*4.4. Real-Time Temperature Rise Prediction of a Traction LIM*

The enclosed structure of the LIM primary, its continuous movement over the reaction rail, and a large gap between the primary and the reaction rail effectively isolate the primary from the heat flux generated in the RR. For all electric and electronic components and devices for which the heat losses can be determined solely from the electric current measurement, such as cables, inductors, relay coils, linear motors, actuators, etc., the maximum nominal temperature rise above ambient can be directly associated with the value of nominal current. However, the device's heat-producing current often varies, due to load changes, ranging from zero to values exceeding the value of the nominal current by a large factor. Such complex load cycles can produce highly variable thermal cycling and result in uncontrolled over-temperature, which negatively impacts the life cycle of a device. During the acceleration and deceleration of a LIM-powered electric traction vehicle, the instantaneous current of the motor exceeds, by a high margin, the nominal thermal value and then decreases below that value during coasting and drops to zero at station stops, therefore undergoing severe thermal cycling. The basic heat equation describes the balance between the dissipated, stored, and radiated heat. The dissipated heat will be partially transferred into the surrounding ambient and partially stored in the heated device by increasing its temperature. So long as the heat power is constant, the temperature rise will achieve a maximum value of Δ*Tmax*, and the thermal transient state will be described by the following equation [91]:

$$
\Delta T = \Delta T\_{\max} \left( 1 - e^{-\frac{t}{\tau}} \right) + \Delta T\_0 e^{-\frac{t}{\tau}} \tag{3}
$$

where Δ*T*<sup>0</sup> is the temperature rise over ambient at *t* = 0, Δ*Tmax* is the maximum temperature rise over ambient for *t* → ∞, and *τ* is a characteristic constant (thermal time constant).

Equation (3) is frequently used to determine an initial temperature rise estimation of a device undergoing thermal cycling by solving it for the thermal cycle average rms current. However, this approach leads to an error in temperature prediction as it does not account for the temperature dependence of a heat power source. This problem was solved in [91] by correcting and then solving the conventional differential equation/algorithm and providing an exact solution that utilized a single heat-run-test data point. The temperature prediction performance of the basic and improved equations was analyzed by calculating the temperature rise of a linear motor subjected to a typical, nominal service duty cycle. The simulation took into consideration the result of a thermal test performed during a thermal qualification of the subject LIM. The simulation results were further compared against the results of temperature measurement taken from a thermal sensor imbedded in a production LIM's winding while operating the LIM-powered vehicle with the nominal load on the existing Vancouver Expo line system. The experimental data in the form of motor temperature and phase current were collected and overlayed with the simulation results. The results (see Figure 15) confirm that the improved algorithm (blue) predicts the LIM temperature rise (green) with much better accuracy than the basic algorithm (red). The measured temperature is still lower than the improved prediction, mainly due to an additional cooling effect resulting from the increased convection of a moving train. Because of a high degree of prediction accuracy and minimal application cost, the improved software algorithm found practical application and had been installed fleet wide.

**Figure 15.** Comparison of different algorithms of temperature rise versus measured temperature results [91].

The efficiency of a traction LIM motor, as has already been mentioned, is rather low and in the case of a loss of forced cooling high losses can drive the temperature of the center of the winding quickly to above the maximum allowed level. Precise detection of this process is important as it allows the train control system to maintain the LIM in operation for as long as possible without compromising the insulation life cycle. Figure 16 shows the typical dependencies of power factor and efficiency for a traction LIM applied on most urban transportation systems in the world. These results were confirmed by experimental measurements taken by one of the authors for the motor, which now operates on a Rapid KL Rail System in Kuala Lumpur, Malaysia. It characterizes similarly built traction LIMs of comparable size and cooling efficiency.

**Figure 16.** Power factor and efficiency curves for one example of a traction LIM.
