Overview of Network Identification by Deconvolution

Introduction

When a device suddenly receives constant extra heating or cooling power (a power step), its temperature does not jump immediately. Instead it follows a characteristic curve that reveals the device’s thermal properties. Network identification by deconvolution (NID) reconstructs those properties from the measured curve and expresses them as an equivalent electric RC network (thermal resistance ↔ electrical resistance, thermal capacity ↔ electrical capacitance).

The main ideas are sketched below.

Step- and impulse-response

  • Step response The thermal impedance

    (1)\[Z_\mathrm{th}(t) = \frac{T_0 - T(t)}{P}\]

    is the temperature rise (relative to the starting temperature \(T_0\)) normalised by the applied power step \(P\).

  • Impulse response Differentiate the step response:

    (2)\[h(t) \;=\; \frac{\partial Z_\mathrm{th}(t)}{\partial t} .\]

    Once \(h(t)\) is known, any time-dependent power profile \(P(t)\) produces

    (3)\[T(t) = T_0 + \int_0^{t} P(t')\,h(t - t')\,\mathrm{d}\t'\]

    via convolution.

    Note

    Differentiation is numerically challenging due to the amplification of measurement noise. See Differentiation for details on specialized algorithms that overcome these challenges.

  • Transfer function The Laplace transform \(Z(s)\) of \(h(t)\) is the impedance of the sought RC network. Knowing \(Z(s)\) numerically is not enough: the individual resistances and capacitances still have to be extracted.

Logarithmic time axis

Thermal transients span microseconds to hours. Calculations therefore switch to a logarithmic time axis

(4)\[\begin{split}z &= \ln\!\bigl(t/t_0\bigr),\\ \zeta &= \ln\!\bigl(\tau/\tau_0\bigr),\end{split}\]

usually with \(t_0=\tau_0=1\;\text{s}\) so that \(z\) and \(\zeta\) are logarithmic times.

Notation: The measured step response expressed on the \(z\)-axis is written

(5)\[a(z) \;=\; Z_\mathrm{th}\!\bigl(e^{z}\bigr).\]

Time-constant spectrum

A one-dimensional heat path can be modelled by a Foster ladder with time-constants \(\tau_i = R_i C_i\). Its logarithmic time-constant spectrum

(6)\[R(\zeta) \;=\; \sum_{i=1}^{n} R_i\,\delta\!\bigl(\zeta-\ln\tau_i\bigr)\]

collects the resistances as Dirac peaks.

Foster network

Fig. 1 Foster network with parallel RC branches.

On the logarithmic axis the step response becomes

(7)\[a(z) \;=\; \int_{-\infty}^{\infty} R(\zeta)\, \bigl(1 - e^{-e^{\,z-\zeta}}\bigr)\,\mathrm{d}\zeta .\]

Differentiation yields the impulse response

(8)\[h(z) \;=\; \int_{-\infty}^{\infty} R(\zeta)\, e^{\,z-\zeta - e^{\,z-\zeta}}\,\mathrm{d}\zeta , \quad\Longrightarrow\quad h \;=\; R \;\ast\; w_z\]

with the kernel

(9)\[w_z(z) \;=\; e^{\,z - e^{\,z}}.\]

Finding \(R(\zeta)\) is therefore a deconvolution problem.

From spectrum to RC network

  1. Discretise the spectrum into bins of width \(\Delta\zeta_i\) centred at \(\zeta_i\).

  2. Foster elements

    (10)\[\begin{split}R_i &= R(\zeta_i)\,\Delta\zeta_i,\\ C_i &= \frac{e^{\,\zeta_i}}{R(\zeta_i)\,\Delta\zeta_i}.\end{split}\]

  3. Cauer network Transform the Foster ladder to a Cauer (series) ladder iteratively (see standard Foster-↔-Cauer formulas). The cumulative heat capacitance versus cumulative thermal resistance obtained from the Cauer ladder is the structure function.

Cauer network

Fig. 2 Foster network with series RC branches.

Inverse route starting from \(Z(s)\)

If only the transfer function is given, the spectrum follows from

(11)\[R(\zeta) \;=\; \frac{1}{\pi}\; \Im\!\bigl[Z\bigl(s = -e^{-\zeta}\bigr)\bigr].\]

Here, \(\Im\!\bigl[Z(s)\bigr]\) is the imaginary part of the transfer function.

Because \(Z(s)\) has poles on the negative real axis, evaluate instead along a line tilted by a small angle \(\delta\)

(12)\[s(\zeta) \;=\; -\bigl(\cos\delta + \mathrm{i}\,\sin\delta\bigr)e^{-\zeta},\]

which broadens the delta peaks and avoids singularities.

With \(R(\zeta)\) known, use the convolution above to recreate \(h(z)\) and \(Z_\mathrm{th}(t)\).

Thermal Structure Functions

The structure function is a strictly one-dimensional representation. Whenever lateral or parasitic heat flows grow significant, assigning its resistances and capacitances to real device layers becomes ambiguous. Typical causes are:

  • asymmetric cooling conditions,

  • strong neighbouring heat sources,

  • inadequate insulation,

  • three-dimensional spreading (cylindrical, conical, spherical).

Even then the spectrum, impulse response and thermal impedance remain well defined, because the Foster elements are mathematical and need not match physical layers.