Distributed Parameters 5 – 2-Port Example

Consider a distributed RC network where:

$\begin{displaymath}\begin{align}Z(s,x)\;&=\;r(x)\;=\;r_o\,\mathit{e}^{2kx}\\\\Y(s,x)\;&=\;g(x)\,+\,s\, c(x)\;=\;c_o(s+\Omega)\,\mathit{e}^{-2kx}\end{align}\end{displaymath}$

Define: $S\;=\;s\,+\,\Omega \qquad \Omega\;&=\;\dfrac{\,g(x)\,}{c(x)}\qquad \tau\;=\;r_o\,c_o$

and let $\xi_o\;=\;\psi_o\;=\;1$

That was the simple part …

The first set of coefficients:

$\begin{displaymath}\begin{align}\xi_1(x)\;&=\;\int_0^x Z(s,x)\,\psi_o\,d x\;=\;\int_0^x r_o\,\mathit{e}^{2kx}\,d x\;=\;r_o\,\frac{\mathit{e}^{2kx}\,-\,1\,}{2\,k}\\\\\psi_1(x)\;&=\;\int_0^x Y(s,x)\,\xi_o\,d x\;=\;\int_0^x S\,c_o\,\mathit{e}^{2kx}\,d x\;=\;-S\,c_o\,\frac{\mathit{e}^{2kx}\,-\,1\,}{2\,k}\end{align}\end{displaymath}$

On to the 2nd:

$\begin{displaymath}\begin{align}\xi_2(x)\;&=\;\int_0^x Z(s,x)\,\psi_1\,d x\;=\;\int_0^x r_o\,\mathit{e}^{2kx}\,\left[\,S\,c_o\,\frac{\,V\,-\,1\,}{2\,k}\,\right]\,\,d x\\\;&=\;\frac{S\,\tau}{\,(2\,k)^2\,}\,\left(\,\mathit{e}^{2kx}\,-\,2\,k\,x\,-\,1\,\right)\\\\\psi_2(x)\;&=\;\int_0^x Y(s,x)\,\psi_1\,d x\;=\;\int_0^x c_o\,\mathit{e}^{2kx}\,\left[\,r_o\,\frac{\,V\,-\,1\,}{2\,k}\,\right]\,\,d x\\\;&=\;\frac{S\,\tau}{\,(2\,k)^2\,}\,\left(\,\mathit{e}^{2kx}\,-\,2\,k\,x\,-\,1\,\right)\end{align}\end{displaymath}$

And on to the 3rd:

$\begin{displaymath}\begin{align}\xi_3(x)\;&=\;\frac{\;S\;\tau\;r_o\;}{\,(2\,k)^2\,}\,\int_0^x \,\left(\,\,1\,+\,\,2\,k\,x\,\mathit{e}^{2kx}\,-\,\mathit{e}^{2kx}\,\right)\,dx\,\;=\;\frac{\;S\;\tau\;r_o\;}{\,(2\,k)^3\,}\,\left[\,\,2\,+\,\,2\,k\,x\,-\,\mathit{e}^{2kx}\,(\,2-\,2\,k\,x\,)\,\right)\,\right]\\\\\psi_3(x)\;&=\;\frac{\;S\;\tau\;c_o\;}{\,(2\,k)^2\,}\,\int_0^x \,\left(\,\,2\,k\,x\,\mathit{e}^{2kx}\,-\,\mathit{e}^{2kx}\,-\,1\,\,\right)\,dx\,\;=\;\frac{\;S\;\tau\;r_o\;}{\,(2\,k)^3\,}\,\left[\,-\,2\,+\,\,2\,k\,x\,+\,\mathit{e}^{2kx}\,\left(\,2\,+\,2\,k\,x\,)\,\right)\,\right]\end{align}\end{displaymath}$

And on to the … oh, never mind. Believe me, it goes on …

Voltages $V_A$ and $V_B$ are now found:

$\begin{displaymath}\begin{align}V_A(s,d)\;&=\;\sum\,\xi_{2n}(d)\;=\;1\,+\,\xi_2(d)\,+\,\xi_4(d)\,+\, ...\\\\&=\;1\,+\,\frac{\;S\;\tau\;}{\,(2\,k)^2\,}\,\left(\,\mathit{e}^{2kx}\,-\,2\,k\,d\,-\,1\,\right)\,+\,\frac{\;S^2\;\tau^2\;}{\,(2\,k)^4\,}\,\left[\,(2\,k\,d)\,\mathit{e}^{2kx}\,+\,3\,2\,(2\,k\,d)\,+\,\frac{\,(2\,k\,d\,)^3\,}{2!}\,\right]\,+\,\cdots\end{align}\end{displaymath}$

Some righteous algebraic nightmares there, eh?

So, continuing on …

The results of a similar procedure for a lossless LC network are:

$\begin{displaymath}A\;=\;D\;=\;\sum \xi_{2n}\;=\;\sum \psi_{2n}\;=\;1\,+\,\frac{\,\left(\,s\,x\sqrt{\,l\,c\,}\,\right)^2\,}{2!}\,+\,\frac{\,\left(\,s\,x\sqrt{\,l\,c\,}\,\right)^4\,}{4!}\,+\, \cdots\;\;\Rightarrow\;\;\mathit{cosh}\,s\,x\,\sqrt{\,l\,c\,}\\\end{displaymath}$

$\begin{displaymath}B\;=\;\sum \xi_{2n-1}\;=\;\sqrt{\,\frac{l}{\,c\,}\,}\,\left[\,s\,x\,\sqrt{\,l\,c\,}\,+\,\frac{\,\left(\,s\,x\sqrt{\,l\,c\,}\,\right)^3\,}{3!}+\, \cdots\,\right]\;=\;\sqrt{\,\frac{l}{\,c\,}\,}\,\mathit{sinh}\left(\,s\,x\,\sqrt{\,l\,c\,}\,\right)\end{displaymath}$

$\begin{displaymath}C\;=\;\sum \psi_{2n-1}\;=\;\sqrt{\,\frac{\,c\,}{l}\,\left[\,s\,x\,\sqrt{\,l\,c\,}\,+\,\frac{\,\left(\,s\,x\,\sqrt{\,l\,c\,}\,\right)^3\,}{3!}\,+\, \cdots\,\right]\;=\;\sqrt{\,\frac{\,c\,}{l}\,}\end{displaymath}$

Note that AD – BC = $\mathit{cosh}^2\,(\,s\,x\,\sqrt{\,l\,c\,})\,-\,\mathit{sinh}^2\,(\,s\,x\,\sqrt{\,l\,c\,})\;=\;1$

That’s good for now. Next: T-Network Taylor Series Expansion

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