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THE BELL SYSTEM TECHNICAL JOURNAL, JULY 195G 



1, 0, then the two-port is potentially unstable since on one side of the 

 intersection both input and output terminations receive power from the 

 two-port. The change in Pi from the minimum value found on the unit 

 circle centered at 1, to Pio divided by Pjo is the criticalness factor, C. 

 A^alues of C greater than unity indicate potential instability. 

 The power input at 1, is 



-PiO 



2hnrh22r — Re{hnh2i) 



2h 



11r 



Using (14) and (20), one obtains 



00 



/i21 



Pio 4/iiir/i22r — 2Re{hi2h2i) 



hnfh 



12«21 



C 



2/l2 



'2hiirfl22r — Ke{hi2'l21J 



= 2 



00 



PiO 



hu 

 h2i 



(20) 



(21) 



(22) 



2h 



22r 



Now if the plane of power input, Fig. G, is parallel to the L-M plane 

 and above it, certainly the point of maximum power gain is the apex of 

 the paraboloid, 1, in Fig. 5. If the plane is incliued but alwaj's above 

 the unit circle centered at 1, certainly the point of maximum power 

 gain is downward along the gradient line which lies above the point 1, 0. 

 This must be so since for any contour of equal power out (a circle of 

 fixed elevation around the paraboloid) the minimum power input (or 

 greatest gain) lies along the line of steepest descent from 1, in Fig. 6. 

 Thus the problem of evaluation of the maximum available gain reduces 

 to the simple problem of finding the abscissa of Fig. 7 where the ratio of 

 ordinates of the parabola and straight line is a maximum. The parabola 



Pl or Po 



PROJECTION IN L-M 



PLANE OF GRADIENT 



LINE OF PLANE 



THROUGH 1,0 



Fig. 7 — Section of paraboloid and inclined plane of Figs. 5 and 6. 



