Looking for Smith chart spiral clockwise locus proof

[avihai]

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.

 

[johnjoe]

https://www.amazon.com/RF-Circuit-Design-Second-Edition/dp/0750685182

good starting point.

 

[biff44]

read up on Foster's Reactance Theory.

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

 

[avihai]

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.

 

[biff44]

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

 

[avihai]

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.

 

[FvM]

I'm looking for a formal proof.

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.

 

[avihai]

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.

 

[FvM]

It happens for any impedance with losses.

How about a RL series circuit?

 

[avihai]

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.

 

[FvM]

It shows spirals with increasing radius.

 

[volker@muehlhaus]

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.

 

[FvM]

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.