36 BELL SYSTEM TECHNICAL JOURNAL 



where 



5" = Cl4 C24 C34+ Cl4 (^24 (^13 + C23) ~l~ C24C'34(Cl2+ C13) + C34 C\i (012 + ^*23) 



+ Gi (Cl2 C23 + ^23 Cu + C12 C13), 



which, upon substituting the value of d, is the sum of 16 terms, each 

 of which is the product of three capacities, every combination of three 

 capacities being included except the four cases in which the three 

 capacities would form a closed circuit. By allowing the capacities 

 to be complex quantities, any admittances are covered by the formulas. 

 If G4 = 



14 _ t--24 _ C-34 _ >-> 



Ci4 C'24 Cz\ C14C24C34 



or the new Y arms present the same ratios as the original Y arms 

 taken alone; that is, the direct capacities C^b^ Cbc of Fig- 4 have 

 no effect on the bridge ratio. Thus the constancy of the bridge ratio 

 holds for all null-impedance bridges regardless of the ratio Z1/Z2 and 

 of the nature of the direct admittances from corners A and C to B. 

 If G4 = and also C24 = C14 and C\i = 0, then 



C\^, = C'-iA = ~ \ Csl = Cn + \ (Ci3 + C23). 



Applying this to Fig. 4, which is possible since the bridge ratio is 

 unity, we find that the three arms of the equivalent Y may be con- 

 sidered as being made up of resistances and capacities in parallel. 

 The resistances are R, R, —R/2 and the associated capacities C, C, 

 — 2C, where R is the original resistance in the Y and C is one-half the 

 sum of the two actual direct capacities from ^ to f^and from -S^ to (?. 

 The equivalent bridge thus obtained has ratio arms made up of 

 ordinary resistances and capacities and therefore Fig. 4 used as a 

 bridge can present no unexpected characteristics; the negative 

 resistances and capacities of the equivalent Y merely affect the current 

 supplied to the bridge. 



An ideal transformer, if such a device existed, might replace the Y, 

 for it would maintain a constant ratio between the currents in the 

 two windings and act as a short circuit when the bridge is balanced. 

 To determine the error when an actual transformer with impedances 

 Zp, Zs, Zps is employed, take the general expression for the ratio 

 of the capacities derived above which is 



Cl4 _ C34 Cu -\- Gj Ci3 

 C24 C'24 ^34 + G4 C23 



Change to admittances by substituting Y for C and G throughout. 

 Assume the transformer replaced by its equivalent conductance star 

 so that 



