### Categorized |Power Design, Power Devices, Power Modules, Thermal Management

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Posted on 14 January 2019

# Thermal stacking of Power Modules

When thermally stacking several heatsinks, in particular in combination with larger power electronics assemblies, the reduction in coolant flow rate resulting from the increased pressure drop and pre-heating of the coolant for the "backward" units has to be considered in the calculations.

Figure 1. a) Individual cooling; b) Thermal stacking with forced air cooling

The following two methods are suitable for calculating pre-heating:

a) Determining a thermal impedance Z th(a-a’) between measuring points at the heatsink 1, 2 and 3
b) Calculating coolant pre-heating; the coolant outlet temperature from the first module is the inlet temperature at the second one, etc.

### Determining an additional thermal impedance

Pre-heating is determined by total power dissipation Ptot(n) of the heat source, the stationary thermal resistance Rth(a-a*), or the transient thermal impedance Zth(a-a*) between two adjacent heatsinks (Figure 1b). To this end, the temperature differences between the heatsink temperatures must be determined at a given power dissipation. As time passes, the second and any other heatsink will get warmer than the unit directly in front. This temperature difference divided by the component power dissipation results in Zth(a-a*). For this part of the transient thermal impedances, 1 R/τ element is sufficient in most cases. The following known correlation applies to component 1 in the direction of coolant flow:

$Z_{th(s-a)1} = \sum^n_{v=1} R_v \cdot \Big ( 1- e^{\frac{-t}{\tau_{thv}}} \Big )$

To cater for component 2, an additional element for the temperature difference between heatsink 1 ("a") and 2 ("a*") is introduced. This pre-heating depends on the losses of component 1, which is why all of the losses must be weighted. If the losses of all heat sources are identical, this may be omitted:

$Z_{th(s-a)2} = \sum ^4_{v=1} R_v \cdot \Big ( 1- e^{\frac {-t}{\tau_{thv}}} \Big ) + \frac {P_{tot1}}{P_{tot2}} \cdot R_{th(a-a^*)} \cdot \Big ( 1- e^{\frac {-t}{\tau_{thv(a-a^*)}}} \Big )$

For component 3 and any additional one, this applies by analogy:

$Z_{th(s-a)3} = \sum ^4_{v=1} R_v \cdot \Big ( 1- e^{\frac {-t}{\tau_{thv}}} \Big ) + \frac {P_{tot1} + P_{tot2}}{P_{tot3}} \cdot R_{th(a-{a^{**}})} \cdot \Big ( 1- e^{\frac {-t}{\tau_{thv(a-a^{**})}}} \Big )$

### Calculating pre-heating for air cooling

The basic idea behind this method is to retain the well proven basic equations of temperature calculation and redetermine the coolant inlet temperature for the "n-th" element only: This can easily be done using the heat retention capability of the coolant, if even through-heating is assumed. Both the specific weight and the heat retention capability are temperature dependent, which is why there is a temperature coefficient for pre-heating.

Figure 2. Thermally stacked, air-cooled layout, split into sectors with different air pre-heating

The general formula is:

$T_a^* = T_a + \Big ( \frac {1}{c_p \cdot \rho} + TC_c \cdot T_a \Big ) \cdot \frac {P_{tot1}}{V_{air}}$

cp:             Specific heat capacity of air [kJ/K/kg]
ρ:               Density of air [kg/m³]
TCc:           Temperature coefficient of the specific heat capacity
Ta*:           Coolant temperature for the second heat source
Ptot1:         Power dissipation of heat source 1

Adapted to an average atmospheric pressure of 1 bar, a basic temperature of 0°C and the conversion to a specification for the volumetric flow rate in [m³/h], the following results:

$T_a^* = T_a + \Big ( \frac {3.6}{1.006 \cdot 1.275 } + 0.01^\circ C^{-1} \cdot T_a \Big ) \frac {^\circ C \cdot m^3}{W \cdot h} \cdot \frac {P_{tot1}}{V_{air}} \cdot K_{Ing}$

Ptot1 [W]:        Power dissipation of heat source 1
Vair [m³/h]:     Volumetric flow rate through the heatsink
King:                 Correction factor for uneven heat profile at the outlet of heatsink 1

Figure 3. Uneven temperature profile of the exiting air over the fan cross section if heat sources are arranged centrally

Figure 4. Temperature difference between incoming and outgoing air as a function of power dissipation PV1 (W) and the volumetric flow rate, Ta = 40°C

### Calculating pre-heating for water cooling

In principle, the same basic equation applies to thermal stacking for liquid cooling as for air cooling. However, it must be borne in mind that the dynamic viscosity changes as a function of the temperature. The heat retention characteristic for a 50:50 water/glycol mixture and a conversion to the volumetric flow rate in [l/min] results in:

$T_a^* = T_a + (0.0174 - 0.000013 \cdot ^\circ C^{-1} \cdot T_a ) \cdot \frac {^\circ C \cdot I}{W \cdot \text {min}} \cdot \frac {P_{tot1}}{V_{H2O}}$

For pure water, the specific heat capacity and the temperature coefficient will change to:

$T_a^* = T_a + (0.0133 - 0.000008 \cdot ^\circ C^{-1} \cdot T_a ) \cdot \frac {^\circ C \cdot I}{W \cdot \text {min}} \cdot \frac {P_{tot1}}{V_{H2O}}$

Figure 5. Temperature difference between water/glycol mixture flowing in/out as a function of the power dissipation and the volumetric flow rate, Ta = 25°C, 50% glycol

Heat Transfer in Power Semiconductor Devices

Cooling Methods for Power Semiconductor Devices

Thermal Modeling of Power Module Cooling Systems

Heat Dissipation and Thermal Resistance in Power Modules

Thermal Equivalent Circuit Diagrams for Junction Temperature Calculations

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