A LADDER NETWORK THEOREM 307 



The first two equivalent networks are of the H type; ® as seven 

 impedances are shown on each, whereas only six are required for 

 complete representation, an arbitrary constant a has been introduced 

 so that the mutual impedance of the uprights may be varied at 

 pleasure. Thus in the first H network, the condition a -]- v = puts 

 all the mutual impedance between the uprights below the crossbar; the 

 condition 1 — v — a = puts it all above. The same type of shift 

 may be made in the second H network. 



The third equivalent is the network of direct impedances ("Cisoidal 

 Oscillations," loc. cit. designation (b)); these are expressed in terms of 

 the transducer parameters with opposite pairs of terminals short- 

 circuited, which following Campbell are denoted by 6"s. Thus 6'i2:i2 is 

 the driving-point impedance between terminals 1 and 2 with terminals 

 3 and 4 short-circuited; 5i2:34 is the ratio of current from 3 to 4 to 

 voltage from 1 to 2 with 1, 2 energized and 3, 4 short-circuited, or its 

 reciprocity theorem equivalent. 



These three equivalents correspond respectively to transformer, T 

 and T transducer equivalent networks. For the first H type the 

 transducer condition that currents into terminals 1 and 2, and 3 and 4, 

 shall be equal and opposite entails zero current in the iJ crossbar, which 

 may be removed, leaving a transformer connection. For the second H 

 type the transducer condition allows grouping the impedances of 

 branches 1 and 2 and their mutual impedance, and of 3 and 4 and their 

 mutual impedance, into single branches, say, branches 1 and 3, which 

 gives the T equivalent network. The reduction of the direct imped- 

 ance network is not so immediate. 



Periodic Ladder Networks 



When the network is periodic, the transducer impedances and 

 current distribution may be expressed completely in terms of the 



The justification of the operation Hes in the fact that, as regards the current half of the 

 subscript, a unit current from 1 to 2 is equivalent by the superposition theorem to 

 unit currents 1 to 3 and 3 to 2 and similarly the voltage 1 to 2 for unit current 1 to 2 

 is the same as the sum of voltages 1 to 3 and 3 to 2. Thus the notation is a shorthand 

 for application of the superposition theorem. Its use is illustrated further in the 

 course of the proof of the theorem. 



^ This is a form of equivalent network falling under designation (c) of the list of 

 equivalents for an arbitrary number of terminals given by G. A. Campbell ("Cisoidal 

 Oscillations," p. 889, loc. cit.), which is described as branches radiating from a common 

 concealed point, one to each of the terminals, with mutual impedances between pairs. 

 This is not a unique representation since the number of elements is redundant, and the 

 set of mutual impedances may be given values appropriate for particular purposes pro- 

 vided that the self-impedances are adjusted correspondingly. In the present applica- 

 tion the mutual impedances of branches to terminals 1 and 4, and 2 and 3, have been 

 set at zero and the mutual impedances of branches 1 and 2 and 3 and 4 in the first H 

 diagram, and of branches 1 and 3, 2 and 4 in the second, have been eliminated in 

 forming the cross bar of the H. 



