[Paper Review] Sampled-Data and Harmonic Balance Analyses of Average Current-Mode Controlled Buck Converter

Chung‐Chieh Fang|arXiv (Cornell University)|Feb 21, 2012

Advanced DC-DC Converters13 references3 citations

TL;DR

This paper proposes a sampled-data and harmonic balance analysis framework for average current-mode controlled buck converters, introducing an exact sampled-data model and a 'lifted' continuous-time model that accurately predicts subharmonic oscillation (PDB) and matches experimental frequency response. The key contribution is a novel S plot that precisely identifies unstable windows in compensator pole placement and guides ramp slope selection to avoid instability.

ABSTRACT

Dynamics and stability of average current-mode control of buck converters are analyzed by sampled-data and harmonic balance analyses. An exact sampled-data model is derived. A new continuous-time model "lifted" from the sampled-data model is also derived, and has frequency response matched with experimental data reported previously. Orbital stability is studied and it is found unrelated to the ripple size of the current-loop compensator output. An unstable window of the current-loop compensator pole is found by simulations, and it can be accurately predicted by sampled-data and harmonic balance analyses. A new S plot accurately predicting the subharmonic oscillation is proposed. The S plot assists pole assignment and shows the required ramp slope to avoid instability.

Motivation & Objective

• To address the limitations of traditional average models in predicting period-doubling bifurcation (PDB) in average current-mode controlled (ACMC) buck converters.
• To resolve the inaccuracy of standard average models when negative real sampled-data poles exist, which indicate oscillatory behavior not captured by single-pole mappings.
• To develop a more accurate modeling framework that preserves orbital stability and matches experimental frequency response.
• To introduce a new design tool—the S plot—that enables precise pole assignment and stability margin assessment for converter designers.

Proposed method

• Derives an exact sampled-data model directly from the switching dynamics of the buck converter, preserving the orbital nature of the periodic solution.
• Introduces a 'lifted' continuous-time model by mapping negative real sampled-data poles to a pair of right-half-plane (RHP) poles and a zero, increasing system dimension to match experimental frequency response.
• Applies harmonic balance analysis to model nonlinearities and predict subharmonic oscillation, using a new S plot defined as S(−1, D, ωs/2) to detect instability boundaries.
• Uses the S plot to identify unstable windows in compensator pole frequency (ωp) and duty cycle (D), with exact and approximate expressions (e.g., Eq. 12, 14–15) for design guidance.
• Validates the models through simulation and comparison with experimental data, showing that the lifted model matches frequency response while average models fail when negative real poles exist.
• Employs matrix-based and harmonic-based formulations (Eqs. 8, 12) to express the S plot condition for PDB, enabling practical design rules.

Experimental results

Research questions

• RQ1 Can sampled-data and harmonic balance models accurately predict period-doubling bifurcation (PDB) in average current-mode controlled buck converters?
• RQ2 Why do standard average models fail to predict PDB when negative real sampled-data poles are present?
• RQ3 Can a 'lifted' continuous-time model with increased system dimension accurately match experimental frequency response and predict PDB?
• RQ4 How can a new S plot be formulated to precisely identify unstable operating regions and guide design parameters like ramp slope and compensator pole location?
• RQ5 What are the quantitative boundaries for instability in terms of duty cycle D and compensator pole ωp, and how do they compare to traditional design rules?

Key findings

• An unstable window for the current-loop compensator pole ωp exists between 0.36ωs and 0.54ωs, where PDB occurs, and this is accurately predicted by the sampled-data and harmonic balance models.
• The S plot, defined as the intersection of S(−1, D, ωs/2) and the ramp slope ˙h(d), correctly identifies instability for D < 0.09 and confirms the unstable window for ωp/ωs ∈ (0.36, 0.54).
• When ωp = 0.55ωs, the sampled-data poles are −0.991, −0.036, 0.882, and 0.986, confirming PDB is avoided, validating the S plot’s predictive accuracy.
• The lifted model has frequency response that matches experimental data and predicts PDB, unlike the standard average model which fails when negative real sampled-data poles exist.
• The average model shows good match with experimental data only under conditions where sampled-data poles are not negative real; this match breaks down when such poles appear.
• The S plot enables practical design: for example, setting ˙h(d) = 185,000 (or Vh = 1.85) ensures stability by operating above the S plot curve, providing a clear design margin.