The calculation of junction temperatures is based on simplified thermal equivalent circuit diagrams (Figure 1) in which three dimensional structures are mapped to one dimensional models. This will inevitably result in errors, since thermal connections between different components inside one housing or on one heatsink are dependent on time as well as on the electric operating point of the components (e.g. distribution of losses between diode and IGBT). More complex models with a matrix composed of coupling elements are impractical and difficult to parameterise. If such effects have to be considered more closely, FEM simulations are preferred to the analytical approaches presented here.

**Figure 1.** *Simplified thermal equivalent circuit diagram of IGBT and freewheeling diode in a power module: for module with copper base plate (left); for module without base plate (right)*

To obtain thermal equivalent circuit diagrams, electrical analogies to thermal parameters are used. The sources of power dissipation P_{tot(T/D)} correspond to current sources, constant temperature levels are represented by voltage sources, and RC elements are used to represent the thermal impedances Z_{th(x-y)}. In power modules with base plate, the transistor and inverse diode are soldered onto a common copper substrate, thus being thermally coupled. For the purpose of simplification, we may assume a common case temperature. The energy losses of all internal components in the module must be dissipated through the thermal impedance Z _{th(c-s)} (Figure 1, left). IGBT losses will heat the base plate and thus the diode, too, even if the diode itself does not produce any losses. Several modules on one heatsink will all contribute to heat build-up, meaning that we may also assume a uniform heatsink temperature in this case.

Modules without base plate only demonstrate low thermal coupling in the module, which only exists at all if there are very small clearances between the heat sources. For the purpose of simplification, we may assume a common heatsink temperature (Figure 1, right). Component losses are directly dissipated to the heatsink, meaning that, together, all of the components heat up the heatsink.

Two equivalent circuit diagrams are quite common when mapping thermal impedance: the ladder type "physical" equivalent circuit diagram (Figure 2, left, "Cauer network") and the chain-type "mathematical" equivalent circuit diagram (Figure 2, right, "Foster network").

**Figure 2.** *Mapping of the Z _{th} four pole blocks by means of thermal equivalent circuit diagrams; left: conductor-type "physical" equivalent circuit diagram (Cauer network); right: chain-type "mathematical" equivalent circuit diagram (Foster network) *

At first glance, both types are equally useful. If a step function response in a black box were "measured", it wouldn't be possible to determine which of the equivalent circuit diagrams was used (Figure 3).

**Figure 3. ***Mapping thermal impedance with the aid of equivalent circuit diagrams*

The smallest error is produced in subsequent calculations if the entire system is measured as a closed system. The time dependent temperature differences of T_{j}, T_{c}, T_{s}, and T_{a} can be used to determine the blocks for thermal impedances Z _{th(j-c)}, Z _{th(c-s)}, and Z _{th(s-a)}, which are represented as four poles. To obtain a ladder type equivalent circuit diagram, the geometrical dimensions and material properties are employed. For each layer in the structure, at least one "ring" is necessary. Subsequent adjustments are often necessary in order to adapt the calculated functions to the measured results. In this way, the physical relation gets somewhat lost. To obtain the chain type equivalent circuit diagram, the factors and time constants for the exponential function are determined by means of formula manipulation:

2 to 3 exponential terms are often sufficient for mapping. The main advantage of the exponential function is that it can be easily used for subsequent temperature calculation. SEMIKRON specifies the power module parameters for Z_{th(j-c)} \Z_{th(j-s)} in the form of 2 to 6 RC elements in the data book for computer aided simulations of the time curve of the junction temperature.

It is possible for both equivalent circuits to line up the sub-blocks determined in the overall system and determine the intermediate temperatures T_{c}, T_{s}. Where things go wrong, however, is when one block from the chain of thermal impedances is to be replaced by another. Heat spreading produces feedback to the preceding blocks. For example, this is the case if parameterisation has been done on a very good water cooler and in the real application the component is mounted on a poor air cooler. Nevertheless, to simplify matters, often only Z_{th(s-a)} is replaced.

Notes on the use of the Foster network:

- The intermediate points within a block (Figure 2, right) do not represent system temperatures.
- The R and C elements cannot be assigned to any physical components.
- The sequence of the RC elements within a block can be interchanged.
- They cannot be connected in series with a resistance.

It is often only the average junction temperatures and their ripples that are decisive for the thermal layout of converters.

**For more information, please read:**

Heat Transfer in Power Semiconductor Devices

Thermal Impedance and Thermal Resistance

Heat Dissipation and Thermal Resistance in Power Modules

Factors Influencing Thermal Resistance of Power Modules