RC Equivalent Circuits

An electrical equivalent circuit is created to provide a transient response which describes the transient thermal impedance of the packaging structure. It employs the well estab- lished thermal-electrical analogy which is stated in Table 4.1.

4.2 Dynamic Compact Thermal Models

Table 4.1: Corressponding physical variables using the electrical-thermal analogy

Thermal ⇐⇒ Electrical

Temperature (K) Voltage (V)

Power (W) Current (A)

Thermal resistance (KW−1_{) Electrical resistance (Ω)}

Thermal capacitance (JK−1_{) Electrical capacitance (F)}

Typically, a transient thermal impedance curve resulting from a step change in power is required to extract a RC network. Power module manufacturers tend to provide nor- malised impedance curves, as shown in Fig. 4.3. Zth(j−c)(t) represents the transient

thermal impedance from junction-to-case. Values ofZth(j−c) can be obtained by multiply-

ing the value of steady state thermal resistance Rth(j−c) by the normalised factor taken

from the curve at the time of interest. Rth(j−c) is specified by a packaging manufacturer

and is typically between 0.2-0.4 K/W for a power module [100]. Unfortunately, manu- facturers tend to provide information which is unreliable. Furthermore, data concerning

Zth(j−c)(t) does not allow for the presence of the power module heatsink.

The transient thermal impedance curve of Zth(j−amb)(t) is needed. This represents

the transient thermal impedance between the junction and ambient conditions, therefore including the heatsink. Fortunately, a cooling/heating curve representing the junction temperatureTj(t) may be generated by simulation software employing numerical methods

or from experimental measurements. From a curve of Tj(t) it is possible to generate a

curve of Zth(j−amb)(t) using Equation (4.2): Zth(j−amb)(t) =

Tj(t)−Tamb(t)

P , (4.2)

where t (s) is time, Tamb(t) (K) is the known ambient temperature and P (W) is the

4.2 Dynamic Compact Thermal Models

Figure 4.3: Graph showing a typical normalised transient thermal impedance curve which would be provided by a power module manufacturer.

An equivalent circuit may then be fitted to Zth(j−amb)(t) in light of the following

approximation: Zth(j−amb)(t) = n X i=1 Ai ³ 1−e−τit ´ (4.3)

whereAi (KW−1) is the pre-exponential coefficient andτi (s) is the time constant. Exam-

ples of this curve fitting approach appear in [24, 26, 28–30]. The two dominant types of RC equivalent circuits are the Foster (partial fraction) model and the Cauer (continued fraction) model.

4.2 Dynamic Compact Thermal Models

4.2.1.1 Foster Model

The Foster model equivalent circuit appears in Fig. 4.4. Its networks element values are related to Equation (4.3) by: Ri =Ai;Ci = _Aτi_i.

This property simplifies determination of the values of the equivalent elements and which can be directly fitted as shown in [101]. Its computational simplicity explains the wide application and popularity of this equivalent network.

Figure 4.4: Foster RC equivalent circuit.

The Foster network is a “black box” approach. It can describe the curve of the junction temperature with any excitation but it has no physical meaning since its RC elements are not directly related to the layers in the structure. Therefore, the Foster model can not be used for physical identification of heat flow through the structure [102–104]. The reason is that the node-to-node capacitances are physically inconsistent, as stated in [25, 27, 105]. For instance, if a pulse of thermal power is injected at the model input the temperature at every internal node would change immediately due to the capacitors forming a series connection between input and the output side. This differs from reality, where there is a time delay before the heat diffuses through the structure. Another weakness of this “black box” approach is that it would be necessary to produce a new graph of the transient junction temperature, and then recalculate the values for all of the RC elements, if an extra layer was added to a structure which had been previously modelled.

4.2 Dynamic Compact Thermal Models

4.2.1.2 Cauer Model

The Cauer model equivalent circuit appears in Fig. 4.5. The Cauer model is the transmis- sion line equivalent circuit [106–108] containing grounded capacitors and floating resistors. It is able to describe the internal heat flow of the structure it is modelling. There is a clear correlation between the RC elements in the equivalent circuit and each physical layer of the power module package. Unfortunately, component manufacturers do not make the value of network parameters readily available.

Figure 4.5: Cauer RC equivalent circuit.

Therefore it is necessary to determine values for the RC elements in the structure, a process known as parameter extraction. This is a long arduous process when using the step response of the Cauer network since it is difficult to achieve mathematically [109]. It occurred with limited success in [110], which required a Laplace transform and aid from a curve fitting algorithms.

The Cauer network can also be obtained by converting from a Foster network [111, 112]. However, this conversion should be used with caution since it is only valid when

Tamb, labelled in Fig. 4.4 and Fig. 4.5, can be considered constant.

Unfortunately, both approaches which are used to obtain Cauer networks invoke some numerical difficulties because there is no one to one relation between any single network element and the terms in Equation (4.3), unlike the Foster network. A further drawback of the Cauer model is that it cannot accurately represent lateral heat spreading [113, 114].

4.2 Dynamic Compact Thermal Models