Английская Википедия: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 Ralph Beebe 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:
- <math>Z = Z_D \frac {1+T_{SC}}{1+T_{OC}} \ , </math>
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]
- <math>Z_{in} = Z^{\infty}_{in} \left[ \frac{1+Z^0_{e}/Z}{1+ Z^{\infty}_{e}/Z}\right]</math>
where:
- <math>Z\ </math> is the impedance of the extra element; <math>Z^{\infty}_{in}</math> is the input impedance with <math>Z\ </math> removed (or made infinite); <math>Z^0_{e}</math> is the impedance seen by the extra element <math>Z\ </math> with the input shorted (or made zero); <math>Z^{\infty}_{e}</math> is the impedance seen by the extra element <math>Z\ </math> with the input open (or made infinite).
Blackman's formula also can be compared with Choma's signal-flow result:[11]
- <math>Z_{SS}=Z_{S0}\left[\frac{1+T_I}{1+T_Z}\right] \ , </math>
where <math>Z_{S0}\ </math> is the value of <math>Z_{SS}\ </math> under the condition that a selected parameter P is set to zero, return ratio <math>T_Z\ </math> is evaluated with zero excitation and <math>T_I\ </math> is <math>T_Z\ </math> 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
References
- ↑ Ошибка цитирования Неверный тег
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