  We have developed a comprehensive theoretical model for predicting the
magnetic field response of a parallel SQUID array in the voltage state. The
model predictions are compared with our experimental data from a parallel SQUID
array made of a YBCO thin-film patterned into wide tracks, busbars and leads,
with eleven step-edge Josephson junctions. Our theoretical model uses the
Josephson equations for resistively shunted junctions as well as the second
Ginzburg-Landau equation to derive a system of coupled first-order nonlinear
differential equations to describe the time-evolution of the Josephson junction
phase differences which includes Johnson noise. Employing the second London
equation and Biot-Savart's law, the supercurrent density distribution is
calculated, using the stream function approach, which leads to a 2D
second-order linear Fredholm integro-differential equation for the stream
function with time-dependent boundary conditions. The novelty of the model is
that it calculates the stream function everywhere in the thin-film structure to
determine during the time-evolution the fluxoids for each SQUID array hole. Our
numerical model calculations are compared with our experimental data and
predict the bias-current versus voltage and the voltage versus magnetic field
response with unprecedented accuracy. The model elucidates the importance to
fully take Meissner shielding and current crowding into account in order to
properly describe fluxoid focussing and bias-current injection. Furthermore,
our model illustrates the failure of the simple lumped-element approach to
describe a parallel SQUID array with a wide thin-film structure.
