A SHIELDED BRIDGE FOR INDUCTIVE IMPEDANCE 163 



inductance L and resistance R paralleled by a capacitance C are 



L — CR~ — ijcrCLr 



L' = 

 R' = 



(1 - oT-CLY + co^C^i?-' ' 



R 



(1 - co'-CL)-' + oS'C'R' ' 



When the bridge is balanced the ec|uivalent series values of each 

 component of the two impedance arms must be equal Respectively to 

 each other. If, however, the two arms have different shunting 

 capacitances, it is evident that this equality will be obtained only by 

 making the values of the two inductive branches of the parallel circuit 

 somewhat different from each other. This difference represents the 

 error introduced by the capacitance unbalance. When the values of 

 the shunting capacitances are small these errors for the purpose of indi- 

 cating their order are sufficiently closely given by the expressions 



ALx = co-Lx-(Cx - Cs) (7) 



and 



ARx = 2co"Lxi?x(Cx - Cs) (8) 



where Cx and Cs are the capacitances shunting the unknown and 

 standard impedance arms, respectively. Reduced to percentages, 

 these expressions become 



ALx (%) = 100co^Lx(Cx - Cs) 

 and 



ARx (%) = 200co'^Lx(Cx - Cs) 



and may also be written 



ALx (%) = 100%^^ (9) 



Cr 



and 



ARx{%) = 200%=-^^ (10) 



Cr. 



where Cr is the value of capacitance that would be required for 

 resonance with inductance Lx at the test frequency. 



These errors are thus proportional to the ratio of the capacitance 

 unbalance to the resonating capacitance of the inductance under test. 

 Ordinarily, values of the latter factor do not go below about 500 mmf. 

 so that in the worst case a difference in capacitance of 0.1 mmf. corre- 

 sponds to errors of 0.02 per cent and 0.04 per cent in inductance and 

 resistance respectively. 



