57 Chapter 4: Agle Modulatio 4.1 Itrodutio to Agle Modulatio This hapter desribes frequey odulatio (FM) ad phase odulatio (PM), whih are both fors of agle odulatio. Agle odulatio has several advatages over AM, suh as oise redutio, iproved syste fidelity ad ore effiiet use of power. However, agle odulatio also has several disadvatages whe opared to AM, iludig requirig a wider badwidth ad utilizig ore oplex iruits. Today, agle odulatio is used extesively for oerial radio broadastig, televisio soud trasissio, ellular radio, ad irowave ad satellite ouiatios systes Agle odulatio results wheever the phase agle, θ of a siusoidal wave is varied with respet to tie ad a be expressed as where (t) = agle-odulated wave V = peak arrier aplitude ω = arrier radia frequey θ (t) = istataeous phase deviatio ( t) = V os[ ω t + θ ( t)] (4.1) where θ (t) is a futio of the odulatig sigal give by: θ ( t) = F[ V si( ω t)] (4.) where ω V = odulatig sigal radia frequey = peak aplitude of the odulatig sigal The differee betwee FM ad PM lies i whih property of the arrier is diretly varied by the odulatig sigal ad whih property is idiretly varied If the frequey of the arrier is varied diretly by the odulatig sigal, FM results If the phase of the arrier is varied diretly by the odulatig sigal, PM results I.e. FM ad PM a be defied as follows: Diret FM is the odulatio proess i whih the frequey of a ostat aplitude arrier is varied diretly proportioal to the aplitude of the odulatig sigal at a rate equal to the frequey of odulatig sigal Diret PM is the odulatio proess i whih the phase of a ostat aplitude arrier is varied diretly proportioal to the aplitude of the odulatig sigal at a rate equal to the frequey of odulatig sigal BENT 3113: Couiatio Priiples
58 4.1.1 Agle Modulatio Represetatio i Frequey ad Tie Doai Figure 4.1 shows a agle-odulated sigal i the frequey doai: Figure 4.1: Agle-odulated sigal i frequey doai The arrier frequey, f is haged whe ated o by the odulatig sigal The agitude ad diretio of the frequey deviatio, is proportioal to the aplitude ad polarity of the odulatig sigal Figure 4. shows a agle-odulated sigal i tie doai: Figure 4.: Agle odulatio i tie doai: (a) Phase hagig with tie; (b) frequey hagig with tie BENT 3113: Couiatio Priiples
59 Figure 4.(a) shows the phase of the arrier is hagig proportioal to the aplitude of the odulatig sigal The phase shift is alled phase deviatio θ. This shift also produes a orrespodig hage i frequey, kow as frequey deviatio f Peak-to-peak frequey deviatio is deteried by: (Figure 4.(b)) p p = 1/ T T i 1/ ax 4. Matheatial Aalysis It is ot apparet fro Equatio (4.1) whether a FM or PM wave is represeted. To differetiate betwee FM ad PM, the followig four ters first eed to be defied ad uderstood 1. Istataeous phase deviatio The istataeous hage i the phase of the arrier at a give istat of tie Istataeous phase deviatio = θ (t) rad (4.3). Istataeous phase The preise phase of the arrier at a give istat of tie Istataeous phase = ω t + θ (t) rad (4.4) 3. Istataeous frequey deviatio The istataeous hage i the frequey of the arrier ad is defied as the first tie derivative of the istataeous phase deviatio Istataeous frequey deviatio = θ ( t) rad/s (4.5) 4. Istataeous frequey The preise frequey of the arrier at a give istat of tie ad is defied as the first tie derivative of the istataeous phase Istataeous frequey = ω = ω + θ ( t) rad/s (4.6) Fro these 4 ters, PM ad FM a the be defied as PM is a agle odulatio i whih θ (t) is proportioal to the aplitude of the odulatig sigal FM is a agle odulatio i whih θ ( t) is proportioal to the aplitude of the odulatig sigal For a odulatig sigal v (t), θ ( t) = Kv ( t) rad (4.7) i θ t) = K v ( t) rad/s (4.8) ( 1 where K ad K1 are ostats ad are the deviatio sesitivities of the phase ad frequey odulators, respetively Substitutig a odulatig sigal v ( t) = V os( ωt), Equatios (4.7) ad (4.8) ito Equatio (4.1) yields BENT 3113: Couiatio Priiples
60 PM: ( t) = V os[ ω t + θ ( t)] ( t) = V os[ ω t + KV os( ω t)] (4.9) FM: As θ ( t) = θ ( t) ( t) = V os[ ω t θ ( t) ] t) = V os[ ω t + K V os( ω t) dt] + ( 1 K1V ( t) = V os[ ω t + si( ω t)] (4.10) ω [Refer Table 7.1, pp 59 of the textbook for further uderstadig] 4.3 FM ad PM Wavefors Figure 4.3 illustrates both FM ad PM of a siusoidal arrier by a sigle-frequey odulatig sigal Figure 4.3: (a) Uodulated arrier; (b) odulatig sigal; () frequey-odulated wave; (d) phase-odulated wave FM ad PM wavefors are idetial exept for their tie relatioship For FM, the axiu frequey deviatio ours durig the axiu positive ad egative peaks of the odulatig sigal For PM, the axiu frequey deviatio ours durig the zero rossigs of the odulatig sigal (i.e. the frequey deviatio is proportioal to the slope or first derivative of the odulatig sigal) BENT 3113: Couiatio Priiples
61 4.4 Modulatio Idex ad Peret Modulatio Coparig Equatios (4.9) ad (4.10), Equatio (4.1) a be rewritte i geeral for as where is alled the odulatio idex 4.4.1 PM ( t) = V os[ ω t + os( ω t)] (4.11) For PM, the odulatio idex is also kow as peak phase deviatio θ ad is proportioal to the aplitude of the odulatig sigal ad is expressed as where = K = V = = θ = KV (radias) (4.1) odulatio idex deviatio sesitivity (radias/volt) peak odulatig sigal aplitude (volt) Therefore, for PM: ( t) = V os[ ω t + KV os( ω t)] 4.4. FM ( t) = V os[ ω t + θ os( ω t)] ( t) = V os[ ω t + os( ω t)] For FM, the odulatio idex is diretly proportioal to the aplitude of the odulatig sigal ad iversely proportioal to the frequey of the odulatig sigal K V 1 1 = = (uitless) (4.13) ω K V f where K 1 = deviatio sesitivity (radias/seod per volt or yles/seod per volt) V = peak odulatig sigal aplitude (volt) ω = radia frequey (radias/seod) f = yli frequey (yles/seod or hertz) Also for FM, the peak frequey deviatio is siply the produt of the deviatio sesitivity ad the peak odulatig sigal voltage. I.e. = K1 V = (uitless) (4.14) f BENT 3113: Couiatio Priiples
6 Therefore, for FM, Equatio (4.10) a also be rewritte as K1V ( t) = V os[ ω t + si( ω t)] f ( t) = V os[ ωt + si( ωt)] f ( t) = V os[ ω t + si( ω t)] [Refer Figure 7.4 ad Table 7. pp 6-63 for further explaatio] 4.4.3 Peret Modulatio Peret odulatio for agle odulatio is deteried i differet aer tha it was with a aplitude odulatio. With agle odulatio, peret odulatio is the ratio of frequey deviatio atually produed to the axiu frequey deviatio allowed, stated i peret for ( atual ) Peret odulatio = 100 (4.15) 4.5 Frequey ad Badwidth Aalysis of Agle-Modulated Waves Frequey aalysis of agle-odulated wave is uh ore oplex opared to the aplitude odulatio aalysis. I phase or frequey odulator, a odulatig sigal produes a ifiite uber of side frequeies pairs (i.e. it has ifiite badwidth), where eah side frequey is displaed fro the arrier by a itegral ultiple of the odulatig frequey 4.5.1 Bessel Futio (ax) Fro Equatio (4.11), the agle-odulated wave is writte as ( t) = V os[ ω t + os( ω t)] Based o above equatio, it is ot obvious the idividual frequey opoets of the agle odulated wave Bessel futio idetities a be used to deterie the side frequeies opoets π os( α + os β ) = J ( ) os( α + β + ) (4.16) = J () is the Bessel futio of the first kid. BENT 3113: Couiatio Priiples
63 Applyig Equatio (4.16) to Equatio (4.11) yields π ( t) = V J ( ) os( ω t + ωt + ) (4.17) = Expadig Equatio (4.17) ( t) = V { J 0 J ( ) os ( ) os( ω t) + J 1 π ( ) os[( ω + ω ) t + ] J π ( ) os[( ω ω ) t ] + [( ω + ω ) t] + J ( ) os[ ( ω ω ) t] +... J ( )...} 1 where (t) = agle-odulated wave = odulatio idex V = peak arrier aplitude J ( ) = arrier opoet 0 J 1( ) = first set of side frequeies displaed fro arrier by ω J ( ) = seod set of side frequeies displaed fro arrier by ω J () = th set of side frequeies displaed fro arrier by ω I other words, agle odulatio produes ifiite uber of sidebad, alled first-order sidebads, seod-order sidebads, ad so o. Also their agitude are deteried by the oeffiiets J ( ), J ( ) ad so o 1 Refer Table 7.3, page 66 of the textbook for the Bessel futio of the first kid for several values of odulatio idex Fro the table, several poits a be dedued: 1. Modulatio idex of 0 produes zero side frequeies. The larger the, the ore sets of side frequeies are produed 3. Values show for J are relative to the aplitude of the uodulated arrier 4. As the dereases below uity, the aplitude of the higher-order side frequeies rapidly beoes isigifiat 5. As the ireases fro 0, the agitude of the arrier J ( ) dereases 6. The egative values for J siply idiate the relative phase of that side frequey set 7. A side frequey is ot osidered sigifiat uless its aplitude is equal or greater tha 1% of the uodulated arrier aplitude ( J 0.01) 8. As ireases, the uber of sigifiat side frequeies ireases. I.e. the badwidth of a agle-odulated wave is a futio of the odulatio idex 0 BENT 3113: Couiatio Priiples
64 4.5. Badwidth Requireet Fro previous setio, it a be see that agle-odulated wave osues larger badwidth tha a aplitude-odulated wave ad the badwidth of a agle-odulated wave is a futio of the odulatig sigal ad the odulatio idex The atual badwidth required to pass all the sigifiat sidebads for a agleodulated wave is equal to two ties the produt of the highest odulatig sigal frequey ad the uber of sigifiat sidebads deteried fro the table of Bessel futio I.e. the iiu badwidth for agle-odulated wave usig the Bessel table: Carso s Rule B = ( f ) Hz (4.18) It is a geeral rule to estiate the badwidth for all agle-odulated systes regardless of the odulatio idex It states that the badwidth eessary to trasit a agle-odulated wave as twie the su of the peak frequey deviatio ad the highest odulatig sigal frequey B = ( + f ) Hz (4.19) For very low odulatio idies ( f is uh larger tha f B = f Hz ) For very high odulatio idies ( is uh larger tha f ) B = Hz Carso s rule is a approxiatio ad gives arrower badwidth tha the badwidth deteried usig Bessel table. Therefore, a syste desiged usig Carso s rule would have a arrower badwidth but a poorer perforae tha a syste desiged usig Bessel table. For odulatio idexes above 5, Carso s rule is a lose approxiatio to the atual badwidth required BENT 3113: Couiatio Priiples