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REVIEW 3 major objections 4 minor 26 references

State space modeling of RLC ladder circuits

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Three RLC ladder circuit types get exact closed-form state-space models, with system matrices A_I and A_II computed analytically for non-uniform component values.

desk verdict The general state-space derivation for these RLC ladders is sound, but the printed equal-component formulas (34a–c) have sign errors that make the models unstable as written; it deserves peer review after the errors are fixed. read the letter →

arxiv 2608.01051 v1 pith:ZZM5HYFF submitted 2026-08-02 cs.CE cs.SYeess.SY

Stephan Scholz This is my paper · ORCID
classification cs.CEcs.SYeess.SY
keywords statespacemodelRLCladdercircuitsnodalanalysissecond-ordersystemlineartime-invariantmatrixinversefinitedifferencestenciltelegrapher'sequation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper derives exact linear time-invariant state space models for three RLC ladder configurations: shunts made of capacitors, resistors, and inductors. After writing the circuit equations as a second-order system $M\ddot z+D\dot z+Sz=G\mu$, it shows that the state-space submatrices are $A_I=-M^{-1}S$ and $A_{II}=-M^{-1}D$, and that both can be written in closed form because $M$ is either an upper triangular matrix or a finite-difference stencil. This turns potentially large ladder networks into ordinary differential equations with explicit matrix structure, bypassing the differential-algebraic equations used by general circuit simulators. Simulations illustrate charging/discharging for the capacitor-shunt ladder and oscillations for the other two.

What carries the argument

The mechanism is the closed-form inversion of the two matrix shapes appearing in the second-order system. The upper triangular matrix $T$ in Eq. (21) has inverse $T^{-1}$ with entries $1/(a_nb_n)$ on the diagonal and $-1/(a_{n+1}b_n)$ on the superdiagonal; the finite-difference stencil $F$ in Eq. (22) has the lower triangular all-ones matrix as inverse. Products $-T^{-1}F$, $-T^{-1}\tilde T$, and $-F^{-1}T$ then generate the tridiagonal $L$, triangular $P$, and dense $Q$ matrices that enter $A_I$ and $A_{II}$, carrying the dynamics of each ladder type.

What would settle it

Build a two-stage ladder ($N=2$) with equal component values, drive it with a smooth input, and compare the state-space solution of Eq. (26) using $A_I=-M^{-1}S$ and $A_{II}=-M^{-1}D$ against a direct numerical solution of the circuit's component laws; any disagreement for positive $r,\ell,c$ would disprove the closed-form construction.

Watch

Extended reading notes

Core claim

The central object is the second-order system $M\ddot z+D\dot z+Sz=G\mu$; the discovery is that for the three ladder types the mass matrix $M$ is invertible in closed form. When $M$ is upper triangular (types 1 and 2), $M^{-1}$ is a bidiagonal finite-difference matrix; when $M$ is the finite-difference stencil $F$ (type 3), $M^{-1}$ is a lower triangular matrix of ones. Multiplying $M^{-1}$ by the damping and stiffness matrices produces a tridiagonal matrix $L=-T^{-1}F$, a triangular matrix $P=-T^{-1}\tilde T$, and a dense matrix $Q=-F^{-1}T$, which form the blocks $A_I$ and $A_{II}$ of the system matrix $A$. This yields an ODE state-space model $\dot x=Ax+B\mu$, $y=C^Tx$ for arbitrary posit

Load-bearing premise

The derivation assumes all voltages and currents are twice continuously differentiable on $[0,T]$ (stated in Section 1), because each ladder type requires differentiating Kirchhoff's voltage law once or twice; the step and derivative-based inputs used in the simulations do not lie in this regularity class, so the closed-form ODEs are classically exact only for smooth inputs.

Editorial extensions

If this is right

  • The state-space models are exact for non-uniform (position-dependent) resistor, inductor, and capacitor values, with no numerical matrix inversion because $M^{-1}$ is known symbolically.
  • Because the closed-form blocks are explicit, spectral analysis, controllability/observability checks, and model reduction can be performed on the matrix structure rather than on black-box numerical matrices.
  • The two matrix shapes mirror discrete integration (triangular) and differentiation (finite-difference stencil), so each ladder type is interpretable as a spatially discretized telegrapher- or diffusion-type system.
  • Simulated step and sweep responses show a clear behavioral split: the capacitor-shunt ladder charges and discharges, while the resistor- and inductor-shunt ladders oscillate with distinct transient signatures.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same inversion strategy should extend to other cascade topologies, including the parallel-in-main-branch ladders the author lists as future work, since they will likely produce the same two matrix shapes.
  • The tridiagonal and dense blocks resemble discrete Laplacian and cumulative-sum operators, suggesting a direct passage to PDE limits and to structure-preserving model reduction for very long ladders.
  • For non-smooth inputs like the step used in Section 3.1, the classical $C^2$ smoothness assumption is violated; a weak or distributional interpretation of the ODEs would connect the derivation to the simulated responses.
  • The equal-component simplifications make the blocks very sparse or diagonal, which could make very large $N$ simulations and real-time implementations especially cheap.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper derives linear time-invariant state-space models for three RLC ladder topologies (capacitor, resistor, and inductor shunts) by applying Kirchhoff's laws and operator notation. The second-order equations are assembled as M z'' + D z' + S z = G mu, with upper-triangular and finite-difference matrix structures. The paper then forms the first-order system x' = A x + B mu, with A_I = -M^{-1} S and A_II = -M^{-1} D, and gives closed-form expressions for the submatrices in both the non-uniform and equal-component cases. Simulations for step and chirp inputs are presented to illustrate the behavior of the three ladder types, and Julia code is provided.

Significance. If the printed formulas are corrected, the paper would be a useful self-contained reference: it gives explicit, parameter-free closed-form state-space matrices for a practically relevant class of ladder circuits, without fitting or numerical inversion. The matrix-shape analysis (upper-triangular versus finite-difference) provides structural insight and the equal-component simplification is pedagogically valuable. The availability of Julia code and reproducible simulations are additional strengths. However, the central closed-form formulas in the equal-component section contain sign errors that make the printed system matrices unstable and inconsistent with the paper's own derivation and simulations; these errors must be fixed before the results can be used as stated.

major comments (3)
  1. [Section 2.2, Eq. (34a-c)] The equal-component state-space formulas have load-bearing sign errors. For Type 1, Eq. (23a) gives M=(lc)T~, D=(rc)T~, S=F, so Eq. (25) gives A_II = -M^{-1}D = -(r/l)I, but Eq. (34a) prints A_II = +(r/l)I. For Type 2, Eq. (23b) gives A_I = -M^{-1}S = -(1/(lc))I, but Eq. (34b) prints A_I = +(1/(lc))I. For Type 3, Eqs. (23c) and (25) imply A_I = -(1/(lc))F^{-1}T~ and A_II = -(r/l)F^{-1}T~; with Q = -F^{-1}T~ these are negative multiples of Q, but Eq. (34c) prints positive multiples of the positive matrix Q~. Consequently each printed A has positive-real-part eigenvalues, contradicting the dissipative second-order equations and the stable simulations in Figs. 6-7. A reader copying Eq. (34) cannot reproduce the claimed dynamics.
  2. [Section 2.2, Inverse Finite Difference Stencil] The definition of Q is internally inconsistent. The text defines Q := -F^{-1}*T, but then gives entries q_{n,k} = b_k * sum_{i=1}^{min(n,k)} a_i, which are the entries of F^{-1}T, not of -F^{-1}T. Equation (33) and Figure 5 likewise show a positive matrix labeled Q. This sign inconsistency propagates directly into the Type 3 state-space matrices in Eq. (34c). The general non-uniform Type 3 formula in this subsection is therefore also ambiguous and needs to be corrected or clarified.
  3. [Section 3.1 and Section 1] The paper assumes in Section 1 that all voltages and currents are twice continuously differentiable, and the derivation for ladder types 2 and 3 requires differentiating the Kirchhoff voltage law once or twice. The step input in Eq. (35), however, is discontinuous, and for types 2 and 3 the state-space input mu is the first or second derivative of u, so the classical smoothness assumption is violated at the switching times. Footnote 8 notes the anti-derivative interpretation but does not resolve the regularity mismatch. The authors should either state a weak/Caratheodory interpretation of the models with discontinuous inputs, or replace the step by a smooth approximation in the simulations.
minor comments (4)
  1. [Eq. (18)] The summation in the first-circuit equation for ladder type 3 is written with lower index n, which is undefined at that point; it should be k = 1 (or n = 1).
  2. [Section 2.2, after Eq. (26a)] The sentence 'we note secondly z''(t) = A_I z(t) + A_II z'(t) + G z(t)' should read '... + tilde G mu(t)', not '... + tilde G z(t)'. Also, 'in vector B we mean a row vector 0 = 0_N' should be 'zero column vector'.
  3. [Eq. (28)] The displayed expression for gamma_n is typeset confusingly: gamma_n = -[alpha_n + beta_n] = -1/(a_n b_n) + -1/(a_{n+1} b_n). This should be -1/(a_n b_n) - 1/(a_{n+1} b_n).
  4. [Introduction and Figure 2] The text refers to 'Fig.1(a)' and 'Fig.1(b)' when describing the three ladder types, but the circuit types appear in Figure 2; please unify the figure references.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained from Kirchhoff's laws with no fitted parameters, and the only self-citation is motivational.

full rationale

The paper's central derivation builds the second-order system M z'' + D z' + S z = G mu directly from Kirchhoff's voltage and current laws, with the matrices M, D, S read off from the differential equations for each ladder type (Eqs. (9)-(18), Table 2). The state-space matrices are then defined as A_I = -M^{-1} S and A_II = -M^{-1} D (Eq. (25)), and the paper computes the inverse matrices in closed form for the two occurring shapes (triangular and finite-difference stencil). Nothing is fitted to data, and no quantity is defined in terms of the target result. The equal-component formulas in Eq. (34) are algebraic consequences of the same definitions; whether they are printed correctly is a correctness question, not a circularity question. The only work by the authors cited, [18], is referenced as an example of dynamic mode decomposition applied to ladder circuits and is not used to justify any derivation step or to rule out alternatives. No uniqueness theorem is imported, and no ansatz is smuggled in through a citation. The regularity limitation (smoothness assumed in Section 1 but step inputs used in Section 3.1 for ladder types 2 and 3) concerns the interpretation of the ODE models for discontinuous inputs; it does not make the derivation circular. The paper is not self-referential in a load-bearing way, so the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No new physical entities or fitted parameters are introduced. The derivation rests on standard circuit laws, ideal component assumptions, and a smoothness assumption that is formally violated by the discontinuous test inputs.

assumptions (4)
  • domain assumption Circuit elements are ideal and nominal: resistors have pure resistance, inductors pure inductance, capacitors pure capacitance.
    Section 1: 'we consider nominal electronic components, i.e. a resistor has only a pure resistance without inductance or capacitance'.
  • domain assumption All voltages and currents are at least twice continuously differentiable on [0,T] and integrable.
    Section 1: 'we consider the currents and voltages to be sufficiently smooth, e.g. twice smoothly differentiable i_n, v_n, z_n in C^2([0,T])'.
  • domain assumption Kirchhoff's voltage and current laws hold for the ladder network.
    Used throughout Section 1 to derive Eqs. (2)-(18).
  • standard math Positive component values r_n, l_n, c_n > 0 guarantee the triangular mass matrices are invertible.
    Positive diagonal products make the upper triangular matrices nonsingular, used in Section 2.2 for M^{-1}.

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Cite this review

Pith. "Pith review of State space modeling of RLC ladder circuits." pith.science (2026). https://pith.science/paper/ZZM5HYFF

@misc{pith2026260801051,
  author       = {Pith},
  title        = {Pith review of: State space modeling of RLC ladder circuits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZZM5HYFF}},
  note         = {Machine review of arXiv:2608.01051}
}
read the original abstract

Large-scale electrical circuits span a wide range of research topics from highly integrated circuits in microelectronics up to power grids for cities and countries. As the complexity increases significantly by each additional component, we find a need to describe large-scale circuits in a manner easy to understand. In this article, we study cascades of simple circuits consisting of resistors, capacitors and inductors, which we call ladder circuits. Such electrical ladders are common in electronics to design filters and they are also used in other disciplines, for example to describe diffusion or wave phenomena. In particular, we derive linear time-invariant state space models for three simple ladder types and we discuss the specific matrix structures of the large-scale second-order differential equations. Furthermore, we exemplify our findings with simulations to unveil the intrinsic dynamical behavior of each ladder type.

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