In amplitude modulation, Power of a carrier wave is represented by : $$P_c = \frac{A_c ^2}{2R}$$

Which is independent of modulation index!

Graph 'percentage of total transmitted power' vs percentage modulation

Question: At what factor is the power of carrier wave dependent on!?


A conventional AM signal can be written as

$$x_{AM}(t)=A_c\big(1+\alpha\; m(t)\big)\cos(\omega_ct)\tag{1}$$

where $\alpha$ is the modulation index and $m(t)$ is the message signal. In Eq. $(1)$ it is assumed that the message signal $m(t)$ is normalized, i.e., $\max_t|m(t)|= 1$. The average power of $x_{AM}(t)$ is


If you assume now that the power $(2)$ is fixed to some value $P$, then the total available power must divided between the carrier term and the message-bearing term in $(2)$. Equating $(2)$ with the available power $P$, and expressing the carrier term $A_c^2/2$ in terms of the modulation index and the average power of the message signal gives


In the figure in your question it looks like they assumed $\overline{m^2(t)}=\frac12$, which for a maximum modulation index $\alpha=1$ results in

$$\frac{A_c^2}{2}=\frac23 P\tag{4}$$

This value corresponds to the right-most point of the curve labeled "carrier".


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