⚡ electrical engineering

State space modeling of RLC ladder circuits

This paper derives linear time-invariant state space models for three types of RLC ladder circuits, analyzes their specific large-scale matrix structures, and uses simulations to reveal their intrinsic dynamical behavior.

Original authors: Stephan Scholz

Published 2026-08-04

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CC BY 4.0

Original authors: Stephan Scholz

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the world of electricity not as a mysterious force, but as a bustling city of tiny travelers. In this city, resistors are like narrow, bumpy streets that slow down the traffic; capacitors are like parking garages that can store cars (electric charge) and release them later; and inductors are like heavy, spinning flywheels that hate to change their speed, making the traffic flow smooth but sluggish to start or stop. When engineers connect these components in a long line, one after another, they create what's called a "ladder circuit." It looks like a ladder lying on its side, with rungs made of these electrical parts.

Why do we care about these electrical ladders? They are the backbone of everything from the tiny filters in your smartphone that block out static noise, to the massive power grids that keep entire countries lit. However, as these ladders get longer and more complex, predicting how electricity will flow through them becomes a mathematical nightmare. It's like trying to predict the exact movement of every single car in a traffic jam that stretches for miles. To solve this, scientists use a special mathematical tool called a "state space model." Think of this as a super-organized spreadsheet that doesn't just track where the cars are right now, but also how fast they are going and how they will react to the next green light, allowing us to simulate the entire system on a computer without building a physical ladder of wires.

In this paper, author Stephan Scholz decides to take a closer look at three specific types of these electrical ladders. He isn't trying to build a new circuit or invent a faster battery; instead, he is acting like a cartographer, drawing a new, clearer map of how these systems behave mathematically. The paper focuses on three distinct "flavors" of ladders, distinguished by what component sits on the "rung" (the shunt) of the ladder: one where the rung is a capacitor, one where it's a resistor, and one where it's an inductor.

Scholz's main discovery is that while all three ladders can be described by the same general rules of physics (Kirchhoff's laws), they each have a unique "personality" in their mathematical structure. By translating the messy physics into a clean, organized format called a "state space model," he reveals that each ladder type creates a different pattern of numbers (matrices) inside the computer. For the capacitor ladder, the math looks like a triangle; for the resistor ladder, it looks like a staircase; and for the inductor ladder, it forms a dense, solid block. These shapes aren't just pretty pictures; they tell us exactly how the electricity will move and change over time.

To test his maps, Scholz ran computer simulations using a ladder made of 10 identical sections, with each section containing a 2 Ω resistor, a 3 F capacitor, and a 5 H inductor. He fed these digital ladders two different types of "traffic" signals. First, he gave them a "step" signal, which is like suddenly flipping a switch to turn the voltage on and then off again. The results showed that the capacitor ladder acted like a slow, gentle sponge, slowly charging up and then slowly draining away. However, the resistor and inductor ladders were much more energetic; they started to vibrate and oscillate, swinging back and forth like a pendulum before settling down.

In a second experiment, he used a "sweep" signal, which is like a sound that slowly changes pitch from low to high and back again. Here, the differences became even clearer. The capacitor ladder acted like a sieve, letting the slow, low-pitch signals through while blocking the fast, high-pitch ones. The resistor ladder, however, seemed to get "kicked" into a frenzy whenever the signal changed speed, creating wild vibrations. The inductor ladder was the most direct, passing the signal through with very little filtering, almost as if the electricity was sliding down a smooth slide.

The paper concludes that while these three ladders might look similar at a glance, their internal mathematical structures are fundamentally different, leading to very different behaviors. Scholz notes that this work is just the beginning. He didn't dive deep into analyzing the stability of these systems or how to control them perfectly; those are jobs for future research. Instead, he provided the foundational blueprints—the specific matrix shapes and equations—that allow other scientists and engineers to build better simulations and understand these complex electrical systems without getting lost in the math. The work is purely theoretical and based on these computer simulations, offering a clearer way to see the hidden dynamics of the electrical world.

Technical Summary: State Space Modeling of RLC Ladder Circuits

Problem Statement
Large-scale electrical circuits, ranging from microelectronics to power grids, present significant modeling challenges due to the increasing complexity introduced by each additional component. While traditional methods often rely on frequency-domain descriptions (complex-valued rational functions) or Differential-Algebraic Equations (DAE) via Modified Nodal Analysis (MNA), these approaches can lack simplicity or require specialized numerical solvers (e.g., Backward Differentiation Formula). There is a need for a straightforward, time-domain description of cascaded RLC circuits that yields Ordinary Differential Equations (ODEs) with specific, exploitable matrix structures. This paper addresses the modeling of three specific types of RLC ladder circuits to derive linear time-invariant state space models that avoid the complexity of DAEs.

Methodology
The author employs standard nodal analysis and Kirchhoff's laws to derive systems of linear ODEs for three distinct ladder configurations, where the shunt component varies:

Type 1: Capacitor as the shunt (Main branch: Resistor + Inductor).
Type 2: Resistor as the shunt (Main branch: Inductor + Capacitor).
Type 3: Inductor as the shunt (Main branch: Resistor + Capacitor).

The modeling process involves:

Derivation of Second-Order Systems: By applying Kirchhoff's voltage and current laws, the author formulates the circuit dynamics as a second-order differential equation system: $M \ddot{z}(t) + D \dot{z}(t) + S z(t) = G \mu(t)$ . Here, $z(t)$ represents the vector of shunt voltages. The matrices $M$ (mass), $D$ (damping), and $S$ (stiffness) exhibit specific structures: either upper triangular shapes (representing spatial integration) or finite difference stencils (representing differentiation).
State Space Transformation: The second-order system is reformulated into a first-order state space representation $\dot{x}(t) = A x(t) + B \mu(t)$ . The state vector $x(t)$ is defined as $[z(t), \dot{z}(t)]^\top$ .
Closed-Form Matrix Inversion: To ensure numerical stability and avoid condition number issues associated with inverting matrix $M$ , the author derives closed-form expressions for the inverse matrices. Depending on the ladder type, the resulting sub-matrices ( $A_I$ and $A_{II}$ ) take the form of tridiagonal matrices, diagonal/triangular matrices, or dense symmetric matrices.
Input Signal Handling: The paper notes that the input signal $\mu(t)$ in the state space model differs from the physical input voltage $u(t)$ depending on the ladder type. For Type 1, $\mu(t) = u(t)$ ; for Type 2, $\mu(t) = \dot{u}(t)$ ; and for Type 3, $\mu(t) = \ddot{u}(t)$ .

Key Contributions

Explicit State Space Derivations: The paper provides explicit, closed-form formulations for the system matrices ( $A, B, C$ ) for three specific RLC ladder topologies, covering both non-uniform and uniform component values.
Matrix Structure Analysis: A detailed analysis of the resulting matrix structures is presented. The author identifies that the system matrices are composed of specific patterns (tridiagonal, triangular, dense symmetric) derived from the inverse of triangular matrices and finite difference stencils.
Avoidance of DAEs: By using standard nodal analysis on these specific ladder structures, the work demonstrates a method to obtain pure ODE state space models, bypassing the need for the more complex DAE formulations typically required for general large-scale circuits.
Simulation Framework: The author implements these models in the Julia programming language (using the Ellasy.jl package) to simulate the dynamical behavior of 10-stage ladders under step and symmetric sweep (chirp) inputs.

Results
Simulations were conducted using equal component values ( $R=2\Omega, C=3F, L=5H$ ) for 10 cascaded stages.

Step Response: Type 1 (Capacitor shunt) exhibited classic charging and discharging dynamics. Types 2 and 3 showed oscillatory behavior. Notably, Type 2 displayed long, intensive oscillations at the final shunt, while Type 3's response decayed rapidly to zero.
Sweep Response: Under a symmetric sweep input, Type 1 demonstrated low-pass filter characteristics, attenuating high-frequency oscillations as the signal propagated. Type 2 showed filtering of high-frequency parts at the input but exhibited "kicked" vibrations at intermediate and final stages during signal transitions. Type 3 showed minimal filtering, with the output signal closely resembling the second derivative of the input.
Dynamical Differences: The simulations confirmed that despite using identical component values, the three ladder types possess distinct intrinsic dynamical behaviors, necessitating individual analysis.

Significance and Claims
The paper positions itself as a focused contribution to the mathematical modeling of electrical ladders, specifically targeting the derivation of state space models with specific matrix structures. The author claims that these models provide a simplified alternative to DAE-based approaches for this specific class of circuits. The work serves as a foundation for further research, explicitly noting that it does not perform in-depth eigenvalue analysis, controllability/observability studies, or control design, but rather provides the structural models required for such future investigations. The author also highlights the potential application of these models in scientific machine learning for system identification, referencing prior work where similar ladder structures were used to compute unknown state space models from data. The paper concludes by acknowledging that the study is limited to these three ladder types and excludes parallel connections in the main branch, leaving those configurations for future contributions.

Technical Summary: State Space Modeling of RLC Ladder Circuits

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