  Ambipolar transport is a commonly occurring theme in semimetals and
semiconductors. Here we present an analytical formulation of the conductivity
for a two-band system. Electron and hole carrier densities and their respective
conductivities are mapped into a two-dimensional unit-less phase space.
Provided that one of the carrier densities is known, the dimensionless phase
space can be probed through magnetoresistance measurements. This formulation of
the two-band model for conductivity is applied to magnetoresistance experiments
on Ca$_3$Ru$_2$O$_7$. While previous such measurements focused on the
low-temperature limit, we cover a broad temperature range and find negative
magnetoresistance in an intermediate interval below the electronic transition
at 48 K. The low-temperature magnetoresistance in Ca$_3$Ru$_2$O$_7$ is
consistent with a two-band structure. However, the model fails to describe the
full temperature and magnetic field dependence. Negative magnetoresistance
found in an intermediate temperature range is, for example, not captured by
this model. We thus conclude that the electronic and magnetic structure in this
intermediate temperature range render the system beyond the most simple
two-band model.
