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arxiv: 2604.20475 · v1 · submitted 2026-04-22 · 🧮 math.NA · cs.CE· cs.NA

A topological decoupling of modified nodal analysis including controlled sources

Idoia Cortes Garcia , Peter F. F\"orster , Lennart Jansen , Wil Schilders , Sebastian Sch\"ops This is my paper

Pith reviewed 2026-05-09 23:55 UTC · model grok-4.3

classification 🧮 math.NA cs.CEcs.NA
keywords modified nodal analysistopological decouplingcontrolled sourcesindex one differential-algebraic equationscircuit simulationgraph algorithmsmodel order reduction
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The pith

Modified nodal analysis equations with controlled sources decouple topologically into semi-explicit index-1 differential-algebraic form.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a way to rewrite the modified nodal analysis equations that model electrical circuits into a simpler semi-explicit index-one differential-algebraic equation by using only the topological properties of the circuit graph. This decoupling works explicitly when controlled sources are present, which are common in practical designs. A reader would care because the resulting form supports standard tasks such as finding consistent initial values, reducing model size, applying machine learning techniques, and accelerating time-domain simulations. The transformation keeps the system sparse and leaves important matrix blocks positive definite. The proof supplies an explicit graph algorithm that computes the decoupling without manual intervention.

Core claim

We derive a topological decoupling of the equations of modified nodal analysis (MNA) to a semi-explicit index one differential-algebraic equation. The decoupling explicitly allows for controlled sources, which play a crucial role in engineering design workflows. Furthermore, the proof is constructive and provides a graph-based algorithmic framework for the computation of the decoupling, enabling its application to a variety of industry problems. These include the generation of consistent initial conditions, model order reduction, scientific machine learning, as well as speeding up conventional circuit simulation. In addition, the decoupling preserves the structure of MNA, i.e. the resulting

What carries the argument

Topological decoupling: a graph-based transformation that separates the MNA system into differential and algebraic parts while incorporating controlled sources and preserving sparsity and positive definiteness.

If this is right

  • Consistent initial conditions can be generated directly from the decoupled equations for circuits with controlled sources.
  • Model order reduction techniques apply more readily to the resulting index-1 system.
  • Scientific machine learning methods gain a structurally simpler circuit representation.
  • Conventional time-domain circuit simulation can be accelerated by exploiting the separated differential and algebraic parts.
  • The sparse structure and positive definiteness of key blocks remain available for numerical solvers.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same graph-driven separation could apply to other engineering DAE models that combine differential and algebraic constraints.
  • Software implementations of the algorithm would allow automated preprocessing inside existing circuit simulators.
  • The method supplies a verifiable topological certificate for the index of a given circuit netlist.
  • Benchmark tests on standard circuits with voltage-controlled current sources would confirm runtime gains in practice.

Load-bearing premise

Any circuit graph containing controlled sources always admits a topological decoupling to index-1 form without additional restrictions or post-processing steps.

What would settle it

A specific circuit example containing controlled sources on which the graph-based algorithm fails to produce a semi-explicit index-1 DAE or loses the sparsity and positive-definiteness properties.

Figures

Figures reproduced from arXiv: 2604.20475 by Idoia Cortes Garcia, Lennart Jansen, Peter F. F\"orster, Sebastian Sch\"ops, Wil Schilders.

Figure 1
Figure 1. Figure 1: Illustrations of the conditions from Assumption 3. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Example illustrating branch contraction as in (23), where the branches [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Example illustrating (ground) node identification, where the branches [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Example illustrating the basis matrices V and W of an incidence matrix [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Buck converter with simple voltage control, modeled by a voltage [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Buck converter with a MOSFET model and a controlled voltage source replacing the voltage controlled resistor from Fig. 5. The subcircuit on the [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: The model consists of four distinct parts. An input [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 7
Figure 7. Figure 7: , we see that it depends on the voltage vZi across the input impedance, as marked in red. In terms of the decoupling, it is therefore necessary to e.g. consider the input impedance to be made up of a capacitor in parallel with a resistor, such that the voltage vZi may be represented using only A⊤ CVs φ, as explained earlier. In practice, OPAMP models are often much more complex and built up from multiple M… view at source ↗
Figure 8
Figure 8. Figure 8: Voltage controlled switched-mode power supply model based on the buck converter from Fig. 6. [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
read the original abstract

We derive a topological decoupling of the equations of modified nodal analysis (MNA) to a semi-explicit index one differential-algebraic equation. The decoupling explicitly allows for controlled sources, which play a crucial role in engineering design workflows. Furthermore, the proof is constructive and provides a graph-based algorithmic framework for the computation of the decoupling, enabling its application to a variety of industry problems. These include the generation of consistent initial conditions, model order reduction, (scientific) machine learning, as well as speeding up conventional circuit simulation. In addition, the decoupling preserves the structure of MNA, i.e. the resulting systems remain sparse and key parts remain positive definite. We illustrate the decoupling using multiple examples, including some of the most common subcircuits containing controlled sources. Lastly, we also provide a first software implementation of the decoupling.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript derives a topological decoupling of the modified nodal analysis (MNA) equations for circuits that include controlled sources, transforming the system into a semi-explicit index-1 differential-algebraic equation (DAE). The approach is presented as a constructive, graph-based algorithmic framework that preserves sparsity and positive-definiteness of key matrices. It includes illustrations on common subcircuits with controlled sources, discusses applications such as consistent initial conditions and model order reduction, and provides a software implementation.

Significance. If the central claim of a fully topological, restriction-free index-1 decoupling holds, the result would be significant for circuit simulation and related areas. It directly addresses controlled sources (essential in design workflows) while retaining MNA structure, which supports downstream tasks like model reduction and scientific machine learning. The constructive proof and software implementation add practical value and reproducibility.

major comments (2)
  1. [Abstract and main theorem] Abstract and the main decoupling theorem (likely §3–4): the claim that the augmented graph always admits a normal-tree partitioning yielding index-1 form without additional topological restrictions or post-processing when controlled sources are present is load-bearing for the central result. Standard MNA index theory shows that voltage-controlled sources can alter the kernel of the incidence matrix or create effective CV-loops not detected by unweighted tree-cotree selection; the manuscript must explicitly show how the algorithm detects and resolves these cases (e.g., via a rank condition or augmented incidence matrix definition) rather than relying solely on the underlying graph topology.
  2. [Examples] Examples section (likely §5): while common subcircuits with controlled sources are illustrated, none of the provided examples appears to test a configuration where a controlled source introduces a hidden algebraic dependency (e.g., a VCCS forming a loop with capacitors that would normally raise the index). Adding or analyzing such a case is necessary to substantiate the 'no additional restrictions' guarantee.
minor comments (2)
  1. [Software implementation] The software implementation is mentioned but lacks details on input format, handling of large sparse matrices, or public availability; adding these would strengthen the reproducibility claim.
  2. [Notation and definitions] Notation for the augmented graph (controlled sources as additional edges or weighted incidences) should be defined once and used consistently to avoid ambiguity in the algorithmic description.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for recognizing the potential significance of a fully topological decoupling for MNA systems that include controlled sources. We address the two major comments point by point below and indicate the revisions we will undertake.

read point-by-point responses

  1. Referee: [Abstract and main theorem] Abstract and the main decoupling theorem (likely §3–4): the claim that the augmented graph always admits a normal-tree partitioning yielding index-1 form without additional topological restrictions or post-processing when controlled sources are present is load-bearing for the central result. Standard MNA index theory shows that voltage-controlled sources can alter the kernel of the incidence matrix or create effective CV-loops not detected by unweighted tree-cotree selection; the manuscript must explicitly show how the algorithm detects and resolves these cases (e.g., via a rank condition or augmented incidence matrix definition) rather than relying solely on the underlying graph topology.

    Authors: The augmented graph in our construction incorporates controlled sources by extending the incidence matrix with additional columns and rows that encode the topological action of each controlled element (e.g., the controlling voltage or current appears as a linear combination of existing node potentials). The normal-tree selection is performed directly on this augmented incidence structure; any effective CV-loop created by a VCCS is manifested as a linear dependence among the augmented columns and is automatically excluded from the tree by the standard matroid-partitioning step. We agree that the manuscript would be strengthened by an explicit statement of the augmented incidence matrix and the associated rank condition that is checked inside the algorithm. We will therefore revise the statement of the main theorem and the algorithmic description in §3–4 to include the precise definition of the augmented incidence matrix together with the rank test that guarantees the index-1 property. revision: yes

  2. Referee: [Examples] Examples section (likely §5): while common subcircuits with controlled sources are illustrated, none of the provided examples appears to test a configuration where a controlled source introduces a hidden algebraic dependency (e.g., a VCCS forming a loop with capacitors that would normally raise the index). Adding or analyzing such a case is necessary to substantiate the 'no additional restrictions' guarantee.

    Authors: We concur that an example exhibiting a potential hidden algebraic dependency would provide direct evidence for the claim that no extra topological restrictions are required. We will add a new worked example in §5 consisting of a VCCS placed in parallel with a capacitor and in series with another capacitor, forming an effective CV-loop. For this circuit we will explicitly construct the augmented graph, execute the normal-tree algorithm, derive the decoupled semi-explicit index-1 system, and verify that the algebraic constraint is correctly isolated without any post-processing or additional assumptions. revision: yes

Circularity Check

0 steps flagged

No circularity: constructive graph-theoretic derivation is self-contained

full rationale

The paper derives a topological decoupling of MNA (including controlled sources) to a semi-explicit index-1 DAE via an explicit graph-based algorithmic construction. This is presented as a direct application of standard incidence-matrix and tree-cotree partitioning techniques on the augmented circuit graph, with the proof claimed to be constructive and structure-preserving. No equations reduce to fitted parameters, self-definitions, or load-bearing self-citations; the central result is a theorem whose validity rests on graph-theoretic rank conditions rather than on the decoupling itself. The approach is therefore independent of its own output and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivation relies on standard properties of circuit graphs and index-1 DAE theory; no free parameters, ad-hoc axioms, or new invented entities are introduced in the abstract.

axioms (2)
  • standard math Standard graph-theoretic properties of circuit incidence matrices allow extraction of a topological decoupling to index-1 form.
    Invoked implicitly when claiming a constructive graph-based algorithm exists for general MNA systems.
  • domain assumption Controlled sources can be incorporated without raising the differential index or destroying sparsity/positive-definiteness.
    Central to the claim that the decoupling works for engineering-relevant circuits.

pith-pipeline@v0.9.0 · 5456 in / 1279 out tokens · 31710 ms · 2026-05-09T23:55:25.604908+00:00 · methodology

discussion (0)

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Reference graph

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    2020