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

The transfer function is a mathematical representation that describes the system&#39;s output for each possible input in the frequency domain.
Consider a general nth-order, linear, time-invariant differential equation. This equation characterizes the system where one variable represents the input, and another represents the output.
Applying the Laplace transform to both sides of this equation results in an algebraic expression.
Assuming that all initial conditions are zero, this equation is further simplified.
The ratio of the output&#39;s Laplace transform to the input&#39;s Laplace transform is called the transfer function.
The transfer function is represented as a block diagram, with the input on the left, the output on the right, and the system transfer function inside the block.
The transfer function&#39;s denominator is identical to the characteristic polynomial of the differential equation.
Consider a first-order differential equation. The transfer function for this equation is calculated by taking the Laplace transform on both sides, assuming zero initial conditions.
Upon simplification, the result is a transfer function representing the system&#39;s response to an input in the frequency domain.

transferfunctie in regelsystemen

Transfer Function in Control Systems

De transferfunctie is een fundamenteel concept in de analyse en het ontwerp van lineaire tijdsinvariante (LTI) systemen. Het biedt een beknopte manier om te begrijpen hoe een systeem reageert op verschillende invoer in het frequentiedomein. Het dient als een brug tussen de differentiaalvergelijkingen in het tijdsdomein die de systeemdynamiek beschrijven en de representatie in het frequentiedomein die eenvoudiger manipulatie en analyse mogelijk maakt.

Om de transferfunctie af te leiden, overweeg een algemene lineaire tijdsinvariante differentiaalvergelijking van de n-de orde van de vorm:

<img alt="Equation1" src="/files/ftp_upload/16095/16095_PT EQUATION_1.svg" style="height: 40px;">


Hier is c(t) de uitvoer, r(t) de invoer en a_i en b_i constante coëfficiënten. Door de Laplace-transformatie op beide zijden toe te passen, ervan uitgaande dat alle beginvoorwaarden nul zijn, kan de differentiaalvergelijking worden omgezet in een algebraïsche vergelijking in termen van s, de complexe frequentie variabele. Als we de termen herschikken, krijgen we:

<img alt="Equation2" src="/files/ftp_upload/16095/16095_PT EQUATION_2.svg" style="height: 22px;">


De overdrachtsfunctie H(s) is gedefinieerd als de verhouding van de uitvoer C(s) tot de invoer
R(s):

<img alt="Equation3" src="/files/ftp_upload/16095/16095_PT EQUATION_3.svg" style="height: 40px;">
Deze uitdrukking laat zien dat de overdrachtsfunctie een rationale functie van s is. De teller is de polynoom gevormd door de invoer coëfficiënten en de noemer is de karakteristieke polynoom van de differentiaalvergelijking.

Deze overdrachtsfunctie geeft aan hoe de uitvoer c(t) van het systeem reageert op een invoer r(t) in het frequentiedomein. De overdrachtsfunctie kan worden weergegeven in een blokdiagram met de invoer R(s) aan de linkerkant, de uitvoer C(s) aan de rechterkant en de overdrachtsfunctie H(s) in het blok. Deze visualisatie vereenvoudigt het begrijpen en analyseren van systeemgedrag, vooral bij complexere systemen.

Transferfunctie in regelsystemen

The transfer function is a mathematical representation that describes the system&#39;s output for each possible input in the frequency domain.
Consider a ...

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Electrical Engineering

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Modeling in Time and Frequency Domain

Modellering in het tijds- en frequentiedomein

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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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