Angle Modulation: Phase and Frequency Deviation Calculation

This solution details the process of finding the maximum phase deviation ($|\phi(t)|_{max}$) and the maximum frequency deviation ($\Delta f$) for the provided angle-modulated signal.

Analyzing the Angle-Modulated Signal

The signal given is:

$x_c(t) = 10\cos[(10^8) \pi t+5\sin 2\pi (10^3)t]$

We compare this to the standard form of an angle-modulated signal:

$A\cos[\omega_c t + \phi(t)]$

From this comparison, we identify the components:

• Amplitude ($A$): 10
• Carrier angular frequency ($\omega_c$): $(10^8) \pi$ rad/s
• Phase modulating function ($\phi(t)$): $5\sin(2\pi (10^3)t)$

Phase Deviation Calculation

The phase deviation is represented by the function $\phi(t)$. For this signal:

$\phi(t) = 5\sin(2\pi (10^3)t)$

To find the maximum phase deviation ($|\phi(t)|_{max}$), we find the peak absolute value of $\phi(t)$. The sine function varies between -1 and 1.

$|\phi(t)|_{max} = |5 \times \sin(2\pi (10^3)t)|_{max}$

$|\phi(t)|_{max} = 5 \times 1 = 5 \text{ radians}$

Frequency Deviation Calculation

The instantaneous angular frequency ($\omega_i(t)$) is found by differentiating the phase term $\left[ \omega_c t + \phi(t) \right]$ with respect to time:

$\omega_i(t) = \frac{d}{dt} \left[ (10^8)\pi t + 5\sin(2\pi (10^3)t) \right]$

Performing the differentiation:

$\omega_i(t) = (10^8)\pi + 5 \times \cos(2\pi (10^3)t) \times \frac{d}{dt}[2\pi (10^3)t]$

$\omega_i(t) = (10^8)\pi + 5 \times \cos(2\pi (10^3)t) \times (2\pi (10^3))$

$\omega_i(t) = (10^8)\pi + (10^4 \pi) \cos(2\pi (10^3)t)$

The frequency deviation ($\Delta\omega(t)$) is the difference between the instantaneous angular frequency and the carrier angular frequency:

$\Delta\omega(t) = \omega_i(t) - \omega_c$

$\Delta\omega(t) = \left[ (10^8)\pi + (10^4 \pi) \cos(2\pi (10^3)t) \right] - (10^8)\pi$

$\Delta\omega(t) = (10^4 \pi) \cos(2\pi (10^3)t)$

The maximum frequency deviation ($\Delta\omega_{max}$) occurs when the cosine term is $\pm 1$.

$\Delta\omega_{max} = |10^4 \pi \times 1| = 10^4 \pi \text{ rad/s}$

To get the frequency deviation in Hertz ($\Delta f$), we divide by $2\pi$:

$\Delta f = \frac{\Delta\omega_{max}}{2\pi} = \frac{10^4 \pi}{2\pi}$

$\Delta f = \frac{10^4}{2} = 5000 \text{ Hz}$

Converting to kHz:

$\Delta f = 5 \text{ kHz}$

Summary of Results

The calculated values are:

• Maximum Phase Deviation ($|\phi(t)|_{max}$): 5 radians
• Maximum Frequency Deviation ($\Delta f$): 5 kHz