On the clockwise rotation of the reflection coefficient with frequency

1. Dec 7, 2013

WhiteHaired

It is well known that the evolution of the input reflection coefficient, $ρ$, of a LTI causal passive system with frequency, $f$, always presents a local clockwise rotation when plotted in cartesian axes $(Re(ρ), Im(ρ))$, e.g. in a Smith chart, as shown in the attached figure.

It must appointed that the local clockwise rotation should not be confused with the derivative of the phase with frequency, which is always negative when the curve encompasses the center of the Smith chart, but it may be positive otherwise (e.g. in a resonant series RLC circuit with R>Z0, where Z0 is the port characteristic impedance). The question here concerns the local rotation, which is always clockwise.

For lossless systems, it may be explained from the Foster’s reactance theorem, “The imaginary immittance of a passive, lossless one-port monotonically increases with frequency”, which has been demonstrated in different ways in literature.

However I couldn’t find any rigorous proof for lossy systems. Books and manuscripts always reference the lossless case and the Foster’s theorem.

Do you know any reference?

In geometry, for a plane curve given parametrically in Cartesian coordinates as $(x(f),y(f))$, the signed curvature, $k$, is

$k=\frac{x'y''-y'x''}{(x^{2}+y^{2})^{3/2}}$

where primes refer to derivatives with respect to frequency $f$. A negative value means that the curve is clockwise. Therefore, the reflection coefficient of a LTI causal passive system with frequency, $f$, has always a negative curvature when plotted in Cartesian coordinates $(Re(ρ), Im(ρ))$, i.e., it satisfies:

$\frac{∂Im(ρ)}{∂f}\frac{∂^{2}Re(ρ)}{∂f^{2}}>\frac{∂Re(ρ)}{∂f}\frac{∂^{2}Im(ρ)}{∂f^{2}}$

or, equivalently,

$\frac{∂}{∂f}\left[\frac{\frac{∂Im(ρ)}{∂f}}{\frac{∂Re(ρ)}{∂f}}\right]>0$→$\frac{∂}{∂f}\left(\frac{∂Im(ρ)}{∂Re(ρ)}\right)>0$

Is this right?
Is there any theorem or property of LTI causal systems from which one may conclude this?
Kramer-Kronig relations or Hilbert transform?
I would appreciate your help on this.

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2. Jun 9, 2014

avihai

Hi.
Did you find any proof? I'm also looking for that.

Thanks.

3. Jun 9, 2014

WhiteHaired

Hi Avihai.
I looked for it in many fundamental books of physics and electromagnetism, but I couldn't find any proof.
Please note that the sign of the inequalities in my last equations is changed. It should read < 0.

Any suggestion?

Thanks.

4. Jun 12, 2014

avihai

Unfortunately I have no suggestions.
I tried to contact Settapong Malisuwan who published a paper concerning a frequency dependent smith chart, but only concerning a microstrip antenna, so for now it is only a private case and not a general. So it is not good enough answer to our problem.
For now, I have no response from him.
this is the paper:
http://www.ijcce.org/papers/262-OC0032.pdf

5. Jun 13, 2014

WhiteHaired

Thank you Avihai. Very interesting the paper from Dr. Malisuwan et al., but, as you said, it's not a general case. They found a useful modification of the Smith chart representation for microstrip circuits, but no general conclussions about the behaviour of the reflection coefficient in frequency when represented in polar coordinates are drawn.