Talk:Distributed Parameter Line Model

Footnotes

Derivation of A₁ and A₂

Based on the boundary conditions at the receiving end of the line ($x=0\,$), i.e.

${\boldsymbol {V}}(0)={\boldsymbol {V_{r}}}\,$

${\boldsymbol {I}}(0)={\boldsymbol {I_{r}}}\,$

The voltage and current equations are as follows:

${\boldsymbol {V}}(0)={\boldsymbol {V_{r}}}=A_{1}e^{{\boldsymbol {\gamma }}(0)}+A_{2}e^{-{\boldsymbol {\gamma }}(0)}\,$

${\boldsymbol {I}}(0)={\boldsymbol {I_{r}}}={\frac {A_{1}e^{\gamma (0)}-A_{2}e^{-\gamma (0)}}{{\boldsymbol {Z}}_{c}}}\,$

Therefore,

${\boldsymbol {V_{r}}}=A_{1}+A_{2}\,$

${\boldsymbol {I_{r}}}={\frac {A_{1}-A_{2}}{{\boldsymbol {Z}}_{c}}}\,$

Substituting $A_{1}={\boldsymbol {I_{r}}}{\boldsymbol {Z}}_{c}+A_{2}\,$ into ${\boldsymbol {V_{r}}}\,$:

${\boldsymbol {V_{r}}}={\boldsymbol {I_{r}}}{\boldsymbol {Z}}_{c}+A_{2}+A_{2}\,$

$\Rightarrow A_{2}={\frac {{\boldsymbol {V_{r}}}-{\boldsymbol {I_{r}}}{\boldsymbol {Z}}_{c}}{2}}\,$

Solving for A₁, we get:

${\boldsymbol {V_{r}}}=A_{1}+{\frac {{\boldsymbol {V_{r}}}-{\boldsymbol {I_{r}}}{\boldsymbol {Z}}_{c}}{2}}\,$

$\Rightarrow A_{1}={\frac {{\boldsymbol {V_{r}}}+{\boldsymbol {I_{r}}}{\boldsymbol {Z}}_{c}}{2}}\,$

Diagonality of Modal Characteristic Impedance Matrix

The modal characteristic impedance matrix is:

$[Z_{c}]=[\gamma ]^{-1}[T_{v}]^{-1}[Z][T_{i}]\,$

If $[\gamma ]\,$ is diagonal, then we shall prove that $[Z_{c}]\,$ is also diagonal. This is done by proving that $[T_{v}]^{-1}[Z][T_{i}]\,$ is diagonal.

We saw that after transformation, the first order differential equations for voltage and current are:

${\frac {d{\boldsymbol {V'}}}{dx}}=[T_{v}]^{-1}[Z][T_{i}]{\boldsymbol {I'}}\,$ ... Equ. (F1)

${\frac {d{\boldsymbol {I'}}}{dx}}=[T_{i}]^{-1}[Y][T_{v}]{\boldsymbol {V'}}\,$ ... Equ. (F2)

We also saw that the square of the modal propagation constants are the eigenvalues of [Z][Y] and [Y][Z], i.e.

$[\gamma ]^{2}=[\lambda ]=[T_{v}]^{-1}[Z][Y][T_{v}]=[T_{i}]^{-1}[Y][Z][T_{i}]\,$ ... Equ. (F3)

If we were to multiply the coefficients of Equations (F1) and (F2), we'd get:

$([T_{v}]^{-1}[Z][T_{i}])([T_{i}]^{-1}[Y][T_{v}])=[T_{v}]^{-1}[Z][Y][T_{v}]\,$ ... Equ. (F4a)

or

$([T_{i}]^{-1}[Y][T_{v}])([T_{v}]^{-1}[Z][T_{i}])=[T_{i}]^{-1}[Y][Z][T_{i}]\,$ ... Equ. (F4b)

We can therefore equate (F4a) and (F4b) with (F3):

$([T_{v}]^{-1}[Z][T_{i}])([T_{i}]^{-1}[Y][T_{v}])=([T_{i}]^{-1}[Y][T_{v}])([T_{v}]^{-1}[Z][T_{i}])=[\gamma ]^{2}\,$ ... Equ. (F5)

We know that the modal propagation constant matrix $[\gamma ]^{2}\,$ is diagonal. Thus for Equation (F5) to hold, then $([T_{v}]^{-1}[Z][T_{i}])\,$ and $([T_{i}]^{-1}[Y][T_{v}])\,$ must also be diagonal. And therefore the modal characteristic impedance matrix is also diagonal.