Answer to Question #90178 in Electrical Engineering for Shivam Nishad

Question #90178
Derive mathematical expression of an A.M. wave and indicate the sideband components. Why is the S.S.B. transmission beneficial ?
1
Expert's answer
2019-06-10T04:56:39-0400

Modulation is a technique which modifies waves of high frequency by the wave of low frequency.

Consider the carrier signal:


Vc(t)=Vcsin(ωct),V_c(t)=V_c\text{sin}(\omega_ct),

the modulating signal:


Vm(t)=Vmsin(ωmt),V_m(t)=V_m\text{sin}(\omega_mt),

and the signal after modulation look like


V(t)=[Vc+Vm(t)]sin(ωct).V(t)=[V_c+V_m(t)]\text{sin}(\omega_ct).

Putting this all together gives


V(t)=Vcsin(ωct)+Vm2cos(ωctωmt)Vm2cos(ωct+ωmt).V(t)=V_c\text{sin}(\omega_ct)+\frac{V_m}{2}\text{cos}(\omega_ct-\omega_mt)-\frac{V_m}{2}\text{cos}(\omega_ct+\omega_mt).

Now multiply the terms with cosines by Vc/VcV_c/V_c:


V(t)=Vcsin(ωct)+V(t)=V_c\text{sin}(\omega_ct)+

+Vm2VcVccos(ωctωmt)Vm2VcVccos(ωct+ωmt),+\frac{V_m}{2}\frac{V_c}{V_c}\text{cos}(\omega_ct-\omega_mt)-\frac{V_m}{2}\frac{V_c}{V_c}\text{cos}(\omega_ct+\omega_mt),

and substitute Vm/VcV_m/V_c with mam_a - the amplitude modulation index.


V(t)=Vcsin(ωct)+Vcma2cos(ωctωmt)Vcma2cos(ωct+ωmt).V(t)=V_c\text{sin}(\omega_ct)+\frac{V_cm_a}{2}\text{cos}(\omega_ct-\omega_mt)-\frac{V_cm_a}{2}\text{cos}(\omega_ct+\omega_mt).

The sideband components are


ωcωm2π=fcfm\frac{\omega_c-\omega_m}{2\pi}=f_c-f_m

and

ωc+ωm2π=fc+fm.\frac{\omega_c+\omega_m}{2\pi}=f_c+f_m.

The single-sideband transmission reduces the bandwidth transmitted and it allows to double the number of channels within the same frequency band.


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