RESISTANCE AND CAPACITANCE IN SERIES 



from now on which greatly facilitates the work. It is the use of they operator. 

 The j operator — The expression for the impedance of a resistance and a 

 capacitance in series is {R^ + Xc'Y'" and to get a numerical answer the 

 processes of squaring, adding, and square-rooting must at some time be 

 carried out. However, in solving networks containing many i?'s and C's, it is 

 possible to delay the appearance of square root terms until the very end of 

 the calculation, and only to have to work them out once. The procedure is to 

 label all the reactances, from the outset, with the letter y, which signifies that 

 they are impedances of a different kind from the resistances. Thereafter the 

 difference can be forgotten, and the theorems advanced for series and parallel 

 connections, Thevenin's theorem, and the star-delta transformation, may be 

 extended to cover the solution of networks containing both resistances and 

 capacitances mixed. Whatever the nature of the problem being solved, one 

 emerges from the calculations with an expression of the form: A -\-jB, and 

 only then is it necessary to square, add, and root to get the numerical answer. 



In Figure 3.18 we see a scale of numbers, and on it a vector which represents 



H 1 1 



-1 



+1 +2 +3 



-A 



+1 +2 +3 



Figure 3.18 



Figure 3.19 



+ 1. Ify operates on +1, (symbolically, y(+l)) it has the eifect of rotating 

 the vector anti-clockwise through an angle of 90 degrees {Figure 3.19). If we 

 allow j to operate again, the vector undergoes a further 90 degrees rotation 

 {Figure 3.20). Symbolically, Figure 3.20 isy{y(+l)}- j obeys the ordinary 

 rules of algebra, so thaty{;(+l)} =y^(+l) or simply y^. 



Looking at Figure 3.20, it is clear that the vector also equals —1, therefore 

 y2 = -l andy = (-1)1/2 



-->3y 

 ...2j 



-2 -1 



0+1 +2 



♦A 



-A -3 



Figure 3.20 



♦1 

 -J 



— 2j 



---2j 



--Ay 

 Figure 3.21 



*2 *3 +A 



(_ 1)1/2 J5 aj^ 'imaginary number', and the vertically directed vector in 

 Figure 3.19 is marking off one division in the 'scale of imaginary numbers'. 

 Fining in this scale, we get Figure 3.21. 



33 



