# Multipoint Thermal Sensing System for Power Semiconductor Devices Utilizing Fiber Bragg Gratings
## 2 FBG Temperature Sensing Principle
### Heat Transfer Model
#### 2.1.2 Multipoint Model  
As depicted in Figure 3, a typical inverter circuit consists of three or more PSDs mounted on a heatsink. These devices encompass IGBTs, SiC MOSFETs, or rectifier modules. Figure 2: Thermal network of a single IGBT module referenced to FBG sensor. In order to monitor the temperature variations regarding these devices, multipoint sensors are required. Installing a standalone sensor for each device would not be economical and eventually make the circuit bulky. Similar to the single-point model, the approach for static thermal network resistance measurement is valid for three-phase inverters if all IGBTs are assumed to experience the same loss, except in the case of a half-bridge operating as a bulk converter [26]. To estimate the mean junction temperature of multiple PSDs mounted on a common heatsink, a thermal resistance network model is employed. This model simplifies the complex thermal behavior of the system into a series and parallel combination of thermal resistances. By considering the thermal resistances of the individual components, including the junction-to-case, case-to-heatsink, and heatsink-to-ambient resistances, the overall thermal resistance between the junction and the ambient is calculated. The mean junction temperature is then determined by considering the total power dissipation, the equivalent thermal resistance, and the average heatsink temperature. This approach provides a valuable tool for thermal management and reliability analysis of power electronic systems [29]. Suppose the sensors are located on the IGBT chips along a fiber cable, where the cable functions as a continuous sensor that can identify temperature changes at each discrete sensing point, and then the equivalent value of \(T_{j}\) for these devices is provided as  
\[T_{j}=\ \Big{[}P_{dt}R_{th(j-a)}\Big{]}n+T_{a} \tag{11}\]  
where \(n\) is the number of PSDs mounted on the heatsink; \(R_{th(j-h)}\) and \(P_{dt}\) are, respectively, the thermal resistance and the sum of power dissipation in each PSD. Now, if the FBG sensor is positioned outside the device's housing to measure \(T_{j}\) of a packed rectifier, the equivalent junction-to-case thermal resistance \(R_{eq(j-c)R}\) of the rectifier is expressed as  
\[R_{eq(j-c)R}\ =\ \bigg{[}\frac{1}{R_{D1}}+\frac{1}{R_{D2}}+\frac{1}{R_{D3}} \ \ldots\ldots\ldots\ldots\frac{1}{R_{DN}}\bigg{]}^{-1} \tag{12}\]  
where \(R_{D1}\) to \(R_{DN}\) represent the parallel combination of the N diodes used for rectification. For an isolated system, the casing temperature for the rectifier stud \(T_{c(R)}\) is a function of the \(P_{dt}\) in the rectifier stud and \(R_{eq(j-c)R}\) and is expressed as  
\[T_{c(R)}\ =\ P_{dt}\ R_{eq(j-c)R} \tag{13}\]  
The thermal resistance junction-to-heatsink for any kth device \(R_{th(j-h),k}\) is provided as  
\[R_{th(j-h),k}=\ \bigg{[}\ \frac{T_{j-k}-T_{s-k}}{P_{dt}N}-R_{th(j-c)}\ \bigg{]}+R_{th(c-h)} \tag{14}\]  
Figure 3: Thermal resistance network of multiple IGBTs. where \(N\) represents the number of separate semiconductors contained within the module stacked onto the heatsink. A three-phase bridge situated on the heatsink, which contains six diodes packed onto a single stud, will simply have \(n=1\) and \(N=6\) for brevity. The overall dissipated power on the heatsink equates to \(P_{di}\ *\ N\).