STUDIES IN RADIO BROADCAST TRANSMISSION Ml 



attenuation or variations in length of path. The relative phase must 

 be determined from equations (6a) and (6b). Knowing the fre- 

 quency variation with time we may by integrating the following 

 equation determine the phase relation at any time (t). 



ei= r2TrF4t, (7) 



02= r2TtF2dt. (8) 



Substituting the general relation for Fi and F^ from equations 

 (6a) and (6b) we have, 



Q,^rFo+fs\nr{t-d/V), (9) 



Q^=£Fo+Jsmr{t-d'/V). (10) 



Evidently the relative phase (AG) will be the difference between 

 these two giving, 



Ae =61-62 = 27r rFodt + 2w / /sin r {t-d/V)dt (11) 



Jo Jo 



-27r CFo dt-2ir f'f sin r {t-d'/V) (12) 



Jo Jo 



which integrated reduces to the form, 



Ae = ^ (cos rt-l) (cos r d' / V- cos r d/ F+ 

 P 



^mrt {s\nrd'/V-s\n rd/V). (13) 



The equation is not in itself very illuminating, but what it tells us 

 generally is that if we represent two frequency modulated waves 

 travelling over paths of different lengths to a distant receiver by 

 rotating vectors, these vectors are constantly shifting their relative 

 position. The magnitude of the shift at any instant is given by the 

 varying angle A6. Due to a change in the angle included by the two 

 vectors their resultant will undergo an amplitude change, the serious- 

 ness of which we will consider later. 



Thus far in the discussion of frequency modulation by means of a 

 rotating condenser we have assumed sinusoidal changes in frequency. 

 The ordinary condenser departs considerably from such a perform- 

 ance. By considering the application of the integral equation for 

 A6 to such a case it will be recognized that the relative space posi- 



