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).

References

1. RB Blackman (1943). "Effect of feedback on impedance". The Bell System Technical Journal. 22 (3): 269–277. doi:10.1002/j.1538-7305.1943.tb00443.x. The pdf file no longer is available from Alcatel-Lucent, but an online version is found at RB Blackman (1943). Effect of feedback on impedance. Retrieved Dec 30, 2014..
2. Dennis L. Feucht (2014). Handbook of Analog Circuit Design. Academic Press. p. 147. ISBN 9781483259383.
3. J. Choma, Jr. (April 1990). "Signal flow analysis of feedback networks". IEEE Transactions on Circuits and Systems. CAS-37 (4): 455–463. Bibcode:1990ITCS...37..455C. doi:10.1109/31.52748. On-line version found at J Choma, Jr. "Signal flow analysis of feedback networks". baidu.com. Retrieved December 31, 2014.
4. RD Middlebrook. "Null double injection and the extra element theorem" (PDF). RDMiddlebrook.com. Blackman is not cited by Middlebrook, but see Eq. 1.4, p. 3 in this discussion of the extra element theorem: Vatché Vorpérian (2002). "Introduction: The joys of network analysis". Fast Analytical Techniques for Electrical and Electronic Circuits. Cambridge University Press. pp. 2 ff. ISBN 978-0521624718.
5. Solomon Rosenstark (1986). "§2.3 Asymptotic gain formula". Feedback amplifier principles. Macmillan USA. p. 16. ISBN 978-0029478103. and Solomon Rosenstark (1974). "A Simplified Method of Feedback Amplifier Analysis". IEEE Transactions on Education. 17 (4): 192–198. Bibcode:1974ITEdu..17..192R. doi:10.1109/TE.1974.4320925. Archived from the original on 2016-06-10. Retrieved 2014-12-20.
6. For a derivation and examples, see Gaetano Palumbo; Salvatore Pennisi (2002). "§3.5 The Blackman Theorem". Feedback Amplifiers: Theory and Design. Springer Science & Business Media. pp. 74 ff. ISBN 9780792376439.
7. For example, see Eq. 8, p. 255 in Paul J Hurst (August 1992). "A comparison of two approaches to feedback circuit analysis" (PDF). IEEE Transactions on Education. 35 (3): 253–261. Bibcode:1992ITEdu..35..253H. doi:10.1109/13.144656.
8. Borivoje Nikolić; Slavoljub Marjanović (May 1998). "A general method of feedback amplifier analysis". ISCAS '98. Proceedings of the 1998 IEEE International Symposium on Circuits and Systems (Cat. No.98CH36187) (PDF). Vol. 3. pp. 415–418. doi:10.1109/ISCAS.1998.704038. ISBN 978-0-7803-4455-6. S2CID 60745907.
9. Dennis L. Feucht (September 15, 2013). "Impedance EET (ZEET)". Middlebrook's Extra Element theorem. EDN Network. Retrieved December 31, 2014.
10. Comparison is made by Dennis L. Feucht (September 15, 2013). "Blackman's Impedance Theorem (BZT)". Middlebrook's Extra Element theorem. EDN Network. Retrieved December 31, 2014.
11. Blackman is not cited by Choma, but see Eq. 38, p. 460 in J. Choma, Jr. (1990). "Signal flow analysis of feedback networks". IEEE Transactions on Circuits and Systems. 37 (4): 455–463. Bibcode:1990ITCS...37..455C. doi:10.1109/31.52748.