Blackman's theorem explained
Blackman's theorem is a general procedure for calculating the change in an impedance due to feedback in a circuit. It was published by Ralph Beebe Blackman in 1943, was connected to signal-flow analysis by John Choma, and was made popular in the extra element theorem by R. D. Middlebrook and the asymptotic gain model of Solomon Rosenstark. Blackman's approach leads to the formula for the impedance Z between two selected terminals of a negative feedback amplifier as Blackman's formula:
where
ZD = impedance with the feedback disabled,
TSC = loop transmission with a small-signal short across the selected terminal pair, and
TOC = loop transmission with an open circuit across the terminal pair. The loop transmission also is referred to as the
return ratio. Blackman's formula can be compared with Middlebrook's result for the input impedance
Zin of a circuit based upon the extra-element theorem:
where:
is the impedance of the extra element;
is the input impedance with
removed (or made infinite);
is the impedance seen by the extra element
with the input shorted (or made zero);
is the impedance seen by the extra element
with the input open (or made infinite).
Blackman's formula also can be compared with Choma's signal-flow result:
where
is the value of
under the condition that a selected parameter
P is set to zero, return ratio
is evaluated with zero excitation and
is
for the case of short-circuited source resistance. As with the extra-element result, differences are in the perspective leading to the formula.
See also
Further reading
- Extending Blackman's formula to feedback networks with multiple dependent sources . Eugene Paperno . IEEE Transactions on Circuits and Systems II: Express Briefs . 59 . 10 . 658–662 . September 2012 . 10.1109/TCSII.2012.2213355 . 10.1.1.695.4656 . 8760900 .
- Book: Ultra Low Power Bioelectronics: Fundamentals, Biomedical Applications, and Bio-Inspired Systems . Rahul Sarpeshkar . §10.7 Driving-point transistor impedances with Blackman's formula . 258 ff . https://books.google.com/books?id=eYPBAyDRjOUC&pg=PA258 . Cambridge University Press . 2010 . 9781139485234.
- Resistance of feedback amplifiers: A novel representation . Amaldo D'Amico . Christian Falconi . Gianluca Giustolisi . Gaetano Palumbo . IEEE Transactions on Circuits and Systems – II Express Briefs . 54 . 4 . April 2007 .