EFFECT OF HARMONICS ox THE TRANSMISSION OF POWER 283 



made by Dr. Duncan in this laboratory, immense distortion of the 

 curves has been found when the induction exceeds 12,000 lines per 

 square centimetre, while the curves are comparatively smooth with only 

 5000; hence I scarcely think it advisable to use more than 5000 for 

 transformers, even though low frequency were used. As to dynamos 

 and motors the limit will depend on the variety of machine used and 

 will not influence the better class very much. 



The fixing of the limit of magnetization of transformers at 5000 

 causes the output with given current to vary inversely as the frequency. 

 As the hysteresis with slow frequency will be less, we may increase the 

 current somewhat to make up for it. As to the exact law, it depends 

 on the relative dimensions of wire and iron. Practically we might 

 estimate for an ordinary transformer that the output varied inversely 

 as the eight-tenth power of the frequency. 



The law that the output varies inversely as the four-tenth power of 

 the frequency assumes that the magnetization increases with decrease 

 of frequency and thus distorts the curves as shown above. 



The immense increase of the size and cost of transformers when dis- 

 tortion of the curve is avoided precludes the use of very low frequencies 

 even were it otherwise desirable. 



It is to be noted that the action of the iron in producing harmonics 

 is directly on the electromotive force, and the amount of current flow- 

 ing will depend on the resistance and the self-induction of the circuit. 

 The resistance, owing to so-called ' skin ' effect, will be greater for the 

 harmonics than for the fundamental period. Self-induction depending 

 on the air will always diminish the harmonics, while if it is due to iron 

 it may either increase or decrease them according to their phase. 



The measurement of the energy supplied by an alternating current is 

 also much complicated by the presence of harmonics. 



Let the current be 



C = A^ sin (bt + <i) + A s sin (3 U -f ?> 3 ) + A & sin (5 bt + ? s ) + 

 and electromotive force 



E = B, sin bt + B 3 sin (3 bt + v'- 8 ) + B, sin ( 5 bt + *.',) + 

 The energy transmitted is, then, per unit of time 



C'CE dt= r'cEd (bt) 



If n is the number of complete periods in the primary term, then b = 

 2;rn and the energy transmitted per second becomes 



\\.A 1 B 1 cos <p + A 3 B, cos O 3 - 8 ) + A, B, cos (cr 5 - <?' 5 ) + etc.] 



