Exact Closed-Form Feedforward Inversion for Dual-Bridge Series Resonant DC/DC Converter via State-Plane Analysis

Technical Summary: Exact Closed-Form Feedforward Inversion for Dual-Bridge Series Resonant DC/DC Converter via State-Plane Analysis

Problem Statement
The Dual-Bridge Series Resonant Converter (DB SRC) is a topology widely used in high-power isolated DC/DC applications, such as electric vehicle chargers and battery storage systems. It utilizes four independent modulation variables: primary duty cycle ($d$), secondary shorting time ($s$), phase shift ($\beta$), and switching frequency ($\omega$). While existing control strategies rely on the First Harmonic Approximation (FHA) to derive feedforward inversion maps (calculating switching parameters from desired commutation angles), the FHA assumes a purely sinusoidal tank current. This assumption neglects higher harmonics, leading to inaccuracies in predicting commutation currents and zero-crossing angles, particularly during above-resonance operation. These errors can result in significant deviations (5–72%) between predicted and actual commutation angles, compromising Zero Voltage Switching (ZVS) margins and control precision.

Methodology
The paper employs exact state-plane trajectory analysis to derive closed-form solutions for the DB SRC. Unlike the FHA, which truncates the system to its fundamental frequency, the state-plane approach models the LC tank dynamics using normalized coordinates where the trajectory under constant applied voltage is a circular arc.

1. Exact Trajectory Construction: The positive half-cycle is decomposed into $N$ intervals, each corresponding to a specific switching state with a constant normalized voltage center ($c_k$) and duration ($\Delta\theta_k$). The state evolution is modeled as a composition of clockwise circular arcs.
2. Steady-State Solution: By enforcing half-wave symmetry ($z_N = -z_0$), the authors derive an exact closed-form expression for the initial state $z_0$ in terms of the switching parameters and the normalized frequency $F_n = \omega/\omega_0$.
3. Transconductance and Zero-Crossing: Exact expressions for output transconductance ($W$) and the precise zero-crossing angles ($\sigma, \delta$) are derived by summing the contributions of each arc segment, accounting for all harmonics.
4. Feedforward Inversion: The core of the methodology involves inverting these exact relationships. The paper derives explicit formulas to calculate the required switching parameters ($d, s, \beta$) for a given set of target commutation angles ($\sigma^*, \delta^*$) and voltage gain ($G$), valid for any normalized frequency $F_n$.

Key Contributions
The paper presents five primary contributions:

1. Exact Closed-Form Inversion Maps: Derivation of exact formulas (Equations 19–21) for computing switching parameters in both Buck and Boost modes. These formulas generalize the FHA results by incorporating the normalized frequency through the term $\Theta = \pi/F_n$.
2. Rigorous Equivalence Proof: A mathematical proof demonstrating that the FHA inversion is a special case of the exact state-plane inversion, valid strictly at the resonance frequency ($F_n = 1$). This establishes the FHA not as an independent approximation, but as the fundamental-frequency limit of the exact solution.
3. Quantitative Error Characterization: Analysis showing that for above-resonance operation ($F_n > 1$), the FHA-based feedforward introduces commutation angle errors ranging from 5% to 72%. The exact model eliminates these errors, ensuring precise alignment of the tank current zero-crossing with switching edges.
4. Partial Rectification Generalization: Extension of the exact inversion to include an additional shorting time ($s_{add}$) for low-power operation, maintaining the closed-form structure suitable for real-time implementation.
5. Controller Architecture Refinement: Proposal of a control architecture that mirrors the parallel nonlinear compensation structure of FHA-based designs but operates on "resonant-time" quantities (physical time intervals in units of $1/\omega_0$) rather than switching-frequency angles. This naturally couples the inner commutation loop with the outer frequency control loop.

Results
Numerical evaluations on a converter with $L=31\,\mu\text{H}$, $C=8.2\,\text{nF}$, and $n=2.2$ confirm the theoretical findings:

• Transconductance Accuracy: At $F_n = 1.05$ (near resonance), the exact and FHA transconductance curves nearly coincide. At $F_n = 2.0$, a visible deviation appears, which the exact model captures accurately.
• Inversion Precision: Table comparisons demonstrate that the exact formulas achieve the target commutation angle $\sigma^*$ precisely. In contrast, the FHA-based inversion yields significant errors (e.g., achieving only 0.028 instead of a target 0.1 at $F_n=1.5$).
• ZVS Margin: The FHA consistently overestimates the commutation angle $\sigma$ by 5–20%, implying that controllers using FHA may operate with lower actual ZVS margins than predicted. The exact model corrects this prediction.
• Frequency Dependence: The aggregated switching parameter $q$ (combining Buck and Boost modes) shows that while the exact and FHA curves coincide at $F_n=1$, they diverge progressively as $F_n$ increases, with the largest deviation occurring near the Buck-to-Boost transition ($G \approx 1$).

Significance
The paper claims that the state-plane framework provides a physically rigorous foundation for DB SRC control that eliminates the harmonic truncation errors inherent in FHA-based designs. By deriving exact, non-iterative, closed-form inversion maps, the work enables real-time implementation of controllers that maintain precise ZVS margins across the entire operating range, including high-frequency, low-power regimes where FHA models fail. The work bridges the gap between geometric state-plane analysis and practical feedforward control, proving that the established FHA method is merely a resonance-specific subset of a more general, exact solution.