A PLL is an electronic circuit that can be used to generate an output signal that is synchronized in frequency and phase with an input signal. The main function of a PLL is to track and synchronize the frequency and phase of a reference signal with an output signal.

Phase Detector

The phase detector receives in input two signals $v1(t)$ and $v2(t)$ and returns a signal $v(t)=K_d.f(\Phi_{ref}-\Phi_{osc})$.

Usual phase detectors are the Balanced Mixer(for sinusoid signals) or XOR comparator(for digital signals).

Low-pass filter

The main objective of the low-pass filter is to generate the voltage used for the VCO. Usually, the transfer function is $F(p)=\frac{1}{1+\tau P}$ with $\tau = RC$ for a RC filter.

VCO

The VCO aims to generate the output signal. The output signal is linear around its working pulsation. Indeed, $\omega_{VCO} = \omega_{1} + K_0(U_{VCO} - U_1)$, with $K_0$ the gain of the VCO.

Extension for frequency synthesizers

In order to synthesize higher frequencies, a frequency divider can be added on the feedback branch. By this way, $f_{VCO}=N.f_{ref}$. $f_{ref}$ is the reference signal, is is called step of synthesis. Indeed, for $N = N+1$. $f_{VCO}=N.f_{ref} + f_{ref}$

Reminder

Useful functions and values:

Transfert function of a linearized PLL without divider:

$$H(p)=\frac{K_d K_0 F(p)}{p + K_d K_0 F(p)} = \frac{\omega _n ²}{\omega _n ² + 2\xi \omega _n p + p²}$$

$$\omega _n = \sqrt{\frac{K_0 K_d}{\tau}} \ \xi = \frac{1}{2} \frac{\omega _n}{K_0 K_d}$$