92 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1951 



As a physical concept the idea of negative impedance is difficult to visual- 

 ize because this type of impedance supplies power to an external circuit 

 rather than dissipates it. This power is supplied either by the application of 

 a reversed voltage or a reversed current. In spite of the difficulty which may 

 be experienced in attempting to visualize the feedback action that takes 

 place inside a negative impedance converter, if its equivalent circuit is known 

 then its stability can be determined readily and its operation as a device 

 for producing negative impedance becomes obvious. 



Stability 



Like any amplifier whose output connects back to its input, the negative 

 impedance converter if not properly terminated can run away with itself and 

 oscillate. Stability can be determined by conventional feedback theory^; 

 fortunately, there is a simpler criterion for determining stability. Consider 

 again the ideal converter. Fig. 1(b). Assume that Zl^ not shown on Fig. 1(b), 

 is an impedance connected to terminals 1 and 2. Consider the circuit mesh 

 formed on the left-hand side of the ideal converter by the connection of Z^ 

 to terminals 1 and 2. Here a negative impedance {—kZji) is seen looking into 

 terminals 1 and 2, and a positive impedance {Zl) is seen looking away from 

 them. The total impedance in this mesh is Zi,— kZ^. If k Zn should equal 

 Zl then the total impedance would be zero; and a voltage inserted in series 

 with this mesh would call for infinite current, a situation obviously impos- 

 sible. Thus it becomes evident that kZ^ should not equal Zl; or, what is the 

 same thing, the ratio UZn/Zl should not equal 1/0 if the system is to be 

 stable. Furthermore, it can be shown that for an ideal converter the ratio 

 UZn/Zl contains the characteristics of the feedback factor (iu/3) of the am- 

 plifier in the converter. In view of this fact, it might be expected that 

 Nyquist's rule* for stability in feedback amplifiers could be paraphrased as 

 follows: To obtain stability in an ideal negative impedance converter the locus 

 oj the ratio UZs/Zl over the frequency range from zero to infinity must not en- 

 close the point 1/0. 



The same general rule for stability can be arrived at by connecting an 

 impedance Zn to terminals 3 and 4 of Fig. 1 (c) and by considering the circuit 

 mesh formed by {Zl/ —^ + Zs. It should be noted in this case that Zl/—^ 

 calls for a flow of current 180 degrees out of phase from that which would 

 flow through Zu/k. This means that where the phase angle of Z^ijk equals 

 that of Zjv the magnitude of Zl/^ must be greater than that of Zat, which is 

 another way of saying that at this phase angle the magnitude of kZs/ZL 

 must be less than unity. 



'H, W. Bode — Book — Network Analysis and Feedback Amplifier Design — D. Van 

 Nostrand Company, Inc. — 1945. 



* H. Nyquist— Regeneration Theory— 5.5. r./.— Jan., 1932. 



