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

La función de transferencia es una representación matemática que describe la salida del sistema para cada posible entrada en el dominio de la frecuencia.
Consideremos una ecuación diferencial general de nésimo orden, lineal e invariante en el tiempo. Esta ecuación caracteriza el sistema donde una variable representa la entrada y otra representa la salida.
La aplicación de la transformada de Laplace a ambos lados de esta ecuación da como resultado una expresión algebraica.
Suponiendo que todas las condiciones iniciales son cero, esta ecuación se simplifica aún más.
La relación entre la transformada de Laplace de la salida y la transformada de Laplace de la entrada se denomina función de transferencia.
La función de transferencia se representa como un diagrama de bloques, con la entrada a la izquierda, la salida a la derecha y la función de transferencia del sistema dentro del bloque.
El denominador de la función de transferencia es idéntico al polinomio característico de la ecuación diferencial.
Consideremos una ecuación diferencial de primer orden. La función de transferencia para esta ecuación se calcula tomando la transformada de Laplace en ambos lados, asumiendo condiciones iniciales cero.
Simplificando, el resultado es una función de transferencia que representa la respuesta del sistema a una entrada en el dominio de la frecuencia.

función de transferencia en sistemas de control

Transfer Function in Control Systems

La función de transferencia es un concepto fundamental en el análisis y diseño de sistemas lineales invariantes en el tiempo (LTI). Ofrece una forma concisa de entender cómo responde un sistema a diferentes entradas en el dominio de la frecuencia. Sirve como un puente entre las ecuaciones diferenciales en el dominio del tiempo que describen la dinámica del sistema y la representación en el dominio de la frecuencia que facilita la manipulación y el análisis.
Para derivar la función de transferencia, considere una ecuación diferencial lineal invariante en el tiempo de orden n general de la forma:
<img alt="Equation1" src="/files/ftp_upload/16095/16095_PT%20EQUATION_1.svg" style="text-align: center; height: 40px;">
Aquí, c(t) es la salida, r(t) es la entrada y a_i y b_i son coeficientes constantes. Aplicando la transformada de Laplace a ambos lados, suponiendo que todas las condiciones iniciales son cero, la ecuación diferencial se puede convertir en una ecuación algebraica en términos de s, la variable de frecuencia compleja. Reordenando los términos, obtenemos:
<img alt="Equation2" src="/files/ftp_upload/16095/16095_PT%20EQUATION_2.svg" style="text-align: center; height: 22px;">
La función de transferencia H(s) se define como la relación entre la salida C(s) y la entrada R(s):
<img alt="Equation3" src="/files/ftp_upload/16095/16095_PT%20EQUATION_3.svg" style="text-align: center; height: 40px;">
Esta expresión muestra que la función de transferencia es una función racional de s. El numerador es el polinomio formado por los coeficientes de entrada y el denominador es el polinomio característico de la ecuación diferencial.
Esta función de transferencia indica cómo responde la salida c(t) del sistema a una entrada r(t) en el dominio de la frecuencia. La función de transferencia se puede representar en un diagrama de bloques con la entrada R(s) a la izquierda, la salida C(s) a la derecha y la función de transferencia H(s) dentro del bloque. Esta visualización simplifica la comprensión y el análisis del comportamiento del sistema, especialmente cuando se trata de sistemas más complejos.

Función de transferencia en sistemas de control

La función de transferencia es una representación matemática que describe la salida del sistema para cada posible entrada en el dominio de la frecuencia.
Considere un ...

Education

Educación

JoVE Core

Electrical Engineering

Ingeniería Eléctrica

Unit 1

Unidad 1

Modeling in Time and Frequency Domain

Modelado en el dominio del tiempo y de la frecuencia

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

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

Description

Transcript

Tags