Rca1948FrequencyModu.. - The New Jersey Antique Radio Club

448 FREQUENCY MODULATION, Volume Iquencies willrotate counterclockwise with angular frequencies of nv,and the vectors with the lower angular frequencies willrotate clockwisewith the same angular frequencies nv. This is pictured in Figure1 (a) for the case where m f= 1.0 and vt = 30°. It is readily seen fromthis figure (as well as from (2) by subtracting m from each angularfrequency) that the even-order sidebands pair off to give a vector suminphase with the carrier vector and the odd-order sidebands pair offto give a vector sum in phase-quadrature with the carrier vector.Thus it is seen either from Figure 1 (a) or from (2) thatJ„ ( I ) = . 7fc5 (a)e-sJj(i)(b)Fig. 1—Vector summation of (a) input signal, and (b) output signal foran arbitrary network (assuming modulation index = 1.0, vt = 30°).e,„ —lJ (m f ) + 2J 2 (m f ) cos 2vt + 2/ 4 (m / ) cos 4vt +•• •+ [2J 1 (ra r ) sin vt + 2J 3 (m f ) sin 3vt + •] 2 (3)] 2and(3 — phase of e lnwith respect to signal of phase oi t = m fsin vt = arc tan[2/j (m f) sin vt + 2/ 3(m f ) sin 3vt +•••][/ (m / ) + 2J 2 (m f ) cos 2vt + 2J i(m f ) cos 4vt + ]The network output voltage is obtained, as indicated by Frantz, bytreating each of the sideband terms as an individual signal and alteringits amplitude by R n (a>) and its phase by 8 n (o>). Thus, the outputvoltage derived from (2)is(4)