Circular polarization

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:

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

Derivation

Let's define an intermediate signal Z shown as follows:

File:Closed Loop Block Deriv.png

Using this figure we can write

$Y(s)=Z(s)G(s)\Rightarrow Z(s)=\frac{Y(s)}{G(s)}$

$X(s)-Y(s)H(s)=Z(s)=\frac{Y(s)}{G(s)}\Rightarrow X(s)=Y(s)\left[1+G(s)H(s)\right]/G(s)$

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

See also

• Federal Standard 1037C
• Open-loop controller

References

• Template:FS1037C

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