Analysis of Key Factors Affecting Case-to-Ambient Thermal Resistance in Thermal Modeling of Power Devices

Abstract

In the application of power converters, the ambient temperature (T_a) experiences significant fluctuations. For the case-to-ambient resistance (R_ca), apart from the influence of the material’s inherent properties, factors such as heat dissipation structure, working environment, and operational state can all have an impact on the R_ca. Notably, while there are a limited number of models that consider environmental changes, existing models for calculating the R_ca predominantly overlook the influence of varying working conditions or boundary conditions on the self-thermal resistance. Based on simulation and experimental analyses, the methods to calculate the R_ca are outlined, and the key factors, inclusive of the coupling effects that influence the R_ca in the thermal modeling of power devices, are thoroughly discussed.

1. Introduction

Literature Review

The accurate calculation of the junction temperature (T_j) of power devices and the establishment of thermal models for power converters heavily depend on the precise values of the case-to-ambient thermal resistance (R_ca) in a power system. As technology continues to advance, power converters become increasingly compact, resulting in narrower spacing between power components. This poses greater challenges for heat dissipation, exacerbating the issue of thermal coupling (mainly including conduction and convection), which is the thermal interaction and influence among devices. And the coupled thermal effects have a significant impact on the performance and reliability of power devices. Additionally, in power converter applications, the ambient temperature (T_a) fluctuates significantly, causing R_ca to vary constantly. This further complicates the establishment of accurate T_j calculation models for power components [1,2,3].

Various factors influence R_ca, including device packaging with its materials, dimensions, and shapes that dictate thermal characteristics and cooling efficiency; operating conditions, such as current, voltage, and power levels that affect heat generation; boundary conditions like ambient temperature, heat sink design, and airflow that directly impact cooling efficiency; and thermal coupling effects among closely spaced power devices, which can redistribute heat and alter individual cooling efficiency [4,5,6]. To determine R_ca, initial basic calculations are conducted using theoretical formulas or experimental measurements, taking into account device packaging, operating conditions, and boundary conditions. When multiple devices are involved, thermal coupling effects are analyzed through simulations and experiments to assess their impact on R_ca. By integrating these findings, a comprehensive evaluation of R_ca’s value and variations under different operating scenarios is conducted, enabling the development of more precise T_j calculation models and optimized thermal designs for power converters [7,8,9].

The calculation formula for T_j in power devices used in converter applications is detailed in [10], comprehensively accounting for the thermal coupling effect and the convection thermal coupling between adjacent semiconductor devices, ensuring reliability across varying environmental temperatures. Additionally, when analyzing the R_ca of these power devices, it is crucial to note that measured R_ca values fluctuate based on changes in device packaging, working conditions, and boundary conditions, as outlined in [11,12]. However, the existing literature often calculates R_ca under limited working conditions, rarely exploring its variation across diverse operational scenarios. This is particularly significant given that ambient temperatures are not static, and the relevant parameters are highly dependent on these temperatures.

The non-uniform distribution of device case temperatures can easily lead to thermal coupling issues between adjacent devices and the concentration of thermal stress within the converter system [13]. Device spacing and operating currents both influence thermal coupling, subsequently affecting the case temperature or junction temperature of the devices. By comprehending the relationship between self-thermal resistance and the coupling thermal resistance among neighboring devices, thermal distribution uniformity can be ensured. This means that the case temperature/junction temperature of the devices can maintain a certain temperature value or range through a deeper understanding of the thermal coupling effect among devices under varying operational conditions. Nevertheless, this aspect is scarcely explored in the existing literature.

Currently, many studies on the thermal model of power devices are focused primarily on the device level and primarily concern thermal conduction within a single device or module. Furthermore, the existing thermal coupling considerations primarily focus on the multiple chips within a single module, overlooking the conduction coupling that occurs between power devices or modules within a power converter system.

In [14], a dynamic ambient profile is thoroughly analyzed, and a corresponding thermal analysis model is established based on this profile. Given that power devices are often situated in outdoor environments, the ambient temperature is not constant, as highlighted in [15]. However, the thermal coupling effect among power devices within the system remains unexamined in [16].

Thermal analysis models for semiconductor devices typically rely on a static ambient temperature, with limited literature focusing on varying environmental ambient profiles. The convective thermal coupling effect among adjacent semiconductor devices is dependent on ambient temperature, resulting in fluctuations in thermal coupling and device junction temperature. Additionally, convective thermal coupling caused by air convection also contributes to a rise in device temperature. However, current thermal analysis models seldom consider the impact of convective thermal coupling.

This paper has been organized into five sections. Section 1 introduces the topic. Section 2 presents the method to calculate the R_ca without considering the thermal coupling between the adjacent power devices. Through thermal coupling simulation and experimentation, Section 3 presents the way to calculate the R_ca, considering the thermal coupling effect. Section 4 comprehensively discusses all the pivotal factors that influence the R_ca, and Section 5 presents the conclusive findings based on the analysis conducted.

2. Analysis and Calculation of the R_ca Ignoring Thermal Coupling

2.1. Analysis of the R_ca for a Single Power Device

The MOSFET, diode, and other semiconductor devices feature intricate and miniaturized structures, making it challenging to accurately measure their junction temperatures [17]. During unpackaged testing, there is a risk of damaging the device’s structure [18], which would alter the actual thermal transfer mechanism and introduce errors in the measurement results. The case temperature (T_c) of these devices is a crucial parameter that reflects their semiconductor performance and serves as a vital technical indicator for heatsink selection. It offers insights into temperature trends and enables the prediction of the device’s T_j through appropriate thermal calculation methods [19]. The thermal calculation equation linking T_j and T_c can be formulated as follows:

$$T_j={ P}_{\mathit{loss}}·R_{\mathit{jc}}+T_c$$

(1)

P_loss represents the power loss of the power device, while T_c denotes the case temperature of the device. The thermal resistance from the junction to the case, commonly referred to as R_jc, is typically provided in the datasheet supplied by the manufacturer.

The thermal resistances of R_ja, R_jc, and R_ca are interrelated, each describing the thermal conduction characteristics of the different parts of a semiconductor device. R_ja represents the total thermal resistance from the semiconductor die to the ambient environment, specifically the resistance from the junction to the ambient temperature. It signifies how the heat generated within the device is conducted to the surrounding environment, serving as a crucial parameter for evaluating the device’s heat dissipation performance. R_jc refers to the thermal resistance between the semiconductor die and the device’s casing, specifically the resistance from the junction to the package housing. It reflects how the heat generated within the device’s internal structure is conducted to the outer shell, making it a crucial factor to consider during the design and application of semiconductor devices; R_ca is typically associated with the cooling mechanism and represents the thermal resistance from the casing to the air or other cooling media. Although the exact definition of R_ca in datasheets or general descriptions is not directly mentioned here, we can infer that it involves the impact of the cooling mechanism on the overall thermal performance of the device. The relationship between these thermal resistances can be expressed as:

R_ja = R_jc + R_ca

(2)

In terms of self-thermal resistance, R_jc is primarily influenced by metal thermal conduction, and its specific values are typically provided in the datasheets furnished by manufacturers. For instance, the maximum R_jc value for the MOSFET (IRF540NPBF) stands at 1.15 °C/W, whereas for the diode (MBR20100CT), it is 2.0 °C/W.

The convection thermal transfer occupies a pivotal position in power systems, especially in the heat exchange process that involves heat sinks and the ambient environment. Convection thermal transfer, on the other hand, refers to the heat exchange between a fluid and a solid wall. This process relies heavily on the fluid’s movement to facilitate thermal transfer [20]. During the operation of power devices, they produce considerable heat, which must be effectively dissipated through cooling mechanisms to avoid overheating and subsequent performance decline. Convection heat transfer, leveraging air currents (whether through natural or forced convection), facilitates the removal of heat, thereby lowering the temperature of the devices. The efficacy of convection heat transfer has a direct bearing on the cooling performance of the devices. An enhancement in convection heat transfer leads to a reduction in device temperature, consequently decreasing the value of thermal resistance (R_ca). This is because thermal resistance serves as a metric for the ease of heat transfer, and a diminished temperature gradient signifies that heat transfer becomes more facile. Consequently, convection thermal transfer is intricately linked to the fluid’s flow conditions.

As illustrated in Figure 1, the case is in black, the baseplate is in white, and the red arrows indicate the direction of heat transfer, the convection thermal transfer characteristics of the same device can vary significantly depending on its location on different hot surfaces.

2.2. Methods to Calculate the Resistance of R_ca

The convection thermal dissipation is influenced by both power losses and the conditions of device thermal dissipation. Notably, the corresponding R_ca is typically omitted from device datasheets, and the values of R_ca are primarily determined by factors such as the surrounding environment and the package structure [21]. These factors can be formally expressed as:

$$R_{\mathit{ca}}={\displaystyle \frac{1}{\mathit{hA}}}$$

(3)

where A represents the convection thermal transfer area, and h denotes the convection thermal transfer coefficient [22].

The value of h is intricately linked to various factors involved in the process of convective thermal transfer. It is influenced not only by the physical attributes of the fluid but also by the shape, size, and configuration of the thermal exchange contact surface. Additionally, it exhibits a strong correlation with the flow velocity of the fluid.

Table 1. Value range of convection thermal transfer coefficient h.

Types of Convection Thermal Transfer	Convection Thermal Transfer Coefficient h/[W/(m2·K)]
Natural convection thermal transfer of air	1~10
Natural convection thermal transfer of water	100~1000
Forced convection thermal transfer of air	10~100
Forced convection thermal transfer of water	100~15,000

presents the values of some commonly utilized coefficients in this context.

Based on the analysis of [23,24], the convection thermal transfer coefficient can be expressed as follows:

$$h\propto {\left({\displaystyle \frac{∆T}{L}}\right)}^{0.25}$$

(4)

where $∆T$ is the temperature difference between T_c and T_a, L is the vertical height of the contact surface.

In the application of converters, heatsinks serve as vital components that effectively facilitate the transfer of heat from semiconductor devices, thereby reducing junction temperatures and safeguarding the integrity of these devices. When designing the structure of the heatsinks and the entire power converter, it is crucial to consider not only the thermal dissipation capacity or coefficient but also the size and installation position of the heatsinks. The designer must make a judicious selection of heatsinks during the design stage to ensure optimal thermal dissipation for the devices.

It is noteworthy that the value of R_ca varies depending on the specific application of the heatsinks. Based on certain standards or results, the designer can select the appropriate type of heatsink for the devices [25]. Consequently, the R_cas are variable until a cooling scheme is determined.

Furthermore, in specific applications, such as air-cooled or water-cooled cooling systems, alterations in the fan speed or the flow rate of cooling water can lead to changes in the corresponding R_ca. Leveraging the relationships between variable thermal resistances outlined in this paper, recommended values for case-to-ambient thermal resistances can be derived. These recommended values, in turn, aid in selecting suitable cooling schemes for power devices.

Figure 2 illustrates the simplified structure of the power devices. The device depicted in Figure 2a lacks a heatsink, resulting in its convection thermal resistance being equivalent to the case-to-ambient thermal resistance. For the purposes of this analysis, we overlook the thermal resistances arising from structural layers such as the solder layer and silicone grease layer. However, if a heatsink is installed on the power device, as shown in Figure 2b, an additional thermal resistance is introduced between the R_jc and the R_ca. This additional resistance is referred to as the R_ch.

In the case of a real packaged power device, numerous layers exist, as evident in Figure 3a. When a heatsink is added, as depicted in Figure 3b, the thermal resistance of the heatsink is positioned between R_jc and R_ca, resulting in the emergence of R_ch. The resistance of R_ch is akin to a conduction resistance, similar to R_jc. Consequently, R_ca transforms into R_ha. For the purposes of this study, the resistances within the solder layers are disregarded. Regardless of whether it is R_ca or R_ha, the focus of this analysis is on convection thermal behavior, and the resistance is environmentally dependent. A deeper understanding of this environmentally dependent resistance will aid in selecting and sizing an appropriate thermal cooling scheme [26].

When selecting the type of power device, the thermal contact area (A) and the vertical height (L) of the contact surface remain as known constants. From Equations (3) and (4), it is evident that the R_ca depends on the temperature difference (ΔT). Furthermore, any changes in power loss will result in alterations to the device’s T_c, subsequently affecting the corresponding convection thermal resistance. Therefore, R_ca can be regarded as a function of both temperature difference and power loss, expressed as:

$$R_{\mathit{ca}}=f\left(∆T,P\right)$$

(5)

where ∆T represents the temperature difference, and P represents the power.

Based on the preceding analysis, the expressions for transient thermal impedance Z_ca(t) and transient thermal capacity C_ca(t) can be derived as follows:

$$\left\{\begin{array}{c}{Z}_{\mathit{ca}}\left(t\right)={\displaystyle \frac{∆T\left(t\right)}{P}}\\ C_{\mathit{ca}}\left(t\right)=\rho c_p\left(t\right)lA\end{array}\right.$$

(6)

where ∆T(t) represents the transient temperature difference, P represents the power, $\rho$ represents the density of the object, $c_p\left(t\right)$ represents the transient specific heat capacity at constant pressure of the object, l represents the length or thickness of the object, and A represents the cross-sectional area of the object.

Additionally, referring to [21,27], the calculation formula for steady-state R_ca in convection thermal transfer has been established.

$$R_{\mathrm{ca}}={\displaystyle \frac{∆T}{P}}$$

(7)

where ∆T represents the temperature difference, and P represents the power.

3. Thermal Coupling Between Power Devices [9]

3.1. Thermal Analysis Considering the Thermal Coupling

Thermal dissipation poses a significant negative impact on the performance and lifespan of semiconductor devices and power converters [28]. As shown in Figure 4, during the operation of power devices, the vast majority of frequent switching losses are dissipated in the form of heat. Due to size constraints, multiple chips within a module are positioned relatively close to each other, leading to heat coupling and mutual influence during the heat transfer process. If heat dissipation is inadequate and heat accumulates excessively, it can easily lead to thermal runaway and even combustion.

For a single IGBT power module, multiple IGBT chips and diode chips are typically packaged together within the module, as illustrated in Figure 4. Conducting a thorough thermal analysis for such a module with numerous chips is imperative, as failure to do so may result in thermal runaway and subsequent burnout of the module, as depicted in the figure. To address this, a FEM analysis model of a three-phase motor inverter (in Figure 5) has been established to analyze the conduction thermal coupling effect between adjacent power chips in various spaces.

The main structure of the IGBT module and the relevant parameters of each key structural layer are shown in

Table 2. Parameters of key structural layers in an IGBT module.

Key Structural Layers	Specific Heat Capacity Cp/[J/(kg*K)]	Thermal Conductivity k/[W/(m*K)]
Chip	700	130
Ceramic substrate	730	35
Copper baseplate	385	400
Heat sink	900	238

. The mesh division method used in the simulation is multi-level division. Multi-level mesh division refers to the adoption of different mesh division methods for different structural layers, namely: finer mesh division is applied to the chip layer, finer mesh is used for other layers such as the ceramic substrate, coarse mesh is used for the baseplate, and relatively coarse mesh division is applied to the heat sink. Additionally, the heat dissipation conditions used in the simulation are set to natural convection, and the finite element simulation is run in steady-state mode.

Using FEM analysis for adjacent IGBT modules, both the temperature distribution and isosurface distribution of the power module can be obtained. Different colors are used to represent different temperatures, while the direction of the body arrows indicates the direction of heat transfer. Based on the simulation results, the temperature at the chip is the highest, marked in bright white. Heat transfers vertically from the chip to the substrate within the module, resulting in a gradual decrease in temperature, as depicted in Figure 6a. As the heat reaches the heatsink, it is transferred both vertically and horizontally. Figure 6b illustrates that the temperature between adjacent modules is also relatively high. This occurs because multiple modules are positioned close to each other on the same heatsink, and the heat is mutually influenced during transverse transmission, resulting in the conduction thermal coupling effect.

3.2. FEM Models for Adjacent Devices of the MOSFET and Diode

The power loss of the MOSFET is set at 5 W, while the diode’s power loss is set to 2 W. Given that the MOSFET’s power loss exceeds that of the diode, the ambient temperature surrounding the MOSFET is correspondingly higher. This difference is clearly reflected in the color tables, which depict the thermal distribution. Through a FEM analysis, the temperature distribution and isosurface distribution of the power devices under varying ambient temperatures are obtained, as illustrated in Figure 7. The color tables in the figures below highlight the distinct differences in the distribution patterns [29].

Drawing upon the analysis of the conduction thermal coupling effect outlined in Section 3.1, it becomes evident that when multiple power modules share a heatsink or numerous devices are interconnected via copper wires on a PCB, a coupling effect arises among these modules/devices. In the realm of medium and low-voltage power converters, MOSFETs and diodes serve as the primary power devices. Once the heat generated by these devices is dissipated into the surrounding air through convection, it undergoes thermal coupling within the ambient atmosphere.

Such thermal coupling effects exert a notable influence on the junction and case temperatures of the devices, ultimately impacting the accuracy of junction temperature predictions and the reliability of both devices and systems [30]. Consequently, it is imperative to establish an analytical model that captures the convection thermal coupling phenomenon between adjacent power devices. This model would provide a valuable tool for enhancing the understanding and prediction of thermal behaviors in complex power converter systems.

3.3. Thermal Coupling Testing Between the Devices

To investigate the impact of thermal convection on devices within a power converter system, a dedicated convection thermal analysis testing platform has been developed [31,32]. This platform focuses on integrating a single semiconductor device—either a MOSFET or a diode—into the circuit, allowing for the examination of case temperature variations by altering the spacing between devices.

The testing devices encompass MOSFETs, diodes, capacitors, inductors, and other components, as shown in Figure 8. Since these devices are not interconnected, they are not subject to thermal conduction effects. Instead, the positioning of the devices and the distances between them govern the convective thermal dissipation. In essence, thermal convection exerts a significant influence on the T_j of the devices.

By varying the working current, the on-state losses of the devices change, leading to alterations in their case temperature. Simultaneously, the remaining devices remain in an open-circuit state. Specifically:

(1)

When testing the MOSFET, it is powered on while in the on-state, and all other devices are left in an open-circuit configuration.

(2)

Similarly, when testing the diode, it is powered on in the on-state while the other devices remain in an open-circuit state.

This approach ensures that the thermal convection effects on individual devices can be accurately isolated and analyzed.

To better analyze convection thermal coupling, the devices under test are not interconnected as a power converter within the circuit. Instead, two key components—a MOSFET and a diode—are evaluated separately within the convection thermal analysis platform. Two distinct testing configurations are employed:

(1)

A PCB with closely spaced devices (designated as Type I), including MOSFETs, diodes, capacitors, inductors, and others. In this configuration, the convective thermal dissipation of the devices is restricted, resulting in significant thermal coupling.

(2)

A PCB with ample spacing (designated as Type II) between MOSFETs, diodes, and other components. This setup provides ample convective thermal dissipation space, minimizing the coupling effect.

The testing setup for these semiconductor devices is depicted in Figure 9. The load currents utilized during testing are 3 A and 6.5 A, respectively. The MOSFET being tested is of the IRF540NPBF type, and the diode is of the MBR2010CT type. Both devices feature a TO-220 packaging format. Thermocouple-type temperature sensors are employed and the sensor probes are placed on the case of the power devices to collect temperature data.

During the convection thermal analysis testing, the operating conditions/currents of the devices are varied. The junction temperatures of the MOSFET and diode, collected from both Type I and Type II configurations across different operating currents, are then compared. This comparison serves to identify the influencing factors of thermal convection and its impact on device temperatures.

In this testing platform, the T_c of the device is recorded under various working conditions using a data logger. A comparison of the testing results for MOSFET and diode under two distinct spacing configurations is presented in Figure 10. Specifically, Figure 10a compares the T_c of MOSFET and the temperature difference across different working currents, while Figure 10b does the same for the diode. Notably, for both MOSFET and diode, the T_c is observed to be higher when tested under Type I (smaller spacing) compared to Type II (larger spacing). Furthermore, for MOSFET, the T_c difference between the two spacing types is 1.9 °C at 3 A and 3.3 °C at 6.5 A. Similarly, for the diode, the T_c difference is significant, reaching 19.0 °C at 3 A and 13.3 °C at 6.5 A (Tested over a duration of 3000 s).

The primary reason for this phenomenon lies in the varying distances among devices and changes in the convection cooling environment, both of which significantly affect the T_c of the devices. Consequently, the convection cooling process is influenced by multiple factors, including power loss/working current, device spacing, and ambient cooling conditions.

3.4. Calculation of the R_ca

Based on Equation (7), the R_ca values can be derived, as depicted in Figure 11. Evidently, the R_ca of the device exhibits non-constant behavior under varying working conditions [20,33,34]. As the loss fluctuates, the case temperature of the devices also varies, leading to corresponding changes in the case-to-ambient thermal resistance. Specifically, as the case temperature rises, the case-to-ambient thermal resistance of both MOSFET and diode decreases, and the reverse occurs when the temperature falls. This underscores the dependency of case-to-ambient thermal resistance on both power losses and the ambient conditions. Eduardo L et al. in [25] demonstrated that case-to-ambient thermal resistance can be a valuable tool in assessing the impact of different cooling techniques. Therefore, during the design phase of power converters, it is imperative to conduct thorough identification and analysis when considering various thermal dissipation solutions. Furthermore, the values of case-to-ambient thermal resistance can aid in the selection of heatsinks, ensuring optimal thermal management.

4. Discussions

Based on the analysis of R_ca, as well as simulation and experimental findings, the primary influencing factors are deliberated upon as follows:

(1)

The ambient temperature

When operating at a certain T_a, the junction or case temperature of power devices rises due to power losses incurred during their operational process. There are two primary methods for determining the ambient temperature used in testing or measurement: the first involves collecting the initial ambient temperature prior to device operation, designated as T_a; the second involves measuring the ambient temperature once the device has reached a steady state (either junction or case temperature), designated as T_a′. These two methods yield distinct ambient temperature values. As the device’s junction and case temperatures rise during operation, the generated heat dissipates into the surrounding air, causing the ambient temperature to increase during this process. Consequently, the ambient temperature obtained through the second method is higher than that of the first.

However, it is noteworthy that for power devices, identical power losses result in the same temperature rise, regardless of the method chosen. After reaching a steady state, the junction or case temperature of the device remains constant. According to the steady-state R_ca calculation Formula (7) outlined in Section 2.2, a larger temperature difference between the case and ambient temperature in the first method corresponds to a higher thermal resistance value. Conversely, a smaller temperature difference in the second method leads to a lower R_ca value. Nevertheless, the thermal resistance network constructed using either ambient temperature collection method will yield identical results for case or junction temperature.

(2)

The case temperature of the power devices

The case temperature of the devices serves as a crucial parameter, effectively reflecting their performance. Leveraging the analysis conducted in this section, the case-to-ambient thermal resistance of the device can be employed to assess the impact of various cooling technologies. Furthermore, adhering to the guidelines outlined in JB/T 9684-2000 [35] for selecting heatsinks for power semiconductor devices, the T_c and R_ca of the devices provides valuable insights for heatsink selection.

(3)

Thermal coupling effect

In power conversion systems, power devices are usually densely arranged, resulting in close distances between devices and, thus, increasing the potential for thermal coupling. Since the thermal generated by one device can be easily transferred to adjacent devices, the temperature distribution within the system becomes complex and difficult to predict.

An increase in temperature can lead to reduced efficiency, increased power loss, and even thermal failure. Therefore, understanding and analyzing these thermal coupling effects is crucial for ensuring optimal performance and reliability of power conversion systems.

Variations in the spacing between adjacent devices lead to changes in thermal coupling resistance, whether it is multiple devices connected on a PCB through copper wires or multiple power modules or devices sharing the same heatsink. Therefore, the coupling thermal resistance or thermal coupling effect is intricately linked to device spacing.

5. Conclusions

In power converters, the method for calculating the R_ca of a single device has been analyzed. Additionally, the simulation comprehensively explored the thermal coupling effects among various chips and components. By establishing an experimental platform for the converter and collecting temperature data, the thermal coupling effects are analyzed, thereby contributing to the determination of the R_ca value. This process identified the key influencing factors of R_ca, serving as a foundation for calculating T_j and thermal modeling, and aiding in the selection of heatsinks.