Phugoids
On the subject of phugoids and attempting to clarify them… firstly, I must stress that I am NOT an Airbus man and am only familiar with publicly available information on the a320 FCS.
My understanding is that the control laws, when in normal law, maintain an attitude compensated “g” (within a limited bank angle range). To a pilot, this means that at centre stick, the aeroplane keeps going where it was pointed. The phugoid mode is basically an exchange of potential and kinetic energy at constant(roughly) angle of attack. So, if the a320 is flying along at a given airspeed at centre stick, an increase in thrust will result in an increase in airspeed along the current flight path. To achieve this, the control laws will pitch the nose down (reducing angle of attack) as airspeed increases. Eventually, drag will balance the thrust and the result is a new trimmed speed along the original flight path. The phugoid motion is, effectively, removed.
In a conventional aeroplane, the increase in thrust will cause an increase in airspeed and, thus, an increase in lift (remember that a stable aeroplane likes to remain at its trimmed angle of attack and there is no control law to modify this). The increase in lift will cause the flight path angle to rise. The increase in flight path angle eventually causes a decrease in airspeed which results in a decrease in lift (still at constant angle of attack) and the aeroplane noses over and begins to increase speed and lift again. Depending on the stability of the phugoid mode, a cyclic variation in airspeed and flight path angle could develop. If the phugoid is stable, the aeroplane would stabilise at a new flight path angle and airspeed, at the original angle of attack.
Now, if we go back to the a320 and consider the angle of attack protection control law. What this does is try to maintain the commanded angle of attack. I imagine that this (without phugoid damping) would return the phugoid behaviour of the a320 to more like that of a conventional aeroplane. Imagine slamming the stick to fully back with the thrust kept constant. The aeroplane would rapidly attain alpha max. Drag and flight path angle would increase, the airspeed would decay, the lift would reduce and the flight path angle would come back down again and off we could go into a potentially undamped phugoid.
What the phugoid damping terms in the alpha control laws seem to be designed to do is keep things nice and stable in terms of flight path, by looking at airspeed trend too. My guess is, at the expense of instant high angle of attack availability, flight path is considered (perhaps optimised) instead. In the case of a rapid pull from a low energy state, "firewalling" the throttles, the control laws make sure flight path doesn’t rapidly come back down again whilst the engines spool up. I also guess that they completely stabilise the phugoid to give the a320 characteristics that I have seen described, i.e. that it stabilises at airspeed for alpha max at full back stick (the eventual flightpath angle depending on the thrust set).
I would hazard a guess that Airbus engineers have done lots of simulation and modelling, with pilots too, to optimise flight path response in full back stick “avoidance” type manoeuvres.