Structure Functions

The lumped Foster- and Cauer-ladders used so far can be viewed as discrete approximations of a one-dimensional distributed RC line. This section shows how classical transmission-line theory maps onto thermal structure functions and how the cumulative and differential structure functions emerge naturally for non-uniform RC lines.

Transmission-Line Basics

A transmission line is a distributed two-port: its series impedance density \(z(x)\) and shunt admittance density \(y(x)\) are smeared continuously along the spatial coordinate \(x\).

../_images/transmission_line.png

Fig. 3 Infinitesimal element of a general transmission line.

For an electrical wire

(37)\[z(x)=r+l\,s, \qquad y(x)=g+c\,s,\]

with \(s\) the complex Laplace variable. The Telegrapher equation

(38)\[\frac{\mathrm{d}^2 V}{\mathrm{d}x^2}-z\,y\,V=0\]

has the general solution

(39)\[V(x)=V_1e^{-\gamma x}+V_2e^{\gamma x}, \qquad \gamma=\sqrt{zy}=\alpha+\mathrm{i}\beta.\]

Here \(\alpha\) describes attenuation, \(\beta\) the phase shift, and

(40)\[Z_0=\sqrt{\frac{z}{y}}\]

is the characteristic impedance. Driving a load \(Z_L\) through a length \(\Delta x\) gives

(41)\[Z_\text{in}= \overline{Z}_0\; \frac{Z_L\cosh(\gamma\Delta x)+\overline{Z}_0\sinh(\gamma\Delta x)} {\overline{Z}_0\cosh(\gamma\Delta x)+Z_L\sinh(\gamma\Delta x)}.\]

Uniform RC Line

For purely diffusive heat flow the electrical analogue keeps resistance in series and capacitance in shunt:

(42)\[z=r,\qquad y=s\,c \;\;\Longrightarrow\;\; \gamma=\sqrt{s\,r\,c}, \quad Z_0=\sqrt{\frac{r}{s\,c}}.\]

On a uniform line the local relations are

(43)\[\mathrm{d}V=-\,r\,I\,\mathrm{d}x, \qquad \mathrm{d}I=-\,c\,\frac{\partial V}{\partial t}\,\mathrm{d}x,\]

which combine to the heat equation

(44)\[\frac{\partial V}{\partial t}= \frac{1}{r\,c}\, \frac{\partial^2 V}{\partial x^2}.\]

Injecting a charge \(Q\) at \(x=0\) gives

(45)\[V(x,t)=\frac{Q/c}{ 2\,\sqrt{\pi t/(r\,c)} }\, \exp\!\Bigl[-x^2/(4t/(r\,c))\Bigr],\]

identical in shape to a thermal Green’s function.

Non-Uniform RC Line ⇔ Structure Function

Real devices do not possess constant material properties; both resistance density \(r(x)\) and capacitance density \(c(x)\) vary along the heat path. Define the cumulative quantities

(46)\[R_\Sigma(x)=\int_0^x r(\xi)\,\mathrm{d}\xi, \qquad C_\Sigma(x)=\int_0^x c(\xi)\,\mathrm{d}\xi.\]

These are exactly the axes of the cumulative structure function obtained from the Cauer ladder in network identification by deconvolution.

Differential and cumulative forms

  • Differential structure function

    (47)\[\sigma\!\bigl(R_\Sigma(x)\bigr)= \frac{c(x)}{r(x)}.\]

    It equals the ratio of local densities expressed against cumulative resistance.

  • Cumulative structure function Also called the Protonotarios–Wing function,

    (48)\[C_\Sigma(R_\Sigma)= \int_{0}^{R_\Sigma}\sigma(R'_\Sigma)\,\mathrm{d}R'_\Sigma \;=\; \int_{0}^{x(R_\Sigma)} c(x)\,\mathrm{d}x.\]

Voltage (temperature) evolution

With spatially varying densities the local balance becomes

(49)\[\frac{\partial V}{\partial t}= \frac{1}{c(x)} \frac{\partial}{\partial x} \Bigl[\frac{1}{r(x)}\, \frac{\partial V}{\partial x}\Bigr].\]

In cumulative-resistance coordinates

(50)\[\frac{\partial V}{\partial t}= \frac{1}{\sigma(R_\Sigma)} \frac{\partial^2 V}{\partial R_\Sigma^2}.\]

A closed-form solution is known only for special profiles (\(\sigma=\text{const.}\) recovers the uniform line), but the equation underlies structure-function analysis: the measured pair \(\bigl(R_\Sigma,C_\Sigma\bigr)\) summarises all one-dimensional non-uniform RC lines that replicate the same driving-point thermal behaviour.

Relation to NID workflow

  1. NID converts the step response to a Foster ladder (time-constant spectrum).

  2. Foster → Cauer transformation produces a series RC ladder whose running sums reproduce \(R_\Sigma\) and \(C_\Sigma\).

  3. Plotting \(C_\Sigma(R_\Sigma)\) yields the cumulative structure function; its slope is the differential structure function \(\sigma\).

Hence the structure function is nothing else than the physical picture of heat flow through a non-uniform RC line.