# Looking for Smith chart spiral clockwise locus proof

#### avihai

Joined May 29, 2014
8
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.

#### timwhite

Joined Apr 10, 2014
50
You may be able to find what you're looking for in "Lithium Batteries: Proceedings of the International Symposium," found here:

#### Papabravo

Joined Feb 24, 2006
20,583
The proof is the straightforward result of applying a conformal mapping from the complex plane to the Smith chart.

#### avihai

Joined May 29, 2014
8
OK. but is it possible to find a formal proof that alpha is always related to the frequency?
I mean, I see the connection but I look for the proof

Thank you

#### Papabravo

Joined Feb 24, 2006
20,583
OK. but is it possible to find a formal proof that alpha is always related to the frequency?
I mean, I see the connection but I look for the proof

Thank you
Before I go and assume what you are talking about I'd like to point out that the Greek letter alpha is used to denote a great many things in science and engineering. What does alpha refer to in this context?

#### avihai

Joined May 29, 2014
8
Before I go and assume what you are talking about I'd like to point out that the Greek letter alpha is used to denote a great many things in science and engineering. What does alpha refer to in this context?
I'm refering to alpha=attenuation constant.

and t_n_k:
there is also a spiral mapping for a lossy line with varying frequency.
I'm looking for a formal proof to this.

#### Papabravo

Joined Feb 24, 2006
20,583
I'm refering to alpha=attenuation constant.
...
Do you mean "attenuation coefficient"? The reason I ask is that constants don't normally have a dependence on frequency.

#### avihai

Joined May 29, 2014
8
Do you mean "attenuation coefficient"? The reason I ask is that constants don't normally have a dependence on frequency.
yes,sorry, of course.

#### KL7AJ

Joined Nov 4, 2008
2,229
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.
The spiral locus only applies if you have a LOSSY transmission line. A perfect transmission line will simply trace circles around the outside perimeter.

However...lacking a formal proof for a lossy transmission line...it's actually pretty easy to demonstrate...even if you can only take intermittent measurements (WRT frequency) with an impedance bridge.

#### avihai

Joined May 29, 2014
8
The spiral locus only applies if you have a LOSSY transmission line. A perfect transmission line will simply trace circles around the outside perimeter.

However...lacking a formal proof for a lossy transmission line...it's actually pretty easy to demonstrate...even if you can only take intermittent measurements (WRT frequency) with an impedance bridge.
Yes, I know that it happens for lossy transmission lines and how to demonstrate it, but I need the formal proof.

Thanks.

#### Papabravo

Joined Feb 24, 2006
20,583
Why haven't you tried my original suggestion?

#### avihai

Joined May 29, 2014
8
The proof is the straightforward result of applying a conformal mapping from the complex plane to the Smith chart.
I was told that this proof is not general to any linear passive impedance so it not good enough.
I think that it should maybe be a proof that starts from fields but I have no idea where to start from.

#### Papabravo

Joined Feb 24, 2006
20,583
I don't see why not. Think about what happens to an inductor as you keep increasing the frequency. Parasitic little capacitors show up between the windings.

Making an inductor look like a capacitor and a capacitor look like an inductor is what the spiral is demonstrating along with some amount of loss to change the radius.

Take an impedance behavior on the Argand (Complex) plane in rectangular coordinates and apply the conformal mapping and see the spiral.

http://en.wikipedia.org/wiki/Jean-Robert_Argand

#### t_n_k

Joined Mar 6, 2009
5,455
The possible issue with a real inductance would be the self resonance condition which may skew the notional spiral.

Also the resistance (loss) component would presumably be effected by skin effect as frequency becomes sufficiently high. In that case the spiralling effect would be outwards with increasing frequency. Not that the trend was indicated by the OP.

I'm yet to fully apprehend the physical system the OP has in mind. Is this an assignment problem with an actual problem statement?? It seemed earlier on it was about a lossy transmission line not terminated in its characteristic impedance. Most of the literature glosses over (ignores?) the modelling of frequency dependent losses in transmission lines, although it is obvious from actual line data that said losses do exist.

Where the issue is of genuine interest the modelling approach often requires the use sophisticated RF modelling tools ....

http://www.simberian.com/AppNotes/ModelingConductorLoss_2007_02.pdf

I note a closely related discussion appeared on edaboard.com.

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#### avihai

Joined May 29, 2014
8
I'm helping a Professor in my university to find that proof.
In any book I checked this phenomenon was ignored and just given as a fact.
A proof for the clockwise rotation with frequency might also help to advance in the research.

#### t_n_k

Joined Mar 6, 2009
5,455
I'm helping a Professor in my university to find that proof.
In any book I checked this phenomenon was ignored and just given as a fact.
A proof for the clockwise rotation with frequency might also help to advance in the research.
Presumably any journal paper published on the subject would acknowledge the source(s) of useful contributions (or indeed formal proofs) obtained through your inquiries here or on other sites.