BRIDGE METHODS OF MEASURING IMPEDANCES 457 



imaginary but not complex, that is, the difference between their phase 

 angles must be 0°, 180° or ± 90°. 



For the case of adjustment by the opposite arm Zab, equation (4) 

 may be written in the form 



RcD -\- jXcD = ZbcZad{Gab — JBab)- (6) 



Then in order that this equation fulfill the requirements of equation 

 (3), the vector product of the fixed arms must be either real or imagi- 

 nary, but not complex, that is, the sum of their phase angles must be 

 0°, 180° or ± 90°. 



In the case of bridges of the type indicated by equation (5), the 

 fixed arms always enter the balance equation as a ratio, and are there- 

 fore called ratio arms, the bridges of this type being called ratio arm 

 bridges. 



In the case of bridges of the type indicated by equation (6), the 

 fixed arms always enter the balance equation as a product, and are 

 therefore called product arms, the bridges of this type being called 

 product arm bridges. 



These two types may be further subdivided according to whether 

 the term involving the fixed arms is real or imaginary. 



It should be pointed out at this time that the fixed arms are fixed 

 in value only to the extent that they are not varied during the course 

 of a measurement. They may be functions of frequency, and may be 

 arbitrarily adjustable to vary the range of the bridge, but they are 

 not adjusted in the course of balancing the bridge. 



Classification of Bridge Types 



The foregoing discussion shows that all simple four arm bridges 

 meeting the requirements specified may be divided into four types. 

 The balance equations of these four types may now be simply derived 

 from the general equations (2) and (4). 



1. Ratio Arm Type — Ratio Real 

 If ZbcIZab is real, then 



= 0^^- 0,5 = 0° or 180°. 

 That is 



ZbcI^ab = Rbc/Rab — Xbc/XaB' (7) 



Substituting equation (7) in equation (5) and separating, 



RadRbc RadXbc /q\ 



KcD ^ — 5 ~ — V ^^' 



-K-AB yi-AB 



