What is the DC motor equation?

[PowerKid]

What is the DC motor equation?

VNOM = Voltage Nominal
RCOIL = Coil Resistance
LCOIL = Coil Inductance
LOAD
MASS
OMEGA = (2 * PI) * (RPM / 60)
RPM

RPV = ((2 * PI) * (RPM / 60)) / VNOM

RPM_SCALE = 60 / (2 * PI)

PI = 3.14159265

RPV = OMEGA / VNOM

RPV = Revolutions Per Volt

What is the equation of RPM based on VNOM, RCOIL, LCOIL, LOAD, MASS, OMEGA, RPV?

DC MOTOR MODEL​

Introduction​

The VSM DC Motor model offers considerably more sophistication than the Active Motor model supplied as standard with ProSPICE. Specifically it correctly models:

• Nominal operating voltage and free running RPM.
• Coil (armature) resistance and inductance.
• Back EMF, such that the motor generates a DC voltage opposing the supply that increases in direct proportion to its angular velocity.
• Load resistance (i.e. friction), specified as a percentage of the maximum (stalled) torque available at the nominal supply voltage.
• Load/Armature inertia, such that the motor will continue to spin after the supply is removed.

Theory​

The DC motor uses a combination of schematic and programmatic modelling techniques. The schematic model is shown below, and demonstrates rather nicely how electrical circuits may be used to simulate mechanical phenomena.

The model corresponds with mechanical reality as follows :

• The armature's properties as a simple inductor are modelled by R1 and L1. This does not accurately model the chopping effect of the commutator, but is sufficient to correctly simulate the load current when stalled, and the back-emf spike that is generated by the armature inductance when the supply is removed.
• Voltage controlled current source G1 is used to measure the current flowing through the armature. The torque developed by the motor is directly proportional to this value. In reality the constant of proportionality involves a number of parameters that for convenience we have subsumed into <VNOM> and the free running angular velocity <RPM>. To avoid overly long formulae, we have then defined <OMEGA> as <RPM> expressed in radians per second, and <RPV> as the ratio <OMEGA> / <VNOM> - a kind of revs. per Volt, if you will.
• The torque current is integrated by capacitor C1 such that the voltage developed across it represents the angular velocity <OMEGA> of the motor in radians per second. C1 has the value of the motor's moment of inertia - the rotational equivalent of its mass. This value has units of gm². The angular velocity voltage is measured by VP1 and fed to the VSM animated rotor model which displays it in the motor's readout panel.
• Voltage controlled current source G2 scales the angular velocity into a current representing degrees of rotation per second, and this value is integrated by capacitor C2 to produce a voltage representing the angular position of the motor. This voltage is measured by real-time voltage probe VP2, and fed to the VSM animated rotor model which then draws the rotor blades at the appropriate angle.
• Since the rotation of the armature occurs within a magnetic field, it acts an electric generator, producing a voltage proportional to the angular velocity. This effect is modelled by the voltage-controlled voltage source E1, which sets up a 'back-emf' voltage opposing the drive current. As the motor's angular velocity approaches the user specified free-running <RPM>, E1 produces a back EMF equal to the motor's nominal running voltage - <VNOM>. At this speed, no current flows in or out of the motor, and it neither accelerates nor decelerates.
• The final touch is resistor R2 which models the mechanical load on the motor. In reality this is a very complicated subject, since the friction acting against a motor's rotation is made up of many complex components. Many of these will produces forces which are non-linear with respect to the motor's velocity and even its absolute position. For the sake of simplicity, we have assumed the frictional force to be directly proportional to velocity, and so this is basically a torque current that subtracts from the torque applied by the armature. The formula for R2 is contrived such that the motor's terminal velocity will reduce in proportion to the load percentage; this effect is independent of <VNOM>, <RPM> or <RCOIL>.
• For accurate modelling of specific loading conditions, the resistor could be replaced by an arbitrary controlled current source with voltage inputs that could be taken from the velocity and/or absolute position voltages.

 

[BradtheRad]

I suppose a comprehensive equation can be found using all those variables. Whoever devised it took on a major challenge trying to inter-relate them.

Motor rpm is determined at the balance of equilibrium between power In versus power Out. Or as a related standpoint: how much is the mechanical resistance, and how much electric power overcomes it? You have load as a variable, and voltage as a variable.

It's hard to use just one equation in order to calculate where is their equilibrium point. Perhaps the math involves simultaneous equations?
Or as an alternative, if we set one of those variables to be unchanging (either load or voltage), then we might run tests to end up with a graph of resulting rpm.

Some conditions are not normal running conditions. For instance, stall the motor and read its current draw. Or run it with no load at various supply voltages. By discovering its parameters when operating at extremes, we might obtain values for parameters which are part of the equation... and thus construct the equation or else understand it better when we find it.