102 BELL SYSTEM TECHNICAL JOURNAL 



two factors by assigning various of the roots to each of the two factors. 



It turns out that physically realizable phase characteristics will be obtained 



if all those roots with positive real parts are assigned to the factor (^4 1 — pB\) 



which appears in (16) when ico is replaced by p, all other roots being assigned 



to the factor (^2 + pB^. 



The physical realizability of the above division of the roots follows from 



pB 

 a theorem which states that -j^ is realizable as the impedance of a two- 



terminal reactance network whenever Ax and B^ are even polynomials in 

 p with real coefficients such that Ax+ pBx has no roots with positive real 

 parts.^ From this theorem and the fact that the evenness oi Ax and Bx 

 causes them to remain unchanged when p is reversed in sign, it follows that 



^— ^ will also be the impedance of a physical two-terminal reactance net- 



Ax 



work whenever Ax — pBx has no roots with negative real parts. Thus, by 



(12) the above division of the roots oi A -\- pB makes tan ( ^ j and tan 



( — ) realizable as the impedances of two-terminal reactance networks. 



These reactance networks and their inverses are merely the arms of unit 



impedance lattices producing the phase characteristics defined by (12). 



The above argument merely shows that each of the two phase-shifting 



networks can at least be realized as a single lattice when tan ( - 1 and 



tan I ^ 1 are determined by the method described. Actually, they can be 



broken into tandem sections directly as soon as the roots of (^1 — pB-^ 

 and {A2 + PB2) have been determined. From (^1 — pB^ , the quantity 

 (^1 + pBi) can be found by merely reversing the signs of the roots. Then 

 by using the principle that the argument of a product is the sum of the 

 arguments of the separate factors, phase-shifting networks can be designed 

 corresponding to various factors or groups of factors as determined from 

 the known roots of (^1 + pBi) and {A2 + pB-i) . There can be a separate 

 section for each real root and each conjugate pair of complex roots.^" 



Determination of a Network Corresponding to a Tcheby- 

 CHEFF Type of Phase Difference Characteristic 



The procedure described above for determining a network corresponding 

 to a general phase difference characteristic is complicated by the necessity 



» See "Synthesis of Reactance 4-Poles which Produce Prescribed Insertion Loss Char- 

 acteristics," Journal of Mathematics and Physics, Vol. XVIII, No. 4, September, 1939— 

 page 276. 



i» See II. W. Bode, "Network Analysis and Feedback Amplifier Design," D. Van 

 Nostrand Company, New York, 1945, Page 239, §11.6. 



