Analytical modeling of resonant converters in frequency domain for wireless power transfer: Continuous current mode (CCM) operation

Sun, 01 May 2022 00:00:00 +0000

Wireless power transfer (WPT) systems move energy across an air gap using a resonant converter: an H-bridge inverter drives a resonant tank (inductors and capacitors) that couples through two loosely-coupled coils, and the received AC is turned back into DC by a diode-bridge rectifier. Getting the steady-state behavior of this converter right — the current and voltage waveforms it settles into — is what lets engineers size components and predict output power before building hardware. The problem is that the diode-bridge rectifier is genuinely non-linear (it switches on and off depending on the polarity of the incoming current), so the waveforms it produces are distorted, not pure sine waves. The industry-standard shortcut, Fundamental Harmonic Approximation (FHA), ignores that distortion and only tracks the fundamental-frequency component. It is fast, but the paper shows it breaks down once the converter is driven away from its resonant point — and that emerging, higher-power-density WPT designs, which deliberately inject extra harmonics, push it into exactly that regime.

Prior frequency-domain models that do account for higher harmonics beyond FHA exist, but each one leans on a simplifying assumption to stay solvable: some fix the source driving the resonant network to an idealized sine wave, while others allow an arbitrary source but force the rectifier’s DC-side output to be a fixed voltage, a square wave, a pre-specified voltage/current source, or — at best — a purely inductive load. Those assumptions break down for more general, realistic WPT loads, such as the paper’s own test hardware, where an output LC filter sits between the rectifier and a resistive load — a combination none of the earlier fixed-output assumptions can fully capture. This paper’s contribution is a model that keeps arbitrary input and output configurations while cutting the model down to a single unknown variable — an “offset angle” describing where the rectifier’s input current crosses zero relative to the inverter’s output voltage — solved through nothing more than matrix operations (no integration or differentiation needed).

Fig. 1 from the paper: overall schematic of the conventional inductive wireless charger, showing the H-bridge inverter, resonant network, and diode-bridge rectifier that the model captures end to end.

The method works in three steps. First, the non-linear diode-bridge rectifier is captured with a harmonic (Fourier-domain) admittance/impedance matrix that relates every harmonic of the AC-side current and voltage to the DC-side output — including a later refinement that adds the diodes’ own forward-voltage drop, which matters most for low-output-voltage or high-frequency designs. Second, the passive resonant network (whatever combination of L’s and C’s the designer chose) is folded in as an ordinary linear two-port network. Combining the two collapses the whole system into one equation with a single unknown: the offset angle. That angle is found by an iterative one-dimensional search — starting from an FHA-based initial guess and refined with a parabolic-interpolation search — which the paper shows converges in under ten iterations, versus gradient-descent methods that can fail outright because the underlying error function isn’t smooth.

To validate the model, the authors built two hardware WPT prototypes — one with series-series (SS) compensation, one with LCC compensation, both based on a dual-coil Qi-style transmitter/receiver — and compared the model’s predicted waveforms against PSIM time-domain simulation and bench measurements.

Fig. 10 from the paper: i_ac and v_H waveforms from experiment, time-domain simulation, the proposed calculation, and FHA, for the SS topology (N = 50, V_in = 11.52 V, R_o = 10 Ω).

The proposed model tracks the measured and simulated waveforms closely, including the current distortion that FHA smooths away — FHA’s calculated peak current comes out roughly 16% above the experimental value. On output-power estimation, FHA is off by about 18% for the SS prototype at full load (about 15% at half load), while the proposed method lands within roughly 1-2% of the measured output power at both load points (the paper’s headline “10% improvement” figure, from the abstract). The match is noticeably looser for the LCC prototype — about 11% output-power error at full load, versus about 3% at half load — which the authors attribute mainly to the small-signal LCR-meter measurement of the coils (taken at only 500 mV excitation) not fully capturing the coupled inductor’s real, non-linear behavior: its effective AC resistance turns out to be roughly five times higher at the third harmonic than at the fundamental frequency, and its behavior also shifts with current level, rather than reflecting a limitation of the frequency-domain method itself. The authors are explicit that the model is currently restricted to continuous current mode (CCM) — it does not yet cover the discontinuous-current regime that shows up under light loads — and that resonant networks with richer harmonic content (like LCC) need a higher-order harmonic matrix, which increases computation cost and is left for future work.

Unified Model of Sediment Transport Threshold and Rate Across Weak and Intense Subaqueous Bedload, Windblown Sand, and Windblown Snow

Thu, 01 Apr 2021 00:00:00 +0000

Grains get pushed along a riverbed, blown across a desert dune, or carried over a snowfield by very different physical processes — yet in each case, a grain that isn’t fully suspended in the fluid moves the same basic way: it hops, rolls, or slides along the bed in a sequence of collisions. This is called nonsuspended sediment transport (NST), and it shapes sedimentary landscapes on Earth and other planetary bodies alike. In reality NST is messy: turbulence, an uneven bed surface, and grain-to-grain variability make individual grain trajectories highly erratic. This paper asks whether, despite that mess, one single deterministic model can still predict both when transport starts (the threshold) and how much material moves once it does (the rate) — across fluvial settings (grains driven by a stream of liquid) and aeolian settings (grains driven by wind) at once.

The model’s key idea is to represent the entire population of grains — including ones that mostly roll or slide rather than hop — by a single, repeating (periodic) saltation trajectory: a grain leaves the bed, follows a ballistic-like path shaped by the mean turbulent flow profile above the bed, and comes back down, where an empirical “rebound law” (fit to grain-bed collision experiments and simulations) sets its average rebound velocity as a function of how it hit the bed. The transport threshold is then defined as the weakest flow for which such a self-sustaining periodic trajectory can still exist, with an extra criterion checking whether a grain has enough energy to roll itself out of the most stable pockets of the bed surface.

The model reduces the entire fluctuating grain motion to a single periodic bouncing trajectory driven by the mean flow profile above the bed (Figure 1).

What makes the result notable is that none of the model’s parameters were tuned to the data it is then tested against — they were fixed beforehand from independent experiments and simulations (grain-bed rebound measurements, earlier discrete-element-method simulations of transport, and separate studies of bed yielding). With those parameters locked in, the model reproduces laboratory measurements and field data of both the transport threshold and the transport rate to within about a factor of two, across density ratios between grain and fluid spanning roughly 2.65 (mineral grains in water) up to around 2000 (mineral grains or snow in air) — that is, across subaqueous bedload, windblown sand, and windblown snow together.

Left: predicted transport-threshold curves versus Galileo number compared with measured threshold data. Right: predicted versus measured dimensionless transport rate. Both panels combine fluvial mineral, aeolian mineral, and aeolian snow data sets (Figure 5c-d).

A striking side finding concerns cohesion. Windblown sand and snow grains are known to stick together via cohesive bonds, yet the model captures their behavior without ever explicitly representing cohesion. This lines up with independent DEM simulations showing that equilibrium aeolian transport is nearly unaffected by how strong those cohesive bonds are — a fairly counterintuitive result that this model helps explain: in the model’s picture, cohesion only matters while a grain is in contact with the bed, and for grains that are mostly saltating (as opposed to rolling), those contacts are so brief that cohesion barely gets a chance to act. The model also yields a simple, physically grounded way to tell apart “bedload” (where a significant fraction of grains roll or slide in sustained contact with the bed) from “saltation” (where that fraction is negligible) — consistent with an independent criterion from the authors’ earlier work — and it further separates viscous from turbulent transport regimes based on how the grain’s hop height compares to the flow’s viscous sublayer thickness.

Physics-Based Modeling | Yonghui Liu

Analytical modeling of resonant converters in frequency domain for wireless power transfer: Continuous current mode (CCM) operation

Unified Model of Sediment Transport Threshold and Rate Across Weak and Intense Subaqueous Bedload, Windblown Sand, and Windblown Snow