Having trouble with Double Null Injection for nEET

[Andrew_K]

Hi,

I'm trying to learn about the Extra Element Theorem, and the N Extra Element Theorem.

In general, this method makes sense to me. But I'm having some trouble understanding Double Null Injection, particularly a few cases that I get when I try to use this method.


1) According to "Fast analytical techniques for electrical and electronic circuits" by Vorperian Vatche, we should never need to deal in terms of the input voltage. However, there are a few cases where I get something like this:



Conveniently, in the books and videos, and even in other books or courses, the circuits are always constructed such that this is never an issue.

2) There are a few situations I can think of where the null injection impedance won't make sense for the given term. For example:



If the "extra element" here is an inductor, this makes perfect sense.

$$i_{t} = 0$$, therefore $$\mathscr{Z} = \frac{v_{t}}{0} = \infty $$

For an inductor, this term will make sense:

$$\frac{L}{\infty} = 0$$

But for a capacitor,

$$C * \infty$$

Which does not make sense.

[Benta]

I survived the first two slides of this:
https://www.coursera.org/lecture/techniques-of-design-oriented-analysis/introduction-to-n-extra-element-theorem-neet-ui2rq
and then killed it.
To me it's just a really convolved way of presenting something bleedingly obvious.
I your place, I wouldn't pursue it. I can imagine that someone finds it "pedagogically valuable", but so is plasticine.
And going to the lengths of introducing special symbols places it... well, I don't know where.

[SuperFungus]

It's been too long since I've messed with this for me to be able to follow your example, but I thought I recall that for the nEET there were "choices" of which pairs of elements to include in each step.  Either choice would usually be equivalent, but it's also possible that one of the choices would introduce a singularity which mean you would have to choose the other pair. Maybe that's what you're hitting?  I was coincidentally thinking I should review my notes on this at some point, if I do that I'll re look at this and maybe have something more helpful to offer.

Something more immediately useful: If you didn't know already, Vorporian has a YouTube channel with some helpful lectures https://www.youtube.com/channel/UCXOe8OLJT92XC56MMvbHUfA

[Andrew_K]

I survived the first two slides of this:
https://www.coursera.org/lecture/techniques-of-design-oriented-analysis/introduction-to-n-extra-element-theorem-neet-ui2rq
and then killed it.
To me it's just a really convolved way of presenting something bleedingly obvious.
I your place, I wouldn't pursue it. I can imagine that someone finds it "pedagogically valuable", but so is plasticine.
And going to the lengths of introducing special symbols places it... well, I don't know where.

Very useful, thanks!

It's a tool to derive simpler transfer functions from circuits without a ton of algebra. Not surprised you don't see the value in this.

[Andrew_K]

It's been too long since I've messed with this for me to be able to follow your example, but I thought I recall that for the nEET there were "choices" of which pairs of elements to include in each step.  Either choice would usually be equivalent, but it's also possible that one of the choices would introduce a singularity which mean you would have to choose the other pair. Maybe that's what you're hitting?  I was coincidentally thinking I should review my notes on this at some point, if I do that I'll re look at this and maybe have something more helpful to offer.

Something more immediately useful: If you didn't know already, Vorporian has a YouTube channel with some helpful lectures https://www.youtube.com/channel/UCXOe8OLJT92XC56MMvbHUfA

I think you're right, I think the first example I had can be resolved by swapping the order.

Still, I'm not so sure about the second one.

It should be 0. I know that from looking at the total transfer function solved from matlab.

I just don't know how this issue is resolved in the neet theorem.

I reached out to Dr. Vorporian on his youtube account. Hopefully he's receptive to questions.

[Benta]

Just set s=0 when using Laplace notation. That'll give you the DC result.
Simple. No need for this kind of complication.

[SuperFungus]

Just set s=0 when using Laplace notation. That'll give you the DC result.
Simple. No need for this kind of complication.

  Sure that's simple, but what does that have to do with nEET?  nEET extracts all the poles and zeros of the circuit in an already factored form and gives you a transfer function describing the circuit performance over the full frequency range.   

[Benta]

nEET extracts all the poles and zeros of the circuit in an already factored form and gives you a transfer function describing the circuit performance over the full frequency range.

I'm always ready to learn. And extracting poles and zeroes in a simpler way than the classical certainly interests me.
Please explain further, I'm all ears and eyes.

The full frequency range was not the OP's topic, BTW. It was DC state.

The palm was somewhat unfriendly and unnecessary, especially from a greenhorn.

[SuperFungus]

I'm not an EET expert but I've dabbled.  The general process for the Extra Element Theorem (EET) is:

1) Designate an element as "extra".  This is often an impedance which you can turn into a short or open to remove from the circuit, but it can also be a gain you take to be infinity.
2) Solve the transfer function of this simplified circuit.  Based on what element you chose as extra this solution may have design significance.  If the extra element is a reactance, then this calculation would be the DC/High Frequency/Mid-band gain depending on which reactance and whether you take it as open or short.  If the "extra element" is a dependent source with infinite gain, then this result is the ideal circuit performance when the gain is infinite etc.
3) Do two more calculations with respect to the extra element in the circuit (Null Double Injection and Driving Point Impedance).  Assemble these two results into a correction factor which extends the simplified transfer function in step 2 to now account for the previously removed Extra Element.  Again, these correction terms have physical meaning.  If the element was a reactance, the results are poles and zeros due to that reactance.  If the element was a gain, this correction factor models how circuit performance is degraded from "ideal" when gain is less than infinite.

The "n-Extra Element Theorm" (nEET) extends the EET from one extra element to many. There is more to it than that, but that's the summary. The idea is instead of one big mess of algebra which you try and factor into a meaningful form at the end of the analysis, you proceed in the analysis in such a way that you intentionally get meaning as you go.

The full frequency range was not the OP's topic, BTW. It was DC state.

No, the OP is solving for one of these intermediate correction factors via Null Double Injection.  It's not the same as the DC state.

It is certainly a different way of looking at circuits, and some of the Null Double Injection calculations take a while to get used to.  But it's a serious technique - certainly not a toy comparable to "plasticine".  If OP has an interest in the technique, I hope he is not discouraged from pursuing his interest further.  I certainly hope to be able to dedicate some more serious study to it in the future, it seems like a very powerful technique.

[SuperFungus]

Still, I'm not so sure about the second one.

It should be 0. I know that from looking at the total transfer function solved from matlab.

I just don't know how this issue is resolved in the neet theorem.

I reached out to Dr. Vorporian on his youtube account. Hopefully he's receptive to questions.

Another theory on the second example:
You're reasoning that $$\frac{v_t}{0} = \infty$$

However, in this case $$v_t = 0$$ as well so you have the indeterminate form $$\frac{0}{0}$$ which you would need to apply L'Hopital's Rule to resolve, but could turn out to be just $$0$$.

There's probably a more direct "Design Oriented Analysis" route to that answer, but I suspect that is the crux of it.

[emece67]

.

[SuperFungus]

I'm not here to dissuade anyone from their informed opinions on the technique - mostly I was trying to cut through some of the misunderstanding and maybe actually answer OPs question. Like I've said, I haven't yet personally mastered the technique, so I'm really not the best one to defend it.  I can agree with you that it doesn't make analysis of complex circuits easy, and it certainly would take some dedicated effort to master the techniques. 

I've watched now the 20 Vorpérian's videos (well, with some FF) and he still does not obtain the transfer function in a factored form, but as a ratio of polynomials, and still after the very same tons of algebra. AFAIK no "factored form" for the solution.

Well, I agree there is no way to get away from the fact that a Laplace domain transfer function is a ratio of polynomials.  The EET results are factored in the sense that the form of the polynomial is directly a pole and a zero associated with the extra element.  I concede that the nEET results seem somewhat more complex, since there is the addition of the cross-coupling terms but I don't know how this could be avoided.  I would venture that the algebra is not "the very same" because each term of the solution can be computed in isolation from the others, and because of the Nulled output or Null-Double Injection conditions of the calculations much more of the circuit can "drop out" while you are doing each computation.

I have revisited this method a few times in these 20 years and, definitely, I find it of no value. The supposed fast method is a mechanical, blind, calculation of open/closed impedances and then an equally mechanical and blind construction of expressions from such previous computations. One finally arrives at a solution, but during the process one has gained no insight on why the circuit works as it works. There is no difference in applying this method or applying, equally mechanically and blindly, the matrix method. Or better, mechanically and blindly letting a computer solve it for you.

There are other techniques and tools that not only allow you to solve the circuit, but also give you knowledge about where to touch the circuit to modify some parameter and what to expect from a circuit. Theorems of Miller, Bartlett, Thevenin, quadrupoles, feedback theory, bode plots, network synthesis theory, control theory,… that's the way. Better walk this path and spend no time in delusions.

I agree that for large circuits for which you'd like to compute a specific response, the numerical methods or simulation are best.  You loose me when you say that EET is a purely mechanical process which gives no insight into how the circuit works, because that is precisely where it excels.  From the start, you pick a reference condition of the circuit and the compute correction terms which model the deviation from that ideal reference state.  Each term along the way is a time constant associated with each reactance, or a cross coupling term.  Since each term is computed from looking at a circuit, opportunities for simplifying approximation are more abundant.  For instance, you may well see that while computing some term two resistances appear in parallel with one much smaller than the other - picking that approximation out of the algebra after taking a huge determinate would pretty much be a non-starter.

EET/nEET isn't meant to be a substitute for other circuit analysis techniques, and I don't really think anyone would suggest someone should just learn the nEET in lieu of any of the things you list.  That said, none of the things you suggest really strike me as incompatible with EET/nEET or as alternatives which would fill the same niche.  You could (and probably often should) apply Thevenin/Norton/Bartlett/Miller theorems in the intermediate computations of a nEET analysis.  Quadripoles (two port networks) is a way of formatting and using 4 transfer functions, but nEET is a way of deriving any individual transfer function so can be useful in producing a 2-port model.  You certainly should analyze your circuits' performance with bode plots and feedback/control theory, but unless you're in a situation where you can do the whole thing numerically you need a transfer function (which the nEET can help with).  Network synthesis is not so much a theory, as it is an entire field of techniques.  Most are approximations limited to a few useful cases, and generally require some sort of analytical model (i.e transfer function) they can fit to a performance target.

Doesn't all these sound to you as marketing BS?

Somewhat exaggerated maybe, but I don't know if I'd really call it "marketing BS".  I don't think there is any subversive profit motive in Vorperian's enthusiasm for the techniques - he probably knows his 18yo circuit analysis books aren't going to crack the NYT best sellers list.  My read is that he's had success with these techniques over his academic (Caltech, VT) and professional career (JPL) and is eager to share them.  Again, if the techniques aren't for you fair enough but it seems like you're implying some sort of less than genuine motives to Vorperian here which seems unnecessary...

[trinacria]

There are three ways to fix an undetermined value.

• Sometimes, you can choose a different combination since the order is arbitrary.
• If not, you can fix it with a temporary dummy resistor.
• Otherwise, you can nest the NEET.

The first two options may require some rearranging, while the third option does not. Also take care to note when the reference state of an element is inverted because it flips the terms upside down. The attached demonstrates everything except nesting for a two-capacitor filter, like your example.