DISTORTION CORRECTION 463 



two physical networks differ in their phase constants but have the 

 same attenuation constants. For one 



tan 



7^/ ^ (V^ + VC2)3 



y 



(32) 



-1 ^iPiQiy- ' 

 and for the other 



tan B" = (V-P^ - ^0^)y r^3) 



where B" has a maximum or minimum depending upon whether y is 

 positive or negative. As a result for the sum 



(B' + B"\ rjT 



tan ^ ■ = VP.T, (34) 



2 

 and for the difference 



IB' -B"\ r^ 

 tan ( 2 ) = ^^23'- (35) 



Now a non-dissipative lattice type network in which Sn is a reactance 

 proportional to y has a formula 



tan (5/2) = M^y, (36) 



where M\ is positive. Comparison of these latter formulae indicates 

 the proof of the theorem and its corollary. 



A simple and useful relation exists between the maximum attenua- 

 tion constant Am occurring at 3; = oo and the maximum or minimum 

 phase constant BJ' of {33) occurring at 3* = ± l/(i'2<22)^'^. It is 



sinh (AJ2) = ± tan BJ'. (37) 



An example Is given by Networks 2a, Appendix IV, and a practical 

 use of this relation will be made in Section 4.2. 



2.84. Composite Networks 



The tandem combination of two or more different sections of 

 constant resistance networks can generally give propagation char- 

 acteristics which are unattainable in a single section. For this reason 

 it is sometimes advantageous to treat such a composite network of 

 two or three simple sections as a single unit. When this is done it 

 will be found that the composite network has attenuation coefficients, 

 if any, which in number may be equal to, greater than, or even less than 

 the sum for the individual networks when considered separately. 



An example of a case in which the number of attenuation coefficients 



