Clockwise rotation of the reflection coefficient w/ frequency

• WhiteHaired
In summary, 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)). This rotation should not be confused with the derivative of the phase with frequency, which can be either positive or negative depending on the circuit. For lossless systems, this can be explained by Foster’s reactance theorem, but there is no rigorous proof for lossy systems. However, it has been shown that the curvature of the reflection coefficient in Cartesian coordinates is always negative, indicating a clockwise rotation. This also applies to the complex impedance. There is currently no known theorem or property that explains this, but it could potentially
WhiteHaired
It is always considered 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. It also applies for the reflection coefficient, since the bilinear transform (from immitance to reflection coefficient) preserves orientation.

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:

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

or, equivalently,

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

The same would apply to the complex impedance Z=R+j*X, (or admittance), i.e., $\frac{∂R}{∂f}\frac{∂^{2}X}{∂f^{2}}<\frac{∂X}{∂f}\frac{∂^{2}R}{∂f^{2}}$ and $\frac{∂}{∂f}\left(\frac{∂X}{∂R}\right)<0$

Is all this right?

Do you know any theorem, property of LTI causal passive systems, energy considerations from which one may conclude this? Kramer-Kronig relations or Hilbert transform?

I would appreciate your help on this.
View attachment 77655

Thank you, not for the moment.

1. What is clockwise rotation of the reflection coefficient with frequency?

Clockwise rotation of the reflection coefficient with frequency refers to the change in magnitude and phase of the reflection coefficient as the frequency of a signal increases. This can be visualized on a Smith chart, where the reflection coefficient moves clockwise along the constant resistance circle as frequency increases.

2. How does clockwise rotation of the reflection coefficient affect a signal?

The clockwise rotation of the reflection coefficient with frequency can cause signal loss and distortion. As the signal frequency increases, the reflection coefficient moves towards the center of the Smith chart, which indicates a higher level of reflection and a weaker signal at the input of a transmission line.

3. What factors contribute to clockwise rotation of the reflection coefficient?

The main factors that contribute to clockwise rotation of the reflection coefficient are impedance mismatch and transmission line length. Impedance mismatch occurs when the characteristic impedance of the transmission line does not match the input or load impedance, causing reflections. Transmission line length can also affect the reflection coefficient due to the time it takes for a signal to travel down the line and reflect back.

4. How can clockwise rotation of the reflection coefficient be minimized?

To minimize clockwise rotation of the reflection coefficient, it is important to match the impedance of the transmission line to the input and load impedances. This can be achieved by using impedance matching techniques, such as using a balun or a matching network. Additionally, using shorter transmission line lengths can also help reduce the amount of rotation.

5. What are the applications of understanding clockwise rotation of the reflection coefficient with frequency?

Understanding clockwise rotation of the reflection coefficient is crucial in the design and analysis of high frequency circuits, such as RF and microwave circuits. It allows engineers to properly match impedances and minimize signal loss, leading to better performance and reliability of the circuit. Additionally, it is important in the fields of antennas, radar systems, and wireless communication systems.

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