PLL as FM demodulator; uses of PLL

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.

 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