Accuracy design considerations

The static inaccuracy of a system is defined in section Small-signal gain and inaccuracy. The static inaccuracy of a negative feedback amplifier is determined by the static inaccuracy of the ideal gain and by that of the servo function. The static inaccuracy of the ideal gain is determined by the inaccuracy of the static transfer of the feedback network. The static inaccuracy of the servo function is determined by the value of the \(DC\) loop gain.

Static inaccuracy of the servo function

The static inaccuracy \(\delta_{DC}\) introduced by the servo function is its difference from unity at DC:

\[\delta_{DC}=S_{DC}-1,\]

where \(S_{DC}\) is the value of the servo function at zero frequency. With the aid of the DC loop gain \(L_{DC},\) we may write

\[\delta_{DC}=\frac{-L_{DC}}{1-L_{DC}}-1=-\frac{1}{1-L_{DC}}.\]

For \(\left\vert L_{DC}\right\vert \gg1\), this can be approximated by

(133)\[\delta_{DC}\approx\frac{1}{L_{DC}}. \label{eq-deltaDC}\]

Hence, for a static inaccuracy of \(-1\%,\) we need a DC loop gain of about \(-100\). Negative feedback will result in a one-sided static inaccuracy, in other words, the servo function will always be less than unity. The error can be made symmetrical by compensating \(50\%\) of the error budget in the ideal gain.

Design conclusion

In general, we may say that a high accuracy requires a large loop gain:

Definition

In order to obtain the static inaccuracy of a feedback amplifier within specifications, the controller must provide a sufficiently large DC loop gain.