PLL lock and capture range; FM demodulation

Lock range and capture range

• Lock range (tracking range): the range of input frequencies over which a PLL that is already locked stays in lock as the input frequency is slowly changed. It is set mainly by the VCO range and loop gain.
• Capture range (acquisition range): the range of input frequencies over which an unlocked PLL can acquire lock. It is limited by the loop filter bandwidth.

Capture range is always less than or equal to lock range: $2f_{capture} \le 2f_{lock}$, both centred on the VCO free-running frequency $f_0$.

FM demodulation using PLL

 FM in   +-------+  e(t)  +------+  v(t) = output
 ------->| Phase |------->| Loop |----+---------->
 s(t)    | det.  |        | LPF  |    |
         +-------+        +------+    |
             ^                        |
             |        +-------+       |
             +--------|  VCO  |<------+
              r(t)    +-------+

Working:

1. Input FM: $s(t) = A_c \sin[2\pi f_c t + \phi_1(t)]$, where $\phi_1(t) = 2\pi k_f \int_0^t m(\tau)d\tau$.
2. The VCO free-running frequency is set to $f_c$. Its output is $r(t) = A_v \cos[2\pi f_c t + \phi_2(t)]$ with $\phi_2(t) = 2\pi k_v \int_0^t v(\tau)d\tau$.
3. The phase detector (multiplier) output, after the loop filter removes the $2f_c$ term, is

$$e(t) \approx \tfrac12 A_c A_v \sin[\phi_1(t) - \phi_2(t)] = \tfrac12 A_c A_v \sin\phi_e(t)$$

4. This error voltage drives the VCO so that its phase follows the input phase. When the loop is locked, $\phi_e$ is small and $\phi_2(t) \approx \phi_1(t)$.
5. Hence

$$2\pi k_v \int_0^t v(\tau)\,d\tau \approx 2\pi k_f \int_0^t m(\tau)\,d\tau \;\Rightarrow\; v(t) \approx \frac{k_f}{k_v}\, m(t)$$

So the VCO control voltage $v(t)$ is directly proportional to the message. The PLL demodulator needs no tuned circuits, is linear, and gives good threshold performance, so it is widely used in IC FM receivers.