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 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]

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

where:

Z\

is the impedance of the extra element;

Z_{{in}}^{{\infty }}

is the input impedance with

Z\

removed (or made infinite);

Z_{{e}}^{0}

is the impedance seen by the extra element

Z\

with the input shorted (or made zero);

Z_{{e}}^{{\infty }}

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]

References

↑ 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. .
↑ Dennis L. Feucht (2014). Handbook of Analog Circuit Design. Academic Press. p. 147. ISBN 9781483259383.
↑ 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.
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.
↑ 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.
↑ 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.
↑ 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.
↑ 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.
↑ Dennis L. Feucht (September 15, 2013). "Impedance EET (ZEET)". Middlebrook’s Extra Element theorem. EDN Network. Retrieved December 31, 2014.
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.
↑ 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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Blackman's theorem

See also

Further reading

References