46 ALTERNATING CURRENTS 



We thus have, dividing the former equation by the latter- 

 cos <j> 



w = w 



COS . COS (0 0) 



If, therefore, is not negligible, then the only case in which the 

 wattmeter reading will be correct is that corresponding to = 9. 

 For values of <f> < 9 the reading of the wattmeter will be too low, and 

 for values of > 9 too high. 



The multiplier K = ^ < ~ ^, which converts the watt- 

 cos 9 . cos (f - 9) 



meter reading into true power, is spoken of as the correcting factor. 

 It may be thrown into various forms. Thus, since 



cos 9 . cos (<j) 9) = cos 0(cos <j> cos 9 4- sin ^ sin 0) 



= cos 2 9 cos 0(1 + tan <f> tan 9) 

 we have 



_ l 6 ^^ 1 + tan 2 



~" 1 -f tan ^ tan ~ 1 -f- tan < tan 



Now, since is always very small, we may write approximately 



K = ^ a = 1 tan (b tan 



1 + tan $ tan 



a form which shows very clearly the rapid increase in the error with 

 decreasing povjer factor cos $. 



Another simple approximate expression for the true power, which 

 may be used when the power factor is very low, has been suggested 

 by Dr. Drysdale.* Using the last form of approximate value for K, 

 we have, since the wattmeter reading is roughly correct, i.e. since 

 roughly w' = VI cos 0, and also sin <j> = nearly 1 for small values of 



COS (ft 



w = w' w' tan <j> tan = w' VI X tan 



The above results may be exhibited very clearly by the aid of the 

 simple vector diagram of Fig. 32, in which I is the vector of current 

 in the main circuit, and V the p.d. vector. If there were no self- 

 inductance in the fine-wire circuit, the current in this circuit would 

 be proportional to V, and the reading to IV cos = IV. By reason 

 of self-inductance, the effective or resultant e.m.f. which maintains 

 the current through the resistance of the fine-wire circuit is not V, 

 but V r , and hence the reading is proportional to IV' r . The error is 

 thus I(V' r V). It is at once evident that : (1) the instrument 

 reading is too low if $ < ; (2) the instrument reading is correct if 



* The Electrician, vol. xlvi. p. 774 (1901). 



