1. Power Components in AM: The total power ($P_t$) in a standard AM signal consists of the carrier power ($P_c$) and the power in the two sidebands ($P_{sb}$):
$$P_t = P_c + P_{sb} = P_c \left(1 + \frac{\mu^2}{2}\right)$$
where $\mu$ is the modulation index.
2. Defining Efficiency ($\eta$): Efficiency is defined as the ratio of useful power (sideband power) to the total power transmitted:
$$\eta = \frac{P_{sb}}{P_t} = \frac{\frac{\mu^2}{2}}{1 + \frac{\mu^2}{2}} = \frac{\mu^2}{2 + \mu^2}$$
3. Role of the Modulation Index: The mathematical expression clearly shows that the efficiency ($\eta$) is purely a function of the
modulation index ($\mu$).
• If $\mu = 0$, efficiency is 0%.
• If $\mu = 1$ (100% modulation), efficiency is $1/(2+1) = 33.33\%$.
This indicates that even under ideal conditions, two-thirds of the power is "wasted" on the carrier, which carries no information, and the maximum efficiency is entirely determined by how deeply the carrier is modulated.