2-B] 



EFFICIENCY. 



57 



hence as the square of El (the line voltage E being constant). 

 These are shown in Fig. 4, in which the curve for total losses is a 

 parabola. 



At any point P on the total loss curve, the loss PA, expressed as 

 a percentage of El, is PA -f- OA, which is the tangent of the angle 

 POA. It is clear that this percentage 

 loss is a minimum (and the efficiency 

 a maximum) for the point P' where 

 the line OP' is tangent to the total 

 loss curve. But at this point P', we 

 have A'B' = B r P r . (From the prop- 

 erties of a parabola, Od is bisected at 

 c.) Hence: For any apparatus hav- 

 ing a constant loss and a variable loss 

 proportional to load current, maximum 

 efficiency occurs at such a load that 

 the constant loss and variable loss 

 are equal. The same result can be 

 shown analytically by obtaining an 

 expression for efficiency, differentiat- 

 ing and equating to zero (See Franklin and Esty's Electrical Engi- 

 neering, L, 137). 



This is true for any apparatus; thus, in a transformer, the effi- 

 ciency is a maximum when the copper loss and constant core loss are 

 equal. Within limits the designer may make the efficiency a maxi- 

 mum at the particular load he desires, giving due consideration to 

 expense and to the uses to which the apparatus is to be put. 



APPENDIX III. 



MODIFICATION FOR VARYING FLUX DENSITIES. 



29. In the foregoing tests, the load run was made with constant 

 field excitation, and hence at constant flux density; the no-load run 

 was made at this same constant flux density. In cases where the 

 flux density varies during the load run (due to a variation in the 

 shunt field current or due to the action of the series field coil in a 

 compound, differential or series wound machine), three (or more) 

 no-load runs should be made at three different flux densities. 



O A El 



FIG. 4. Total losses repre- 

 sented by a parabola ; P' is 

 point of maximum efficiency. 



