274 BELL SYSTEM TECHNICAL JOURNAL 



Feedback during Impedance Measurements 



While the feedback computed from impedance measurements by formula 

 (4) or (5) is the normal feedback, the feedback during the impedance meas- 

 urements may be quite different, due to the impedance of the impedance 

 measuring circuit. Referring to Fig. 1 we see that the feedback during 

 measurement is by definition 



Ua, 



where Z is the impedance of the impedance measuring circuit. By equa- 

 tions (1) this is easily reduced to 



_, ZpFsh + ZFop />-\ 



"' = z. + z (^^ 



Under the conditions to which formula (4) applies 



F. = -^ (7) 



It is clear therefore that even if Fa- satisfies Nyquist's Stability Criterion, 

 Z may be of such a character that Fz violates that criterion. In that case 

 it will be impossible to make the impedance measurements. 



Contrariwise, if Fn violates Nyquist's Stability Criterion, it is possible 

 to choose Z so that Fz satisfies that criterion and make it possible to meas- 

 ure the impedance. Substituting (4) into (7) we find that a sufficient but 

 not necessary condition in order that \Fz\< 1 is that 



\Z\ > \Z^\ + 2\Zp\ 

 Under the conditions to which formula (5) applies 



F.= -~ (8) 



'+f 



Similar observations may be made with respect to (8) as were made with 

 respect to (7). Substituting (5) into (8) we find that a sufficient but not 

 necessary condition in order that |Fz|< 1 is that 



_L> J_ + J_ 



izriz.riz,i 



Feedback Control of Impedance 



The application of the relationship (3) to the feedback control of im- 

 pedance may be illustrated by a few concrete examples. 



