# 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 $\pi$ line impedance $\boldsymbol{Z}$ and admittance $Y_{C} \,$ need to be adjusted such that:

$\left[ \begin{matrix} A & C \\ B & D \end{matrix} \right] = \left[ \begin{matrix} \left( 1 + \frac{\boldsymbol{Z'} \boldsymbol{Y'}}{2} \right) & \boldsymbol{Z'} \\ \\ \boldsymbol{Y'} \left( 1 + \frac{\boldsymbol{Z'} \boldsymbol{Y'}}{4} \right) & \left( 1 + \frac{\boldsymbol{Z'} \boldsymbol{Y'}}{2} \right) \end{matrix} \right] = \left[ \begin{matrix} \cosh (\boldsymbol{\gamma} l) & \boldsymbol{Z}_{c} \sinh(\boldsymbol{\gamma} l) \\ \\ \frac{1}{\boldsymbol{Z}_{c}} sinh (\boldsymbol{\gamma} l) & \cosh(\boldsymbol{\gamma} l) \end{matrix} \right] \,$

Where $\boldsymbol{Z'}$ and $\boldsymbol{Y'} \,$ are the adjusted line impedance and admittance respectively

$l \,$ is the length of the line (m)
$\boldsymbol{\gamma} = \sqrt{\boldsymbol{zy}}$ is the propagation constant ($m^{-1}$)
$\boldsymbol{Z}_{c} = \sqrt{\boldsymbol{\frac{z}{y}}}$ is the characteristic impedance ($\Omega$)

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

$\boldsymbol{Z} = \boldsymbol{z} l$
$\boldsymbol{Y} = \boldsymbol{y} l$

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

So let's consider the C term:

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

Similarly, consider the A term:

$A = \left( 1 + \frac{\boldsymbol{Z'} \boldsymbol{Y'}}{2} \right) = \cosh (\boldsymbol{\gamma} l) \,$

Re-arranging the above, we get:

$\frac{\boldsymbol{Y'}}{2} = \frac{\cosh (\boldsymbol{\gamma} l) - 1 }{\boldsymbol{Z'}}$

Substituting in $\boldsymbol{Z'} = \boldsymbol{Z}_{c} \sinh(\boldsymbol{\gamma} l)$:

$\frac{\boldsymbol{Y'}}{2} = \frac{\cosh (\boldsymbol{\gamma} l) - 1 }{\boldsymbol{Z}_{c} \sinh(\boldsymbol{\gamma} l)}$

Using the hyperbolic half-angle identity $\tanh \frac{x}{2} = \frac{\cosh x - 1}{\sinh x}$:

$\frac{\boldsymbol{Y'}}{2} = \frac{\tanh \left( \frac{\boldsymbol{\gamma} l}{2} \right)}{\boldsymbol{Z}_{c}}$
$= \left[ \frac{\tanh \left( \frac{\boldsymbol{\gamma} l}{2} \right)}{\sqrt{\frac{\boldsymbol{z}}{\boldsymbol{y}}}} \right] \left( \frac{\boldsymbol{y} l}{\boldsymbol{y} l} \right)$
$= \left[ \frac{\tanh \left( \frac{\boldsymbol{\gamma} l}{2} \right)}{\sqrt{\boldsymbol{zy}} l} \right] \boldsymbol{y} l$
$= \left[ \frac{\tanh \left( \frac{\boldsymbol{\gamma} l}{2} \right)}{\boldsymbol{\gamma} l} \right] \boldsymbol{Y}$
$= \left[ \frac{\tanh \left( \frac{\boldsymbol{\gamma} l}{2} \right)}{\frac{\boldsymbol{\gamma} l}{2}} \right] \frac{\boldsymbol{Y}}{2}$

### Alternative Representation of Adjusted Line Parameters

From the above section, we derived the following adjusted line parameters for the equivalent $\pi$ line:

$\boldsymbol{Z'} = \left[ \frac{\sinh(\boldsymbol{\gamma} l)}{\boldsymbol{\gamma} l} \right] \boldsymbol{Z}$
$\frac{\boldsymbol{Y'}}{2} = \left[ \frac{\tanh \left( \frac{\boldsymbol{\gamma} l}{2} \right)}{\frac{\boldsymbol{\gamma} l}{2}} \right] \frac{\boldsymbol{Y}}{2}$

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

$\boldsymbol{Z}_{c} = \sqrt{\frac{\boldsymbol{z}}{\boldsymbol{y}}}$
$= \frac{\boldsymbol{z}}{\sqrt{\boldsymbol{zy}}}$
$= \frac{\boldsymbol{z}}{\boldsymbol{\gamma}}$

Or using similar logic:

$\boldsymbol{Z}_{c} = \frac{\boldsymbol{\gamma}}{\boldsymbol{y}}$

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

$\boldsymbol{Z'} = \left[ \frac{\sinh(\boldsymbol{\gamma} l)}{\boldsymbol{\gamma} l} \right] \boldsymbol{z}l$
$= \left[ \frac{\sinh(\boldsymbol{\gamma} l)}{\frac{\boldsymbol{z}}{\boldsymbol{Z}_{c}} l} \right] \boldsymbol{z}l$
$= \boldsymbol{Z}_{c} \sinh(\boldsymbol{\gamma} l)$

Similarly, we can represent $\frac{\boldsymbol{Y'}}{2}$ as follows:

$\frac{\boldsymbol{Y'}}{2} = \frac{1}{\boldsymbol{Z}_{c}} \tanh \left( \frac{\boldsymbol{\gamma} l}{2} \right)$