A 10kW carrier is simultaneously modulated by two modulating signals corresponding to a modulation index of 40% and 30%, respectively. The total radiated power will be:
11.25 kW
Understanding AM Modulation and Total Radiated Power
This question asks us to find the total radiated power of an Amplitude Modulated (AM) carrier when it is simultaneously modulated by two different signals. We are given the carrier power and the individual modulation indices for each signal.
Calculating Total Modulation Index for Multiple Signals
When a carrier wave is modulated by more than one signal simultaneously, the total effect of modulation is represented by a combined modulation index. This total modulation index ($m_t$) is calculated as the square root of the sum of the squares of the individual modulation indices ($m_1, m_2, \dots$).
The formula for the total modulation index with multiple modulating signals is:
$$ m_t = \sqrt{m_1^2 + m_2^2 + m_3^2 + \dots} $$
In this specific problem, we have two modulating signals with modulation indices $m_1 = 40\%$ and $m_2 = 30\%$. We need to convert these percentages to decimal values:
- $m_1 = 40\% = \frac{40}{100} = 0.4$
- $m_2 = 30\% = \frac{30}{100} = 0.3$
Now, we can calculate the total modulation index ($m_t$):
$$ m_t = \sqrt{m_1^2 + m_2^2} $$
Substituting the values:
$$ m_t = \sqrt{(0.4)^2 + (0.3)^2} $$
$$ m_t = \sqrt{0.16 + 0.09} $$
$$ m_t = \sqrt{0.25} $$
$$ m_t = 0.5 $$
So, the total modulation index is 0.5.
Determining Total Radiated Power in AM
The total radiated power ($P_t$) in an AM wave is related to the carrier power ($P_c$) and the modulation index ($m$). The formula for total radiated power is:
$$ P_t = P_c \left(1 + \frac{m^2}{2}\right) $$
When dealing with simultaneous modulation by multiple signals, we use the total modulation index ($m_t$) in this formula:
$$ P_t = P_c \left(1 + \frac{m_t^2}{2}\right) $$
We are given that the carrier power $P_c = 10 \text{ kW}$ and we calculated the total modulation index $m_t = 0.5$. Now, we can substitute these values into the formula:
$$ P_t = 10 \left(1 + \frac{(0.5)^2}{2}\right) $$
$$ P_t = 10 \left(1 + \frac{0.25}{2}\right) $$
$$ P_t = 10 \left(1 + 0.125\right) $$
$$ P_t = 10 \left(1.125\right) $$
$$ P_t = 11.25 \text{ kW} $$
Therefore, the total radiated power is 11.25 kW.
Comparing with Options
Let's compare our calculated total radiated power with the given options:
- Option 1: 10 kW
- Option 2: 12.5 kW
- Option 3: 11.25 kW
- Option 4: 10.25 kW
Our calculated value of 11.25 kW matches Option 3.
Revision Table: Key Concepts
| Concept | Formula / Description |
|---|---|
| Carrier Power ($P_c$) | The power contained in the unmodulated carrier wave. |
| Modulation Index ($m$) | A measure of how much the modulated variable of the carrier signal varies around its unmodulated level. For AM, it indicates the depth of modulation. |
| Total Modulation Index ($m_t$) for Multiple Signals | $ m_t = \sqrt{m_1^2 + m_2^2 + \dots} $ where $m_i$ are individual modulation indices. |
| Total Radiated Power ($P_t$) for AM | $ P_t = P_c \left(1 + \frac{m^2}{2}\right) $ where $m$ is the modulation index (or $m_t$ for multiple signals). |
| Sideband Power ($P_{sb}$) | The power contained in the sidebands. $P_{sb} = P_t - P_c = P_c \frac{m^2}{2}$. |
Additional Information on AM Power
In Amplitude Modulation, the total power of the transmitted signal is distributed between the carrier component and the sidebands. The carrier power ($P_c$) remains constant regardless of the modulation, while the power in the sidebands ($P_{sb}$) depends on both the carrier power and the modulation index ($m$).
The total power ($P_t$) is the sum of the carrier power and the sideband power:
$$ P_t = P_c + P_{sb} $$
The sideband power is given by:
$$ P_{sb} = P_c \frac{m^2}{2} $$
Substituting this into the total power equation:
$$ P_t = P_c + P_c \frac{m^2}{2} $$
$$ P_t = P_c \left(1 + \frac{m^2}{2}\right) $$
This confirms the formula used to calculate the total radiated power. When multiple signals modulate the carrier, the total modulation index represents the effective modulation depth that determines the total power increase in the sidebands.
Important Questions from Amplitude Modulation
The highest modulation frequency typically used in AM broadcast is
Which of the following is NOT the advantage of amplitude modulation?
Bandwidth requirement for Amplitude modulated wave is:
The frequency range of AM radio is:
In an amplitude modulated system, a sinusoidal carrier signal of 1 MHz is modulated by a 10 kHz sinusoidal signal. If the lower sideband and the carrier are suppressed and if the amplitude modulated signal is sampled for further processing, what should be the minimum sampling frequency for baseband sampling?
