REFLEX OSCILLATORS 527 



the frequency of oscillation changes infinitely rapidly with load. Still 

 further change results in the formation of loops. Further change results in 

 expansion of loops so that they overlap, giving more than three intersections 

 with the electronic admittance line. 



Loops may exist for very low standing wave ratios if the line is sufficiently 

 long. Admittance plots for low standing wave ratio are very nearly cy- 

 cloidal in shape; those for higher standing wave ratios are similar to cycloids 

 in appearance but actually depart considerably from cycloids in exact form. 



By combining the expression for the near resonance admittance of a tuned 

 circuit with the transmission line equation for admittances, the expression 

 for these admittance curves is obtained. Assuming the termination to be 

 an admittance I'V which at frequency wo is do radians from the resonator, 



1 -\-j{Yt/Ml) tan 0o(l + Aco/wo) 



The critical relation of parameters for which a cusp is formed is important, 

 for it divides conditions for which oscillation is possible at one frequency 

 only and those for which oscillation is possible at two frequencies. This 

 cusp corresponds to a condition in which the rate of change with frequency 

 of admittance of the mismatched line is equal and opposite to that of the 

 circuit. This may be obtained by letting Yt be real. 



Yt/Ml > 1, do = nir where n is an integer. 



The standing wave ratio is then 



a = Yt/Ml . (9.26) 



The second term on the right of (9.25) is then 



\1 +_;o- tan ^oAco/coo/ 

 For very small values of Aco we see that very nearly 



72 = MlW - i(cr2 - l)0oAco/a'o] • (9.28) 



Thus for the rate of change of total admittance to be zero 



2Mh = Ml{c' - 1)60 



% = 2{Mj,/ML)(a' - 1) 



= 2Q^/{a' - 1) . (9.30) 



Thus, the condition for no loops, and hence, for a single oscillating frequency, 

 may be expressed 



00 < IQeHo" - 1) (9.31) 



