10.3 Frequency Modulation (FM), Phase Modulation & Discriminators

10.3 Frequency Modulation (FM), Phase Modulation & Discriminators

Board Examination Focus: Frequency deviation ($\Delta f$), modulation index ($m_f = \Delta f / f_m$), Carson's bandwidth rule ($BW = 2(\Delta f + f_m)$), Bessel function sideband distribution, pre-emphasis/de-emphasis time constants ($75\ \mu\text{s}$), and Phase-Locked Loop (PLL) demodulation are core PRC board exam competencies.

Angle modulation is a category of analog modulation wherein the angle (argument) of a high-frequency sinusoidal carrier wave is varied in accordance with an information signal. Angle modulation encompasses Frequency Modulation (FM) and Phase Modulation (PM). Unlike Amplitude Modulation, angle modulation maintains a constant peak carrier envelope amplitude, providing high noise immunity and superior fidelity at the expense of wider occupied bandwidth.

Fundamentals of FM and PM

An unmodulated carrier wave is expressed as $v_c(t) = V_c \cos[\theta(t)] = V_c \cos(\omega_c t + \phi_0)$. In angle modulation, the instantaneous phase angle $\theta(t)$ changes dynamically:

1. Phase Modulation (PM): Instantaneous phase deviation $\Delta\theta(t)$ is directly proportional to the modulating signal voltage $v_m(t)$: $\theta(t) = \omega_c t + k_p v_m(t) = \omega_c t + k_p V_m \cos(\omega_m t)$ Where $k_p$ is the phase deviation constant ($\text{radians/Volt}$).

2. Frequency Modulation (FM): Instantaneous frequency deviation $\Delta f(t)$ is directly proportional to modulating signal voltage $v_m(t)$: $f(t) = f_c + k_f v_m(t) = f_c + k_f V_m \cos(\omega_m t) = f_c + \Delta f \cos(\omega_m t)$ Where $k_f$ is the frequency deviation constant ($\text{Hz/Volt}$), and peak frequency deviation is $\Delta f = k_f V_m$.

Integrating instantaneous frequency to obtain instantaneous phase yields the general time-domain FM wave equation:

$v_{\text{FM}}(t) = V_c \cos\left[\omega_c t + \left(\frac{\Delta f}{f_m}\right) \sin(\omega_m t)\right] = V_c \cos[\omega_c t + m_f \sin(\omega_m t)]$

FM Modulation Index ($m_f$) and Deviation Ratio

The FM modulation index ($m_f$) is defined as the ratio of peak frequency deviation to modulating signal frequency:

$m_f = \frac{\Delta f}{f_m}$

Unlike AM, where $m \le 1.0$, the FM modulation index $m_f$ can be much greater than $1.0$.

For commercial FM broadcast transmissions with multiple modulating frequencies, the Deviation Ratio (DR) is defined using maximum allowed frequency deviation and maximum modulating audio frequency:

$\text{DR} = \frac{\Delta f_{\text{max}}}{f_{m,\text{max}}} = \frac{75\text{ kHz}}{15\text{ kHz}} = 5.0$

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FM Spectrum and Bessel Functions

Expanding $v_{\text{FM}}(t)$ using Fourier series analysis reveals an infinite number of sideband pairs spaced at harmonics of $f_m$ around carrier $f_c$:

$v_{\text{FM}}(t) = V_c \sum_{n=-\infty}^{\infty} J_n(m_f) \cos[(\omega_c + n \omega_m) t]$

Where $J_n(m_f)$ represents a Bessel function of the first kind of order $n$ evaluated at modulation index $m_f$.

           Bessel Function Carrier and Sideband Amplitudes vs m_f
   Relative Amplitude
     1.0 |-----\ (J0 Carrier)
     0.8 |      \
     0.6 |       \       /---\ (J1 First Sideband)
     0.4 |        \     /     \
     0.2 |         \   /       \
     0.0 +----------\-/---------\------------------- m_f
                   2.404       5.52        8.65
                 (1st Null)  (2nd Null)  (3rd Null)

Key Properties of FM Bessel Spectrum

1. Carrier Nulls: The carrier component $J_0(m_f)$ drops to zero at specific modulation indices: $m_f = 2.404, 5.52, 8.65, 11.79$. Measuring carrier nulls on a spectrum analyzer is the standard laboratory technique for calibrating FM frequency deviation.
2. Infinite Sidebands: Theoretical FM bandwidth is infinite, but sideband amplitudes drop rapidly for $n > m_f + 1$.
3. Constant Total Power: Total power in an FM wave remains constant regardless of modulation index: $P_t = P_c = V_c^2 / (2R)$. Power is redistributed from carrier into sidebands.

AM vs FM vs PM Comparison

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Carson's Bandwidth Rule & Pre-Emphasis Networks

In 1922, John R. Carson demonstrated that 98% of total power in an FM signal is contained within a finite bandwidth given by Carson's Rule:

$BW = 2 (\Delta f + f_m) = 2 f_m (m_f + 1)$

For commercial FM broadcasting in the Philippines and North America:

• Maximum frequency deviation: $\Delta f_{\text{max}} = \pm 75\text{ kHz}$
• Maximum modulating audio frequency: $f_{m,\text{max}} = 15\text{ kHz}$
• Carson's Bandwidth: $BW = 2 (75\text{ kHz} + 15\text{ kHz}) = 2 \times 90\text{ kHz} = 180\text{ kHz}$
• Standard channel spacing: $200\text{ kHz}$ (includes $20\text{ kHz}$ guard bands)

Pre-Emphasis and De-Emphasis Networks

In FM transmission, high-frequency audio components suffer from a triangular noise voltage distribution where noise power increases with frequency ($N(f) \propto f^2$).

To restore high-frequency Signal-to-Noise Ratio:

• Pre-emphasis Network (Transmitter): High-pass $R-C$ network that boosts high-frequency audio signals above a cutoff frequency $f_c$ prior to modulation.
• De-emphasis Network (Receiver): Matching low-pass $R-C$ network that attenuates high frequencies by an identical amount after demodulation, restoring flat audio response while reducing high-frequency noise.

Standard time constant ($\tau = R C$):

• North America & Philippines: $\tau = 75\ \mu\text{s} \implies f_c = \frac{1}{2\pi \tau} = \frac{1}{2\pi \times 75 \times 10^{-6}\text{ s}} \approx 2122\text{ Hz} \ (2.12\text{ kHz})$
• Europe & Asia-Pacific: $\tau = 50\ \mu\text{s} \implies f_c \approx 3183\text{ Hz} \ (3.18\text{ kHz})$

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FM Demodulators & Phase-Locked Loops (PLL)

FM demodulator circuits convert frequency variations into proportional voltage variations.

Classical Discriminator Circuits

1. Slope Detector: Simple LC tuned circuit detuned from carrier; poor linearity and susceptible to amplitude fluctuations.
2. Foster-Seeley Discriminator: Uses a center-tapped tuned transformer to produce phase-dependent output voltages; highly linear but requires prior amplitude limiter stage.
3. Ratio Detector: Variant of Foster-Seeley featuring inverted diodes; inherently suppresses AM noise, eliminating separate limiter stage requirement.

Phase-Locked Loop (PLL) FM Demodulator

Modern receivers utilize Integrated Circuit Phase-Locked Loops (PLL) for ultra-linear FM detection without bulky tuned inductors.

 FM Input ---> [ Phase Detector ] ---> [ Low-Pass Filter ] ---> Audio Output (v_o)
                     ^
                     |
                     +------ [ Voltage-Controlled Osc (VCO) ] <---+

• Phase Detector: Compares phase of incoming FM signal with VCO output, generating an error voltage.
• Low-Pass Filter (LPF): Removes high-frequency carrier ripple, producing a smooth control voltage.
• Voltage-Controlled Oscillator (VCO): Adjusts its frequency to track incoming FM deviation. The LPF control voltage required to maintain lock is an exact replica of the original modulating audio signal!

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Step-by-Step Worked Example: FM Bandwidth and Modulation Index

Problem: An FM broadcast transmitter is modulated by a $10\text{ kHz}$ sine wave ($f_m = 10\text{ kHz}$) with a peak modulating voltage that produces a peak frequency deviation of $\Delta f = 50\text{ kHz}$. Calculate: (a) FM modulation index ($m_f$), (b) Bandwidth using Carson's Rule, and (c) Pre-emphasis cutoff frequency for $\tau = 75\ \mu\text{s}$.

Solution:

1. Calculate modulation index ($m_f$): $m_f = \frac{\Delta f}{f_m} = \frac{50\text{ kHz}}{10\text{ kHz}} = 5.0$
2. Calculate Carson's Bandwidth ($BW$): $BW = 2 (\Delta f + f_m) = 2 (50\text{ kHz} + 10\text{ kHz}) = 2 \times 60\text{ kHz} = 120\text{ kHz}$ Alternative formula check: $BW = 2 f_m (m_f + 1) = 2 \times 10\text{ kHz} \times (5 + 1) = 20 \times 6 = 120\text{ kHz}$.
3. Calculate pre-emphasis cutoff frequency ($f_c$): $f_c = \frac{1}{2\pi \tau} = \frac{1}{2\pi \times 75 \times 10^{-6}\text{ s}} = \frac{1}{4.7124 \times 10^{-4}} \approx 2122.07\text{ Hz} = 2.12\text{ kHz}$