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

A função de transferência é uma representação matemática que descreve a saída do sistema para cada entrada possível no domínio da frequência.
Considere uma equação diferencial geral de nésima, linear e invariante no tempo. Esta equação caracteriza o sistema onde uma variável representa a entrada e outra representa a saída.
A aplicação da transformada de Laplace a ambos os lados desta equação resulta em uma expressão algébrica.
Assumindo que todas as condições iniciais são zero, esta equação é ainda mais simplificada.
A razão entre a transformada de Laplace da saída e a transformada de Laplace da entrada é chamada de função de transferência.
A função de transferência é representada como um diagrama de blocos, com a entrada à esquerda, a saída à direita e a função de transferência do sistema dentro do bloco.
O denominador da função de transferência é idêntico ao polinômio característico da equação diferencial.
Considere uma equação diferencial de primeira ordem. A função de transferência para esta equação é calculada tomando a transformada de Laplace em ambos os lados, assumindo condições iniciais zero.
Após a simplificação, o resultado é uma função de transferência que representa a resposta do sistema a uma entrada no domínio da frequência.

função de transferência em sistemas de controle

Transfer Function in Control Systems

A função de transferência é um conceito fundamental na análise e no design de sistemas lineares invariantes no tempo (LTI). Ela oferece uma maneira concisa de entender como um sistema responde a diferentes entradas no domínio da frequência. Ela serve como uma ponte entre as equações diferenciais do domínio do tempo que descrevem a dinâmica do sistema e a representação do domínio da frequência que facilita a manipulação e a análise.
Para derivar a função de transferência, considere uma equação diferencial linear invariante no tempo de ordem n geral da forma:
<img alt="Equation1" src="/files/ftp_upload/16095/16095_PT%20EQUATION_1.svg" style="height: 40px;">
Aqui, c(t) é a saída, r(t) é a entrada e a_i e b_i são coeficientes constantes. Aplicando a transformada de Laplace a ambos os lados, assumindo que todas as condições iniciais são zero, a equação diferencial pode ser convertida em uma equação algébrica em termos de s, a variável de frequência complexa. Reorganizando os termos, obtemos:
<img alt="Equation2" src="/files/ftp_upload/16095/16095_PT%20EQUATION_2.svg" style="height: 22px;">
A função de transferência H(s) é definida como a razão entre a saída C(s) e a entrada R(s):
<img alt="Equation3" src="/files/ftp_upload/16095/16095_PT%20EQUATION_3.svg" style="height: 40px;">
Esta expressão mostra que a função de transferência é uma função racional de s. O numerador é o polinômio formado pelos coeficientes de entrada, e o denominador é o polinômio característico da equação diferencial.
Esta função de transferência indica como a saída c(t) do sistema responde a uma entrada r(t) no domínio da frequência. A função de transferência pode ser representada em um diagrama de blocos com a entrada R(s) à esquerda, a saída C(s) à direita e a função de transferência H(s) dentro do bloco. Esta visualização simplifica a compreensão e a análise do comportamento do sistema, especialmente ao lidar com sistemas mais complexos.

Função de transferência em sistemas de controle

A função de transferência é uma representação matemática que descreve a saída do sistema para cada entrada possível no domínio da frequência.
Considere um ...

Education

JoVE Core

Electrical Engineering

Engenharia Elétrica

Unit 1

Unidade 1

Modeling in Time and Frequency Domain

Modelagem no Domínio do Tempo e da Frequência

KEY TERMS AND CONCEPTS

TERMOS E CONCEITOS-CHAVE

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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