VOLTAGE RATIO IN CONVERTERS 239 



may be immediately obtained by an additional very simple graphical 

 construction, which is as follows. At R erect the perpendicular RS, 

 and join OS. With O as centre and radius OS describe the arc ST, 

 intersecting OP at T, and on OT as diameter describe the semi- 



OT 

 circle OUT. Then, since OT = OS = </% OR, we have ~ 



(JK _ ^ or Qrp __ _ -g ut ^ sem i c i rc i es being similar 

 20R x^ 2 



and similarly situated with respect to 0, it follows that the ratio of any 

 two secants drawn from and making the same angle with OP will 



also be /=.. Thus OU = ^=, i.e. OU gives the r.m.s. value of the 



alternating voltage between the slip-rings.* 



A simple formula, immediately evident by a reference to the 

 diagram, may also be used for calculating the voltage ratio. For 

 we have 



___ OQ OPcosPOQ OP . 



r.m.s. value of voltage = OU = -.- = - ,- - /= sin ORQ 



180 

 and since ORQ in degree measure is equal to v-, 



continuous voltage . 90 

 r.m.s. value of slip-ring voltage = -- -f= - - . sin r- . (1) 



the value of d being given by 



T3O1G* T)l tch. 



distance between points of connection of slip-rings = - 



Using this formula, or the graphical construction just explained, 

 we have, calling the continuous voltage 100, the following values of 

 the alternating (r.m.s.) voltage for a single-, two-, and three-phase 



(1 2 \ 



-5=1,1 and = respectively J 



Continuous voltage. Single-phase. Two-phase. Three-phase. 



100 707 707 61-2 



These values, it must be remembered, are only approxiinate, not 

 only because we have taken no account of the resistance drop in the 

 converter, but because the assumed sine law of the magnetic flux 

 distribution around the armature periphery is more or less departed 

 from in practice. 



* This elegant construction is due to Mr. 0. J. Ferguson (Electrical World and 

 Engineer, vol. xliv. p. 733 (1U04)). 



