  The requirement for solving nonlinear algebraic equations is ubiquitous in
the field of electric power system simulations. While Newton-based methods have
been used to advantage, they sometimes do not converge, leaving the user
wondering whether a solution exists. In addition to improved robustness, one
advantage of holomorphic embedding methods (HEM) is that, even when they do not
converge, roots plots of the Pad\'e approximants (PAs) to the functions in the
inverse-$\alpha$ plane can be used to determine whether a solution exist. The
convergence factor (CF) of the near-diagonal PAs applied to functions expanded
about the origin is determined by the logarithmic capacity of the associated
branch cut (BC) and the distance of the evaluation point from the origin.
However the underlying mechanism governing this rate has been obscure. We prove
that the ultimate distribution of the PA roots on the BC in the complex plane
converges weakly to the equilibrium distribution of electrostatic charges on a
2-D conductor system with the same topology in a physical setting. This, along
with properties of the Maclaurin series can be used to explain the structure of
the CF equation We demonstrate the theoretical convergence behavior, with
numerical experiments.
