Talk:Single-Phase Line Models

Footnotes

Derivation of Adjusted Line Parameters for Equivalent $\pi$ Model

In order to get the same ABCD parameters as the distributed parameter line, the nominal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} line impedance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{Z}} and admittance $Y_{C}\,$ need to be adjusted such that:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left[{\begin{matrix}A&C\\B&D\end{matrix}}\right]=\left[{\begin{matrix}\left(1+{\frac {{\boldsymbol {Z'}}{\boldsymbol {Y'}}}{2}}\right)&{\boldsymbol {Z'}}\\\\{\boldsymbol {Y'}}\left(1+{\frac {{\boldsymbol {Z'}}{\boldsymbol {Y'}}}{4}}\right)&\left(1+{\frac {{\boldsymbol {Z'}}{\boldsymbol {Y'}}}{2}}\right)\end{matrix}}\right]=\left[{\begin{matrix}\cosh({\boldsymbol {\gamma }}l)&{\boldsymbol {Z}}_{c}\sinh({\boldsymbol {\gamma }}l)\\\\{\frac {1}{{\boldsymbol {Z}}_{c}}}sinh({\boldsymbol {\gamma }}l)&\cosh({\boldsymbol {\gamma }}l)\end{matrix}}\right]\,}

Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{Z'}} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\boldsymbol {Y'}}\,} are the adjusted line impedance and admittance respectively

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l \, } is the length of the line (m)

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\boldsymbol {\gamma }}={\sqrt {\boldsymbol {zy}}}} is the propagation constant (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m^{-1}} )

${\boldsymbol {Z}}_{c}={\sqrt {\boldsymbol {\frac {z}{y}}}}$ is the characteristic impedance (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega} )

We now want to determine the adjusted impedance and admittance in terms of their original values so that we can easily convert a nominal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} line into an equivalent Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} line.

Firstly, it should be noted that the uppercase parameters represent total values whereas the lowercase parameters are per-length values, i.e. the relationship between upper and lower cases parameters is as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{Z} = \boldsymbol{z} l }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{Y} = \boldsymbol{y} l }

(Note that the conductance G is assumed to be 0 in the nominal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} model, hence Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\boldsymbol {y}}=j\omega C} S/m)

So let's consider the C term:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle C={\boldsymbol {Z'}}={\boldsymbol {Z}}_{c}\sinh({\boldsymbol {\gamma }}l)}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle =\left[{\sqrt {\frac {\boldsymbol {z}}{\boldsymbol {y}}}}\sinh({\boldsymbol {\gamma }}l)\right]\left({\frac {{\boldsymbol {z}}l}{{\boldsymbol {z}}l}}\right)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \left[ \frac{\sinh(\boldsymbol{\gamma} l)}{\sqrt{\boldsymbol{zy}} l} \right] \boldsymbol{z} l }

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle =\left[{\frac {\sinh({\boldsymbol {\gamma }}l)}{{\boldsymbol {\gamma }}l}}\right]{\boldsymbol {Z}}}

Similarly, consider the A term:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A=\left(1+{\frac {{\boldsymbol {Z'}}{\boldsymbol {Y'}}}{2}}\right)=\cosh({\boldsymbol {\gamma }}l)\,}

Re-arranging the above, we get:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\boldsymbol{Y'}}{2} = \frac{\cosh (\boldsymbol{\gamma} l) - 1 }{\boldsymbol{Z'}} }

Substituting in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{Z'} = \boldsymbol{Z}_{c} \sinh(\boldsymbol{\gamma} l) } :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\boldsymbol{Y'}}{2} = \frac{\cosh (\boldsymbol{\gamma} l) - 1 }{\boldsymbol{Z}_{c} \sinh(\boldsymbol{\gamma} l)} }

Using the hyperbolic half-angle identity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tanh \frac{x}{2} = \frac{\cosh x - 1}{\sinh x} } :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\boldsymbol{Y'}}{2} = \frac{\tanh \left( \frac{\boldsymbol{\gamma} l}{2} \right)}{\boldsymbol{Z}_{c}} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \left[ \frac{\tanh \left( \frac{\boldsymbol{\gamma} l}{2} \right)}{\sqrt{\frac{\boldsymbol{z}}{\boldsymbol{y}}}} \right] \left( \frac{\boldsymbol{y} l}{\boldsymbol{y} l} \right) }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \left[ \frac{\tanh \left( \frac{\boldsymbol{\gamma} l}{2} \right)}{\sqrt{\boldsymbol{zy}} l} \right] \boldsymbol{y} l }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \left[ \frac{\tanh \left( \frac{\boldsymbol{\gamma} l}{2} \right)}{\boldsymbol{\gamma} l} \right] \boldsymbol{Y} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \left[ \frac{\tanh \left( \frac{\boldsymbol{\gamma} l}{2} \right)}{\frac{\boldsymbol{\gamma} l}{2}} \right] \frac{\boldsymbol{Y}}{2} }

Alternative Representation of Adjusted Line Parameters

From the above section, we derived the following adjusted line parameters for the equivalent Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} line:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{Z'} = \left[ \frac{\sinh(\boldsymbol{\gamma} l)}{\boldsymbol{\gamma} l} \right] \boldsymbol{Z} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\boldsymbol{Y'}}{2} = \left[ \frac{\tanh \left( \frac{\boldsymbol{\gamma} l}{2} \right)}{\frac{\boldsymbol{\gamma} l}{2}} \right] \frac{\boldsymbol{Y}}{2} }

Knowing that the expression for characteristic impedance can be manipulated as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{Z}_{c} = \sqrt{\frac{\boldsymbol{z}}{\boldsymbol{y}}} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac{\boldsymbol{z}}{\sqrt{\boldsymbol{zy}}} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac{\boldsymbol{z}}{\boldsymbol{\gamma}} }

Or using similar logic:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{Z}_{c} = \frac{\boldsymbol{\gamma}}{\boldsymbol{y}} }

Using the above expressions, we can represent Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{Z'}} in an alternative manner as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{Z'} = \left[ \frac{\sinh(\boldsymbol{\gamma} l)}{\boldsymbol{\gamma} l} \right] \boldsymbol{z}l }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \left[ \frac{\sinh(\boldsymbol{\gamma} l)}{\frac{\boldsymbol{z}}{\boldsymbol{Z}_{c}} l} \right] \boldsymbol{z}l }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \boldsymbol{Z}_{c} \sinh(\boldsymbol{\gamma} l) }

Similarly, we can represent Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\boldsymbol{Y'}}{2} } as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\boldsymbol{Y'}}{2} = \frac{1}{\boldsymbol{Z}_{c}} \tanh \left( \frac{\boldsymbol{\gamma} l}{2} \right) }