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How do I go about finding the load resistance and load inductance of a brushed DC electric motor? I would guess that:

Load resistance = Operating voltage / Load current

These values you can find in the datasheet. To find motor inductance, you operate the motor with a "nominal load" so that it uses an average current with the same value as the "load current" from the datasheet. You then need to take a scope plot of the PWM control signal and motor current from the current sensor (isense). Because inductance is defined by:

V = L di/dt

where V = the rising voltage of the PWM signal
di = the change in isense
dt = the change in time where the PWM signal is High

inductance calculation

Then solve for L which would be load inductance. Is this method correct?

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You seem to be asking about the resistance that a DC motor represents as the load on an electronic speed controller. That resistance is very low and is not operating voltage divided by load current. Armature Resistance = (Operating Voltage - Back EMF) / Load Current. The back EMF is proportional to motor speed. The armature inductance could be determined somewhat as you suggest, but with a low voltage applied and the motor shaft clamped down to prevent is from turning, commonly known as "locked-rotor" or "blocked-rotor."

I believe the rotor resistance is also determined using a locked-rotor test. In both cases, there may be some variation with different armature positions due to the way in which the brushes transition between commutator segments. It may also be desirable to determine the brush resistance separately.

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