Hi Nate. Thanks for your post.
Could you clarify a couple of your statements please?
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Pmech ~ Pelec (the mechanical (input) power of the motor is approximately proportional to the electrical (input) power of the motor)
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What do you mean by the phrase "the mechanical (input) power of the motor"? The only input power to the motor is electrical.
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Pmech ~ RPM^2 (the mechanical power is proportional to the square of the rotational speed of the armature)
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What mechanical power are you referring to here?
For a constant load torque, the mechanical output power of the motor is proportional to the rpm, not the square of the rpm.
For a constant applied voltage and a varying load torque, the mechanical output power first increases and then decreases as the load torque is varied from 0 to stall.
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RPM ~ V (the rotational speed of the armature is proportional to the input voltage)
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The speed is
proportional to the applied voltage only when the load torque is zero and the no-load current is small enough to be ignored. Since my question was about not ignoring the behavior of the no-load current, using this relation would be circular reasoning.