CONSTANT PHASE DIFFERENCE 101 



Similarly, if attention is focused on the phase shifts of the individual 

 phase-shifting networks rather than on the phase difference, the following 

 odd rational functions can be introduced: 



-(f) = t 



(12) 



tan 



©■ 





in which Ai, Bi, A2, and B2 are additional even polynomials in co. This 

 requires 



(13) 



It also requires 



I' = arg(^i + zco5i) 



arg (.42 + ioi^'i). 



(14) -^^ = arg (A, - icB,). 



Since the argument of a product is the sum of the arguments of the sep- 

 arate factors, (13) and (14) require 



(15) ^ = ^^^' = ^^g (^- + ^■'^^2)(^i - ico^i). 



This permits us to write 



(16) (A2 + io:B2){Ai - ic^B,) = H(A + ic^B) 

 in which ^ is a real constant. 



When tan ( - j is prescribed, a corresponding polynomial of the form 



{A -\r io}B) can readily be derived. The problem is then to factor it into 

 the product of two polynomials (.42 + 100^2) and (.4i — iwBi) such that 

 Ai, Bi, A2, and Bo determine physically realizable phase shifts through 

 (12). Two factors of the general form (A2 + icioB2) and (.4i — iooBi) can 

 readily be obtained in a number of ways. The only question is how to obtain 

 them in such a way that the corresponding phase characteristics will be 

 physical. A procedure meeting this requirement is described below. 



The variable co is first replaced in (.4 + iuB) by p representing ico. This 

 leaves a polynomial in p with real coefficients, since A and B represent 

 polynomials in co-, while p~ represents -co''. Suppose all the roots of the poly- 

 nomial A -\- pB are determined. Then this polynomial can be split into 



