156 Dr. Sumpner on the Measurement 



as, and do not possess anything like the range of, the corre- 

 sponding direct-current instruments. A wattmeter o£ the 

 right range is not always available, and in such cases the 

 three- voltmeter method, or some modification of it, has often 

 proved a convenient substitute. 



The errors arising in practice in measuring phase by the 

 three-voltmeter method are serious for low -power factor 

 circuits, that is for values of 6 approaching 90 degrees, 

 but are not so important when high-power factors have to be 

 measured. The perfection of the method in theory, and its 

 limitations in practice, are exactly comparable with the 

 determination of the angles of a triangle from measurements 

 of its three sides. If these sides are measured accurately 

 the angles can be correctly calculated in all cases, but for 

 given percentage errors made in estimating the sides, the 

 resulting error made in calculating the angle will largely 

 depend on the shape of the triangle. Thus, in fig. 1, the 



Fisr. 1. 



*& 



& 



angle will be determined much less accurately from measure- 

 ments of the sides of the triangle OY 1 Y 2 , where OVj and 

 OV 3 are very different in magnitude, than from equally 

 accurate measurements of the sides of OV 1 V 3 , in which OV x 

 and OV 3 are supposed to be nearly of equal length. The 

 measurement will be most accurate, for given percentage 

 errors in the sides, if the length of the perpendicular VjV^ 

 on the line OV 2 can be measured, since then the value of sin 6 

 will be known as accurately as the ratio of V X V 4 to OVx can 

 be determined. If the sides of the figure represent voltages 

 in different phases, the measurement of the ratio in question 

 can in this case be determined as accurately as two voltmeter 

 readings can be read and compared. 



If we denote by v x and r 2 the lengths of the sides of the 

 triangle forming the angle 6. and if the third side of the tri- 

 angle opposite 6 is represented by v 9 it can easily be shown 

 that 



cos0=l-i<£ 2 , . . . . . . (1) 



