56 Notes on the Theory of the Transformer, 



1. A simple harmonic current through an iron transformer 

 will produce a secondary electromotive force and current, or 

 both, which contain not only the fundamental period hut the 

 higher odd harmonics. 



2. This effect is not due to hysteresis, but to the variation 

 in the magnetic permeability. 



3. The harmonics increase with the increase in magnetiza- 

 tion of the iron and nearly vanish as the magnetization 

 decreases, although it is doubtful if they ever quite vanish. 

 Hence, an increase of resistance will decrease the harmonics. 



4. In the method of introducing the hysteresis into the 

 equations, it enters as an addition to the resistance in the 

 term "Rax + Aw6, where It is the resistance, a x the maximum 

 current, A the coefficient of hystersis, which is dependent 

 upon the amount of magnetization of the iron, n the number 



o_ 

 of turns of wire, and &== .-5= is 2-7T divided by the time of a 

 complete period. 



The introduction of the hysteresis into the ordinary 

 equations, therefore, presents little or no difficulty. 



Many observers have noted that the current curve in a 

 transformer was not a sine curve and Prof. Ayr ton has shown 

 the presence of the odd harmonics, but gives no explanation. 

 Mr. Fleming has attributed them to hysteresis, but I believe 

 the present paper gives the first true explanation. 



Problem 3. — To find the work of hysteresis. Let the 

 resistance, It, be zero. The work done will then be the 

 integral of the current times the electromotive force, or 



w =]y^ dt 



dt 

 the integral to be taken for one period of the current. 



IV 





 iv= j [a sin (bt + e l )y+ Bn y d £ + Gn 3 3 tf^jL + l dt 



A T 



w = & j-a v 



All the other terms are zero. 



In a unit of time the energy absorbed is 



Aofj 



~Y' 



Steinmetz has found by experiment that this varies as the 

 1^ power of the magnetic induction. Of course the present 

 theory gives nothing of this but only suggests a way of 

 introducing the hysteresis into calculations of this nature. 



