of Small Differences of Phase. 159 



The former equations illustrate a method of determining 

 <t> bv observing v, i\, and v 2 , while the latter equation indi- 

 cates a method of determining sin <j> from these measurements 

 together with an observation of r 3 . It is to be noted that the 

 conductor APC need not necessarily be noninductive. It 

 should be of the same inductive character all through, and such 

 that it is possible to adjust the point P connected with the 

 voltmeter. The point P will then move along the vector AC 

 (see fig. 3) as the adjustment is varied. Moreover there is no 

 disadvantage in choosing a small noninductive resistance 

 A B such that the voltage v± is small compared with v 2 , pro- 

 vided always that suitable voltmeters are available for making 

 the requisite measurements. One of the disadvantages of 

 the three-voltmeter method for the measurement of power 

 factor, is that the noninductive resistance AB should absorb 

 about the same voltage as the conductor BC, and hence a 

 greater amount of power than the load under test. If the 

 bridge method just described is used, the load B C may absorb 

 several hundred volts, while v x and v may each be less than 

 a volt. Where AB is negligible compared with B C, it of 

 course follows that yfr is small compared with 0, and that Q 

 and <p can in general be considered the same. Now in order 

 to determine one of the angles by the above method, all that 

 is really needed is to have two voltages, one like AB in phase 

 with the current, and the other in phase with either BC or 

 AC. One of these voltages should be fixed in magnitude, 

 and the other adjustable. Such voltages can be obtained 

 with all requisite accuracy for most purposes by the aid of 

 current and voltage transformers. The former must have 

 its primary in the main circuit, and its secondary closed 

 through a non-inductive resistance, the voltage on which will 

 be in phase with AB. The latter must be a constant potential 

 transformer with its primary across AC or BC, and its 

 secondary closed through a noninductive resistance, the 

 voltage on which will be in phase with that on the primary. 

 The secondary voltage need not be greater than corresponds 

 with the length AP in the figure. The two secondary circuits 

 must be connected together at one point to determine a 

 common potential represented by the point A in fig. 3. Such 

 an arrangement makes the method adaptable to loads absorb- 

 ing high potentials, or taking heavy currents. The phase- 

 differences introduced by such transformers, if of good design, 

 can in practice be reduced to about a tenth of a degree, and 

 hence, for values of 6 such as are usual, no appreciable error 

 will be introduced into the measurement by them. 



The writer has tested this two-voltmeter method in all the 



