⚡ electrical engineering

Simulation of Switching Converters on the Level of Averaged Voltages and Currents

This paper proposes an algorithm for simulating switching converters in both continuous and discontinuous conduction modes by utilizing an averaged circuit model with a "switching cell" concept and reconstructing instantaneous waveforms through quasi-steady-state and linear ripple approximations, as demonstrated on buck, boost, buck-boost, and flyback converters.

Original authors: Aleksandra Lekić, Predrag Pejović

Published 2026-04-22

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CC BY 4.0

Original authors: Aleksandra Lekić, Predrag Pejović

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 you are trying to watch a high-speed movie of a power converter (a device that changes electricity, like the charger in your phone). The problem is, the "movie" is playing at 100,000 frames per second. Every single frame shows a switch flipping on and off, creating tiny, jagged spikes in voltage and current.

If you try to simulate this on a standard computer using traditional methods, it's like trying to count every single grain of sand on a beach to figure out the shape of the dunes. The computer gets overwhelmed, gets stuck (convergence problems), and takes forever to finish the calculation.

The Solution: The "Blurry Photo" Approach

The authors of this paper, Aleksandra Lekić and Predrag Pejović, came up with a clever shortcut. Instead of trying to simulate every single grain of sand (every tiny switch flip), they decided to simulate the average shape of the dune.

Here is how their method works, broken down into simple analogies:

1. The "Switching Cell" (The Magic Box)

Think of a power converter as a complex machine made of different parts. The authors realized that the most chaotic part is the "switching cell"—the specific group of components where the switch flips.

The Analogy: Imagine a busy intersection with traffic lights. Instead of tracking every single car's exact position every millisecond, you treat the intersection as a "Magic Box." You only care about the average flow of traffic coming in and going out.
The Innovation: They created a mathematical "Magic Box" (called a Switching Cell) that describes the average behavior of these switches. This box works whether the traffic is heavy (Continuous Mode) or if the road sometimes empties out completely (Discontinuous Mode).

2. The "Smoothed-Out" Simulation

Once they have this "Magic Box," they simulate the circuit using averaged values.

The Analogy: Instead of drawing a jagged, saw-tooth wave (which represents the real, fast-switching electricity), they draw a smooth, straight line that represents the average energy.
Why it helps: Calculating a smooth line is incredibly fast for a computer. It avoids the "traffic jams" (convergence errors) that happen when computers try to calculate millions of tiny, jagged spikes.

3. Adding the "Ripple" Back In (The Secret Sauce)

You might ask: "If we only look at the average, how do we know the real voltage spikes?"
This is the paper's second trick. After the computer calculates the smooth, average line, the algorithm uses two simple rules to "paint" the details back on:

Quasi Steady State: It assumes the average level is stable for a moment.
Linear Ripple Approximation: It assumes the "spikes" (ripples) on top of the average line are just straight, simple slopes (like a triangle).
The Analogy: Imagine you are drawing a mountain range. First, you draw the smooth, rolling hills (the average). Then, you quickly sketch simple, straight lines on top of the hills to represent the jagged rocky peaks (the ripples). You didn't draw every rock, but you still have a picture that looks like the real mountain.

4. The Result: Fast and Accurate

The authors tested this on common power converters (like Buck, Boost, and Flyback converters).

The Speed: They simulated 5 milliseconds of real-time operation. A traditional computer might take minutes or hours to do this because it's counting every grain of sand. Their method did it in 50 milliseconds (less than a blink of an eye).
The Output: The computer gave them the smooth average line and the reconstructed jagged spikes, along with the highest and lowest points, all in a fraction of a second.

Why Does This Matter?

In the real world, engineers need to design power supplies for everything from laptops to electric cars.

Old Way: Engineers use slow, general-purpose tools that often crash or take too long, especially when the power supply is turning on or off (transients).
New Way: This algorithm is a specialized, high-speed tool. It lets engineers test designs instantly, see how the power behaves, and tweak the numbers without waiting for the computer to "think."

In Summary:
The paper presents a "smart shortcut" for simulating power electronics. Instead of counting every single switch flip (which is slow and confusing), they calculate the average flow first, then use simple geometry to reconstruct the fast details. It turns a slow, messy calculation into a fast, clean, and accurate prediction.

1. Problem Statement

The authors identify a significant bottleneck in the simulation of power electronics circuits using general-purpose circuit simulators (like SPICE). These tools rely on continuous and differentiable device models, whereas power electronic switches exhibit discontinuities in their constitutive equations. This mismatch frequently leads to:

Convergence problems: The numerical solvers struggle to handle the rapid switching transitions.
Inefficiency: Simulating long-term transients or steady-state behavior requires excessive computational time because the solver must resolve every individual switching cycle.
Design limitations: These issues reduce the effectiveness of simulation during the design phase, particularly when analyzing control loops.

The goal of the research is to develop a specialized, fast simulation algorithm that avoids these convergence issues while still providing accurate instantaneous waveform data for both Continuous Conduction Mode (CCM) and Discontinuous Conduction Mode (DCM).

2. Methodology

The proposed algorithm is based on circuit averaging applied at the "switching cell" level, combined with waveform reconstruction techniques.

A. The Switching Cell Concept

The core of the method is modeling the converter not as a full circuit with individual switches, but as a network of "switching cells."

Three-Terminal Device: Switching cells are treated as three-terminal devices (Input, Output, Common).
Families: The paper defines two families:
- Cell A: Used for basic converters (Buck, Boost, Buck-Boost). It includes a synchronous version (two bidirectional switches) and a standard version (switch + diode).
- Flyback Cell: A modified cell incorporating a transformer to handle galvanic isolation.
Averaged Model: Instead of simulating the switching action, the algorithm uses an averaged model of the cell. This model calculates the average currents and voltages over a switching period ( $T_S$ ) based on the duty cycle ( $d$ ) and the operating mode (CCM or DCM).

B. Mathematical Formulation

Discretization: The averaged circuit is discretized using the trapezoidal integration rule. This transforms the differential equations of the circuit into a linear system of algebraic equations ($Ax = z$) for each time step.
Mode Detection: The algorithm predicts the operating mode (CCM vs. DCM) at the beginning of each step based on the inductor current.
- In CCM, the averaged circuit is linear.
- In DCM, the circuit becomes nonlinear, requiring iterative adjustments of the matrix elements within the time step.
Equation System: The system solves for node voltages, source currents, and additional variables required by the switching cell model.

C. Waveform Reconstruction

While the core simulation solves for averaged values, the paper introduces a method to reconstruct instantaneous waveforms:

Quasi-Steady State Approximation: Assumes the average values change slowly compared to the switching frequency.
Linear Ripple Approximation: Assumes the ripple (AC component) of the inductor current and capacitor voltage is linear within the switching period.
Reconstruction: By combining the calculated average values with the linear ripple slopes, the algorithm constructs the instantaneous waveforms, allowing for the extraction of peak-to-peak ripple, RMS values, and minima/maxima.

3. Key Contributions

Unified CCM/DCM Algorithm: Unlike previous averaging methods that often struggled with DCM, this approach generalizes the switching cell concept to handle both modes seamlessly by adjusting the cell equations based on the inductor current zero-crossing.
Switching Cell Generalization: The paper extends the switching cell concept (previously used for basic converters) to include flyback converters with transformers, requiring a specific modification to the cell model to account for galvanic isolation.
Efficient Instantaneous Reconstruction: The method provides a computationally cheap way to generate instantaneous waveforms from averaged data, avoiding the need to simulate every switching edge.
Implementation: The algorithm was implemented in Python (using the matplotlib and PyLab environments), demonstrating that high-level interpreted languages can achieve high performance for power electronics simulation when the underlying mathematical model is optimized.

4. Simulation Results

The authors validated the algorithm by simulating four specific converters:

Buck Converter (Synchronous): Simulated with a controlled switch and a bidirectional switch.
Buck Converter (Diode): Simulated with a controlled switch and a diode.
Flyback Converter: Simulated in both synchronous and diode configurations.

Key Findings:

Accuracy: The simulated average output voltages and inductor currents matched theoretical predictions (e.g., $V_{OUT} = 5V$ for the Buck converter, $V_{OUT} = 20V$ for the Flyback).
Transient Analysis: The algorithm successfully captured startup transients.
Waveform Reconstruction: The reconstructed instantaneous waveforms (inductor current and capacitor voltage) accurately displayed the ripple and matched the expected linear slopes in both CCM and DCM.
Performance: The simulation of a 5ms transient (covering 500 switching periods at 100kHz) completed in 50ms on a standard PC (Intel Core 2 Duo). This represents a significant speedup compared to traditional time-domain simulators that would need to resolve every microsecond of the switching cycle.

5. Significance and Conclusion

The paper presents a robust and efficient alternative to general-purpose circuit simulators for power electronics design.

Speed and Stability: By avoiding the numerical stiffness associated with switching discontinuities, the method eliminates convergence failures and drastically reduces simulation time.
Design Utility: The ability to quickly obtain average values, ripple parameters, and reconstructed waveforms makes it highly suitable for control loop design and optimization.
Future Work: The authors suggest further development in formalizing the equation system generation, improving the accuracy of numerical integration for highly nonlinear circuits, and extending the method to converters with complex feedback control systems.

In summary, this work offers a specialized, mathematically rigorous framework for simulating switching converters that balances computational efficiency with the detailed waveform analysis required for modern power electronics engineering.