of Small Differences of Phase. lt)l 



for the measurement of alternating voltages. Suppose the 

 alternator used for the tests is made to drive a synchronous 

 motor constructed with at least four poles and having on its 

 rotating spindle a commutator with as many metallic sectors 

 as there are current reversals in the time of a revolution. 

 Let half of the sectors (the first, third, &c.) be in metallic 

 connexion with each other, and the other sectors be insulated. 

 A pair of brushes suitably spaced on the commutator will 

 then be in metallic connexion during half the period, and 

 insulated from each other for the other half. But the instant 

 at which the brushes are connected will not be the same as 

 that at which the current reversal takes place, except for 

 special positions of the brushes. If the brushes are adjusted to 

 this position, and the alternating voltage to be measured has its 

 terminals connected to the brushes with a direct current volt- 

 meter in circuit, this instrument will measure half the mean 

 value of the ordinate of the positive portion of the alternating- 

 wave of potential, while the true value of the alternating 

 voltage will be the square root of mean square of this ordinate. 

 The sectors are supposed to be all equal to each other, and 

 the width of the gap between successive sectors should be 

 small compared with the circumferential width of the sectors, 

 though for wave-forms approximately sinuous the gap width 

 can be an appreciable fraction of the sector width without 

 causing any appreciable effect on the readings. Theory 

 shows that if the gap width is a small fraction e of the sector 

 width, the fractional error of the reading is represented by 

 7r 2 e 2 /8 for sine currents. It follows that if € is one per cent, 

 the error is only 1/80 per cent., while e has to be 9 per cent, 

 before the error amounts to one per cent. The sensitiveness 

 of the arrangement might have been doubled if the commit- 

 tator had been constructed so as to reverse the voltage at 

 every half period by connecting alternate sectors to two slip- 

 rings. But further sensitiveness was not required, and hence 

 only the simplest form of commutator was tried. 



For each wave-form there is a definite value/ for the ratio 

 of the square root of mean square of the ordinates to the 

 arithmetical mean. For sine waves this ratio is 



/=-L=Hl, 



For such waves the reading of the direct-current voltmeter 

 would have to be multiplied by 2"22 to get the reading of an 

 alternate-current instrument arranged so as to measure 

 the voltage directly. For other wave-forms the multiplier 

 would be 'If, where/ may be called the form factor of the 

 Phil. Man. S. 6. Vol. 9. No. VJ. Jan. 1905. M 



