Meta-material

Large Surface LC-resonant Metamaterials: from Circuit Model to Modal Theory and Efficient Numerical Methods

L. Krähenbühl1,2, R. Scorretti1,2 A. Bréard1,2, C. Vollaire1,2J.-M. Guichon2,3, O. Chadebec2,3, G. Meunier2,3, A. Urdaneta-Calzadilla3, V. C. Silva2,4, C.A.F. Sartori2,4

1Univ Lyon, ECLyon, Ampère, CNRS, France – 2LIA Maxwell, CNRS (France)/CNPq (Brazil) 3Univ Grenoble Alpes, Grenoble INP, G2Elab, CNRS, France – 4EP-USP, LMAG, São Paulo, Brazil

The MaSuRe project objectives Equations

Modal approach (very large number of cells)

We study the harmonic magnetodynamic behavior (low frequency) of a resonant surface metamaterial, made up of many identical, regularly arranged LC resonant cells. - How does it work? - Why is power transfer possible also with

large excentricity of the receiver? - How can we explain the antiresonance - Find an efficient numerical method for very

large numbers of cells (e.g. 1000 × 1000)

Perspectives: - Effective numerical homogenization - Review of possible manufacturing methods - Development of a specific SMPS - Experimental measurements of a very large

structure

metamaterial

source

receiver

holder

i

j

𝑀𝑖𝑖

𝑀𝑖𝑟 𝑀𝑖𝑠

Front (planar coil) inductance 𝐿 = 𝑀𝑖𝑖

resistor R

Back (SMC capacitor C )

This work is supported by the French-Brazilian USP/COFECUB program

under the grant 173/18 « MaSuRe »

ℱ −𝒋𝜔

𝑅 + 1𝒋𝜔𝐶 + 𝒋𝜔ℳ𝑘

∞

𝝓𝑘𝑠

𝜙𝑖𝑠

𝑰𝑘

k-transfert function k : −n … n

ℳ𝑘∞ 𝜙𝑖 ⊕

𝜙𝑖,𝑡𝑡𝑡

𝜙𝑖𝑠

ℱ−1 𝝓𝑘

ℳ𝑘𝑟∞ 𝜙𝑟(𝑥𝑟)

⊕ 𝜙𝑟,𝑡𝑡𝑡 𝑥𝑟

𝜙𝑠𝑟 𝑥𝑟

ℱ−1 𝝓𝑘𝑟

receiver

meta-material

ℱ−1 𝐼𝑖

Source flux(xi : cell position i)

𝜙𝑖𝑠

Source flux(k : spatial fr. fk=k/L)

𝝓𝑘𝑠

∑ k=-n…n

∑ k=-n…n

𝑅 +1𝒋𝜔𝐶

+ 𝒋𝜔ℳ𝑘∞ 𝑰𝑘 = −𝒋𝜔 𝝓𝑘𝑠

𝑘 = −𝑛…𝑛

Reference solution (circuit equations for 𝑵 cells, full 𝑁 × 𝑁 matrix):

𝑅 +1𝒋𝜔𝐶

𝐼𝑖 + 𝒋𝜔� 𝑀𝑖𝑖𝐼𝑖𝑁

𝑖=1= −𝒋𝜔 𝑀𝑖𝑠𝐼𝑠 + 𝑀𝑖𝑟𝐼𝑟 𝑖 = 1 …𝑁

Space modal development of cell currents and fluxes (discrete values i):

𝑋𝑖𝑠 = � � 𝑿𝑘𝑙,𝑠𝒆−𝒋2𝜋𝑘𝑥𝑖𝐿𝑥

𝑛𝑥

𝑘=−𝑛𝑥

𝒆−𝒋2𝜋𝑙𝑦𝑖𝐿𝑦

𝑛𝑦

𝑙=−𝑛𝑦

Exact modal equation: Note: for a 2D 𝑁𝑥 × 𝑁𝑦 metamaterial , 𝑁 = 𝑁𝑥 × 𝑁𝑦 ⟹ matrix dimension = 𝑁𝑥 × 𝑁𝑦

2

Example: 1000 cells × 1000 cells ⟹ full 106 × 106 matrix. For a 1D problem (𝑁 × 𝑁 matrix, 𝑁 = 2𝑛 + 1):

� 𝑅 +1𝒋𝜔𝐶

+ 𝒋𝜔ℳ𝑘,𝑖 𝑰𝑘𝒆−𝒋2𝜋𝑘𝑥𝑖𝐿

𝑛

𝑘=−𝑛= −𝒋𝜔� 𝝓𝑘𝑠𝒆

−𝒋2𝜋𝑘𝑥𝑖𝐿 𝑛

𝑘=−𝑛

Asymptotic solutions for infinite dimensions (𝑁 equations with one unknown):

𝑅 +1𝒋𝜔𝐶

+ 𝒋𝜔ℳ𝑘∞ 𝑰𝑘 = −𝒋𝜔 𝝓𝑘𝑠 𝑘 = −𝑛… 𝑛,𝑛 → ∞

2n+1 resonance freq. :

𝑓𝑘 =1

2𝜋 ℳ𝑘∞.𝐶

quality factor: 𝑅 ℳ𝑘

∞ 𝐶⁄

Solution cells i=1,N Flux

Current

induced flux (modal)

source flux (cell i)

ℳ𝑘,𝑙∞ = ℳ(𝑓𝑠𝑥, 𝑓𝑠𝑦) = lim

𝑛→∞� 𝑀𝑖𝑖𝒆

−𝒋2𝜋𝑘𝑥𝑗−𝑥𝑖𝐿𝑥 𝒆

−𝒋2𝜋𝑙𝑦𝑗−𝑦𝑖𝐿𝑦

𝑛

𝑖=−𝑛

Variations of ℳ𝑘𝑙𝑖𝑖 as a function of the relative position of the cell (i,j) on a 2D metamaterial

15x15 cells, 2D-mode k=l=0 (% of the modal reference ℳ00

∞)

Modal mutual inductance

source flux (receiver)

induced flux (modal)

induced flux

induced currents

total flux

Why is the modal solution not exact ? 1) ℳ𝑘,𝑙

𝑛 ≠ℳ𝑘,𝑙∞ near edges (finite material)

2) numerical solution 𝑛 ≠ ∞ : foldover distorsion infinite anti-periodic excitation

Flux: resonant solutions on the observation surface

|𝜙𝑟,𝑡𝑡𝑡| 𝑥𝑟

Cell currents: typical behavior for: 1. res. freq. ↖↑ 2. non-res. freq. ↙ 3. intermediate case ↓

𝐼𝑖

1D : analytic values for ℳ𝑘∞

Data used for this example:

Results

Reduction methods 1) Modal solutions - use only the most energetic modes. Typically 𝑁 → 𝑁

10⁄ Equivalent to filter (neglect) the higher space frequencies 2) Circuit equations - typically (𝑁 × 𝑁) → (𝑁 10⁄ × 𝑁

10⁄ ) by using: - rough mesh representation of cell currents 𝐼𝑖 to reduce the number of unknowns - same linear combination of equations to reduce the number of circuit equations. Equivalent to filter (neglect) the higher space frequencies 3) Mutual inductance matrix 𝑴𝒊𝒋 - "Exact" values (analytic values, FEM or PEEC values) only for neighbour cells i, j - far cells: just distance-dependent values

Displacement of the anti-resonance area with frequency changes

𝐼2 𝜔 = � 𝐼𝑖 𝜔2

𝑁

Meta-material global frequency behavior:

Receiver - global frequency behavior:

𝜂 𝑥𝑟 ,𝜔 = 1 + � 𝑀𝑟𝑖𝐼𝑖 𝜔𝑁

𝑖=1𝑀𝑟𝑠𝐼𝑠�

𝜂ℓ2(𝜔) =1ℓ� 𝜂

2𝑥,𝜔 𝑑𝑥

𝑥𝑟𝑟+ℓ

𝑥𝑟𝑟

Meta-material Receiver Test with ANSYS HFSS

Source Receiver

met

a-m

ater

ial

𝒇𝟏

𝒇𝟐 > 𝒇𝟏

𝒇𝟑 > 𝒇𝟐

PEEC method for 2D structure 12×12 cell test with Altair Flux PEEC

∘

1D model

Possible power transfer with large excentricities

receiver source

reference (circuit model) reduced circuit model reduced modal solution