# Derive the Power Relation for Single Tone AM Wave and Multiple-tone AM Wave

## Definition

It may be observed from the expression of AM wave that the carrier component of the amplitude modulated wave has the same amplitude as unmodulated carrier.

In addition to carrier component , the modulated wave consists of two sideband components. It means that the modulated wave contains more power than the unmodulated carrier.

However, since the amplitudes of two sidebands depend upon the modulation index, it may be anticipated that the total power of the amplitude modulated wave would depend upon the modulation index also. In this section, we shall find the power contents of the carrier and the sidebands.

As we know the general expression of AM wave is given as :

s(t) = A cos ω_{c}t + x(t) cos ω_{c}t ………………..(1)

The total power P of the AM wave is the sum of the carrier power P_{C} and sideband power P_{S}.

## Carrier Power

The carrier power P_{C }is equal to the mean-square (ms) value of the carrier term A cos ω_{c}t i.e.

P_{C }= mean square value of A cos ω_{c}t

## Sideband Power

The sideband power P_{S} is equal to the mean square value of the sideband term x(t) cos ω_{c}t i.e.

P_{S }= mean square value of x(t) cos ω_{c}t

In AM generation , a Band pass filter (BPF) or a tuned circuit tuned to carrier frequency ω_{c} is used to filter out the second integral term.

Therefore,

However, the total sideband power P_{S} is due to the equal contributions of the upper and lower sidebands. Hence, the power carried by the upper and lower sidebands will be

## Power of a Single-Tone Amplitude-Modulated (AM) Signal

### Definition

We know that the power content of the AM signal when modulating-signal is a random signal and may consist of several frequency components. Likewise we can find power content of single-tone Amplitude Modulated (AM) signal.

### Mathematical Expression

Let us consider that a carrier signal A cos ω_{c}t is amplitude-modulated by a single-tone modulating signal x(t) =V_{m }A cos ω_{m}t.

Then the unmodulated or carrier power

P_{C }= mean square (ms value)

## Power Content In Multiple-tone Amplitude Modulation (AM)

### Definition

A multiple-tone amplitude modulation is that type of modulation in which the modulating signal consists of more than one frequency components.

### Mathematical Expression

Let us consider that a carrier signal A cos ω_{c}t is modulated by a baseband or modulating signal x(t) which is expressed as :

x(t) = V_{1} cos ω_{1}t + V_{2} cos ω_{2}t + V_{3} cos ω_{3}t ………………….(9)

We know that the general expression for AM wave is

s(t) = A cos ω_{c}t + x(t) cos ω_{c}t ………………..(10)

Putting the value of x(t), we get

s(t) = A cos ω_{c}t + [V_{1} cos ω_{1}t + V_{2} cos ω_{2}t + V_{3} cos ω_{3}t ] cos ω_{c}t

or,

The expression for AM wave in equation (12) can further be expanded as under:

s(t) = A cos ω_{c}t + m_{1} A cos ω_{c}t cos ω_{1}t + m_{2} A cos ω_{c}t cos ω_{2}t + m_{3} A cos ω_{c}t cos ω_{3}t

Now we know that the total power in AM is given as

P_{t} = carrier power + sideband power

P_{t} = P_{C} + P_{S }…………(13)

The carrier power P_{C }is given as

## Total or Net Modulation Index for Multiple-Tone Modulation

### Definition

Let us consider that m_{t} is the total or net modulation indexes for a multiple-tone modulation.

### Mathematical Expression

We know that for a multiple-tone modulation, the total power is expressed as :

Where, m_{1} , m_{2} , …… m_{n} are the modulation indexes for different modulating signals.

The power for AM wave is also expressed as

This is the desired expression for the total or net modulation index.