⚡ electrical engineering

Stationarity-Based Reformulation of Fibonacci-Based Nearest-Level Modulation for a Nine-Level Asymmetric Cascaded H-Bridge Inverter

This paper demonstrates that the previously iterative Fibonacci-based nearest-level modulation for a nine-level asymmetric cascaded H-bridge inverter can be analytically reduced to a scalar fixed-point problem, proving that the rational ratio 13/55 yields a stationary solution with total harmonic distortion (8.9023%) virtually identical to the original 60-iteration result.

Original authors: Anderson William Simões Borges, Sérgio Vidal Garcia Oliveira, Daniel Gustavo Castellain

Published 2026-09-23

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CC BY 4.0

Original authors: Anderson William Simões Borges, Sérgio Vidal Garcia Oliveira, Daniel Gustavo Castellain

Original paper licensed under CC BY 4.0 (https://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

In the world of high-power electricity, getting energy from a battery or a solar panel to a motor or a city grid often involves a tricky balancing act. The goal is to create a smooth, flowing wave of voltage that mimics the natural rhythm of the power grid. However, the devices that do this work, known as inverters, are built from switches that can only turn on or off. When these switches chop up the electricity, the result is often a jagged, stepped waveform that looks more like a staircase than a smooth curve. These jagged edges create unwanted noise in the system, called harmonic distortion, which can overheat equipment and waste energy. To fix this, engineers use a method called nearest-level modulation. Imagine trying to draw a smooth curve using only a ruler and a set of fixed-height steps; the challenge is to decide exactly where to place each step so the resulting shape stays as close to the smooth curve as possible. The fewer steps you take, the more jagged the line, but taking too many steps makes the machine switch on and off too fast, wasting energy and creating heat. Finding the perfect balance is a complex puzzle that engineers have been solving for decades.

A recent study by researchers at the Universidade do Estado de Santa Catarina in Brazil revisits a specific solution to this puzzle that was proposed for a nine-step inverter. The original method, developed by Kumar and colleagues, used a clever, step-by-step search to find the best places for the switches to change state. They started with a standard arrangement and then tweaked four specific angles one by one, checking the quality of the wave after every tiny adjustment. After sixty rounds of these manual tweaks, they arrived at a set of angles that produced a very clean wave with low distortion. The researchers in the new study asked a simple but profound question: was this long, sixty-step search actually necessary? They wondered if the final answer they found was just a lucky guess or if there was a hidden, simpler rule governing the whole process that the search had stumbled upon without realizing it.

To find the answer, the team took the final set of angles from the original study and worked backward. They discovered that the four angles, which looked like independent numbers, were actually all connected by a single, simple relationship. It turned out that the entire complex search could be described by just one number. When they tested this idea, they found that if you force all four angles to follow this single rule, the quality of the wave remains exactly the same as the result from the long search. The researchers proved mathematically that for this specific type of inverter, any point where the wave quality is at its best must follow this one-number rule. This means the original sixty-step search was essentially walking in circles around a path that could have been drawn with a single straight line. The complex, multi-dimensional problem was actually a one-dimensional problem all along.

The team then looked for a simple, whole-number fraction that could represent this single rule, hoping to find a connection to the famous Fibonacci sequence, a pattern of numbers where each number is the sum of the two before it. They found that the fraction thirteen divided by fifty-five matched the optimal setting almost perfectly. Using this simple fraction, they could calculate the exact switch positions without needing to run any complex searches or simulations. When they built a computer model of the inverter to test this new, direct method, it produced a wave with a distortion level of 8.9023 percent. This was virtually identical to the 8.9033 percent achieved by the original, laborious sixty-step search. The difference was so small it was barely measurable, proving that the new method reached the same destination without the long journey.

The significance of this finding is not that the new method makes the electricity cleaner than before; the original method was already excellent. Instead, the discovery changes how engineers understand the problem. It shows that the search for the best switch positions was structurally redundant. The researchers demonstrated that the complex, multi-step optimization was unnecessary because the solution was constrained to a simple, predictable path. By identifying this hidden simplicity, they replaced a tedious, trial-and-error process with a direct calculation. This does not mean the original researchers were wrong, but rather that their final result was part of a much larger, simpler pattern that had been waiting to be recognized. The study confirms that for this specific nine-level inverter, the best way to control the switches is not a complex negotiation between four different variables, but a single, elegant rule derived from a simple fraction.

Technical Summary: Stationarity-Based Reformulation of Fibonacci-Based Nearest-Level Modulation for a Nine-Level Asymmetric Cascaded H-Bridge Inverter

Problem Statement
This study revisits a previously proposed Fibonacci-series-based Nearest-Level Modulation (F-NLM) strategy for a nine-level asymmetric Cascaded H-Bridge (CHB) inverter with a 1:3 DC-source ratio. The original method, reported by Kumar et al., utilized a sequential, iterative search to adjust four switching angles ( $\alpha_1$ to $\alpha_4$ ) to minimize Total Harmonic Distortion (THD). The process involved perturbing angles in steps associated with Fibonacci numbers over 60 iterations, culminating in a specific set of angles yielding an analytical THD of approximately 8.9028%.

The unresolved question addressed in this paper is whether the final solution found by this iterative procedure represents a genuinely four-dimensional optimization result or if the angles are constrained by a lower-dimensional structure inherent to the THD objective function. Specifically, the authors investigate whether the four switching angles remain independent at the interior stationary solution or if they satisfy a latent scalar relationship that renders the multi-angle search structurally redundant.

Methodology
The authors employ a combination of inverse mapping, analytical derivation, and numerical verification:

Inverse Mapping: The final switching angles reported by Kumar et al. were inverted using the standard NLM sinusoidal relation ( $\alpha_i = \sin^{-1}((i-0.5)q)$ ). This process calculated an effective scalar parameter $q_i$ for each angle. The dispersion of these $q_i$ values was analyzed across the reported iterations to determine if they converged toward a single scalar value.
Stationarity Analysis: The authors derived the necessary conditions for the interior stationary points of the analytical THD criterion used in the original study. They formulated the THD function in terms of angle aggregates and applied calculus to find the gradient conditions ( $\partial D/\partial \alpha_i = 0$ ).
Deterministic Reformulation: Based on the derived stationarity conditions, the authors proposed a direct, non-iterative formulation. They identified a finite Fibonacci ratio ( $q_F = 13/55$ ) as a rational surrogate for the optimal scalar $q$ . This ratio was used to generate switching angles directly without iterative search.
Numerical Verification: To validate the analytical findings, the study utilized PSIM simulations. Three modulation strategies were tested on an identical nine-level CHB model: (i) conventional NLM, (ii) the final iterative angles from Kumar et al., and (iii) the proposed deterministic $q_F = 13/55$ formulation. The simulations used ideal switching devices to isolate the modulation effects from hardware non-idealities.

Key Contributions
The paper makes four primary contributions, emphasizing structural characterization over new performance metrics:

Inverse Mapping Evidence: The study demonstrates that the final reported angles from the iterative search map to a scalar region ( $q \approx 0.2368$ ) with negligible dispersion (0.055%), suggesting the solution lies on a one-dimensional manifold rather than a four-dimensional space.
Stationarity Theorem: The authors prove (Theorem 1) that for the specific analytical THD functional used, every interior stationary point must satisfy the relation $\sin(\alpha_i) = (i - 0.5)q$ . This mathematically reduces the multi-angle stationary problem to a scalar fixed-point condition, regardless of the number of levels $n$ .
Direct Deterministic Synthesis: The paper introduces a direct Fibonacci-based realization using the ratio $q_F = 13/55$ . This provides a compact rational representation of the optimal scalar, allowing for the immediate calculation of switching angles without the need for the 60-step iterative search.
Dimensionality Reduction: The work clarifies that the sequential multi-angle search in the original method was structurally redundant for finding the interior stationary solution, as the objective function itself constrains the solution to a scalar trajectory.

Results

Analytical Comparison: The deterministic formulation using $q_F = 13/55$ yields an analytical THD of 8.902300%. This is nearly identical to the low-THD stationary solution found via unrestricted numerical minimization ( $q^* \approx 0.236386$ , THD $\approx$ 8.902299%) and the final iterative result from Kumar et al. (8.902832%). The difference between the deterministic and iterative results is approximately 0.0005 percentage points.
Simulation Verification: PSIM simulations at 50 Hz confirmed the analytical findings. The deterministic $q_F$ formulation produced a THD of 8.9023%, while the published iterative angles produced 8.9033%. Both represented a reduction of approximately 4.89% compared to conventional NLM (9.3602%).
Voltage Utilization: Both the iterative and deterministic methods increased the normalized fundamental voltage component by approximately 3.5% compared to conventional NLM, a geometric consequence of the optimized transition timing.

Significance and Claims
The authors explicitly state that the primary contribution is not a further reduction in THD beyond what was achieved by the original iterative method. Instead, the significance lies in the characterization and dimensional reduction of the stationary set.

The paper claims that the "near-identical" distortion values between the deterministic and iterative approaches serve as evidence of equivalence, proving that the complex iterative search can be bypassed in favor of a direct, deterministic calculation based on the Fibonacci ratio $13/55$ . The work reframes the problem from a multi-variable optimization task to a scalar fixed-point problem, offering a more transparent mathematical understanding of the F-NLM behavior for the nine-level asymmetric CHB inverter. The authors note that these results are restricted to the specific analytical THD functional and ideal waveform assumptions considered in the study.

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