## What is a transfer function?

A transfer function (also known as system function or network function) of a system, subsystem, or component is a mathematical function that modifies the output of a system in each possible input. They are widely used in electronics and control systems.

## What is a formula of transfer function?

For all possible input values, the transfer function represents the relationship between the output signal and the input signal of the control system. A reference input called stimulus or cause for any control system operates through a transfer operation (i.e., a switch function) to produce an effect that leads to a controlled output or response.

Thus, the cause and effect relationship between the single output and single input is related through a **transfer function.**

The block diagram of the transfer function of the system is as follows:

$\mathrm{Transfer}\mathrm{function}=\frac{\mathrm{Laplace}\mathrm{transform}\mathrm{function}\mathrm{output}}{\mathrm{Laplace}\mathrm{transform}\mathrm{function}\mathrm{input}}$

In a Laplace transform $T\left(s\right)$, if the input is represented by $X\left(s\right)$ in the numerator and the output is represented by $Y\left(s\right)$ in the denominator, then the transfer function equation will be

$T\left(s\right)=\frac{Y\left(s\right)}{X\left(s\right)}$

The transfer function model is considered an appropriate representation of the dynamic timing system line. In a control system, the way the system behaves when using the input will cause the output to change.

So, the overall transfer function of the system multiplied by the input function gives the output function of the system.

## Operation of transfer function

There is a simple process of determining the transfer function:

- In the system, the Laplace transform is performed on the system statistics, and the initial condition is zero.
- Specify system output and input.
- Finally, take the ratio of the output Laplace to transform to the input Laplace transform, that is, the required overall transfer function.

The output and input of the control system do not have to be in the same category. For example, in an electric motor, the input is an electrical signal, and the output is a machine signal because electric power is required to rotate the engine. Similarly, in an electric generator, the input is a machine signal, and the output is an electrical signal, as the power of the equipment is required to generate electricity in the generator.

But to analyze statistical data, all types of symbology must be expressed in the same way. This is done by converting all types of signals into their Laplacian form. The Laplace transform function is decomposed into the Laplace transform function, and the transfer function of the system is expressed in the form of Laplace.

## Terms related to the transfer function of a system

Terms of the transfer function of the subcircuit to determine the entire transfer function of the electrical system:

### Series connection of transfer functions

If two subcircuits are connected in series, the general transfer function is a product of the transfer functions of the subcircuits.

The series connection of transfer functions is shown in the figure below:

The formula for series connection of transfer functions:

$H\left(s\right)=\frac{Y\left(s\right)}{X\left(s\right)}={H}_{1}\left(s\right)\xb7{H}_{2}\left(s\right)$

In the equation, numerator Y(s) is an output, and denominator X(s) is input.

### Parallel connection of transfer functions

If the two subcircuits are connected in parallel, and their effects are shortened, then the total transfer function is the sum of the transfer functions of the sub-circuits.

The formula for parallel connection of transfer functions:

$H\left(s\right)=\frac{Y\left(s\right)}{X\left(s\right)}={H}_{1}\left(s\right)+{H}_{2}\left(s\right)$

In the equation, numerator Y(s) is an output, and denominator X(s) is input.

## Feedback transfer function

The entire transfer function of the response system is sometimes called the closed-loop transfer function, $\frac{forwardgain}{1+openloopgain}$.

Open loop gain is $G\left(s\right)\xb7H\left(s\right)$

Closed loop transfer function $\frac{Y\left(s\right)}{X\left(s\right)}=\frac{G\left(s\right)}{1\pm G\left(S\right)\xb7H\left(S\right)}$.

## Poles and zeros

The pole and the zero-point system are the focus of this module, which provides the characteristic information principle to determine the answer. This is very helpful because it allows the control system designer to understand how design parameters can gain acceptance response features. Using an explicit and false enablement method called root locus is to use a design method in which the designer can change the design parameters to a value that leads to an acceptable response and confirm the formation by analyzing the system response time. This module is a continuation of the Laplace transform module and introduces the concepts of transmission services, poles, and zero systems.

Transfer functions for a circuit have a ratio of polynomials of *s*. Polynomials are factored to create a factored form of the transfer function.

*K *is a static state. Each *s *digit that results in numerators is zero and is called zeros. These are all viewed comprehensively. Also, each value of *s *leading to the denominator being zero is called a pole. The reason for the term pole is that the value of the transfer function goes up or down to infinity when *s *is equal to one of these values. The poles and zeros of the transfer function are used to determine the number of circuit signals, such as the stability and response of the response control system.

## Advantages of the transfer function

- The Laplace transform can convert complex time governing equations into simple algebraic form in the transfer function.
- Provides a mathematical transfer function model for the entire system and every part of it. The output response of any reference input can be easily determined using the confirmed transfer function.
- It helps determine important system parameters such as poles, zeros, etc.
- The system stability can be easily analyzed using the transfer function.
- It helps to relate output with input.
- Converting from state-space form to a transfer function is straightforward because the form of the transfer function is different.

## Disadvantages of the transfer function

- It doesn't apply to non-linear systems.
- The initial conditions aren't taken into consideration because the effects generated by them are neglected.

## Common Mistakes

Remember that the term "transfer function analysis" is also used in the analysis of domain frequency systems using conversion methods such as Laplace transform; it means the output amplitude as the frequency function of the input signal.

## Context and Applications

In each of the expert exams for undergraduate and graduate publications, this topic is huge and is mainly used in the following context:

- Bachelor of Technology in the Electrical and Electronics Department
- Bachelor of Science in Physics
- Master of Science in Physics

## Related Concepts

- Lag lead compensator
- response of the second-order system
- Lag compensator
- Continuous-time response of the first-order system
- Lead compensator
- Convolution theorem

## Practice Problems

**Q1** A major part of the automatic control theory applies to ________.

- non-linear systems
- casual systems
- time-variant systems
- linear time-invariant systems

**Correct option**- (d)

**Explanation**- The linear time-invariant (LTI) system provides the same output for the same input regardless of when the input is delivered. The LTI system is also used to predict long-term system behavior.

**Q2 **The traffic light system is an example of a/an ________.

- closed-loop system
- open-loop system
- closed-loop system and open-loop system
- None of these

**Correct option**- (b)

**Explanation**- The traffic lights will light up according to the set time and sequence, depending on the time. Time tracking is controlled by a relay that runs at a predetermined time. It does not depend on the speed of the road.

**Q3** Transfer function of a system is defined as:

- Z-transformer
- Fourier transform
- Laplace transform
- All of these

**Correct option**- (c)

**Explanation**- Assuming that the initial conditions are zero, the transfer function is defined as a measure of the Laplacian transformer from the output to the laplacian input.

**Q4** The transfer function of a system can used to study its ________.

- Transient behavior
- Steady-state behavior
- Transient behavior and steady-state behavior
- None of these

**Correct option**- (c)

**Explanation**- The transfer function is defined as the Laplace transform output response value in the input response. Laplace transform can be used to the transient state behavior and the steady-state behavior of a system.

**Q5 **The output of the feedback control system should be a function of ________.

- reference and output
- input
- feedback signal
- None of these

**Correct option**- (a)

**Explanation**- The feedback control system has the characteristic of reducing the error, that is, by making a difference between the output and the required output.

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