What is the relationship between spool stroke and flow rate, and what is partial stroke good for?

As the spool advances, the metering area opened grows and the flow rate passing through increases; but the relationship is not perfectly linear. In the overlap region flow is zero, in the middle region the area opens rapidly, and as the spool approaches full stroke the area saturates and additional travel does not increase flow. Partial stroke is the basic way to drive an actuator slowly and under control.

In a proportional valve, flow rate depends on two things: the metering area opened and the pressure differential across the valve. The area is set by spool position, but the flow passing through varies with the square root of the pressure differential. That is why speed changes if the load changes at the same spool position.

The regions of the stroke:

The benefit of using partial stroke is that slow movement can be obtained without throttling pump flow rate: the excess oil returns through the relief valve or the center passage, and only the desired flow goes to the actuator. The cost is that the pressure drop across the throttled oil turns into heat. Working at very low speed for long periods raises the oil temperature; in such cases a pressure compensated valve or speed control is more efficient.

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