Matching a hydraulic pump to a PMDC motor

Matching a gear pump to a permanent magnet DC (PMDC) motor presents a challenge because the motor speed decreases linearly with torque, but analysis is easy because both the pump curve and the motor curve are perfectly linear.

As discussed in an earlier post, manufacturer's pump curves can be adapted to predict torque requirements and volumetric efficiency at any combination of rpm, viscosity, and pressure. This information is required before proceeding because the volumetric efficiency of the pump will be strongly affected by rpm, which for a PMDC motor will decrease with load (pressure).

Motor manufacturers publish motor curves which are in the form of rpm and current draw as a function of flow rate. Be aware that the "motor constant" approach in electrical engineering textbooks is wrong: this assumes that current is zero at zero torque; in practice, current draw is not zero at zero load due to i2R, hysteresis, and windage losses. However, both rpm and current as a function of torque are linear functions. Consider the following motor curve from Allied Motion:

Units of torque are in oz*in.

Solving for Motor Curves

Take two data points from the motor curve such as above, for rpm "N" and current "I" at two values of torque "T", and solve as follows:

Where c1 is the no-load rpm and c2 is the slope of the rpm vs. torque curve; c3 is the no-load current and c4 is the slope of the current vs. torque curve. Note the correction to c1 for varying supply voltage, which is discussed below. Obviously, these equations will need to be corrected to keep the units consistent.

Once these constants have been determined, the motor curves can be plotted. In this case the curves are plotted at 12, 8, and 6 volts.

I have plotted a single current curve on the same graph, obviously this is scaled differently than the rpm curves.

Working from this, power out = torque * omega, power in = volts * amps, which allows motor efficiency to be determined.

Effect on Motor Curves of varying supply voltage

These curves are for a stated supply voltage, in this example 12 V. Note the high current that this type of motor can draw: the electrical resistance of the wiring harness and the tolerance on the supply voltage must be taken into account, as these can easily consume a few volts at high loads, which will have a disproportionate effect on the performance of the pump. Also be aware that the motor performance curves are subject to tolerances: consult with the supplier. The effect of changing the supply voltage is to change the no-load rpm, which varies linearly with voltage. The slope of the voltage vs. torque curve is unchanged, and the current vs. torque curve can be treated as identical, independent of voltage.

You heard correctly: at 8, 12, or 14 Volts, you can assume that the above current vs. torque curve does not change. The reason for this is simply the physics of torque creation: more current = more force on a moment arm, i.e. torque.

Max recommended operating point of PMDC motors

Before proceeding further, you must take into account that PMDC motors are not designed to be loaded steady-state above their peak efficiency. Therefore, use the rpm and current curves from the motor curve to solve for the unknown constants of the linear functions, and then plot the the efficiency and power output curves. Design your system to not load the motor above its peak efficiency point.

Producing a Pump Curve for the System

Your goal is to produce the flow rate vs. pressure curve for the pump-motor system. Start by choosing a candidate motor and pump. I'm assuming you have used the previous post to determine the pump displacement, friction variable, and leakage resistance at the viscosity of your desired fluid. Since the resulting flow rate vs. pressure curve is linear, you only need perform this calculation at two points.

The approach we will be taking is to understand that motor power output is used to produce 3 things:

1. Useful work
2. Internal leakage
3. Viscous friction

The power consumed by internal leakage is handled automatically, since the leakage fluid is pumped to the output, where it leaks back to input. So the power consumed is simply the pressure * the total flow rate, where the total flow rate is simply the pump displacement  * the rpm. However, we must take internal leakage into account when calculating the actual volume produced by the pump.

To simplify the math, use zero pressure as one of the data points since 100% of the motor torque is to combat viscous friction, with zero leakage losses.