1414 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1957 



By transmission at dc we mean the ratio of the dc component of the 

 steady state output voltage to the intensity of the applied direct current. 

 Thus we refer to (26) and set coo = and n = 0. Suppose the lossless 

 networks Ni and A^2 are designed so that their transfer impedance Zu 

 is of the Butter worth type, that is 



I Zuiju) p = 



1 + C02^' 



where for our purposes AI is a large integer. 



In the following sum, which is the denominator of (26) when coo = 

 n = 



2 E Z{jkc,.)So{jko::), 



k=—<x 



(where co,. > 2 since the cutoff of the networks N occurs at w = 1), the 

 terms corresponding to values of /c f^ will make a contribution that 

 vanishes as M -^ x . This is a consequence of the following facts : 



(a) Re[Z(jA-coJ] = | Zuijkuis) \ , since the networks A^i and A^2 are 

 dissipationless. Hence for k ^ and as M ^ ^o Re[Z(yAws)] -^ 0, 



(b) lm[Z{jko:S\ = -Im[Z(-iA-a;.)], 



(c) So{jo}) is real. 



Thus the imaginary part of the products Z{jkus)So(jkws) cancel out and 

 the real part (for k 7^ 0) decreases exponentially to zero as ilf -^ oc . 

 Hence for sufficiently large M the denominator of (26) may be made as 

 close to two as desired. 



It is easy to check that the numerator of (26) reduces to 7o , the in- 

 tensity of the applied direct current. Therefore the ratio of A^ , the dc 

 component of the output voltage to /o may be made as close to one-half 

 as desired. 



VI. A SIMPLE EXAMPLE 



Since the approximate formulae derived in Section IV are somewhat 

 unfamiliar it seems proper to consider in a rather detailed manner a 

 simple example.* 



Consider the system of Fig. 5. Assume that the current source applies 

 a constant current to the system and assume that the steady state is 

 reached. For simplicity let hR = E. 



The steady-state behavior of the voltages ^^2(0 and e:i{t) = €4(1) is 



* In addition, the limiting case of the sampling rate — > <», i.e., T" — > 0, is treated 

 in Appendi.x II. 



