Just to add some more bits of info for this
Yes the equiv cct for a brushed-DC (as well as a brushless) is a R-L-E load
The R is effectively static (although temp will play a part in it a bit).
The L isn't really it will have some variation across commutation (can be viewed as none-changing for Brushed-DC, for brushless this change can be for a well designed machine 20% change across commutation zones)
The E however, is a speed dependent potential
There are two constants with machines Kt and Ke (for an ideal machine they are equal, in practice not far off)
Kt is the torque constant T = Kt*i
Ke is the backEMF constant, E = Ke*w (w = speed of rotor in rad/s)
at zero-speed no potential is developed, with increase speed the machine acts like a generator and a potential is developed.
say you have this 3V battery for a stalled rotor and the terminal resistance is 3Ohms, then 1Amp will flow (not 3Amps :wink
since it is DC the inductor will eventially not play a part.
Say we start spinning AND the rotor is loaded, if we spin at say 3000rpm with the same 3V battery and say the backEMF at thios speed is 1V, it means we only have 2V available to force current in (ie a maximum of 0.7Amps) THUS if we want to spin at 3000RPM we can ONLY do so IF the load is such that it doesn't require 0.7A
We can spin at 3000rpm un-loaded and say only 50mA is being drawn. Why so low? because of the inductance and commutation. A machine will only draw as much current as it need to meet the Torque-speed & power-speed of that operating point.
The inductance will limit the di/dt BUT more current will cause the rotor to spin faster, a faster spinning rotor will cross a commutation point sooner THUS the level that the current reaches is dependent on this
Start to load the machine and the rotor will slow down, this means current has a chance to flow (in that commutation zone) for longer giving it time to build to a higher level (to produce more torque) result is the speed catchs up to the 3000rpm BUT more current is being drawn
IF the load is too great for what the machine can deliver at that speed (part of the Torque-speed curve) the speed will collapse to meet that torque