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Document EMT line parameter models
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‎GridKit/Model/EMT/Parameters/Effects/Carson/README.md‎

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# Carson Model
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`Carson` computes frequency-dependent earth-return resistance and inductance
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matrices for overhead conductors using the Deri-Semlyen approximation to
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Carson's homogeneous-earth return term.
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## Model Parameters
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Symbol | Units | JSON | Description | Note
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------ | ----- | ---- | ----------- | ----
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`tower` | [-] | - | Static conductor tower | [`Tower`](../../Geometry/Tower/README.md)
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$\sigma_g$ | [S/m] | `earth_conductivity` | Earth conductivity | $\sigma_g\ge0$
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$\varepsilon_g$ | [F/m] | `earth_permittivity` | Earth permittivity | $\varepsilon_g\ge\varepsilon_0$
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$\mu_0$ | [H/m] | - | Vacuum permeability | constant
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$\varepsilon_0$ | [F/m] | - | Vacuum permittivity | constant
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### Parameter Validation
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```math
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\sigma_g\ge0,\qquad
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\varepsilon_g\ge\varepsilon_0,\qquad
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\omega > 0,\qquad
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(\gamma_g^{\mathrm r})^2+(\gamma_g^{\mathrm i})^2 > 0
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```
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The connected tower model owns its geometry validation.
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### Model Derived Parameters
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For conductor pair $(i,j)$,
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```math
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\begin{aligned}
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d_{ij} &= |x_i-x_j| \\
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s_{ij} &= h_i+h_j \\
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\Delta_g &= (\gamma_g^{\mathrm r})^2+(\gamma_g^{\mathrm i})^2 \\
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p^{\mathrm r} &= \frac{\gamma_g^{\mathrm r}}{\Delta_g} \\
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p^{\mathrm i} &= -\frac{\gamma_g^{\mathrm i}}{\Delta_g} \\
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H_{ij}^{\mathrm r} &= s_{ij}+2p^{\mathrm r} \\
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H_{ij}^{\mathrm i} &= 2p^{\mathrm i} \\
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A_{ij} &= d_{ij}^2+(H_{ij}^{\mathrm r})^2-(H_{ij}^{\mathrm i})^2 \\
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B_{ij} &= 2H_{ij}^{\mathrm r}H_{ij}^{\mathrm i} \\
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M_{ij} &= A_{ij}^2+B_{ij}^2 \\
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\theta_{ij} &= \operatorname{atan2}(B_{ij},A_{ij}) \\
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\Lambda_{ij} &= \frac{1}{4}\ln M_{ij}-\ln D'_{ij}
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\end{aligned}
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```
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## Model Variables
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### Internal Variables
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#### Differential
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None.
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#### Algebraic
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Symbol | Units | Description | Note
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------ | ----- | ----------- | ----
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$\eta_g$ | [1/m$^2$] | Magnitude of earth-return propagation radicand | real
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$\gamma_g^{\mathrm r}$ | [1/m] | Real earth-return propagation constant component | real
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$\gamma_g^{\mathrm i}$ | [1/m] | Imaginary earth-return propagation constant component | real
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$\mathbf{R}^{\mathrm{carson}}$ | [$\Omega$/m] | Carson earth-return resistance | $\mathbb{R}^{K\times K}$
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$\mathbf{L}^{\mathrm{carson}}$ | [H/m] | Carson earth-return inductance | $\mathbb{R}^{K\times K}$
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### External Variables
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#### Differential
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None.
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#### Algebraic
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None.
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## Model Equations
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### Differential Equations
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None.
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### Algebraic Equations
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```math
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\begin{aligned}
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0 &= -\eta_g^2
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+ (\omega^2\mu_0\varepsilon_g)^2
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+ (\omega\mu_0\sigma_g)^2 \\
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0 &= -2(\gamma_g^{\mathrm r})^2
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+ \eta_g - \omega^2\mu_0\varepsilon_g \\
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0 &= -2(\gamma_g^{\mathrm i})^2
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+ \eta_g + \omega^2\mu_0\varepsilon_g \\
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0 &= -4\pi R_{ij}^{\mathrm{carson}}
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- \omega\mu_0\theta_{ij} \\
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0 &= -2\pi L_{ij}^{\mathrm{carson}}
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+ \mu_0\Lambda_{ij}
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\end{aligned}
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```
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## Initialization
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Initialize $\eta_g$, $\gamma_g^{\mathrm r}$, $\gamma_g^{\mathrm i}$,
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$\mathbf{R}^{\mathrm{carson}}$, and $\mathbf{L}^{\mathrm{carson}}$ from the
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algebraic equations at the current $\omega$.
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## Model Outputs
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Symbol | Units | Description | Note
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------ | ----- | ----------- | ----
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$\sigma_g$ | [S/m] | Earth conductivity | real
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$\varepsilon_g$ | [F/m] | Earth permittivity | real
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$\eta_g$ | [1/m$^2$] | Magnitude of earth-return propagation radicand | real
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$\gamma_g^{\mathrm r}$ | [1/m] | Real earth-return propagation constant component | real
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$\gamma_g^{\mathrm i}$ | [1/m] | Imaginary earth-return propagation constant component | real
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$\mathbf{R}^{\mathrm{carson}}$ | [$\Omega$/m] | Carson earth-return resistance | $\mathbb{R}^{K\times K}$
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$\mathbf{L}^{\mathrm{carson}}$ | [H/m] | Carson earth-return inductance | $\mathbb{R}^{K\times K}$

‎GridKit/Model/EMT/Parameters/Effects/GeometricInductance/README.md‎

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# GeometricInductance Model
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`GeometricInductance` computes the external geometric series inductance matrix
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for the physical conductors.
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## Model Parameters
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For $K$ physical conductors:
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Symbol | Units | Description | Note
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------ | ----- | ----------- | ----
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`tower` | [-] | Static conductor tower | [`Tower`](../../Geometry/Tower/README.md)
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`conductor` | [-] | Static conductor data | [`Conductor`](../../Geometry/Conductor/README.md)
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$\mu_0$ | [H/m] | Vacuum permeability | constant
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### Parameter Validation
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The tower and conductor models must pass their own validation.
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### Model Derived Parameters
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The tower model provides $h_i$, $D_{ij}$, and $D'_{ij}$. The conductor model
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provides $r_i$.
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```math
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\left(\boldsymbol{\Lambda}^{\mathrm{geo}}\right)_{ij} =
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\begin{cases}
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\ln\dfrac{2h_i}{r_i}, & i=j \\
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\ln\dfrac{D'_{ij}}{D_{ij}}, & i\ne j
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\end{cases}
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```
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## Model Variables
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### Internal Variables
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#### Differential
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None.
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#### Algebraic
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Symbol | Units | Description | Note
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------ | ----- | ----------- | ----
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$\mathbf{L}^{\mathrm{geo}}$ | [H/m] | External geometric inductance | $\mathbb{R}^{K\times K}$
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### External Variables
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#### Differential
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None.
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#### Algebraic
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None.
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## Model Equations
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### Differential Equations
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None.
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### Algebraic Equations
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```math
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\mathbf{0} = -2\pi\mathbf{L}^{\mathrm{geo}}
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+ \mu_0 \boldsymbol{\Lambda}^{\mathrm{geo}}
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```
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## Initialization
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None beyond the static tower and conductor initialization.
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## Model Outputs
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Symbol | Units | Description | Note
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------ | ----- | ----------- | ----
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$\mathbf{L}^{\mathrm{geo}}$ | [H/m] | External geometric inductance | $\mathbb{R}^{K\times K}$

‎GridKit/Model/EMT/Parameters/Effects/InsulatorLeakage/README.md‎

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# InsulatorLeakage Model
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`InsulatorLeakage` computes an empirical direct shunt conductance effect for
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the physical conductors.
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## Model Parameters
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For $K$ physical conductors:
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Symbol | Units | JSON | Description | Note
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------ | ----- | ---- | ----------- | ----
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$g_i^{\mathrm{leak}}$ | [S/m] | `leakage_conductance` | Insulator leakage conductance to ground | $g_i^{\mathrm{leak}}\ge0$
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### Parameter Validation
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```math
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g_i^{\mathrm{leak}}\ge0,\qquad i=1,\dots,K
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```
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### Model Derived Parameters
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For a vector $\mathbf a\in\mathbb{R}^K$, let
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$\operatorname{diag}(\mathbf a)$ denote the diagonal matrix with $\mathbf a$ on
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the main diagonal.
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## Model Variables
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### Internal Variables
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#### Differential
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None.
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#### Algebraic
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Symbol | Units | Description | Note
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------ | ----- | ----------- | ----
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$\mathbf{G}^{\mathrm{leak}}$ | [S/m] | Insulator leakage shunt conductance | diagonal
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### External Variables
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#### Differential
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None.
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#### Algebraic
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None.
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## Model Equations
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### Differential Equations
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None.
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### Algebraic Equations
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```math
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\mathbf{0} = -\mathbf{G}^{\mathrm{leak}}
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+ \operatorname{diag}(\mathbf{g}^{\mathrm{leak}})
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```
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## Initialization
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Validate the leakage conductance vector.
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## Model Outputs
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Symbol | Units | Description | Note
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------ | ----- | ----------- | ----
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$\mathbf{G}^{\mathrm{leak}}$ | [S/m] | Insulator leakage shunt conductance | $\mathbb{R}^{K\times K}$

‎GridKit/Model/EMT/Parameters/Effects/README.md‎

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# Effect Models
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`Effects` contains the physical effect models that assemble full-conductor
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per-unit-length overhead-line data:
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```math
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\mathbf{R}'(\omega),\quad \mathbf{L}'(\omega),\quad
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\mathbf{G}'(\omega),\quad \mathbf{C}'(\omega)
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```
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These models are frequency-domain parameter models, not EMT network components.
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They consume geometry, conductor, and earth data and produce algebraic signals
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for the response models. They are evaluated at the current system parameter
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$\omega$; sweep endpoints, sampling, and linear or logarithmic grids belong to
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the application or study driver.
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Model | Description
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----- | -----------
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[`SeriesImpedance`](SeriesImpedance/README.md) | Builds full-conductor per-unit-length series resistance and inductance
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[`GeometricInductance`](GeometricInductance/README.md) | Computes external geometric inductance entries
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[`SkinEffect`](SkinEffect/README.md) | Computes conductor-internal skin-effect resistance and inductance entries
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[`ShuntAdmittance`](ShuntAdmittance/README.md) | Builds full-conductor per-unit-length shunt conductance and capacitance
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[`ShuntPotential`](ShuntPotential/README.md) | Computes potential-derived zero conductance and capacitance
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[`InsulatorLeakage`](InsulatorLeakage/README.md) | Computes direct shunt leakage conductance
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[`Carson`](Carson/README.md) | Computes Carson earth-return resistance and inductance entries
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## Assembly
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Output | Built From
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------ | ----------
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$\mathbf{R}'$ | Internal skin-effect resistance and Carson earth-return resistance
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$\mathbf{L}'$ | Internal skin-effect inductance, geometric inductance, and Carson earth-return inductance
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$\mathbf{G}'$ | Shunt potential terms and optional direct leakage terms
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$\mathbf{C}'$ | Shunt potential terms
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The aggregate [`Overhead`](../README.md#overhead-aggregate) model wires these
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effect outputs into the response models.

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