42 TRANSMISSION LINE FORMULAS 



involve hyperbolic or trigonometrical functions, and so do 

 not require any mathematical tables, the only operations 

 being multiplication and addition. The series can be car- 

 ried to any accuracy desired by merely using enough of the 

 terms, which diminish very rapidly when commercial fre- 

 quencies are involved. 



The fundamental formulas as expressed by convergent 

 series have been rearranged, and some new convergent 

 series have been added, to make the formulas as tabulated 

 in this chapter directly applicable to the exact solution of 

 all the problems treated by the K formulas. Exactly 

 the same final formulas in A, B, C, D, etc., are used with 

 the convergent series as with the K formulas. 



Unlike the K formulas, which are expressed in the 

 simplest algebraical form, the convergent series involve 

 the use of complex numbers, that is, numbers containing 

 the well-known "j" terms. No difficulty should be expe- 

 rienced on this account, however, as the rules for using 

 complex quantities are quite straightforward, and even 

 one who has never worked with them should be able to 

 make use of the formulas described in this chapter by 

 closely following the instructions. 



Each of the complex quantities, (A + JB), (P JQ), 

 Z = (r + jx) I* Y = (g +jb) /, etc., is composed of two 

 parts, the first, a so-called "real" term, and the second, a 

 j term. In adding complex numbers, the j terms must 

 be kept separate from the others. Thus 



4+j 5 added to 7 +73 = n +78. 



In multiplying two complex quantities, the simple rules 



* The notation Z = (r +jx)l, etc., is used in accordance with the 

 resolution adopted by the International Electrotechnical Commission in 

 Turin, Sept., 1911. 



