Derivation of the expreesion for amplitude modulation signal

The main Element of Modulation input and output signal:
(i) Modulating Signal
(ii) Modulated Signal
(iii) Carrier Signal

We Know That,
ec= Ec cos wCt
ec= Carrier voltage
Ec= Carrier Amplitude
Wc= 2Пfc ( Angular Velocity at carrier frequency fc)

es= Es cos wst
es= Carrier voltage
Es= Carrier Amplitude
Ws= 2Пfs
Modulation Factor= m,
Es= mEc

So, Amplitude of AM wave
= Ec+es
= Ec+Es cos wst
= Ec(1+cos wst)

Current Voltage of AM Wave,
= Amplitude X Cos wct
= Ec(1+cos wst) X cos wct
= Ec cos wct+ Ec cos wct cos wst
= Ec cos wct + mEc/2{2 cos wct cos wst}
= Ec cos wct + mEc/2 { cos (wc+ws)t+cos (wc-ws)t }
= Ec cos wct + mEc/2 cos (wc+ws)t + mEc/2 cos (wc-ws)t
c = Ec cos 2Пfct+ mEc/2 cos 2П (fc+fs)t+ mEc/2 cos 22П (fc-fs)t

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