A lumped parameter, state space model for a tokamak transformer including the slow flux penetration in the plasma (skin effect transformer model) is presented. The model does not require detailed or explicit information about plasma profiles or geometry. Instead, this information is lumped in system variables, parameters and inputs. The model has an exact mathematical structure built from energy and flux conservation theorems, predicting the evolution and non-linear interaction of plasma current and internal inductance as functions of the primary coil currents, plasma resistance, non-inductive current drive and the loop voltage at a specific location inside the plasma (equilibrium loop voltage). Loop voltage profile in the plasma is substituted by a three-point discretization, and ordinary differential equations are used to predict the equilibrium loop voltage as a function of the boundary and resistive loop voltages. This provides a model for equilibrium loop voltage evolution, which is reminiscent of the skin effect. The order and parameters of this differential equation are determined empirically using system identification techniques. Fast plasma current modulation experiments with random binary signals have been conducted in the TCV tokamak to generate the required data for the analysis. Plasma current was modulated under ohmic conditions between 200 and 300 kA with 30 ms rise time, several times faster than its time constant L/R approximate to 200 ms. A second-order linear differential equation for equilibrium loop voltage is sufficient to describe the plasma current and internal inductance modulation with 70% and 38% fit parameters, respectively. The model explains the most salient features of the plasma current transients, such as the inverse correlation between plasma current ramp rates and internal inductance changes, without requiring detailed or explicit information about resistivity profiles. This proves that a lumped parameter modelling approach can be used to predict the time evolution of bulk plasma properties such as plasma inductance or current with reasonable accuracy; at least under ohmic conditions without external heating and current drive sources.