Conductors

LineCableModels.Engine.InternalImpedance

Compute the inner, outer, and transfer surface impedances of a tubular conductor.

Formulas

Formula{:schelkunoff1934}

Surface impedances of a homogeneous tubular conductor of inner radius $r_\mathrm{in}$ and outer radius $r_\mathrm{ex}$, from the cylindrical solution of the field inside the metal [1], with the modified Bessel functions $I_\nu$ and $K_\nu$:

\[\begin{aligned} z_\mathrm{inner} &= \frac{s \mu}{2 \pi w_\mathrm{in}} \frac{I_0(w_\mathrm{in}) K_1(w_\mathrm{ex}) + K_0(w_\mathrm{in}) I_1(w_\mathrm{ex})}{D}, \\ z_\mathrm{outer} &= \frac{s \mu}{2 \pi w_\mathrm{ex}} \frac{I_0(w_\mathrm{ex}) K_1(w_\mathrm{in}) + K_0(w_\mathrm{ex}) I_1(w_\mathrm{in})}{D}, \\ z_\mathrm{transfer} &= \frac{1}{2 \pi r_\mathrm{in} r_\mathrm{ex} \sigma D}, \end{aligned}\]

where $D = I_1(w_\mathrm{ex}) K_1(w_\mathrm{in}) - I_1(w_\mathrm{in}) K_1(w_\mathrm{ex})$, $w = m r$, $m = \sqrt{s \mu \sigma}$, $\mu = \mu_0 \mu_r$ and $\sigma = 1 / \rho$. Here $s$ is the complex frequency [1/s], $r$ is the radius of the corresponding conductor surface [m], $m$ is the radial wavenumber [1/m], $\sigma$ is the conductivity [S/m], $\rho$ is the resistivity [Ω·m], $\mu$ is the permeability [H/m] and $\mu_r$ is the relative permeability [dimensionless]. The constant $\mu_0$ is vacuum_permeability. The modified Bessel functions $I_\nu$ and $K_\nu$ have order $\nu$.

A solid conductor, with $r_\mathrm{in} = 0$, has no inner surface: its inner and transfer impedances are zero, and its outer impedance is $s \mu I_0(w_\mathrm{ex}) / (2 \pi w_\mathrm{ex} I_1(w_\mathrm{ex}))$. The implementation evaluates the Bessel functions in their exponentially scaled form.

Identifiers of the internal-impedance formulas.

struct Formula{ID, O<:LineCableModels.Commons.FormulationOptions} <: LineCableModels.Engine.AbstractInternalImpedance

Represent a formula for the surface impedances of a tubular conductor. Its options form a record of type FormulationOptions{Formula{ID}}, with ID the formula identifier.

• options::LineCableModels.Commons.FormulationOptions: Options of the formula.

Formula(
    ::Val{ID};
    kw...
) -> LineCableModels.Engine.InternalImpedance.Formula

Build the formula named by ID with its declared defaults and supplied options.

Arguments

• Val{ID}: formula identifier from FORMULAS.

Keywords

• kw...: options declared by defaults for Formula{ID}.

Returns

• A Formula with a typed option record.

Errors

• Throws ArgumentError for an unknown identifier or option.

impedance(
    _::LineCableModels.Engine.InternalImpedance.Formula{:schelkunoff1934},
    r_in::Real,
    r_ex::Real,
    rho::Real,
    mu_r::Real,
    s::Complex{T<:Real}
) -> NamedTuple{(:inner, :outer, :transfer), <:Tuple{Any, Any, Any}}

Return the inner, outer, and transfer surface impedances of a tubular conductor by the formula Formula{:schelkunoff1934}, whose equations are in the section "Formulas" of InternalImpedance.

Arguments

• Formula{:schelkunoff1934}: cylindrical surface-impedance formula.
• r_in: inner radius [m].
• r_ex: outer radius, in [m].
• rho: resistivity [Ω·m].
• mu_r: relative permeability [dimensionless].
• s: complex frequency, j 2π f, in [1/s].

Returns

• inner: impedance of the inner surface [Ω/m].
• outer: impedance of the outer surface, in [Ω/m].
• transfer: transfer impedance between the two surfaces [Ω/m].

Errors

• Throws DomainError unless the radii are finite with 0 ≤ r_in < r_ex, rho and mu_r are positive and finite, and s is finite and nonzero.

LineCableModels.Engine.Resistivity

Compute conductor resistivity as a function of temperature.

Formulas

Formula{:linear}

Resistivity of a conductor at the temperature $\theta$, by the linear correction of its reference value [9]:

\[\rho(\theta) = \rho_0 \left[1 + \alpha (\theta - T_0)\right],\]

where $\rho_0$ is the resistivity rho of the material at the reference temperature $T_0$, the field T0 [°C], in [Ω·m], $\alpha$ is the temperature coefficient alpha [1/K], and $\theta$ is the conductor temperature [°C]. The law holds within 150 K of $T_0$ and returns an infinite resistivity unchanged.

Identifiers of the resistivity formulas.

struct Formula{ID, O<:LineCableModels.Commons.FormulationOptions} <: LineCableModels.Engine.AbstractResistivity

Represent a temperature law for conductor resistivity. Its options form a record of type FormulationOptions{Formula{ID}}, with ID the formula identifier.

• options::LineCableModels.Commons.FormulationOptions: Options of the formula.

Formula(
    ::Val{ID};
    kw...
) -> LineCableModels.Engine.Resistivity.Formula

Build the formula named by ID with its declared defaults and supplied options.

Arguments

• Val{ID}: formula identifier from FORMULAS.

Keywords

• kw...: options declared by defaults for Formula{ID}.

Returns

• A Formula with a typed option record.

Errors

• Throws ArgumentError for an unknown identifier or option.

resistivity(
    _::LineCableModels.Engine.Resistivity.Formula{:linear},
    material::Material{T<:Real},
    temperature::Real
) -> Any

Return the resistivity of material at temperature by the formula Formula{:linear}, whose equation is in the section "Formulas" of Resistivity.

Arguments

• Formula{:linear}: linear temperature law.
• material: material of the conductor.
• temperature: temperature of the conductor [°C].

Returns

• Resistivity [Ω·m].

Errors

• Throws DomainError when the temperature difference from T0 is nonfinite or at least 150 K.
• Throws DomainError when the factor in brackets is nonpositive or nonfinite.