What is the difference between first order, second order and third order phase locked loops?
I found this link to an amazing document that spells out the fine details up to 4th order filters.

Filter order just refers to the number of poles used to filter the output of the phase comparator so it provides a smooth DC error voltage to the VCO.


1st order filters are actually just the VCO's filter characteristics, requiring a minimum amount of time to settle down (zero phase) to a change in frequency or phase tracking. The raw phase comparator output voltage is fed to the VCO (voltage controlled oscillator) with only the noise spikes filtered out. This type offers fast tracking of frequency changes and locks quickly to the newest setting, but can have erratic output until it locks onto a new frequency.
A 2nd order filter has 1 RC stage, either passive or using an op-amp for a sharper roll-off. It is a tiny bit slower in locking onto a new frequency (zero phase) but less erratic in settling down and being stable. Recommended for most all PLL designs.
A 3rd order filter uses an optional op-amp and double RC networks. It settles down slower than the others but tolerates FSK/QFSK/QPSK better by staying stable even with complex modulation schemes. The RC networks have to be tuned for a given range of baud rates so a actual change in bit rate is followed as quickly as possible.
The PLL loop must always be able to find and lock onto a new carrier frequency rather quickly or loss of data occurs, forcing a resending of data packets or sending a EOF/EOL/EOT command first. Fortunately fast MPU's can emulate or have the entire PLL function blocks built in, so the use of analog filters and discrete PLL circuits are rare. Use QPSK as a search term and you will find plenty of support IC's and ready-to-use modules. Be careful of any 'special' software or license agreements.


For more details of FSK and QPSK.