Reflection coefficient bigger than one

[trichop]

s11 reflection coefficient

I got a simple but awkward problem.
Suppose you have a simple circuit, a source with complex impedance connected to a load with also complex impedance.
You want to calculate S11 which in our case equals to reflection coefficient.
Γ = (Zload -Zo)/(Zload + Zo)
However, in this way I get an S11 magnitude bigger than one. (at conjugate matching, for example Zload = 20 + 40j, Zo = 20 - 40j)

Could this be reasonable?
I simulated the same network on ADS and for conjugate matching it gives me
mag(S11) of almost zero.

Any idea?

 

[jian]

transmission coefficient bigger smaller than 1

Hi, trichop:

Basically, transmission line (TLN) theory is no longer exact when you have complexc Zc. You will get this kind of problems when you use complex Zc for normalization. It might be ok when the Im(Zc) is relatively small and you may not see |G|>1 for passive structures. However, the problem is there. The fundamental reason is that TLN theory is no longer exact. You can read the Appendix in IE3D's user's manual about it.

Complex impedance is encountered by RFID designers all the time. People naturely thought they could use complex Zc for normalization for such a case. The results might be ok when the Z_antenna_input_impedance is close the conjugate of Z_source_impedance. However, when the Z_antenna_input_impedance is far away from the Z_source_impedance, you may get |G| > 1 easily. It is unreasonable and you should avoid using complex Zc.

Regards.

 

[biff44]

in what case reflection coefficient is complex?

You are misusing this equation. Z0 is the characteristic impedance of the transmission line leading up to the reference plane that you want to make the measurement at, for example 50 ohms. The transmission line does not have a Z0 of 20-j40 ohms!

 

[radiohead]

trichop has a point. In fact when the source impedance and load impedance is complex, but with positive real parts, no reflection gain should be possible. The reason is that you lost a complex conjugate operator on the reference impedance. As most of the time this is 50 ohm, you don't care about it. In this case you do. The correct formula is (with * the conjugate operator):

\[\Gamma = \frac{{Z{}_{load} - Z_{source} ^* }}{{Z{}_{load} + Z_{source} }}\]

When Zload is equal to Zsource*, you have match and Gamma equals zero. In other cases Gamma should always be smaller than one.

If you want to learn more about generalized S-parameters, read the original papers from Kurokawa. “Power waves and the scattering matrix,” IEEE Trans. Microwave Theory and Tech., pp. 194-202, March 1965

He also had a second paper about generalized S-parameters of which I can't find the reference right now.

 

[jian]

Hi:

Using the formula

G = ( Zload - Zsource* ) / ( Zload + Zsource),

you will be able to avoid |G| > 1 for passive structures. It is certainly more reasonable. However, there are still cases you can't get reasonable results. For example, if Zload = j X or the load is a pure inductor or capacitor, you will find |G| is not equal to 1. It means an inductor or capacitor creates loss (indicated by |G| < 1).

Anyway, formula does not resolve the problem which is more fundamental for complex Zc.

Best regards,

 

[kspalla]

In the above formula on point is missing, it is related to magnitude not the angle.
let say ZL and ZO which ever has the high magnitude will be used as first parameter and the next one will get subtracted from it.
This way it is always <1.

 

[radiohead]

Jian, you're right that the interpretation of G is no longer intuitive, but it is still correct if you use generalized S-parameters.

It is mathematically correct, but I agree that you can argue whether or not the formulation is useful in practical applications.

 

[jian]

Hi, Radiohead:

I believe complex Zc is not just causing problem in practical applications. It does have problem mathematically. If you assume Zc, the incident wave and the reflected wave are separated. In some sense, the power going into the system is:

Pnet = Pinc - Pref
Pinc = | inc * inc |
Pref = | ref * ref |

When you have complex Zc, the abvoe formulas are no longer valid. You can try to find the power using the Poynting vector formula. You will find the incdient power and reflected power can't be separated.

Dig into it more, you will find the fundamental problem is that incident wave and reflected wave are no longer travelling in one direction (back and forth). It has the behaviour similar to spherical wave or cylindrical wave. You just no longer can define the reflection coefficient based upon incident wave and reflected wave. In fact, the whole idea of separate the power into incident wave and reflected wave along a waveguide (or TLN) is no longer precise when you have to use complex Zc. You can still use it to get some reasonable results for most cases if the Im(Zc) is not serious. However, you should keep in mind that waveguide (or TLN) theory is no longer precise in such a case. Anyway, there is no case without complex Zc. Using waveguide and TLN theory based upon real Zc is a good approach anyway. Regards.