# Closed-loop transfer function

A closed-loop transfer function in control theory is a mathematical expression (algorithm) describing the net result of the effects of a closed (feedback) loop on the input signal to the circuits enclosed by the loop.

## Overview

The closed-loop transfer function is measured at the output. The output signal waveform can be calculated from the closed-loop transfer function and the input signal waveform.

An example of a closed-loop transfer function is shown below:

The summing node and the G(s) and H(s) blocks can all be combined into one block, which would have the following transfer function:

${\displaystyle {\dfrac {Y(s)}{X(s)}}={\dfrac {G(s)}{1+G(s)H(s)}}}$

## Derivation

We define an intermediate signal Z shown as follows:

Using this figure we write:

${\displaystyle Y(s)=G(s)Z(s)}$
${\displaystyle Z(s)=X(s)-H(s)Y(s)}$
${\displaystyle Y(s)=G(s)(X(s)-H(s)Y(s))=G(s)X(s)-G(s)H(s)Y(s)}$
${\displaystyle Y(s)+G(s)H(s)Y(s)=G(s)X(s)}$
${\displaystyle Y(s)(1+G(s)H(s))=G(s)X(s)}$
${\displaystyle \Rightarrow {\dfrac {Y(s)}{X(s)}}={\dfrac {G(s)}{1+G(s)H(s)}}}$