IMPEDANCES IN SERIES 19 



and the phase angle of the total impedance is given by 



_j Zi sin 0i + ZQ sin a -f Z 8 sin 8 4- 

 Zi cos 0i -{- Z a cos a + Z8 cos 8 + . . . 



If, instead of being given the impedances Z\, Z a , etc., and their 

 phase angles 1} a . . . , we are given the values of the resistances 

 r\* fa f*a an( i of the reactances Vi, V a , V 8 . . . , then 

 clearly 



total impedance = V^ + r a + r B + . . .) a + (Vi + V a + V 8 + ...)* 



With a series arrangement of impedances, it is, therefore, per- 

 missible to add the resistances arithmetically in order to obtain the 

 total resistance ; and to add the reactances algebraically in order to 

 obtain the total reactance. But it is not permissible to add the 

 impedances arithmetically ; this latter addition must be carried out 

 vectorially, i.e. in accordance with the polygon law. 



As a result, we find that in the case of a series circuit such as the 

 one shown in Fig. 13, if Vi, V 2 , V 8 denote the r.m.s. values of the p.d.'s 

 across AB, BC, and CD respectively, 

 and V the r.m.s. value of the p.d. , 



across the entire circuit AD, then in P*"~ V,~" > T f '~Vf *7*"~ V 3 ~ *1 

 general Vi + V a + V 8 > V, since the ' 

 sum of the sides of any open polygon 

 is in general greater than the closing 

 side of the polygon. In the special 

 case in which tii = 2 = 3 , we have Fia - 13. Series Arrangement of 

 Vi + V 2 + V 8 = V, the polygon in Impedances, 



this case degenerating into a straight line. 



It may be well to point out that although the sum of the r.m.s. 

 values of the p.d.'s in the case considered is in general different from 

 the r.m.s. value of the total p.d., yet at every instant the sum of the 

 instantaneous valws of the p.d.'s must necessarily equal the instan- 

 taneous value of the total p.d. (the sum of the projections of the sides 

 of an open polygon being always equal to the projection of the 

 closing side of the polygon). 



9. Parallel Arrangement of Impedances. 



Let a number of impedances, Z\, Z a , Z 8 . . . , having phase 

 angles 0i, 2 , 8 . . . , be connected in parallel between two points, 

 and let it be required to find their joint or parallel impedance. 



The quantity which is common to all the branch circuits is the 

 p.d. It is, therefore, suggested to take the p.d. vector as the vector 



