So, you want to know how to calculate the actual winding temperature of a Parker permanent magnet (PM) servo motor? In the course of applying rotary motors in applications, it sometimes becomes desirable to figure out the actual winding temperature given certain data.

### Setting the context

First off, we will define some terms:

### Calculating the winding temperature

First, calculate the resistance of the winding at the measured or assumed RMS current from the drive to the motor. The equation for this is:

R_{hot} = R_{25C} *(0.90175 + 0.00393T_{w})

*Equation 1 *

This will give the value for R_{hot} in terms of the unknown value, T_{w} - our desired term.

Next, the losses due to shaft rotation must be calculated. These losses are the case losses, W_{c}, and are due to the motor friction and motor damping. Fail safe brake and gearbox losses can also add to the case losses - but will be omitted from this example.

W_{c }= (FT * w) + (B * w^{2})

*Equation 2 *

The friction and damping values are supplied by Parker.

Once the case losses are calculated, the resistive losses should be calculated. This equation is:

W_{r} = 1.5(I_{rms})^{2} * R_{hot}

*Equation 3 *

W_{r} will be given in terms of T_{w} once we plug in the equation for R_{hot}.

W_{r} = 1.5(I_{rms})^{2} * (R_{25C} *(0.90175 + 0.00393T_{w})

*Equation 4 *

Insert all known values and simplify. Equation 4 will be reduced to a simple form with T_{w} still as an unknown. The final equation for the winding temperature is as follows:

T_{w} =[(R_{th c-a}) *(W_{c} + W_{r}) + (R_{th w-c}) * W_{r}] +25

*Equation 5 *

Plug in the simplified Equation 4 for W_{r} and the thermal resistance values supplied by Parker. Simplify. Now you can easily solve for the winding temperature, T_{w}.

If you prefer, you can put everything in terms of T_{w} and plug in the values at the end to solve, but the equation gets long and complicated.

Also, if you are able to reliably measure the case temperature, you can increase the accuracy of the winding temperature calculation and simplify the Equation 5 to the following:

T_{w} = T_{c} + (R_{th c-a}) *(W_{r}) + 25

*Equation 6 *

At that point, the case temperature would include all losses and cooling due to convection and conduction.

### Example calculation

Find the winding temperature (T_{w}) for the BE232D rotating at 5000RPM with RMS current of 1.8A_{rms}

Plugging in the 7.72 for R_{25C} and simplifying, Equation 1 becomes:

R_{hot} = R_{25C} *(0.90175 + 0.00393T_{w})

R_{hot} = 6.962 + 0.03034T_{w}

Converting 5000 rpm to radians/second gives us 523.6rad/s.

Plugging this and the friction and damping values into Equation 2 yields:

W_{c} = (FT * w) + (B * w^{2})

W_{c} = (.014123 * 523.6) + (.00003278*523.62)

W_{c} = 16.377 W

Find the resistive losses for the RMS current of 1.8A from Equation 3 and 4:

W_{r} = 1.5(I_{rms})^{2} * R_{hot}

W_{r} = 4.86 * (6.962 + 0.03034T_{w})

W_{r} = 33.84 + 0.147T_{x }

Finally, use Equation 5 to put it all together and solve for T_{w}:

T_{w} =[(R_{th c-a}) *(W_{c} + W_{r} )+ (R_{th w-c}) * W_{r}] +25

T_{w} =[(1.02*(16.377 + 33.84 + .147T_{w})) + (.56 * (33.84 + .147T_{w}))] +25

This simplifies to:

T_{w} = 95.17 + .2323T_{w}

T_{w} = 124^{o}C

The speed and the current happen to be the rated speed and current for the BE232D motor.

If you would like to calculate the winding temperature for your motor, you will need to contact your local Parker distributor for some of the unpublished data required to perform the calculations. Let them know to look on the Parker Distributor Extranet Product Pages for this unpublished data.

Article contributed by Jay Schultz, Product Manager-Motors, Parker Electromechanical Automation North America. Originally published on ParkerMotion Blog June 26, 2013.

Other articles by the Electromechanical Team:

What You Should Know About Frameless Motors

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