Line Constants Math

line_constants_math.ipynb

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   "source": [
    "import numpy as np\n",
    "from math import pi\n",
    "from math import log"
   ]
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    "# Example structure\n",
    "freq = 60\n",
    "ri = 0.5540\n",
    "gmr = 0.0375\n",
    "res = 0.0263\n",
    "hi = 40"
   ]
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    "## Formulation 1\n",
    "Formulation1 is from: http://www.ece.mtu.edu/faculty/bamork/ee5200_F10/LineConstantsHKH.pdf\n",
    "\n",
    "Self impedance is split into an internal part and external part:\n",
    "\n",
    "\\begin{equation}\n",
    "Z_s = Z_i + Z_e\n",
    "\\end{equation}\n",
    "\n",
    "Internal Impeadance is:\n",
    "\\begin{equation}\n",
    "Z_i = R_i + j \\omega \\frac{\\mu}{8 \\pi}\n",
    "\\end{equation}\n",
    "\n",
    "The last term may be writen as:\n",
    "\\begin{equation}\n",
    "j \\omega \\frac{\\mu}{8 \\pi} = j \\omega \\frac{\\mu_0}{2 \\pi}\\frac{\\mu_r}{4}\n",
    "\\end{equation}\n",
    "\n",
    "External Self impedance:\n",
    "\\begin{equation}\n",
    "Z_e = j \\omega \\cdot ln \\frac{2 h + p}{r}\n",
    "\\end{equation}\n",
    "\n",
    "where $p$ is the skin depth of the ground.  \n",
    "\n",
    "\\begin{equation}\n",
    "p = \\sqrt{\\frac{\\rho}{j \\omega \\mu_0}}\n",
    "\\end{equation}\n",
    "\n",
    "Where $\\rho$ is the the soil resistivity in $[\\Omega m]$\n",
    "\n",
    "Assume that the ground permeability is equal to the permeability in free space $\\mu = \\mu_0 = 4 \\pi \\cdot 10^{-7} [H/m]$\n",
    "by definition\n",
    "\n",
    "### Generalized self impedance\n",
    "The inductive part of the internal and external impedances can be merged.  Skipping the math we end up with\n",
    "\n",
    "\\begin{equation}\n",
    "Z_s = R_i + \\frac{\\omega \\cdot \\mu_o}{8} + \\frac{j \\omega \\cdot \\mu_0}{2 \\pi} \\cdot ln \\frac{D_j}{r^{'}}\n",
    "\\end{equation}\n",
    "\n",
    "Where $r^{'} = GMR$ and \n",
    "\n",
    "\\begin{equation}\n",
    "D_j = 660 \\cdot \\sqrt{\\frac{rho [\\Omega m]}{f [Hz]}} [m]\n",
    "\\end{equation}\n",
    "\n",
    "I don't think Glover and Sarma uses this generalization."
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    "### Formulation 1: Mutual Impedance\n",
    "For low frequencies the mutual impedance is\n",
    "\n",
    "\\begin{equation}\n",
    "Z_m = \\frac{\\omega \\cdot \\mu_0}{8} + j\\omega \\cdot \\frac{\\mu_0}{2\\pi} \\cdot ln \\frac{D_j}{D^{'}} \n",
    "\\end{equation}\n",
    "\n",
    "Where $D^{'}$ is the distance between the conductor $i$ and $j$"
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   "source": [
    "The series impedance matrix is symmetrical.  With no nuetral/ground wires the matrix is:\n",
    "\n",
    "\\begin{equation}\n",
    "Z =\n",
    "\\begin{bmatrix}\n",
    "    Z_{sa} & Z_{mab} & Z_{mac} \\\\\n",
    "           & Z_{sb}  & Z_{mbc}  \\\\\n",
    "           &         & Z_{sc} \n",
    "\\end{bmatrix}\n",
    "\\end{equation}\n",
    "\n",
    "If the distance between all phases are the same and we assume a perfectly transposed system:\n",
    "\n",
    "### Ideal case\n",
    "\\begin{equation}\n",
    "Z =\n",
    "\\begin{bmatrix}\n",
    "    Z_{s}  & Z_{m} & Z_{m} \\\\\n",
    "           & Z_{s} & Z_{m}  \\\\\n",
    "           &       & Z_{s} \n",
    "\\end{bmatrix}\n",
    "\\end{equation}\n",
    "\n",
    "The positive sequence impedance is:\n",
    "\n",
    "\\begin{equation}\n",
    "Z_{pos} = Z_s - Z_m \n",
    "\\end{equation}\n",
    "\n",
    "The influence of ground disappears in the positive sequence and after much math we have:\n",
    "\n",
    "\\begin{equation}\n",
    "Z_{pos} = R_i + j \\omega \\cdot \\frac{\\mu_0}{2\\pi} ln \\frac{D^{'}}{r^{'}}\n",
    "\\end{equation}\n",
    "\n",
    "I think this case also includes the generalization above which is not used in Glover and Sarma."
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    "### Practical case\n",
    "However, the distance between phases is always differenc in practice so the positive sequence impedance becomes:\n",
    "\n",
    "\\begin{equation}\n",
    "Z_{pos} = \\frac{1}{3} * (Z_a + Z_b + Z_c)\n",
    "\\end{equation}"
   ]
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    "## Formulation 2\n",
    "Formulation2 is from https://fenix.tecnico.ulisboa.pt/downloadFile/395137455925/Resumo_ingles.pdf\n",
    "\n",
    "Self impedance is given by\n",
    "\n",
    "\\begin{equation}\n",
    "Z_{ii} = X_{ii} + Z_c + Z_g\n",
    "\\end{equation}\n",
    "\n",
    "Where the self-reacttance of the conductor *i* is given by\n",
    "\n",
    "\\begin{equation}\n",
    "X_{ii} = j \\omega L_{ii}\n",
    "\\end{equation}\n",
    "\n",
    "And the self inductance is calculated by:\n",
    "\n",
    "\\begin{equation}\n",
    "L_{ii} = \\frac{\\mu_0}{2\\pi} ln \\frac{2h_i}{r_i}\n",
    "\\end{equation}\n",
    "\n",
    "Where $h_i$ is the average height of the conductor above earth.\n",
    "\n",
    "And $\\mu_0$ is the magnetic permeability constant in the a vacuum:\n",
    "\n",
    "\\begin{equation}\n",
    "\\mu_0 = 4 \\pi \\cdot 10^{-7} [H/m]\n",
    "\\end{equation}"
   ]
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    "# Self reactance\n",
    "w = 2*pi*freq\n",
    "u0 = 4*pi*1e-7\n",
    "Lii = u0 / 2*pi * log(2*hi/ri)\n",
    "Xii = complex(0, w*Lii)"
   ]
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   "source": [
    "The mutual impedance between conductors $i$ and $j$ is given by\n",
    "\n",
    "\\begin{equation}\n",
    "Z_{ij} = X_{ij} + Z_{gm}\n",
    "\\end{equation}\n",
    "\n",
    "The mutual reactance is defined by:\n",
    "\n",
    "\\begin{equation}\n",
    "X_{ij} = j \\omega L_{ij}\n",
    "\\end{equation}\n",
    "\n",
    "And the mutual inductance is defined by:\n",
    "\n",
    "\\begin{equation}\n",
    "L_{ij} = \\frac{\\mu_0}{2\\pi} ln \\frac{D^{'}_{ij}}{D_{ij}}\n",
    "\\end{equation}\n",
    "\n",
    "Where $D_{ij}$ is the distance between conductor $i$ and $j$\n",
    "\n",
    "Where $D^{'}_{ij}$ is the distance between conductor $i$ and the image of conductor $j$"
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