94 ELEMENTS OF ELECTRICAL ENGINEERING. 



Substituting the values of E l and E^ from (i) and (ii) in (iii) and solving for 

 /, we have : 



C 1 



Substituting this value in (i) and (ii) we have : 



(r.+jxjE 



l ~>i+ ' 2 +/(*i+* 2 ) 



and 



These equations (v) and (vi) express the electromotive forces ^ and -" a in 

 terms of the known quantities E, r lt r v x\ and x y 



In order to make actual numerical calculations, of E l for example, the second 

 member of (v) must be separated into its components, that is, into real and imaginary 

 parts, and then the numerical value of E l is found by taking the square root of the 

 sum of the squares of these components, and the tangent of the angle between E and 

 E l (the angle by which E l leads E in phase) is equal to the y-component of E l 

 divided by the real component of E v 



Thus, multiplying numerator and denominator of (v) by [^ -J- r^ j(x l +-* 2 )1> 

 / is removed from the denominator, and we have : 



F F r i r i r 

 '>i + ".) 8 



The first term of this expression is the component of JE l parallel to E and the 

 second term, dropping j, is the component of E^ perpendicular to E, Taking the 

 square root of the sum of the squares of these components we have : 



Numerical value of E^ V r \ + * __ y^g ^ii) 



Taking the ratio of the two components we have : 



Tangent of angle from E forwards to E^ = * 1 ~- 1 2 ^ (ix) 



r \\ r \ + r t) ~r x i\ x i~\~ x i) 



A negative value of this tangent indicates that E^ is behind E in phase. 



45. General outline of complex quantity method. The above discussion 

 illustrates the general use of complex quantity in alternating-current problems, and 

 the procedure is always essentially the same, namely: First, the formulation of the 

 problem by equations expressing the relation between current and voltage in elementary 

 circuits, and by equations expressing the known sum of certain voltages when 

 elementary circuits are in series, or the known sum of certain currents when elementary 

 circuits are in parallel ; Second, the transformation of these complex equations so as to 

 express the unknown quantities in terms of their components as in equation (vii) 

 Art. 44 ; Third, the derivation of equations (not involving complex quantity) express- 

 ing numerical values and phase angles in terms of the given data ; and, Fourth, the 

 actual calculation of numerical results. 



