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:

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