Talk:Distributed Parameter Line Model

Footnotes

Derivation of A₁ and A₂

Based on the boundary conditions at the receiving end of the line (Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x=0\,} ), i.e.

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\boldsymbol {V}}(0)={\boldsymbol {V_{r}}}\,}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\boldsymbol {I}}(0)={\boldsymbol {I_{r}}}\,}

The voltage and current equations are as follows:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\boldsymbol {V}}(0)={\boldsymbol {V_{r}}}=A_{1}e^{{\boldsymbol {\gamma }}(0)}+A_{2}e^{-{\boldsymbol {\gamma }}(0)}\,}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\boldsymbol {I}}(0)={\boldsymbol {I_{r}}}={\frac {A_{1}e^{\gamma (0)}-A_{2}e^{-\gamma (0)}}{{\boldsymbol {Z}}_{c}}}\,}

Therefore,

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\boldsymbol {V_{r}}}=A_{1}+A_{2}\,}

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

Substituting Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A_{1}={\boldsymbol {I_{r}}}{\boldsymbol {Z}}_{c}+A_{2}\,} into ${\boldsymbol {V_{r}}}\,$:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\boldsymbol {V_{r}}}={\boldsymbol {I_{r}}}{\boldsymbol {Z}}_{c}+A_{2}+A_{2}\,}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Rightarrow A_{2}={\frac {{\boldsymbol {V_{r}}}-{\boldsymbol {I_{r}}}{\boldsymbol {Z}}_{c}}{2}}\,}

Solving for A₁, we get:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\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:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle [Z_{c}]=[\gamma ]^{-1}[T_{v}]^{-1}[Z][T_{i}]\,}

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\gamma] \, } is diagonal, then we shall prove that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [Z_c] \, } is also diagonal. This is done by proving that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [T_{v}]^{-1} [Z] [T_{i}] \, } is diagonal.

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d \boldsymbol{V'}}{dx} = [T_{v}]^{-1} [Z] [T_{i}] \boldsymbol{I'} \, } ... Equ. (F1)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ([T_{v}]^{-1} [Z] [T_{i}]) ([T_{i}]^{-1} [Y] [T_{v}]) = [T_{v}]^{-1} [Z] [Y] [T_{v}] \, } ... Equ. (F4a)

or

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ([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):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ([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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\gamma]^{2} \, } is diagonal. Thus for Equation (F5) to hold, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ([T_{v}]^{-1} [Z] [T_{i}]) \, } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ([T_{i}]^{-1} [Y] [T_{v}]) \, } must also be diagonal. And therefore the modal characteristic impedance matrix is also diagonal.