In the first installment of this series, "Calculating a practical hotspot safety margin", we discussed how two-parameter thermal models assume that the entire motor, including windings, exhibits one dynamic temperature. Servomotor manufacturers and users most commonly apply this oversimplified thermal model to make duty-cycle calculations -- and most manufacturers publish only one value for any given motor's winding-to-ambient thermal resistance R_th with the thermal time constant. For this reason, motor users must make duty-cycle calculations using this two-parameter model, unless they test and measure operating motor temperatures in house, which is unlikely.

An argument could be made that with the right sensors to protect a motor from overheating, the simplified model is acceptable. However, as we'll explore in this installment, even this approach has limitations.

Motor ratings

Data sheets for typical brushless dc (BLDC) servomotors normally include continuous or safe-operating-area torque-speed curves plus intermittent peak curves. A safe-operating-area torque-speed curve defines a torque-speed boundary within which a motor can operate safely and indefinitely without exceeding its maximum continuous operating temperature when powered by a specified drive under a specific ambient condition.

Published peak torque-speed curves typically specify at least a 2:1 peak-to-continuous torque ratio, or even a 4:1 or 5:1 ratio where a motor is rated for it. However, commanding a servomotor to output peak torque greater than its 1x maximum continuous value for too long causes overheating.

Case in point: 4x peak torque corresponds to 16x power dissipation in the motor's electrical winding. Why? As torque output increases linearly with current, electrical resistance power dissipation amounts to I²R where I equals current and R equals winding resistance.

In short, commanding a servomotor to output peak torque is normal (and allowed) but duty cycle must be kept to below 100% or the motor windings can overheat and even fail.

Thermal models

Used for over 50 years, the two-parameter thermal model assumes that any given motor has one dynamic operating temperature, and one winding-to-ambient thermal resistance R_th (¢XC/watt) in parallel with thermal capacitance C_th (joule/°C). It's a model analogous to a simple RC electrical circuit: Solving this two-parameter thermal model for both constant power dissipation heatup and zero power dissipation cooldown shows that the motor both heats up and cools down predictably and exponentially with thermal time constant T (sec) where T = R_thC_th. For this reason, servomotor data sheets generally report both R_th and T -- to allow the calculation of a motor's two-parameter thermal capacitance.

A four-parameter thermal model is more modern and accurate in predicting winding temperature when current supplied to a servomotor exceeds 1x continuous current. Measurements show that even within a motor's windings, there can be significant temperature differences.

Furthermore, thermodynamics dictate that for heat power to flow from within the motor towards its outer exposed surface area (and ultimately into the surrounding ambient environment) there must be a temperature gradient within the motor, and between it and the environment.

Depending on motor size and operating temperature, there can be as much as a 30°C to 50°C temperature difference between the motor's electrical winding and its exposed outermost surface area -- too significant to ignore. The motor's winding has its own dynamic operating temperature and thermal resistance and a thermal time constant that differs from the rest of the motor.

Our 60-mm example motor exhibits winding and case heatup with 1x constant power dissipation.

The four measured parameter values (and four-parameter thermal model) for one particular 60-mm-diameter servomotor are shown in Fig. 1. With 1x constant power dissipation, winding temperature begins rising immediately from 25°C ambient temperature. However, the case (and motor body) exhibits a temperature-rise time lag. As we'll explore shortly, this lag in central to the limitations of temperature sensors used for protection against overheating -- particularly if the sensor isn't attached directly to the winding.

For our 60-mm servomotor, winding temperature ultimately stabilizes at its rated 130°C maximum continuous value; case temperature stabilizes at 100°C when surrounded by 25°C ambient air temperature. However, this 30° gradient presents another problem: Where should temperature sensors be installed, and what type should be used? On-winding installation may seem best, but as we'll detail, this arrangement exposes the system to nuisance shutdowns.