BRIDGE METHODS OF MEASURING IMPEDANCES 455 



Thus, if ZcD is the unknown impedance, equation (1) evaluates it 

 in terms of the other three impedances. Equation (1) is a vector 

 equation and therefore the value of Zcd both in magnitude and phase, 

 or both components of it when considered as a complex quantity, may 

 be obtained from this equation. 



Although the above equations and subsequent discussion are based 

 primarily on the use of impedances, it should be remembered that all 

 of these relations may be obtained in the same general form if the 

 bridge arms are considered as admittances. 



The Bridge Requirements 



If the impedances of equation (1) are replaced by the complex equiva- 

 lents R + jX, then 



, .^ (Rbc -\- JXbc){Rad -\- JXad) /^n 



K-CD +J^CD = 5 _i_ -v ^^^ 



From this equation Rcd and Xcd may be evaluated in terms of the 

 other six quantities. Thus, if each component of the impedances of 

 three arms is known, each component of the fourth impedance in 

 terms of the other six components can be determined. 



In obtaining the balance, any or all of the six component impedances 

 occurring in the right hand side of equation (2) may be adjusted. Since 

 there are two unknown quantities to be determined, at least two of 

 these components must be adjusted. From the standpoint of sim- 

 plicity and speed in operation and in order to keep the cost of the 

 circuit to a minimum, it is desirable that not more than two of the 

 known components be adjustable. It is also essential that the choice 

 be such that a variation of one adjustable standard balance one 

 component of the unknown, irrespective of the other component. 

 In other words Rcd should be balanced by one known standard, this 

 value of the standard being independent of the magnitude of Xcd, 

 and, in turn, Xcd should be balanced by another standard, the value 

 of which should be independent of the magnitude of Rcd- This con- 

 dition of independent adjustment for the two components is essential 

 for satisfactory operation of the bridge, since it allows the balance to 

 be made more rapidly and systematically, and a given setting of one 

 standard always corresponds to the same value of one component of 

 the unknown, independent of the magnitude of the other component, 

 thus allowing the calibration of each of the adjustable standards in 

 terms of the unknown component which it measures. 



To meet this requirement, the two components for use as adjustable 

 standards should be so chosen that, when equation (2) is reduced to 



