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:

${\displaystyle Z=Z_{D}{\frac {1+T_{SC}}{1+T_{OC}}}\ ,}$

where Z_D = impedance with the feedback disabled, T_SC = loop transmission with a small-signal short across the selected terminal pair, and T_OC = 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 Z_in of a circuit based upon the extra-element theorem:^[4][9][10]

${\displaystyle Z_{in}=Z_{in}^{\infty }\left[{\frac {1+Z_{e}^{0}/Z}{1+Z_{e}^{\infty }/Z}}\right]}$

where:

${\displaystyle Z\ }$ is the impedance of the extra element; ${\displaystyle Z_{in}^{\infty }}$ is the input impedance with ${\displaystyle Z\ }$ removed (or made infinite); ${\displaystyle Z_{e}^{0}}$ is the impedance seen by the extra element ${\displaystyle Z\ }$ with the input shorted (or made zero); ${\displaystyle Z_{e}^{\infty }}$ is the impedance seen by the extra element ${\displaystyle Z\ }$ with the input open (or made infinite).

Blackman's formula also can be compared with Choma's signal-flow result:^[11]

${\displaystyle Z_{SS}=Z_{S0}\left[{\frac {1+T_{I}}{1+T_{Z}}}\right]\ ,}$

where ${\displaystyle Z_{S0}\ }$ is the value of ${\displaystyle Z_{SS}\ }$ under the condition that a selected parameter P is set to zero, return ratio ${\displaystyle T_{Z}\ }$ is evaluated with zero excitation and ${\displaystyle T_{I}\ }$ is ${\displaystyle T_{Z}\ }$ 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

• Mason's gain formula

Further reading

• Eugene Paperno (September 2012). "Extending Blackman's formula to feedback networks with multiple dependent sources" (PDF). IEEE Transactions on Circuits and Systems II: Express Briefs. 59 (10): 658–662. CiteSeerX 10.1.1.695.4656. doi:10.1109/TCSII.2012.2213355. S2CID 8760900.
• Rahul Sarpeshkar (2010). "§10.7 Driving-point transistor impedances with Blackman's formula". Ultra Low Power Bioelectronics: Fundamentals, Biomedical Applications, and Bio-Inspired Systems. Cambridge University Press. pp. 258 ff. ISBN 9781139485234.
• Amaldo D'Amico; Christian Falconi; Gianluca Giustolisi; Gaetano Palumbo (April 2007). "Resistance of feedback amplifiers: A novel representation" (PDF). IEEE Transactions on Circuits and Systems – II Express Briefs. 54 (4).