THE POLYPHASE SYSTEM. 



in which W and W" are the respective wattmeter readings and a, b, <?/ and e" 

 represent the effective values of the respective voltages and currents.* Now when 

 is 60, that is when the power factor of each receiving circuit is ^, the angle 30 -f- # 

 is 90, and W reduces to zero ; and 

 when 6 is greater than 60 the angle 

 30 -f 6 is greater than 90, and W 

 becomes negative. That is, when the 

 power factor of the similar receiving 

 circuits is greater than ^, both watt- 

 meter readings are positive, when the 

 power factor is equal to ^ one wattmeter 

 reading becomes zero, and the other in- 

 dicates the total power, and when the 

 power factor is less than ^, one watt- 

 meter reading becomes negative and the 

 other reading is greater than the total 

 power. 



60. Relative copper economy 

 of two-phase and three-phase 

 transmission lines. As a basis 

 of comparison of the copper 

 economy let us consider a four- 

 wire two-phase transmission 

 line and a three-wire three- 

 phase transmission line of the same length with the same voltages 

 between mains in each case. Let E be the common value of 

 the voltages at the receiving end, and let it be required to find 

 the relative amounts of copper required to transmit the same amount 

 of power (unity power factor) with the same loss of power in the 

 mains in each case. Let R 2 be the resistance of each of the 

 four wires of the two-phase line, let R^ be the resistance of 

 each of the three wires of the three-phase line, let / 2 be the 

 current in each of the four wires of the two-phase line, and 

 let 7 3 be the current in each of the three wires of the three-phase 

 line. Then 2EI 2 is the power delivered by the two -phase line 

 and \/$Ef s is the power delivered by the three-phase line, and, 

 since the total power is assumed to be the same in each case, we 

 have 27 2 = l/3^/ s or 



* It is contrary to established conventions to use small letters to represent effective 

 values, and it is done here for the sake of brevity. Small letters stand for instantan- 

 eous values throughout this article except in the above two equations for W and W" . 



Fig. 106. 



