Example of how to partition phase impedance matrix to calculate the zero sequence mutual impedance.

mutualz_math.ipynb

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    "# Mutual Impedance Calculation"
   ]
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    "Lets start with a the phase impedance matrix for a two circuit structure.  We'll ignore the shield wires for this example.  \n",
    "\n",
    "Circuit 1 has the phases $(a, b, c)$\n",
    "\n",
    "Circuit 2 has the phases $(a', b', c')$\n",
    "\n",
    "$$\n",
    "Z_p = \\begin{bmatrix}\n",
    "    Z_{aa} & Z_{ab} & Z_{ac} & Z_{a'a} & Z_{a'b} & Z_{a'c}  \\\\\n",
    "    Z_{ba} & Z_{bb} & Z_{bc} & Z_{b'a} & Z_{b'b} & Z_{b'c}  \\\\\n",
    "    Z_{ca} & Z_{cb} & Z_{cc} & Z_{c'a} & Z_{c'b} & Z_{c'c}  \\\\\n",
    "    Z_{aa'} & Z_{ab'} & Z_{ac'} & Z_{a'a'} & Z_{a'b'} & Z_{a'c'}  \\\\\n",
    "    Z_{ba'} & Z_{bb'} & Z_{bc'} & Z_{b'a'} & Z_{b'b'} & Z_{b'c'}  \\\\\n",
    "    Z_{ca'} & Z_{cb'} & Z_{cc'} & Z_{c'a'} & Z_{c'b'} & Z_{c'c'}  \\\\\n",
    "\\end{bmatrix}\n",
    "$$"
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    "This matrix can be partitioned into the sub-matrices\n",
    "\n",
    "$$\n",
    "Z_p=\n",
    "\\left[\n",
    "\\begin{array}{c|c}\n",
    "A & B \\\\\n",
    "\\hline\n",
    "C & D\n",
    "\\end{array}\n",
    "\\right]\n",
    "$$\n",
    "\n",
    "Where partitions $A, D$ represent the self-impedance of circuits 1 and 2 \n",
    "\n",
    "And partiion $B, C$ represent the mutual coupling between circuits 1 and 2"
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    "## Mutual Impedance\n",
    "To calculate the mutual coupling between circuit 1 and 2 we use partition $B$ or $C$ (either parition gives the same result).\n",
    "\n",
    "$$\n",
    "Z_{p12} = \\begin{bmatrix}\n",
    "    Z_{aa'} & Z_{ab'} & Z_{ac'} \\\\\n",
    "    Z_{ba'} & Z_{bb'} & Z_{bc'} \\\\\n",
    "    Z_{ca'} & Z_{cb'} & Z_{cc'} \\\\\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "This is the phase matrix which we must convert to the sequence components using the transform:\n",
    "\n",
    "$$\n",
    "A_s = \\begin{bmatrix}\n",
    "    1 & 1 & 1 \\\\\n",
    "    1 & a^2 & a \\\\\n",
    "    1 & a   & a^2 \\\\\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "So:\n",
    "\n",
    "$$\n",
    "Z_s = As^{-1} \\cdot Z_p \\cdot A_s\n",
    "$$\n",
    "\n",
    "The resulting 3x3 matrix is in the format with $$Z0m$$ representing the zero sequence mutual coupling between the circuits:\n",
    "\n",
    "$$\n",
    "Z_s = \\begin{bmatrix}\n",
    "    Z0m & 0 & 0 \\\\\n",
    "    0 & 0 & 0 \\\\\n",
    "    0 & 0 & 0 \\\\\n",
    "\\end{bmatrix}\n",
    "$$"
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    "## Conversions\n",
    "Remember that this impedance is ohms per meter.  Get the impedance for the entire line:\n",
    "\n",
    "$$\n",
    "Z0m_{\\Omega} = Z0m * len\n",
    "$$\n",
    "\n",
    "We convert to per-unit impedance for the modeling tool.  Remember that if the two circuits have different voltages bases we convert to a common base.\n",
    "\n",
    "$$\n",
    "Z0m_{pu} = \\frac{Z0m_{\\Omega}}{Z_{base}} \n",
    "$$"
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