Talk:Distributed Parameter Line Model

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Footnotes

Derivation of A1 and A2

Based on the boundary conditions at the receiving end of the line (), i.e.

The voltage and current equations are as follows:

Therefore,

Substituting into :

Solving for A1, we get:

Diagonality of Modal Characteristic Impedance Matrix

The modal characteristic impedance matrix is:

If is diagonal, then we shall prove that is also diagonal. This is done by proving that is diagonal.

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

... Equ. (F1)
... Equ. (F2)

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

... Equ. (F3)

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

... Equ. (F4a)

or

... Equ. (F4b)


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


... Equ. (F5)


We know that the modal propagation constant matrix is diagonal. Thus for Equation (F5) to hold, then and must also be diagonal. And therefore the modal characteristic impedance matrix is also diagonal.