Talk:Single-Phase Line Models

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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 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]