### Thermally Modeling a Four-Parameter Servomoter

## by Courtney Cline

Submitted : Spring 2015

In this project, a schematic diagram of a four-parameter motor at Exlar Corporation was given. From the schematic diagram, an equation for the change of winding temperature as a function of time and an equation for the change of case temperature as a function of time were asked to be found. First, differential equations describing the schematic diagram were made for the case temperature (T_{c}) and winding temperature (T_{w}) relative to the ambient temperature (T_{0}). The differential equations were then put into Laplace Transform Notation. Then, the inverse Laplace Transform for each expression was found through partial fraction expansion and referencing the Standard Table of Laplace Transform. Finally, the time dependent solutions for the winding temperature and case temperature were obtained.

Thus, the time dependent solutions show that the values of thermal resistance (R), thermal capacitance (C), and power dissipation (P) will determine the certain temperature the motor must not exceed to avoid overheating. From there, a duty cycle can be established that will run the motor to keep the temperature below a certain level. In regards to the duty cycle, it was found experimentally that the winding’s thermal time constant (t_{w}) is about one-tenth of the case’s thermal time constant (t_{c}) because the winding of the motor heats up quickly while the case dissipates the heat slowly. Therefore, the winding must first not exceed a certain temperature to avoid overheating. Overheating will cost engineering companies a lot of time and money because the internal parts of the motors will be damaged. Thus, to gain a clearer understanding of the temperature and time limitations of a motor and to improve motors in the future, equations that solve for the change in winding temperature and the change in case temperature as functions of time are necessary.

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