Multiphase induction machines are investigated as competitive magnet-free candidates for high torque-density, overload-capable, and wide torque-speed range applications. These benefits are enabled by torque-enhancing harmonic injection and phase-pole changes. However, such operational modes may subject the different harmonic planes to cross-saturation. Consequently, a well-performing drive requires an accurate model. This paper models the inter-plane cross saturation between the excited harmonic planes in the main flux path and the rotor slot-bridges. It is valid even for unsynchronized harmonic magnetic fields manifesting themselves during current control as low-frequency beats in the voltage amplitude. An advanced Gamma-model also including the skin effect of the rotor is proposed and validated against experiments. The results indicate that the proposed model represents the behavior of a 15-kW variable phase-pole machine more accurately than a constant parameter Gamma-model. This warrants better field orientation and torque output predictions in future drives.