1410 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1957 



We shall now show that the zeroth approximation may be conveniently 

 ol)tained from the block diagram of Fig. 4. 



4.2 Description of the Block Diagram 



All the blocks of the block diagram are unilateral and their correspond- 

 ing transfer functions are defined in the following. Capital symbols repre- 

 sent ^-transform of the corresponding time functions, thus /o(p) is the 

 £-transform of in(/). 



Referring to Fig. 1, 



Z,,{v) = 



E,{v) 

 hip) 



lr=0 



Thus Znip) represents the transfer impedance of A^i when its output is 

 open-circuited (i.e., Ir = 0). Since A''! and N2 are identical we also have, 

 from /?i = 1 and reciprocity, Znip) = Ei/Ir, where Ir is the cur- 

 rent entering A? . 



The impulse modulator is periodically operated every T seconds, 

 and has the property that if its input is a continuous function f{t) its 

 output is a sequence of impulses: 



x;/(0 5(/ - i-T). 



The transfer function Sx{p) is defined by 



>Si(p) = £[si(0] = - 



Wo 



p" + coo 



cosh -? e-'"\ 



(14) 



Let Z{p) be the driving point impedance at the terminal pair (2) of A^ . 

 It is also that of Nt since Ai and A'2 are assumed to l)e identical. 



Let V{p) be the output of the first block, then, by definition, V{p) = 

 Zn{p)h ■ Let v{f} be the corresponding time function. The voltage v{t) 



Z,2 



v(p) 



IMPULSE 

 MODULATOR 



<^ 



-fO 



CS,(p) 



2Z(p) 



-12 



note: 



all blocks are unilateral 



Fig. 4 — Zeroth-approximation block diagram. 



