﻿1030 Prof. V. Karapetoff on General Equations 



Separating the real and the imaginary parts, we get 



M ] /C 3 = r 1 r J (24) 



and r 3 Mi = (L 1 -Mi)r 4 (25) 



Eq. (25) may be also written in the form 



Li/M^l + fo/n) (25 a) 



Item JS r o. 8. This frequency bridge was described bv 

 Mr. Cone, ibid. p. 1749. Eq. (2) gives :— 



Vi = r 4 [r 1 -;/(»p i )]''(3+i«C^. . . . (26) 



Separating the real and the imaginary parts, we get 



^'2/^=^1 + 02^/0, (27) 



l/(ml\) = €oC 2 nr 2 . ..... (28) 



The last equation gives 



^CAnr^l (29) 



from which the unknown frequency may be computed. The 

 following special case is of practical interest. In eq. (27) 

 put r 3 = r i and r 2 = 2?v; then C 1 = 2C 2 , and eq. (29) becomes 



uC l r 1 =, ( oG 2 r 2 = l (30) 



As is mentioned above, the general equation (D), or any 

 of its particular forms, (1) to (8), may be used for the 

 derivation of new forms of the bridge. Take for example 

 the simplest case, that of eq. (1). It may be written in the 

 form 



fa +,Fs) fa +>' 2 ) = (n +>*i) fa +jx, v ). . . (31) 



Separating the real and the imaginary parts, we get 



r s r 2 — x z x 2 = 7-!^— x x x± (32) 



x s r 2 + r 3 x 2 = x l r i + r l x 4i (33) 



We may put, if we so choose, r 3 r 2 =^.r^r 4i that is, require 

 the bridge to be first balanced with direct current. Then 

 eq. (32) becomes x 3 x 2 = x 1 x 4 , and these two conditions, 

 together with eq. (33), may be used to investigate various 

 possible bridge combinations with resistances and induc- 

 tances. The condition r 3 r , 2 = r 1 r 4 may be dropped and eqs. 

 (32) and (33) used for an investigation of various bridge 

 arrangements containing resistance, inductance, and capacity. 

 One or two of the x's may be put equal to zero, with a 

 resulting simplification in the algebraic relationships. In a 

 similar manner, eqs. (2) to (8) may be resolved into their 



