306 BELL SYSTEM TECHNICAL JOURNAL 



where the summations in the last expression extend over all the 

 sections; this impedance then is simply the parallel impedance of the 

 sides taken in their entirety. 



The current in branch k of line 2, designated by Ik on Fig. \A, for any 

 condition of energization is expressed in terms of the currents in the 

 same branch and in the same direction' for unit current supplied 

 between terminals 1 and 2, and 3 and 4 [terminals 3, 4 (1, 2) open, 

 respectively]], designated by 4: 12 and 4:34, respectively, by the following 

 equation : 



h = V{h + U) - ik:l2lvh - (1 - V)h'] - H-.ulvh - (1 - V)h'], (2) 



where /i, I2, I3 and li are currents flowing out of the network from the 

 respective terminals, and v is the current in side 2 for unit current 

 between short-circuited transducer terminals, as given on Fig. IB. 

 Thus Ik is a linear function of currents 4:i2 and 4:34, as stated in the 

 second half of the theorem. 



Three types of networks completely equivalent to any ladder 

 network satisfying the condition of the theorem are shown on Fig. IB. 

 The transducer impedances employed in the representation by these 

 networks are the driving-point impedances between transducer termi- 

 nals 1 and 2, and 3 and 4 [terminals 3, 4 (1, 2) open, respectively]] and 

 the corresponding transfer impedance between the ends of the trans- 

 ducer. These impedances are designated Zn.n, Zzh-.za and Zmisi, 

 following a notation for Neumann integrals used by G. A. Campbell.'* 

 For present purposes the notation has the advantage of putting into 

 evidence the terminals between which current is supplied and the 

 terminals between which voltage is measured; thus Zi2:i2 may be read 

 as the voltage drop from 1 to 2 for unit current from 1 to 2 (terminals 

 3, 4 open), Zi2:34 the voltage drop from 3 to 4 for unit current from 1 to 2 

 under the same conditions.^ By the reciprocity theorem Zi2:34 = ^34:12. 



^ "Mutual Impedance of Grounded Circuits," Bell System Technical Journal, 2, 

 1-30 (Oct. 1923). 



* Further, the subscripts may be handled algebraically to give results following 

 from the superposition theorem. For this purpose the numbers in each part of the 

 two-part subscript are taken as separated by a minus sign and the colon is taken as a 

 sign of multiplication; thus: 



■2l2:i2 = ■^(l-2)(l-2) = Zii + Z22 — 2Zi2, 



the last expression being formed by writing out the indicated product and separating 

 the terms. The equation expresses the fact that the impedance of a circuit may be 

 subdivided into the self-impedances of sides (real or fictional) associated with its 

 terminals minus twice their mutual impedance. Moreover, any additional sub- 

 scripts desired may be intercalated by adding and subtracting the same numeral; 

 the expansion of bracketed terms then gives a relation between circuit impedances; 

 thus: 



Z(l_2)(l-2) = Z((i_3) + (3_2))((1_3)+C3-2)1 

 = Z(i3+32)(13+32) 

 = Zir-lS + ^32:32 + 2Z\y.s2- 



