47 

 NOTES ON THE THEORY OF THE TRANSFORMER 



[Johns Hopkins University Circulars, No. 99, pp. 104, 105, 1892; Philosophical 

 Magazine |51, XXXIV, 54-57, 1892 ; Electrical World, XX, 20, 1892] 



As ordinarily treated the coefficient of self and mutual induction of 

 transformers is assumed to be a constant and many false conclusions 

 are thus drawn from it. 



I propose to treat the theory in general, taking account of the hyster- 

 esis as well as the variation in the magnetic permeability of the iron. 1 



The quantity p as used by Maxwell is the number of lines of magnetic 

 induction enclosed by the given conductor. This will be equal to the 

 number of turns of the wire into the electric current multiplied by the 

 magnetic permeability and a constant. But the magnetic permeability 

 is not a constant but a function of the magnetizing force, and hence we 

 must write 



p Bny + C(nyY + D(ny} b + etc. 



Where B, C, etc., are constants, n is the number of turns and y the 

 strength of current. 



In this series only the odd powers of y can enter in order to express 

 the fact that reversal of the current produces a negative magnetization 

 equal in amount to the direct magnetization produced by a direct cur- 

 rent. This is only approximately true, however, and we shall presently 

 correct it by the introduction of hysteresis. It is, however, very nearly 

 true for a succession of electric waves. 



To introduce hysteresis, first suppose the current to be alternating so 

 that y = c sin (bt -f- e) where t is the time and e the phase. The intro- 

 duction of a term A cos (U -\- e) into the value of the number of lines 

 of induction will then represent the effect very well. But the current 

 is not in general a simple sine curve and so we must write 



y = a x sin (bt + e^ + a 2 sin (2bt + e 2 ) + a s sin (3bt + e 3 ) + . 



1 The problem is treated by the method of magnetic circuit first applied by me to 

 iron bars in my paper on 'Magnetic Distribution' (Pliil. Mag., 1875), and afterwards 

 to the magnetic circuit of dynamos at the Electrical Conference at Philadelphia in 

 1884. I also used the same method in my paper on magnetic permeability in 1873. 



