Two-parameter motor models assume that the entire motor exhibits one dynamic temperature. Four-parameter thermal models are just as easy to use, but far more accurate — and better for calculating realistic hotspot safety margins.

In today's world of motion control, system designers and application engineers look for the highest performance and efficiency from the smallest, most rugged, and least costly servomotors possible. Different calculations are recommended for determining the optimum servomotor for an application.

Although no servomotor delivers infinite torque at any temperature, sometimes they're run to output the maximum peak torque allowed by the manufacturer for the longest possible time. It's in such situations that a servomotor's electrical winding can quickly overheat and even burn up.

To prevent overheating, some servomotors are listed with a high hotspot temperature safety margin, which is defined as the difference between the winding's maximum allowable hotspot temperature and the maximum continuous winding temperature.

Winding temperature and torque output

Most data sheets for both brush and brushless dc (BLDC) servomotors publish the value for each motor's maximum continuous winding temperature, plus corresponding maximum continuous current input and torque output — along with factors that affect the total ambient condition. The latter includes drive electronics, ambient temperature, any heat sinks, and so on. If the maximum continuous current is never surpassed, and the total ambient condition is equivalent to that specified by the manufacturer, a motor's maximum continuous winding temperature won't be exceeded. However, that's not the way a servomotor typically operates.

Instead, servomotors are often commanded to provide a dynamic motion profile containing one or more time periods during which the motor outputs peak torque greater than its maximum continuous value. This is why manufacturers also specify a peak torque for each motor. Depending on the manufacturer and model, a motor's peak-to-continuous-torque ratio typically ranges from 2:1 to 7:1.

Although it's normal for a servomotor to output peak torque in excess of its 1× maximum continuous value, if it does so for too long, then the motor's electrical winding will overheat and could even burn up. For this reason, during times of peak torque output, the motor's duty cycle should be limited to less than 100% — and with a peak torque that far exceeds the 1× maximum continuous value, duty cycle should be lowered even further.

Two-parameter thermal model

For more than 50 years, servomotors have been thermally characterized by what's generally called the two-parameter thermal model. Here, data sheets from most manufacturers for both brush and BLDC servomotors list a value for the motor's winding to ambient thermal resistance R_th (°C/W) plus the corresponding thermal time constant τ (sec) that allows calculation of motor thermal capacitance C_th (Joules/°C) as:

Using this two-parameter thermal model, both manufacturers and motor users attempt to size and select the optimum motor for each application. In fact, many manufacturers have even developed motor sizing programs — into which users supply all necessary application data, and the software determines which motor is optimal for the application.

The only flaw with these software packages is that no manufacturers are willing to size and recommend a competitor's motor; they (and their software) only report which of their motors is most suitable for an application.

For this reason, to effectively compare different manufacturers' motors (and find out which servomotor provides maximum performance density), motor users must size and compare available servomotors on their own.

Two publications have set the standard for servomotor sizing: The Electro-Craft Engineering Handbook (Electro-Craft Corp., DC Motors, Speed Controls, Servo Systems: First Edition, October, 1972) and a paper by S. Noodleman & B. Patel, Duty Cycle Characteristics for DC Servo Motors. (This was Paper TOD-73-30 presented at the 1972 IEEE/IAS Conference in Philadelphia.)

These classic references also outline the most common approach used today for calculating required duty cycle and dynamic motion profile, to ensure that motors won't overheat.

The Electro-Craft Handbook lists the first step in this sizing process: Accurately specify the dynamic motion profile to ensure that the motor won't overheat when executing the required motion profile (see Fig. 1) along with the total ambient condition — ambient temperature, heat sinks, forced cooling — in which the motor operates.

Next, along with motor engineering specifications, we must determine the peak torque and velocity required of the motor during the most demanding portions of the dynamic motion profile. (Again see Fig. 1.) This peak operation point is then entered onto the motor's combined intermittent and continuous torque-speed curves as shown in Fig. 2.

Finally, using the two-parameter thermal model in combination with a time-averaged power dissipation technique, we calculate the root-mean-square (RMS) torque and velocity for the entire motion profile — and then enter this RMS operation point into the combined torque-speed curves of Fig. 2.

If a peak operation point doesn't lie within the continuous torque-speed curve boundary, the selected motor is sure to fail, and a more robust and powerful motor must be chosen.

Conversely, an RMS operation point the lies within the boundary of the continuous torque-speed curve doesn't necessarily guarantee reliable operation. In real-world applications, maximum allowable winding temperatures are sometimes exceeded (and UL 1446 violated) even though the oversimplified two-parameter thermal model reports acceptable winding temperatures.