Q.A 10 kW carrier wave is amplitude modulated at 80% depth of modulation by a sinusoidal modulating signal. Calculate the total power, sideband power and transmission efficiency of the AM wave.

With the carrier power and modulation depth known, $P_T = P_C(1 + m_a^2/2)$ gives the total power, $P_T - P_C$ the sideband power, and $\eta = m_a^2/(2 + m_a^2)$ the transmission efficiency. …

[!TLDR]

For $P_C = 10$ kW and $m_a = 0.8$: $P_T = 13.2$ kW, $P_{SB} = 3.2$ kW (1.6 kW each) and $\eta = 24.2\%$.

The total AM power exceeds the carrier power because the sidebands add power that grows with the modulation depth:

The useful (information-bearing) power lies wholly in the sidebands, $P_{SB} = P_T - P_C$, and the transmission efficiency measures the fraction of total power carried by the sidebands:

Given $P_C = 10$ kW and $m_a = 80/100 = 0.8$:

$P_T = 10\times10^3\left(1 + \frac{(0.8)^2}{2}\right) = 10\times10^3\times1.32 = 13.2\ \text{kW}$ …