of Small Differences of Phase. 157 



where /, r' 2 — d~ , 7 



This is true whatever the value of 6, but for small values 6 1 

 becomes identical with <£. and we have in circular measure 



V va: 2 \A,v 2 V w 2 y 



Some years ago the writer tested the phase-difference between 

 the primary and secondary voltages of a small equal-ratio 

 transformer by joining a low-reading hot-wire voltmeter to 

 two terminals, one on each coil, and by connecting the other 

 two terminals with a wire. Some numbers taken from an 

 old note-book and referring to a test with the secondary on 

 open circuit give rj = 72 : r 2 = 73'8 ; i-=l*9. Here d=l'S y 

 and the value of 6 works out to be 0*0084 radian, or 0*48 

 degree, for which cos 6 is 0*99996. But recent tests on the 

 same transformer by a more accurate method have proved 

 that this estimate of 6 is no less than 24 times too great, 

 that its real value is only 0*00037 radian, or 0*021 degree, 

 and that consequently the value of cos 6 only differs from 

 unity by 7 parts in a hundred million. 



The explanation of the error made in the earlier test is 

 simply the difficulty of determining the difference d with 

 sufficient accuracy when it is about the same magnitude as v 

 and small compared with i\ or v 2 . If, assuming i\ correct, 

 v 2 had been read 73*9 instead of 73*8, the angle 6 would have 

 worked out to be zero, so that an error of only about one- 

 tenth per cent, in reading one of the voltmeters completely 

 accounts for the difference between the two results. This 

 was not noticed at the time, and as the test was not repeated, 

 the error was not discovered till long afterwards. 



If in fig. 1 the voltage OV 2 is adjustable without alteration 

 of phase, the voltmeter measuring the side opposite 6 can be 

 made to give a minimum reading by altering OV 2 . The 

 minimum reading will be ViV 4 perpendicular to OV 4 , and 

 we can determine sin 6 = vjv l as accurately as we can read the 

 voltmeters. 



Thu> the phase relation of the voltages of two coupled 

 alternator- can be determined by a two-voltmeter method in 

 this way, by -imply adjusting the excitation of one of them. 

 this one being preferably run on open circuit. But in any 

 case, by shunting the larger of the two voltages forming the 

 angle 6 with a non-inductive resistance, it is possible, by 

 tapping this resistance at various points, to get a minimum 

 reading of the voltmeter v, the ratio of which to the unshunted 



