386 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1951 



dual transformation is performed, each element being transformed into a 

 new element which is its dual. The transformed elements are not, however, 

 connected together in the same way as were the original ones. Elements in 

 parallel are transformed into elements in series and vice versa. Nodes trans- 

 form into loops and loops into nodes. 



There are cases when finding the dual of a network is not as straightfor- 

 ward as the reader might infer from the above. Complications may arise 

 when the network contains mutual inductance or non-Hnear elements, or 

 if the network cannot be drawn on a flat surface without crossovers. Some 

 of these questions are discussed by Bode.^ 



Duality 



Table 1 shows side by side a number of network elements and the duals 

 of these elements related through the transformation resistance r. The table 

 also shows the duals of a few simple networks. It is the purpose of this 

 section to show by means of examples how these dual relationships are 

 established. 



One network element is the dual of another provided the role of current 

 in one is played by voltage in the other, and vice versa. Consider what this 

 means in the case of a capacitance in which current and voltage are related 

 by the equation 



(2) 6 = ^ u 



jC(a 



Interchanging the roles of current and voltage means replacing e in this 

 equation by i'r and replacing i by e'/r. The value of r determines how many 

 volts across the condenser are to correspond to an ampere through its dual. 

 Making the indicated substitutions gives 



(3) i' = ^ ' 



jr'^Cif) 



This, however, is the kind of equation which relates the current through 

 an inductance to the voltage across it. It is seen, therefore, that the dual of 

 a capacitance C is an inductance of value given by 



(4) L' = r'^C. 



In the dual transformation of a network every capacitance in the original 

 network will be transformed in this way into an inductance in the dual 

 network. Also, if ec and ic represent the voltage across a capacitance and 



* Bode, loc. cit. 



