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Explain amplitude modulation. Derive the voltage equation of an amplitude modulated wave. |
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Answer» Solution :Amplitude modulation In AM, the amplitude of the carrier WAVE is varied in accordance with the modulating signal, while frequency of the carrier wave remains constant. The given figure shows the principle of amplitude modulation. Fig `(a)` shows the audio electric signal, fig. `(b)` shows carrier wave of constant amplitude and fig. `(c )` shows the amplitude modulated wave.   Mathematical analysis Let the carrier `(c )` and modulating `(m)` waves be represented as `e_(c)=E_(c)sinomega_(c)t...(1)` `e_(m)=E_(m)sinomega_(m)t...(2)` From the definition of AM, the maximum amplitude `E_(C)` of the carrier will have to be made proportional to the instantaneous value of the modulating wave `E_(m)sinomega_(m)t.` This is shown in fig. `(a)`. `i.e.E(t)` or `(E_(c))_(AM)` `E_(c)+K_(a)e_(m)` `E(t)=E_(c)+K_(a)E_(m)sinomega_(m)t` `E(t)=E_(c)(1+(K_(a)E_(m))/(E_(c))sinomega_(m)t)...(3)` `K_(a)` is propotionality constant which determines the maximum variation in amplitude for a given signal voltage `E_(m)`.  The instantaneous voltage of the resulting AM wave is `e=E(t)sinomega_(c)t` `=E_(c)(1+m_(a)sinomega_(m)t)sinomega_(c)t,...(4)` where `m_(a)=(K_(a)E_(m))/(E_(c))...(5)` is called the modulation index or modulation factor or depth of modulation. The modulation index when multiplied by `100,` gives percentage of modulation. Eq.`(4)` can be expressed as `e=E_(c)sinomega_(c)t+(m_(a)E_(c))/(2)cos(omega_(c)-omega_(m))^(t)` `-(m_(a)E_(c))/(2)cos(omega_(c)+omega_(m))^(t),...(6)` where the trigonometrical realtion `sinAsinB=(1)/(2)[cos(A-B)-cos(A+B)]` has been used. Side bands. In eq.`(6),` we have three terms, one is the ORIGINAL carrier frequency, `omega_(c)`, and the other two represent  the same and the different of the carrier and modulation frequencies. The carrier envelope produced by amplitude modulation in shown in Fig. `(b)`. The sum  `omega_(c)+omeha_(m)` is the frequency of the upper side band (USB) and the difference `omega_(c)-omega_(m)` is the frequency of the LOWER side band (LSB). For modulation by complex wave forms of SPEECH or music, a great many such side frequency PAIRS will exist, one pair for each frequency component, these side frequency groups are called side bands. The band width required for transmission of an amplitude modulated radio- frequencies signal is equal to twice the highest ( of all the frequencies present ) modulating frequency Fig. `(c )` shows the frequency apectrum as is indicated by eq. `(6).` of these, the central frequency `i.e.,` the carrier, has the highest amplitude and the other two are disposed symmetrically about it, having amplitude which are equal to each other, but which can never exceed half the carrier amplitude. |
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