  In this study, we present a comprehensive analysis of the electroweak
sphaleron formalism and its application to electroweak phase transition (EWPT)
patterns in extensions of the Standard Model scalar sector with electroweak
multiplets. We offer an equivalence proof for different choices for the form of
sphaleron configurations; construct the previously unestablished
high-dimensional $\text{SU}(2)$ sphaleron transformation matrix; and revisit
the required boundary conditions needed for solving the sphaleron field
equations. We then investigate the leading order sphaleron dynamics in the
context of a multi-step EWPT. We showcase two distinct analytical approaches
for extending the $\text{SU}(2)$ scalar multiplet to the standard model (SM)
under differing EWPT scenarios, and perform an explicit calculation of the
sphaleron energy using a septuplet example. In the context of a single-step
EWPT leading to a mixed phase, we find that the additional multiplet's
contribution to the sphaleron energy is negligible, primarily due to the
prevailing constraint imposed by the $\rho$ parameter. Conversely, in a
two-step EWPT scenario, the sphaleron energy can achieve significantly high
values during the initial phase, thereby markedly preserving baryon asymmetry
if the universe undergoes a first-order EWPT. In both cases, we delineate the
relationship between the sphaleron energy and the parameters relevant to dark
matter phenomenology.
