Some of the main reasons for a phase plug are to:
a) Reduce path length differences between driver diaphragm and horn
throat
b) Cancel resonant modes in the front (throat) cavity (only the
important first few)
c) Reduce front cavity volume so that [Hf] may be extended (second
resonance)
d) Assist in determining (controlling) dispersion pattern
The formulas below will help with assessing some of the above.
[fp] = [c]/(2*[pi]*[h]*[T]  high pass cutoff frequency for phase plug
(Hz)
and
[W] = 4*([pi]^2)*([f]^2)*(X^2)*[p0]*[c]*[Sd]*[T] – acoustic power at
[St] (Watts)
where,
[c] = 345 m/s – sound velocity
[h]  diaphragm to phase plug clearance
[T] = [Sd]/[St]  compression ratio
[f]  reference frequency ([f] > [fc]) (Hz.)
[X]  diaphragm excursion
[p0] = 1.18 kg/m^3  air density
And if you are wondering where you take your diaphragm reference point
from, if you’ve done it right then [h] will be a constant and so can be
taken from any point along the inside edge. If a constant has not been
achieved then reading will be the mean diaphragm to inside edge
clearance.
The Power equation is acoustic watts
d) can include assisting in dispersion pattern by allowing the use of
near parallel and hence tight dispersion expanding horn walls that would
normally dictate a large throat section. The phase plug would be used to
occupy volume in the throat and reduce its area down to that dictated by
the mouth and length for the chosen fo.

