522 ANTENNAS AND RF COMPONENTS 



used in the past to express system performance. Explicitly, db difference is 

 the difference between the oneway antenna relative power curves in decibels 

 at extreme positions of the conical scan cycle. While this term is less 

 frequently used now (per cent fundamental modulation more accurately 

 describes the tracking performance) the term db difference is often seen in 

 specifications. For the exponential beam approximation per cent modula- 

 tion equals 



Per cent modulation = °^^ X (db difference) X 100 per cent. (10-9) 



The above relation is derived for the nonsinusoidal waveform as the 

 difference between maximum and minimum amplitudes divided by the 

 average value. As the modulation is nearly sinusoidal for small target 

 error, the db difference is very nearly so related to the per cent fundamental 

 modulation in this region. 



To illustrate the utility of the Bessel approximation to the narrow-beam 

 pattern, the relation between directivity and beamwidth may be calculated. 

 It is necessary to restrict this to narrow beams to make the integration 

 practical, and as has been mentioned, because the beamwidth becomes 

 undefined for wide beams. From the definition of directivity and for 

 narrow beams 





].m 



(bdY 



(10-10) 



47r 



where |x is the power density at unit distance from a unit power source 

 radiating omnidirectionally. The proportionality constant K may be 

 determined by integrating^ the pattern over all space with a narrow-beam 

 approximation, giving 



where 9a and 0^ are beamwidths in degrees along the major and minor axes 

 of the beam. 



Further use of the Bessel representation is made in calculating tracking 

 performance out to the sidelobe region. The fundamental and second 

 harmonic modulation calculations are made as outlined in the reference of 

 footnote 6. The results may be shown as in Fig. 10-4. Here {Et) is target 

 error, which is normalized to antenna beamwidth (9). 



^Higher Transcendental Functions, Bateman Manuscript Project, California Institute of 

 Technology, Volume 2, p. 51. 



