Insulated Gate Bipolar Transistor Junction Temperature Estimation Technology for Traction Inverters Using a Thermal Model

4.1. Inverter Circuit Structure

The inverter used in this study is a three-phase-power, module-based inverter consisting of IGBTs and diodes. For the junction temperature estimation experiments, the inverter modules were connected in series using an open-end configuration, allowing greater flexibility in current and duty settings for each phase.

To accurately measure the junction temperature of the IGBTs and diodes, an infrared thermal camera was adopted. In addition, to improve the accuracy of the thermal camera measurements, the IGBT and diode surfaces were treated with black paint to minimize reflection, as shown in Figure 6 (left). Meanwhile, the layout of the NTC sensors on the heatsink is depicted in Figure 6 (right). The overall setup of the infrared thermal camera is illustrated in Figure 7.

The NTC sensors were placed near the four corners (top-left, bottom-left, top-right, and bottom-right) of the IGBT on the heatsink. The average of these four NTC temperature readings were then used as the base temperature. In conventional power modules, a single NTC sensor is typically placed in a fixed location. However, because the distances between this single sensor and each IGBT junction differ, the sensor’s temperature reading may not accurately represent the overall junction temperature.

To overcome these limitations, this study strategically placed four additional NTC sensors near the four corners of the IGBT on the heatsink. The average of their readings was applied as the base temperature, denoted as $T_{a v g}$ in (8). This approach improves the estimation accuracy by considering the cross-coupling effects between the junctions.

4.3. Data Preprocessing Procedure

This study implemented a systematic preprocessing procedure to analyze temperature data collected from an infrared camera and NTC sensors. For the external NTC sensors, the temperature data were automatically collected from the step point where power loss begins, using a DAQ (Data Acquisition System) integrated into the system. The collected temperature data were averaged across multiple NTC sensor readings for each time sample, and this average was used as the reference temperature. For the infrared camera’s temperature data, the starting point of the data was reset by detecting the moment when the rate of temperature change exceeded a threshold value. This process ensured accurate identification of the step point of the temperature variation. The following equation was used for this purpose:

$Δ T_{i} = T_{i} - T_{i - 1}, if Δ T_{i} > threshold, then mark t_{i}$

(19)

Here, $Δ T_{i}$ represents the rate of temperature change at time $i$ , while $T_{i}$ and $T_{i - 1}$ denote the temperature values at time $i$ and $i - 1$ , respectively. This equation was used to detect the first time point $t_{i}$ at which the temperature change exceeded the specified threshold. The detected time $t_{i}$ was then set as the new starting point for the data.

Additionally, the thermographic data and the NTC sensor data were sampled at different intervals. Specifically, the thermographic camera recorded the surface temperatures of each component (HT, LT, HD, LD) at an average interval of approximately 20 ms, whereas the NTC sensors measured the heatsink temperature at 100 µs intervals and provided an averaged value each second. To align these two datasets on a common time base, a linear interpolation was performed, resulting in a unified time resolution of 0.1 ms. This interpolation ensured a consistent comparison of temperature data collected from the infrared camera and the NTC sensors.

4.5. Computational Complexity Analysis of Foster Network Orders

As introduced in (17) and (18), the partial temperature rises

Δ t_{i j (k)}

and the total junction temperature

Δ T_{i}

can be computed for an O-th-order Foster network, where

O

denotes the model order. While higher-order models can capture faster transient phenomena, they inevitably demand greater computational resources. To illustrate this increase in complexity, we let “4” represent the number of power semiconductor components (for example, two IGBTs and two diodes). Under this setup, the multiplication count per time step can be expressed as

$M u l t i p l i c a t i o n C o u n t = 2 \times 4 \times 4 \times O .$

(20)

Here, the multiplication count scales linearly with the model order and quadratically with the number of components.

$A d d i t i o n C o u n t = 2 \times 4 \times 4 \times o r d e r + 3 \times 4,$

(21)

where the extra term ( $3 \times 4$ ) corresponds to row-wise summation operations needed to accumulate the partial temperatures from multiple components. Finally, the total operation count is

$T o t a l O p e r a t i o n C o u n t = 4 \times 4 \times o r d e r + 3 \times 4,$

(22)

reflecting both updates and summations. Because the model order $O$ appears linearly in each expression, the number of multiplications and additions both grow linearly with $O$ . Furthermore, the factor of four (HT, LT, HD, LD) indicates that the total operation count also depends on the number of components in a quadratic manner.

Table 1 summarizes the operation counts for selected model orders. Although higher orders can offer more accurate representation of rapid thermal transients, an excessively large model order may become impractical for real-time applications. Consequently, a second-order Foster network can provide a balanced compromise between modeling fidelity and computational efficiency.

Moreover, as will be discussed in Section 5.2, we conducted a comparative analysis of the Mean Absolute Error (MAE) across different model orders. Our findings showed that the first-order model suffered from notably higher errors, while the second-order model significantly reduced the MAE. Beyond the second order, additional accuracy gains were marginal, whereas computational demands continued to increase linearly with the model order. These observations reinforce the practicality of adopting a second-order Foster network for the time scales (several seconds to tens of seconds) addressed in this study.

4.6. RC Parameter Extraction Process

In this study, the electrical circuit shown in Figure 8 was established to ensure efficient data acquisition and reliable current control. The circuit includes a three-phase open-end structure with an R-L load and two independent IGBT inverters. One is the target inverter (open model), and it maintains the fixed duty setup, while the other inverter regulates the corresponding current to the desired value by means of a PI current controller applied to the RL circuit. Additionally, the two inverters share the DC-link power, enabling bidirectional current flow (e.g., +45 A and −45 A). This bidirectional flow is essential for identifying the conduction regions of each IGBT and diode. Overall, this design effectively demonstrates the proposed current control method and highlights its role in achieving accurate and reliable current regulation within the presented system.

Figure 9 illustrates the thermal distribution when power loss is concentrated on a specific junction in one phase leg inverter. Under each condition, 93% of the current flows through a specific junction, while the remaining 7% flows through its complementary device, allowing an analysis of the heat generation characteristics. For the 93% 45A (HT) condition, 93% of the current is concentrated on HT, where the primary power loss occurs. In LD, only 7% of the current flows, resulting in minimal heat generation. Similarly, for the 7% −45A (LT) condition, 93% of the current is concentrated on LT, with the primary power loss occurring there, while HD carries 7% of the current, generating minimal heat. In the 93% −45A (HD) condition, 93% of the current is concentrated on HD, where the primary power loss occurs. LT carries 7% of the current, resulting in minimal heat generation. For the 7% 45A (LD) condition, 93% of the current flows through LD, concentrating the primary power loss there, while HT carries 7% of the current, generating minimal heat. This method of concentrating current on a specific device minimizes the influence of other variables, enabling the analysis of the device’s pure thermal characteristics and facilitating the accurate extraction of RC parameters.

Additionally, the current path changes depending on the direction of the current, and the device experiencing power loss varies accordingly. When the current flows in the positive direction, the main current path is formed through HT and LD, and power loss occurs only in HT and LD. Conversely, when the current flows in the negative direction, the main current path is formed through LT and HD, and power loss occurs only in LT and HD. In the previously mentioned thermal impedance matrix equation (Equation (4)), the condition where the power loss of a specific device becomes zero is defined based on the direction of the current. When the current flows in the positive direction, the main current path is formed through HT ( $p_{1}$ ) and LD ( $p_{4}$ ), resulting in $p_{2}$ (LT) and $p_{3}$ (HD) being zero.

Conversely, when the current flows in the negative direction, the main current path is formed through LT (

p_{2}

) and HD (

p_{3}

), with

p_{1}

(HT) and

p_{4}

(LD) becoming zero. In this way, the primary devices experiencing power loss change depending on the current direction, which is expressed as different systems of equations in the thermal impedance matrix. When the current flows in the positive direction (I+), the current primarily flows through HT and LD, with significant power loss occurring in HT and LD. At this time, the temperature increment (

Δ T

) is calculated by the power losses in HT and LD. When the current flows in the positive direction, the temperature increment of HT (

Δ T_{(I +) 1}

) is given as follows:

$Δ T_{(I +) 1} = Z_{11} \cdot P_{(I +) 1} + Z_{14} \cdot P_{(I +) 4}$

(23)

Here,

Z_{11}

represents the self-heating of HT, while

Z_{14}

represents the cross-heating effect where heat generated in LD is transferred to HT. Conversely, when the current flows in the negative direction (I-), the current primarily flows through LT and HD, with significant power loss occurring in LT and HD. In this case, the temperature increment is calculated by the power losses in LT and HD. When the current flows in the negative direction, the temperature increment of HT (

Δ T_{(I +) 1}

) is given as follows:

$Δ T_{(I -) 1} = Z_{12} \cdot P_{(I -) 2} + Z_{13} \cdot P_{(I -) 3}$

(24)

Here,

Z_{12}

represents the cross-heating effect where heat generated in LT is transferred to HT, while

Z_{13}

represents the cross-heating effect where heat generated in HD is transferred to HT. Consequently, for all experimental cases, the complete set of equations is listed as follows, when the current flows in the positive direction:

$Δ T_{(I +, D 1) 1} = Z_{11} \cdot P_{(I +, D 1) 1} + Z_{14} \cdot P_{(I +, D 1) 4}, Δ T_{(I +, D 2) 1} = Z_{11} \cdot P_{(I +, D 2) 1} + Z_{14} \cdot P_{(I +, D 2) 4} Δ T_{(I +, D 1) 2} = Z_{21} \cdot P_{(I +, D 1) 1} + Z_{24} \cdot P_{(I +, D 1) 4}, Δ T_{(I +, D 2) 2} = Z_{21} \cdot P_{(I +, D 2) 1} + Z_{24} \cdot P_{(I +, D 2) 4} Δ T_{(I +, D 1) 3} = Z_{31} \cdot P_{(I +, D 1) 1} + Z_{34} \cdot P_{(I +, D 1) 4}, Δ T_{(I +, D 2) 3} = Z_{31} \cdot P_{(I +, D 2) 1} + Z_{34} \cdot P_{(I +, D 2) 4} Δ T_{(I +, D 1) 4} = Z_{41} \cdot P_{(I +, D 1) 1} + Z_{44} \cdot P_{(I +, D 1) 4}, Δ T_{(I +, D 2) 4} = Z_{41} \cdot P_{(I +, D 2) 1} + Z_{44} \cdot P_{(I +, D 2) 4}$

(25)

When the current flows in the negative direction, the resultant equations are as follows:

$Δ T_{(I -, D 1) 1} = Z_{12} \cdot P_{(I -, D 1) 2} + Z_{13} \cdot P_{(I -, D 1) 3}, Δ T_{(I -, D 2) 1} = Z_{12} \cdot P_{(I -, D 2) 2} + Z_{13} \cdot P_{(I -, D 2) 3} Δ T_{(I -, D 1) 2} = Z_{22} \cdot P_{(I -, D 1) 2} + Z_{23} \cdot P_{(I -, D 1) 3}, Δ T_{(I -, D 2) 2} = Z_{22} \cdot P_{(I -, D 2) 2} + Z_{23} \cdot P_{(I -, D 2) 3} Δ T_{(I -, D 1) 3} = Z_{32} \cdot P_{(I -, D 1) 2} + Z_{33} \cdot P_{(I -, D 1) 3}, Δ T_{(I -, D 2) 3} = Z_{32} \cdot P_{(I -, D 2) 2} + Z_{33} \cdot P_{(I -, D 2) 3} Δ T_{(I -, D 1) 4} = Z_{42} \cdot P_{(I -, D 1) 2} + Z_{43} \cdot P_{(I -, D 1) 3}, Δ T_{(I -, D 2) 4} = Z_{42} \cdot P_{(I -, D 2) 2} + Z_{43} \cdot P_{(I -, D 2) 3}$

(26)

Equations (25) and (26) are rewritten in shorter form using a matrix form. the matrix equations are useful for computation in numerical analysis programs. When the current flows in the positive direction, the matrix form of (25) is as follows:

$\begin{matrix} [\begin{matrix} Δ T_{(I +, D 1) 1} & Δ T_{(I +, D 2) 1} \\ Δ T_{(I +, D 1) 2} & Δ T_{(I +, D 2) 2} \\ Δ T_{(I +, D 1) 3} & Δ T_{(I +, D 2) 3} \\ Δ T_{(I +, D 1) 4} & Δ T_{(I +, D 2) 4} \end{matrix}] = [\begin{matrix} Z_{11} & Z_{14} \\ Z_{21} & Z_{24} \\ Z_{31} & Z_{34} \\ Z_{41} & Z_{44} \end{matrix}] \cdot [\begin{matrix} P_{(I +, D 1) 1} & P_{(I +, D 2) 1} \\ P_{(I +, D 1) 4} & P_{(I +, D 2) 4} \end{matrix}] \end{matrix}$

(27)

When simplified, it is expressed in the following general equation:

$\begin{matrix} Δ T_{I +} = Z_{I +} \cdot P_{I +} \end{matrix}$

(28)

In the above equation,

Z_{I +}

, which corresponds to the first and fourth columns of the thermal impedance matrix, is calculated as follows:

$\begin{matrix} Z_{I +} = Δ T_{I +} \cdot P_{I +}^{- 1} \end{matrix}$

(29)

When the current flows in the negative direction, the matrix form of (26) is as follows:

$[\begin{matrix} Δ T_{(I -, D 1) 1} & Δ T_{(I -, D 2) 1} \\ Δ T_{(I -, D 1) 2} & Δ T_{(I -, D 2) 2} \\ Δ T_{(I -, D 1) 3} & Δ T_{(I -, D 2) 3} \\ Δ T_{(I -, D 1) 4} & Δ T_{(I -, D 2) 4} \end{matrix}] = [\begin{matrix} Z_{12} & Z_{13} \\ Z_{22} & Z_{23} \\ Z_{32} & Z_{33} \\ Z_{42} & Z_{43} \end{matrix}] \cdot [\begin{matrix} P_{(I -, D 1) 2} & P_{(I -, D 2) 2} \\ P_{(I -, D 1) 3} & P_{(I -, D 2) 3} \end{matrix}]$

(30)

When simplified, it is expressed as the following general equation:

$\begin{matrix} Δ T_{I -} = Z_{I -} \cdot P_{I -} \end{matrix}$

(31)

In the above equation,

Z_{I -}

, which corresponds to the second and third columns of the thermal impedance matrix, is calculated as follows:

$\begin{matrix} Z_{I -} = Δ T_{I -} \cdot P_{I -}^{- 1} \end{matrix}$

(32)

Using this method, the time-varying data of each junction in the thermal impedance matrix was curve-fitted to the second-order form of the following equation, which represents the step response of the Foster thermal network, to extract

R

and

τ^{- 1}

$z_{i j} (t) = \sum_{k = 1}^{n} R_{i j (k)} (1 - e x p (- \frac{t}{τ_{i j (k)}}))$

(33)

Table 2 shows the

R

and

τ^{- 1}

values extracted for each junction of the thermal impedance matrix through curve fitting. Figure 10 visually compares the original data with the modeled data obtained through curve fitting for each element of the thermal impedance matrix.

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Kijung Kong www.mdpi.com

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Insulated Gate Bipolar Transistor Junction Temperature Estimation Technology for Traction Inverters Using a Thermal Model

4.1. Inverter Circuit Structure

4.3. Data Preprocessing Procedure

4.5. Computational Complexity Analysis of Foster Network Orders

4.6. RC Parameter Extraction Process

Greenberg

4.1. Inverter Circuit Structure

4.3. Data Preprocessing Procedure

4.5. Computational Complexity Analysis of Foster Network Orders

4.6. RC Parameter Extraction Process

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