Suppose I have a DC motor and I put a load on it (in fact, the loading part isn't quite necessary for my question) but I have a load on the motor shaft and I connect it to a power supply. I can calculate what the final rotational speed of the motor would be given all the parameters (voltage of power supply, armature resistance, ke, etc) and I know that the torque of the motor would eventually equal to the loading torque (if loaded, 0 ideally if not loaded) and the torque would get driven down with time due to the back emf from the motor, but my question is this: what would be the start up torque of the motor? If there was no load on the motor, then a finite start up torque would be needed to accelerate the motor shaft due to its rotational inertia, but what would this value be?
I can derive a torque speed characteristic for the motor because I have all its parameters, so would the torque just on start up be the same as the motor's stall torque (because both lie at omega = 0 on the torque-speed characteristic curve)? The same goes for loading: what would be the torque just as the motor starts (not just to start the motor)? I have a section here in my lecture notes that's implying the startup torque is equal to the loading torque but that makes no sense to me: as the motor shaft accelerates, the back emf would drive the current (and therefore torque) down, so surely the torque on startup should be greater than the load torque (otherwise it'd be impossible for the motor shaft to accelerate). I guess my question is, is there a way to calculate the torque on startup using DC analysis and the specifications of the motor? Thanks for all your help.