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 .^[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. 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 (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.
^ ^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.
^ 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.
^ 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. Bibcode:1992ITEdu..35..253H. 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. Vol. 3. pp. 415–418. doi:10.1109/ISCAS.1998.704038. ISBN 978-0-7803-4455-6.
^ Dennis L. Feucht (September 15, 2013). "Impedance EET (ZEET)". Middlebrook's Extra Element theorem. EDN Network. Retrieved December 31, 2014.
^ ^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.
^ 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 Systems. 37 (4): 455–463. Bibcode:1990ITCS...37..455C. doi:10.1109/31.52748.

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[Blackman-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..

[Feucht-2] Dennis L. Feucht (2014). Handbook of Analog Circuit Design. Academic Press. p. 147. ISBN 9781483259383.

[Choma-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.

[Middlebrook-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.

[Rosenstark-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.

[Palumbo-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.

[Hurst-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.

[Nicolic-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. Vol. 3. pp. 415–418. doi:10.1109/ISCAS.1998.704038. ISBN 978-0-7803-4455-6.

[FeuchtD-9] Dennis L. Feucht (September 15, 2013). "Impedance EET (ZEET)". Middlebrook's Extra Element theorem. EDN Network. Retrieved December 31, 2014.

[FeuchtD1-10] 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.

[Choma1-11] 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 Systems. 37 (4): 455–463. Bibcode:1990ITCS...37..455C. doi:10.1109/31.52748.

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Blackman's theorem

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