In amplitude modulation, the amplitude of the carrier voltage is varied according to (i.e. proportional to) the instantaneous value of the modulating voltage. Let’s see the amplitude modulation derivation.

## Amplitude Modulation Derivation

Thus, let the instantaneous value of the modulating voltage be given by,

…….(1)

Where Vm is the amplitude and is the angular frequency of the modulating voltage or the signal.

………(2)

The above equation giving the carrier voltage v_{c} we may for the sake of convenience take phase angle as zero since is not involved in amplitude modulation process. The above equation may be written as,

…..(3)

On amplitude modulation, amplitude of the carrier no longer remains constant but varies with time as given by the following expression:

………(4)

Where is the change in carrier amplitude caused by amplitude modulation, ka being a constant.

Then the instantaneous value of amplitude modulated carrier voltage is given by,

…….(5)

Where ma is the modulation index or modulation factor or depth of modulation and is given by,

…….(6)

Percent modulation equals 100xm_{a}

### Waveform of amplitude modulated voltage

Figure 1(a) gives the waveform of unmodulated carrier voltage. Figure 1(b) gives the waveform of sinusoidal modulating voltage. Figure 1(c) gives the waveform of amplitude modulated carrier voltage. From figure 1(c) we find that the frequent of the carrier remains unchanged after modulation but its instantaneous amplitude varies according to the instantaneous value of the modulating voltage v_{m}.

From equation (5) we see that,

…….(7)

Also, ……(8)

Combining equation (7) and (8) we get,

…….(9)

Equation (9) enables us to experimentally determine m_{a}. Thus, we apply the modulated carrier to the y deflection plates of a CRO and apply a suitable time base voltage to the x-deflection plates to get a steady pattern, similar to that in figure 1(c). On measuring and , we may use equation (9) to calculate the modulation index m_{a}.

### Frequency Components Produced in Amplitude Modulation

Equation (5) may be expanded to give,

……..(10)

Equation 10 shows that a sinusoidal carrier voltage on amplitude modulation by another sinusoidal voltage contains the following three frequency terms:

- Original carrier voltage of angular frequency
- Upper sideband term of angular frequency .
- Lower sideband term of angular frequency

The two side band terms are located in frequency spectrum on either side of the carrier frequency at frequency interval of from carrier as shown in figure 2. Further the magnitude of each sideband term is . Thus, if the modulation index , each sideband term has amplitude equal to .

Amplitude modulation thus shifts the signal of frequency from audio frequency level to the level of the carry frequency . Further, the signal appears in the form of two sideband terms symmetrically placed relative to the carrier frequency . Each sideband term carries the complete intelligence (or message) contained in the original modulation signal. Thus, the signal exists at two frequencies in the amplitude modulated carrier.

Thus, far we have assumed the modulating signal to be signal frequency. In practice, however, the modulating voltage has a complex waveform and may be represented by a band of frequency components of different amplitudes and phases as shown in figure 3 by g(w). Each frequency term in the modulating signal produces on modulating a pair of sideband terms. The complete modulating signal then procedures two sidebands symmetrically placed about the carrier as shown in figure 3 where and are the lowest and the highest frequencies in the modulating signal.

Bandwidth: When carrier is amplitude modulated by a band of frequencies extending say from frequency f_{1} to f_{2}, then the bandwidth occupied by the amplitude modulated carrier equals 2f_{2}.

### Power Relations in Amplitude Modulated Voltage

Equation 10 shows that the carrier component of the amplitude modulated wave has the same amplitude as the unmodulated carrier. But the amplitude modulated voltage contains lower and upper sideband components as well carrying power of say P_{LS} or P_{US} respectively. Thus, total power in the modulated voltage is,

….(11)

The first term on the right-hand side of equation (11) is the unmodulated carrier power and is given by,

…..(12)

Each sideband term has peak value of and rms value of . Hence power in each sideband term is,

……(13)

Substituting equations (12) and (13) into equation (11) we get,

…(14)

Or ……..(15)

Equation (15) shows that power P_{t} in the modulating wave is times the unmodulated carrier power P_{C}.

For m_{a} = 1, modulated carrier power equals 1.5P_{C}. This is the maximum power in the amplitude modulated carrier.

* Carrier Current on Modulating: *In practice, it is relatively easy to measure the modulated and unmodulated carrier currents. We may then calculate the modulated index m

_{a}from the measured values of these two currents.

Let I_{C} and I_{t} be the rms values of unmodulated and modulated carrier currents and let R be the resistance into which these currents flow.

Then, ……(16)

But from equation (15),

……(17).

Hence,

Or, …..(18)

* Modulation by Several Sine Waves: *In most applications, carrier is simultaneously modulated by several sinusoidal modulating voltages. We may calculate the resulting modulated carrier power. Procedure consists in calculating the total modulating index and then substituting it in equation (15) to calculate the total power P

_{t}.

Equation of (15) may be written as,

……..(19)

Where P_{SB} is the total sideband power and is given by,

……….(20)

When several sine waves simultaneously modulate the carrier, the carrier power P_{C} remains unaltered but the total sideband power P_{SBT} is now the sum of the individual sideband, powers. Thus, we get,

…. ………(21)

On combining Equations (20) and (21) we get,

……(22)

Hence,

…. ….(23)

However, the total modulation index must not exceed unity or else distortion will result.

### Example related to amplitude modulation derivation

**Example 1: A sinusoidal carrier voltage of frequency 1 MHz and amplitude 60 volts is amplitude modulated by a sinusoidal frequency 10 KHz producing 50% modulation. Calculate the frequency and amplitude of upper and lower sideband terms.**

* Solution:* Frequency of upper sideband = 1000 KHz + 10 KHz = 1010 KHz

Frequency of lower sideband = 1000 KHz – 10 KHz = 990 KHz.

Amplitude of each sideband term = volts.

**Example 2: A sinusoidal carrier voltage of amplitude 100 volts is amplitude modulated by a sinusoidal voltage of frequency 5 KHz resulting in maximum carrier amplitude amplitude of 130 volts. Calculate the modulation index.**

**Solution:**

**Example 3: A sinusoidal carrier voltage of frequency 1000 KHz is amplitude modulated by a sinusoidal voltage of frequency 20 KHz resulting in maximum and minimum modulated carrier amplitude of 130 and 90 volts respectively. Calculate**

**(a) frequencies of lower and upper sideband terms (b) unmodulated carrier amplitude (c) modulated index and (d) amplitude of each sidebad term.**

**Solution:**

For (a)

Lower sideband frequency = (1000-20) = 980 KHz

Upper sideband frequency = (1000 + 20) = 1020 KHz

For (b) Unmodulated carrier amplitude

For (c) Modulated index

For (d) Amplitude of each sideband term

**Example 4: The rms value of carrier voltage is 80 volts. Calculate its rms value when it has been amplitude modulated to a depth of (a) 30% and (b) 50%.**

**Solution:**

Or,

Hence,

For (a)

volts

For (b)

volts

**Example 5: The rms value of a carrier voltage is 100 volts. After amplitude modulation by a sinusoidal audio voltage, the rms value of the carrier voltage become 112 volts. Calculate the modulating index.**

**Solution:**

Hence,

**Example 6: Unmodulated RF carrie power of 20 kW sends a current of 20 Amperes through an antenna. On amplitude modulation by another sinusoidal voltage, the antenna current increases to 24 Amperes. Calculated (a) the modulation index and (b) carrier power after modulation.**

**Solution:**

Let the rms current before and after modulation be I_{O} and I respectively.

Then,

Carrier power after modulation is,

**Example 7: The rms value of a carrier voltage after amplitude modulation to a depth of 50% by another sinusoidal voltage is 60 volts. Calculate the rms value of carrier voltage when amplitude modulated to a depth of 70%.**

**Solution:**

With 70% modulation, rms value of modulated carrier voltage is

**Example 8: A broadcast transmitter radiates 5 kW when the percentage modulation is 50%. Calculate the total carrier power when the modulation has been reduced to 30%.**

**Solution:**

With 30% modulation,