The linear plasma response to a resonant error field (EF) is formalized by the so-called ‘delta-prime’, here simply denoted by , a quantity representing the amplitude and phase of the current sheet induced at the resonant surface mostly by the electron fluid rotation. A semi-analytical computation of in the framework of two-fluids drift magneto-hydrodynamic is carried out by exploiting a formalism similar to those of previous works (Fitzpatrick 2022 Phys. Plasmas29 032507; Park 2022 Phys. Plasmas29 072506). However, instead of assuming the high-poloidal-beta ordering of these derivations, a conventional large aspect ratio, low-beta ordering is here adopted. On the basis of , we estimate the EF penetration threshold in terms of plasma density, magnetic field strength and machine size, by inspection of the torque balance equation at the resonant surface. This is done for the ohmic tokamak, by exploiting the experimental scaling laws obtained in this configuration for several kinetic quantities entering the model. The results are compared to the prediction from the single-fluid as obtained in a previous analysis (Zanca 2025 Nucl. Fusion65 056023). No significant difference is found between the two-fluids and the single-fluid EF thresholds. Instead, the modeling of the neoclassical poloidal-flow damping time and of the momentum confinement time , two quantities that enter the torque balance equation beside , has a significant impact on the EF threshold. In particular, by identifying with the ion energy replacement time, a positive scaling of the EF threshold with the machine size is obtained, in agreement with recent experimental analysis (Logan et al 2020 Nucl. Fusion60 086010), in addition to a linear dependence on density: , with the major radius. Instead, the customary identification of with the total energy confinement time leads to a negative scaling with the machine size and to a weaker density dependence.