Frequency Modulation Index Calculation

The question asks us to determine the frequency modulation index for a VHF (Very High Frequency) carrier signal. We are given the frequency deviation produced and the frequency of the modulating signal.

Understanding Frequency Modulation Index

In frequency modulation (FM), the frequency of the carrier wave is varied in proportion to the amplitude of the modulating signal. The frequency modulation index, often denoted by $\beta$ (beta) or $m_f$, is a crucial parameter that indicates the extent of frequency variation relative to the modulating signal's frequency. It tells us how much the carrier frequency deviates from its unmodulated value for a given modulating signal.

The formula to calculate the frequency modulation index ($\beta$) is given by:

\[ \beta = \frac{\Delta f}{f_m} \]

Where:

• $\Delta f$ (Delta f) is the maximum frequency deviation (the maximum change in the carrier frequency from its center frequency).
• $f_m$ (f_m) is the maximum frequency of the modulating signal (also known as the modulating frequency).

The unit for the frequency modulation index is radians, although it is a dimensionless quantity as it's a ratio of two frequencies.

Given Parameters for Modulation Index

Let's list the values provided in the problem statement:

• Frequency deviation ($\Delta f$) = 50 kHz
• Modulating signal frequency ($f_m$) = 100 Hz

Calculating the Frequency Modulation Index

Before we use the formula, it's important to ensure that both frequencies are in consistent units. We will convert kHz to Hz.

• Convert frequency deviation from kHz to Hz: \[ \Delta f = 50 \text{ kHz} = 50 \times 1000 \text{ Hz} = 50000 \text{ Hz} \]
• The modulating signal frequency is already in Hz: \[ f_m = 100 \text{ Hz} \]

Now, we can substitute these values into the formula for the frequency modulation index:

\[ \beta = \frac{\Delta f}{f_m} \]

\[ \beta = \frac{50000 \text{ Hz}}{100 \text{ Hz}} \]

\[ \beta = 500 \]

Thus, the frequency modulation index is 500 radians.

Conclusion on Frequency Modulation Index

The calculated frequency modulation index is 500 radians. This value represents how widely the carrier frequency changes in response to the 100 Hz modulating signal, given the 50 kHz frequency deviation.