In an amplitude modulator circuit, the carrier wave is given by, C(t) = 4 sin (20000πt) while modulating signal is given by, m(t) = 2 sin (2000πt). The values of modulation index and lower side band frequency are:

Concept:

Modulation index is a measure of modulation done in extent on carrier signal.

M is the amplitude of modulating signal

C is the Carrier wave

The frequency of carrier wave

From question, the carrier wave is given as:

c(t) = 4 sin (20000πt)

The general formula of the carrier wave in since wave form is:

c(t) = C(ωc + φ)

Where:

‘Carrier frequency’ in Hertz is equal to 

‘C’ is the carrier amplitude

‘φ’ is the phase of the signal at the start of the reference time

Calculation:

Both ‘C’ and ‘φ’ can be omitted to simplify the equation by changing C to ‘1’ and φ to ‘0’.

So, the given carrier wave is:

c(t) = 4 sin (20000π(1) + 0)

From question, modulation signal/ messenger wave is given as:

m(t) = 2 sin (2000πt)

The general formula of the messenger wave in since wave form is:

m(t) = M sin (ωm + φ)

Where:

‘Modulating signal frequency’ in Hertz is equal to 

‘M’ is the modulation amplitude

‘φ’ is the phase of the signal at the start of the reference time

Both ‘C’ and ‘φ’ can be omitted to simplify the equation by changing C to ‘1’ and φ to ‘0’.

So, the given modulation wave is:

m(t) = 2sin (2000π(1) + 0)

Now, the modulation index ‘m’ is given by the formula:

On substituting the values,

∴ m = 0.5

The lower side band frequency are lower frequency range of the given signal.

The frequency of carrier wave is:

On substituting the values,

∴ f_c = 10000 Hz

The frequency of modulation wave is:

On substituting the values,

∴ f_m = 1000 Hz

The lower band frequency is:

⇒ f_c - f _m = 10000 - 1000

∴ f_c - f _m = 9000 Hz = 9 kHz