  We report the measurement of cumulants ($C_n, n=1\ldots4$) of the net-charge
distributions measured within pseudorapidity ($|\eta|<0.35$) in Au$+$Au
collisions at $\sqrt{s_{_{NN}}}=7.7-200$ GeV with the PHENIX experiment at the
Relativistic Heavy Ion Collider. The ratios of cumulants (e.g. $C_1/C_2$,
$C_3/C_1$) of the net-charge distributions, which can be related to volume
independent susceptibility ratios, are studied as a function of centrality and
energy. These quantities are important to understand the quantum-chromodynamics
phase diagram and possible existence of a critical end point. The measured
values are very well described by expectation from negative binomial
distributions. We do not observe any nonmonotonic behavior in the ratios of the
cumulants as a function of collision energy. The measured values of $C_1/C_2 =
\mu/\sigma^2$ and $C_3/C_1 = S\sigma^3/\mu$ can be directly compared to lattice
quantum-chromodynamics calculations and thus allow extraction of both the
chemical freeze-out temperature and the baryon chemical potential at each
center-of-mass energy.
