# Reflection Coefficient when conjugate matching

1. ## Reflection Coefficient when conjugate matching

I have a question about the conjugate matching.

When we calculate the reflection coefficient of the circuit, we use Γ=(ZL-Zg)/(ZL+Zg).

What if Zg is conjugate matching we ZL which means Zg=ZL*, and Zg=R+jX, ZL=R-jX?

The Γ is not 0.

I think the conjugate matching make sure that there is no power reflect back to source generator so the Γ should be 0. I know why we need the conjuagate matching. All I need is to solve this math problem.

Thanks

2. ## Re: Reflection Coefficient when conjugate matching

Your calculations are right, Γ is not 0. Here is most interesting links I've found to help with your question:
1. http://www.analog-rf.com/match.shtml
2. http://www.thefullwiki.org/Maximum_power_theorem
3. http://www.ece.rutgers.edu/%7Eorfanidi/ewa/ch13.pdf

From [1]:
Complex conjugate matching produces the maximum small signal transfer of power from a source that is not a transmission line to a load at the sacrifice of:...
From [2]:
Maximizing power transfer versus power efficiency
The theorem was originally misunderstood (notably by Joule) to imply that a system consisting of an electric motor driven by a battery could not be more than 50% efficient since, when the impedances were matched, the power lost as heat in the battery would always be equal to the power delivered to the motor. In 1880 this assumption was shown to be false by either Edison or his colleague Francis Robbins Upton, who realized that maximum efficiency was not the same as maximum power transfer. To achieve maximum efficiency, the resistance of the source (whether a battery or a dynamo) could be made close to zero. Using this new understanding, they obtained an efficiency of about 90%, and proved that the electric motor was a practical alternative to the heat engine.
From[3]:
Conjugate matching is not the same as reflectionless matching, which refers to matching the load to the line impedance,ZL=Z0, in order to prevent reflections from the load.
to be honest, I do not have good understanding on this topic. I think that reading on basic theory like in [3] may help. Maybe someone with better understanding will clarify your question.

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3. ## Re: Reflection Coefficient when conjugate matching

Originally Posted by Georgy.Moshkin
Your calculations are right, Γ is not 0. Here is most interesting links I've found to help with your question:
1. http://www.analog-rf.com/match.shtml
2. http://www.thefullwiki.org/Maximum_power_theorem
3. http://www.ece.rutgers.edu/%7Eorfanidi/ewa/ch13.pdf

From [1]:

From [2]:

From[3]:

to be honest, I do not have good understanding on this topic. I think that reading on basic theory like in [3] may help. Maybe someone with better understanding will clarify your question.
Thanks Georgy.

I am thinking another way to explain this is situation. If we think about the wave theory, PL=P+-P-. P+= V+^2/R and P-= V-^2/R. The V+could have higher level than voltage at the output of generator, where v+ is at insert voltage the load and v- is reflect voltage at the load. therefore, even Γ is not zero, there are still half of the power transmitted to the load.

4. ## Re: Reflection Coefficient when conjugate matching

A simple example of conjugate matching is a RL series load. Conjugate matching can be achieved by placing a series C to the generator with same amount R of real impedance. Means you put the capacitor in resonance with the load L. At the load terminals you have a large reflection coefficient, most energy in the system is circulating between L and C although the generator R is perfectly matched to the load R.

Reflection coefficient is no useful parameter to describe conjugate matching.

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5. ## Re: Reflection Coefficient when conjugate matching

Originally Posted by SDRookie
When we calculate the reflection coefficient of the circuit,
we use Γ=(ZL-Zg)/(ZL+Zg).
This is a reflection coefficient for Voltage Wave.

Originally Posted by SDRookie
What if Zg is conjugate matching we ZL which means Zg=ZL*, and Zg=R+jX, ZL=R-jX?
The Γ is not 0.
Reflection coefficient for Voltage Wave is not zero.

Originally Posted by SDRookie
I think the conjugate matching make sure that there is no power reflect back to source generator so the Γ should be 0.
Reflection coefficient for Power Wave is zero.

Port Impedance=R+j*X

Then calculate S11 using RF simulator.
Which case do you think you can get S11=0 ?

Gamma_V = {Zin - Zref} / {Zin + Zref}
Gamma_P = {Zin - conjugate(Zref)} / {Zin + Zref}

Power Wave formulation is most used in RF world.
For example, S-parameters of Keysight ADS are based on Power Wave formulation.

RF simulator give Gamma_P as S11.

See followings.

Originally Posted by FvM
Reflection coefficient is no useful parameter to describe conjugate matching.
Wrong.
This is a reflection coefficient for Voltage Wave.

Reflection coefficient in Power Wave formulation is useful parameter to describe conjugate matching.

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6. ## Re: Reflection Coefficient when conjugate matching

More information see "Microwave Network Design Using the Scattering Matrix-Janusz A. Dobrowolski-ARTECH HOUSE pp-75-77

7. ## Re: Reflection Coefficient when conjugate matching

Like others mentioned in the linked previous thread https://www.edaboard.com/showthread....vs-power-waves, the terms "power wave" and "power wave reflection coefficient" aren't commonly used in the majority of RF text books.

I see that they have been apparently introduced in the 1965 IEEE paper Power Waves and the Scattering Matrix by Kurokawa. They are surely useful when calculating power transfer with complex termination. Thanks for sketching the concept.

8. ## Re: Reflection Coefficient when conjugate matching

S11 is {Zin - conjugate(Zref)} / {Zin + Zref}
in RF simulator or internal post processing in VNA(Vector Network Analyzer).
It is never {Zin - Zref} / {Zin + Zref}

Many people don't know this.

Code:
```//****************************************************************************
/*BEGIN_DOC
FUN_NAME:zin()
FUN_DESCRIPTION: Given a reflection coefficient and the reference impedance, this measurement returns the input impedance looking into the measurement ports.
RETURNED_VALUE: Integer, real or complex
CATEGORY: S-Parameter
SYNTAX:  z = zin(Sii, Z)
EXAMPLE: zIN = zin(S11, 50.0)
ARGUMENT
ARG_NAME: Sii
ARG_DESCRIPTION: complex reflection coefficient.
ARG_DEFAULT: None
ARG_RANGE: (-inf:inf)
ARG_TYPE: Complex
ARG_REQUIRED: YES
ARGUMENT
ARG_NAME: zRef
ARG_DESCRIPTION: reference impedance
ARG_DEFAULT: 50.0
ARG_RANGE: (-inf:inf)
ARG_TYPE: Integer, real or complex
ARG_REQUIRED: NO
DEFINED IN: \$HPEESOF_DIR/expressions/ael/network_fun.ael
EXTERNAL: yes
AUTHOR: Agilent Technologies
DATE: Unknown
END_DOC*/
//****************************************************************************
defun zin(sii,zRef)
{
decl zi = if (zRef == NULL) then 50.0 else zRef;
return (conj(zi)+zi*sii)/(1-sii);
}```

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10. ## Re: Reflection Coefficient when conjugate matching

Originally Posted by volker@muehlhaus
ADS2016 give S11 as Voltage-Reflection Coefficient ?

Until ADS2009U1 since MDS, S11 was Power-Reflection Coefficient.
When did it change ?

This might be due to trend of current ADS target application, SI(Signal Interity) or High Speed Data Transmission.
Conjugate matching is not important in such applications.

I'm very curious about other simulators such as Microwave Office, Eagle, dead GoldenGate etc.

11. ## Re: Reflection Coefficient when conjugate matching

Originally Posted by pancho_hideboo
ADS2016 give S11 as Voltage-Reflection Coefficient ?
Yes, obviously. That is also consistent with the S11 definition that our RF/microwave textbooks are teaching. I haven't seen your conjugate S11 definition before.

12. ## Re: Reflection Coefficient when conjugate matching

Originally Posted by volker@muehlhaus
I haven't seen your conjugate S11 definition before.
Power Wave Formulation in ADS is well known in FAQ.
See attached document(https://www.edaboard.com/attachment....9&d=1531391256).

S-Parameters in both Agilent ADS2009U1 and Cadence Spectre(Version 17.1.0.270) are defined by Power Wave Formulation.

S11 is {Zin - conjugate(Zref)} / {Zin + Zref}

Originally Posted by volker@muehlhaus
Yes, obviously.
That is also consistent with the S11 definition that our RF/microwave textbooks are teaching.
Show me result of "zin(0.0, 50+j*50)" in Data Display Window.

If function zin() of measurement expression is also defined by Voltage Wave formulation, it has to give 50+j*50.

I don't have valid support contract now, so I can not access Keysight EEsof Knowledge Center.

If you can access Keysight EEsof Knowledge Center, search any change or update about S-parameter definition for complex Zref.

13. ## Re: Reflection Coefficient when conjugate matching

Originally Posted by pancho_hideboo
Show me result of "zin(0.0, 50+j*50)" in Data Display Window.
Here it is ...

"Given a reflection coefficient and the reference impedance, this measurement returns the input impedance looking into the measurement ports."
I wonder what the highlighted section really means. It doesn't say this is the impedance of the DUT, the wording is a bit strange.

But anyway, I've been a happy RF/µW engineer for decades now without power wave reflection coefficient definition, using real impedance reference values with the non-conjugate equation from textbooks

14. ## Re: Reflection Coefficient when conjugate matching

Originally Posted by volker@muehlhaus
But anyway, I've been a happy RF/μW engineer for decades now
without power wave reflection coefficient definition,
using real impedance reference values with the non-conjugate equation from textbooks
I noticed that your usage of "S1P_Eqn" is wrong in https://www.edaboard.com/attachment....5&d=1531330031

Z[1] is a "Port reference impedance" not input impedance.

S[1,1]=Blank and Z[1]=50-j*50 for "S1P1".
So input impedance is 50+j*50 not 50-j*50.
Power Wave formulation is used here.

S-parameters are defined by Power Wave Formulation even in ADS2016.

Confirm it by using "Z1P_Eqn" instead of "S1P_Eqn".
See https://www.edaboard.com/attachment....1&d=1531400101

Originally Posted by volker@muehlhaus
"Given a reflection coefficient and the reference impedance,
this measurement returns the input impedance looking into the measurement ports."

I wonder what the highlighted section really means.
It doesn't say this is the impedance of the DUT,
the wording is a bit strange.
zin() is surely for Power Wave formulation even in ADS2016.

Power Wave Formulation are used for both simulation and post-processing even in ADS2016.

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15. ## Re: Reflection Coefficient when conjugate matching

Originally Posted by pancho_hideboo
I noticed that your usage of "S1P_Eqn" is wrong in https://www.edaboard.com/attachment....5&d=1531330031
Z[1] is a "Port reference impedance" not input impedance.
You are correct, my testcase was wrong!

16. ## Re: Reflection Coefficient when conjugate matching

Originally Posted by volker@muehlhaus
You are correct, my testcase was wrong!
I've been a happy RF/μW engineer for decades invariably
with power wave reflection coefficient definition,
using complex impedance reference values with the conjugate equation from RF expert textbooks

17. ## Re: Reflection Coefficient when conjugate matching

Originally Posted by pancho_hideboo
using complex impedance reference values with the conjugate equation from RF expert textbooks
Maybe that topics is teached differently in Japan, due to Kurokawa's efforts? FvM and I have been doing our PhD at the same university in Germany where voltage/current waves were teached.

I haven't used the power waves concept, so I can't estimate the practical value. Here's a publication from IEEE micrwave magazine where the author is a bit sceptal:
"Much like a lava lamp, the power waves are comforting to look at, but do not offer the engineer the tools required for analytic design."

https://www.nist.gov/sites/default/f...eory-proof.pdf

But I agree that i seems to solve the discrepancy from post #1, so thanks for poiting to this topic!

18. ## Re: Reflection Coefficient when conjugate matching

Originally Posted by volker@muehlhaus
I've also seen this. They suggest that Pseudowaves "have long been a cornerstone of microwave circuit theory" but I've only found a couple publications that mention pseudowaves or pseudo s parameters (aside from the author's other works), and the first use of the term came 27 years after Kurokawa's paper. Meanwhile they refer to power waves as a "new addition" to network theory. It looks to me like Williams is pushing a pet theory. If he weren't affiliated with NIST, I'd assume he's a hack.

19. ## Re: Reflection Coefficient when conjugate matching

Originally Posted by volker@muehlhaus
Maybe that topics is teached differently in Japan, due to Kurokawa's efforts?
No.

Generalized S-parameters where port impedances are complex value are not taught at all even in Japan.

For example, see the following standard text book on microwave engineering.
https://www.wiley.com/en-sg/Microwav...-9780470631553

Many people don't know difference between Power Wave Formulation and Pseudo Wave Formulation even in Japan.

From practical point of view, S-parameters with complex value reference impedance are not needed at all.

As far as my opinion, generalized S-parameters where port impedances are complex value are no more than mathematical extension.

However important fact is that Generalized S-parameters are defined by "Power Wave Formulation" in almost all commercial RF Circuit Simulators.
They give S11=0 at conjugate matched condition.

Internal post processing in actual VNA(Vector Network Analyzer) also use "Power Wave Formulation".

We can see descriptions of "Power Wave Formulation" in any vendor's documents.
-Keysight
-Anritsu
-Wiltron(Now Anritsu)

Many people don't know this fact.

Attached are results of Agilent GoldenGate-4.3.8
S-Parameters of Agilent GoldenGate-4.3.8 are also defined by Power Wave Formulation.

Originally Posted by mtwieg
They suggest that Pseudowaves "have long been a cornerstone of microwave circuit theory"
but I've only found a couple publications that mention pseudowaves or pseudo s parameters (aside from the author's other works),
and the first use of the term came 27 years after Kurokawa's paper.
No.
You can see many references about "Pseudo Wave Formulation" in EM simulation therory books.

Keysight Momentum use "Power Wave Formulation" as sama as ADSsim.
But Keysight EMPro use "Pseudo Wave Formulation".

Almost all circuit simulators use "Power Wave Formulation".
But this is not true for EM simulator.

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