160 Dr. Sumpner on the Measurement 



various forms above indicated, and found it satisfactory. It 

 is not so simple or so accurate as the wattmeter method 

 when an instrument of the right range is available, and when 

 the value of to be determined is large. But its great 

 adaptability to loads of all kinds makes it a useful laboratory 

 method. Under ordinary circumstances, a hot-wire voltmeter 

 reading up to 1 volt can be used for taking the two measure- 

 ments v and vi, while the voltage v 2 on the load can be taken 

 by the instrument necessarily used on the circuit. It is then 

 only necessary to find two noninductive resistances, one 

 such as to absorb about one volt when traversed by the load 

 current, and the other suitable for the voltage v 2 or v 3 , and 

 having a portion of it (taking about a volt) of such a structure 

 that a voltmeter can be connected to it at various points. 



It is, however, when the phase-difference to be determined 

 is small that the method is most useful and most accurate, 

 while the wattmeter method altogether fails. The vector 

 diagram, shown in fig. 3, applies whatever the frequency or 

 wave form of the current, and even when the current is 

 unidirectional but varying in strength. In a test in which a 

 hand-regulated direct-current arc was put in series with a 

 noninductive resistance, it was found, with ^ = 70 volts and 

 v 2 = 40 volts, that the minimum value of v was less than 0*1 volt. 

 A lower voltage could not be measured with any certainty 

 with the hot-wire voltmeter used. It follows from this test, 

 that the power factor of the direct-current arc cannot differ 

 from unity by more than 5 parts in a million, and the phase- 

 difference <f> between the voltage and the current works out 

 to be about 0*003 radian. A similar test, made on the small 

 equal-ratio transformer already referred to, with the primary 

 subjected to over 80 volts, gave as minimum voltage v, a value 

 estimated to be only 0*03 volt. This would correspond with 

 a phase-difference of only 0*00037 radian, or 0*021 degree, 

 and a value of cos <f> differing from unity by only seven parts 

 m a hundred million. 



Voltmeters for measuring minute alternate voltages are 

 not procurable. For small phase-angles v will be small 

 compared with either v x or v 2 . If the apparatus to be tested 

 is such that the application of high voltages to it would either 

 do it injury or alter its working conditions, the method 

 becomes impracticable when the angles to be measured are 

 small. 



In order to overcome the difficulty arising from the absence 

 of sensitive alternate-current voltmeters, the w<riter has adopted 

 the drastic method of commutation. By such means it is 

 possible to make use of sensitive direct-current instruments 



