206 THE MAGNETIC CIRCUIT [ART. 63 



In the symbolic notation this is represented as ix\2 r l, where x\% 

 = '2-fLi2 is the reactance corresponding to Z/i 2 ', 7 2 is the vector 

 of the effective value of the current in the wire B, and / signifies 

 that the vector i 2 '7 2 is in leading quadrature with the vector 1%. 

 Consequently, the voltage drop E\ in the wire A, equal and 

 opposite to the induced e.m.f ., is 



...... (132) 



When the currents are given, 7 2 and 7 3 can be expressed in the 

 usual way through their components, and the drop E\ is then 

 expressed through its components as e\+je'\. The reactances 

 x\2 and 13' are taken from the available tables, for the specified 

 frequency and the appropriate spacings, or else they can be 

 calculated using the value of L' from eq. (125) . 



The voltage drop EI in eq. (132) can be represented as if it were 

 due to an equivalent reactance x\ and an equivalent resistance r\ 

 in the phase A (the latter in addition to the actual ohmic resistance 

 of the wire). This is possible when 1 2 and 1$ can be expressed in 

 terms of 7 1; and is especially convenient whenever the phase differ- 

 ence between these currents and their ratio is constant. Namely, 

 let the current 7 2 lead the current Ii in phase by </>i 2 electrical 

 degrees (Fig. 49)'. Then 



), . . . (133) 



where (7 2 /7i) is the ratio of the effective values of the currents, 

 apart from their phase relation. Multiplying the vector Ii by 

 (7 2 /7i) changes its magnitude to that of 7 2 , while multiplying it by 

 (cos <j>\2 +/ sin <i 2 ) turns it counter-clockwise by <i 2 degrees. By 

 analogy we also have that 



/3 = (/3/7i)7 1 (cos^i 3 +/sin^ 13 ). . . . (134) 



Both $12 and $13 are measured counter-clockwise. Substituting 

 these values into eq. (132) and separating the real from the 

 imaginary part we get 



EI =7i[(7 2 /7i)i 2 ' sin <i 2 + (7 3 /7i)zi 3 ' sin <i 3 ] 



-i!i[(l2/li)xi2cos<t> 12 + (l3/li)x l3 'cos<t> 13 ]. . . (135) 



Thus, the drop EI is the same as if it were caused by a fictitious 

 reactance 



XT! = - (/2/7iW cos fa - (I 3 /Ii)xi B ' cos 18 , . (136) 



