90 ELEMENTS OF ELECTRICAL ENGINEERING. 



If / is not the reference axis we have, from equation (2') : 



which, substituted in equation (15) gives 



(iii) 



If we are to build up a consistent system of algebra on equations (i), (ii), and (15), 

 it is necessary to assign to the letter j a meaning which will not only enable it to 

 serve as a distinguishing mark for the j-component in such expressions as (i), (ii), 

 and (15), but which will make equation (iii) give the correct expressions for the com- 

 ponents of E. That is, we must have : 



which comes from equation (iii) by substituting fi~\-je u for E and carrying out the 

 multiplication indicated in the right hand member of (iii). It can be shown, however, 

 by ordinary trigonometry that the components of a line whose length is IV R 2 -\- X 2 

 and which is 6 degrees ahead of /, where 6 is the angle whose tangent is X\ R t are 



and \ (v) 



e ll =Xi l -\-Ri n \ 



Therefore the term j 2 Xi u in the above expression (iv) must be equal to Xi UJ or, 

 in other words, we must have : 



/t = -I (vi) 



It should be stated in explanation of the derivation of equation (vi) from (iv) and 

 (v) that a complex equation is always equivalent to two simple equations ; those parts 

 of the two members which do not involve the unit / are equal to each other, and 

 those parts which contain j as a factor are equal to each other. 



In equation (15) the expression R -\-jX stands as a multiplier and it is called a 

 direct operator. If, however, we solve equation (15) for /we have: 



In this equation the expression R -\-jX stands as a divisor and in this case it is called 

 an inverse operator. If we multiply numerator and denominator of the right hand 

 member of equation (vii) by R jX, remembering that j* = I we have : 



- 



or 



According to this equation E X RI(R 2 + X 2 } should be the component of / par- 

 allel to E, and X X/(X* + X 2 ) should be the component of / 90 ahead of 

 E, or EXX/(J? 2 + X 2 ) should be the component of / 90 behind E. In 

 fact these are the correct expressions for the components of /. 



