THEORY OF NEGATIVE IMPEDANCE CONVERTER 



91 



same result from a circuit viewpoint as dividing a positive impedance by 

 — 1. A positive impedance multiplied by —i is a series type negative im- 

 pedance. A positive impedance divided by —i is a shunt type of negative 

 impedance. 



A practical (i.e., real or actual) converter circuit can be represented by 

 Fig. 1(d). A vacuum tube circuit contains positive impedance elements. 

 Some of these will show up in the equivalent circuit on the left-hand side 

 of the ideal converter; others will show up on the right-hand side of the ideal 

 converter. This will be clarified in the discussion of the El circuit which 

 follows. The equivalent circuit of any practical negative impedance converter 

 of this type can be represented by the equivalent circuit of Fig. 1 (d) which 

 shows the positive impedance elements associated with the vacuum tube in 



*z, kz 



kZ^ kZz -kZ2 -kZ^ 



kZ^ 



? — VW 



-kZ^ 



■VA — \ 



kZ^ >-kZ^ 



[c) 



c 

 ■k:^ 



Fig. 2 — Equivalent circuits. 



the form of two equivalent networks (iVi and N2) arranged one on each side 

 of the ideal converter (C) having a transformation ratio of —k:l. 



It should be noted that should the series arms of i\^i equal k times the 

 series arms of N2 and should the same relationship exist for the shunt arms 

 then the effect of these networks is cancelled, except for power dissipation, 

 and Fig. 1(d) is equivalent to Fig. 1(a). This is illustrated in Fig. 2 where Xi 

 and iY2 have been represented by equivalent T networks as shown specifi- 

 cally in Fig. 2(a). Network X2 can be multiplied by — ^ and transformed to 

 the left-hand side of the ideal converter. Fig. 2(b). The adjacent series arms 

 of these two networks cancel each other as shown in Fig. 2 (c) . The shunt arms 

 go to infinity and the other series arms also cancel leaving Fig. 2(d). 



It is not possible to cancel Xi and N2 perfectly in a practical circuit design. 

 But over the frequency range of interest this could be closely approxunated 

 by making all impedances shunting the ideal converter as large as possible, 

 and by cancelling all resistances in series in .Yi by a resistance added in 

 N2. 



