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:

Z = Z_D \frac {1+|T_{SC}|}{1+|T_{OC}|} \ ,

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]

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

where:

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

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

Z_{SS}=Z_{S0}\left[\frac{1+T_I}{1+T_Z}\right] \ ,

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

Further reading

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. 1 2 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. 1 2 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".
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