744 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1953 



or in more convenient fonn as, 



where 





The above expression contains four variables, namely F/G, (Qk), {2Q5) 

 and p, so that the complex admittance plane representation will now 

 have to be restricted to particular values of p. One such plot, for p = 

 J^, appears in Fig. 11. It is similar to that of Fig. 7 except that only the 

 positive half has been shown since equation (3.12) is symmetrical about 

 the conductance axis; in addition, the parameter, (Q/c)^, has been carried 

 to the point of critical coupling only. i.e.. the formation of a cusp, and 

 not beyond. The frequency contours are again seen to be straight Unes 

 crossing the horizontal axis at a value of conductance equal to the in- 

 put conductance for the condition of critical coupling and 2Q8 = 0. 

 Equation (3.12) shows that the susceptance term will be zero for 



(3.13) 



The value of conductance corresponding to condition (a) is given by 

 g = 1 -\- piQhf and the value corresponding to condition (b) by gr = 

 1 + 1 /p. It is interesting to note that this latter value which determines 

 the point of intersection of the frequency contours, as well as of the 

 admittance plot for critical coupling, with the conductance axis, is 

 independent of the actual values of Q and the degree of coupling and 

 only dependent upon p, the ratio of Q's. Thus in Fig. 11, which con- 

 stitutes a plot for p = 0.5, the value of conductance at whieh all the 

 above named contours meet is given hy g = 1 + 1/0.5 = 3. 



From what has been said before we know that at critical coupling the 

 admittance locus forms a cusp intersecting the conductance axis at 

 ^ = 1 + \/p at the frequency, 2Qb = 0. Substituting this value of 

 2Q6 into the conductance term of equation 3.12 and equating to 1 + 1/p 

 yields 



1 + pmy = 1 + -- . 

 p 



