Rca1948FrequencyModu.. - The New Jersey Antique Radio Club

492 FREQUENCY MODULATION, Volume ISince the denominators of (75) and (76) are real, the real andimaginary parts are determined by the scalar productJ ln (r):A„*(r)in the integrals in the numerators.terms of E n (r), which is real, isThis scalar product, expressed inJ 0n^ K„, r) • AttKir)* (r) = /„„«' ( Mon , r) e-w*.(78)Consequently, for the optimum real part of this expression and thereforethe optimum frequency shift,lag the electric field intensity by 90 degrees.the electron current should lead orFor maximum decrease infrequency (75) shows that the electron current must lead E n by 90degrees, while for maximum increase in frequency the electron currentshould lag E n by 90 degrees. This is analogous to the situation incircuit theory.Figures 10(b)The vector diagrams for these two cases are shown inand 10(c), respectively.Formulas (76) and (78) show that optimum loading takes placewhen the current is in phase with E n. This is the case when energy istransferred from the electro-magnetic field to the electrons. If thecurrent were 180 degrees out of phase with E,„ energy would betransferred from the electrons to the field. In this case one wouldhave a generator and a negative Q.If the time phase of the current is a function of position in thecavity, as it usually will be, the net frequency shift is, of course, givenby the integral.In addition to the optimum time-phase relationships, (75) and (76)show that the greatest frequency change or loading for a given currentoccurs when that current is parallel to the electric field and located inthat region of the cavity where the electric field is the largest. Thisis the region of largest interaction of field and electi'ons.Calculation of CurrentThe frequency shift and loading will be computed for a rectangularcavity for a special way of introducing the electrons. We shall assumethat before the electrons are introduced the mode of oscillation is suchthat the electric field is parallel to the x direction only, as shown inFigure 11. The electric field for this mode is given by