VARIABLE FREQUENCY ELECTRIC CIRCUIT THEORY 521 

 thus defining the remainder Rn. Then (29) becomes 



I == Eexpli C ^t + id) 



. . Ci d . Cn d"" 



1 "T TT 3 [-••■ + 



1! doic n\ dcoc"' 



Yiio^e) 



+ E exp (iuct + id) C i?„(coc, w)F(i'co)g»"'^co. (31) 



•J — 00 



In practice it is usually desirable to take « = 1. 

 Now the infinite integral 



D(t) = r* Rniooc, co)F(iw)e»"'(/aj (32) 



must be kept small if the finite series in (31) is to be an accurate repre- 

 sentation of the current /. While it is not in general computable, we 

 see that, in order to keep it small, Rn(o:c, cj) must be small over the 

 essential range of frequencies of F(io}). In cases of practical im- 

 portance we shall find (see Appendix 1) this range is from co = — X 

 to o) = + X. 



If the transducer introduces a large phase shift, the linear part of 

 which is predominant in the neighborhood of to = ooc, it is preferable 

 to express the received current / in terms of a "retarded" time. To 

 do this, return to (25) and write 



F(icoc + ico) = I Y{icoc + ic^)\e-^t>, (33) 



(j) = ojcT + wr + /3(co) + dc\ 



^(0) = ^'(0) = 0, 

 so that 



/ = E exp {ioicf + id') r I Y{ioic + ^"co) |e-^^(")E(*"w)e''"'Vw, (34) 



1/ — 00 



where /' = / — r is the "retarded" time and 6' = d — dc. Formula 

 (34) is identical with (25) but is expressed in the "retarded" time. 

 Now we can expand the function 



I F(t'co, + ^)|e-^^("> 

 in powers of w ; thus 



where 



(^ 1 + co^J I F(ic..)| + E r„(co.)co", 



rnM =-,\^\ Y{io:c + ic^) 1 e-^^^-) I 

 nl [ dcoc J u 



=0 



