POTENTIAL ANALOGUE METHOD OF NETWORK SYNTHESIS 



357 



For the cable function we introduce polar coordinates, p = pe**', in the 

 />-plane, so that 



Fi{p) = kpi/2 e'V/2, 



Vi{p) = kp'^' cos y, ^iip) = kp'i^ sin J^, 



and it is easy to see that the equipotentials are parabolas in the />-plane, as 

 illustrated in Fig. 22a. Along the equipotential V i = k ■\/a the stream 

 function is 



^^Xp) = ± k\/p - a 



where the positive sign refers to that part of the parabola which lies above 

 the real />-axis and the negative sign to the part below the real />-axis. The 

 closure of the contour at infinity is shown in Fig. 22b. 



(b) 



Fig. 22 — The cable transmission function K y/p ; (a) equipotential contours are parab- 

 olas, (b) contour closed at infinity through a circular arc. 



If charge is placed on the equipotential in such a way that ^t/lir repre- 

 sents the integrated charge density, then the correct potential and stream 

 function will be produced everywhere to the right of the contour and the 

 potential to the left of the contour will be the constant k\^a. To keep the 

 contour from crossing the Im />-axis we must take a = 0. Then the parabola 

 degenerates into the negative real />-axis and charge is distributed with inte- 

 grated density function 



Q{p) = --V~P 



on the axis. 



The lumped charge approximation consists of placing zeros at points 

 pn = —Pn where Q(p„) = w — J; i.e. zeros are to be placed at 



pn = - (n - 2)^^, 



n =, 1, 2, 3 



