i. **Mathematical Equation**

The instantaneous values of modulating signal and carrier signal can be represented as given below.

**Instantaneous value of modulating signal**

$$e_m=E_m\sin w_mt$$

where $e_m=\text{instantaneous amplitude} \\
E_m =\text{maximum amplitude} \\
w_m =2πf_m=\text{angular frequency and} \\
f_m =\text{frequency of modulating signal}$

**Instantaneous value of carrier signal**

$$e_c=E_c\sin w_ct$$
where $e_c=\text{instantaneous amplitude} \\
E_c =\text{maximum amplitude} \\
w_c =2πf_m=\text{angular frequency and} \\
f_c =\text{frequency of carrier signal}$

**Instantaneous value of amplitude modulating signal**

Using above given mathematical expression for modulating and carrier signals, we can create a new mathematical expression for the complete modulated wave, as given below
$$E_{AM} =E_c+e_m \\
=E_c+E_m\sin w_mt$$

The instantaneous value of the amplitude modulated wave can be given as

$$e_{AM}=E_{AM} \sinϴ \\
=E_{AM} \sin w_ct \\
=(E_c+E_m\sin w_mt)\sin w_ct$$

ii. **AM waveform**

In the figure shown below, the carrier wave has positive and negative half cycles. Both these cycles are varied according to the information to be sent.

The carrier then consists of sine waves whose amplitudes follow the amplitude variations of the modulating wave.

The carrier is kept in an envelope formed by the modulating wave.
From the figure, the amplitude variation of the high frequency carrier is at the signal frequency and the frequency of the carrier wave is the same as the frequency of the resulting wave.

iii. **AM amplitude and power spectrum**

The AM wave equation has three frequency components.

a. First term represents unmodulated carrier at frequency $f_c$.

b. The second term represents sideband of amplitude $\dfrac{mE_c}{2}$ located at $f_c-f_m$ and

c. The third term represents sideband of amplitude $\dfrac{mE_c}{2}$ located at $f_c+f_m$

Figure shows this spectrum

Both the sidebands are located at frequencies $f_m$ from carrier $f_c$.

Both the sidebands have same amplitude and hence they contain same power.

iv. **Modulation Coefficient**

In undistorted AM to occur, the modulating signal voltage $V_m$ must be less than the carrier voltage $V_c$.

Therefore the relationship between the amplitude of the modulating signal and the amplitude of the carrier signal is important.

This relationship, known as the modulation coefficient m, is the ratio

$$m=\dfrac{V_m}{V_c}$$

These are the peak values of the signals, and the carrier voltage is the unmodulated value.

v. **Transmission Power**

A modulated wave has more power transmission than had by the carrier wave before modulating. The total power transmitted components in amplitude modulation can be written as:

$$P_{total}=p_{carrier}+P_{LSB}+P_{USB}$$

Considering additional resistance like antenna resistance R.

$$P_{carrier}=[\dfrac{\dfrac{V_c}{\sqrt {2}}}{R^2}]=\dfrac{V^2_c}{2R}$$

Each side band has a value of $m/2V_c$ and r.m.s value of $mV_c/2√2$. Hence power in LSB and USB can be written as

$$P_{LSB}=P_{USB}=(mV_c/2 \sqrt{2})^2/R=m^2/4*V^2C/2R=m_2/4P_{carrier} \\P_{total}=V^2_c/2R+[m^2/4*V^2C/2R]+[m^2/4*V^2C/2R]=V^2_c/2R(1+m^2/2)=P_{carrier}(1+m^2/2)$$

In some applications, the carrier is simultaneously modulated by several sinusoidal modulating signals. In such case, the total modulation index is given as

$$Mt=\sqrt{(m1^2+m2^2+m3^2+m4^2+....)}$$

If IC and IT are the r.m.s values of unmodulated current and total modulated current and R is the resistance through which these current flow, then

$$\dfrac{P_{total}}{P_{carrier}}=\bigg(\dfrac{It.R}{Ic.R}\bigg)^2=\bigg(\dfrac{It}{Ic}\bigg)^2 \\
\dfrac{P_{total}}{P_{carrier}}=\bigg(1+\dfrac{m^2}{2}\bigg) \\ \dfrac{It}{Ic}=1+\dfrac{m^2}{2}$$