Rca1948FrequencyModu.. - The New Jersey Antique Radio Club

_TRANSMISSION THROUGH A NETWORK 449 — v)t + 6>_i}] + J 2 (m>) [R 2 sin {(a> + 2v) t + 9 2 )+ JB_ 2 sin{(a> -2i;)t + fl_ 2 }] +••. (5)If (5) is pictured on the counterclockwise rotating reference frame, itis seen (as indicated in Figure 1(b) for same conditions as 1(a) withhypothetical network characteristics) that{R J (m f ) cos 8 + Jiiwif) [R 1cos (vt + 8 X ) — i2_ xcos (vt — dx ) ]+ J 2 (m t ) [R 2 cos (2vt + 6. 2 ) + #_ 2 cos (v — d_ )'\ 2 H }-+ {R J (m f ) sin 8 + J 1(m f ) [R tsin (vt + 6-s) + R_ tsin (vt — 0^ ]+ J 2 (m f ) [R 2 sin (2v£ + 6 2 )- R_ 2 sin (2vt - 0_ 2 ) ]•+• •Yand(6)jBphase of e out with respect to signal of phase a t = arc tanR J (m f ) sin 6 + J 1(m f ) [R lsm (vt + 6^ +R_ 1sm (vt — 8_ 1 )] -^ 1R J Q(m f) cosd + J 1(m f ) [J^cos (vt + 6-s)— -B_ 1cos (vt — 8_ 1 ) ] +•i„-j(7)Using (6) and (7) the point-by-point amplitude and phase of theoutput voltage as a function of vt can be obtained either by algebraicor graphical calculation. Sufficient terms are retained to give, thedesired accuracy with more terms being necessary when m fis large.If the series indicated in (6) can be summed up analytically, then aclosed analytic expression for the output is possible.If the network-gain characteristic is symmetrical so that R_ n= R n ,and the network phase characteristic is skew symmetrical so thatB_ n= — 6 n (which is the case treated in detail by Frantz) then (6)and (7) simplify to{R J (m f ) + 2R 2J 2(m f) cos (2vt + 6 2 ) + 2RJ 4 (m f) cos (Avt + 4 )+••Y+ {2R 1J 1(m f) sin (vt + 9 X ) + 2R RJ 3(m f) sin (3vt + 3 ) H } 2and (8)/? = arc tan[2R 1 J 1(m f) sin (vt + 6 X ) + 2R ?>J. i(m f) sin (Svt + 6 3 )•+]"R J (m f ) +2R,J.,(m f) cos (2vt + &