Understanding the Concept:
In angle modulation, the modulation index represents the maximum phase deviation produced by the modulating signal.
• Phase Modulation (PM): The modulation index \( \beta_p \) is defined directly as the peak phase deviation:
\[
\beta_p = \Delta \phi = k_p \cdot A_m
\]
Where \( k_p \) is the phase deviation constant and \( A_m \) is the peak amplitude of the message signal.
• Frequency Modulation (FM): The modulation index \( \beta_f \) is defined as the ratio of peak frequency deviation to the frequency of the message signal:
\[
\beta_f = \frac{\Delta f}{f_m} = \frac{k_f \cdot A_m}{f_m}
\]
Where \( k_f \) is the frequency deviation constant, \( A_m \) is the message peak amplitude, and \( f_m \) is the message frequency in Hz.
Step 1: Extract Parameters from given equations
From the message signal equation \( m(t) = 2\cos(4000\pi t) \):
• Peak amplitude of modulating signal, \( A_m = 2\,\text{V} \)
• Modulating angular frequency, \( \omega_m = 4000\pi\,\text{rad/s} \)
• Modulating frequency in Hz, \( f_m = \frac{\omega_m}{2\pi} = \frac{4000\pi}{2\pi} = 2000\,\text{Hz} \)
Given deviation constants:
\[
k_p = 3\,\text{rad/V}
\]
\[
k_f = 3000\,\text{Hz/V}
\]
Step 2: Calculate Phase Modulation Index (\( \beta_p \))
Using the direct linear dependence formula for PM:
\[
\beta_p = k_p \cdot A_m
\]
Substituting the known values:
\[
\beta_p = 3 \times 2 = 6\,\text{rad}
\]
Step 3: Calculate Frequency Modulation Index (\( \beta_f \))
First, find the maximum frequency deviation \(\Delta f\):
\[
\Delta f = k_f \cdot A_m = 3000\,\text{Hz/V} \times 2\,\text{V} = 6000\,\text{Hz}
\]
Now, divide by the modulating signal frequency \(f_m\):
\[
\beta_f = \frac{\Delta f}{f_m} = \frac{6000}{2000} = 3
\]
Thus, the values of \( \beta_f \) and \( \beta_p \) are 3 and 6 respectively.