72 BELL SYSTEM TECHNICAL JOURNAL 



terms of I and P by the equation 



P - j^jT/o - 2/0 + (7 - 70) 1 + (/ - 7«)/27o- ^ ^ 



If the system contained no internal irregularities within the loaded 

 cable itself and also no terminal irregularity at the far end, p would 

 of course be zero. There are three types of irregularities here to be 

 considered: section-irregularities, coil-irregularities, and the terminal- 

 irregularity. Each of these types will be considered separately, with 

 the ultimate object of constructing, by superposition, an approximate 

 formula for p in terms of all of the existing irregularities. 



First, consider the typical section-irregularity, situated in section 

 No. r and consisting in the admittance-deviation ^^ yr = Yr — Y of 

 the admittance Yr of this section from its nominal value Y. The 

 admittance-increment yr may evidently be regarded as situated any- 

 where within the section. However, for the present purpose it is most 

 conducive to simplicity of thought to regard yr as situated just beyond 

 the nominal mid-point of the section, namely the point which is at a 

 distance of half a normal, or "regular," section from the initial end 

 of the section ; for then it is immediately evident that the admittance 

 of the portion of the system beyond the nominal mid-point will deviate 

 from the mid-section iterative admittance 77 by an amount approxi- 

 mately '^ equal to yr, and hence that the corresponding reflection 

 coefficient fr pertaining to that mid-point will, in accordance with 

 (108), be given (approximately) by the formula 



f = yi = y-/^^ . (110) 



^' 2H + yr 1 + yr/2H ^' ""^ 



Due to the presence of the internal admittance-increment > in section 

 No. r, the admittance W of the whole system (Fig. 13) at its initial 

 end will deviate somewhat from the mid-section iterative admittance 

 77; the admittance-deviation 1^-77 will be denoted by y/, and the 

 corresponding reflection coefficient of the system will be denoted by 

 fr', so that, in accordance with (108), 



t'--^ = y-'^^^ . (Ill) 



^' - 277 + yr' 1 + yr'/2H ^ 



" Here r = 1, 2, • • • w — 1; for of course the nominal values of Yo and Y„ are 

 each F/2, and hence yo = Fo - Y/2 and yn = F„ - F/2. With these qualifications 

 dulv observ^ed, formula (110) is valid for r = and r = w as well as for r = 1, 2, 

 • ■ ■ n — \. As seen below, a'o is to be regarded as situated at the initial end of 

 section No. 0, and t„ at the far end of section No. n. 



" "Approximately," because yr is distributed; "exactly," if 3V were localized. 



