FM demodulation using PLL

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

  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 $\sin\phi_e \approx \phi_e$:

 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.