User:DJ7BA

~ This page was started, as Constant314 had deleted the derivation of two different Gammas on the page "Impedance Matching".

He gave this "reason": Not a valid derivation since it presumes the result for optimum current. But not needed. {{WP:NOTTEXT]].

My reply talk:

Please don't confuse the necessarily presumed maximum power transfer theorem with the derivation, resulting in a rating figure Γ for mismatch, the theorem doesn't provide.

That rating figure is the result of the derivation, not the maximum power transfer theorem itself again.

That would be presuming the result: "It must be true - else it wouldn't be true" is what you mean. That "presumed result" logic is really not present in this case.

You deleted two Γ Coefficient derivations, and a "beware" warning to not confuse these truely different coefficients - as obviously often is done - see the deleted references for ATIS and IEC glossary.

Where else - if not in "Impedance matching" - that already presents both Γ equations and the accompanying circuit, could the obviously necessary warning be better placed?

"Equating both Γ coefficients is an unproven assumption" is a strong statement, demanding counter-proof by derivation itself. So derivation - for the sake of reliable info

in wiki - is a must. "not needed" is no sufficient excuse.

Γ{{sub|SLIM}} to my best knowledge was never derivated in wiki - right? So why delete a necessary, never before presented derivation and at the same time let the unproven assumption,

that Γ{{sub|SLIM}} = RC,  go on unproven even though the deleted derivation proves ATIS and IEC glossary as being wrong?

Does wiki indeed hinder true relations challenging false ones? This is what that deletion does.

Of course, there are many ways to present a derivation. The one given is not the only one possible. But it is a surpriingly easy to understand one, as it doesn't need much complex math.

But you need to understand what relative difference between actual and optimum value of one of the the circuit's quantities for a clearly defined goal means. That's not asking too much, or?

See the wikilink.

Below find an alternative, but it is somewhat more demanding. It is based on the undisputable maximum available real power P{{sub|avl}} from a source having a complex source impedance, on one hand,

and on real power P{{sub|L}} calculation in the complex terminating load impedance 's real part R{{sub|L}} on the other hand. It uses these Powers, instead of currents in the other Γ{{sub|SLIM}} derivation deleted.

I am certain, the deleted one is easier to understand for most people. So I prefer it.

If you study it carefully, you will see the same, as you say "presumption" (Max. power transfer theorem) used as a given fact and the same (but more demanding, as Γ{{sub|SLIM}} is squared for power) complex Ohm's law etc.

Also it uses - of course, as Ohm's law needs that - the current I, implicitely included in $$P_L = \frac {|V_S|^2}{|Z_S + Z_L|^2} \cdot R_L$$.

Instead of I{{sub|opt}} it implicitely uses Voltage V{{sub|opt}} by saying: $$P_{avl} = \frac {|V_S|^2}{2} \cdot \frac {1}{Z_S + {Z_S}^*}$$ and as we all know that $$I^2 = \frac {U^2}{R} = I^2 \cdot R$$, that (P{{sub|avl}} or I{{sub|opt}}) cannot really be the point in your deletion, can it.

So: What is the difference with respect to "presuming the result for optimum (in this case) voltage"?

In a nutshell: Both are using optimum quantities given because of the Maximum Power Transfer Theorem, and both apply usual electronic complex Ohm's law etc.

One uses maximum available power, the other one uses the optimum current, but both derivations, of course, yield the same equation.

Here is the derivation as done by Augsburg University for Applied Sciences, Prof. Dr.-Ing. Reinhard Stolle. He kindly did it on special request for me, at a time when we had no other Γ{{sub|SLIM}} derivation yet.

If you blame Prof. Dr. Reinhard Stolle with not knowing what are the accepted priciples of a valid derivation, that comes close to an insult. Please don't do that. Wiki doesn't permit such. Else prove it with

the validity of proof that you yourself set as a standard by deleting a logically and physically good, 100% valid proof.

We both can presume that the maximum power transfer theorem is correct in what it states - don't we?

We both can use it for calculating an optimum value, be it I{{sub|opt}} or be it P{{sub|avl}}. Both are optimum values of quantities used together with Ohm's law etc. Nothing special. No presuming of the result.

Resulting is an equation that permits rating of the quality of a given (mis)match by numbers, that is Γ. This is not included in the presumed maximum power transfer theorem, that ONLY knows the optimum,

but not the suboptimum figure. The page title is "Impedance matching" showing a need to match mismatched circuits. The Γ equations derived will help to evaluate the matching efforts before purchasing parts.

This is what the presumed maximum power transfer theorem does not provide. It desctribes just that one - optimum - match. Nothing more that that. The Γ equations derived do that. Their result is more

than the presumed theorem. The fact that the presumed theorem will result in optimum current or optimum (= full available) power for a certain load impedance, is not a bad derivation method, but physics.

P{{sub|avl}} is the availabler power from a source. It occurs at perfect conjugate match Z{{sub|L}} = Z{{sub|S}}*.

$$P{avl} = \frac {|V_S|^2}{4 R_S}$$, or using $$R_S = \frac {1}{2} \cdot (Z_S + {Z_S}^*)$$

$$P_{avl} = \frac {|V_S|^2}{2} \cdot \frac {1}{Z_S + {Z_S}^*}$$

Real power transferred to the load is:

$$P_L = \frac {|V_S|^2}{|Z_S + Z_L|^2} \cdot R_L$$, or using $$R_L = \frac {1}{2} \cdot (Z_L + {Z_L}^*)$$

$$P_L = \frac {|V_S|^2}{2} \cdot \frac {(Z_L + {Z_L}^*)}{|Z_S + Z_L|^2}$$

If not all available power is transferred as useful power to the load, some of it is unused:

$$P_{unused} = P_{avl} - P_L$$.

The ratio of unused to available power is the square of |Γ{{sub|SLIM}}|.

$$|\Gamma_{SLIM}|^2 = \frac {P_{avl} - P_L}{P_{avl}} = 1 - \frac {P_L}{P_{avl}}$$

$$= 1 - \frac {(Z_L + {Z_L}^*) \cdot (Z_S + {Z_S}^*)}{|Z_S + Z_L|^2}$$

$$= 1 - \frac {Z_L Z_S + {Z_L}^*{Z_S}^*+{Z_L}^*Z_S+Z_L{Z_S}^*} {|Z_S + Z_L|^2}$$

$$= \frac {(Z_S+Z_L)({Z_S^* +{Z_L}^*) - (Z_LZ_S + {Z_L}^*{Z_S}^* + {Z_L}^*Z_S + Z_L{Z_S^*})}}{|Z_S + Z_L|^2}$$

$$= \frac {|Z_S|^2 + |Z_L|^2 + Z_S{Z_L}^* +Z_L{Z_S}^* - Z_LZ_S -{Z_L}^* {Z_S}^* - {Z_L}^*Z_S - Z_L{Z_S}^*}{|Z_S + Z_L|^2}$$

$$= \frac {|Z_S|^2 + |Z_L|^2 - Z_LZ_S -{Z_L}^*{Z_S}^* }{|Z_S + Z_L|^2}$$

and because $$|Z_L - {Z_S}^*|^2 = (Z_L - {Z_S}^*)({Z_L}^* - Z_S) = |Z_S|^2 + |Z_L|^2 - Z_SZ_L - {Z_S}^*{Z_L}^*$$

$$\left| \Gamma_{SLIM} \right|^2 = \frac {|Z_L - {Z_S}^*|^2} {|Z_S + {Z_L}^*|^2}$$

$$|\Gamma_{SLIM}| = \left| \frac {Z_L - {Z_S}^*}{Z_S + Z_L} \right|$$

Anything perhaps not correct? Then prove it please.