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

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Posted on 24 July 2020

# Thermal Modeling of Power Module Cooling Systems

The extensive heat build-up in power modules caused by forward, switching, and blocking losses has to be dissipated by means of heatsinks. These heatsinks provide an expanded surface for convection and radiation, spreading the heat flow as well as reducing the intensity of transient thermal processes. Due to their internal insulation, all of the power modules in a system can be mounted onto one common heatsink, which may also take on some function in the design construction, too (case, chassis etc.). Heat dissipation through a heatsink works on the principle that the heat is dissipated to the coolant either by direct heat conduction or via a heat carrier. Coolants may be air, water, aqueous mixtures, or (less frequently) insulation oil, which is circulated by the effect of gravity or using fans or pumps.

$Q = \alpha \cdot A \cdot \Delta T = P_{tot}$

This result in the definition of thermal resistance gives

$R_{th(s-a)} = \frac {\Delta T_{(s-a)}}{P_{tot}} = \frac {1}{\alpha \cdot A}$

(Q - dissipated heat quantity, α - heat-transfer coefficient, A - heat transfer area)

Since Ts was measured at a certain point in order to determine ΔT(s-a), the value for Rth(s-a) only applies to this measuring point of ambient temperature. Other measuring points will result in different values. To simplify analysis, a uniform heatsink temperature for all components is often assumed in other layouts containing several modules. Losses occurring in all of the heat sources on the heatsink are dissipated through the common Rth(s-a).

The equation above for Rth shows that a high number of fins makes sense in order to increase the dissipation area. What must be ensured, however, is that the flow conditions are set in a way which will not excessively reduce the flow speed and hence the heat transfer coefficient α. This explains, for example, why there are different optimization criteria for heatsinks with natural and forced air cooling. In line with better heat penetration on the heatsinks as a result of higher power dissipation, homogenous heat distribution, or more even heat spread, the effective heat exchange area Aeff increases and the efficiency of the heatsink increases or Rth(s-a) decreases. Swirling the coolant as much as possible greatly increases the value of α, which also contributes to the reduction of Rth(s-a).

If a sudden increase in power dissipation at t = 0 from P = 0 to P = Pm occurs, the transient thermal impedance characteristic of the heatsink Zth(s-a) versus time t is split up into several time constants. The total thermal impedance characteristic Z th(j-a) of the assembly may be determined by adding the thermal impedance characteristics of the power module and the heat transfer to the heatsink. The Zth curves can be described as the sum of n exponential functions using the following equations:

$\Delta T_{(s-a)} (t) = P_m \cdot \sum_{v=1}^n R_{thv} \cdot \Big ( 1 - e^{\frac{-t}{\tau_{thv}}} \Big )$

$Z_{th(s-a)} (t) = \frac {\Delta T_{(s-a)} (t)}{P_m}$

It thus follows that

$Z_{th(s-a)} (t) = P_m \cdot \sum_{v=1}^n R_{thv} \cdot \Big ( 1 - e^{\frac{-t}{\tau_{thv}}} \Big )$

The number ν of summands and the values for Rthn and τν are chosen such that a sufficient approximation of the curve shape can be produced with a reasonable amount of calculations involved, irrespective of the physical structure. The basis for determining values of Rth and τth is the existing Zth curve. Mathematical programs and spreadsheets such as Excel (→ Solver) are able to solve equation systems with a large number of unknown elements by setting the sum of error squares to zero (0) for a large number of interpolation nodes, as shown in the following equation.

$\Bigg ( Z_{th(s-a)} (t_n) - \sum R_{thv} \cdot \Big ( 1-e^{\frac {-t_n}{\tau_thv}} \Big ) \Bigg )^2 = 0$

In order to rule out absurd solutions, a constraint must define Rth and τth > 0. The number of Rthth pairs can be increased until the desired accuracy has been achieved. In most cases, 3...5 pairs are sufficient.

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