---
title: "FM demodulation using PLL"
subject: "Communication Systems"
chapter: 3
key: "semester_6/communication_systems:fm-demod-pll"
source: https://www.bctengineer.com/q/semester_6/communication_systems/fm-demod-pll
site: BCT Engineering
---

# FM demodulation using PLL

Communication Systems · Chapter 3: Angle Modulation

- Asked in: 2075 Bhadra (CS I), 8 marks; 2067 Mangsir (CS I), 10 marks (asked 2 times)
- Video: [FM Demodulation by Phase Lock Loop (Basics, Block Diagram, Working & Advantages) Explained](https://www.youtube.com/watch?v=30DaodObd9Q) — Engineering Funda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/fm-demod-pll
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-3

## Question

Describe the process of demodulation of FM using PLL.

## Short answer

A PLL locks a voltage-controlled oscillator to the incoming FM signal. The phase detector error, filtered by a low-pass loop filter, forces the VCO to track the input instantaneous frequency. In lock, the filter output voltage is directly proportional to the original message, v(t) = (k_f/k_v)m(t).

## Answer

A **phase-locked loop (PLL)** is a negative-feedback system whose voltage-controlled oscillator (VCO) tracks the phase of the input signal. When the input is an FM wave, the voltage that steers the VCO is proportional to the message, so the PLL acts as an FM demodulator.

#### Block diagram

```text
  FM in   +-----------+   e(t)  +--------+  v(t)
 s(t) --->|  Phase    |-------->|  Loop  |---+---> output
          | detector  |         | filter |   |    ~ m(t)
          +-----------+         | H(f)   |   |
               ^                +--------+   |
               |   r(t)   +-------+          |
               +----------|  VCO  |<---------+
                          +-------+
```

#### Blocks

1. **Phase detector:** a multiplier comparing input and VCO output; its output depends on the phase difference.
2. **Loop filter:** a low-pass filter that removes the double-frequency term and controls loop bandwidth and stability.
3. **VCO:** free-running at $f_c$ when $v = 0$; its frequency changes by $k_v$ Hz per volt.

#### Analysis

Input FM signal:

$$
s(t) = A_c\sin[\omega_ct + \phi_1(t)], \qquad \phi_1(t) = 2\pi k_f\int_0^t m(\tau)\,d\tau
$$

VCO output (90° offset so the error is zero at lock):

$$
r(t) = A_v\cos[\omega_ct + \phi_2(t)], \qquad \phi_2(t) = 2\pi k_v\int_0^t v(\tau)\,d\tau
$$

Multiplier output:

$$
s(t)r(t) = \frac{A_cA_v}{2}\left\{\sin[2\omega_ct + \phi_1 + \phi_2] + \sin[\phi_1 - \phi_2]\right\}
$$

The loop filter removes the $2\omega_c$ term. The error signal is

$$
e(t) = K_m\sin\phi_e(t), \qquad \phi_e = \phi_1 - \phi_2
$$

where $K_m = A_cA_v/2$ (times the multiplier gain).

#### Linear model (loop in lock)

In lock $\phi_e$ is small, so $\sin\phi_e \approx \phi_e$:

```text
 phi1 -->(+)--> K_m --> H(f) --+--> v(t)
          ^ -                  |
          |                    |
         phi2 <--[2 pi kv / s]-+
```

$$
v(t) = K_m\,[\phi_e(t) * h(t)], \qquad \phi_2(t) = 2\pi k_v\int v\,dt
$$

In the frequency domain,

$$
\Phi_e(f) = \frac{\Phi_1(f)}{1 + L(f)}, \qquad L(f) = \frac{K_mk_vH(f)}{jf}
$$

When the loop gain $|L(f)| \gg 1$ over the message band, $\Phi_e \approx 0$, so $\phi_2(t) \approx \phi_1(t)$:

$$
2\pi k_v\int v(t)\,dt = 2\pi k_f\int m(t)\,dt \;\Rightarrow\; v(t) = \frac{k_f}{k_v}\,m(t)
$$

So the loop filter output is a scaled copy of the message.

#### Physical explanation

- If the input frequency rises, the phase error grows; the error voltage increases and pushes the VCO frequency up until it matches.
- Thus the control voltage always follows the instantaneous input frequency, $f_c + k_fm(t)$, i.e. it follows $m(t)$.

#### Lock and capture range

- **Lock range:** frequency range over which a locked loop stays locked.
- **Capture range:** range over which an unlocked loop can acquire lock (smaller than lock range).
- The peak deviation $\Delta f$ must lie within the lock range.

#### Advantages

- No tuned LC circuits; easy IC realization (e.g. NE565).
- Better (lower) FM threshold than a limiter–discriminator.
- Good linearity and tracking of slow carrier drift.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which set lists the PLL blocks in the correct loop?
   A. Phase detector, loop filter, VCO fed back to the detector
   B. Limiter, discriminator, VCO
   C. Mixer, IF amplifier, detector
   D. Oscillator, balanced modulator, BPF
   Answer: A. A multiplier phase detector, low-pass loop filter and VCO form a feedback loop. (Key point: Blocks of PLL)

2. What does the multiplier output become after the 2ωc term is removed?
   A. K sin(φ1 + φ2)
   B. K cos(2ωc t)
   C. K m(t)
   D. Km sin(φ1 − φ2) = Km sin φe
   Answer: D. The low-frequency part depends on the phase error φe = φ1 − φ2. (Key point: Error signal)

3. Why can sin φe be replaced by φe?
   A. Because in lock the phase error is small
   B. Because the VCO is linear in frequency
   C. Because the message is small
   D. Because the loop filter is ideal
   Answer: A. A locked loop keeps φe small, so sin φe ≈ φe. (Key point: Linear model)

4. What happens when |L(f)| >> 1 over the message band?
   A. The loop unlocks
   B. φe ≈ 0, so φ2 ≈ φ1 and v(t) = (kf/kv) m(t)
   C. The VCO stops
   D. The output becomes the integral of m(t)
   Answer: B. Φe = Φ1/(1 + L) is nearly zero, so the VCO phase tracks the input and v is a scaled copy of m. (Key point: Loop gain effect)

5. Which is correct about lock and capture range?
   A. Capture range is larger than lock range
   B. The two are always equal
   C. Capture range is smaller than lock range, and Δf must lie within lock range
   D. Lock range only matters for AM
   Answer: C. An unlocked loop acquires lock over the smaller capture range; a locked loop stays locked over the larger lock range. (Key point: Lock and capture range)

## Related in Communication Systems

- [PLL lock and capture range; FM demodulation](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Apll-lock-capture-fm-demod) — Communication Systems, chapter 3
- [Amplitude limiter; FM demodulation using PLL](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Alimiter-discriminator-pll-fm) — Communication Systems, chapter 3, asked 2×
- [Phase error in coherent DSB-SC detection; AM demodulation by PLL](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aphase-error-dsbsc-am-pll) — Communication Systems, chapter 2
- [PLL as FM demodulator; uses of PLL](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Apll-fm-demod-uses) — Communication Systems, chapter 3
- [PLL components and AM demodulation using PLL](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Apll-components-am-demod) — Communication Systems, chapter 2, asked 3×
- [Limiter-discriminator FM demodulation; need for pre-emphasis](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Alimiter-discriminator-pre-emphasis) — Communication Systems, chapter 3