Examples of power module junction temperature calculations for various operating conditions are given below. The following calculations make use of thermal equivalent circuit diagrams, a method that can be used in certain situations to simplify junction temperature calculations. For more complex situations, other calculation techniques must be employed.

__Junction temperature during stationary operation (mean value analysis)__

After losses have been calculated, temperatures during stationary operation can be calculated with the aid of the thermal resistances R_{th} (= final value of the Z_{th} curves). Temperature calculation is performed starting with the ambient temperature from the outside to the inside.

**Figure 1.** *Temperature calculation under stationary conditions*

If there is more than one source of power loss on a heatsink, the individual losses of all n_{1} components are added up (e.g. 6 IGBT and 6 freewheeling diodes of a 3-phase inverter); this total loss is used to calculate the heatsink temperature:

Modules with base plate show good thermal coupling between their components and the thermal resistance R_{th(c-s)} is specified for the entire module, which is why all sources of power loss n_{2} in the module are added up to calculate the case temperature (e.g. 2 IGBT and 2 freewheeling diodes in a half-bridge module):

In some examples, this modelling will result in excessive temperatures if just one IGBT and its spatially divided freewheeling diode is used in a half-bridge module, or if the IGBT and parallel inverse diode produce time shifted losses in inverter mode. If R_{th(c-s)} is only specified for a single component (e.g. the IGBT), the thermal coupling of the copper base plate with the inverse diode is neglected. In this case, modelling in combination with simultaneous losses in the IGBT and diode will produce an error with temperatures that are too low.

The junction temperature is finally calculated from the losses of the single component and its thermal resistance to the case for modules with base plate, or to the heatsink for modules without base plate. For an IGBT this is, for example,

or

All semiconductor losses are temperature dependent, meaning that chip temperature and losses are coupled. In the simplest case, losses are calculated at the maximum junction temperature. This will ensure that you are on the safe side, since most losses increase in line with temperature. In an improved procedure, losses are determined iteratively at the calculated junction temperature. For this loop that can easily be programmed, the starting value is the power dissipation at ambient temperature (see the loop below). This value allows for an initial approximation of the junction temperature. At this temperature, a new, more precise loss value is produced. After 3 to 4 iterative loops, the final value will have been reached in most cases.

For k=1 to 10 (T_{j(0)}=T_{a} )

**Figure 2.** *Loop used to calculate temperature-dependent semiconductor losses for a module without base plate*

__Junction temperature during short-time operation__

Higher current loads are permitted during short-time operation of the power semiconductors than specified in the datasheets for continuous operation. What must be ensured here, however, is that the maximum junction temperature that develops under the defined conditions does not exceed the limit value for T_{j(max)}. T_{j} is calculated with the aid of the thermal impedance Z_{th}. For pulses in the millisecond range, it is sufficient for Z_{th(j-c)} to be considered at a constant case temperature T_{c}. In the range up to 1s, it is possible to work with the module impedance Z _{th(j-s)} = Z _{th(j-c)} + Z _{th(c-s)} and constant heatsink temperature. In longer pulse sequences, the total impedance Z _{th(j-a)} = Z _{th(j-s)} + Z_{ th(s-a)} should be used.

### Single Phase

**Figure 3.** *Time curve of power dissipation and junction temperature in the event of a single power dissipation pulse*

The junction temperature change at the moment t_{1}, following a single power dissipation pulse, is calculated using the following formula:

The following applies to the junction temperature curve during the cooling phase:

t_1) = P \cdot \sum_{v=1}^n R_{thv} \cdot \Big ( 1- e^{\frac {-t}{\tau_{thv}}} \Big ) - P \cdot \sum_{v=1}^n R_{thv} \cdot \Big ( 1- e^{\frac {-(t-t_1)}{\tau_{thv}}} \Big ) "/>

The formulae assume a fixed reference temperature.

### One-time sequence of m power dissipation pulses

**Figure 4.** *Time curve of power dissipation and junction temperature in the event of a one-off sequence of m power dissipation pulses*

The junction temperature change at the moment t_{1} is to be calculated in the same way as for the single pulse. For a junction temperature change at the moment t_{2}, the following applies:

Junction temperature change at the moment t_{m} :

### Junction temperature at constant pulse sequence

In order to calculate the mean and maximum junction temperature for periodically recurring power dissipation loads, the Z_{th(j-c)} curve of the transistor and diode in periodic pulse operation can be used as specified in datasheets.

**Figure 5.** *Temperature curve for periodic pulse load*

The average junction temperature T_{j(av)} results from the product of stationary heat resistance R_{th} and the average total loss P_{av} . This value is obtained by averaging the loss energies per pulse over the entire length of a period T. The maximum junction temperature at the end of the loss pulse P_{(tp)} results from

For pulse durations in the millisecond range, significant temperature swings occur. After some 100 ms, a strong stationary temperature difference will already have developed between the chip and case.

__Junction temperature at fundamental frequency__

The calculation of the junction temperature determined by the fundamental frequency of the converter output current is only efficient if performed using computer aided methods. Both thermal system and electrical system per pulse duration have to be calculated in detail in order to integrate the IGBT and diode junction temperature over a sine half-wave. The thermal model essentially corresponds to a simplified thermal equivalent circuit of IGBT and freewheeling diode in a power module simulating the thermal impedances by means of exponential functions.

Switching losses per pulse may be calculated based on stored characteristics if the current converter parameters such as DC link voltage and instantaneous load current are given. The instantaneous junction temperature is entered into the calculation via the temperature coefficients. Figure 6 shows the time characteristic of the power dissipation and the average power dissipation in an IGBT, as well as the resulting junction temperature characteristics for different fundamental output frequencies as the result of a simulation.

**Figure 6.** *Simulated junction temperature and power dissipation characteristics of a 1200 V / 50 A-IGBT in inverter mode for different fundamental output frequencies; [37] v _{d} = 540 V; i_{1eff} = 25 A, f_{s} = 8 kHz; cos ϕ = 0.8; m = 0.8; T_{h} = 50°C*

In this example, the maximum junction temperature exceeds the average value by just about 4-5 K at a frequency of 50 Hz. For low frequencies, the average junction temperature is no longer permitted to determine the thermal layout of the system because of its far higher maximum value. Therefore, the minimum frequency at rated power output is a critical parameter for inverters besides overcurrent. When the frequency decreases, the maximum temperature will rise despite constant mean losses. An explanation for this is the fact that the thermal time constants of power semiconductors are below one second and thus within the range of normal inverter output frequencies. When frequencies are high, the thermal capacitances of the semiconductor (case) still average the temperature characteristic. When frequencies are low (<2...5 Hz), the junction temperature follows the power dissipation. This results in high temperature fluctuations around the mean temperature. Consequently, the permissible output current RMS value for a given power module will decrease at a given heatsink temperature and switching frequency.

A particularly critical case with regards to the thermal stress of power modules is the voltage and frequency controlled starting process of a three-phase motor drive supplied by an inverter. Figure 7 shows a suitable simulation example.

**Figure 7.** *Starting process of a three-phase motor drive (Parameters as in Figure 4)*

If you do not use a circuit simulation but rather use the given formulae, inverter operation at an output frequency of f_{out} = 0 Hz is a limiting case. If mean values of a period are used for calculations, the period will be infinite at f = 0 Hz. "Infinite" applies if the dwell time in this condition is far greater than the thermal time constants of the system. Besides the marginality of the formula definition, it is extremely problematic to generally assume an even distribution of losses on the heatsink, whereas in this case it is only one of the 6 switches that produces the major part of the losses. Theoretically, one switch might switch a direct current at the very moment when the current and losses reach their maximum. It is therefore the better choice to calculate this operating condition with the aid of a chopper circuit (step-down converter).

**For more information, please read:**

Heat Transfer in Power Semiconductor Devices

Thermal Equivalent Circuit Diagrams for Junction Temperature Calculations

Thermal Impedance and Thermal Resistance