：

<img alt="Equation1" src="/files/ftp_upload/16095/16095_PT%20EQUATION_1.svg" style="height: 40px;">

ここで、c(t) は出力、r(t) は入力、a_i と b_i は定数係数です。すべての初期条件がゼロであると仮定して、両辺にラプラス変換を適用すると、微分方程式は、複素周波数変数 s に関する代数方程式に変換できます。項を並べ替えると、次のようになります：


<img alt="Equation2" src="/files/ftp_upload/16095/16095_PT%20EQUATION_2.svg" style="height: 22px;">

伝達関数 H(s) は、出力 C(s) と入力 R(s) の比として定義されます：


<img alt="Equation3" src="/files/ftp_upload/16095/16095_PT%20EQUATION_3.svg" style="height: 40px;">

この式は、伝達関数が s の有理関数であることを示しています。分子は入力係数によって形成される多項式であり、分母は微分方程式の特性多項式です。

この伝達関数は、システムの出力 c(t) が周波数領域で入力 r(t) にどのように応答するかを示します。伝達関数は、左側に入力 R(s)、右側に出力 C(s)、そしてブロック内に伝達関数 H(s) を配置したブロック図で表すことができます。この視覚化により、特に複雑なシステムを扱う場合に、システムの動作の理解と分析が簡単になります。

伝達関数は、周波数領域の可能な各入力に対するシステムの出力を記述する数学的表現です。
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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

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 coefficients a_i and b_i remain fixed, which makes the system linear and time-invariant.

The transfer function is found by applying the Laplace transform to both sides of the differential equation. This step assumes zero initial conditions. The result is an algebraic equation in terms of s, the complex frequency variable.

After rearranging the terms, the transfer function H(s) is defined as C(s) divided by R(s), where C(s) is the output in the Laplace domain and R(s) is the input in the Laplace domain. This form shows that H(s) is a rational function of s. Its numerator comes from the input coefficients, and its denominator is the characteristic polynomial of the differential equation.

This representation shows how the output c(t) responds to an input r(t) in the frequency domain. It can also be drawn as a block diagram, with R(s) entering on the left, C(s) leaving on the right, and H(s) inside the block. That diagram makes system analysis easier, especially for more complex systems.

Transfer Function from Input to Output

transfer function in control systems

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 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.

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 transfer functions connect a system’s input and output using Laplace transforms, zero initial conditions, and a simple block diagram.

Transfer Function from Input to Output | 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

Transfer Function from Input to Output

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