# 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:

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

## Derivation

We define an intermediate signal Z shown as follows:

Using this figure we write:

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