Efficiency also depends on the motor mechanical load. A common type of load is a "square-law" load, typical of fans and pump impellers:

T_L = a× w ²

A PMS or brush motor has a torque-speed curve described by the equation:

where the stall (maximum) torque is

and the no-load (maximum) speed is

L is the mechanical flux linkage, the motor parameter that relates its mechanical and electrical quantities, and R is the motor winding resistance.

The motor torque can be plotted along with the fan load torque, as a load line. Where the two intersect is the operating point for the motor-fan mechanical "circuit." For a concrete example, assume the following parameters:

L  = 50 V/krpm = 477.465 mV× s, R = 75 W , V_s = 150 V

Furthermore, for the fan-load parameter, a, three values will be assigned:

a₀ = 5 μN× m× s², a₁ = 20 μN× m× s², a₂ = 50 μN× m× s²

Three fan curves are then plotted with the motor curve, as shown below.

The operating-point speed can be found by equating motor and fan torque and solving for w _L. The result is:

For the three a parameters, the load speeds are:

w_L(a₀) = 2.181 krpm, w_L(a₁) = 1.484 krpm, w_L(a₂) = 1.061 krpm

For which load is the motor operating most efficiently? An efficiency equation can be derived as the mechanical (output) power divided by the electrical (motor input) power. The mechanical power is:

The motor torque is L × i = T_L. Then the motor current is: i = T_L /L  = a× w ²/L . Electrical power is:

 Substituting for i and reducing,

The efficiency formula can now be written as

When efficiency is plotted as a function of speed, with a as parameter, the curves shown below result.

The efficiencies at w _L(a_x) are:

h (a₀) = 0.727, h (a₁) = 0.495, h (a₀) = 0.354

What conclusions can be drawn from these results? As the motor speed increases, efficiency decreases. The higher the frequency at which a fan load intersects the motor curve, the quicker the efficient drops off with speed. For a given mechanical pumping power,

where Q is the volumetric flow rate of fluid pumped, and D P is the pressure drop across the pump. The pressure-flow-rate plots for pumps look similar to PMS motor torque-speed curves. For an optimal system, the pump flow-rate as a function of speed is needed, and the pump efficiency with speed. Then the overall system efficiency can be optimized. But for the motor, the slower the more efficient. A large-radius, slow impeller maximizes motor efficiency.