TRIC: A Thermal Resistance and Impedance Calculator for Electronic Packages †

: This paper presents the Thermal Resistance and Impedance Calculator (TRIC) tool devised for the automatic extraction of thermal metrics of package families of electronic components in both static and transient conditions. TRIC relies on a solution algorithm based on a novel projection-based approach, which—unlike previous techniques—allows (i) dealing with parametric detailed thermal models (pDTMs) of package families that exhibit generic non-Manhattan variations of geometries and meshes, and (ii) transforming such pDTMs into compact thermal models that can be solved in short times. Thermal models of several package families are available, and dies with multiple active areas can be handled. It is shown that transient thermal responses of chosen packages can be obtained in a CPU (central processing unit) time much shorter than that required by a widely used software relying on the ﬁnite-volume method without sacriﬁcing accuracy.


Introduction
The thermal analysis of electronic devices, circuits, and systems has always been an activity of utmost relevance in the semiconductor industry and academia. A thermally aware design can be achieved with the aid of numerical simulations, which are very challenging in terms of CPU time and memory storage, if a high level of accuracy is desired. This has stimulated many research groups to develop tools relying on suitable algorithms to accelerate the solution process (e.g., [1][2][3]).
For the specific case of packaged components, the preferred approach is to build boundary condition independent (BCI) compact thermal models (CTMs) to alleviate the computational burden without perceptible accuracy loss [4][5][6][7]; an interesting review of BCI CTMs for electronic parts is offered in [8].
Standardized procedures have been introduced to allow a fair comparison among packaged components in terms of thermal performances. More specifically, Joint Electron Device Engineering Council (JEDEC) metrics [9] are evaluated and included in the product datasheets. In order to satisfy the request for JEDEC thermal metrics of chosen package families, the authors developed the Thermal Resistance Advanced Calculator (TRAC) tool, the features of which were initially sketched in [10] • providing an exhaustive picture of the TRIC features; • reporting and discussing a larger number of results, including those obtained for the newly included package families (eQFN-mr and PowerSSO); • adding a detailed comparison (only touched upon in [24]) in terms of accuracy and CPU time with the commercial finite-volume (FV) software FloTHERM [25]; showing a simulated temperature map at a chosen time instant for a multi-source case of practical relevance.
The remainder of the paper is articulated as follows. In Section 2, TRIC is described and the details concerning all the thermally-modeled packages are provided. Section 3 probes into the solution algorithm. The numerical results and the main findings, as well as the comparison with FloTHERM, are shown and discussed in Section 4. Conclusions are then given in Section 5.

TRIC Features
TRIC, like the former release TRAC, is suited to automatically extract the JEDEC metrics ϑ JA , Ψ JB , Ψ JCtop , ϑ JB , ϑ JCtop , and ϑ JCbottom in four ambients [9] for each specimen in a package family. The ambients mainly differ in terms of thermal path followed by the heat generated within the HS and emerging from the die; more specifically, the ambient to evaluate ϑ JCbottom requires a cold plate in intimate contact with the package backside; the plate is located over the top surface when aiming to compute ϑ JCtop ; in the ambient for determining ϑ JB , a cold ring surrounds the package; no cooling Energies 2020, 13, 2252 3 of 16 systems are adopted in the ambient common to ϑ JA , Ψ JB , and Ψ JCtop . To this end, a purely conductive pDTM is defined for each family of packages immersed in a specific ambient, within which the boundary conditions (BCs) are calibrated following the JEDEC environment specifications. For instance, for the evaluation of ϑ JCbottom , an extremely high heat transfer coefficient was applied to the bottom surface to describe the thermal path from the die to the heat sink. The geometry, assumed to be Manhattan (whereas its variation can also be non-Manhattan), is modeled by subdividing the domain into rectangular parallelepipeds (also referred to as cells or, more picturesquely, as bricks) with edges parallel to the x, y, and z axes, for which dimensions, material properties, and heat generation are provided. All data on geometry and properties are stored in a parameter vector p varying in a set P. The pDTM includes information to automatically generate a Cartesian mesh for all specimens belonging to a family of packages in the chosen ambient.
It is worth noting that the pDTMs are created by resorting to reasonable simplifications (coming from the vendor's experience) that allow a marked reduction in computational burden while negligibly affecting the simulation accuracy. More specifically, in all ambients, the board over which the package is mounted is modeled with a single finely-meshed parallelepiped with a thermal conductivity adjusted to account for the aggregate effect of metal traces and vias, the detailed representation of which would have led to a far too complex problem. In a similar fashion, the pins in the eQFN and eQFN-mr package families, as well as the leads in the pLQFP and PowerSSO families, were thermally represented by a rectangular parallelepiped, the thermal conductivity of which was determined through a weighted averaging procedure.
The thermal metrics can be automatically evaluated through a graphical user interface, in which a chosen specimen in a package family is determined by selecting the corresponding set of parameters. Package size and thickness, pad size, lead count, die size, and thickness are examples of the data that the user can input.
As mentioned in [11], a preliminary convergence analysis of the 3-D mesh discretization of the constructed pDTMs was performed for selected packages; in particular, the calculated thermal metrics were monitored by increasing the degrees of freedom (DoF) until a negligible mesh sensitivity was observed. Then the discretization leading to about 0.1% inaccuracy was chosen to avoid unnecessarily onerous too-fine meshes.
So far, the pDTMs of many package families are available. The list, along with the main geometrical features, is reported below.
• eQFPs, which are surface mount integrated circuit packages with a flat rectangular body, and leads extending from all the four sides. In particular, both the eLQFP and eTQFP variants are available, which differ in terms of body thickness (1.4 and 1 mm, respectively). The square epad structure, which is a standard lead frame wherein the die pad is depressed down to the package bottom face, represents a valuable solution to ease the heat dissipation from die to board. Bottom side views of the above-reported packages are shown in Figure 1, while examples of DTMs are depicted in Figure 2. Thermal models of further electronic components can be created with relatively little effort. Bottom side views of the above-reported packages are shown in Figure 1, while examples of DTMs are depicted in Figure 2. Thermal models of further electronic components can be created with relatively little effort.  Differently from TRAC, in TRIC the evaluation of the temperature field under transient conditions can be enabled for any profile of dissipated power at the HSs. In addition, TRIC allows coping with die layouts integrating circuitries with multiple separate active areas (i.e., HSs), for which the assumption of a uniform power distribution over the whole die surface would have been unacceptably inaccurate. As a result, TRIC can be used with many practical aims. As an example, by emulating the dissipation over the die with only one HS, the thermal impedance can be computed by determining the thermal response to a power step. Moreover, power profiles typically encountered  Bottom side views of the above-reported packages are shown in Figure 1, while examples of DTMs are depicted in Figure 2. Thermal models of further electronic components can be created with relatively little effort.  Differently from TRAC, in TRIC the evaluation of the temperature field under transient conditions can be enabled for any profile of dissipated power at the HSs. In addition, TRIC allows coping with die layouts integrating circuitries with multiple separate active areas (i.e., HSs), for which the assumption of a uniform power distribution over the whole die surface would have been unacceptably inaccurate. As a result, TRIC can be used with many practical aims. As an example, by emulating the dissipation over the die with only one HS, the thermal impedance can be computed by determining the thermal response to a power step. Moreover, power profiles typically encountered layouts integrating circuitries with multiple separate active areas (i.e., HSs), for which the assumption of a uniform power distribution over the whole die surface would have been unacceptably inaccurate. As a result, TRIC can be used with many practical aims. As an example, by emulating the dissipation over the die with only one HS, the thermal impedance can be computed by determining the thermal response to a power step. Moreover, power profiles typically encountered in real applications, like the antilock braking system (ABS) in vehicles, injection, etc., can be taken into account.

TRIC Solution Algorithm
The parametric MOR techniques implemented in TRAC are only suited to deal with Manhattan variations of geometry/mesh. A Manhattan transformation is a geometry transformation that converts the x, y, and z coordinates into coordinates X, Y, and Z given by with f x , f y , and f z continuous and monotonic functions. Conveniently, contrary to TRAC, TRIC exploits a two-step approach to cope with Manhattan pDTMs associated with non-Manhattan variations of geometry/mesh. This is, for example, the case (i) of pLQFPs, where the relative position of the leads changes by varying the die thickness ( in real applications, like the antilock braking system (ABS) in vehicles, injection, etc., can be taken into account.

TRIC Solution Algorithm
The parametric MOR techniques implemented in TRAC are only suited to deal with Manhattan variations of geometry/mesh. A Manhattan transformation is a geometry transformation that converts the x, y, and z coordinates into coordinates X, Y, and Z given by with fx, fy, and fz continuous and monotonic functions. Conveniently, contrary to TRAC, TRIC exploits a two-step approach to cope with Manhattan pDTMs associated with non-Manhattan variations of geometry/mesh. This is, for example, the case (i) of pLQFPs, where the relative position of the leads changes by varying the die thickness (   The functioning principle of TRIC can be described as follows. The DTM of a specimen in a family of packages immersed in a standard environment, automatically extracted by TRIC using the FV method, has the following form: in which ( ) In the first step, a basis for the projection of (2) is determined by Algorithm 1 exploiting Algorithm 2. In the second step, using this projection basis, a fast solution of (2) is provided for a chosen value of p by Algorithm 3, which again exploits Algorithm 2.  in real applications, like the antilock braking system (ABS) in vehicles, injection, etc., can be taken into account.

TRIC Solution Algorithm
The parametric MOR techniques implemented in TRAC are only suited to deal with Manhattan variations of geometry/mesh. A Manhattan transformation is a geometry transformation that converts the x, y, and z coordinates into coordinates X, Y, and Z given by with fx, fy, and fz continuous and monotonic functions. Conveniently, contrary to TRAC, TRIC exploits a two-step approach to cope with Manhattan pDTMs associated with non-Manhattan variations of geometry/mesh. This is, for example, the case (i) of pLQFPs, where the relative position of the leads changes by varying the die thickness (   The functioning principle of TRIC can be described as follows. The DTM of a specimen in a family of packages immersed in a standard environment, automatically extracted by TRIC using the FV method, has the following form: in which ( ) In the first step, a basis for the projection of (2) is determined by Algorithm 1 exploiting Algorithm 2. In the second step, using this projection basis, a fast solution of (2) is provided for a chosen value of p by Algorithm 3, which again exploits Algorithm 2.  The functioning principle of TRIC can be described as follows. The DTM of a specimen in a family of packages immersed in a standard environment, automatically extracted by TRIC using the FV method, has the following form: in which ϑ(t, p) is the N(p) rows vector with the DoF of temperature rise at each time instant t, M(p) is the N(p)-order mass matrix, K(p) is the N(p)-order stiffness matrix, G(p) is the N(p) × M power density rectangular matrix, and P(t) is the M rows vector of source powers.
In the first step, a basis for the projection of (2) is determined by Algorithm 1 exploiting Algorithm 2. In the second step, using this projection basis, a fast solution of (2) is provided for a chosen value of p by Algorithm 3, which again exploits Algorithm 2.  (2) for Θ m (σ, p) usingΘ m (σ, p) as initial guess 3 add (p, Θ m (σ, p)) to space S m (σ) for Ξ times do pick a random value p in P apply Algorithm 2 4 compute residual ρ of (2) forΘ m (σ, p) if ρ > ε do break In Algorithm 1, at line 1, a set of complex frequency values, proper for characterizing the thermal behavior of the family of packages, is chosen following [13]. At line 2, the detailed thermal problem in the complex frequency domain in which g m is the m-th column of G and is numerically solved by an iterative solver. A multigrid solver is used, and the number of iterations is reduced by assuming as initial guess theΘ m (σ, p) estimation. At line 3, the solutions Θ m (σ, p) are added to S(σ). At line 4, the residual ρ is determined substituting Θ m (σ, p) withΘ m (σ, p) in (3).  In Algorithm 3, for a chosen value of p, a space S(p) is determined. Projecting (2) onto such a space, the CTM ensues, the response of which to any power profile is computed for approximating ϑ(t, p). It must be remarked that in commercial numerical codes the complexity for the transient solutions is proportional to (n x × n y × n z ) α × nt , where α is in the range 1 ÷ 1.5 (depending on the iterative method adopted); n x , n y , and n z are the numbers of grid points along x, y, and z, and n t is the number of time instants in which the problem has to be solved. Conveniently, the corresponding complexity in TRIC is proportional to n t and independent of n x , n y , and n z .

Numerical Results
Except for the multi-source analysis (Section 4.4), the intrinsic symmetry of the packages under test allowed meshing and simulating only a quarter of each structure, thus mitigating the computational burden; the missing portions were virtually restored by applying adiabatic BCs (i.e., zero heat flux) over the planes of symmetry. Figure 5 shows the static thermal metrics ϑ JA and ϑ JCtop corresponding to eLQFPs and pLQFPs for two horizontal package sizes, namely (a) 10 × 10 and (b) 14 × 14 mm 2 , the epad size being 6 × 6 and 7.2 × 7.2 mm 2 for cases (a) and (b), respectively. The metrics are determined by TRIC for various sizes of the square die, i.e., 2 × 2, 3 × 3, 4 × 4, 5 × 5, 6 × 6 mm 2 , the latter only for case (b). It can be inferred that the presence of the pad, which eases the heat removal, yields a sizable beneficial impact on ϑ JA , whereas the closer proximity of the die to the top of the package in pLQFPs prevails over the cooling action of the epad in terms of ϑ JCtop . In Algorithm 3, for a chosen value of p, a space S(p) is determined. Projecting (2) onto such a space, the CTM ensues, the response of which to any power profile is computed for approximating ( ) It must be remarked that in commercial numerical codes the complexity for the transient solutions is proportional to (nx × ny × nz) α × nt , where α is in the range 1 ÷ 1.5 (depending on the iterative method adopted); nx, ny, and nz are the numbers of grid points along x, y, and z, and nt is the number of time instants in which the problem has to be solved. Conveniently, the corresponding complexity in TRIC is proportional to nt and independent of nx, ny, and nz.

Numerical Results
Except for the multi-source analysis (Section 4.4), the intrinsic symmetry of the packages under test allowed meshing and simulating only a quarter of each structure, thus mitigating the computational burden; the missing portions were virtually restored by applying adiabatic BCs (i.e., zero heat flux) over the planes of symmetry. Figure 5 shows the static thermal metrics ϑJA and ϑJCtop corresponding to eLQFPs and pLQFPs for two horizontal package sizes, namely (a) 10 × 10 and (b) 14 × 14 mm 2 , the epad size being 6 × 6 and 7.2 × 7.2 mm 2 for cases (a) and (b), respectively. The metrics are determined by TRIC for various sizes of the square die, i.e., 2 × 2, 3 × 3, 4 × 4, 5 × 5, 6 × 6 mm 2 , the latter only for case (b). It can be inferred that the presence of the pad, which eases the heat removal, yields a sizable beneficial impact on ϑJA, whereas the closer proximity of the die to the top of the package in pLQFPs prevails over the cooling action of the epad in terms of ϑJCtop.

Thermal Impedances
One of the main features of TRIC not available in TRAC is the possibility to determine the transient thermal responses of a specimen of a package family for any profile of the power sources.

Thermal Impedances
One of the main features of TRIC not available in TRAC is the possibility to determine the transient thermal responses of a specimen of a package family for any profile of the power sources. In particular, the thermal impedances-often used to characterize the dynamic thermal behavior of components-can be evaluated as the thermal responses (temperature rises normalized to power) to a power step applied at t = 0 to a die modeled with only one HS. Such a capability is witnessed through the analysis of a large set of cases. Figure 6 shows the junction-to-ambient thermal impedance Z THJA = ϑ JA (t) of a 10 × 10 mm 2 eTQFP equipped with a 6 × 6 mm 2 epad for four different die sizes. The simulations allow quantifying the favorable influence of a large die, which benefits from a lower power density. Static conditions are reached at about 400 s, regardless of die size. Figure 7 illustrates the Z THJA of a 10 × 10 mm 2 pLQFP with a 4 × 4 mm 2 die for three different pad sizes. All curves coincide for short times (<0.1 s), where the heat emerging from the die has not hit the pad yet. For this case, (i) a complex evolution with an inflection takes place due to the involved package geometry: the heat propagates through the pad, the mold, and then reaches the leads, which are in direct contact with the board, and (ii) the impedances flatten at 1200 s, as induced by the absence of the cooling epad action. Figure 8 reports the Z THJA of 9 × 9 mm 2 single-row eQFN packages for three combinations of epad and die sizes. Here the positive influence of a bigger epad (7 × 7 mm 2 instead of 5.7 × 5.7 mm 2 ) for the same die size is evident; again, the impedances overlap for short times. Similar to the study conducted for the eTQFP family, also for eQFN packages the thermal impedance is reduced and delayed for bigger dies. Figure 9 confirms the cooling impact for medium/long times of a larger package body and/or a larger epad for eQFN-mr packages with two rows of pins. Lastly, the dynamic thermal behavior of a 10.3 × 7.5 mm 2 PowerSSO-36 with a 4.09 × 3.17 mm 2 die is determined for four different epad sizes in Figure 10. In particular, the thermal impedances -often used to characterize the dynamic thermal behavior of components -can be evaluated as the thermal responses (temperature rises normalized to power) to a power step applied at t = 0 to a die modeled with only one HS. Such a capability is witnessed through the analysis of a large set of cases. Figure 6 shows the junction-to-ambient thermal impedance ZTHJA = ϑJA(t) of a 10 × 10 mm 2 eTQFP equipped with a 6 × 6 mm 2 epad for four different die sizes. The simulations allow quantifying the favorable influence of a large die, which benefits from a lower power density. Static conditions are reached at about 400 s, regardless of die size. Figure 7 illustrates the ZTHJA of a 10 × 10 mm 2 pLQFP with a 4 × 4 mm 2 die for three different pad sizes. All curves coincide for short times (<0.1 s), where the heat emerging from the die has not hit the pad yet. For this case, (i) a complex evolution with an inflection takes place due to the involved package geometry: the heat propagates through the pad, the mold, and then reaches the leads, which are in direct contact with the board, and (ii) the impedances flatten at 1200 s, as induced by the absence of the cooling epad action. Figure 8 reports the ZTHJA of 9 × 9 mm 2 single-row eQFN packages for three combinations of epad and die sizes. Here the positive influence of a bigger epad (7 × 7 mm 2 instead of 5.7 × 5.7 mm 2 ) for the same die size is evident; again, the impedances overlap for short times. Similar to the study conducted for the eTQFP family, also for eQFN packages the thermal impedance is reduced and delayed for bigger dies. Figure  9 confirms the cooling impact for medium/long times of a larger package body and/or a larger epad for eQFN-mr packages with two rows of pins. Lastly, the dynamic thermal behavior of a 10.3 × 7.5 mm 2 PowerSSO-36 with a 4.09 × 3.17 mm 2 die is determined for four different epad sizes in Figure 10.

Comparison with FloTHERM
First and foremost, it must be underlined that the user-friendly graphical interface of TRIC allows avoiding the long, painstaking, and prone-to-error geometry/mesh construction process often required by conventional numerical tools, thus markedly lowering the pre-processing effort and time. As far as the accuracy and efficiency of TRIC are concerned, they were estimated by comparison with the widely used commercial software FloTHERM. The tools share the same geometry simplifications and BCs. The favorable matching between the thermal metrics determined by TRAC and the FV software was already shown in [11] for eLQFPs, eTQFPs, and eQFN packages; the slight discrepancy (typically <2%, the maximum value being around 3%) was mainly attributed to the different mesh styles of the simulators. Evidence of the good agreement is also provided in Figure 11, which shows the (static) maps of temperature rise over ambient (Tamb = 20 °C) determined for three eTQFPs with dies dissipating 1 W in the ϑJA-related ambient; in particular, (a) corresponds to a 14 × 14 mm 2 package size with a 9 × 9 mm 2 epad and a 3 × 3 mm 2 die; (b) to a 10 × 10 mm 2 package with a 6 × 6 mm 2 epad and a 4 × 4 mm 2 die; (c) to a 10 × 10 mm 2 package with a 6 × 6 mm 2 epad and a 2 × 2 mm 2 die.

Comparison with FloTHERM
First and foremost, it must be underlined that the user-friendly graphical interface of TRIC allows avoiding the long, painstaking, and prone-to-error geometry/mesh construction process often required by conventional numerical tools, thus markedly lowering the pre-processing effort and time. As far as the accuracy and efficiency of TRIC are concerned, they were estimated by comparison with the widely used commercial software FloTHERM. The tools share the same geometry simplifications and BCs. The favorable matching between the thermal metrics determined by TRAC and the FV software was already shown in [11] for eLQFPs, eTQFPs, and eQFN packages; the slight discrepancy (typically <2%, the maximum value being around 3%) was mainly attributed to the different mesh styles of the simulators. Evidence of the good agreement is also provided in Figure 11, which shows the (static) maps of temperature rise over ambient (T amb = 20 • C) determined for three eTQFPs with dies dissipating 1 W in the ϑ JA -related ambient; in particular, (a) corresponds to a 14 × 14 mm 2 package size with a 9 × 9 mm 2 epad and a 3 × 3 mm 2 die; (b) to a 10 × 10 mm 2 package with a 6 × 6 mm 2 epad and a 4 × 4 mm 2 die; (c) to a 10 × 10 mm 2 package with a 6 × 6 mm 2 epad and a 2 × 2 mm 2 die.
The accuracy ensured by TRIC under transient conditions can be inferred from Figure 12, which shows the favorable matching with data computed by FloTHERM for three packages belonging to the eTQFP family. Again, the discrepancy is below 3% within the whole time range. It is worth noting that the CPU time required to obtain an impedance by TRIC for a typical number of 2 × 10 6 grid points is about 1 min on a workstation with an Intel Xeon E5-2630 v4 @ 2.2 GHz equipped with a 64 GB RAM, whereas more than 15-20 minutes are needed when using FloTHERM. This notable gain in terms of efficiency is expected to take place with respect to most popular simulators based on numerical methods. The accuracy ensured by TRIC under transient conditions can be inferred from Figure 12, which shows the favorable matching with data computed by FloTHERM for three packages belonging to the eTQFP family. Again, the discrepancy is below 3% within the whole time range. It is worth noting that the CPU time required to obtain an impedance by TRIC for a typical number of 2 × 10 6 grid points is about 1 min on a workstation with an Intel Xeon E5-2630 v4 @ 2.2 GHz equipped with a 64 GB RAM, whereas more than 15-20 minutes are needed when using FloTHERM. This notable gain in terms of efficiency is expected to take place with respect to most popular simulators based on numerical methods. Figure 11. Temperature rise maps for three eTQFPs dissipating 1 W, as calculated by TRIC (left) and FloTHERM (right): (a) 14 × 14 mm 2 package size with a 9 × 9 mm 2 epad and a 3 × 3 mm 2 die; (b) 10 × 10 mm 2 package with a 6 × 6 mm 2 epad and a 4 × 4 mm 2 die; (c) 10 × 10 mm 2 package with a 6 × 6 mm 2 epad and a 2 × 2 mm 2 die.

Multi-Source Analysis
Unlike TRAC, TRIC allows also simulating realistic packages that integrate multiple HSs, as evidenced through the following illustrative examples.
First, a 9 × 9 mm 2 eQFN package is considered, with a 5 × 5 mm 2 epad and a 4 × 4 mm 2 die, the latter including four 1 × 1 mm 2 active areas (i.e., HSs), each dissipating 0.5 W. As sketched in Figure  13, four positions of the HSs are chosen to describe practical layouts. The static temperature rise over

Multi-Source Analysis
Unlike TRAC, TRIC allows also simulating realistic packages that integrate multiple HSs, as evidenced through the following illustrative examples.
First, a 9 × 9 mm 2 eQFN package is considered, with a 5 × 5 mm 2 epad and a 4 × 4 mm 2 die, the latter including four 1 × 1 mm 2 active areas (i.e., HSs), each dissipating 0.5 W. As sketched in Figure 13, four positions of the HSs are chosen to describe practical layouts. The static temperature rise over ambient was monitored in five critical points, namely, at the die center (∆T center ) and at the centers of the HSs (∆T 1 , ∆T 2 , ∆T 3 , ∆T 4 ). Results corresponding to the four layouts are reported in Table 1, along with the maximum temperature rise over the whole die (∆T max ). Again, a fairly good agreement with the temperature maps determined by FloTHERM (not shown here) was obtained, the discrepancy between the maxima being <3%. The data plainly illustrate how the temperature field over the die modifies depending on the specific layout. The main findings are (i) all the HSs share the same temperature in cases #1 and #3 for symmetry reasons; (ii) as expected, in layout #1 ∆T center = ∆T max due to the concurrent influence of all the HSs, while (iii) ∆T max is reached near the die side in layout #2. This simple analysis shows that TRIC can be effectively exploited to identify the most thermally efficient layout. In addition, the information gained on the temperature field over the die is also important to properly place temperature sensors.

Multi-Source Analysis
Unlike TRAC, TRIC allows also simulating realistic packages that integrate multiple HSs, as evidenced through the following illustrative examples.
First, a 9 × 9 mm 2 eQFN package is considered, with a 5 × 5 mm 2 epad and a 4 × 4 mm 2 die, the latter including four 1 × 1 mm 2 active areas (i.e., HSs), each dissipating 0.5 W. As sketched in Figure  13, four positions of the HSs are chosen to describe practical layouts. The static temperature rise over ambient was monitored in five critical points, namely, at the die center (ΔTcenter) and at the centers of the HSs (ΔT1, ΔT2, ΔT3, ΔT4). Results corresponding to the four layouts are reported in Table 1, along with the maximum temperature rise over the whole die (ΔTmax). Again, a fairly good agreement with the temperature maps determined by FloTHERM (not shown here) was obtained, the discrepancy between the maxima being <3%. The data plainly illustrate how the temperature field over the die modifies depending on the specific layout. The main findings are (i) all the HSs share the same temperature in cases #1 and #3 for symmetry reasons; (ii) as expected, in layout #1 ΔTcenter = ΔTmax due to the concurrent influence of all the HSs, while (iii) ΔTmax is reached near the die side in layout #2. This simple analysis shows that TRIC can be effectively exploited to identify the most thermally efficient layout. In addition, the information gained on the temperature field over the die is also important to properly place temperature sensors. Figure 13. Representation of four layouts corresponding to a four-source die. The temperaturesensing points are located at the center of the die and at the centers of the HSs.

ABS Source Profile
The TRIC capability to cover complex die layouts and transient power profiles is demonstrated by simulating the die temperature dictated by a realistic ABS power profile. The examined package is a 14 × 14 mm 2 eTQFP with an 8 × 8 mm 2 epad and a 30 mm 2 die, the circuitry over which presents eight active areas (HSs), as depicted in Figure 14a. The geometry of the system and the external BCs were adapted to this specific application. Figure 14b illustrates the TRIC interface with the probes (placed at the centers of the HSs) where the temperatures are taken. Figure 14c shows the evolution of the temperature rises over ambient from 0 to 20 s. Figure 14d reports the temperature rise field at the most thermally critical time instant, i.e., 2.5 s. is a 14 × 14 mm 2 eTQFP with an 8 × 8 mm 2 epad and a 30 mm 2 die, the circuitry over which presents eight active areas (HSs), as depicted in Figure 14a. The geometry of the system and the external BCs were adapted to this specific application. Figure 14b illustrates the TRIC interface with the probes (placed at the centers of the HSs) where the temperatures are taken. Figure 14c shows the evolution of the temperature rises over ambient from 0 to 20 s. Figure 14d reports the temperature rise field at the most thermally critical time instant, i.e., 2.5 s.

Conclusions
In this paper, a tool denoted as Thermal Resistance and Impedance Calculator (TRIC) has been presented. TRIC allows the automatic extraction of thermal metrics of package families of electronic components under both static and transient conditions. It exploits a solution algorithm based on a novel projection-based approach, which allows dealing with non-Manhattan geometry and mesh variations in the parametric detailed thermal model (pDTM) of a package family. The pDTMs of many relevant package families have been included, and dies with multiple active areas can be handled. An extensive simulation campaign, focused on cases of practical interest, has been performed. A comparison between TRIC and the FV program FloTHERM has been carried out, with the aim of validating the accuracy and assessing the efficiency of the proposed tool; the main findings can be summarized as follows: (i) the discrepancy in terms of thermal metrics calculated by the simulators amounts at most to 3% and is mainly ascribable to the different mesh styles; (ii) thanks to its advanced solution algorithm, TRIC allows obtaining a reduction in CPU time by a factor of 15-20 with respect to FloTHERM when simulating a transient thermal response. Owing to the above reasons, TRIC can Funding: This research received no external funding.

Conflicts of Interest:
The authors declare no conflict of interest. central processing unit epad exposed pad QFP quad flat package eLQFP exposed-pad low-profile (thick) QFP eTQFP exposed-pad thin QFP pLQFP full-plastic low-profile (thick) QFP QFN quad flat no-leads package eQFN exposed-pad QFN eQFN-mr multi-row eQFN PowerSSO package belonging to the Small Outline family ABS antilock braking system