Consider a transmission line circuit like the following but where \$Z_{o2}\$ was also unique (say \$Z_{o2}=7\,\Omega\$):

enter image description here

I know how I would find \$l_1\$ and \$l_2\$ using smith charts:

$$Re\left\{Y_n(z=-l_1)\right\}=\frac{Z_{o1}}{Z_o}=0.5\rightarrow l_1=\left(0.106\lambda,\,0.394\lambda\right)$$

$$Y(z=-l_1)=Y_n(z=-l_1)\cdot Y_{o1}$$

$$Y_n(z=-l_2)=\frac{-Im\left\{Y(z=-l_1)\right\}}{Y_{o2}}=0.161j\rightarrow l_2=(0.161\lambda,\,0.349\lambda)$$

but, since I would like to write a program for calculating this, I need to be able to solve this from the governing equations.

Starting on page 230 in this book, the author describes the equations used for matching a circuit where \$Z_o=Z_{o1}=Z_{o2}\$ but how would I solve this problem in the more general case?


1 Answer 1


More generally you could use this equation: -

enter image description here

So, the two t-lines will each present an impedance at the node where they join and that therefore becomes a parallel impedance that is then presented as the load on the output of the t-line coming from the left.

Formula Image from here.

  • \$\begingroup\$ Thanks! Say I am comfortable calculating l1 and just want to use this eq to find l2; could you help get me started on that? \$\endgroup\$
    – Landon
    Nov 20, 2020 at 18:02
  • 1
    \$\begingroup\$ @Landon are we done here? \$\endgroup\$
    – Andy aka
    Dec 30, 2022 at 10:45

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