--- title: "PLL as FM demodulator; uses of PLL" subject: "Communication Systems" chapter: 3 key: "semester_6/communication_systems:pll-fm-demod-uses" source: https://www.bctengineer.com/q/semester_6/communication_systems/pll-fm-demod-uses site: BCT Engineering --- # PLL as FM demodulator; uses of PLL Communication Systems · Chapter 3: Angle Modulation - Asked in: 2064 Shrawan (CS I), 6+2 marks - 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/pll-fm-demod-uses - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-3 ## Question Show that PLL can be used to demodulate FM signal with a neat diagram. What are the basic uses of PLL? ## Short answer PLL demodulates FM by locking VCO phase to input FM signal; the VCO control voltage becomes the demodulated message. Basic uses include FM/FSK demodulation, carrier recovery for coherent detection, frequency synthesis with dividers, clock and bit-timing recovery, and frequency multiplication/division. ## Answer A **phase-locked loop (PLL)** is a negative-feedback system in which a voltage-controlled oscillator (VCO) is forced to track the phase (and so the frequency) of the input signal. When it tracks an FM wave, the VCO control voltage is the message. ```text FM in +--------+ e(t) +-------+ v(t) output s(t) --->| Phase |----->| Loop |---+----> m(t) |detector| |filter | | +--------+ | H(f) | | ^ +-------+ | | r(t) +-------+ | +----------| VCO |<---+ +-------+ ``` #### Proof that PLL demodulates FM 1. Input FM: $s(t) = A_c\sin[2\pi f_c t + \phi_1(t)]$, with $\phi_1(t) = 2\pi k_f\int_0^t m(\tau)d\tau$. 2. VCO output (free-running at $f_c$): $r(t) = A_v\cos[2\pi f_c t + \phi_2(t)]$, with $\phi_2(t) = 2\pi k_v\int_0^t v(\tau)d\tau$. 3. Phase detector (multiplier) output: $$ s(t)\,r(t) = \tfrac12 A_cA_v\sin[\phi_1 - \phi_2] + \tfrac12 A_cA_v\sin[4\pi f_c t + \phi_1 + \phi_2] $$ The loop filter rejects the $2f_c$ term, leaving $$ e(t) = \tfrac12 A_cA_v\sin\phi_e(t), \qquad \phi_e = \phi_1 - \phi_2 $$ 4. **Linearised model:** when locked, $\phi_e$ is small, so $\sin\phi_e \approx \phi_e$. The loop adjusts $v(t)$ to keep $\phi_e$ near zero. Then $$ \phi_2(t) \approx \phi_1(t) \;\Rightarrow\; 2\pi k_v\int_0^t v(\tau)\,d\tau \approx 2\pi k_f\int_0^t m(\tau)\,d\tau $$ 5. Differentiating: $$ v(t) \approx \frac{k_f}{k_v}\,m(t) $$ So the loop-filter output is proportional to the message: the PLL is an FM demodulator. For good tracking the loop gain must be high and the loop bandwidth must cover the message bandwidth. #### Basic uses of PLL - FM and FSK demodulation - Carrier recovery for coherent (synchronous) detection, e.g. stereo pilot, DSB-SC, PSK - Frequency synthesisers (with a divider in the feedback path) in radios and phones - Clock and bit-timing recovery in digital receivers - Frequency multiplication/division and tracking filters ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In the PLL FM demodulator, what does the loop filter do to the multiplier (phase detector) output? A. Converts the FM wave to AM before detection B. Removes the difference term and keeps the 2fc term C. Removes the 2fc term, leaving a signal proportional to sin of the phase error D. Amplifies the 2fc term to drive the VCO Answer: C. The product has a difference-phase term and a 2fc term; the loop filter rejects the 2fc term, leaving e(t) proportional to sin(phi1 - phi2). (Key point: Phase detector output) 2. Why can sin(phi_e) be replaced by phi_e when the PLL is locked? A. Feedback keeps the phase error phi_e small B. The message has zero amplitude C. The loop filter has unity bandwidth D. The VCO frequency equals twice the carrier Answer: A. Negative feedback drives the phase error near zero, so sin(phi_e) is approximately phi_e and the loop is linear. (Key point: Linearised locked condition) 3. What is the loop-filter output v(t) when the PLL tracks an FM wave? A. v(t) is approximately the carrier at fc B. v(t) is approximately the integral of m(t) C. v(t) is approximately (kv/kf) times the derivative of m(t) D. v(t) is approximately (kf/kv) m(t) Answer: D. Locking gives phi2 = phi1, so kv*integral(v) = kf*integral(m); differentiating gives v = (kf/kv) m(t). (Key point: Demodulated output) 4. Which is a basic use of a PLL? A. Generating the sampling impulse train for flat-top sampling only B. Removing quantization noise in PCM C. Carrier recovery for coherent detection D. Increasing the amplitude of the message Answer: C. PLLs recover carriers for synchronous detection, besides FM/FSK demodulation, frequency synthesis and clock recovery. 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