We develop a theoretical model for a fluidic current source consisting of a via, a detour channel, and a push-up type micro-valve. The model accurately describes the non-linear behaviour of this type of device, which has been previously measured experimentally. We show how various structural parameters and material properties of the device influence the saturated flow rate and the minimum driving pressure required for the device to function as a current source. Conversely, the model can be used to design a fluidic current source with a desired saturated flow rate and low operational pressure. The present model can be straightforwardly applied to microfluidic circuits composed of many functional autoregulatory devices.