ELECTRIC CIRCllT THEORY 87 



to a unit r.ni.f. applied to the initial section (;/=()). I-Voni (232) 

 and (2;^4) it follows at once that 



1 / '^''' 



kjo Jo 



Integrating the second member by parts and noting that A",{o)-0, 

 this reduces to 



^'•=11 ^^e~^J J-in{T)dT . (235) 



which is the indicial admittance formula for the quasi-distortionless 

 low-pass filter, or artificial line. 



Before discussing these formulas, it is of interest to derive the 

 formula for A", by power series expansion. Formula (233) can be 

 written 



1 /aj,\2"+i 1 1 



^"4(?i 



This can be expanded in a series in inverse powers of p; thus 



_1 I /C0_A2«+l_2»-f2 /C0<:x2,+3 



^'■ = k2^"\\p) 2-1! \p ) 



(2n + 3) (2w+4) /a;A2"+^_ 

 "^ 2^2! \p ) 



Replacing I'p" by t"/nl in accordance with the Heaviside Rule we get 



2 [ 1 /co,/\2" + i 2» + 2 fc^J\^"+'^ 



^"~ k |(2«+1)!V 2) i:(2;/ + 3)!V2 J 



(2>^ + 3)J2« + 4)/c^\2"+5_ 1 (23.5-a) 



^ 21(2w+5)! \2j 1 



This can be recognized as the powder series expansion of (234). 



The artificial cable is also of interest and practical importance. 

 In this structure the series impedance is a resistance R and the shunt 

 impedance is a capacity C, so that 



Zi=R, l/z2=pC, 



z,Z2 = R/pC, z,/z.2=pRC, (236) 



p=pRC/4. 



