Question E8B02:
How does the modulation index of a phase-modulated emission vary with RF carrier frequency?  

Choices:  
A: It increases as the RF carrier frequency increases  
B: It decreases as the RF carrier frequency increases  
C: It varies with the square root of the RF carrier frequency  
D: It does not depend on the RF carrier frequency    

Correct Answer is  D: It does not depend on the RF carrier frequency  

To understand phase modulation (PM), remember: the PM modulation index is the peak phase deviation in radians. It depends on how much you wiggle the phase, not on the RF carrier frequency.

1) Why the correct answer is correct (D)
- In PM, the transmitted signal is:
  $$s(t) = A_c \cos\big(2\pi f_c t + k_p\, m(t)\big)$$
  where $k_p$ is the phase sensitivity (rad/volt) and $m(t)$ is the modulating signal.
- For a single-tone modulation $m(t) = A_m \cos(2\pi f_m t)$, the PM modulation index is the peak phase deviation:
  $$\beta_{PM} = \Delta \phi = k_p A_m$$
- Note that $f_c$ (the RF carrier frequency) does not appear. Therefore, the modulation index does not depend on RF carrier frequency.

2) Why the other choices are incorrect
- A: Increases with RF carrier frequency — Incorrect. No $f_c$ term in $\beta_{PM} = k_p A_m$.
- B: Decreases with RF carrier frequency — Incorrect for the same reason.
- C: Varies with the square root of RF carrier frequency — Also incorrect; there is no functional dependence on $f_c$.

3) Useful formulas and rules of thumb
- PM modulation index (single tone):
  $$\beta_{PM} = \Delta \phi = k_p A_m \quad (\text{radians})$$
- Instantaneous frequency in PM:
  $$f_i(t) = f_c + \frac{1}{2\pi}\frac{d}{dt}\big(k_p m(t)\big)$$
- For a single tone $m(t) = A_m \cos(2\pi f_m t)$, the PM frequency deviation is:
  $$\Delta f_{PM} = k_p A_m f_m$$
- Relationship to FM index:
  $$\beta_{FM} = \frac{\Delta f}{f_m} \quad \Rightarrow \quad \beta_{PM} = \frac{\Delta f_{PM}}{f_m} = k_p A_m = \Delta \phi$$
- Carson’s rule for PM bandwidth (single tone):
  $$B \approx 2\big(\Delta f_{PM} + f_m\big) = 2\big(k_p A_m f_m + f_m\big)$$

Key takeaway: In PM, the modulation index is set by phase deviation (and thus by modulating amplitude and modulator sensitivity), not by the RF carrier frequency.