  We consider the evolution of current-carrying cosmic string networks
described by the charge-velocity-dependent one scale (CVOS) model beyond the
linear equation of state regime, specifically focusing on the Witten
superconducting model. We find that, generically, for almost chiral currents,
the network evolution reduces dynamically to that of the linear case, which has
been discussed in our previous work. However, the Witten model introduces a
maximum critical current which constrains the network scaling behaviour during
the radiation era when currents can grow and approach this limit. Unlike the
linear model, only if the energy density in the critical current is comparable
to the bare string tension will there be substantial backreaction on the
network evolution, thus changing the observational predictions of
superconducting strings from those expected from a Nambu-Goto network. During
the matter era, if there are no external sources, then dynamical effects dilute
these network currents and they disappear at late times.
