Understanding AM Antenna Current Variation

In Amplitude Modulation (AM), the total power transmitted by a station depends on both the carrier wave's power and the modulation applied. This impacts the antenna current. When the modulation depth (or modulation index, denoted by '$m$') increases, the overall signal amplitude increases, leading to a higher antenna current.

The relationship between the antenna current ($I$) in an AM transmitter and the modulation depth ($m$) is given by a specific formula that considers the carrier current ($I_c$):

• $I$ represents the total antenna current.
• $I_c$ represents the antenna current when there is no modulation (carrier wave only, $m=0$).
• $m$ is the modulation depth (a value between 0 and 1, or 0% and 100%).

AM Antenna Current Formula Explained

The formula connecting antenna current and modulation depth is:

$$ I = I_c \sqrt{1 + \frac{m^2}{2}} $$

This formula shows that the antenna current ($I$) increases as the modulation depth ($m$) increases because the term $\frac{m^2}{2}$ increases.

Calculating Antenna Current for 90% Modulation

We are given the antenna current at 50% modulation and need to find it at 90% modulation.

Step 1: Initial Conditions and Carrier Current Calculation

First, let's identify the given values:

• Initial modulation depth, $m_1 = 50\% = 0.5$
• Initial antenna current, $I_1 = 12$ A
• Final modulation depth, $m_2 = 90\% = 0.9$

We can use the initial condition to find the carrier current ($I_c$), although it's more efficient to use ratios.

Using the formula with the initial values:

$$ I_1 = I_c \sqrt{1 + \frac{m_1^2}{2}} $$

$$ 12 = I_c \sqrt{1 + \frac{(0.5)^2}{2}} $$

$$ 12 = I_c \sqrt{1 + \frac{0.25}{2}} $$

$$ 12 = I_c \sqrt{1 + 0.125} $$

$$ 12 = I_c \sqrt{1.125} $$

From this, we can express $I_c$ as:

$$ I_c = \frac{12}{\sqrt{1.125}} $$

Step 2: Calculating New Current at 90% Modulation

Now, we use the formula again for the final condition ($m_2 = 0.9$) to find the new antenna current ($I_2$):

$$ I_2 = I_c \sqrt{1 + \frac{m_2^2}{2}} $$

Substitute the expression for $I_c$ we found:

$$ I_2 = \left( \frac{12}{\sqrt{1.125}} \right) \sqrt{1 + \frac{(0.9)^2}{2}} $$

$$ I_2 = \left( \frac{12}{\sqrt{1.125}} \right) \sqrt{1 + \frac{0.81}{2}} $$

$$ I_2 = \left( \frac{12}{\sqrt{1.125}} \right) \sqrt{1 + 0.405} $$

$$ I_2 = \left( \frac{12}{\sqrt{1.125}} \right) \sqrt{1.405} $$

To simplify the calculation, we can rearrange it as:

$$ I_2 = 12 \times \frac{\sqrt{1.405}}{\sqrt{1.125}} $$

$$ I_2 = 12 \times \sqrt{\frac{1.405}{1.125}} $$

$$ I_2 = 12 \times \sqrt{1.24888...} $$

$$ I_2 = 12 \times 1.11753... $$

$$ I_2 \approx 13.41 $$

Final Answer and Explanation

The calculated antenna current when the modulation depth is increased to 90% is approximately 13.4 A. This increase from 12 A is expected because a higher modulation depth signifies a larger amplitude variation, thus increasing the overall power and current drawn by the antenna.