Looking for Smith chart spiral clockwise locus proof

Hi.

I am looking for a formal proof for the spiral locus of an impedance when increasing the frequency.
Can anybody refer to somewhere it is proved?
Any book or formal suggestions will be great. I didn't find it in the basic microwave engineering books.

Thanks a lot.
Avihai.

http://www.amazon.com/RF-Circuit-Des.../dp/0750685182

good starting point.

read up on Foster's Reactance Theory.

But with metamaterials, I think the theory has some holes in it.

Yes, this is exactely the hole that I look for.
for lossy transmission lines there is also that spiral mapping when frequency varies.

Yeah it spirals, but the radius of the spiral should decrease when the frequency goes up, or the line length increases

I know that this is happening. I'm looking for a formal proof.
Any idea?

Thanks.

For which statement?

I know that some impedances show a rotation in s-plane, but not necessarily a spiral. Others have fixed coordinates or move linearly. So obviously you have to specify a class of impedances with the claimed behaviour.

I'm looking for the proof that the impedance spiral radius is decreasing with frequency.
It happens for any impedance with losses.

How about a RL series circuit?

What do you mean?
I am not sure, but I think that it happens as well.
This is also written in many places as a fact but never proved by any book.

It shows spirals with increasing radius.

I would not call it a circle. For series R-L we have half a turn in the Smithchart.

That's right. I was referring to a RL-series circuit connected through a transmission line. The "spirals" previously discussed in this threads are only possible with additional delay.