- Thermal Model Definition: A thermal model of a motor is defined as a simplified representation to calculate heat generation and dissipation in a motor.
- Heat Generation (p1): This is the amount of heat produced inside the motor, measured in watts.
- Heat Dissipation (p2): Heat is transferred to the cooling medium, also measured in watts.
- First Order Differential Equation: This equation calculates temperature rise over time, helping predict motor heating and cooling.
- Heating and Cooling Curve: This curve shows how the motor’s temperature changes during operation, crucial for understanding thermal behavior.
An operating electrical motor converts its electrical and mechanical losses into heat. Thermal modeling of a motor estimates how that heat raises component temperatures and leaves through the cooling paths. Losses and heat flow depend strongly on motor geometry, materials, speed, load and cooling. The simple thermal model below treats the whole machine as one uniform thermal mass connected to ambient by one thermal conductance. It is useful for first-order heating estimates and protection settings, but it cannot predict every winding, rotor or bearing hot spot in an electric motor.
At time t, define these lumped-model quantities:
p1 = total motor loss, or heat-generation rate, in watts.
p2 = heat-transfer rate to the cooling medium in watts.
W = effective mass of the represented motor parts in kilograms.
h = effective specific heat in joules per kilogram per oC.
A = effective cooling surface in m2.
d = effective heat-transfer coefficient in watts per m2 per oC.
θ = mean temperature rise above the cooling medium in oC.
Over a short interval dt, the stored thermal energy changes by C dθ.
Energy conservation gives stored heat = generated heat minus heat transferred to the cooling medium.
Therefore, C dθ = p1dt – p2dt…………….(i)
For a linear heat-transfer approximation, p2 = θdA…………….(ii)
Substitution gives the first-order differential equation:
Here C = Wh is thermal capacity in joules/oC, while D = dA is thermal conductance in watts/oC. Their ratio C/D is the thermal time constant in seconds.
For constant losses and constant model parameters, solving the first-order equation gives an exponential temperature response:
The initial temperature at t = 0 determines the integration constant K in equation (iii). The result can also be written as the final temperature rise plus the initial-to-final difference multiplied by e raised to -t divided by the thermal time constant.
The heating curve approaches its steady temperature exponentially. With the same constant parameters, one thermal time constant reaches about 63.2% of the final rise. Cooling is also exponential, although a stopped self-cooled motor can have a longer cooling time constant because its fan is no longer moving air. This one-node thermal modeling of a motor should be checked against manufacturer data or measurements when hot-spot accuracy matters.
