Theory behind the T3Ster software: the NID method

The time-constant spectrum

The theoretical background of the evaluation of the T3Ster measurement results is based on a new representation of the distributed RC networks. In this representation the behavior of the thermal system (considered as an infinite distributed RC network) is described by convolution equations. Here we present this theory very briefly. For further details of the theory, see the references.

In the following calculation the t time and the w angular frequency will be substituted by their natural logarithm:

(1)

The R(z) time-constant density function will be defined in order to represent the RC circuits either in lumped-element or in distributed circuit case. This function gives the intensity of the terms of different time-constants in the response. We can interpret this function as a special kind of a spectrum, which depicts the occurrence and the relative intensity of the different time-constants in the circuit response. Sometimes we will use the alias-name time-constant spectrum as well.

The R(z) function is a sum of Dirac-d 's in case of a lumped element network where the response consists of terms of discrete time-constants in a finite number:

(2)

t_i are the time-constants of the poles, K_i are the corresponding magnitudes, p is the number of the poles. The R(z) spectrum is a continuous function in case of an infinite distributed network. In this case the definition is

(3)

The R(z) time-constant spectrum is related both to the time and the frequency responses. The relation is of convolution-type in both cases. In the time-domain

(4)

where a(z) is the unit-step response (i.e. the thermal transient measured by the T3Ster equipment), Ä is the convolution operator and the w_t(z) function is defined by

(5)

In the frequency-domain, for the Z(w) impedance function the following equations can be applied:

(6)

where

(8)

and

(9)

Unfortunately we do not have a straightforward way to execute the deconvolution. In most cases this operation is extremely ill-defined, and as a consequence the smallest inaccuracy or noise in the input function (in the network response in our case) makes the result completely useless. There are various methods to overcome these difficulties and the solutions are usually tailored to the individual problems.

A usual solution is the Fourier-domain inverse filtering. As it was shown previously the responses of the network are obtained by convolution integrals, as

(10)

where m(x) is a response of the network, R(x) is the time-constant spectrum and w(x) is one of the three weighting-functions of (5), (8) and (9). Turning into the Fourier domain we have the corresponding formula as

(11)

where the capital letters represent the Fourier transform functions whereas F is their "frequency" variable. This equation contains multiplication instead of convolution.

It must be emphasized that the m(x) ® M(F) transformation is not the same as the usual transformation into the frequency-domain since the variable x is not the time but ln(time) or ln(frequency). The frequency F can be interpreted as the number of waves per frequency-decade or time-decade on the logarithmic R(z) function.

The Fourier domain, the space of the F frequencies is well suited to execute the deconvolution (the inverse of convolution) since (11) yields

(12)

Thus theoretically we can obtain the required function by a simple division. Since the higher F frequency components of W(F) are quite small the division enhances the higher frequency values of M(F) extremely, resulting in an unwanted enhancement of the noise. This enhanced high-frequency noise can be as large as or even larger than the useful part of the function and can completely hide them. This fact constitutes the ultimate limit of the resolution or accuracy of the deconvolution.

The relation between the noise level and the resolution limit of the deconvolution is detailed in [6] for the network identification problem. Here we recall only some results of these investigations. Let us examine for instance the identification using Eq.(6). If we have m(x) with an accuracy of 10^-8 (which is not impossible in case of a response produced by simulation) the possible resolution of the approximate R(z) function is 0.66 octave. This means that a single line of a discrete-line spectrum is broaden to a finite-width peak the half-value width of which is approx. 0.66 octave.

In case of evaluating measured time domain response functions (heating up or cooling down curves obtained by T3Ster) the deconvolution process - that results in the R(z) time-constant spectra - is performed by a method called Bayes-iteration, since it is better suited for the evaluation of measurement results than deconvolution in the Fourier-domain.

The last problem to be solved is to find a procedure suitable to generate a lumped element equivalent network in the knowledge of the formerly determined R(z) function. The R(z) function of the distributed networks is continuous if their length can be taken to be infinite. Owing to the finite resolution, identification of lumped networks results in continuous R(z) functions as well. In order to build lumped models we have to approximate these continuous functions by a set of discrete spectrum lines. This approximation always involves a trade-off. On one hand, it is practical to keep the number of these lines as low as possible in order to minimize the size of the model network. On the other hand the error of the approximation should remain below an allowable limit.

Let us discuss the time-constant density function in the case when R(z)³ 0 (case of driving point impedances). A possible approach is the direct discretization of the R(z) function. The most straightforward way for this is the equidistant placement of the poles on the logarithmic z or W axis. If the relevant part of the function is in the [z_a,z_b] region of the z axis (see Figure 1) then the location of the poles is given by

i=1…N

(13)

where N is the pole number of the approximation, and

(14)

The magnitudes of the discrete spectrum lines can be calculated by the following simple expression:

(15)

Figure 1: Discretization of the time-constant spectrum

Possessing now the discrete-line R(z) spectrum the construction of the RC-ladder model network (Figure 2 ) is a routine task of the linear network theory.

In the case of transfer functions (response measured at a location different to the dissipation excitation) a special problem appears typically: the magnitudes are partly negative. An easy way to cope with this problem is to realize the positive and the negative part of R(z) separately, by two distinct RC one-ports and finally to add the output variables of these two subnetworks with the appropriate sign (see Figure 3 ).

Figure 2: Single Cauer-network for driving-point impedances

Figure 3: Twin Cauer-ladder for transfer impedances (thermal couplings)

When evaluating T3Ster measurement results the ultimate goal is to obtain as detailed information about the chip-to-ambient heat-conduction path as possible.

Once the time-constant spectrum is extracted from a measured transient response, the next step of evaluation procedure is to generate a lumped RC ladder model of 100-200 stages. This is an intermediate step; the resulting RC ladder is used to calculate the so called structure functions only, they are not aimed as compact models for simulation purposes.