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XXXVII. Note on the Resistance and Self-induction of 

 Branched Circuits. By Prof. A. Andekson, M.A* 



ON page 134 of Fleming's 'Alternate Current Transformer,' 

 vol. i., the values are given of the effective resistance 

 and inductance of a system of conductors connected in parallel, 

 and acted on by an impressed electromotive force varying 

 according to a simple harmonic law. The conductors are 

 supposed to have no mutual induction. The results are taken 

 from a paper by Lord Rayleigh in the Philosophical Magazine 

 of May 1886, to which the reader is referred. 



The following method of obtaining these results, although 

 it really does not differ much from that given in Lord 

 Rayleigh's paper, will perhaps be more easily understood. 



Let the resistances of the conductors be R x , R 2 , R 3 , . . . R„, 

 the coefficients of self-induction L 1? L 2 , L 3 , . . . L n , and sup- 

 pose the impressed electromotive force to be E sin pt. The 

 currents in the branches may be denoted by I x sin (pt — 0i), 

 I 2 sin (pt — 2 ), . . . I n sin (pt— n ). Hence, if i denote the 

 total current, we have 



i = sin pt 21 cos — cosp^Slsinfl. 

 Now, since 



E sin pt = Rjli sin (pt — 1 )+pJj 1 I l cos (jpt—6i) 



= R 2 I 2 sin (pt — 9 2 ) +joL 2 I 2 cos (pt—0 2 ) 



= . . ., 



we have 



E = Rilj cos 0i 4-^LiIi sin 1} 



= p^ili cos i — Rjli sin # x , 



T "' ER X . pEL, 



IlCOS ^=E7+pL7' TlSm ^ = R7+7L? ; 



and similar expressions hold for I 2 cos 2 , I2 sin 2 , &c. 

 Hence 



R L 



or 



i = AE sin pt— pBE cos pt, where A is written for 



2u2-— -an, and B for 2™ 



whence 



R 2 +/L 2 ' -*R< +^L 2 ' 



* Coniiiiimicated bv the Author. 



