J-fraction Route to the Cauer Ladder

Polynomial long-division and Lanczos both produce the Cauer (Stieltjes) continued fraction directly. A third family of algorithms—due to Khatwani and Sobhy—first casts the impedance into a J-fraction and converts that to the desired S-fraction. This detour avoids explicit high-order divisions and relies only on scalar recurrences.

Starting point: rational impedance

After deconvolution the driving-point impedance is known in rational form

(63)\[Z(s)= \frac{\alpha_0+\alpha_1s+\dots+\alpha_{N-1}s^{N-1}} {\beta_0+\beta_1s+\dots+\beta_{N-1}s^{N-1}+\beta_Ns^{N}} .\]

The goal is to rewrite (63) as the Cauer (S-) fraction of length 2 N whose coefficients are the thermal resistances and thermal capacitances.

J-fraction representation

Both Khatwani’s and Sobhy’s schemes build the continued fraction

(64)\[Z(s)= \cfrac{a_1^{2}}{s-b_1-\cfrac{a_2^{2}}{s-b_2-\ddots- \cfrac{a_N^{2}}{s-b_N}}}\!,\]

known as a J-fraction. The auxiliary parameters \(a_i^2\) and \(b_i\) are obtained by recursively expanding the simpler form

(65)\[Z(s)= \cfrac{1}{ H_1 s+h_1+ \cfrac{1}{ H_2 s+h_2+\ddots+ \cfrac{1}{H_N s+h_N}}}\]

with \(H_i , h_i\) computed either from the Markov parameters (Khatwani) or by Sobhy’s direct iteration. This is referred to as the \(H\)–\(h\) continued fraction.

For \(i > 1\) the conversion is

(66)\[\begin{split}\begin{aligned} a_1^{2}&=\frac{1}{H_1}, &\; a_i^{2}&=-\frac{1}{H_i H_{i-1}},\\ b_1&=-\frac{h_1}{H_1}, &\; b_i&=-\frac{h_i}{H_i}. \end{aligned}\end{split}\]

From J– to S-fraction

The J-fraction (64) must be rearranged into the Stieltjes or S-fraction

(67)\[Z(s)=\cfrac{1}{ c_1 s+\cfrac{1}{ c_2+\cfrac{1}{ c_3 s+\ddots+\cfrac{1}{c_{2N}}}}}\!,\]

whose \(c_k\) are exactly the Cauer \(R'_k,\,C'_k\) in alternating order (\(c_{2k-1}=C'_k\), \(c_{2k}=R'_k\)).

Initial step

(68)\[c_1=\frac{1}{a_1^{2}},\qquad c_2=-\frac{a_1^{2}}{b_1}.\]

Recursive update for \(i>0\)

(69)\[\begin{split}\begin{aligned} c_{2i+1}&= \frac{1}{c_{2i-1}c_{2i}^{\,2}a_{i+1}^{2}},\\[4pt] c_{2i+2}&= -\frac{c_{2i}}{1+c_{2i+1}c_{2i}b_{i+1}} . \end{aligned}\end{split}\]

Iterate until \(i=N-1\) to obtain all \(2N\) coefficients.

Algorithm outline

J-fraction Algorithm for Foster-to-Cauer Conversion

Input: \(\{\alpha_k\}\), \(\{\beta_k\}\) (numerator / denominator of \(Z(s)\))

Output: \(\{R'_k, C'_k\}\) for \(k = 1 \ldots N\) (Cauer ladder)

Compute \(\{H_k, h_k\}\) (using Khatwani or Sobhy formulae)
Convert \((H, h) \rightarrow (a^2, b)\) using (66)
Initialize \(c_1, c_2\) using (68)
for \(i = 1 \ldots N-1\) do
1. Compute \(c_{2i+1}, c_{2i+2}\) using (69)
end for
Map \(c_1, c_2, c_3, \ldots\) to \(C'_1, R'_1, C'_2, R'_2, \ldots\)