Blackman's theorem
Blackman's theorem is a general procedure for calculating the change in an impedance due to feedback in a circuit. It was published by R.B. Blackman in 1943,[1] 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.[2][3][4][5] 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.[6] The loop transmission also is referred to as the return ratio.[7][8] Blackman's formula can be compared with Middlebrook's result for the input impedance Zin of a circuit based upon the extra-element theorem:[4][9][10]
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:[11]
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.[10]
See also
Further reading
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References
- ↑ Lua error in package.lua at line 80: module 'strict' not found. The pdf file no longer is available from Alcatel-Lucent, but an online version is found at Lua error in package.lua at line 80: module 'strict' not found..
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
- ↑ Lua error in package.lua at line 80: module 'strict' not found. On-line version found at Lua error in package.lua at line 80: module 'strict' not found.
- ↑ 4.0 4.1 Lua error in package.lua at line 80: module 'strict' not found. Blackman is not cited by Middlebrook, but see Eq. 1.4, p. 3 in this discussion of the extra element theorem: Lua error in package.lua at line 80: module 'strict' not found.
- ↑ Lua error in package.lua at line 80: module 'strict' not found. and Lua error in package.lua at line 80: module 'strict' not found.
- ↑ For a derivation and examples, see Lua error in package.lua at line 80: module 'strict' not found.
- ↑ For example, see Eq. 8, p. 255 in Lua error in package.lua at line 80: module 'strict' not found.
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
- ↑ 10.0 10.1 Comparison is made by Lua error in package.lua at line 80: module 'strict' not found.
- ↑ Blackman is not cited by Choma, but see Eq. 38, p. 460 in Lua error in package.lua at line 80: module 'strict' not found.
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