Determine the depth of a transformer
Is there a better way to calculate the depth?


I would ask you this question: -

What primary inductance do you want?


It is the primary inductance that dictates how much off-load current your transformer takes from the AC supply and how much your core may be saturating. Clearly it shouldn't be tens of amps and, at the other end of the scale, going to something really low like sub 1 mA would be impractical. So you need to decide this. A general idea for 50 Hz and 230 volts would be about 10 henries.

So, given you have a target primary inductance you can achieve that by having a high permeability core with fewer turns or, a lower permeability core with more turns. However, a higher permeability core will saturate at a lower level of off-load ampere-turns so it's something you need to weigh up and getting this right is really important.

Some folk will answer this differently to me because I come from more of a ferrite-transformer background rather than an AC power background. But the principles involved are just the same.

The cross sectional area (\$A\$) of the core is the key as well as the mean length (\$\ell\$)of the magnetic core and these are inter-related with inductance, number of turns (\$N\$) and permeability (\$\mu\$): -



So now, if you have primary inductance you can determine the H-field in the core because this is: -

$$ \dfrac{amperes\cdot turns}{length}$$

And from all of this (and knowing the BH curve of your core material) you can see how much the core may saturate. If, for the inductance value you have chosen the core saturates too much you can either: -


Reduce the permeability of the core and wind more turns to get back to the same inductance (this is the route taken if copper losses are not enough to be problematic). Note that the benefit here is that (say) halving the permeability doubles the flux density handling but only requires \$\sqrt2\$ more turns to restore inductance hence, given the turns have increased by \$\sqrt2\$ the practical flux density handling has increased by \$\sqrt2\$.
Increase the cross sectional area of your core and possibly wind fewer turns to make the flux density less (this is the route taken if your power throughput is beginning to be limited by the copper losses). More cross sectional area means higher inductance so you can reduce the number of turns to restore the original value of inductance and, fewer turns means a smaller H-field and lower saturation levels.