100 BELL SYSTEM TECHNICAL JOURNAL 



in which 5 represents the total variation of the phase difference jS in the 

 approximation range. Similarly, the average value /3a of )3 in the approxi- 

 mation range is given by^ 



(7) tan m = ^:(L±Vl^). 



1 - £/Vl - k\ 

 If the phase variation 5 is reasonably small, (6) and (7) can be replaced 

 by the approximate relationships 



sin (^a) ,2 ,. 



6 = — ^ — ki radians 



/3a 



tanj^^') = I' v^l - k\.' 



A still further modification is obtained by replacing k\ by the quantity 16(/i, 

 which is an approximate equivalent when kl is small, and by then replacing 

 q\ by the equivalent q" of (5). This gives 



(9) 5 = 8 sin (/3„)5" 



tan r| j = V \/l - 165". 



When combined with (3) and tabulations of sin~^(^) vs logio(9) , these 

 formulae can be used to compute 5 when the parameters coi, C02, /3a and n are 

 prescribed. Curves of 5 are plotted against ^2/^1 in Fig. 2, assuming ^a to 

 be 90 degrees. 



Determination' of a Network Corresponding to a General 

 Phase Difference Function 



Since tan ( - 1 must be an odd rational function of co, it can be expressed 

 in the form 



(10) tan (f) = "1 



\Z/ A. 



in which A and B are even polynomials in co. This requires 



(11) ^ = arg {A + iu^B). 



' More exactly, /3„ is the average of the maximum and minimum values of (3 occurring 

 in the range of api)roximation. 



' In the important sjjecial case in which the average phase dilTerence /3a is 90°, this 



expression for tan ( ) is exact rather than approximate. 



