A Junction Temperature Prediction Method Based on Multivariate Linear Regression Using Current Fall Characteristics of SiC MOSFETs

Abstract

The junction temperature (T_j) is a key parameter reflecting the thermal behavior of Silicon carbide (SiC) MOSFETs and is essential for condition monitoring and reliability assessment in power electronic systems. However, the limited temperature sensitivity of switching characteristics makes it difficult for traditional single temperature-sensitive electrical parameters (TSEPs) to achieve accurate estimation. To address this challenge and enable practical thermal sensing applications, this study proposes an accurate, application-oriented T_j estimation method based on multivariate linear regression (MLR) using turn-off current fall time (t_fi) and fall loss (E_fi) as complementary TSEPs. First, the feasibility of using current fall time and current fall energy loss as TSEPs is demonstrated. Then, a coupled junction temperature prediction model is developed based on multivariate linear regression using t_fi and E_fi. The proposed method is experimentally validated through comparative analysis. Experimental results demonstrate that the proposed method achieves high prediction accuracy, highlighting its effectiveness and superiority in MLR approach based on the current fall phase characteristics of SiC MOSFETs. This method offers promising prospects for enhancing the condition monitoring, reliability assessment, and intelligent sensing capabilities of power electronics systems.

1. Introduction

Silicon carbide power devices offer advantages such as high-power density [1], excellent high-temperature tolerance, low on-state resistance [2], and high switching frequency [3], making them well-suited for high-power-density power electronics systems. In recent years, they have been increasingly adopted in applications such as rail transportation, aerospace, and wind energy conversion systems [4].

As SiC power device technology continues to mature, reliability concerns in practical applications are becoming increasingly prominent [5]. Under typical operating conditions in power electronics systems, both the absolute junction temperature and its dynamic fluctuations are critical factors influencing device reliability.

Figure 1 illustrates the distribution of failure modes. Research shows that approximately 31% of power electronics system failures are attributed to power semiconductor device failures, with nearly 60% of these directly linked to thermal factors [6]. Thermal stress (including overheating and temperature fluctuations) is the primary cause of failure in power semiconductor devices, resulting in both transient failures (such as thermal breakdown, bond wire breakage, and solder melting) and aging failures (including bond wire fatigue and solder layer degradation) [7]. Furthermore, the failure rate of power semiconductors approximately doubles for every 10 °C increase in junction temperature [8]. These findings demonstrate the urgent need for effective junction temperature estimation techniques that not only support reliability assessments but also enable integration into practical thermal sensing systems for real-time device health monitoring.

Currently, the methods for measuring junction temperature in power MOSFETs can be broadly categorized into four types: physical contact methods based on sensors, optical methods, thermal resistance network model-based approaches, and temperature-sensitive electrical parameter methods [6]. The physical contact method is invasive and has a slow response time [9,10]; the optical method is limited in practical applications due to its high cost and difficulty in applying it to opaque packaging structures [11,12]. The thermal resistance network model-based approach is a commonly employed method for junction temperature estimation. However, it requires accurate measurement of the module case temperature and the development of a precise thermal resistance network model to be effective [13]. The accuracy of this method largely depends on the accuracy of the thermal impedance network model. Key thermal parameters necessary for constructing a high-precision model are often difficult to obtain, and the model accuracy is easily affected by device aging. Compared with the above three methods, the TSEP method has significant advantages such as fast response speed, high implementation feasibility, and high accuracy [14,15,16]. Commonly used temperature-sensitive electrical parameters (TSEPs) include on-state resistance or voltage drop, switching delay, leakage current, gate threshold voltage, and peak gate current, among others. Real-time tracking of junction temperature is achievable through the measurement of these indicators, and this method has demonstrated promising results, particularly in silicon (Si)-based devices [15,17,18,19].

However, for SiC MOSFETs, certain TSEP-based approaches often fail to deliver optimal monitoring performance and can be problematic for online implementation. First, the inherently high switching frequency of SiC MOSFETs complicates the precise measurement of switching delay indicators; second, due to their excellent high blocking capability, SiC MOSFETs exhibit extremely low leakage current at 25 °C (approximately 1 μA level), making precise measurement difficult [20]; third, although methods based on-state resistance remain applicable, their temperature sensitivity is relatively low (approximately 0.2 mΩ/°C), which limits their ability to meet high-accuracy requirements [21].

N. Baker et al. [22,23] employed the gate peak current as a TSEP, which can be obtained by measuring the voltage drop across the external gate drive resistor. However, this method is susceptible to measurement inaccuracies due to the influence of the gate drive circuit and varying operating conditions. Z. Zhang et al. [24] proposed an online monitoring method for junction temperature prediction based on the turn-off delay time of SiC MOSFETs. F. Yang et al. [25] utilized the turn-on delay time as a temperature-sensitive electrical parameter and further considered the impact of device aging on its effectiveness. However, the switching delay time of SiC MOSFETs exhibits inherently low temperature sensitivity, and, to enhance this sensitivity, both studies [24,25] employed relatively large gate drive resistances, which may in turn affect the device’s switching characteristics. Ying Wang et al. [26] proposed an online junction temperature prediction method based on the falling edge duration of the drain voltage, utilizing auxiliary circuitry to monitor the relevant characteristics of the drain voltage transient. However, this parameter is highly susceptible to measurement errors caused by the parasitic elements of the SiC MOSFET and requires high-precision instrumentation for accurate detection.

Previous studies have demonstrated that the voltage slew rate (dv/dt) and voltage rise time (t_rv) during the turn-off process are commonly employed for junction temperature estimation in IGBT devices. Lingfeng Shao et al. [27] proposed a hybrid model based on the voltage rise time and voltage rise loss (E_rv) during the IGBT turn-off process. However, the turn-off characteristics are highly susceptible to variations in load conditions, which may compromise the accuracy of the junction temperature prediction model. Moreover, the applicability and temperature sensitivity characteristics of this method for SiC MOSFETs still require further experimental validation.

To address the issue of insufficient temperature sensitivity in the switching characteristics of SiC MOSFETs and the difficulty of accurately predicting junction temperature using traditional single TSEPs, this paper proposes an MLR method for junction temperature estimation based on t_fi and E_fi. This method is designed not only to improve estimation accuracy but also to enable practical integration into temperature-sensing systems for real-time thermal monitoring of power semiconductor devices. The structure of this paper is as follows: Section 2 first establishes a mathematical model for the current fall stage of SiC MOSFETs, theoretically derives the temperature-sensitive characteristics of t_fi and E_fi during the current fall stage, and verifies these characteristics through simulation; Section 3 details the construction method and process of the MLR junction temperature prediction model based on t_fi and E_fi; Section 4 conducts experimental research using a double-pulse test (DPT) platform, in which both the turn-off characteristics and current fall phase parameters of SiC MOSFETs are measured, and corresponding MLR models are then constructed for comparison and validation. Experimental results demonstrate the accuracy and effectiveness of this junction temperature prediction method; and Section 5 summarizes the key findings and concludes this paper. This work provides a promising foundation for the development of compact, low-cost junction temperature sensors based on SiC MOSFETs.

2. Analysis of the Temperature-Dependent Characteristics of Parameters During the Current Fall Phase

2.1. Structure and Equivalent Circuit Model of SiC MOSFETs

Taking a SiC MOSFET packaged in a TO-247-3L package as an example, the internal structure diagram and the corresponding equivalent circuit of the discrete device are presented in Figure 2. As shown in Figure 2a, the TO-247-3L is one of the most commonly used discrete packaging structures for SiC MOSFETs. In this package, the drain terminal on the backside of the chip is connected to the lead frame through a solder layer, while the gate and source terminals on the front side are connected to their respective leads via aluminum bond wires. The entire assembly is then encapsulated in a plastic molding compound. However, due to the use of metal bond wires and a planar chip layout, the current path from drain to source is relatively long. As a result, the parasitic inductances associated with the internal package and external leads become non-negligible and can significantly influence the device’s high-speed switching performance.

Figure 2b illustrates the equivalent circuit diagram corresponding to the parasitic inductance components of the TO-247-3L package shown in Figure 2a. In this diagram, G, D, and S denote the intrinsic gate, drain, and source terminals of the SiC MOSFET die. The nodes labeled g, d, and s represent the gate, drain, and source terminals after accounting for the internal bonding wires. The nodes g′, d′, and s′ further represent the respective terminals after incorporating both the internal bonding wire inductance and the parasitic inductance introduced by the external package leads. L_G(int), L_D(int), and L_S(int) represent the equivalent parasitic inductances of the internal bonding wires within the device package, while L_G(ext), L_D(ext), and L_S(ext) denote the equivalent parasitic inductances associated with the external package leads and PCB trace connections.

Figure 3 presents the double-pulse test circuit for SiC MOSFETs, incorporating the key parasitic parameters. In Figure 3, V_DC represents an ideal voltage source equivalent to the DC bus; I_L denotes an ideal current source equivalent to the load inductance; C_L accounts for the parasitic capacitance associated with the load inductance; D_H is an ideal SiC Schottky Barrier Diode (SBD); C_J represents the equivalent junction capacitance of the SiC SBD; C_GS, C_GD, and C_DS represent the gate-to-source, gate-to-drain, and drain-to-source capacitances of the SiC MOSFET, respectively; L_D(int) and L_S(int) denote the parasitic inductances associated with the drain and source terminals introduced by the device packaging; R_G(int) is the intrinsic gate resistance of the SiC MOSFET; R_G(ext) represents the external gate drive resistance; L_G represents the parasitic inductance of the gate loop; L_D(ext) and R_loop denote the equivalent parasitic inductance and stray resistance, respectively, in the conduction path from the positive terminal of the DC bus to the drain terminal of the SiC MOSFET; and L_S(ext) corresponds to the parasitic inductance in the return path from the source terminal of the SiC MOSFET to ground.

Figure 3 provides a detailed depiction of the parasitic parameters associated with each component, along with clearly defined current flow directions. Here, i_D denotes the drain current, i_G represents the gate current, and i_GS, i_GD, and i_DS represent the displacement currents through C_GS, C_GD, and C_DS, respectively. i_CH refers to the channel current of the SiC MOSFET, and I_L is the load current.

2.2. Modeling of the Drain Current Fall Phase

Figure 4 presents the equivalent circuit for the drain current fall phase (t₃–t₄) under double-pulse testing. The corresponding experimental turn-off waveforms of the SiC MOSFET are depicted in Figure 5. At time t₃, the drain voltage clamps to the bus voltage, preventing further rise. Consequently, the gate-source voltage begins to decrease. With the MOSFET operating in the saturation region, the drain current commences its decay. Simultaneously, the load current commences commutation to the freewheeling diode until i_D reaches zero. Throughout this interval, v_GS continues to fall towards the threshold voltage V_TH, governed by the expression:

$$v_{\mathrm{GS}}=V_{\mathrm{DRV}}-R_{\mathrm{G}}{i}_{\mathrm{G}}-L_{\mathrm{G}}{\displaystyle \frac{\mathrm{d}{i}_{\mathrm{G}}}{\mathrm{d}t}}-L_{\mathrm{S}}\left({\displaystyle \frac{\mathrm{d}{i}_{\mathrm{D}}}{\mathrm{d}t}}+{\displaystyle \frac{\mathrm{d}{i}_{\mathrm{G}}}{\mathrm{d}t}}\right)$$

(1)

$$i_{\mathrm{G}}=C_{\mathrm{GS}}{\displaystyle \frac{\mathrm{d}{v}_{\mathrm{GS}}}{\mathrm{d}t}}+C_{\mathrm{GD}}\left({\displaystyle \frac{\mathrm{d}{v}_{\mathrm{GS}}}{\mathrm{d}t}}-{\displaystyle \frac{\mathrm{d}{v}_{\mathrm{DS}}}{\mathrm{d}t}}\right)$$

(2)

Since the gate current i_G is significantly smaller than the drain current i_D, it is neglected in the derivation of the circuit equations.

$$v_{\mathrm{GS}}=V_{\mathrm{DRV}}-R_{\mathrm{G}}{i}_{\mathrm{G}}-L_{\mathrm{S}}{\displaystyle \frac{\mathrm{d}{i}_{\mathrm{D}}}{\mathrm{d}t}}$$

(3)

By combining the above equations, the governing equation for this stage is obtained:

$$R_{\mathrm{G}}{C}_{\mathrm{ISS}}{\displaystyle \frac{\mathrm{d}{v}_{\mathrm{GS}}}{\mathrm{d}t}}=R_{\mathrm{G}}{C}_{\mathrm{RSS}}{\displaystyle \frac{\mathrm{d}{v}_{\mathrm{DS}}}{\mathrm{d}t}}-v_{\mathrm{GS}}-L_{\mathrm{S}}{\displaystyle \frac{\mathrm{d}{i}_{\mathrm{D}}}{\mathrm{d}t}}+V_{\mathrm{DRV}}$$

(4)

$$v_{\mathrm{DS}}=V_{\mathrm{DC}}-L_{\mathrm{loop}}{\displaystyle \frac{\mathrm{d}{i}_{\mathrm{D}}}{\mathrm{d}t}}-R_{\mathrm{loop}}{i}_{\mathrm{D}}$$

(5)

$$i_{\mathrm{D}}=g_{\mathrm{fs}}\cdot \left(v_{\mathrm{GS}}-V_{\mathrm{TH}}\right)+C_{\mathrm{GD}}{\displaystyle \frac{\mathrm{d}{V}_{\mathrm{GD}}}{\mathrm{d}t}}+C_{\mathrm{DS}}{\displaystyle \frac{\mathrm{d}{V}_{\mathrm{DS}}}{\mathrm{d}t}}$$

(6)

The expression for the drain current fall rate is derived as follows:

$${\displaystyle \frac{\mathrm{d}{i}_{\mathrm{D}}}{\mathrm{d}t}}=-{\displaystyle \frac{g_{\mathrm{fs}}(V_{\mathrm{p}}-V_{\mathrm{EE}})}{R_{\mathrm{G}}(C_{\mathrm{GS}}+C_{\mathrm{GD}})+g_{\mathrm{fs}}{L}_{\mathrm{S}}}}$$

(7)

The expression for the drain current i_D during the interval t₃ to t₄ is derived as follows:

$$i_{\mathrm{D}}(t)=I_{\mathrm{L}}-{\displaystyle \frac{g_{\mathrm{fs}}(V_{\mathrm{p}}-V_{\mathrm{EE}})}{R_{\mathrm{G}}(C_{\mathrm{GS}}+C_{\mathrm{GD}})+g_{\mathrm{fs}}{L}_{\mathrm{S}}}}(t_4-t_3)$$

(8)

The duration of this interval is then calculated as follows:

$$t_{\mathrm{fi}}=t_4-t_3=I_{\mathrm{L}}{\displaystyle \frac{R_{\mathrm{G}}(C_{\mathrm{GS}}+C_{\mathrm{GD}})+g_{\mathrm{fs}}{L}_{\mathrm{S}}}{g_{\mathrm{fs}}(V_{\mathrm{p}}-V_{\mathrm{EE}})}}$$

(9)

The power dissipation during this switching phase is given by the following:

$$E_{\mathrm{fi}}={\displaystyle {\int }_{t_3}^{t_4}\left[v_{\mathrm{DS}}\left(t\right)\cdot i_{\mathrm{D}}\left(t\right)\right]\mathrm{d}t}={\displaystyle \frac{1}{2}}{V}_{\mathrm{DC}}\cdot I_{\mathrm{L}}\cdot (R_{\mathrm{G}}{C}_{\mathrm{ISS}}+g_{\mathrm{fs}}{L}_{\mathrm{S}})\cdot \ln {\displaystyle \frac{V_{\mathrm{DRV}}+V_{\mathrm{TH}}}{V_{\mathrm{TH}}}}$$

(10)

2.3. Temperature-Dependent Characteristics Analysis of t_fi and E_fi During the Current Fall Phase

The transfer characteristics of a SiC MOSFET describe the relationship between the drain current and the gate-source voltage. This relationship is typically represented by a transfer characteristic curve, whose slope corresponds to the device’s transconductance. A steeper slope indicates higher transconductance, reflecting greater sensitivity of the drain current to changes in gate voltage. Figure 6 presents the transfer characteristic curves of the SiC MOSFET model NTHL020N090SC1 from the ON Semiconductor. Within a certain range, the slope of the curve increases with the gate-source voltage, indicating enhanced transconductance. A comparison of the curves at different temperatures reveals that, with rising temperature, the entire transfer curve shifts upward, and the threshold voltage decreases progressively. This indicates that higher temperatures facilitate device turn-on, as a lower gate voltage is required to achieve the same drain current.

The transfer characteristic curve illustrates the relationship between the gate-source voltage v_GS and the drain current i_D under a constant drain-source voltage. The slope at each point on the curve reflects the device’s transconductance, which characterizes the amplification capability of the MOSFET. Among these parameters, the gate threshold voltage and transconductance are particularly significant in the output characteristics of SiC MOSFETs, both exhibiting strong temperature dependence. The threshold voltage can be expressed by Equation (11):

$$V_{\mathrm{TH}}={\displaystyle \frac{\sqrt{4{\epsilon }_{\mathrm{s}}kT{N}_{\mathrm{A}}ln(N_{\mathrm{A}}/n_{\mathrm{i}})}}{C_{\mathrm{ox}}}}+{\displaystyle \frac{2kT}{q}}ln\left({\displaystyle \frac{N_{\mathrm{A}}}{n_{\mathrm{i}}}}\right)$$

(11)

In this equation, n_i denotes the intrinsic carrier concentration, while n and p represent the concentrations of electrons and holes, respectively. k is the Boltzmann constant, T is the absolute temperature in kelvins, N_A is the acceptor doping concentration, ε_s is the relative permittivity of the semiconductor, C_OX is the characteristic capacitance of the oxide layer, and q is the unit charge. The intrinsic carrier concentration of SiC increases with temperature, and this rise predominantly influences the variation in the threshold voltage. As a result, the threshold voltage of SiC MOSFETs exhibits a negative temperature coefficient, decreasing as the temperature increases.

In the transfer characteristic curve of a MOSFET, the transconductance represents the ratio of the change in drain current i_D to the change in gate-source voltage v_GS, reflecting the degree to which the gate-source voltage controls the drain current. It can be mathematically expressed as shown in Equation (12):

$$g_{\mathrm{fs}}={\displaystyle \frac{Z{\mu }_{\mathrm{ni}}{C}_{\mathrm{OX}}}{L_{\mathrm{CH}}}}(v_{\mathrm{GS}}-V_{\mathrm{TH}})$$

(12)

In this equation, Z represents the channel width, μ_ni denotes the electron mobility in the inversion channel, v_GS is the gate-to-source voltage, and L_CH represents the channel length. The electron mobility in the inverted channel, μ_ni, can be expressed as a combination of several components derived from typical scattering mechanisms, including bulk lattice scattering (μ_B), surface roughness scattering (μ_sr), surface phonon scattering (μ_ph), and interface state scattering (μ_it):

$${\mu }_{\mathrm{ni}}^{-1}={\mu }_{\mathrm{B}}^{-1}+{\mu }_{\mathrm{it}}^{-1}+{\mu }_{\mathrm{ph}}^{-1}+{\mu }_{\mathrm{sr}}^{-1}$$

(13)

The inversion channel electron mobility of SiC MOSFETs increases with temperature up to approximately 600 K [28]. On the other hand, the threshold voltage of SiC MOSFETs decreases with rising temperature, so the transconductance of SiC MOSFETs increases with rising temperature.

Based on the analysis and modeling of the current fall phase, the duration of this phase can be expressed as follows:

$$t_{\mathrm{fi}}=I_{\mathrm{L}}{\displaystyle \frac{R_{\mathrm{G}}(C_{\mathrm{GS}}+C_{\mathrm{GD}})+g_{\mathrm{fs}}{L}_{\mathrm{S}}}{g_{\mathrm{fs}}(V_{\mathrm{p}}-V_{\mathrm{EE}})}}$$

(14)

By differentiating the current fall time with respect to the junction temperature T_j, the following expression is obtained:

$${\displaystyle \frac{\mathrm{d}{t}_{\mathrm{fi}}}{\mathrm{d}{T}_{\mathrm{j}}}}={\displaystyle \frac{I_{\mathrm{L}}{g}_{\mathrm{fs}}(V_{\mathrm{TH}}-V_{\mathrm{EE}})(\frac{1}{L_{\mathrm{S}}{g}_{\mathrm{fs}}}+R_{\mathrm{G}}\frac{(C_{\mathrm{GS}}+C_{\mathrm{GD}})}{L_{\mathrm{S}}}-1)}{{(g_{\mathrm{fs}}{V}_{\mathrm{TH}}+I_{\mathrm{L}}-g_{\mathrm{fs}}{V}_{\mathrm{EE}})}^2}}{\displaystyle \frac{\mathrm{d}{g}_{\mathrm{fs}}}{\mathrm{d}{T}_{\mathrm{j}}}}-{\displaystyle \frac{I_{\mathrm{L}}(R_{\mathrm{G}}(C_{\mathrm{GS}}+C_{\mathrm{GD}})+g_{\mathrm{fs}}{L}_{\mathrm{S}})g_{\mathrm{fs}}}{{(g_{\mathrm{fs}}{V}_{\mathrm{TH}}+I_{\mathrm{L}}-g_{\mathrm{fs}}{V}_{\mathrm{EE}})}^2}}{\displaystyle \frac{\mathrm{d}{V}_{\mathrm{TH}}}{\mathrm{d}{T}_{\mathrm{j}}}}$$

(15)

Based on the above analysis, it can be concluded that the transconductance g_fs exhibits a positive temperature coefficient, whereas the threshold voltage V_TH exhibits a negative temperature coefficient. Therefore, the following relationship can be derived:

$${\displaystyle \frac{\mathrm{d}{t}_{\mathrm{fi}}}{\mathrm{d}{T}_{\mathrm{j}}}}>0$$

(16)

Therefore, the current fall time during the turn-off process exhibits a positive temperature coefficient, indicating that the current fall duration of the SiC MOSFET increases progressively with rising temperature.

The energy loss during the current fall phase can be expressed as follows:

$$E_{\mathrm{fi}}={\displaystyle {\int }_{t_3}^{t_4}\left[v_{\mathrm{DS}}\left(t\right)\cdot i_{\mathrm{D}}\left(t\right)\right]\mathrm{d}t}={\displaystyle \frac{1}{2}}{V}_{\mathrm{DC}}\cdot I_{\mathrm{L}}\cdot (R_{\mathrm{G}}{C}_{\mathrm{ISS}}+g_{\mathrm{fs}}{L}_{\mathrm{S}})\cdot \ln {\displaystyle \frac{V_{\mathrm{DRV}}+V_{\mathrm{TH}}}{V_{\mathrm{TH}}}}$$

(17)

Similarly, differentiating the energy loss E_fi during the current fall phase with respect to the junction temperature T_j yields the following:

$${\displaystyle \frac{\mathrm{d}{E}_{\mathrm{fi}}}{\mathrm{d}{T}_{\mathrm{j}}}}={\displaystyle \frac{1}{2}}{V}_{\mathrm{DC}}{I}_{\mathrm{L}}\cdot \left[L_{\mathrm{S}}\cdot {\displaystyle \frac{\mathrm{d}{g}_{\mathrm{fs}}}{\mathrm{d}{T}_{\mathrm{j}}}}\cdot \ln \left({\displaystyle \frac{V_{\mathrm{DRV}}+V_{\mathrm{TH}}}{V_{\mathrm{TH}}}}\right)-{\displaystyle \frac{(R_{\mathrm{G}}{C}_{\mathrm{ISS}}+L_{\mathrm{S}}{g}_{\mathrm{fs}})\cdot V_{\mathrm{DRV}}\cdot \frac{\mathrm{d}{V}_{\mathrm{TH}}}{\mathrm{d}{T}_{\mathrm{j}}}}{V_{\mathrm{TH}}(V_{\mathrm{DRV}}+V_{\mathrm{TH}})}}\right]$$

(18)

From the above analysis, the transconductance g_fs exhibits a positive temperature coefficient, while the threshold voltage V_TH shows a negative temperature coefficient. As a result, the derivative of E_fi with respect to junction temperature T_j is positive, dE_fi/dT_j > 0.

Therefore, the energy loss during the current fall stage of the turn-off process exhibits a positive temperature coefficient, indicating that the current fall energy loss of the SiC MOSFET increases progressively with rising junction temperature.

From the above analysis, it can be seen that, during the turn-off process, the current fall time and current fall loss both increase with the rise in junction temperature, exhibiting a positive correlation with junction temperature.

2.4. Simulation Analysis of Temperature-Dependent Characteristics

Based on the above analysis, it can be concluded that the parameter characteristics during the current fall stage of SiC MOSFETs in practical circuit applications are significantly influenced by the junction temperature and exhibit a monotonic trend with temperature variation. These characteristics can therefore serve as TSEPs for predicting the junction temperature of SiC MOSFETs.

This section presents simulation results conducted using LTspice software XVII. The parasitic inductance parameters of both the power and gate drive circuits are treated as adjustable variables, while all other parameters are set according to actual operating conditions. The simulations aim to verify the impact of varying junction temperatures on the current fall time and current fall loss. The detailed parameters of the SiC MOSFET double-pulse test circuit used in the simulation are summarized in

Table 1. Parameters of the simulated SiC MOSFET circuit.

Parameter	Value
Bus Voltage VDC (V)	600
Load Current IL (A)	10–30
Load Inductance Lload (μH)	460
Gate Drive Voltage VDRV (V)	+18/−3
External Gate Resistor RG(ext) (Ω)	4.5
Internal Gate Resistor RG(int) (Ω)	10.5
Loop Stray Resistance Rloop (Ω)	0.1
Source Parasitic Inductance LS (nH)	10
Gate Parasitic Inductance LG (nH)	15
Drain Parasitic Inductance LD (nH)	20

.

A double-pulse simulation circuit was implemented in LTspice XVII, with the device model’s junction temperature set at five discrete levels: 30 °C, 60 °C, 90 °C, 120 °C, and 150 °C. The current fall time t_fi and current fall loss E_fi parameters were extracted and recorded from the simulation results. Figure 7 presents the simulated relationships among junction temperature, load current, and current fall time during the turn-off process of the SiC MOSFET. The results align well with the theoretical analysis, demonstrating that the current fall time increases with both rising junction temperature and load current during turn-off.

Figure 8 illustrates the simulated relationship between junction temperature, load current, and current fall energy loss during the turn-off process. The simulation results are consistent with the theoretical analysis, indicating that the current fall energy loss increases with both higher junction temperatures and greater load currents.

Based on the above simulation analysis, it can be observed that, during the current fall phase of the SiC MOSFET turn-off process, both the current fall time t_fi and the associated energy loss E_fi increase with rising junction temperature. The simulation results further demonstrate that t_fi and E_fi exhibit a linear correlation with T_j, effectively reflecting the junction temperature variations in the SiC MOSFET.

Based on the aforementioned theoretical and simulation analyses, it can be concluded that both t_fi and E_fi exhibit a positive correlation with junction temperature during the turn-off process of the SiC MOSFET. This confirms their feasibility as TSEPs. Accordingly, this paper proposes an MLR approach to estimate the junction temperature based on t_fi and E_fi.

3. Junction Temperature Estimation Method Based on MLR During the Current Fall Phase of SiC MOSFETs

Through experimental testing on a calibration platform, temperature-sensitive electrical parameter data under various operating conditions and junction temperatures can be obtained. By fitting a three-dimensional surface model involving current fall time t_fi, current fall energy loss E_fi, load current I_L, and junction temperature T_j, MLR models for t_fi-T_j-I_L and E_fi-T_j-I_L relationships can be established, as expressed in Equations (19) and (20):

$$t_{\mathrm{fi}}={\alpha }_1{T}_{\mathrm{j}}+{\beta }_1{I}_{\mathrm{L}}+{\eta }_1$$

(19)

$$E_{\mathrm{fi}}={\alpha }_2{T}_{\mathrm{j}}+{\beta }_2{I}_{\mathrm{L}}-{\eta }_2$$

(20)

The general form of the multiple linear regression model is defined as shown in Equation (21):

$$Y={\beta }_0+{\beta }_1{X}_1+{\beta }_2{X}_2+\dots +{\beta }_{\mathrm{m}}{X}_{\mathrm{m}}$$

(21)

This equation defines the structure of a multiple linear regression model. In this context, Y denotes the dependent variable, while X₁, X₂, ..., X_m represent the independent variables. The partial regression coefficients β_j (j = 1, 2, ..., m) represent the average change in Y when X_j increases or decreases by one unit while keeping the other independent variables constant.

Simulation results demonstrate that the TSEPs increase with rising junction temperature T_j and load current I_L. In this context, t_fi and E_fi serve as dependent variables, while T_j and I_L are considered independent variables. Accordingly, a multiple linear regression model was established to describe the relationship between the temperature-sensitive parameters and the junction temperature and load current. However, for the purpose of junction temperature prediction, it is essential to construct a regression model in which T_j is the dependent variable and the remaining parameters serve as independent variables, as expressed in Equation (22).

$$T_{\mathrm{j}}=\alpha I_{\mathrm{L}}+\beta t_{\mathrm{fi}}+\eta$$

(22)

However, since there is no direct linear causal relationship between I_L and T_j in this model, the fitting accuracy is limited, and the explanatory power for the dependent variable is relatively weak. Therefore, it is necessary to eliminate the influence of load current to improve the accuracy of the temperature-sensitive electrical parameter method when applied to junction temperature prediction of SiC MOSFETs. By coupling multiple TSEPs, the effect of load current can be eliminated through matrix operations, enabling the extraction of the junction temperature. The implementation process is as follows:

$$\left[\begin{array}{l}{t}_{\mathrm{fi}}\\ E_{\mathrm{fi}}\end{array}\right]=\left[\begin{array}{cc}{\alpha }_1& {\beta }_1\\ {\alpha }_2& {\beta }_2\end{array}\right]\left[\begin{array}{l}{T}_{\mathrm{j}}\\ I_{\mathrm{L}}\end{array}\right]+\left[\begin{array}{l}{\eta }_1\\ {\eta }_2\end{array}\right]$$

(23)

$$\left[\begin{array}{l}{T}_{\mathrm{j}}\\ I_{\mathrm{L}}\end{array}\right]={\left[\begin{array}{cc}{\alpha }_1& {\beta }_1\\ {\alpha }_2& {\beta }_2\end{array}\right]}^{-1}\left[\begin{array}{l}{t}_{\mathrm{fi}}\\ E_{\mathrm{fi}}\end{array}\right]+\left[\begin{array}{c}{\eta }_3\\ {\eta }_4\end{array}\right]$$

(24)

Then, the following expression can be obtained:

$$T_{\mathrm{j}}={\alpha }_3{t}_{\mathrm{fi}}-{\beta }_3{E}_{\mathrm{fi}}-{\eta }_3$$

(25)

An MLR model for predicting the junction temperature of T_j-t_fi-E_fi was established based on the three-dimensional data of t_fi-T_j-I_L and E_fi-T_j-I_L. By incorporating multiple TSEPs, the influence of load current I_L was effectively decoupled, thereby enhancing the accuracy of junction temperature estimation for SiC MOSFETs using electrical parameter-based methods.

Therefore, an MLR method for junction temperature prediction based on the current fall characteristics of SiC MOSFETs is proposed, as illustrated in Figure 9. Initially, a DPT platform was established to facilitate model construction. To ensure that the parasitic parameters in the calibration setup closely match those in the actual operating circuit, both circuits should be designed with similar structural configurations and PCB layouts.

Experimental validation was performed using a DPT platform, with an oscilloscope employed to capture the temperature-dependent characteristics of the current fall phase during the turn-off process. The operating conditions for the SiC MOSFET were set, including the junction temperature T_j, DC bus voltage, and load current. The junction temperature was controlled via an intelligent temperature regulation unit. The t_fi and E_fi of the SiC MOSFET during the turn-off process were measured under various operating conditions and different junction temperatures. The relationship between junction temperature and TSEPs was fitted using MATLAB R2023b or Origin 2021 software. Based on the coupling of multiple TSEPs in the current fall phase, an MLR model was established to predict the junction temperature of the SiC MOSFET. Finally, by substituting the measured values of t_fi and E_fi into the established T_j-t_fi-E_fi junction temperature prediction model, the junction temperature T_j of the SiC MOSFET can be accurately estimated. This implementation not only enhances temperature estimation accuracy but also provides a practical foundation for developing compact, real-time junction temperature sensing solutions in SiC-based power electronic systems.

4. Experimental Validation

4.1. Experimental Platform

Figure 10 illustrates the DPT platform developed in this study for characterizing SiC MOSFETs. The main components of the platform include the following: (1) an intelligent temperature control chamber; (2) the DPT circuit; (3) a high-power DC power supply; (4) an infrared thermometer; (5) auxiliary power supplies; (6) high-voltage differential probes and high-bandwidth current probes; (7) a digital storage oscilloscope; and (8) load inductor.

The NTHL020N090SC1 SiC MOSFET was selected as the device under test (DUT). The DUT was mounted on a temperature-controlled heating stage equipped with an insulation pad. The stage temperature was varied from 30 °C to 150 °C in 20 °C increments (30 °C, 50 °C, 70 °C, 90 °C, 110 °C, 130 °C, 150 °C). At each temperature, the DUT was allowed to thermally stabilize for 5 min prior to measurement. Switching characteristics were then measured at load currents ranging from 10 A to 30 A in 5 A steps.

A constant-temperature heating duration of 5 min ensures sufficient thermal conduction within the chip, allowing the case temperature (T_c) and junction temperature T_j of the device to closely approximate the set temperature of the heating platform. Furthermore, to minimize the influence of self-heating during the calibration process, each pulse signal is limited to only a few microseconds. The heat generated by the chip during this extremely short period is negligible and does not significantly affect the junction temperature. Once the MOSFET chip reaches the target temperature, the corresponding voltage and current waveforms are captured using a digital oscilloscope, and the measurement data are subsequently recorded for analysis.

In addition, the platform adopts the same gate driver circuit design as that used in actual power electronic converters and allows configurable experimental conditions to emulate real operating points. This ensures that the switching characteristics obtained from experiments essentially reflect the behavior of the device under test within practical converters under identical driving conditions, thereby maximizing the comparability of the extracted parameters and the relevance of the model calibration data. This experimental consistency also enhances the applicability of the proposed method for integration into embedded temperature-sensing systems, where the extracted TSEPs can serve as internal indicators for real-time junction temperature monitoring without requiring additional temperature sensors or circuitry.

Table 2. Parameters of the SiC MOSFET double-pulse test platform.

Parameter	Value
Bus Voltage VDC (V)	600
Load Current IL (A)	10–30
Load Inductance Lload (μH)	460
Gate Drive Voltage VDRV (V)	+18/−3
External Gate Resistor RG(ext) (Ω)	4.5
Internal Gate Resistor RG(int) (Ω)	10.5

presents the operating conditions of the SiC MOSFET double-pulse test platform.

4.2. Experimental Results and Analysis

As illustrated in Figure 11a,b, the variation of t_fi with junction temperature under different load currents and its variation with load current at different junction temperatures are depicted, respectively. The experimental results demonstrate that the current fall time during the turn-off phase increases approximately linearly with both rising junction temperature and load current. Based on these observations, a multiple linear regression model for t_fi-T_j-I_L can be established.

As shown in Figure 12, a three-dimensional surface model of t_fi-T_j-I_L is constructed and fitted using Matlab R2023b/Origin 2021 software based on multiple linear regression.

When t_fi is treated as the dependent variable, the resulting regression model is expressed in Equation (26), with a goodness of fit R² of 98.08%. Conversely, when T_j is taken as the dependent variable, the corresponding regression model is given in Equation (27), with an R² value of 88.88%.

$$t_{\mathrm{fi}}=0.06439T_{\mathrm{j}}+0.84329I_{\mathrm{L}}+29.25011$$

(26)

$$T_{\mathrm{j}}=-11.72593I_{\mathrm{L}}+13.90505t_{\mathrm{fi}}-397.30461$$

(27)

Similarly, E_fi was measured under varying junction temperatures and load currents, with the experimental results presented in Figure 13. Figure 13a,b illustrate the variation of E_fi with junction temperature at different load currents, and with load current at different junction temperatures, respectively. The results indicate that the current fall loss during the turn-off phase increases approximately linearly with both junction temperature and load current. Accordingly, a multiple linear regression model for E_fi as a function of T_j and I_L was established.

As illustrated in Figure 14, the three-dimensional surface model of E_fi with respect to T_j and I_L is presented.

A multiple linear regression model was fitted using Matlab R2023b/Origin 2021 software. When E_fi is treated as the dependent variable, the resulting regression model is given in Equation (28), yielding a goodness of fit R² of 99.04%. Conversely, when T_j is taken as the dependent variable, the corresponding regression model is shown in Equation (29), with a significantly lower goodness of fit R² of only 28.91%.

$$E_{\mathrm{fi}}=0.038287T_{\mathrm{j}}+32.82411I_{\mathrm{L}}-264.59014$$

(28)

$$T_{\mathrm{j}}=-27.70953I_{\mathrm{L}}+0.84418E_{\mathrm{fi}}+284.27306$$

(29)

The multiple linear regression models derived from fitting the t_fi-T_j-I_L and E_fi-T_j-I_L datasets indicate that both t_fi and E_fi increase linearly with rising junction temperature and load current. These models, with t_fi and E_fi as dependent variables and T_j and I_L as independent variables, demonstrate a high goodness of fit, effectively capturing the temperature- and current-dependent variations in the electrical parameters. However, the fitted MLR models exhibit varying degrees of goodness of fit. The R² value for the T_j-I_L-t_fi three-dimensional model is 88.88%, whereas that for the T_j-I_L-E_fi model is only 28.91%. Therefore, relying on a single temperature-sensitive electrical parameter to estimate the junction temperature may introduce substantial errors, resulting in reduced prediction accuracy.

Since switching losses in SiC MOSFETs primarily occur during the voltage rise and current fall intervals of the turn-off process, the total energy dissipated in these intervals is defined as the turn-off loss E_off, and the corresponding duration is defined as the turn-off time t_off. In the following sections, three-dimensional models of t_off and E_off are established, followed by the development of a junction temperature prediction model. A comparative analysis of their performance is also presented.

As shown in Figure 15a,b, the variations of t_off with junction temperature under different load currents, and with load current under different junction temperatures, are presented, respectively. The experimental results indicate that the turn-off time increases approximately linearly with increasing junction temperature. Figure 16 presents the three-dimensional surface model of t_off-T_j-I_L. When t_off is treated as the dependent variable, the corresponding multiple linear regression model is given in Equation (30), yielding a goodness of fit R² of 73.85%. Conversely, when T_j is considered the dependent variable, the fitted model is shown in Equation (31), with an R² value of 73.53%.

$$t_{\mathrm{off}}=0.10455T_{\mathrm{j}}+0.075I_{\mathrm{L}}+98.56246$$

(30)

$$T_{\mathrm{j}}=-0.53864I_{\mathrm{L}}+7.18183t_{\mathrm{off}}-685.43913$$

(31)

Similarly, E_off was characterized under various junction temperatures and load currents, with the experimental results presented in Figure 17. Figure 17a,b, respectively, illustrate the variation of E_off with junction temperature under different load currents, and the variation of E_off with load current under different junction temperatures. The experimental results indicate that the current fall time during the turn-off process increases approximately linearly with both junction temperature and load current. Figure 18 presents the three-dimensional surface model of E_off-T_j-I_L. When E_off is treated as the dependent variable, the fitted multiple linear regression model is presented in Equation (32), yielding a goodness of fit R² of 99.43%. When T_j is considered the dependent variable, the corresponding regression model is shown in Equation (33), with a goodness of fit of 45.95%.

$$E_{\mathrm{off}}=0.61383T_{\mathrm{j}}+47.91057I_{\mathrm{L}}-289.88571$$

(32)

$$T_{\mathrm{j}}=-38.34555I_{\mathrm{L}}+0.80036E_{\mathrm{off}}+277.79661$$

(33)

The fitted multiple linear regression models of t_off-T_j-I_L and E_off-T_j-I_L indicate that both t_off and E_off exhibit an approximately linear increase with rising junction temperature. The regression models established with t_off and E_off as dependent variables and T_j and I_L as independent variables demonstrate a high goodness of fit, effectively capturing the variations in TSEPs induced by changes in junction temperature and load current. However, when T_j is treated as the dependent variable, the goodness of fit of the regression models is relatively low. Specifically, the goodness of fit R² for the T_j-I_L-t_off model is 73.85%, while that for the T_j-I_L-E_off model is only 45.95%.

Based on the fitting results of the junction temperature prediction models developed for the turn-off process and the current fall stage using TSEPs, it is evident that models relying on a single temperature-sensitive electrical parameter to back-calculate the junction temperature from operating conditions exhibit limited goodness of fit and poor prediction accuracy. To eliminate the influence of load current on junction temperature prediction, multiple TSEPs were coupled in the multiple linear regression model. Figure 19 presents the T_j-t_fi-E_fi and T_j-t_off-E_off three-dimensional models, respectively.

These models incorporate the TSEPs from the current fall stage and the turn-off process to construct and fit improved junction temperature prediction models. When junction temperature T_j is used as the dependent variable, and t_fi and E_fi are the independent variables, the resulting multiple linear regression model is presented in Equation (34), yielding a goodness of fit R² of 94.11%. Similarly, when T_j is modeled as a function of t_off and E_off, the corresponding regression model is given in Equation (35), with a goodness of fit R² of 75.41%.

$$T_{\mathrm{j}}=16.96642t_{\mathrm{fi}}-0.43547E_{\mathrm{fi}}-605.07921$$

(34)

$$T_{\mathrm{j}}=7.23313t_{\mathrm{off}}-0.01057E_{\mathrm{off}}-694.1756$$

(35)

The goodness of fit comparison of the temperature prediction models established using a single temperature-sensitive electrical parameter and multiple linear regression based on coupled TSEPs is shown in

Table 3. Goodness of fit of each TSEP.

3D Model	Goodness of Fit R2
tfi-Tj-IL	88.88%
Efi-Tj-IL	28.91%
Tj-tfi-Efi	94.11%
toff-Tj-IL	73.53%
Eoff-Tj-IL	45.95%
Tj-toff-Eoff	75.41%

.

As shown in the comparison results of the fitting accuracy of the three-dimensional model of TSEPs in Table 2, the proposed multiple linear regression method for junction temperature prediction based on the current fall phase characteristics of SiC MOSFETs demonstrates high accuracy, with an improvement of 5.23% compared to using the single temperature-sensitive electrical parameter t_fi. Furthermore, coupling temperature-sensitive parameters from the turn-off process also enhances prediction accuracy relative to single-parameter models, thereby confirming the effectiveness and feasibility of the selected parameters. Additionally, comparison of the experimental results reveals that, for a junction temperature prediction model based on multiple TSEPs, its prediction accuracy depends on the combined contribution of each TSEP. The greater the temperature sensitivity of the individual TSEPs, the higher the prediction accuracy achieved by their coupled junction temperature model. Specifically, the T_j-t_fi-E_fi junction temperature prediction model established using coupled TSEPs from the current fall phase achieves higher accuracy than the T_j-t_off-E_off model based on the turn-off process, indicating its greater suitability for junction temperature estimation in SiC MOSFETs.

Table 4. Summary of TSEPS for junction temperature measurement.

TSEP	Ref.	Device	Linearity	Real-Time Implementation	Properties
Gate resistance	[18,21]	MOSFET/IGBT	Medium	Medium	Highly susceptible to noise.
Turn-on/off delay time	[19,25,26]	MOSFET/IGBT	Good	Hard	A large gate resistance needs to achieve detectable resolution.
Turn-on/off di/dt	[20]	MOSFET	Low	Hard	Needs very high detection resolution and accuracy.
On-resistance	[22]	MOSFET	Medium	Easy	An additional high-precision drain-source voltage sensing circuit is required.
Gate turn-on peak current	[23,24]	MOSFET	Medium	Hard	Good for slow switching application.
Voltage rise time and voltage rise loss	[28]	IGBT	Medium	Medium	When applied to MOSFETs, the linearity is insufficient, resulting in a lower goodness of fit for the model.
The proposed method	/	MOSFET	Good	Medium	When applied to MOSFETs, the linearity is better, leading to a higher goodness of fit for the model.

presents a comparison of TSEPS for junction temperature measurement. Compared with other methods, the proposed approach exhibits better linearity and higher goodness of fit when applied to SiC MOSFETs. It requires no additional auxiliary circuitry, is well-suited for online monitoring, and does not interfere with the intrinsic characteristics of the circuit. Furthermore, the influence of load current is eliminated, thereby improving the accuracy of the temperature-sensitive electrical parameter-based junction temperature estimation model. These features not only validate the model’s estimation accuracy but also highlight its potential for integration into compact and low-cost embedded temperature-sensing systems for SiC-based power electronics.

5. Conclusions

This paper proposes an MLR method for junction temperature prediction based on the current fall characteristics of SiC MOSFETs. First, the feasibility of utilizing current fall time and current fall energy loss as TSEPs is demonstrated through theoretical analysis and simulation. Subsequently, an MLR junction temperature estimation model is constructed based on the three-dimensional mapping relationships between t_fi-T_j-I_L, E_fi-T_j-I_L, and T_j-t_fi-E_fi. Experimental results demonstrate that coupling the two TSEPs, t_fi and E_fi, effectively mitigates the influence of load current on model accuracy, thereby enhancing the precision of the MLR junction temperature prediction model. Compared to using t_fi as the sole temperature-sensitive electrical parameter, the model accuracy improved by 5.23%. Additionally, compared to the T_j-t_off-E_off prediction model based on turn-off process characteristics, the proposed T_j-t_fi-E_fi model demonstrates superior suitability for junction temperature estimation in SiC MOSFETs. This study not only validates the feasibility and effectiveness of the proposed approach but also demonstrates its potential for practical deployment in compact, low-cost junction temperature sensing systems. The model’s non-intrusive nature and high accuracy make it well-suited for real-time thermal monitoring, health management, and reliability assessment in SiC-based power electronic applications.