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

La fonction de transfert est une représentation mathématique qui décrit la sortie du système pour chaque entrée possible dans le domaine fréquentiel.
Considérons une équation différentielle générale d’ordre n, linéaire et invariante dans le temps. Cette équation caractérise le système où une variable représente l’entrée et une autre représente la sortie.
L’application de la transformée de Laplace aux deux côtés de cette équation donne une expression algébrique.
En supposant que toutes les conditions initiales sont nulles, cette équation est encore simplifiée.
Le rapport entre la transformée de Laplace de la sortie et la transformée de Laplace de l'entrée est appelé fonction de transfert.
La fonction de transfert est représentée sous la forme d’un diagramme, avec l’entrée à gauche, la sortie à droite et la fonction de transfert du système à l’intérieur du bloc.
Le dénominateur de la fonction de transfert est identique au polynôme caractéristique de l'équation différentielle.
Considérons une équation différentielle du premier ordre. La fonction de transfert de cette équation est calculée en prenant la transformée de Laplace des deux côtés, en supposant des conditions initiales nulles.
En cas de simplification, le résultat est une fonction de transfert représentant la réponse du système à une entrée dans le domaine fréquentiel.

fonction de transfert dans les systèmes de contrôle

Transfer Function in Control Systems

La fonction de transfert est un concept fondamental dans l'analyse et la conception des systèmes linéaires invariants dans le temps (LTI). Elle offre un moyen concis de comprendre comment un système répond à différentes entrées dans le domaine fréquentiel. Elle sert de pont entre les équations différentielles du domaine temporel qui décrivent la dynamique du système et la représentation du domaine fréquentiel qui facilitent la manipulation et l'analyse.

Pour dériver la fonction de transfert, considérons une équation différentielle linéaire invariante dans le temps d'ordre n de la forme :

<img alt="Equation1" src="/files/ftp_upload/16095/16095_PT%20EQUATION_1.svg" style="text-align: center; height: 40px;">

Ici, c(t) est la sortie, r(t) est l'entrée et a_i et b_i sont des coefficients constants. En appliquant la transformée de Laplace aux deux côtés, en supposant que toutes les conditions initiales sont nulles, l'équation différentielle peut être convertie en une équation algébrique en termes de s, la variable de fréquence complexe. En réorganisant les termes, nous obtenons :

<img alt="Equation2" src="/files/ftp_upload/16095/16095_PT%20EQUATION_2.svg" style="text-align: center; height: 22px;">

La fonction de transfert H(s) est définie comme le rapport entre la sortie C(s) et l'entrée R(s) :

<img alt="Equation3" src="/files/ftp_upload/16095/16095_PT%20EQUATION_3.svg" style="text-align: center; height: 40px;">

Cette expression montre que la fonction de transfert est une fonction rationnelle de s. Le numérateur est le polynôme formé par les coefficients d'entrée et le dénominateur est le polynôme caractéristique de l'équation différentielle.

Cette fonction de transfert indique comment la sortie c(t) du système répond à une entrée r(t) dans le domaine fréquentiel. La fonction de transfert peut être représentée dans un diagramme en blocs avec l'entrée R(s) à gauche, la sortie C(s) à droite et la fonction de transfert H(s) à l'intérieur du bloc. Cette visualisation simplifie la compréhension et l'analyse du comportement du système, en particulier lorsqu'il s'agit de systèmes plus complexes.

Fonction de transfert dans les systèmes de contrôle

La fonction de transfert est une représentation mathématique qui décrit la sortie du système pour chaque entrée possible dans le domaine fréquentiel.
Considérez un ...

Education

Enseignement

JoVE Core

Electrical Engineering

Génie électrique

Unit 1

Unité 1

Modeling in Time and Frequency Domain

Modélisation dans les domaines temporel et fréquentiel

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

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 Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

Transfer Function from Input to Output

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