Anisotropic Media (3D)

The anisotropic media formulations supported in the Solver are diagonalised tensor, full tensor, complex tensor and Polder tensor (for ferrites).

Note: Only passive media are supported. Passive media can be either lossless or lossy.¹

Diagonalised Tensor

The permittivity along the UU, VV and NN axes are described by diagonal tensor:

(1)

ε = ε_{0} ε_{r} = [\begin{matrix} ε_{u u} & 0 & 0 \\ 0 & ε_{v v} & 0 \\ 0 & 0 & ε_{n n} \end{matrix}]

The permeability along the UU, VV and NN axes are described by diagonal tensor:

(2)

μ = μ_{0} μ_{r} = [\begin{matrix} μ_{u u} & 0 & 0 \\ 0 & μ_{v v} & 0 \\ 0 & 0 & μ_{n n} \end{matrix}]

Full Tensor

The permittivity along the UU, UV, UN, VU, VV, VN, NU, NV and NN axes are described by the dyadic tensor:

(3)

ε = ε_{0} ε_{r} = [\begin{matrix} ε_{u u} & ε_{u v} & ε_{u n} \\ ε_{v u} & ε_{v v} & ε_{v n} \\ ε_{n u} & ε_{n v} & ε_{n n} \end{matrix}]

The permeability along the UU, UV, UN, VU, VV, VN, NU, NV and NN axes are described by the dyadic tensor:

(4)

μ = μ_{0} μ_{r} = [\begin{matrix} μ_{u u} & μ_{u v} & μ_{u n} \\ μ_{v u} & μ_{v v} & μ_{v n} \\ μ_{n u} & μ_{n v} & μ_{n n} \end{matrix}]

Complex Tensor

The permittivity along the UU, UV, UN, VU, VV, VN, NU, NV and NN axes are described by the dyadic tensor:

(5)

ε = ε_{0} ε_{r} = ε_{0} [\begin{matrix} ε_{r_{u u}} & ε_{r_{u v}} & ε_{r_{u n}} \\ ε_{r_{v u}} & ε_{r_{v v}} & ε_{r_{v n}} \\ ε_{r_{n u}} & ε_{r_{n v}} & ε_{r_{n n}} \end{matrix}]

The permeability along the UU, UV, UN, VU, VV, VN, NU, NV and NN axes are described by the dyadic tensor:

(6)

μ = μ_{0} μ_{r} = μ_{0} [\begin{matrix} μ_{r_{u u}} & μ_{r_{u v}} & μ_{r_{u n}} \\ μ_{r_{v u}} & μ_{r_{v v}} & μ_{r_{v n}} \\ μ_{r_{n u}} & μ_{r_{n v}} & μ_{r_{n n}} \end{matrix}]

To create the full permittivity and permeability tensors, create up to nine dielectrics constituting the medium properties along the UU, UV, UN, VU, VV, NU, NV and NN axes.

If no linear dependencies exist between two axes, add a zero (0) entry.

Important:

An entry in the tensor may be a complex number, pure real number or a pure imaginary number.
An entry may not be 0.

Polder Tensor

The ferrimagnetic² material is described by the permittivity tensor (where the static magnetic field is orientated respectively along the U, V and N axis):

(7)

ε = ε_{0} ε_{r} = ε_{0} [\begin{matrix} ε_{r} (1 - j \tan δ) & 0 & 0 \\ 0 & ε_{r} (1 - j \tan δ) & 0 \\ 0 & 0 & ε_{r} (1 - j \tan δ) \end{matrix}]

The ferrimagnetic material is described by the permeability tensors (where the static magnetic field is orientated respectively along the U, V and N axis):

(8)

μ = μ_{0} μ_{r} = [\begin{matrix} μ_{0} & 0 & 0 \\ 0 & μ & j_{κ} \\ 0 & - j_{κ} & μ \end{matrix}] (X directed)

(9)

μ = μ_{0} μ_{r} = [\begin{matrix} μ & 0 & j_{κ} \\ 0 & μ_{0} & 0 \\ - j_{κ} & 0 & μ \end{matrix}] (Y directed)

(10)

μ = μ_{0} μ_{r} = [\begin{matrix} μ & j_{κ} & 0 \\ - j_{κ} & μ & 0 \\ 0 & 0 & μ_{0} \end{matrix}] (Z directed)

Where $μ$ and $κ$ elements of the permeability tensor are given by

(11)

μ = μ_{0} (1 + \frac{ω_{0} ω_{m}}{ω_{0}^{2} - ω^{2}})

(12)

κ = μ_{0} \frac{ω ω_{m}}{ω_{0}^{2} - ω^{2}}

and where,

operating frequency: $ω$

Lamor (precession) frequency: $ω_{0} = μ_{0} γ H_{0}$

forced precession frequency: $ω_{m} = μ_{0} γ M_{s}$

gyromagnetic ratio: $γ$

magnetic bias field: $H_{0}$

DC saturation magnetisation: $M_{s}$ .

To account for magnetic loss, the resonant frequency can be made complex by introducing a damping factor ( $α$ ) into Equation 11 and Equation 12. The damping factor and the field line width ( $Δ H$ ), the width of the imaginary susceptibility curve against the bias field at half its peak value, are related by

(13)

α = \frac{μ_{0} γ Δ H}{2 ω}

Note: The Polder tensor is defined using CGS³ units in terms of:

saturation magnetisation (Gauss): $4 π M_{s}$
line width (Oersted): $Δ H$
DC bias field (Oersted): $H_{0}$
field direction.

¹ A lossless passive medium allows fields to pass through the medium without attenuation. In a lossy passive medium, a fraction of the power is transformed to heat, as an example.

² D. M. Pozar, “Theory and Design of Ferrimagnetic Components” in “Microwave Engineering”, 2nd ed., New York: Wiley, 1997, ch 9, pp. 497-508

³ CGS is the system of units based on measuring lengths in centimetres, mass in grams and time in seconds.