Polynomial Long-Division

The Foster ladder obtained from the time-constant spectrum is compact and convenient for simulation, yet every branch lacks a direct physical meaning. To retrieve layer-by-layer thermal parameters we convert the network into the Cauer ladder – a series chain in which each \(R'_k,\,C'_k\) pair maps to a consecutive portion of the heat path.

The conversion is an exercise in network synthesis: rewrite the Foster impedance

(58)\[Z(s)=\sum_{i=1}^{n}\frac{R_i}{1+sR_iC_i}\]

as the Stieltjes continued fraction

(59)\[Z(s)= \cfrac{1}{ sC'_1+ \cfrac{1}{ R'_1+ \cfrac{1}{ sC'_2+ \cfrac{1}{ R'_2+\ddots+ \cfrac{1}{sC'_n+\cfrac{1}{R'_n}} } } } }.\]

Polynomial Long-Division algorithm

The standard procedure extracts the Cauer elements one at a time via repeated polynomial division (Euclid algorithm).

Notation

  • Foster elements \(R_i,\,C_i\)

  • Cauer elements \(R'_k,\,C'_k\)

  • \(Z_k(s)\) – remaining impedance after the first k−1 Cauer pairs have been stripped (start with \(Z_n(s)=Z(s)\)).

Recursive step

For each iteration \(k = n, n-1, \ldots, 1\):

  1. Split \(Z_k(s)=p_k(s)/q_k(s)\) with deg \(q_k\) = deg \(p_k\) + 1.

  2. Divide

    (60)\[\frac{q_k(s)}{p_k(s)} \;=\; sC'_k\;+\;\frac{1}{R'_k}\;+\;\frac{r_k(s)}{p_k(s)}.\]

    The quotient coefficients give immediately

    (61)\[C'_k=\text{(coefficient of }s), \qquad R'_k=\bigl[\text{constant term}\bigr]^{-1}.\]

  3. Form new impedance

    \begin{aligned} p_{k-1}(s)&=-\,R'_k\,r_k(s),\\ q_{k-1}(s)&=\dfrac{p_k(s)}{R'_k}+r_k(s),\\ Z_{k-1}(s)&=\dfrac{p_{k-1}(s)}{q_{k-1}(s)}. \end{aligned}

  4. Stop when \(p_0(s)=0\); all Cauer elements have been found.

Why the division always works

Because each branch in the Foster sum (58) adds one real pole but no new zeros, deg \(q_k\) is exactly one higher than deg \(p_k\) (see the pole–zero discussion in the previous section). Hence the quotient of polynomials always reduces to a linear term plus a remainder, making the extraction of a single RC pair straightforward.

Numerical considerations

  • Coefficient growth – Polynomial coefficients explode rapidly (typically beyond 50–100 terms). Use arbitrary-precision arithmetic once double-precision runs out of mantissa bits.

  • Conditioning – Rescale \(s\) (e.g. by the dominant time-constant) before the first division to keep coefficients near unity.

When to stop early?

If only the front section of the structure function is required, terminate the division once the cumulative resistance exceeds the target layer. The remaining high-order Foster branches can stay in parallel; accuracy at long times is preserved while numerical effort is reduced.

Summary

The Foster-to-Cauer transformation is nothing more than repeated Euclidean division of two polynomials whose degrees differ by one. Each quotient yields one thermal resistance and one thermal capacitance; the remainder becomes the dividend of the next step. Starting from the positive-real Foster impedance guarantees convergence and keeps every intermediate network physically meaningful.