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Blackman's theorem

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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

  • 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. doi:10.1109/TCSII.2012.2213355.
  • 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 Trans Circuits & Systems – II:Express Briefs. 54 (4).{{cite journal}}: CS1 maint: multiple names: authors list (link)

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". The Internet Archive; archive.org. 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: 455–463. On-line version found at J Choma, Jr. "Signal flow analysis of feedback networks". baidu.com. Retrieved December 31, 2014.
  4. ^ a b 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. doi:10.1109/TE.1974.4320925.
  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. doi:10.1109/13.144656.
  8. ^ Borivoje Nikolić, Slavoljub Marjanović (May 1998). "A general method of feedback amplifier analysis" (PDF). Proceedings of the 1998 IEEE International Symposium on Circuits and Systems: ISCAS '98. 3: 415–418. doi:10.1109/ISCAS.1998.704038.
  9. ^ Dennis L. Feucht (September 15, 2013). "Impedance EET (ZEET)". Middlebrook’s Extra Element theorem. EDN Network. Retrieved December 31, 2014.
  10. ^ a b 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. "Signal flow analysis of feedback networks". {{cite journal}}: Cite journal requires |journal= (help)
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