  Given a $n\times n$ random matrix $X_n$ with i.i.d. entries of unit variance,
the circular law says that the empirical spectral distribution (ESD) of
$X_n/\sqrt{n}$ converges to the uniform measure on the unit disk. Let $M_n$ be
a deterministic matrix that converges in $*$-moments to an operator $x_0$. It
is known from the work by \'{S}niady and Tao-Vu that the ESD of
$X_n/\sqrt{n}+M_n$ converges to the Brown measure of $x_0+c$, where $c$ is
Voiculescu's circular operator. We obtain a formula for the Brown measure of
$x_0+c$ which provides a description of the limit distribution. This answers a
question of Biane-Lehner for arbitrary operator $x_0$.
  Generalizing the case of circular and semi-circular operators, we also
consider a family of twisted elliptic operators that are $*$-free from $x_0$.
For an arbitrary twisted elliptic operator $g$, possible degeneracy then
prevents a direct calculation of the Brown measure of $x_0+g$. We instead show
that the whole family of Brown measures are the push-forward measures of the
Brown measure of $x_0+c$ under a family of self-maps of the plane, which could
possibly be singular. We calculate explicit formula for the case $x_0$ is
selfadjoint. In addition, we prove that the Brown measure of the sum of an
$R$-diagonal operator and a twisted elliptic element is supported in a deformed
ring where the inner boundary is a circle and the outer boundary is an ellipse.
  These results generalize some known results about free additive Brownian
motions where the free random variable $x_0$ is assumed to be selfadjoint. The
approach is based on a Hermitian reduction and subordination functions.
