Lossless MTL Matrix to Pi-Ladder

Procedure for lowering one declared lossless two-line LC matrix into a finite coupled pi ladder.

Node role: Procedure.

Reader task: Convert one eligible lossless two-line \(\mathbf L,\mathbf C\) artifact into a finite coupled \(\pi\) ladder.

Contract Value
Operation Multiply per-unit-length entries by section length and stamp the corresponding branch elements.
Input Labeled \(2\times2\) \(\mathbf L\) and Maxwell \(\mathbf C\), line orientation, total length, and section count.
Boundary Both matrices use the same terminal basis, reference conductor, units, frequency, and provenance.
Output A finite ladder whose physical length and reconstructed Maxwell capacitance match the input.
Why Provide a circuit model for distributed simulation without changing matrix meaning.
Concept details Multiconductor RLGC Matrices owns basis, modal, Maxwell, sign, and artifact semantics.

Why the ladder has these elements

A Maxwell capacitance matrix represents energy in terminal voltage differences, while an inductance matrix relates oriented series currents to flux linkage. The ladder must lower these two meanings differently. Endpoint half-shunts keep one section’s electric energy from being counted twice at an internal boundary.

This-page derivation / engineering choice. Start from the explicitly lossless, common-basis matrix model in the matrix-semantics page. Integrating one section gives its element values; symmetric endpoint splitting selects a finite approximation, not a universal circuit layout.

1 — Choose the section length

For total coupled length \(\ell\) and \(N\) equal sections,

\[ \Delta z=\frac{\ell}{N}. \]

The derivation assumes the declared lossless, common-basis input model. Use the matrix interpretation checklist to reconstruct that model; consumer eligibility remains with its owning contract. Here \(\ell\) is only the line-owned region. If an end coupler or attachment is extracted separately, stop the ladder at its declared cut plane; do not include the same physical length in both models. See localized electrostatic component modeling.

2 — Convert matrix entries into section elements

This-page derivation. Multiply each per-length coefficient by the section length, and expand Maxwell energy into ground and cross-line branches. The series mutual sign remains attached to the current orientation.

For one section,

Physical element Section value
line-1 series self inductance \(L_{11}\Delta z\)
line-2 series self inductance \(L_{22}\Delta z\)
mutual inductance \(L_{12}\Delta z\), with the declared branch-orientation sign
line-1 capacitance to reference \((C_{11}+C_{12})\Delta z\)
line-2 capacitance to reference \((C_{22}+C_{21})\Delta z\)
cross-line capacitance \(-C_{12}\Delta z\) for a reciprocal pair

The row sums are the shunts to the reference conductor. The negative Maxwell off-diagonal becomes the positive cross-line capacitor.

3 — Stamp the pi section

For each section, stamp

\[ \boxed{ \mathbf C_{\rm left} =\mathbf C_{\rm right} =\frac{\Delta z}{2}\mathbf C_{\rm Maxwell}, \qquad \mathbf L_{\rm series}=\Delta z\,\mathbf L. } \]

Convert each endpoint Maxwell matrix into reference shunts and cross-line capacitors using the branch mapping in the matrix-semantics node. The series matrix is one coupled-inductor pair between the left and right line nodes of that section.

At an interior boundary shared by two uniform sections, add the right half of the first section and the left half of the second. Their sum is \(\mathbf C_{\rm Maxwell}\Delta z\). A physical end boundary receives only one half-section stamp for the line-owned region. A separate local component then adds its complete terminal block at that cut plane.

Spatial pairing must match the physical layout:

  • co-directed lines pair corresponding section boundaries;
  • oppositely directed lines pair reversed boundaries and reverse the mutual inductive branch orientation.

4 — Check the lowering

Engineering use. Length, branch reconstruction and orientation are exact accounting statements. Response refinement evaluates the finite approximation; any meaning of “materially” and any tolerance belongs to the receiving task, not a new rule on this page.

The ladder is usable only if all four checks pass:

  1. total section length equals \(\ell\);
  2. reconstructed Maxwell capacitance equals the input section matrix;
  3. branch orientation matches the declared current directions; and
  4. the target response converges as \(N\) increases.

If the response moves materially under refinement, the ladder is not yet a design result.

Field Value
Status Seed
Used by Multiconductor RLGC Matrices as its finite-circuit lowering procedure.
Review need Confirm the orientation map and refinement evidence against the current Workbench consumer.