394 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1951 



in the manner illustrated in a previous section (see Fig. 2). In doing this, 

 replace 



^pby -ic, 



ipby—Vey 



Vg by —ie , and • 



ighy—Ve. 



This gives a set of Kirchoff equations which apply to the dual circuit 

 and it remains only to find, by inspection, a circuit which satisfies them. 

 It will often be found that, in order to satisfy these equations, it is necessary 

 to introduce a phase reversing transformer. Several examples of this method 

 of procedure will be given in sections to follow. In the appendix a great 

 many more examples of vacuum tube circuits and their transistor duals 

 are shown. 



Gyrators and Duality 



Tellegen^ has shown that in principle it is possible to make a new kind of 

 passive 4-pole circuit element to which he has given the name "ideal gy- 

 rator". This device is characterized by the 4-pole equations 



ei — Ri2 and 

 62 = —Rii . 



Though such a device is not known to have been reaUzed in a practical 

 physical form as yet, its properties are so closely related to duahty as to 

 be worth mentioning here. 



The following interesting properties can readily be deduced from the 

 equations above. First, signals are transmitted through the device in one 

 direction without phase reversal, while signals transmitted in the other 

 direction are reversed in phase. Second, if an impedance Z is connected 

 across the output terminals of an ideal gyrator, the impedance seen at the 

 input terminals is I^/Z. This means that the ideal gyrator has the property 

 of transforming any two-terminal network into its dual. Also a three-termi- 

 nal network can be converted into its dual by connecting one gyrator to the 

 input terminals of the network and another to the output terminals. These 

 gyrators must be so poled that no phase reversal is produced in either direc- 

 tion by the action of the two together. 



This means that the dual of a vacuum tube triode can be obtained by 

 using a triode plus two gyrators as shown in Table I and, of course, the dual 

 of a transistor can be obtained by using a transistor plus two gyrators. Also, 



* The Gyrator, A New Electric Network Element, B.D.H. Tellegen, Phillips Research 

 Reports, 1948, pp. 81-101. 



