264 BELL SYSTEM TECHNICAL JOURNAL 



Dropping the subscript .v, the differential equation for the line in terms of 

 the impedance Z may be written^ 



i?^=y^(i?2-Z^) (2) 



ax V 



R = {L/Cf (3) 



V = (LC)-'" (4) 



R is the nominal characteristic impedance, and v is the nominal phase 

 velocity, which is constant for many tapered lines with the same dielectric 

 material separating the conductors throughout their length. In such lines, 

 if the dielectric is air or vacuum, v is c, the velocity of light. 



It should not be surprising that (2) is of the first order. Although there 

 are two boundary^ conditions, the impedances terminating the right and 

 left ends of the line, there are two impedances, that looking toward the right 

 and that looking toward the left. The impedance looking toward the right 



—fwm 



Zx 



C6.1C z^,^ 



x+ax 



Fig. 1 



is unaffected by the left end termination, and that looking toward the left 

 is unaffected by the right end termination. 



As R is real, it may be seen from (2) that the only case in which the im- 

 pedance can equal the nominal characteristic impedance R at all points 

 is for R constant. This tells us that the characteristic impedance of any 

 lossless tapered line is complex. For ver\' gradually tapering lines the 

 characteristic impedance differs from the nominal characteristic impedance 

 principally by a small imaginary component. 



The simplest solution of (2) is of course that for a uniform line, with R 

 a constant which will be called Ro. In this case (2) can be integrated 

 directlv, giving the familiar result 



~ = tanh (ju^x/v + K) (5) 



Ro 



^ It is interesting to note that the equation for admittance Y can be obtained by re- 

 placing Z by Y and R by (l/R) = G in (2). 



