Q.The current of an AM transmitter is 8 A when only the carrier is sent, but it increases to 8.65 A, when the carrier is amplitude modulated. Find the percentage modulation. Determine the antenna current when the depth of modulation is 0.75.

The total antenna current relates to the carrier current by $I_t = I_C\sqrt{1 + m_a^2/2}$; solving for $m_a$ gives the percentage modulation, and re-substituting the new depth gives t …

[!TLDR]

From $8.65 = 8\sqrt{1 + m_a^2/2}$, $m_a = 0.58$ (58%); at a depth of 0.75, $I_t = 8\sqrt{1 + 0.75^2/2} = 9.056$ A.

When a carrier is amplitude modulated, the antenna carries extra sideband power, so the total (modulated) current rises above the carrier-only current according to

This is first solved for the modulation index using the measured currents, then evaluated again at the required depth.

Given $I_C = 8$ A and $I_t = 8.65$ A:

$\frac{I_t}{I_C} = \sqrt{1 + \frac{m_a^2}{2}} \implies \left(\frac{8.65}{8}\right)^2 = 1 + \frac{m_a^2}{2}$ …