A Review on Transient Thermal Management of Electronic Devices

As electronic devices continue to undergo intensifying miniaturization [1,2], power densities persist in being on a rising trend. Heat fluxes over 10² W/cm² are currently generated by microprocessors [3]. Moreover, hot spots in chips involve localized heat fluxes of 1 kW/cm² or more, leading to excessive local temperatures [4,5]. Rising user demand for augmented computational performance and functionalities has been fueling the development of high-power microprocessors. Apart from IC devices, power semiconductor devices such as insulated gate bipolar transistors (IGBTs) and laser diode arrays also generate heat fluxes above 1 kW/cm² [6–9]. Operation at excessive temperatures impairs device performance and reliability and ultimately causes their failure. The different failure modes of microelectronic devices, namely mechanical, electrical, and corrosion, are linked to high operating temperatures [10]. Pecht and Gu discussed the various failure mechanisms identified in electronic products. Fatigue due to temporal temperature oscillations leads to failure at locations such as the die-attach, wire bonds, solder leads, bond pads, and vias [11]. Similarly, in the case of power electronic devices, high-frequency switching operation results in periodic temperature swings. The resulting thermomechanical stresses lead to the failure of the constituent semiconductor components and thereby undermine device reliability. Transient thermal management techniques that reduce temperature swings, for such applications involving time-varying power loads, therefore become inevitable.

There has been considerable progress in implementing various cooling techniques to address the thermal management challenges associated with high heat flux devices. A summary of these techniques is listed in Table 1. As can be inferred, developments in single-phase, two-phase cooling, jet impingement, and spray cooling have enabled the dissipation of ultrahigh heat fluxes [9,23,34,35,38,39].

While much research work in conventional thermal management deals with cooling under steady-state device operation, most microelectronic devices seldom operate under steady-state conditions. Figures 1(a) and 1(b) demonstrates the temporal variation in CPU workloads of a D2461 Siemens—Fujitsu server. The power consumption of the 4 GHz AMD Athlon 64 dual-core processor can be seen to vary continuously with time as it performs an online music search and download operation [40].

Another development in the microelectronic industry is the growing shift toward portable handheld devices such as smartphones and tablets [41,42]. These devices exhibit long durations of low-power consumption for regular applications and short durations of high-power consumption during process-intensive applications such as video calling/recording [43–45].

Compared to microprocessor chips, a more pronounced transient thermal behavior is exhibited by power semiconductor devices. Rapid fluctuations in junction temperatures are experienced during high-frequency switching operations. Highly transient thermal characteristics are also witnessed in high-power semiconductor laser diode arrays [46].

Cooling solutions that work well under steady-state conditions may not offer desired thermal characteristics when devices are operated under transient workloads. The primary focus of steady-state cooling systems is to reduce the overall thermal resistance (R) of the package so that maximum power can be dissipated without exceeding the maximum temperature limit (T_j,max) of the device. The thermal capacitance of the package (C), though, is not considered. Such a policy of minimizing R without optimizing C results in the following drawbacks:

1. Highly efficient compact cooling devices with low R usually have low C [47,48]. Packages with low C exhibit sudden changes in temperature in response to pulsed workloads. The consequent thermal fatigue effects can be detrimental to device reliability.

2. Steady-state cooling systems developed based on T_j,max is often over-designed as the average heat dissipation rates of electronic devices are usually lower during operation.

3. Any attempt to augment device performance, for a package designed with a specific R, is constrained by power limitations governed by T_j,max.

The limitations of steady-state-based cooling systems under pulsed workloads were demonstrated by Jankowski and McCluskey [48], who examined the transient thermal response of a semiconductor power electronics package when cooled using five different configurations (cases 1–5 as illustrated in Fig. 2(a)). The overall thermal resistances (R) and thermal capacities (C) of the configurations were in decreasing orders of their extents of cooling integration. As seen from Fig. 2(b), the maximum temperatures of cases 3 and 5 were, respectively, 30% and 207% higher than that of case 1, while their temperature swings were higher by 106% and 401% as the period of the pulses approached their thermal time constants (τ = RC). From this study, it is evident that highly efficient steady-state cooling solutions with low thermal resistances (R) and thermal capacities (C) are not capable of mitigating thermal transients during the pulsed load operation, especially when the frequency of the pulses is comparable to the thermal time constant of the package. A similar observation was made by Meysenc et al. [47], who compared the transient thermal responses of an integrated microchannel liquid heat exchanger, a liquid cold plate, a forced convection heat sink, and a natural convection heat sink (in increasing order of thermal resistance (R) and thermal capacity (C)). The results of this study are discussed in detail in Sec. 2.2. It becomes pertinent to employ dynamic thermal management techniques that can mitigate thermal transients.

In this review, emerging transient thermal management requirements are identified, and different transient cooling techniques reported in the literature are examined. Based on their heat dissipation capabilities and time scales of thermal response, viable cooling solutions for electronic devices such as microprocessors, IGBTs, and high-power laser diode arrays are recommended.

The transient thermal issues associated with different microelectronic devices vary in terms of their operating conditions and the time scales involved. In this section, the transient thermal management challenges associated with microprocessor chips, IGBTs, and high-power laser diode devices are examined in detail.

While the ongoing trend toward diminishing transistor sizes is driven by performance enhancements associated with smaller transistors, transistor downscaling is limited by rising leakage currents, and the proportion of static power to the total power consumed is rising considerably [49]. This also leads to the problem of dark silicon, where the active area of the chip is minimized to limit the amount of heat generation. Processors are, therefore, thermally constrained to operate below their rated peak performance levels [49].

where T_j,max, and T_ambient are the maximum allowable junction temperature and ambient temperature, respectively, while R_tot is the total thermal resistance between the junction and ambient. TDP signifies the maximum power that can be dissipated by the chip within the limit of T_j,max.

For steady-state thermal designs not utilizing the thermal capacitance of the cooling system, device operation within the limits of TDP prevents the utilization of the available thermal headroom, which constrains device performance, whereas when the thermal capacitance of the cooling system is used as a buffer, the device can be made to run temporarily at workloads exceeding TDP before the junction temperature reaches T_j,max. One of the initial demonstrations of this approach was made by Cao et al. [50]. They employed liquid and solid thermal energy storage systems to facilitate the operation of portable computing devices under short power bursts. Such a mode of operation is currently used in Intel's Turbo-Boost [51] and AMD's Turbo-Core [52] technologies. This technology was also studied extensively by Raghavan [49], who termed it as computational sprinting.

Based on user experience, computational workloads can be classified into throughput workloads and bursty workloads [53–55]. While throughput workloads involve continuous computations for sustained user experience (examples: gaming, streaming, graphics processing), bursty workloads involve intermittent computations (examples: browsing, interactive programs) where there are short periods of computational bursts followed by long durations of idle activity. Bursty workloads are becoming increasingly prevalent in portable device chips as such devices primarily involve user-interactive applications. For bursty workloads, implementing a steady-state thermal design with thermal throttling leads to an increase in latency and thereby reduces the quality of service (QoS). If the device is instead subjected to short-duration computational bursts surpassing the TDP, its performance can be augmented considerably without reaching T_j,max limits. QoS can accordingly be enhanced by improving user responsiveness [55].

Rotem et al. [54] demonstrated Intel's Turbo-Boost technology in a quad-core Intel^® Core^TM 2 Duo 2860QM processor having a rated TDP of 45 W. The thermal responses of the chip under steady-state and transient (turbo) workloads were compared for a notebook device (large form factor) and a portable device (small form factor), as shown in Figs. 3(a) and 3(b), respectively. To facilitate turbo-operation, the thermal capacitance of the device (consisting of the chip, heat sink, and enclosure) was employed as a buffer. Under turbo-operation, the notebook chip offered a performance enhancement of 34% was obtained over the 20 s duration compared to steady-state operation (as shown in Fig. 3(a)).

Similarly, Raghavan et al. [53] presented the concept of computational sprinting that enhances the user experience for interactive applications. Whenever a demand for increased user responsiveness arises, all processor cores are made to run simultaneously (parallel sprinting) such that high performance is delivered. To prolong the sprint duration, Raghavan et al. [53] employed PCM to limit the rise in T_j over a brief period of time during which the PCM melts. Higher computational performance can be sustained when the duration of computational sprinting is extended by employing a PCM to store thermal energy.

Owing to their high operating voltage and current capacities as well as their high switching speeds, IGBTs are extensively used as a power converter, power inverter, and uninterrupted power supply (UPS) devices. They are primarily employed for traction applications and power generation, and management (solar PVs, wind energy, power grid) [56].

The electrical and thermal characteristics of IGBTs exhibit transient behavior as these devices involve high-frequency switching operations. Frequencies can reach up to 150 kHz under soft-switching conditions [57]. As the switching frequency increases, the switching losses and heat dissipation rates also increase [58,59]. The cyclic variation of junction temperature with periodic on-off power conditions causes thermomechanical stresses in the IGBT die components. Such stresses affect device reliability and reduce the number of cycles to failure.

Held et al. [60] described two types of failure mechanisms caused by the temperature swings during power cycling, namely lifting of bond wires and solder joint fatigue. Fast power cycling tests demonstrated that the number of cycles to failure decreases with junction temperature swings (ΔT_j) in a logarithmic manner, as shown in Fig. 4. Moreover, for a given ΔT_j, the number of cycles to failure reduces with increasing medium junction temperature (T_m). IGBT modules consist of direct bonded copper (DBC) substrates containing silicon chips. These chips are connected to the DBC by metallic bond wires (made of aluminum). The attachments between the silicon chips and DCB as well as that between the DCB and base plate are established by solder joints. During power cycling, the varying junction temperatures cause thermomechanical stresses that arise due to the difference in thermal expansion coefficients between the silicon chips and the metallic bond wires. Consequently, cracks are formed in the bond wires, which results in their failure. Once a bond wire lifts, the electric current distribution to the silicon chips via the remaining bond wires becomes nonuniform. This leads to even higher temperatures across the chips, which exacerbate thermomechanical crack propagation in the remaining bond wires. Following this, failure of the solder joints commences due to thermal fatigue. Thermomechanical stresses also give rise the chip metallization. This phenomenon, called reconstruction, increases the metallization resistance and leads to higher collector-emitter voltages. As a result, heat dissipation rates and hence the chip temperatures become greater. It, therefore, becomes imperative to reduce ΔT_j for improving the reliability of IGBTs.

The swing in T_j depends on the IGBT switching frequency and the output frequency relative to the thermal time constant of the IGBT module. As the output frequency reduces and becomes comparable to the thermal time constants, ΔT_j becomes more significant [57,61]. ΔT_j also increases with switching frequency [61]. Typically, IGBT modules are remotely cooled using a heat sink mounted onto the base plate. Air-cooling, forced convection liquid cooling, two-phase cooling, jet impingement, and spray cooling are the conventionally employed cooling techniques [56].

While the cooling performances of these techniques have been enhanced in terms of lowering the thermal resistance, the thermal capacities of the cooling systems are often not considered for controlling the fluctuations in T_j. Meysenc et al. [47] examined the transient response of different IGBT cooling systems, namely an integrated microchannel liquid heat exchanger, a liquid cold plate, a forced convection heat sink, and a natural convection heat sink, under a periodic load. It was identified that the thermal time constants of the cooling systems decreased with increasing cooling performance. While the integrated microchannel heat exchanger reduced the average T_j on account of its low thermal resistance, its poor thermal inertia led to higher swings in T_j. As shown in Fig. 5, both the liquid cold plate and microchannel heat exchanger offer relatively similar cooling performance below 10 s. However, the cold plate has a thermal resistance twice as high as the microchannel heat exchanger. For such power cycle frequencies, using a liquid cold plate becomes more feasible in cost and manufacturability.

In their experimental study on the transient thermal characteristics of a liquid-cooled SiC power electronic module, Mandrusiak et al. [62] identified that temperature swings became severe when the frequency of the electric current excitation was comparable to the thermal time constant of the module. The thermal time constant of the die (∼100 ms) being closest to the power supply frequency (∼10 Hz) had the most significant influence on the module's transient thermal response. With a large thermal time constant of 10 s, the cold plate had the least influence on the transient thermal sensitivity.

Insulated gate bipolar transistors cooling systems must therefore be designed such that the module thermal time constants are sufficiently higher than the power cycle periods to lessen the temporal variation in T_j. As IGBT switching frequencies and output frequencies are generally in the order of tens of kHz and tens to hundreds of Hz, respectively, cooling solutions should be able to respond within a time scale range of 10⁻² to 10¹ s.

Semiconductor laser diodes are being used extensively for telecommunications, solid-state laser optical pumping, industrial machining, image scanning, laser printing, medical surgery, and military applications. With optical efficiencies generally in the range of 30-50%, laser diode bars involve high-power densities exceeding 1 kW/cm² [46,63,64]. Such heat losses arise due to optical conversion inefficiencies during the lasing process, bulk thermal resistances, and contact resistances. The rise in temperature with heat generation increases the threshold current and reduces the slope efficiency. Higher threshold currents, in turn, cause greater heat generation [46]. Increasing temperature also causes the peak wavelength of the emitted laser to rise (redshift), a phenomenon called spectral shift. Wavelengths can increase at a rate of 0.27 nm/°C for an 808 nm semiconductor laser array [65]. Such wavelength shifts are particularly detrimental when semiconductor laser diode arrays are used as pump sources for solid-state laser devices. Increasing temperature can also result in spectral broadening. This happens when nonuniform temperatures across the laser diode, usually higher at the active regions in the center of the diode bar while lower at the edges, lead to lower red-shifting of the laser beams generated by the emitters located at the edges. Carter et al. [66] developed a thermal model to analyze the transient thermal response of a quasi-continuous wave GaAs laser diode. It was shown that temperature oscillations were highest at the active regions (with responses in microseconds) of the diode emitters while they got dampened toward the substrate (with responses in milliseconds) due to thermal inertia effects. Also, high laser diode bar temperatures generate thermal stresses that cause mechanical expansion and variations in the refractive indices of the optical elements. These effects impair the laser beam quality [46].

Laser diode bars involve highly transient operating conditions as laser beams are usually emitted in short-duration pulses. Such quasi-continuous waves consist of high-intensity pulses with periods of 10 μs–1 ms. Consequently, temperature variations of about 1–10 °C can occur in microseconds [67].

To improve the performance and reliability of laser diode arrays, thermal system designs should strive to limit temperature oscillations in the active areas. While a number of thermal management techniques such as flow boiling [64], microchannel liquid cooling [68–70], thermoelectric cooling [71,72], spray cooling [73,74], and jet impingement [75] have been demonstrated to dissipate the high heat fluxes generated by laser diode arrays, they are based on the steady-state cooling objective of minimizing thermal resistance. To address the transient thermal challenges of laser diode arrays, cooling systems should tune both the thermal resistance and thermal capacitance to reduce temperature fluctuations and provide a response in a timescale of milliseconds.

In Secs. 3.1–3.5, research developments associated with different transient thermal management techniques are reviewed.

Although two-phase forced convection in microchannel heat sinks has demonstrated high heat power dissipation rates with nearly uniform spatial device temperatures, they perform well mostly during a steady-state operation involving stable boiling conditions. Operating such devices under certain heat flux and mass flux conditions trigger boiling instabilities that result in early dry-out followed by CHF. The pressure drop-mass flux curve under a fixed heat flux, two-phase operation in the region where the curve has a negative slope results in excursive or Ledinegg instabilities. If the pressure drop supply curve (of the pump) has a less negative slope than the pressure drop demand curve, the two-phase operation becomes unstable and can shift to a superheated condition. Excursive instabilities can also lead to pressure drop oscillations. These flow fluctuations arise due to compressible volumes located upstream of the microchannels [76–78]. Two-phase flow stability also depends on the prevalent two-phase flow regime in the microchannels. Boiling instabilities are more pronounced under a slug flow regime where elongated vapor slugs expand rapidly in the microchannels and cause vapor backflow. With progression to an annular flow regime at higher heat fluxes, boiling stability improves. A marked rise in instabilities is seen near CHF conditions. The use of two-phase microchannel heat sinks for transient electronics cooling is less reported in the literature.

Zhang et al. [77] demonstrated that two-phase microchannel cooling could be used for transient heat load applications when flow boiling instabilities are actively controlled. Two active flow instability control techniques were considered to enable stable boiling operation under varying heat inputs. For the first type of feedback control, the inlet valve opening was adjusted, while for the second type, the supply pump was controlled. A considerable reduction in flow oscillations was attained upon initiating active control of either the inlet valve opening or supply pump. When the inlet valve opening was adjusted using active feedback control, the system pressure increased by 1.01 kPa in about 1.7 s when the heat input was raised from 1 kW to 1.5 kW. The thermal capacity of the channel walls was found to have a significant effect on the transient two-phase flow characteristics. While the frequency of pressure-drop flow oscillations was passively lowered with increasing wall thermal capacity, its amplitude initially increased and then decreased with wall thermal capacity. When subjected to stepwise heat inputs, minimal wall temperature and flow oscillations were obtained at high wall thermal capacity conditions. In this work, it was also shown that the working fluid selected for two-phase microchannel cooling should have a smaller liquid-to-vapor density ratio so that vapor bubbles depart with smaller sizes (relative to the channel size). This can assist in reducing boiling instabilities. Despite its higher latent heat and thermal conductivity, pure water can result in more considerable two-phase flow instabilities as the vapor bubble sizes are much greater, whereas organic coolants such as HFE-7100 and R-134a, demonstrated to have smaller liquid-to-vapor densities than water (at a given temperature), can mitigate instabilities.

Another class of working fluids viable for two-phase microchannel cooling is self-rewetting alcohol/water binary mixtures that offer smaller bubble departure diameters and improved rewetting on account of solutal and thermal Marangoni effects. Such coolants have been demonstrated to reduce boiling instabilities when compared to pure water. The water vapor bubbles generated in microchannels tend to have relatively larger bubble diameters, and they expand rapidly in the confined microchannel passages. In contrast, bubble sizes are smaller for alcohol/water binary mixtures and do not coalesce so readily [79,80].

Active control of two-phase microchannel flow instabilities was also implemented by Bhide et al. [81], who introduced flow pulsations in the two-phase loop using a solenoid valve. When compared to flow boiling of water in a trapezoidal microchannel without active control, pressure-drop fluctuations were seen to reduce considerably when the pulsed flow was introduced in the loop. Two-phase flow stability improved as the frequency of the flow pulsation increased.

While two-phase microchannel cooling can be used under transient heat input conditions when flow boiling instabilities are actively controlled, the effectiveness of this technique under more rapidly varying workloads, for instance, during the pulsed power inputs associated with IGBTs and laser diode arrays, as well as during turbo-operation in microprocessors, needs to be explored.

Phase change materials (PCMs) are traditionally used for applications requiring thermal energy storage (such as for solar thermal energy storage, offsetting peak cooling/heating loads for HVAC systems, and spacecraft thermal systems [82,83]). Based on their material composition, PCMs are generally classified as inorganic, organic, or metallic. As PCMs can suppress temperature rises during power surges, they have received much interest in the transient thermal management of electronic devices, in particular mobile devices.

Vesligaj and Amon [84] assessed the thermal response of a technical information assistant (TIA) wearable computer involving a PCM-based thermal control unit (TCU). When subjected to a power duty cycle, the TIA with TCU showed reductions in the maximum surface temperature and temperature swing compared to the TIA without TCU.

Hodes et al. [85] investigated the use of two types of PCMs, namely Thermasorb-122 and n-alkane tricosane, for the thermal management of a mobile handset. The device operating time could be extended considerably, by up to five times and three times using tricosane and Thermsorb-122, respectively, when compared to the configuration without PCM.

A computational study was undertaken by Krishnan and Garimella [86] to evaluate the transient thermal performance of a PCM-based energy storage system (Eicosane) subjected to pulsed heat inputs. The transient thermal response for 12 cases involving different pulsed power cycles, container aspect ratios, and heater locations was simulated. It was identified that the container with aspect ratio 1 subjected to bottom heating resulted in the highest thermal energy storage and the lowest maximum temperatures, whereas the configurations with top heating offered the poorest thermal performance.

To enhance the thermal performance of PCM-based transient cooling, Krishnan et al. [87] developed a novel hybrid heat sink in which PCMs were integrated with a plate-fin heat sink. The tips of the thin aluminum fins of the hybrid heat sink were immersed in a PCM. Compared to the heat sink operated without PCM, the hybrid heat sink enabled continuous heat dissipation with temporal fin temperatures maintained in a narrower temperature range under high and low air-cooling conditions.

While pure PCMs can suppress device temperature rises during power surges, their poor thermal conductivities lead to a nonuniform melting process. This, in turn, results in uneven temperature distribution. To address this drawback, some investigators studied the impact of integrating herringbone type graphite nanofibers [88] and extended surfaces [43,89] with low-conductivity PCMs for transient thermal performance. They found that the PCM heat sinks prolonged the duration to steady-state conditions and lowered the peak temperature. Kandasamy et al. [90] examined the effectiveness of employing heat sinks embedded with PCMs to cool mobile electronic devices. Jaworski [91] integrated a lauric acid PCM with a pin fin heat spreader to promote heat penetration into the PCM. It was computationally demonstrated that the PCM-based pin-fin heat sinks lowered the temperature excursion rate compared to the non-PCM pin-fin heat sink.

In their experimental work, Yoo and Joshi [92] examined the use of metallic PCM-based finned heat sinks for the thermal management of transient workloads. Aluminum plate and pin-fin heat sinks, with Inalloy 158 PCM (50 Bi/27Pb/13Sn/10Cd) embedded internally in the fin structures, were considered. The plate-fin heat sink with PCM was able to lower the steady-state temperature compared to its non-PCM counterpart. In the case of fan integrated heat sinks, the PCM-based plate and pin fin heat sinks required shorter fan operating times compared to their non-PCM counterparts.

Shao et al. [45] examined the transient thermal response of a silicon thermal test chip (TTC) with alloy PCM integrated into a Samsung Galaxy S3 smartphone. The temperature response of the TTC-based smartphone was compared with the original smartphone under single and periodic short-duration power bursts. The ability of the TTC-based smartphone to lower the peak temperature and temperature swing by 16 °C and 21.3 °C, respectively, demonstrated the effectiveness of using a metallic alloy PCM for such power cycles.

The effectiveness of using low melting point gallium PCM for managing thermal shocks of the order of 100 W/cm² was explored by Yang et al. [93]. For a fixed fin height and fin width fraction, an increase in the fin number resulted in a greater extent of phase change and temperature uniformity in the PCM.

From the above studies, it is seen that organic and inorganic PCMs involve slower thermal responses that lead to longer durations for heating/cooling during melting/resolidification. As will be shown later in Sec. 4.1, the organic PCMs reviewed in this section are found to have response times >10 s/° C under heat fluxes in the range of 0.1–10 W/cm². This is due to their lower thermal conductivities and nonuniform melting characteristics. The superior thermophysical characteristics of metallic PCMs make them suitable candidates for transient thermal management applications. This is discussed in detail by Shamberger and Bruno [94]. Figure 6 illustrates the higher phase change heat transfer rates into metallic PCMs compared to nonmetallic PCMs (salt hydrates and paraffin). The figure of merit (FOM_q), as defined by Shamberger [95] in a previous study, was found to be primarily dependent on the thermal conductivity (k_l) and volumetric latent heat of fusion (L_v) of the PCM such that FOM_q ∼ $klLv$⁠. Accordingly, the significantly higher FOM_q associated with metallic PCMs signifies quicker heat removal from the heat source compared to nonmetallic PCMs. As will be discussed in Sec. 4.1, some of the metallic PCMs reviewed in this section are found to offer response times < 0.1 s/°C even at high heat fluxes (10–100 W/cm²). This demonstrates their potential to serve as compact thermal buffer devices that can mitigate short-duration power bursts passively. Another important aspect of PCM-based transient thermal management systems is the melting temperature associated with the PCM. Considering the case of maintaining safe device operating temperatures of about 80 °C, it is desirable to have PCMs with melting temperatures in the range of 60–70 °C. Metallic PCMs, such as those studied by Shao et al. [45] and Yoo and Joshi [92], appear to be suitable for transient electronics cooling applications as they involve melting temperatures of 58 °C and 70 °C, respectively.

Nevertheless, the larger weights of metallic PCMs compared to nonmetallic PCMs put them at a disadvantage despite their superior thermal performance. Boteler et al. [96] addressed this tradeoff by developing a module in which a high thermal conductivity metallic PCM, Gallium, is combined with a lightweight organic PCM, PT-29, such that their time scales of thermal response are matched. In such a hybrid module, the heat dissipated by the chip is initially removed rapidly by adjacent Ga (higher k) as it undergoes a phase change (primary melting) following which the unabsorbed heat is transferred to Pt-29 (low k) that undergo a slower phase change process (secondary melting). Among the hybrid Cu/Ga/Pt-29 configurations, case 11 (Ga/Pt-29) offered lower temperatures than pure Cu (for about 89 s) while it weighed 50% and 25% lesser than pure Cu and Ga, respectively.

Heat pipes/vapor chambers are passive thermal transport devices that effectively transfer heat over short distances from a heat source to a heat sink using phase change heat transfer. As heat transfer in heat pipes/vapor chambers occurs via phase change, nearly isothermal conditions are maintained across these devices [97,98]. Heat pipes/vapor chambers are used for thermal transport applications in spacecraft, waste heat recovery, HVAC, solar thermal energy storage, and electronics cooling [99].

More lately, heat pipes/vapor chambers are increasingly being used to cool portable electronic devices such as laptops, tablets, and smartphones [100]. This is being made possible due to the development of high-performance ultrathin heat pipe devices integrated into such small form-factor devices. Ultrathin flat heat pipes and loop heat pipes with thicknesses in the ranges of 0.35–0.6 mm and 0.6 mm–1.2 mm, respectively, were demonstrated for practical applications [100–102]. As portable devices involve user-interactive applications that lead to intermittent short computational bursts followed by long idle times, transient thermal analyses of heat pipes/vapor chambers under varying workloads become important.

Patankar et al. [44] studied the transient thermal performance of vapor chambers compared to metal spreaders in terms of the governing mechanisms involved. A performance metric was defined to ascertain the transient thermal performance of the vapor chamber under different vapor core thicknesses and heating power inputs. From this analysis, it was found that vapor chambers with vapor core thicknesses in the range of 50–100 μm outperformed the copper spreader for time scales greater than 5 s, while those with vapor core thicknesses in the range of 200–300 μm offered better thermal performance when operated at time scales either below 2–4 s or above 30–50 s.

In their subsequent work, Patankar et al. [103] developed guidelines on the design of vapor chambers for transient operation, particularly the thicknesses of the vapor core, wall, and wick, as well as the choice of working fluid. Vapor chambers used for transient operation should possess high thermal capacity and high in-plane vapor core thermal conductivity. While thicker vapor cores were found to perform better under steady-state conditions, an optimum vapor core thickness was found to exist for transient operating conditions. The numerical study also determined working fluid choice that resulted in a superior vapor chamber thermal performance under transient conditions. A high vapor core thickness (small wall thickness and minimum wick thickness based on capillary limit) and vapor phase figure of merit were earlier shown to improve the vapor chamber steady-state performance due to the greater in-plane conductance of the vapor core. However, both high in-plane vapor core conductance and total vapor chamber thermal capacity were required to enhance transient thermal performance for transient conditions. This included the volumetric thermal capacity of the working fluid's liquid phase in addition to the vapor core thickness and vapor phase figure of merit. The vapor chamber's transient thermal performance with different vapor core thicknesses was compared between two working fluids, namely water, and methanol. Compared to water, methanol resulted in a smaller peak to mean temperature at the optimum vapor core thickness on account of the larger thermal capacities and the higher vapor phase figure of merit.

The transient response of a flat copper heat pipe subjected to heating power inputs exceeding the capillary limit was studied by Baraya et al. [104]. The temperature response of the 150 mm (L) × 9 mm (W) × 0.62 mm (H) heat pipe was then examined when a heating power of 10 W, higher than the capillary limit (of 5.1 W), was supplied for 10 s from an initial steady-state heating power of 3 W. Sharp increases in the evaporator and condenser temperatures were seen when the heating power was stepped up. An inflection point and plateau were observed in the transient temperature profiles of the evaporator and condenser, respectively, 5.25 s after the 10 W pulse was supplied. These points were identified to indicate the occurrence of dry-out in the heat pipe. When the heating power input was reduced to the initial steady-state value of 3 W, the evaporator temperature was 12 °C higher than the temperature before the supply of the 10 W input. This temperature hysteresis was an indication that dry-out had taken place. For the pulsed heat input, which is below the time to dry-out duration, no inflection point was seen in the evaporator temperature profile. There was no temperature hysteresis after the heating power was lowered to the initial steady-state level. This demonstrated that the heat pipe could be operated above the capillary limit for short durations without reaching dry-out.

To address the transient thermal management challenges associated with starter-alternators used in automobiles, Harmand et al. [105] proposed flat heat pipes. In their numerical study, the transient thermal performance of the flat heat pipe device was compared with a solid copper plate and hollow copper plate. Under a periodic power supply, the flat plate heat pipe offered a lower peak temperature and evaporator temperature rise compared to the hollow copper plate, and smaller evaporator-condenser temperature differences compared to the solid copper plate. Moreover, being lighter, the flat heat pipe was identified to be advantageous for transient electronics cooling applications.

In subsequent work, the transient thermal response of vapor chambers involving nanofluid-based working fluids was studied by Hassan and Harmand [106]. They observed computationally that the Cu-based nanofluid offered the lowest peak temperatures compared to the CuO- and Al₂O₃-based nanofluids. The temperature responses of the nanofluid-based vapor chambers were compared under different volumetric fractions, nanoparticle diameters, wick porosities, and thickness, with a copper block. Lower peak temperatures were obtained using the nanofluids. This observation became pronounced with decreasing Cu nanoparticle diameter and increasing volumetric fraction. Compared to the vapor chamber with pure water, the use of Cu nanofluids was seen to enhance evaporation and condensation mass flow rates. These mass flow rates were found to increase with smaller nanoparticle diameters, wick thickness and porosity. The improved evaporation rates with the nanofluids were attributed to their higher thermal conductivities. A larger liquid velocity in the wick was obtained with reducing Cu nanoparticle diameter, wick thickness, and porosity. The liquid velocities for the nanofluid with the smallest Cu nanoparticles (10 nm diameter) were greater than those for pure water.

The startup and shut-down thermal characteristics of a flat copper heat pipe were studied by Wang and Vafai [107] through experiments and numerical analysis. Higher convective heat transfer coefficients at the condenser outer surface led to a reduction in the maximum wall temperatures and the duration to reach a steady-state. While the maximum temperature rise increased with an increase in heat flux or a decrease in the convective heat transfer coefficient, heat flux had a more pronounced effect on the temperature rise.

Variable conductance heat pipes (VCHPs) offer the capability of maintaining device temperatures within a range by altering the thermal conductance in response to changes in heating power input and ambient conditions. Their use for the precision temperature control of photonics systems was demonstrated by Cleary et al. [108]. As presented in Fig. 7(a), the VCHP is similar to a conventional constant conductance heat pipe except that it contains a noncondensable gas reservoir after the condenser section. Based on the vapor saturation pressure inside the vapor core, the noncondensable gas expands into or withdraws from the condenser section, thereby blocking or exposing the condenser area, respectively. By changing the condenser area available for heat transfer, the thermal conductance of the heat pipe is altered. For the passive operation of the VCHP only configuration, evaporator temperatures were reasonably controlled for varying heating power inputs at a fixed ambient temperature of 65 °C. When the VCHP was operated in active mode (involving a heated reservoir), the evaporator temperature remained near 70 °C for much of the ambient temperature range.

On similar lines, Liu et al. [109] developed a silicon vapor chamber whose thermal resistance could be tuned passively to facilitate its use as a thermal switch for applications involving nonuniform and transient heat loads. The device, as presented in Fig. 7(b), is similar to a conventional vapor chamber with the addition of noncondensable gas (NCG) in the vapor core. As the NCG pressure was increased, the device thermal resistance became higher. This behavior became more pronounced with decreasing heat input. At higher heat input, the NCG thermal resistance weakened as the vapor mass fraction gradient became greater. For the NCG pressure of 12 kPa, the device thermal resistance was reduced by four times as the heat input was lowered from its minimum to maximum level. Such a vapor chamber could thereby be used as a protective device against surges in heating power. The device was also able to clamp the temperature difference between the evaporator and condenser such that it remained unchanged with heat input.

In order to control the amount of heat transfer and its direction, Zhou et al. [110] employed a vapor chamber device as a thermal diode and a thermal switch. A 2 cm (L) × 2 cm (W) × 6.3 cm (H) copper vapor chamber with a super-hydrophilic evaporator and super-hydrophobic condenser was used for this purpose. The working fluid used was water. When the heat was supplied to the super-hydrophilic section, phase change occurred, and the condenser liquid droplets returned by bouncing off the superhydrophobic condenser surface. However, when the heat was supplied to the superhydrophobic surface, the condensed liquid could not return from the super hydrophilic surface. As a result, thermal resistance was higher as the heat was transported through the vapor core without sustained phase change. In effect, the vapor chamber functioned as a thermal diode. The vapor chamber was also configured to function as a thermal switch by means of adjusting the saturation temperature to the desired level. When the evaporator temperature reached the set saturation temperature, phase change occurred, and the vapor chamber was in an “on” state. Conversely, when the evaporator temperature was lower than the set saturation temperature, the vapor chamber was in an “off” state. The desired vapor saturation temperature was attained by controlling the amount of residual noncondensable gas in the vapor core and the amount of charged working fluid.

A thermal switch-based active thermal management system for power electronic devices was developed by Yang et al. [111]. The thermal switch, involving a 1 mm x 5 mm x 30 mm channel containing a liquid metal droplet (Galinstan), was integrated with single and dual GaN devices attached to a PCB. Heat was dissipated from the system in two paths, namely via the front end of the GaN devices subjected to forced convection liquid cooling and via the rear end of the PCB cooled by natural convection. By adjusting the position of the liquid metal droplet, the system could operate in thermally “on” or “off” states. When the droplet made contact with metal conductors connected to the single GaN device, the thermal switch was “on” and about 70% or more of the heat input was dissipated through the frond-end. Conversely, when the droplet was made to break contact with the metal conductors, the switch was “off” and heat dissipation via the front end was reduced. For the dual GaN device system, the thermal switch was able to achieve uniform junction temperatures for both GaN devices by adjusting the droplet's position relative to them. With a switching speed of 1 Hz, the thermal switch demonstrated the potential to offer quick thermal responses under transient loads.

Researchers have considered complementing the high thermal energy storage capacity of PCMs with the high thermal conductivity of a heat pipe/vapor chamber by combining the two to form composite heat pipes. While most configurations reported in the literature involve PCM integrated with the heat pipe externally, a few involve PCM embedded within the vapor chamber. Both configurations are examined in Secs. 3.4.1 and 3.4.2.

Li et al. [112] studied the prospect of combining a liquid metal PCM (LMPCM), gallium with a flat heat pipe (FHP) to improve the heat storage effectiveness of the PCM. It was demonstrated that the LMPCM-FHP modules offered smaller temperature changes with time as well as the lowest peak temperatures at steady-state. High heat spreading facilitated by the FHP enabled heat to diffuse more effectively into the entire PCM. Under normal and heavy usage operations, the FHP-LMPCM configuration was able to maintain stable temperatures in the ranges of 30–59 °C and 22.5–31 °C, respectively.

The combination of a flat heat pipe and PCM was explored by Weng et al. [113] and Zhuang et al. [114]. Under different periodic power inputs, it was seen that the composite heat pipe was able to lower the maximum to minimum temperature compared to the conventional heat pipe (without PCM). A similar study based on copper heat pipe combined with a PCM was assessed by Behi et al. [115] through experimental and computational studies.

Yang et al. [116] developed a novel cooling device that combined a low melting point metal PCM (LMPM), E-BiInSn, with high thermal conductivity flat heat pipes (FHP) containing finned plates at the condenser sections. Under periodic heating power inputs (1000 W for 10 min followed by 15 min of no power), temperatures for the LMPM—FHP device (with the air-cooling radiator) were maintained within 50 °C–85 °C as opposed to temperature swings of 20–160 °C associated with the non-PCM FHP device.

In their experimental work, Robak et al. [117] employed heat pipes to enhance the melting and solidification rates of a paraffin PCM (n-octadecane). Compared to the PCM-only module, the heat pipe- and fin-assisted PCM modules improved charging and discharging thermal performance.

While PCMs embedded with metallic fins or carbon foams offer an enhanced thermal conductivity, the total mass of the PCM-based heat sink increases as the metallic fins or carbon foams account for a considerable part of the heat sink mass. To improve the phase change performance of PCMs without increasing the total mass of the PCM-based heat sink, Yun et al. [118] integrated a paraffin PCM within a vapor chamber (as depicted in Fig. 8(a)). Microencapsulated PCM beads, laid inside the vapor chamber, assisted in storing thermal energy and wick the condensed liquid to the evaporator section. At temperatures exceeding the melting point of the PCM, heat supplied to the vapor chamber was stored in the PCM. When the temperature dropped below its melting point, the PCM would release its stored heat energy to the vapor chamber working fluid. The PCM vapor chamber was able to serve both as a heat spreader as well as a heat storage device, depending on the operating conditions. Compared to a PCM heat sink embedded with aluminum fins, the PCM-vapor chamber configuration weighed 52.9% lower for the same quantity of PCM.

Similarly, Lee et al. [119] embedded PureTemp25^® PCM into an aluminum vapor chamber, in ten aluminum drawers stacked horizontally with a spacing of 0.32 cm between them. As shown in Fig. 8(b), the space between the drawers served as vapor conduits. Screen mesh that lined the vapor chamber and drawer walls facilitated the wicking of condensed liquid to the evaporator section. The device enabled operation under heat spreading or heat storage modes.

A novel integrated vapor chamber—thermal energy storage device (VCTES) was developed by Kota [120] for the thermal management of pulsed heat loads. The VCTES was purposed to serve as the heat sink of a spray cooling module. When the electronic device generated power pulses, vapor generated during two-phase cooling in the spray cooling module was directed to the VCTES, where it condensed upon transferring its latent heat of vaporization to a PCM in the condenser section. The heat stored by the PCM was then dissipated to the ambient during the time between the power pulses. While the liquid–vapor phase change process in the spray cooling module facilitated rapid heat transfer from the heat source, the solid–liquid phase change process during vapor condensation enabled slower heat dissipation to the ambient over the long idle state period. Experiments were performed on a VCTES involving four vertically oriented TES columns that were partially within the vapor chamber and partially extended into the ambient. A graphite foam (Pocofoam^®) embedded with paraffin wax (Polywax^®) constituted each TES column. The temperature response of the system was examined under heat fluxes of 40 W/cm² generated in 16 s and 32 s pulses. Compared to a VCTES with no PCM, the VCTES with PCM was able to lower vapor temperature rise.

Engerer [67] demonstrated that temperature overshoots were best suppressed during rapid cooling with high initial rates. Compared to conventional cooling techniques such as pool boiling, single-phase and two-phase microchannel cooling, jet impingement, and spray cooling, flash boiling was identified to be a promising candidate that can offer rapid cooling rates in 100 ms–10 s.

Flash boiling occurs when a fluid is suddenly depressurized below its saturation pressure. During this process, the fluid gets superheated and enters a metastable state. Once nucleation is initiated (either by heterogeneous or homogeneous nucleation), the liquid begins to undergo rapid phase-change [121–126]. The phenomenon of flash boiling has been studied for understanding boiling liquid exploding vapor expansion (BLEVE) accidents associated with pressurized vessels that rupture [123,124], for improving fuel atomization to enhance combustion efficiencies in engines [127], and more recently, for rapid cooling of electronic devices involving power pulsations [46,128–130].

Engerer et al. [129] demonstrated the potential of using this phenomenon for the transient cooling of electronic devices involving high power pulsed loads. In their experimental work, it was shown that the transient flash cooling process takes place rapidly within a time span of 1 s. The cooling device was able to maintain temperatures within ± 5 °C, even at a high heat flux of 104 W/cm². In earlier work, Engerer et al. [128] studied flash boiling of methanol in chambers with and without porous foam structures. These high thermal conductivity structures were provided to increase the convective heat transfer area and bubble nucleation sites. Two types of porous foams were considered—a graphitic foam and a carbon-boron-nitrogen foam (CBN). A maximum temperature drop of 21 °C was attained in 65 ms using the CBN foam.

Flash boiling was also demonstrated by Shah et al. [130] to be a promising cooling solution for addressing the high power densities associated with silicon interconnect fabric (Si-IF) based devices. In their experimental study, flash cooling of a Si-IF-based device involving a single copper terminal block was examined using methanol as the working fluid. The transient thermal response of the copper tube and silicon wafer were examined under four flash cooling pulse timings. As the flash cooling pulses were applied more frequently, the average silicon wafer temperature as well as the temperature variations of the silicon wafer and copper tube became lower.

In the flash boiling studies reported previously, methanol was used as the working fluid because of its large latent heat of vaporization and boiling point close to standard temperature and pressure conditions [67].

In this section, the different transient thermal management techniques examined in the Sec. 3 are compared in terms of their time scales of thermal response. This is followed by a recommendation of the package thermal resistance (R) and capacitance (C) suitable for cooling transiently operated microprocessor chips, IGBTs, and high-power laser diode arrays.

The suitability of a given transient thermal management technique for a given application should be determined by how quickly or slowly it can dissipate a given pulsed load. For high-frequency power pulsations, the cooling package should be able to quickly remove the heat from the heat source during the heat loading interval, store it, and thereafter release the stored thermal energy to the ambient during the heat unloading interval, such that the system returns to its original low-temperature state in time to receive the subsequent heat pulse. If a slow thermal response cooling system is used for such an application, insufficient time available to discharge the stored heat to the ambient will result in a net accumulation of thermal energy within the system. Nevertheless, packages with slower thermal response times would be suitable for applications involving moderate or low-frequency power pulsations. Table 2 compares the heat flux, temperature increment and decrement ranges, and the corresponding time intervals for different transient thermal management approaches under thermal charging (heating power “on”) and discharging (heating power “off”). Thermal response time, τ_r, defined as the duration per unit change in temperature (t_ΔTpeak/ΔT_peak), is thereafter estimated to signify the nature of the thermal response that the cooling system exhibits when subjected to heat loading/unloading. Here, t_ΔTpeak pertains to the transient temperature response of the reported thermal management system. It is the duration over which the peak junction temperature increases from minimum to maximum when the heat input is turned “on.” Conversely, it is the duration over which the peak junction temperature drops from maximum to minimum when the heat input is turned “off.” The time durations t_on and t_off provided in Table 2 are the durations over which the heating power is “on”/“off,” respectively. ΔT_peak is the maximum-to-minimum temperature difference during the heating power “on” and “off” phases.

The thermal response times of various transient thermal management techniques under thermal charging and discharging are presented in Figs. 9(a) and 9(b), respectively, with respect to the base heat flux. It can be seen that heat pipes/vapor chambers, metallic and hybrid PCMs, flash boiling, and actively controlled two-phase microchannel cooling offer faster responses to heat input. Metallic PCMs and vapor chambers show response times within 0.1 s/°C while flash boiling, actively controlled two-phase microchannel cooling, and hybrid PCMs demonstrate response times between 0.1 and 2.5 s/°C. From the standpoint of actual applications, such as microprocessors, IGBTs, and laser diode arrays, which involve high heat fluxes >100 W/cm²_, metallic PCM and flash boiling appear to be promising cooling solutions with fast response. The majority of the organic PCMs and combined PCM (organic/metallic)—heat pipes/vapor chambers exhibit slower thermal response times (>10 s/°C) with lower heat dissipation rates in the range of 0.1–75 W/cm². Table 3 classifies the different transient cooling techniques into groups based on their thermal response time and maximum heat flux ranges.

It follows from the Sec. 4.1 that the thermal response time of a transient cooling package is governed by the thermal resistance (R) and thermal capacitance (C) of the system. In particular, the thermal time constant of the system (τ = RC) relative to the time period of the power pulsations, dictates the amplitude and time period of the junction temperature under a pulsed load. The goal of a transient thermal management package should be to reduce junction temperature swings as well as maintain the temperature amplitude within maximum limits. This objective is of paramount importance from the standpoint of thermal reliability. As discussed in Sec. 2.2, the system must be designed to have a thermal time constant (τ) that is sufficiently higher than the time period of the power pulsations.

Figure 11(b) shows the applied transient heat pulse, while Fig. 11(c) presents the corresponding temperature responses for the three Cases. In case 1, the high R and low C result in a rapid temperature excursion (large ΔT_j) during the heating cycle (heater “on”) and in an incomplete dissipation of the stored thermal energy during the cooling cycle (heater “off”). As a result of the thermal energy accumulated in each heating cycle, the peak junction temperature can be seen to rise with each subsequent cycle whereas for case 2 involving low R and high C, the temperature rise during each heating cycle is suppressed. Moreover, there is a lesser accumulation of thermal energy during the cooling cycle as the stored heat is dissipated more readily. Case 3, which involves a low R and low C, exhibits a similar thermal response as case 1. The peak junction temperature increases rapidly under a pulsed power input on account of its small thermal capacity.

For applications in which only the junction temperature is limited, the package should be designed with a low thermal resistance (R) and high thermal capacitance (C). This will facilitate the reduction of the maximum temperature amplitude as well as suppress temperature increment rates. Such a thermal design will suit applications such as data server microprocessors, IGBTs, and high-power laser diode arrays. However, for hand-held computing devices such as smartphones and tablets, both junction and skin temperatures, T_j and T_s, need to be limited. In fact, maintaining T_s,max within the level of human comfort (∼45 °C) is more challenging than the regulation of T_j,max in these applications. As such devices are increasingly being operated in transient mode on account of the heavy usage of responsive-based applications, a higher C will be able to lower the rise in T_j over the duration of a power burst. Moreover, a higher R will limit heat dissipation to the ambient such that T_s does not exceed T_s,max. To achieve a high system level R, it should be noted that the thermal resistance between the cooling element (which temporarily stores the thermal energy) and device skin needs to be made high to reduce heat dissipation to the ambient. The thermal resistance between the cooling element and skin can be increased by the use of air gaps or low-k insulation layers (such as aerogel) [131,132], whereas the local thermal resistance between the heat source and cooling element should be minimized so that the heat pulse can be rapidly transferred to the cooling element where it is stored temporarily before being dissipated to the ambient via the skin.

A desirable solution is to have a thermal system that enables tuning of its R and C based on the given pulsed load condition. The use of variable conductance heat pipes/vapor chambers [108,109], whose R and C can be varied by adjusting the pressure of the noncondensable gas, is an attractive approach for transient heat load applications. Similarly, the use of vapor chambers with internal thermal energy storage appears to be promising on account of their ability to operate in heat spreading and heat storage modes [119].

Thermal design guidelines on appropriate R and C combinations for different applications are depicted in Fig. 12. Thermal systems for microprocessor-based portable devices should be designed with sufficiently high R and high C values (indicated by I). In the case of IGBTs, high-power semiconductor laser diode arrays, and microprocessors for data centers, thermal system designs should employ low R and high C (as indicated by II) such that the thermal time constant, τ = RC, is higher than the period of the pulsed load. It should be noted that this is applicable for cooling solutions in which the working fluid/cooling medium remains within the control volume of interest. This may not be applicable for cooling systems such as liquid metal thermal switches in which the working fluid/cooling medium moves in and out of the control volume. For low-frequency power loads that are more or less steady-state in nature, thermal systems should have low R and C (indicated by III). Applications requiring thermal insulation can use systems with high R and low C (indicated by IV).

Traditional thermal management of microelectronic devices based on steady-state heat transfer often results in over-designed cooling systems that are sized based on peak loads. They can also result in undesirably large temperature swings when employed under pulsed heat load conditions. Large temperature variations with time cause thermo-mechanical fatigue effects that jeopardize device reliability. In this review, the need for a paradigm shift toward the development of cooling solutions based on transient thermal characteristics is outlined initially. Thereafter, the thermal management challenges and requirements associated with three types of microelectronic devices, namely microprocessors, IGBTs, and high-power semiconductor laser diode arrays, are identified in detail. The transient thermal management capabilities and developments associated with actively controlled two-phase microchannel cooling, PCMs, heat pipes/vapor chambers, combined PCM—heat pipe/vapor chamber configurations, and flash boiling are then assessed extensively. The effectiveness of these techniques for microprocessor chips, IGBTs, and high-power laser diodes is evaluated based on their thermal response times and the maximum heat flux dissipated. Metallic/hybrid PCMs and flash boiling are found to be suitable cooling solutions for applications involving short-duration high-power pulses. An important aspect of transient thermal management is the appropriate selection of the overall thermal resistance (R) and capacitance (C) of the cooling package. The thermal time constant must be designed to be larger than the time period of the power cycles in order to reduce temperature variations. Cooling systems offering tunable R and C show much promise for future transient thermal management applications as they can dynamically adapt to time-varying workloads.

The Institute Postdoctoral Fellowship funding provided by IIT Bombay is gratefully acknowledged by the authors.

• Institute Postdoctoral Fellowship IIT Bombay (Funder ID: 10.13039/501100005808).

C =

thermal capacitance (J/K)

CHF =

critical heat flux

CPU =

central processing Unit

h =

heat transfer coefficient (W/m² K)

IGBT =

insulated gate bipolar transistor

k =

thermal conductivity (W/mK)

L =

latent heat of fusion (J/Kg)

PCM =

phase change material

Q =

heating power (W)

q″ =

heat flux (W/cm²)

R =

thermal resistance (K/W)

sp =

single pulse

t =

time (s)

T =

temperature (°C)

TDP =

thermal design power

Greek Symbols

ρ =

density (kg/m³)

τ =

thermal time constant (s)

τ_r =

thermal response time (s/°C)

Subscripts

∞ / amb =

ambient

i =

initial

in =

input

j =

junction

m =

melting

max =

maximum

out =

output

s =

skin

r =

response

th =

thermal

Superscript

“ =

per unit area