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.

transfer function in control systems

The transfer function is a fundamental concept in the analysis and design of linear time-invariant (LTI) systems. It offers a concise way to understand how a system responds to different inputs in the frequency domain. It serves as a bridge between the time-domain differential equations that describe system dynamics and the frequency-domain representation that facilitates easier manipulation and analysis.
To derive the transfer function, consider a general nth-order linear time-invariant differential equation of the form:
<img alt="Linear differential equation, \(a_n \frac{d^n c(t)}{dt^n}\), symbolic representation, mathematical context." src="/files/ftp_upload/16095/16095_PT%20EQUATION_1.svg" style="text-align: center; height: 40px;" data-altupdatedat="1762173717000" data-alt="Equation1" />
Here, c(t) is the output, r(t) is the input, and ai and bi are constant coefficients. Applying the Laplace transform to both sides, assuming all initial conditions are zero, the differential equation can be converted into an algebraic equation in terms of s, the complex frequency variable. Rearranging terms, we get:
<img alt="Polynomial equation illustrating control systems analysis; mathematical formula representation." src="/files/ftp_upload/16095/16095_PT%20EQUATION_2.svg" style="text-align: center; height: 22px;" data-altupdatedat="1762173717000" data-alt="Equation2" />
The transfer function H(s) is defined as the ratio of the output C(s) to the input
R(s):
<img alt="Transfer function equation; G(s) expression; control systems diagram." src="/files/ftp_upload/16095/16095_PT%20EQUATION_3.svg" style="text-align: center; height: 40px;" data-altupdatedat="1762173717000" data-alt="Equation3" />
This expression shows that the transfer function is a rational function of s. The numerator is the polynomial formed by the input coefficients, and the denominator is the characteristic polynomial of the differential equation.
This transfer function indicates how the system's output c(t) responds to an input
r(t) in the frequency domain. The transfer function can be represented in a block diagram with the input R(s) on the left, the output C(s) on the right, and the transfer function H(s) inside the block. This visualization simplifies understanding and analyzing system behavior, especially when dealing with more complex systems.

Transfer Function in Control Systems

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

KEY TERMS AND CONCEPTS

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

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

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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...

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state space representation

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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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