Gamma-SLIM

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Gamma-SLIM, the Source Load Impedance Mismatch Coefficient

Source load impedance mismatch is characterized by a Coefficient denoted $\Gamma _{SLIM}$ to avoid confusion with the Reflection Coefficient $RC$ or $\Gamma$ on a mismatched transmission line.

Equating both "more broadly" or beause of "more universal principles" (whatever that means) is a widespread, yet unproven assumption . It yields impossible $|\Gamma |>1$ in passive lumped element circuits. The assumption is carried on into $|\Gamma |$ dependant Standing Wave Ratio $SWR$ and Return Loss $RL$ equations , yielding impossible $SWR<1$ and negative Return Loss $RL$.

The cause of the problem is disregard of inconsistent optimizing goals for:

1. Reflection suppression on complex load impedance terminated, complex characteristic impedance referenced, transmission lines and
2. Real power transfer between complex source impedance and complex load impedance.

It indicates a need to better understand $\Gamma$. This article describes $\Gamma$ as relative difference. Based on that, derivation of Source Load Impedance Mismatch Coefficient $\Gamma _{SLIM}$ and Reflection Coefficient $RC$ is given.

General Definition of Gamma (Γ)

Factor $\Gamma$ is a unitless ratio. It is the relative difference of a quantity's actual value in proportion to a suitable (i.e. nominal, optimal for a specified goal, same month last year, or whatever) reference value.

For example: A $C_{ref}=220\;pF$ (nominal) capacitor, actually measuring $C_{actual}=231\;pF$, has a factor $\Gamma _{C}=0.05$ $(=5\%)$ relative difference, referenced to it's (in this case nominal) value $C_{ref}$.

Relative difference : $\mathrm {\Gamma } _{x}={\frac {\Delta _{x}}{x}}={\frac {x_{ref}-x_{actual}}{x_{ref}}}=1-{\frac {x_{actual}}{x_{ref}}}$

Like percentages, a coefficient (=factor) like $\Gamma$ is not bound to any particular phenomenon (i.e. reflection). It has two values of same magnitude $\Gamma _{x}$ , but different direction depending on the trend, if increase of $x_{actual}$ will increase or decrease $\Gamma _{x}$.

In many applications only magnitude $0\leq |\Gamma _{x}|\leq 1$ is meaningful:

Magnitude of Gamma: $\mathrm {|} \Gamma _{x}|=\left|1-{\frac {x_{actual}}{x_{ref}}}\right|$

The Role of the Goal

As matching is an optimizing attempt, the reference value $x_{ref}$ is a quantity's optimum value for a specified goal (like maximized real power transfer between source and load - or - reflection suppression on a transmission line. These two conflicting goals cause $\Gamma$ not to share a common impedance mismatch equation (unless in a special case $Z_{S}$ be real only, so $Z_{S}={Z_{S}}^{*}=R_{S}+j0\;\Omega$).

Source Load Impedance Mismatch

Γ_SLIM derivation by actual and optimum currents

$I_{actual}={\frac {V_{S}}{Z_{S}+Z_{L}}}$

According to the maximum power transfer theorem, the optimum goal, maximum real power $P_{L}$ in $R_{L}$, is achieved for perfect conjugate complex match: $Z_{L}={Z_{S}}^{*}$

With $R_{L}=R_{S}$ and $X_{L}=-X_{S}$, $I_{opt}={\frac {V_{S}}{2\cdot R_{S}}}$

$\mathrm {\Gamma } _{SLIM}=1-{\frac {I_{actual}}{I_{opt}}}=1-\displaystyle {\frac {\displaystyle {\frac {V_{S}}{Z_{S}+Z_{L}}}}{\displaystyle {\frac {V_{S}}{2\cdot R_{S}}}}}=1-{\frac {2\cdot R_{S}}{Z_{L}+Z_{S}}}={\frac {R_{L}-R_{S}+j(X_{L}+X_{S})}{R_{L}+R_{S}+j(X_{L}+X_{S})}}={\frac {Z_{L}-{Z_{S}}^{*}}{Z_{L}+Z_{S}}}$

$\Theta _{\Gamma _{SLIM}}=arg\left(\displaystyle {\frac {Z_{L}-{Z_{S}}^{*}}{Z_{L}+Z_{S}}}\right)\cdot \displaystyle {\frac {180^{\circ }}{\displaystyle \pi }}$, This angle $\Theta _{\Gamma _{SLIM}}$ - other than angle $\Theta _{RC}$ - cannot be used in the Smith Chart's $\Gamma$ Plane, as it is inconsistent with the chart's conformal mapping relation $\Gamma ={\frac {Z_{1}-Z_{0}}{Z_{1}+Z_{0}}}$ .

For real power transfer, an (equivalent to) Standing Wave Ratio SWR and Return Loss RL calculation, the magnitude only is needed:

$|\Gamma _{SLIM}|=\left|{\frac {Z_{L}-{Z_{S}}^{*}}{Z_{L}+Z_{S}}}\right|$

Γ = RC derivation by actual and optimum currents

Mismatch causing reflections on a transmission line

$I_{actual}={\frac {V_{S}}{Z_{L}+Z_{S}}}$

According to the description of Reflection Coefficient: A signal source with internal impedance $Z_{S}$, possibly followed by a transmission line of characteristic impedance $Z_{0}=Z_{S}$, is represented by its Thévenin equivalent at the terminal point.

A terminating load impedance $Z_{L}$ causes no reflections on the line, if it is identical to the line's characteristic impedance $Z_{0}=Z_{S}$: It presents the same load as a hypothetical infinite length of this line, where no reflections can occur.

The optimum goal, complete reflection suppression, is achieved for identical impedance: $Z_{L}={Z_{S}}$.

With $R_{L}=R_{S}$ and $X_{L}=X_{S}$, $I_{opt}={\frac {V_{S}}{2\cdot Z_{S}}}$

$\Gamma =RC=1-{\frac {I_{actual}}{I_{L_{opt}}}}=1-{\frac {\frac {V_{S}}{Z_{L}+Z_{S}}}{\frac {V_{S}}{2\cdot Z_{S}}}}=1-{\frac {2\cdot Z_{S}}{Z_{L}+Z_{S}}}={\frac {Z_{L}-Z_{S}}{Z_{L}+Z_{S}}}$

$|RC|=|\Gamma |=\left|{\frac {Z_{L}-Z_{S}}{Z_{L}+Z_{S}}}\right|$

$\Theta _{RC}=arg\left({\frac {Z_{S}-{Z_{L}}}{Z_{S}+Z_{L}}}\right)\cdot {\frac {180^{\circ }}{\pi }}$ This angle can be used in the $\Gamma$ Plane of the Smith Chart because of same conformal mapping.

The Relevance of the Reference

The Smith Chart - and especially a top quality Smith Chart program like SimSmith - is an important tool for calculating complex impedance RF circuits and systems. While probably most users have the goal of maximum real power transfer, there are also transmission lines involved. Realistic transmission lines are subject to a (though comparatively small) capacitive component in the transmission line's characteristic line impedance. So eventually reflection suppression is of interest, and one is tempted to use the coplex characteristic line impedance as system impedance - that is, as reference center of the chart. Normalizing the center to "1" by dividing impedances over the complex system impedance doesn't change it's complex reference nature. In a complex centered Smith Chart's Gamma plane things are not referenced to real impedances and Γ = RC = 0 + j 0 will not represent real match, as would be needed for real power maximizing. What can be done? Use two different Smith Charts with different reference impedance for the different goal.

For source Load matching: If the source is complex, one would have a similar situation. There is, however, a nice workaround: Moving the source reactive part to the load side. Why can that be done? $\Gamma _{SLIM}={\frac {R_{L}-R_{S}+j(X_{L}+X_{S})}{R_{L}+R_{S}+j(X_{L}+X_{S})}}$ only uses the sum $X_{L}+X_{S}$ in both, numerator and denomionator, never one of them alone. So moving $X_{S}$ to the load side will not change $\Gamma _{SLIM}$. Now we have new components for the same circuit: ${Z_{S}}'=R_{S},\;\;{Z_{L}}'=Z_{L}+jX_{S}=R_{L}+j(X_{L}+X_{S})$. So the new center is real only, but because of the trick Γ is the same: The Smith Chart's Gamma plane, as now real referenced, can be used for power matching, too!

Recommended

Excellent derivation of Gamma for power transfer in both directions between Antenna and Transceiver

Excellent Smith Chart program: Look for latest update of this great Freeware.

Acknowledgements

Thanks for different Gamma derivations personally done for me, to Prof. Dr.-Ing. Reinhard Stolle, University of Applied Sciences, Augsburg, DL5DU

Thanks for encouragement and approval to my former fellow Student and holder of numerous patents in RF engineering, Prof. Dr.-Ing. Habil. Dr. h. c. mult. Ulrich Lothar Rohde, N1UL, DJ2LR, DL1R, HB9AWE, V25UL

Thanks for his article and email response, first mentioning to me the difference between RC and Γ_SLIM to Dr.-Ing. habil. Walter Doberenz, DL1JWD,

Special recognition to my supportive amateur radio friends in DARC e.V., OV T-19, Friedberg

References

This article "Gamma-SLIM" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Gamma-SLIM. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

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