736 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1953 



which may be readily derived from eq. 3.6. Note that for {Qkf = 1, 

 the impedance level at midband has dropped to half the value obtained 

 mth a single resonator or (Qkf = 0. The degree of coupling corre- 

 sponding to (QkY = 1 is noteworthy for reasons other than the one just 

 mentioned. They will be discussed in the following section. 



3.1.2 Input Admittance Plot in g-b Plane j Fig. 7. The graphical repre- 

 sentation of input admittance in the complex admittance plane is of 

 particular usefulness in the analysis of the coupled resonator reflex 

 klystron. For the moment, however, we shall restrict the discussion to 

 a consideration of the passive circuit only. 



Each solid line in Fig. 7 is the locus of the admittance vector for a 

 particular tightness of coupling. For {Qkf = the locus is a straight 

 Une parallel to the susceptance axis as described earlier for the case of a 

 single resonator. For (Qk) =0.3 the shape of the locus approximates 

 a circle with its center at the origin; this, of course, being true only over 

 a restricted frequency range. As the coupling to the secondary resonator 

 is progressively tightened, the locus is seen to bulge out in the direction 

 of increasing conductance until, for (Qk) = 1, it forms a cusp. This 

 condition will henceforth be referred to as ''critical coupling."* Coup- 

 ling even tighter causes the formation of loops of increasing size. 



For the overcoupled case, (Qk) > 1, the admittance-vector locus 

 crosses the conductance axis three times, with the first and third crossings 

 coincident and independent of (Qk)^ and the second crossover a function 

 of (Qk) . The location of these intersections with the ^-axis may be 

 determined by equating the susceptance to zero, i.e., from equation 

 (3.5), 



^'['-r^J- 



Hence the crossing to the extreme right occurs for 2Q8 = while the 

 first and third interesections correspond to 



2Qd = ±ViQky - 1. 



To determine the size of the loop we substitute 2Q8 = into the expres- 

 sion for the conductance, i.e., equation (3.4) and obtain, 



oL.^ = 1 + my, 



'2Qi-^ 



* It should be noted that the term "critical coupling" as applied to the transfer 

 characteristics of coupled tuned circuits, though also occuring for (Qk)^ = 1, 

 assumes a different significance in that it describes the condition of maximum 

 flatness in response and optimum phase linearity. (See reference 6.) In the case of 

 the coupled resonator reflex klvstron, "critical coupling" forms the transition 

 between stable performance and load hysteresis as will be shown later. 



