1892.] 



Transformers. 



457 



most general case is this : — Take the internal resistance of the sth 

 coil to be r s ; let it have a condenser outside of capacity K s in series 

 with a resistance p s " and let there be a resistance pj in parallel with 

 p s " and the condenser. Then, in the general expression, instead of H s 

 we must nse 



r s + (p/ P ,"K,e + />/)/{ (p/ + Ps ")K s e + 1 }. 



Having f ound O s , we must remember that the part passing through 

 Ps is (/>/'K^ + l)0^/{(/!>/ + /)/ / )K^ + l}. Everywhere, when any resist- 

 ance r is spoken of, if there be also a coefficient of self-induction I (or 

 in case it is an internal resistance r s that is spoken of, should there 

 be magnetic leakage), we must use instead of r the expression r + lO. 

 Our formulae, in case there is magnetic leakage, in case there is self- 

 induction, or in case there are condensers, really require that a com- 

 plicated operation 



a + he + cfl 2 + dd z + eO* +/6 5 + &c. 

 a + b'6 + c'O' 1 + d'& + e'O* +/6 5 + &c. 



should be performed upon a function of the time. Now the functions 

 with which engineers deal are periodic functions, and the evanescent 

 parts of the answers may be left out of consideration. As any 

 periodic function of the time is the sum of simple harmonic func- 

 tions, we have only to perform the above operations upon functions 

 like sin (hi -f m) . But it is obvious that the complicated operation 

 when performed on sin (ht + m) reduces to 



(q-cF + efr 4 — &c.) + (b-dk' i +fk*-&c) 

 (a - c'k 2 +e'& 4 — &c.) + (6'— d' W +/'& 4 - Ac.) 6 ' 



an operation which is very easy to perform. 



Writing it in the form (p + qO)l(oc + (30), the result of operating 

 upon sin(&£ + m) is to convert it into 



Hence the most complicated cases are readily worked out whei 

 numerical examples are taken. Engineers are in the habit of speak- 

 ing of the epoch m as a lead when it is positive, and as a lag when it 

 is negative. The reciprocal of the periodic time is called the 

 frequency, and fc is 2irX frequency. 



IV. Law connecting A and I : — 



1. If p,, the magnetic permeability of the iron, is constant, let a 

 stand for 47r« / wlO _9 /X where « is the area of cross-section of the iron 

 n square centimetres, and A, is the average length of the induction 



