Episode #45: Servo Motor Speed & Torque Curve Formulas Explained

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The Motion Control Show

In the last episode I talked about how to understand a servo motor’s speed/torque curves.  Now I'm going to go a little bit further and explain how that is developed.  If you have any questions, reach out to us here at this email address and website.  I’m Corey Foster at Valin Corporation.  I hope this helps.

There's the KT or Torque Constant.  When current passes through a conductor, the copper wire, that is located in the magnetic field, it creates a force that is exerted on the conductor.  Now, if that conductor is held steady, the magnets are going to move.  If the magnets are held steady, the conductor is going to move.  In the case of a motor, the conductors, the copper wires, are held steady and therefore the magnets move, and that's how we control the turning of the rotor inside of a motor.  This is the product of the field strength times the conductor length times the current magnitude.  It comes in units of either ounce-inches per amp or Newton-meters per amp depending if you're looking at English or metric. 

There's also a Ke Voltage Constant that is defined by when a force is applied to that conductor located in the magnetic field, causing it to move, a voltage is induced in that conductor.  Again, this is the product of the field strength, the conductor length, and this time the conductor velocity.  It ends up with units of volts per 1000 RPM or volts per Radian per second, again depending on English or metric.

There's a definite relationship between the two.  Both of them are proportional to the same motor parameters, so there are essentially two different ways of measuring the motor operation in a reciprocal relationship, and here are the reciprocations here.  The volts are units of the peak of the sine wave.  Since the KE, the voltage, is the easier of the two to measure, the KT is typically derived from the KE measurement.

There are definitely some losses that have to be taken into account when measuring the KT and KE and seeing how much power goes through a motor and how efficient the motor is.  There's the I2R losses.  There are eddy currents that are in the stator’s iron structure.  There's hysteresis.  Some of those losses increase exponentially with the velocity of the motor.  Others are affected by the stator design, the nature of the stator iron, the air gap, etc.  So that's why you get some variations in the motor designs and how well they perform. 

There's this thing called a thermal resistance.  The total thermal resistance, Theta, is the sum of the winding-to-the-case, the case-to-mounting-surface, and the mounting-surface-to-ambient thermal resistances.  The winding-to-the-case is a function of the internal motor design.  The Theta case-to-the-mounting is usually very low and determined by how the pilot is mounted to the surface connection.  Then there's the mounting-to-ambient, which is the most variable and it's determined by the mechanical design of the application.  I referred to this in the last episode where the motor could be in a box and it has no air cooling.  But, you have to look at the size, the material and at the cooling.  You have to look at all the assumptions that come into play here.  If the parameters of a particular application are known, you end up with a particular equation that looks like this.

Now the KE Voltage Constant, if we were to graph it for a particular motor, you might have a certain amount of torque here.  Maybe this is ounce-inches over here, 250 ounce-inches, and then you have a speed of maybe 6000 RPM.  This is what the KE voltage defines.  This is what the KT Torque Constant defines where you get a really high speed here, but a lower torque here.  What the motor is capable of is the overlap and intersection of the two.  So here is the continuous capability of that motor.  That is defined by this thermal, or the KT Torque Constant, but it's defined by the thermal properties here, but it's limited by the voltage properties here.  This is what can be done all day, every day.  This is 100% duty cycle. 

The peak current is really defined by some assumptions by the designers.  Typically in the industry, it's three times the continuous.  So, if we plot a 3X curve here, this is 3 times the continuous.  We could potentially go a little bit higher, but it's a really, really short amount of time the motor can produce that.  This is typically what the industry is settled on.  Sometimes you see 2X curves instead, but that's going to be the peak region capabilities there.  What you end up with is looking like this speed/torque curve here.  That's pretty typical with continuous here, with a peak here, and you have a slope that is defined by the BackEMF here.  The faster a motor goes, it turns into a generator.  And as the motor spins up, it creates the BackEMF.  That's the voltage that it produces.  When that voltage meets the voltage going in, that's the maximum speed.  The maximum voltage going in to the motor is the bus voltage of the drive.  Let's say it's 120 VAC, which gets rectified to 170 VDC.  When that motor spins fast enough in order to get that 170 VDC BackEMF, the motor can't go any faster because there's no more potential from the drive.

I hope that helps.  I'm Corey Foster at Valin Corporation.  Reach out to us here at this email address and website.  We're happy to help.

If you have any questions or are just looking for some help, we're happy to discuss your application with you.  Reach out to us at (855) 737-4716 or fill out our online form.