AC-bias in tape heads: could someone explain?
I can't give you equations, but I can tell you that the main purpose of AC Bias applied to the record head of an analog tape recorder is to overcome the inherent hysteresis of the magnetic oxide contained in the binder on the recording tape.

I'm no longer competent to give you an extremely detailed explanation - I last worked with analog tape recorders more than 20 years ago and this kind of memory is one of those "Use it or Lose it" type of things.

In general, we used to supply more bias than necessary to improve the distortion figure that we could get from analog tape - the number that is still burned into my brain is 3dB of over-bias.

I do remember playing the amount of over-bias for the various tape brands that we used - Ampex 406 and 456 tape was the most common tape in use by the recording studios and broadcasters that I was working with at the time.  We spent days messing with bias and equalization to find the best compromise between distortion and S/N ratio.  This was massively fun work way back then but I wouldn't care to repeat it now.

[Edit]

I went looking for the calibration tapes that I used the most recently and found their home page: the company is Magnetic Reference Laboratories and much documentation can can be found at their home page.  I haven't reviewed this documentation in several decades and can't tell you which file contains the information that you are looking for but it is a great place to start looking.

These people are what I would consider to be the modern-day experts in regards to analog magnetic tape recording.

Look further down the page for technical documentation.

MRL

[Edit 2]

One of the links on the above page is to a PDF that does a great job of describing what AC Bias is and why it works. The link is: AC Bias