326 



THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1951 



cancelled by the contribution from the opposite negative charge. Thus, 

 by our analogy, da/do) vanishes, and the constant phase delay network 

 has a constant gain at all frequencies. 



For the charge spacing illustrated in Fig. 5 the poles of the transmission 

 function are located at the positive charge points, p'^ = —a-\- ivb, v = 

 — w, • • • 0, • • • w, while the infinite loss points are located at the negative 

 charge points, p^ = +o + ivh. Thus the required transmission function is 



— a — ifxb 



F(p) = constant + log H 



-m p -{- a — ifjib 



(29) 



REAL p 



(a) DISTRIBUTED CHARGE (b) LUMPED CHARGES 



NATURAL MODES Py= -a+it^b 



^ = -m,---,o,---,+m 

 27r. 



INFINITE LOSS POINTS pj,= +a+L^b^ 



APPROXIMATE PHASE OF NETWORK At\ ^_ 27r., -^"^^ o.m 27r .. 

 FREQUENCIES WELL INSIDE "PLATES" / /-* " b i)iN ^ u/ 



Fig. 5 — The condenser plate analogue; (a) distributed charge, (b) lumped charges. 



If we allow the number of charges to become infinite, but still with con- 

 stant spacing by the infinite product may be recognized as the ratio of sine 

 or cosine functions,* 



sm, cos 



F = constant + log 



(^•') 



sm, cos 



(^"') 



(30) 



*See, for instance, B. O. Pierce's "Short Table of Integrals," page 96, equations 

 816, 817. 



