POTENTIAL ANALOGUE METHOD OF NETWORK SYNTHESIS 



353 



In band-pass filters, whose positive frequencies of interest extend between 

 two finite values, coo < oj < coi , we must be able to use a contour of the type 

 shown in Fig. 20a. This consists of two disjoint closed curves, one above and 

 one below the real axis (real p). The physical requirements are satisfied 

 if the curves are symmetric about the real />-axis, but as usual it is advan- 

 tageous to make them symmetric also about the real co-axis. For then, if a 

 point py lies on one of the curves, the point — pv will lie on the other. This 

 makes it possible to map the disjoint contour C on a single closed curve C2 

 in the ^^-plane, the ^'-plane of Fig. 20b, by means of the transformation 

 P = ^/y- The single contour C2 may now be mapped on the unit circle in 



(a") p PLANE (b) y PLANE = p^ PLANE (C) w PLANE 



Fig. 20— Contours for a band-pass filter; (a) disjoint contour symmetric about both 

 axes in ^-plane, (b) single contour in /)2-plane, (c) unit circle in tt'-plane. 



the w-plane. Fig. 20c, by means of a second transformation y = ri(w). 

 Combining the transformations we have 



p = VFiW 



(99) 



P = TiW, 



as the transformation which maps C on Ci . 



The conditions on the function Ti(w) are the same as in (63) except that, 

 since C2 is in the left half of the y-plane and does not cut the positive real 

 axis, the fourth condition must be replaced by a similar requirement. 



ri(+ 1) — ri(— 1) is real and positive. 



Now the presence of the square root in the transformation (99) may 

 introduce branch points in the w-plane corresponding to the branch points 



