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If experimental data is obtained from a proportional solenoid valve, specifically flow vs current (assuming inlet pressure is constant), the fit is nearly quadratic in nature.

Andersen's Analysis and Design of Pneumatic Systems predict that the flow area of an orifice is linearly proportional to the flow rate for constant inlet pressure. But for a valve, the magnetic characteristics must also contribute a relationship between applied current and displacement of the valve poppet which changes flow area of the orifice.

Is the electromagnetic element the main factor leading to the quadratic flow vs current relation, or are other factors involved?

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