Jules: Created page with "== Footnotes == === Derivation of Adjusted Line Parameters for Equivalent $\pi$ Model === In order to get the same ABCD parameters as the distributed parameter li..."

2020-11-22T07:35:51Z

Created page with "== Footnotes == === Derivation of Adjusted Line Parameters for Equivalent <math>\pi</math> Model === In order to get the same ABCD parameters as the distributed parameter li..."

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== Footnotes ==

=== Derivation of Adjusted Line Parameters for Equivalent <math>\pi</math> Model ===

In order to get the same ABCD parameters as the distributed parameter line, the nominal <math>\pi</math> line impedance <math>\boldsymbol{Z}</math> and admittance <math>Y_{C} \,</math> need to be adjusted such that:

: <math> \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] \, </math>

Where <math>\boldsymbol{Z'}</math> and <math>\boldsymbol{Y'} \,</math> are the adjusted line impedance and admittance respectively
:: <math> l \, </math> is the length of the line (m)
:: <math> \boldsymbol{\gamma} = \sqrt{\boldsymbol{zy}} </math> is the propagation constant (<math>m^{-1}</math>)
:: <math> \boldsymbol{Z}_{c} = \sqrt{\boldsymbol{\frac{z}{y}}} </math> is the characteristic impedance (<math>\Omega</math>)

We now want to determine the adjusted impedance and admittance in terms of their original values so that we can easily convert a nominal <math>\pi</math> line into an equivalent <math>\pi</math> 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:

: <math> \boldsymbol{Z} = \boldsymbol{z} l </math>

: <math> \boldsymbol{Y} = \boldsymbol{y} l </math>

(Note that the conductance G is assumed to be 0 in the nominal <math>\pi</math> model, hence <math> \boldsymbol{y} = j \omega C </math> S/m)

So let's consider the C term:

: <math>C = \boldsymbol{Z'} = \boldsymbol{Z}_{c} \sinh(\boldsymbol{\gamma} l)</math>
:::: <math>= \left[ \sqrt{\frac{\boldsymbol{z}}{\boldsymbol{y}}} \sinh(\boldsymbol{\gamma} l) \right] \left( \frac{\boldsymbol{z} l}{\boldsymbol{z} l} \right) </math>
:::: <math>= \left[ \frac{\sinh(\boldsymbol{\gamma} l)}{\sqrt{\boldsymbol{zy}} l} \right] \boldsymbol{z} l </math>
:::: <math>= \left[ \frac{\sinh(\boldsymbol{\gamma} l)}{\boldsymbol{\gamma} l} \right] \boldsymbol{Z} </math>

Similarly, consider the A term:

: <math> A = \left( 1 + \frac{\boldsymbol{Z'} \boldsymbol{Y'}}{2} \right) = \cosh (\boldsymbol{\gamma} l) \, </math>

Re-arranging the above, we get:

: <math> \frac{\boldsymbol{Y'}}{2} = \frac{\cosh (\boldsymbol{\gamma} l) - 1 }{\boldsymbol{Z'}} </math>

Substituting in <math>\boldsymbol{Z'} = \boldsymbol{Z}_{c} \sinh(\boldsymbol{\gamma} l) </math>:

: <math> \frac{\boldsymbol{Y'}}{2} = \frac{\cosh (\boldsymbol{\gamma} l) - 1 }{\boldsymbol{Z}_{c} \sinh(\boldsymbol{\gamma} l)} </math>

Using the [http://en.wikipedia.org/wiki/Hyperbolic_function#Identities hyperbolic half-angle identity] <math> \tanh \frac{x}{2} = \frac{\cosh x - 1}{\sinh x} </math>:

: <math> \frac{\boldsymbol{Y'}}{2} = \frac{\tanh \left( \frac{\boldsymbol{\gamma} l}{2} \right)}{\boldsymbol{Z}_{c}} </math>

:: <math> = \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) </math>

:: <math> = \left[ \frac{\tanh \left( \frac{\boldsymbol{\gamma} l}{2} \right)}{\sqrt{\boldsymbol{zy}} l} \right] \boldsymbol{y} l </math>

:: <math> = \left[ \frac{\tanh \left( \frac{\boldsymbol{\gamma} l}{2} \right)}{\boldsymbol{\gamma} l} \right] \boldsymbol{Y} </math>

:: <math> = \left[ \frac{\tanh \left( \frac{\boldsymbol{\gamma} l}{2} \right)}{\frac{\boldsymbol{\gamma} l}{2}} \right] \frac{\boldsymbol{Y}}{2} </math>

=== Alternative Representation of Adjusted Line Parameters ===

From the above section, we derived the following adjusted line parameters for the equivalent <math>\pi</math> line:

: <math>\boldsymbol{Z'} = \left[ \frac{\sinh(\boldsymbol{\gamma} l)}{\boldsymbol{\gamma} l} \right] \boldsymbol{Z} </math>

: <math> \frac{\boldsymbol{Y'}}{2} = \left[ \frac{\tanh \left( \frac{\boldsymbol{\gamma} l}{2} \right)}{\frac{\boldsymbol{\gamma} l}{2}} \right] \frac{\boldsymbol{Y}}{2} </math>

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

: <math>\boldsymbol{Z}_{c} = \sqrt{\frac{\boldsymbol{z}}{\boldsymbol{y}}} </math>
:: <math> = \frac{\boldsymbol{z}}{\sqrt{\boldsymbol{zy}}} </math>
:: <math> = \frac{\boldsymbol{z}}{\boldsymbol{\gamma}} </math>

Or using similar logic:

:: <math>\boldsymbol{Z}_{c} = \frac{\boldsymbol{\gamma}}{\boldsymbol{y}} </math>

Using the above expressions, we can represent <math>\boldsymbol{Z'}</math> in an alternative manner as follows:

: <math>\boldsymbol{Z'} = \left[ \frac{\sinh(\boldsymbol{\gamma} l)}{\boldsymbol{\gamma} l} \right] \boldsymbol{z}l </math>
:: <math> = \left[ \frac{\sinh(\boldsymbol{\gamma} l)}{\frac{\boldsymbol{z}}{\boldsymbol{Z}_{c}} l} \right] \boldsymbol{z}l </math>
:: <math> = \boldsymbol{Z}_{c} \sinh(\boldsymbol{\gamma} l) </math>

Similarly, we can represent <math> \frac{\boldsymbol{Y'}}{2} </math> as follows:

: <math> \frac{\boldsymbol{Y'}}{2} = \frac{1}{\boldsymbol{Z}_{c}} \tanh \left( \frac{\boldsymbol{\gamma} l}{2} \right) </math>

Talk:Single-Phase Line Models - Revision history

Jules: Created page with "== Footnotes == === Derivation of Adjusted Line Parameters for Equivalent \pi Model === In order to get the same ABCD parameters as the distributed parameter li..."

Jules: Created page with "== Footnotes == === Derivation of Adjusted Line Parameters for Equivalent $\pi$ Model === In order to get the same ABCD parameters as the distributed parameter li..."