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arxiv: 2410.18217 · v2 · submitted 2024-10-23 · 📡 eess.SY · cs.SY

Accurate Analytical Modeling of Small-Size Rotary Transformers for Wound-Rotor Resolvers

Saeed Hajmohammadi , MohammadSadegh KhajueeZadeh , Farid Tootoonchian , Sajjad Mohammadi This is my paper

Pith reviewed 2026-05-23 18:49 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords rotary transformerwound-rotor resolvermagnetic equivalent circuitleakage inductancemagnetizing inductanceflux fringingair gap effectssecondary voltage
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The pith

An analytical magnetic equivalent circuit model for miniature rotary transformers derives magnetizing and leakage inductances by including flux fringing and air gap effects to predict secondary voltage more accurately than conventionalIdeal

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that small rotary transformers used in wound-rotor resolvers deviate from ideal transformer behavior because leakage inductance grows comparable to magnetizing inductance when core size shrinks and air gaps appear. It builds an equivalent circuit model that analytically calculates both inductances while accounting for fringing flux paths. This matters for engineers designing compact resolver systems, where reliable excitation transfer predictions reduce the need for repeated full numerical simulations or prototypes. The model is checked against three-dimensional finite element analysis and measurements on a physical prototype under no-load and operating conditions.

Core claim

The paper claims that a magnetic equivalent circuit which incorporates reluctances for the main flux path, fringing flux, and leakage paths yields closed-form expressions for magnetizing and leakage inductances in small-size rotary transformers, producing secondary voltage predictions that match three-dimensional finite element results and prototype measurements more closely than models that ignore fringing and air-gap effects.

What carries the argument

Magnetic equivalent circuit that separates main, fringing, and leakage flux paths to compute magnetizing and leakage inductances analytically.

If this is right

  • Designers can adjust core dimensions and gap sizes analytically to achieve a target voltage transfer ratio without repeated 3D simulations.
  • Excitation signal transfer in miniature resolvers can be characterized with quantified leakage effects rather than assumed ideal turns ratios.
  • Initial performance estimates for compact resolver systems become available before physical prototypes are built.
  • The same inductance formulas can be reused across families of similar rotary transformers that differ only in scale or gap length.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The fringing-accounting approach may transfer to analytical modeling of other small air-gap devices such as miniature motors or sensors where leakage dominates.
  • If the circuit remains accurate at still smaller scales, it could shorten design cycles for next-generation compact rotary electromagnetic components.
  • Similar separation of flux paths might improve leakage predictions in linear variable differential transformers that also operate with small air gaps.

Load-bearing premise

A simplified magnetic equivalent circuit can capture enough of the three-dimensional flux distribution and fringing in small rotary transformers with unavoidable air gaps to replace full numerical field solutions for design work.

What would settle it

Three-dimensional finite element analysis or measurements on the fabricated prototype showing secondary voltage errors equal to or larger than those from a conventional ideal-transformer model would falsify the claim of improved accuracy.

Figures

Figures reproduced from arXiv: 2410.18217 by Farid Tootoonchian, MohammadSadegh KhajueeZadeh, Saeed Hajmohammadi, Sajjad Mohammadi.

Figure 2
Figure 2. Figure 2: and Table I show the geometry and geometrical dimension data of this configuration for large- and small-size transformers. C. Bode Diagram Bode diagram can be used to confirm (4)-(9); therefore, the rotor-to-stator voltage ratio can be written as (10)-(12) for each circuit. In this case, 𝑧𝑙 is a 10 Ohm resistance and 𝑠 = 𝑗𝜔. Consistent with [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Bode diagrams for circuit models of transformer: 𝐻𝑇 (Fig. 1a), 𝐻𝑇𝑟1 (Fig. 1b), and 𝐻𝑇𝑟2 (Fig. 1c). TABLE II PARAMETERS OF CIRCUIT MODELS Model Parameters Axial Small-Size Axial Large-Size 𝑙𝑠𝑠 3.322mH 4.693H 𝑙𝑠𝑟 2.968mH 4.625H 𝑙𝑟𝑟 3.348mH 4.695 H 𝑙𝑙 0.6915mH 0.137H 𝑙𝑚 2.631mH 4.556H 𝑛 0.887 0.985 𝑡𝑢𝑟𝑛 𝑛𝑢𝑚𝑏𝑒𝑟 𝑟𝑎𝑡𝑖𝑜 1 1 = [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Bode diagrams for transformer circuit model in Fig. 1a, where 𝐻𝑇𝑟2 denotes exact transformer ratio, and 𝐻𝑇𝑟2𝑝 denotes turn number ratio: (a) small-size transformer, and (b) large-size transformer [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: is the magnetic circuit, which takes into account the fringe reluctance. In this regard, the magnetization inductance (𝑙𝑚) can be written as follows, where 𝑅𝑚 is the reluctance of the main magnetic flux, and 𝑅𝑡𝑠, 𝑅𝑡𝑟, 𝑅𝑦𝑠, 𝑅𝑦𝑟, 𝑅𝑐𝑠, and 𝑅𝑐𝑟 are the tooth, yoke, and corner reluctances of the stator and rotor. Here, the indices of 𝑡, 𝑦, and 𝑐 symbolize the tooth, yoke, and corner. Moreover, 𝑅𝑔,𝑒𝑞 denotes the… view at source ↗
Figure 6
Figure 6. Figure 6: Paths of integrations and flux tubes of the rotary transformer [PITH_FULL_IMAGE:figures/full_fig_p004_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Modeling the airgap fringe reluctances includes (1) 𝑅𝑠ℎ,𝑜 , (2) 𝑅𝑓,𝑜 , (3) 𝑅𝑠ℎ,𝑖 , and (4) 𝑅𝑓,𝑖 . It is worth to mention that 𝐿𝑥 differs between outer and inner regions, where it is 2𝜋𝑟𝑖 and 2𝜋(𝑟𝑖 + 𝑤𝑡𝑠), respectively. Analogous treatment can be regarded to other regions. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p005_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: (a) Flux leakage lines, and (b) leakage magnetomotive force (MMF) diagram and magnetic field strength. , , [PITH_FULL_IMAGE:figures/full_fig_p005_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The magnetic flux density and magnetic field strength for (a) 𝐼𝑠 = 𝐼𝑟 and (b) 𝐼𝑠 > 𝐼𝑟 [PITH_FULL_IMAGE:figures/full_fig_p006_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The schematic of the mesh grid. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p006_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: (a) Leakage inductances from 3D-FEA, the investigated methodology in [8], and the suggested MEC in this study. (b) Magnetization inductances from 3D-FEA, MEC with fringing effect, and MEC without fringing effect. TABLE III COMPARISON OF EXACT AND ADJUSTED TRANSFORMER RATIOS FOR DIFFERENT TURN NUMBER RATIOS Model Parameters 𝑇𝑢𝑟𝑛 𝑛𝑢𝑚𝑏𝑒𝑟 𝑟𝑎𝑡𝑖𝑜 0.5 1 2 𝑙𝑠𝑠 830.567𝜇H 3.322mH 3.322mH 𝑙𝑠𝑟 1.484mH 2.968mH 1.484mH… view at source ↗
Figure 13
Figure 13. Figure 13: The prototyped (a) rotary transformer has the same geometrical dimensions as the small-size transformer given in Table I, supplying a (b) resolver as a load [PITH_FULL_IMAGE:figures/full_fig_p007_13.png] view at source ↗
Figure 12
Figure 12. Figure 12: Bode diagrams for different turn number ratios (a) 0.5, (b) 1, and (c) 2, where 𝐻𝑇𝑟2 denotes the exact transformer ratio, 𝐻𝑇𝑟2𝑝 denotes the turn number ratio, and 𝐻𝑇𝑟2𝑝𝑝 denotes the adjusted transformer ratio for transfer functions of transformer [PITH_FULL_IMAGE:figures/full_fig_p007_12.png] view at source ↗
Figure 15
Figure 15. Figure 15: The results of measurement, stator- and rotor-side voltages, for airgaps of 0.6mm (left) and 1.2mm (right): (a) No-load condition (b) resolver with 𝑅𝐿 = 9 Ω and 𝑙𝐿 = 2.289 𝑚𝐻 as a load [PITH_FULL_IMAGE:figures/full_fig_p007_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: The results of 3D-FEA, stator- and rotor-side voltages, for airgaps of 0.6mm and 1.2mm: on-load and no-load [PITH_FULL_IMAGE:figures/full_fig_p008_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: The results of the circuit model in Fig. 1c, including stator- and rotor-side voltages, based on the suggested magnetic model for the calculation of leakage and magnetization inductances, as well as the suggested transformer ratio adjustment, for air gaps of 0.6mm and 1.2mm: on-load and no-load. TABLE IV COMPARISON OF THE SUGGESTED METHODOLOGY, 3D FEA, AND EXPERIMENTAL MEASUREMENT Airgaps Rotor-Side Volta… view at source ↗
read the original abstract

Rotary transformers are commonly used in wound rotor resolvers to transfer excitation signals to the rotating winding without mechanical contact. In many analyses, the rotary transformer is modeled as an ideal transformer, where the voltage transfer ratio is assumed to be equal to the turns ratio. However, in miniature rotary transformers used in compact resolver systems, leakage inductance can become comparable to the magnetizing inductance due to reduced core dimensions and unavoidable air gaps, leading to deviations from the ideal voltage transfer behavior. This paper presents an accurate equivalent circuit model for miniature rotary transformers employed in wound rotor resolvers. The proposed model analytically derives the magnetizing and leakage inductances using a magnetic equivalent circuit that accounts for flux fringing and air gap effects. The model is validated through three dimensional finite element analysis and experimental measurements on a fabricated prototype under both no load and resolver excitation conditions. The results demonstrate improved prediction accuracy of the secondary voltage compared with conventional models, enabling more reliable characterization of excitation transfer in compact resolver systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper proposes an analytical equivalent circuit model for small-size rotary transformers in wound-rotor resolvers. It derives magnetizing and leakage inductances from a magnetic equivalent circuit (MEC) that incorporates flux fringing and air-gap effects, then validates the model against 3D FEA and prototype measurements under no-load and resolver excitation conditions, claiming improved secondary-voltage prediction relative to conventional ideal-transformer models.

Significance. If the quantitative validation holds, the work supplies a practical design tool for compact resolvers where leakage inductance becomes comparable to magnetizing inductance; this could reduce reliance on full 3D FEA during early-stage sizing while remaining grounded in standard magnetic-circuit methods.

major comments (2)
  1. [Abstract / Validation] Abstract and validation sections: the central claim of 'improved prediction accuracy' is stated without any reported quantitative error metrics (RMS deviation, percentage error on secondary voltage or inductance values) between the proposed MEC model, conventional models, 3D FEA, and measurements; this information is load-bearing for the accuracy assertion.
  2. [MEC derivation] MEC derivation (assumed §3): the treatment of fringing and effective air-gap length is described only at a high level; if these quantities involve an adjustable factor rather than a fully closed-form expression, the model is not strictly parameter-free and the 'analytical' characterization requires explicit clarification.
minor comments (1)
  1. [Figures] Figure captions and axis labels should explicitly state whether plotted voltages are peak, RMS, or normalized to the ideal turns ratio.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below and will revise the manuscript to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract / Validation] Abstract and validation sections: the central claim of 'improved prediction accuracy' is stated without any reported quantitative error metrics (RMS deviation, percentage error on secondary voltage or inductance values) between the proposed MEC model, conventional models, 3D FEA, and measurements; this information is load-bearing for the accuracy assertion.

    Authors: We agree that explicit quantitative error metrics are required to substantiate the accuracy claim. Although comparative results are shown, the revised manuscript will include RMS deviations and percentage errors for secondary voltage and inductance values across the proposed MEC, conventional models, 3D FEA, and measurements. revision: yes

  2. Referee: [MEC derivation] MEC derivation (assumed §3): the treatment of fringing and effective air-gap length is described only at a high level; if these quantities involve an adjustable factor rather than a fully closed-form expression, the model is not strictly parameter-free and the 'analytical' characterization requires explicit clarification.

    Authors: The fringing factors and effective air-gap lengths are obtained from standard closed-form magnetic-circuit expressions (Carter coefficient and fringing-flux formulas) with no adjustable parameters. The revised Section 3 will present the explicit formulas and derivation steps to confirm the model is fully analytical. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper applies standard magnetic equivalent circuit analysis to derive magnetizing and leakage inductances from geometry, fringing, and air-gap parameters, then validates the resulting voltage predictions against independent 3D FEA and physical prototype measurements. No step reduces a claimed prediction to a fitted input by construction, invokes a self-citation as the sole justification for a uniqueness claim, or renames an empirical pattern as a derived result. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Model rests on standard magnetic circuit theory applied to this geometry; no new entities introduced and limited free parameters beyond geometry-derived quantities.

free parameters (1)
  • Fringing factor or effective air-gap length
    Likely introduced or calibrated to account for flux fringing in the analytical expressions for inductances.
axioms (1)
  • domain assumption Magnetic equivalent circuit approximation accurately captures dominant flux paths and leakage in the miniature rotary transformer geometry
    Core modeling premise invoked to enable closed-form inductance expressions instead of full 3D field solution.

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