56 ALTERNATING CURRENTS 



Dividing the second of the last two equations by the first, we 

 find- 

 tan 30 tan = w 1 w 2 

 or 



By using this formula, we are enabled to calculate the power 

 factor cos of the balanced three-phase system from the readings of 

 the two wattmeters. The formula is frequently referred to by con- 

 tinental writers as the " tangent formula." It must be carefully 

 noted, however, that the tangent formula is based on the assumption 

 of sine waves of p.d. and current, f and cannot be relied upon when 

 the wave-forms are greatly distorted. 



If the power factor of a balanced three-phase system is unity, the 

 load being entirely non-inductive, and = 0, then it is evident, 

 either from the vector diagram of Fig. 39, or from the tangent formula 

 just established, that Wi = W 2 , i.e. the readings of the two watt- 

 meters are equal. This is, in fact, the only case in which the two 

 instruments give identical readings. 



As the angle of lag 6 increases, it is evident from Fig. 39 that, 

 the currents being assumed to retain their original values, the 

 reading of Wi will increase, and that of W 2 decrease, until 6 becomes 

 equal to 30. At this stage, the reading of Wi reaches a maximum 

 value for a given value of the line currents. Any further increase 

 of will cause both readings to decrease (it is evident from the 

 diagram that the reading of W 2 will decrease more rapidly than that 

 of Wi). When 6 reaches 60, we have 30 + 60 = 90, and the 

 current I 3 is in quadrature with "VY, hence W 2 will give a zero 

 reading. A still further increase in 8 will cause the reading of W 2 

 to become negative. 



A very special case of a balanced three-phase load is that in 

 which the load is star-connected, with an accessible neutral point (a 

 star connection is very common, but the neutral point is not generally 

 accessible). A single wattmeter may then be used to measure the 

 power. The current coil of the wattmeter is introduced into one of 



* This may also be written in the form 



'-i 



-W 



tan 6 = V3 W 2 



' + 1 



W 2 



t It may be shown that, \vith distorted wave-forms, the star p.d.'s of a three-phase 

 system cannot, in general, be represented by three co-planar vectors. Hence the 

 ordinary graphical method fails (see, in this connection, a paper by E. Orlich in 

 Elektrotechniclie Zeitschrift, vol. xxiv. p. 59). 



