Talk:Dual impedance

Merge

This could easily be merged with the article on Duality_(electrical_circuits). The current article is more detailed but less comprehensive than that one.84.227.254.143 (talk) 06:28, 30 March 2014 (UTC)

Y=Z

Jordsan, what you have written, "the dual of Z is the admittance Z'=Y", is not correct, or at least adds no meaning. Z' is not an admittance, it is an impedance. The admittance that is numerically equal to Z' has an impedance of Z. Thus, it is equivalent to Z, not the dual of it. Take the case of an inductance, L. The impedance of this one-port is Z=jωL. The dual impedance of this is Z'=1/(jωL), which is in the form of the impedance of a capacitor. The capacitance of this capacitor is numerically equal to L. The admittance of this dual circuit is Y'=1/Z'=jωL. Y' is still representing a capacitor and is still the dual of Z. Thus, the admittance Y' (the dual of Z) is numerically equal to the impedance Z. SpinningSpark 14:37, 29 September 2015 (UTC) @Jordsan: SpinningSpark 14:38, 29 September 2015 (UTC)

Spninningspark, given that impedance and admittance are defined as

$\ Z={\frac {V}{I}}$, $\ Y={\frac {I}{V}}$ respectively,

stating that

$\ Y=Z$

creates confusion for everybody and, anyway you look at it, is not correct either.

I see your point and I am able to follow your line of thought as you explained above. A more appropriate way of stating those ideas is on your last edit, where it is said that $\ Y'=Z$. I would just add that $\ Y'=Z$ and $\ Z'=Y$ are equivalent statements. As you should be able to see from your last sentence "Thus, the admittance Y' (the dual of Z) is numerically equal to the impedance Z.", it is possible to equally correctly rewrite it as "Thus, the impedance Z' (the dual of Z) is numerically equal to the admittance Y." @Spinningspark: Jordsan (talk) 17:28, 5 October 2015 (UTC)

Well it's no more incorrect than writing Z' = 1/Z. That does not make sense in dimensional analysis either, until one assigns the units ohms-squared to the constant. Once dimensions are assigned to the constant, then Y'=Z is just as good as Z'=1/Z. SpinningSpark 17:34, 5 October 2015 (UTC)

Incorrect rejection of edits

@Spinningspark edits are incorrect and confusing. They are also not consistent.

In the beginning of the page, taking all equations together, it clearly states Z'=1/Z=Y=I/V and Y'=1/Y=Z=V/I.

In general, introducing a "reverse impedance and admittance" is silly when we already have the relation Z=1/Y. The whole page should just use impedance and admittance as is standard in electrical engineering. That would also fit with https://en.wikipedia.org/wiki/Duality_(electrical_circuits)

In dual circuits:

V <-> I

Z <-> Y

etc.

Now, aside from this unnecessary algebraic inverse, there are clear physical errors in the table, specifically the figure captions. One cannot simply equate R=G and Z=Y, no matter how you arrange it. By the pages own definition at the top, Z=V/I and Y=I/V, impedance and admittance cannot be the same, even if you normalize and scale it. While the (inverse) impedances are correct, the figure captions for the dual passive elements are imprecise/misleading and should simply read:

Resistor R <-> Conductor G ... G!=R but is actually the inverse. A 5 Ohm resistor must be depicted by a 0.2 S conductor as its dual, not 5 S as the current caption suggests. Both value and unit of the element change! Conductor G <-> Resistor R ... same as above, 1 mS conductor is dual of 100 Ohm resistor

Inductance L <-> Capacitance C ... here the numerical value stays the same, but the unit still changes. Capacitance C <-> Inductance L ... a 10 mF capacitor becomes a 10 mH inductor, so C!=L since a physical variable is value + unit!!

And now come the serious errors with the series and parallel connections:

1st caption says: Y=Z1+Z2 (wrong!) while Z'=1/(Z1+Z2) (=Y)

Both caption and figure are actually wrong, since a series connection of impedances becomes are parallel connection of admittances! But Y=Y1||Y2=Y1Y2/(Y1+Y2) = 1/(Y1/Y1Y2 + Y2/Y1Y2) = 1/(1/Y2 + 1/Y1) = 1/(Z1+Z2), so the dual impedance is not wrong.

2nd figure and caption says: 1/Y=1/Z1+1/Z2 (wrong!)

It's again parallel imp. become series admittances with NOT the same values (normalized/scaled or not). Y=Z'=Y1+Y2=1/Z1+1/Z2

Simple proof with DC and 2 series resistors only:

R1=Z1=400 Ohm

R2=Z2=100 Ohm

Z1+Z2 (series impedance)

input voltage/drop source 10 V DC

Clearly the voltages split into 8 V over R1 and 2 V over R2 with a current of 20 mA flowing.

In the dual circuit, current and voltage shift role (voltage "flows" and current "drops"). Taking the (incorrect) approach of G=R and going to a parallel of the impedances as the pictures and captions suggest:

G1=Z1=400 S

G2=Z2=100 S

Z1||Z2

input "flow" source 10 V DC

Voltage now "flowing" 2 V into G1 and 8 V into G2, with a parallel current "drop" of 800 A. Hmm, that doesn't seem match up, no matter how you put or scale it ...

However, if we do it correctly with G=1/R and parallel admittances:

G1=Y1=2.5 mS

G2=Y2=10 mS

Y1||Y2 (parallel admittance)

input "flow" source 10 V DC

Voltage "flow" divided in 8 V into G1 and 2 V into G2, with parallel current "drop" of 20 mA. Oh look at that, a duality!

Similarly, the active elements in the end of the table are imprecise. One cannot state V=I, which is only numerically correct, not in the units.

And before the normalization/scaling is pointed at, it doesn't make sense either, or rather it is physically and mathematically incorrect. Beginning of page says Z'=1/Z. That is a pretty clear cut formula. Unit of impedance is Ohms, [Z] = 1 Ohm. So [Z']= 1/Ohm.

Then the scaling formula says Z'/R0 = R0/Z. So in terms of units we have 1/(Ohm*Ohm) = 1 Ohm/Ohm ... which makes the equation physically invalid. Scaling in ANY equation means multiplying BOTH sides with a scaling value. One can't just multiply one side and divide the other ... that's elementary algebra. — Preceding unsigned comment added by 192.38.90.64 (talk) 12:00, 24 August 2020 (UTC)

Plainly, admittance is not equal to impedance, but an admittance can by numerically equal to an impedance. This is the dual relationship and what this article is trying to show. In the case of the dual of two series impedances Z1 and Z2, the dual is two parallel admittances numerically equal to Z1 and Z2. So it absolutely is true that the dual admittance Y is numerically equal to the sum of the original impedances values Z1+Z2. Z1 and Z2 in the second equation are admittances, not impedances.

The definition Z'=1/Z is made rational in units by assigning the units ohms squared to the "1". This can be viewed as the reference impedance is equal to 1. The relationship of units is only made explicit in the scaled version of Z'=Z0²/Z. SpinningSpark 15:14, 24 August 2020 (UTC)

Take a look at how this textbook annotates the dual diagrams (if you can access it). Or this book which has R=1/R' in an equation. That is pretty much following what we are doing here. SpinningSpark 15:53, 24 August 2020 (UTC)

Alright, I've taken a look at the books. None of them mention what you claim, i.e. none of them write G=R or Y=Z like in the figure captions. In fact, they explicitly write Z=1/Y and Z'=1/Z=Y. The latter book has R=1/R' as you write, which is NOT the same as R=G, but rather R'=G. If you really want to cling onto the inverse notation, the figure captions for the elements should use the inverse form like that, however to be precise inductance and capacitance would need the jω to be included since that inverts, too, i.e. jωL <-> (jωL)'=1/jωC. But I guess that would confuse even more.

Let's take a look at your cited example from the first book. Figure 3.5, an RC ladder and its capacitive dual another RC ladder with shunt and series switched. First we note, that unlike what the Wiki page states, the capacitor does NOT become an inductor, but stays a capacitor. That's because it is the special case of a capacitive dual for cascades that is analog to the normal dual. The Wiki page and table don't list that distinction and it is also beyond the scope. But let's check it for a 2 stage RC cascade:

RC ladder has capacitive dual of CR ladder (switching numeric values) and general dual of GL ladder following the relations in the table, but INVERTING numeric values with G!=R but rather 1/R and generally Z=1/Y!

Circuit: https://i.imgur.com/QFsAPqB.png, Results: https://i.imgur.com/3ULgx5V.png

All spectra on top of each other, all 3 circuits are equivalent. The bottom dual is the general one following the table on this Wiki page, BUT considering Z=1/Y. The numerical values for the elements are in fact not the same. (Side note: while C takes the same value as a dual of L, it is not numerically identical since the actual element values are of course impedance and admittance, where jωL <-> 1/jωC, so numerically the inverse).

Let us simply turn it into a DC R-R 2 stage ladder and see what happens when keeping the numerical values the same (incorrect) or properly applying Z=1/Y.

Circuit + result: https://i.imgur.com/h9sy1Pm.png

Clearly keeping impedance and admittance numerically the same as the Wiki page figure captions suggest is incorrect. The short circuit output current does not match the open circuit output voltage. Only by using the numeric inverse, Y=1/Z, does the dual circuit yield a dual behavior.

Let's check the math of "In the case of the dual of two series impedances Z1 and Z2, the dual is two parallel admittances numerically equal to Z1 and Z2. So it absolutely is true that the dual admittance Y is numerically equal to the sum of the original impedances values Z1+Z2." for this example. R1 is in series with R3||(R2+R4). Let's look at only the numerical values, so Z1=100 and Z2=152.78 (sorry, didn't pick nicer numbers). This transforms into a parallel circuit with Y1||Y2. With numerically equal, we'd get the incorrect second circuit. The numeric values must actually be inverse, not equal, leading the correct 3rd dual circuit, with a dual admittance of Y=10m||6.545m=3.956m ... which is NOT numerically equal to 252.78, but its inverse.

Last not least, again it's the figure captions that are wrong and misleading. Incidentally, the examples at the bottom of the page omit any resistive components, which would show these errors. I don't see how anyone can claim the captions are correct, when for example the parallel admittance caption says Y=Z1+Z2 but right next to it it says Z'(=Y)=1/(Z1+Z2). Clearly that math doesn't add up. Now excuse me while I continue preparation of lectures next week where I also repeat duality, in the hopes the students don't rely on incorrect and confusing Wiki pages like this. --192.38.90.64 (talk) 09:55, 25 August 2020 (UTC)

Z'≠Y (or Y' as we should more properly write). Z'=1/Y'. SpinningSpark 12:05, 25 August 2020 (UTC)

What now? You are making less and less sense. The article clearly defines Z'=1/Z. Naturally, Y'=1/Y follows. And the first sentence also introduces Y'=Z. So clearly Z'=Y is correct. And it even follows from your second formula. In even just that one sentence, the equations you give are not consistent, just like the figure captions in the article. You can't just go around and redefine variables as you go along and claim things like "[impedances] Z1 and Z2 in the second equation are admittances".

One should also note how you clearly ignore the mathematical and calculated proofs given above. Please demonstrate how "Y is numerically equal to the sum of the original impedances values Z1+Z2" matches the actual circuits given. Also kindly demonstrate where exactly your references write things like "Y=Z1+Z2" or "R=G" etc. --192.38.90.64 (talk) 13:15, 25 August 2020 (UTC)

Please, stop jumping off at the deep end and give me some credit of understanding the subject. Can we not agree that priming all the quantities that represent duals will completely address your objection? I cannot see that the problem is any more than that. SpinningSpark 13:31, 25 August 2020 (UTC)

To be clear, I agree with you that there is some inconsistency in notation. My objection to your edit is that the right hand half of the table is for the dual circuits and the expressions should be for the dual impedance (admittance). Your edit gave expressions for the reference impedance (admittance). Can we clean up the notation without moving from that format? SpinningSpark 14:01, 25 August 2020 (UTC)

I suggest the figure captions should simply try to avoid (incorrectly) doubling the equations already given by the Z and Z' column (where it's correct). That would remove most of the confusion. So in essence, something like:

Resistor R <-> Conductor G

... G <-> ... R

... L <-> ... C

... C <-> ... L

Series impedances <-> Parallel admittance (or Parallel dual impedances)

Parallel impedances <-> Series admittance (or Series dual impedances)

... V <-> ... I

... I <-> ... V

Same table with all incorrect or confusing aspects removed. Only problem that remains are the figures for the admittance, since they read Z but should read Y, or keeping your preference, Z'. That way, you would also get rid of all mentioning of Y in any equation or figure, could even remove the equation at the top (not the sentence). Relating everything to impedance or dual impedance and omitting any further mention of admittance would definitely benefit the page. Can you edit the figures? --192.38.90.64 (talk) 15:17, 25 August 2020 (UTC)

Essentially the table would the clearly define: "The dual of a Resistor (figure column) with value R (Z column) is a conductor (figure column) with a value of 1/R (Z' column)", or "The dual of an inductor with value jwL is a capacitor with value 1/jwL" and "The dual of a series with value Z1+Z2 is a parallel connection with value Z'1||Z'2=Z'1Z'2/(Z'1+Z'2)=1/(1/Z'1+1/Z'2)=1/(Z1+Z2)" etc. — Preceding unsigned comment added by 192.38.90.64 (talk) 15:26, 25 August 2020 (UTC)

I don't agree with some of that. Using Y brings out some of the essential nature of the dual circuit – the isomorphism of the equeations of the circuit and its dual. Z=Z1+Z2 → Y'=Z1+Z2 brings this out clearly in a way that Z'=1/(Z1+Z2) does not. Further, it is clearly stated that the dual of an impedance is an admittance, so an equation for admittance could reasonably be expected. SpinningSpark 08:25, 26 August 2020 (UTC)

Then use that. Y'=Z1+Z2=1/Y1+1/Y2 and Z'=1/(Z1+Z2) is the same equation. The accompanying figure is still incorrect though, since the value for the parallel elements is not Z1 and Z2 but their inverse, i.e. 1/Z1=Z'1 and 1/Z2=Z'2. You cannot assign the elements on the left and right the same values, they must be inverse, just like the rest of the elements above the series/parallel circuit. Based on the book you quoted, voltage over Z2 in the series connection should be numerically equal to current through Z'2 in the parallel connection, when driving with a numerically equal voltage/current, respectively. Let's set the sources to unity: V2=Z2/(Z1+Z2)*1V should be numerically equal to I2=Z'1/(Z'1+Z'2)*1A, which yields Z2/(Z1+Z2)=Z'1/(Z'1+Z'2). Setting Z'1=Z1 as in the figure, does not solve the equation. Setting Z'1=1/Z1 as it should be, however does solve the equation: Z2/(Z1+Z2)=1/(1+Z1/Z2)=1/Z1/(1/Z1+1/Z2)=Z'1/(Z'1+Z'2)!! So the figure on the right must use Z'1 and Z'2 or 1/Z1 and 1/Z2 or Y1 and Y2, but NOT Z1 and Z2. And the caption can then read Y'=Z1+Z2 or Y=1/(Z1+Z2) or Z'=1/(Z1+Z2) or ... --192.38.90.64 (talk) 08:53, 26 August 2020 (UTC)