⚡ electrical engineering

Simulation of Switching Converters Using Linear Capacitor Voltage and Inductor Current Prediction and Correction

This paper proposes a fast and accurate large-signal simulation algorithm for constant-frequency switching converters that predicts and corrects inductor currents and capacitor voltages to determine the duty ratio, thereby avoiding convergence issues with a fixed number of circuit solutions per switching period.

Original authors: Aleksandra Lekić, Vujo Drndarević

Published 2026-04-22

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Aleksandra Lekić, Vujo Drndarević

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 drive a car that has a very bumpy, jerky engine. Every time you press the gas pedal, the engine revs up and down thousands of times a second. If you tried to film this car with a camera that takes a picture every millisecond, you'd get a massive amount of data, but your computer would crash trying to process all those tiny, rapid movements.

This is exactly the problem engineers face when simulating switching converters (the electronic circuits that power your phone charger, laptop, or electric car). These circuits switch on and off incredibly fast to control voltage. Traditional simulation methods try to calculate every single "on" and "off" switch, which is like trying to count every grain of sand on a beach to measure the beach's weight. It's accurate, but it takes forever and often causes the computer to get stuck (a "convergence problem").

The New Approach: The "Smart Predictor"

The authors of this paper, Aleksandra Lekić and Vujo Drndarević, propose a clever shortcut. Instead of watching every single bump in the road, they decided to look at the average speed of the car and predict where it's going next.

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

1. The "Two-Step Dance" (Prediction and Correction)

Imagine you are playing a game of catch with a friend.

The Prediction (The Throw): You guess where your friend will be in the next second based on where they are now. You throw the ball to that spot.
The Correction (The Catch): Your friend catches the ball, realizes they were actually a little to the left, and tells you, "Hey, I was actually here!" You then adjust your next throw to account for that new information.

The authors' algorithm does exactly this with electricity:

Predict: It guesses what the voltage and current will be at the end of the next cycle.
Calculate: It uses that guess to figure out how much power needs to be switched (the "duty ratio").
Correct: It checks the math, sees if the guess was slightly off, and fixes the numbers before moving to the next cycle.

Because they only do this two times per cycle (once to guess, once to fix), the computer doesn't have to do thousands of calculations. It's like taking a photo of the car's average speed every second instead of a video of every vibration.

2. The "Swing Set" (The Switching Cell)

To make this work, they treat the complex circuit as a simple "Swing Set" (which they call a Switching Cell).

In a real circuit, the electricity flows through switches, diodes, and inductors in complicated patterns.
The authors simplified this into a standard "block" with a few simple rules. Whether the swing is moving fast (Continuous Mode) or stopping briefly (Discontinuous Mode), the math for the "Swing Set" remains the same. This avoids the computer getting confused about which "mode" the circuit is in, which is a common reason simulations fail.

3. The "Traffic Cop" (The Regulator)

The circuit needs a brain to decide how much power to let through. This is the Regulator.

Sometimes the regulator has "hard limits," like a speed governor that says, "No matter what, you cannot go faster than 60 mph."
The authors' method handles these limits easily. It predicts the speed, checks the limit, and adjusts the throttle immediately. Other simulators often crash when they hit these "hard limits" because they try to calculate the exact moment the speed hits 60 mph, which is mathematically messy.

Why This Matters: The Results

The authors tested their method against other popular simulation tools (like PETS and PLECS) and a very precise but slow method called "EXACT."

The Race: Imagine a 200-meter race.
- The EXACT method (the old, precise way) took 394 seconds (almost 7 minutes) to finish.
- The PLECS method took 22 seconds.
- The New Method finished in 0.95 seconds.

It was hundreds of times faster than the precise method and significantly faster than the others, all while maintaining about 95% accuracy.

The Bottom Line

This paper introduces a "smart guess-and-check" system for simulating power electronics. Instead of trying to calculate every tiny, fast switch in a circuit (which is slow and prone to errors), it looks at the average behavior, makes a quick guess, corrects it, and moves on.

The Analogy:
If traditional simulators are like trying to count every single raindrop in a storm to measure the total rainfall, this new method is like looking at the water level in a bucket and saying, "Based on the last minute, the bucket will be full in 10 seconds." It's fast, it avoids the headache of counting drops, and it gets the job done so engineers can design better, faster electronics without waiting days for a computer to finish its work.

1. Problem Statement

Transient simulation of power electronic circuits, particularly switching converters, presents significant challenges for general-purpose simulators.

Convergence Issues: Standard simulators often rely on the Newton-Raphson iterative method to solve nonlinear equations. This frequently leads to convergence failures, especially when dealing with fast-switching elements or "hard nonlinearities" (e.g., peak current limiting, duty ratio limiting).
Computational Cost: Reducing the simulation time step to force convergence often fails to resolve the issue and drastically increases computational time.
Limitations of Existing Methods:
- Piecewise-linear simulators avoid convergence issues but introduce complexity in determining the correct operating segment of nonlinear elements.
- Averaging methods (like state-space averaging) are efficient but often struggle with discontinuous conduction modes (DCM) or require complex multiplication matrices to maintain spectral accuracy.
- Exact solving methods (solving differential equations precisely) are highly accurate but computationally prohibitive for large-signal transient analysis.

2. Methodology

The authors propose a novel algorithm for large-signal transient simulation based on switching cell averaging combined with a predictor-corrector integration method.

A. Switching Cell Averaging

Instead of modeling the entire converter, the method focuses on a "Switching Cell" (Type Cell A), a three-terminal, two-port element containing an inductor, a switch, and a diode.

Unified Model: The algorithm uses a single set of averaged equations (Eqs. 4–7) that apply to both Continuous Conduction Mode (CCM) and Discontinuous Conduction Mode (DCM).
Assumptions: The method assumes quasi-steady state and linear ripple approximation, allowing for the estimation of inductor current and capacitor voltage ripples without prior knowledge of the exact duty cycle.

B. Predictor-Corrector Integration (PECE)

To determine the duty ratio ( $d$ ) without a priori knowledge, the algorithm employs a Prediction-Evaluation-Correction-Evaluation (PECE) scheme:

Prediction: Uses the Forward Euler method to predict the capacitor voltage ( $v_C$ ) and inductor current ( $i_L$ ) for the current switching period based on previous values.
Evaluation (Step 1): The predicted values are used to calculate the duty ratio ( $d$ ) via the regulator logic (e.g., PWM or peak current limiting). This duty ratio is then used to solve the converter circuit equations (Modified Nodal Analysis).
Correction: The capacitor voltage and inductor current are corrected using the Trapezoidal integration formula to improve accuracy.
Final Evaluation (Step 2): The corrected values are used to recalculate the duty ratio and perform a final circuit evaluation.

C. System Decomposition

The algorithm divides the system into two parts: the Converter and the Regulator.

The regulator is discretized using bilinear transformation with a sampling time equal to the switching period ( $T_S$ ).
Feedback is established through predicted/corrected state variables and the calculated duty ratio.
Fixed Computational Load: The method performs exactly two circuit solutions per switching period (one for prediction, one for correction), ensuring a fixed and low computational cost.

3. Key Contributions

Convergence-Free Simulation: By using pre-calculated switching cell equations and a fixed-step predictor-corrector approach, the algorithm eliminates the need for iterative Newton-Raphson solving, thereby avoiding convergence problems associated with hard nonlinearities.
Unified CCM/DCM Handling: The proposed switching cell model seamlessly handles both conduction modes without requiring complex mode-switching logic or special matrices.
Computational Efficiency: The algorithm reduces the system matrix size by separating the converter and regulator and limits the number of matrix inversions to two per switching period.
Instantaneous Waveform Reconstruction: While the core simulation operates on averaged values, the method allows for the reconstruction of instantaneous waveforms (inductor current and capacitor voltage) by superimposing calculated ripples onto the averaged results.

4. Results and Performance

The proposed algorithm was implemented in Python 2.7.6 and tested against three other methods: PETS (piecewise-linear), PLECS (commercial averaging), and EXACT (exact differential equation solving).

Test Cases:

Regulated Buck Converter: Simulated with a soft-start, duty ratio limiting (0.85), and current limiting (4A). An input voltage step change (20V to 40V) was applied.
Switch Mode Rectifier: A complex circuit known to cause multiple zero-crossing issues in other simulators, tested with and without a filter.

Performance Metrics (Table I):

Speed: The proposed method was significantly faster than all competitors.
- Buck Converter (200ms transient): Proposed method took 0.95 seconds, compared to PETS (10.02s), PLECS (22.21s), and EXACT (394.93s).
- Rectifier (800ms transient): Proposed method took ~0.41 seconds, while EXACT took 42.05s. PETS failed to simulate the rectifier with the filter included due to zero-crossing errors.
Accuracy: When compared to the EXACT method, the absolute error was estimated to be up to 5%, which is considered excellent for fast transient simulation purposes.
Robustness: The proposed method successfully simulated circuits where PETS and PLECS failed or required reduced precision due to zero-crossing detection issues.

5. Significance

This paper presents a highly efficient solution for the transient simulation of power electronic converters, specifically addressing the trade-off between speed and accuracy.

Practical Application: It enables rapid design verification of converters with complex control loops and hard nonlinearities (like current limiting) without the risk of simulation divergence.
Scalability: The fixed number of calculations per period makes the algorithm highly scalable for complex systems where switching frequency is much higher than the line frequency ( $f_S \gg f_{line}$ ).
Accessibility: The ability to run such simulations in an interpreted language (Python) with performance exceeding compiled commercial tools demonstrates the efficacy of the underlying mathematical approach.

In conclusion, the authors successfully developed a robust, fast, and accurate simulation algorithm that overcomes the convergence and speed limitations of existing tools, making it a valuable method for the analysis of regulated switching converters.