A SHIELDED BRIDGE FOR INDUCTIVE IMPEDANCE 155 



effects equivalent to shunting capacitances of the order of 10 to 15 

 mmf. for all frequencies up to 3000 cycles. Since individual coils 

 made according to this method may differ in their effective capacitances 

 by as much as five mmf., some adjustment of these capacitances (as 

 well as of the resistance) is required in order to make them suitable 

 for use as the required ratio arms. Assuming that this is done by 

 adding to the coil having the lower value a small capacitance of suit- 

 able constancy, it may be concluded that two coils so balanced would 

 be suitable for use at frequencies up to 3000 cycles. 



In arriving at the requirements for the more extended frequency 

 range of this bridge, the necessary phase-angle balance was first con- 

 sidered. Designating by Lx and Rx the inductance and effective 

 resistance of the impedance being tested, and by Ls and Rs, the corre- 

 sponding components of the standard impedance required to balance 

 it in a bridge circuit having ratio arms of exactly equal resistances R 

 but shunted by slightly different capacitances, Ci and C2, and assuming 

 that the quantities are such that co'^R'Cr and orR^Ci are small in 

 comparison with unity, the equation for balance is 



{Rx +jwLx){R - jwCiR') = (Rs + jivLs){R - jwGi?-), 



which reduces to 



Rx = i?s + iv'RiC^Ls - CiLx) (1) 



and 



Lx = Ls- RiC.Rs - C^Rx). (2) 



Neglecting second order effects, these can be written 



Rx = Rs + w-'RLxiC^ - Ci) (3) 



and 



Lx = Ls - RRxiCo - Ci). (4) 



If the readings Rs and Ls are taken as the values of the unknown 

 resistance and inductance, respectively, it is evident that errors as 

 given by the last terms of these equations will be present. The 

 percentage errors in the two cases are as follows: 



ARx (%) = 100a;2i?(Co - Ci) 1^ 



= 100coi?(C2 - Ci) tan 6, (5) 



ALx (%) - 100R(C2 - C:) f^ . (6) 



Lx 



