The Bell System Technical Journal 



Vol. XXII October, 1943 ■ No. 3 



Effect of Feedback on Impedance 



By R. B. BLACKMAN 



THE impedance of a network is defined as the complex ratio of the alter- 

 nating potential difference maintained across its terminals by an ex- 

 ternal source of electromotive force, to the resulting current flowing into 

 these terminals. If the network contains active elements such as vacuum 

 tubes, the resulting current (or potential difference if the input current is 

 taken as the independent variable) may be due in part to the excitation of 

 the active elements. The definition of impedance does not discriminate 

 between the part of the current (or potential difference) due directly to the 

 external source of electromotive force and the part due to the excitation of 

 the active elements by the external source. Hence the impedance will in 

 general depend upon the degree of activity of the active elements. 



These observations were made early in the development of feedback 

 amplifiers by H. S. Black^ who made two important uses of the effect of 

 feedback on impedance. In the first place it afforded a method of measur- 

 ing feedback which has some advantages over the method which involves 

 opening the feedback loop, providing proper terminations for it and meas- 

 uring the transmission around it. In the second place the effect of feedback 

 on impedance was used to control the impedances presented by a feedback 

 amplifier to the external circuits connected to it. 



Relations between impedance and feedback were derived by Black and 

 others for a number of specific feedback amplifier configurations. In some 

 cases these relations turned out to be very simple. For the most part, how- 

 ever, these relations were so complicated that they defied reduction to a 

 common form.- The difficulty seems to have been due, in part at least, to 

 the attempt to formulate the relationship, in each case, in terms of the nor- 

 mal feedback of the amplifier. In some cases the dilficulty seems to have 

 been due partly also to the valid, but, as it turns out, irrelevant observation 

 that the feedback is affected by the impedance of the measuring circuit as 



iH. S. Black, "Stabilized Feedback Amplifiers", B.S.T.J., January, 1934. 



2 Shortly after the general relationship between feedback and impedance was derived, 

 it was independently established by H. \V. Bode and J. M. West by examination of a 

 variety of feedback amplifier designs. The generality of the relationship was also in- 

 dependently proved for amplifiers with a single feedback path by J. G. Kreer and by C. 

 H. Elmendorf. 



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