Q.The antenna current of AM transmitter is 15 A when un modulated but rises to 18 A when modulated. Calculate the depth of modulation.

The total antenna current of an AM transmitter relates to the carrier current by $I_t = I_c\sqrt{1 + m_a^2/2}$; solving for the modulation index with $I_c=15\,\text{A}$ and $I_t=18\,\text{A}$ gives the depth …

[!TLDR]

From $I_t = I_c\sqrt{1+m_a^2/2}$: $(18/15)^2 = 1.44 = 1 + m_a^2/2$, so $m_a^2 = 0.88$ and $m_a = 0.938$ (about $93.8\%$).

This Karnataka 2nd PUC Electronics AM problem uses the antenna-current method to find the modulation index. Modulating a carrier adds sideband power, which raises the total radiated (antenna) current above the unmodulated carrier value. The relation is

where $I_c$ is the unmodulated carrier current and $I_t$ the total current when modulated.

Step 1 - Form the current ratio:

Step 2 - Substitute and solve:

$1.44 = 1 + \frac{m_a^2}{2} \;\Rightarrow\; \frac{m_a^2}{2} = 0.44 \;\Rightarrow\; m_a^2 = 0.88 \;\Rightarrow\; m_a = 0.938.$ …