  A numerical model based on a lumped circuit element approximation for a
bi-superconducting quantum interference device (bi-SQUID) operating in the
presence of an external magnetic field is presented in this paper. Included in
the model is the novel ability to capture the resultant behaviour of the device
when a strong electric field is applied to its Josephson junctions by utilising
gate electrodes. The model is used to simulate an all-metallic SNS (Al-Cu-Al)
bi-SQUID, where good agreement is observed between the simulated results and
the experimental data. The results discussed in this work suggest that the
primary consequences of the superconducting field effect induced by the gating
of the Josephson junctions are accounted for in our minimal model; namely, the
suppression of the junctions super-current. Although based on a simplified
semi-empirical model, our results may guide the search for a microscopic origin
of this effect by providing a means to model the voltage response of gated
SQUIDs. Also, the possible applications of this effect regarding the operation
of SQUIDs as ultra-high precision sensors, where the performance of such
devices can be improved via careful tuning of the applied gate voltages, are
discussed at the end of the paper.
