Input impedance of a "Ladder" transmission line

[Marcus95]

Homework Statement

My electronics&physics lecture notes contain the following side note:
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"A ladder transmission line comprises an alternating sequence of segments of two different transmission lines both of length $l$ with characteristic impedance $Z1$ and $Z2$. If the line is constructed such that its input impedance remains unchanged when another pair of $Z1$ and $Z2$ segments is added, the input impedance of the ladder transmission line obeys the following relationship:
$$i Z_{in}^2(Z_1+Z_2) + Z_{in} (Z_1^2-Z_2^2) - iZ_1Z_2(Z_1+Z_2) = 0. $$
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Homework Equations

$$ Z_{in} = Z_0 \frac{Z \cos kl + i Z_0 \sin kl}{Z_0 \cos kl + i Z \sin kl} . $$
$$V = IR$$

The Attempt at a Solution

Now to me this formula appears to be very much pulled out of thin air, so I have naturally attempted to prove it. First I considered using

$$ Z_{in} = Z_0 \frac{Z \cos kl + i Z_0 \sin kl}{Z_0 \cos kl + i Z \sin kl} . $$

but then realized that this formula is derived for when we have a single reflecting boundary only, so might not be applicable. The second approach which could be taken would be to write down the equations for both transmitted and reflected waves in all regions and hence find the input impedance for the case of 2 segments and for 4. However, this would require solving 8 simultaneous equations...

Anybody who knows how this formula is proved in a nicer way?

 

[rude man]

Right now I can only suggest an approach, may look at it in more detail later:
There is some ambiguity as to whether the total line is of finite or infinite length; this should be resolved; but in either case:

Determine the ABCD matrix for line 1 of length l.
Same for line 2.
Multiply the two matrices and raise this matrix to the n th power where n is a positive integer. n is finite for a finite-length line and → ∞ for an infinite-length one. Call this the "resultant" matrix.
Write the expression for Z_in in terms of the resultant matrix's ABCD parameters.
Substitute in your given equation to show its correctness.

Of course this assumes you've covered 2-terminal networks. I wouldn't know another approach though.

 

[Marcus95]

Of course this assumes you've covered 2-terminal networks. I wouldn't know another approach though.

Thank you for your reply! To be honest I have not studied networks in any great detail at all (ie I have no clue what an ABCD matrix is)... this is just an physics EM course which has a 10 page section on transmission lines where this comes up. :/

 

[rude man]

Thank you for your reply! To be honest I have not studied networks in any great detail at all (ie I have no clue what an ABCD matrix is)... this is just an physics EM course which has a 10 page section on transmission lines where this comes up. :/

As you might judge from the paucity of replies (namely mine only) this is not an elementary problem, and significant exposure to networks is most likely required.