Talk:Miller theorem

Some history

A transimpedance amplifier explained by Miller's idea in May 14, 1992. The handwritten text is translated below (think of E and U as of V; the small rectangles represent resistors).

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An ideal I-to-U converter (a possible explanation by opposite voltage)The current-sensing resistor R_I creates "disturbing" voltage drop U_R (it is needed but disturbing; there is a technical contradiction). We can neutralize it by "antivoltage" U_anti (an inversed copy of U_R) that is subtracted from U_R. This may be performed automatically by an op-amp A that changes U_anti so that the difference U_R - U_anti = 0 (the op-amp copies U_R).

I have started this page with a lot of excitement because I had the good fortune to realize on my own this great circuit idea in the early 90's (you can see on the genuine yellowed sheet on the right how I have discerned this idea in the circuit of the popular transimpedance amplifier). Actually, I realized even the more general idea of modifying circuit attributes (not obligatory impedance) by including an additional source (not obligatory proportional). First, I became aware of its "proportional voltage" version of modifying impedance by additional voltage (Miller theorem) and extracted six "golden rules" for obtaining modified impedance (increased, infinite, negative with current inversion, decreased, zeroed and negative with voltage inversion). They allowed me to understand all the popular circuits described in the Applications section and to realize the great circuit phenomena (Miller effect, virtual ground, bootstrapping, negative impedance, etc.), on which they were based. A bit later, I realized the dual "proportional current" version of modifying impedance by additional current (dual Miller theorem) that helped me to understand and explain to students such exotic circuits with negative impedance as load cancellers, Howland current source and Deboo integrator.

Since 2006, I have been trying to incorporate my insights about this modifying principle in Wikipedia pages dedicated to these circuits and phenomena (Virtual ground, Negative resistance, Negative impedance converter, Current-to-voltage converter, Voltage-to-current converter, Charge amplifier, Operational amplifier applications, Gyrator, etc). Although the idea was extremely simple, clear and intuitive - modifying impedance by connecting an additional proportional voltage source in series to the main input voltage source (or by connecting an additional proportional current source in parallel to the main input current source) and it showed the basic ideas behind all these popular circuits, my insertions were qualified as original research and they were removed many times. Imagine it turned out that connecting a battery in series to a resistor was supposed to be an original research! To help understanding, I was placed links to plenty of resources (circuit stories, flash movies, etc.) located on my site of circuit-fantasia.com; all they were removed as well. As a result, the truth about these legendary circuits and phenomena was buried in the history of these Wikipedia pages...

Then I started Circuit Idea wikibook and began creating mirror stories about these great circuit ideas. To give a chance to Wikipedia readers to find out the truth, I placed links to this sister Wikimedia project at the end of the according Wikipedia pages; but imagine even these humble links were removed! It turned out Wikibooks was not so sisterly project for wikipedians as it was proclaimed...

But the truth is truth; it cannot be removed or hidden, it is immortal, it is eternal... More and more often, we have been coming upon this powerful idea in various Wikipedia pages (see for example, discussions about gyrators and integrators). In the page about Miller effect, I noticed some interesting thoughts (maybe proposed by Roger) about the possibility to generalize the Miller effect that deeply impressed me. To this very moment, I had had (from my student years) a scanty notion of it as some nasty circuit phenomenon increasing stray capacitances. I was inspired as we were already not only two - the TRUTH and I; we were already three - the TRUTH, Miller and my humble person:) So, I wrote an emotional proposal about this aspect of generalized Miller effect and inserted explanations in the article. Unfortunately, the Anglo-Saxon coldness confronted my southern Slav warmth again:) and my insertions were brushed aside as too general...

Finally, the conversation between Oli Filth and Roger about the scope of the Miller effect, the explanations of Roger about Miller theorem and the suggestion of Oli Filth for creating a special article about the theorem made me start this page. Now I have the feeling that I have finally managed to find a place where to expose my insights about this great phenomenon... Circuit dreamer (talk, contribs, email) 11:04, 31 July 2010 (UTC)

Key points of exposing Miller theorem

How to understand the theorem

The key of understanding Miller theorem is to see in this arrangement three elements - two voltage sources and one impedance. The first (left on the top figure) source is main (driving, input, independent); the second (right) source is additional (supplementary, auxiliary, dependent). Combining two of these elements into one composed element can help us to see the basic idea behind the theorem. First, we may see that the two voltage sources are connected in series through the common ground; so, we may think of them as of a new composed voltage source supplying the impedance. Second, we may notice that the actual impedance and the second voltage source are connected in series; so, may think of them as of a modified actual impedance or as of a new virtual impedance. The role of the second voltage source in this arrangement is to change (modify) the effective voltage, the current and finally the impedance seen from the side of the first voltage source by adding or subtracting its own voltage from the input voltage (by "helping" or "impeding" the input source). Similarly, the dual Miller theorem can be understand by seeing in its arrangement three elements - two current sources and one impedance.

How to present the theorem

To present this powerful idea to readers in the most reliable way, it is extremely important to consider and compare the two situations - without and with connected additional voltage source V₂. Indeed, the first arrangement is exactly the humble Ohm's circuit (a voltage source V₁ supplying the impedance Z) but this auxiliary case is necessary only to realize the role of the second voltage source V₂.

Generalizing the theorem

This idea is extremely simple, clear and intuitive; it is not only a specific electric circuit idea; it is a general idea that we can see everywhere around us. It shows how to change in an artificial way the resistive properties of passive objects by attaching an additional power source, how to inject energy into these arrangements making the objects more or less passive and even active (negative). Here is a mechanical example: imagine the engine of your car accidentally stops running during the movement and you press the brake pedal; you (V₁) will experience noticeable opposition (impedance Z) as the vacuum booster does not work (V₂ = 0). When the engine runs and you press the brake pedal again, you will experience decreased impedance (Z_in < Z) as the vacuum booster "helps" you by multiplying and adding the force that your foot applies to the master brake cylinder (V₂ > 0). Note an important difference: the vacuum amplifier is not connected between you and the brake (the standard amplifying arrangement); it is connected in parallel to you! In cars, this arrangement is made for safety reasons.

Relations between wikipedians also obey Miller theorem:) In our case, creating this page I (V₁) encounter various difficulties (Z). You (V₂) may do nothing (V₂ = 0), you might help me (adding V₂) or you may oppose me (subtracting V₂); as a result, I will experience normal, decreased or increased difficulties. In the most general form, the two versions of this idea are referred to as synergism (cooperation, collaboration, assistance, team-work, Wikipedia idea, etc.) and antagonism (contention, opposition, resistance, impediment, etc.)

The subordination

Well, let's build a hierarchic classification of these phenomena to see where Miller theorem and Miller effect stay.

GENERAL MODIFYING PRINCIPLE (modifying the impedimental properties of passive objects by additional power source)

Electrical modifying principle (modifying circuit attributes by additional electrical source)

Modifying impedance by additional electrical source

Modifying impedance by additional proportional electrical source

Modifying impedance by additional proportional voltage source (Miller theorem)

Connecting the voltage source with an opposite direction to the input voltage source (voltage subtraction)

- increased impedance

- infinite impedance

- negative impedance with current inversion

Connecting the voltage source with the same direction to the input voltage source (voltage addition)

- decreased impedance (Miller effect)

- zeroed impedance (Miller effect)

- negative impedance with voltage inversion

Modifying impedance by additional proportional current source (dual Miller theorem)

Connecting the current source with an opposite direction to the input current source (current subtraction)

- decreased impedance

- zeroed impedance

- negative impedance with voltage inversion

Connecting the current source with the same direction to the input current source (current addition)

- increased impedance

- infinite impedance

- negative impedance with current inversion

As you can see, Miller theorem and its dual are special cases of modifying principle and Miller effect is a special case of Miller theorem. Circuit dreamer (talk, contribs, email) 11:04, 31 July 2010 (UTC)

Is it an original research?

Yes, it is... but in the common sense of this term, not in the specific Wikipedia OR sense. Really, I have exposed here and in the article my insights about modifying principle and my interpretations of Miller's theorems. As you have seen, they are more general and fully comprise the two theorems. Actually, I have filled the theorems with a new content - more general (modifying principle) and more specific (the 12 rules for modifying impedance); before, the theorems were only tools for creating equivalent circuits. So, maybe I should work out them in a separate Cyril's modifying principle or Cyril's golden rules for modifying circuit attributes (of course, this is only a humor!:) and to make Miller theorem and Miller effect subpages of this more general page (again just a humor!:)? Of course these great ideas are neither Cyril's nor Miller's nor somebody else's; they belong to NATURE; they are NATURE'S ideas!

But I have managed to reduce the two Miller's theorems and their implementations to the most elementary electrical circuit concepts and circuits (Ohm's and Kirchhoff's laws, series and parallel connected sources, etc.) that are well known for everyone from basic physics courses and from common sense. So, these clear, evident and simple explanations based on the human common sense should not be treated as original research in the Wikipedia OR sense; they do not need to be referenced! To defend this cause, a have extracted the key thoughts from the article below:

1. There are two voltage sources with voltages V₁ and V₂ and one element with impedance Z (apparent from basic electricity course).
2. The two voltage sources are connected in series (obvious); the impedance and the second voltage source are connected in series as well (obvious). In any amplifier the input voltage and the output voltage are connected in series (apparent). In amplifiers with parallel feedback, the feedback resistor and the output voltage are connected in series (apparent); as an example, in a transimpedance amplifier the resistor and the op-amp output are connected in series.
3. The second (output) voltage source depends linearly on the first (input) voltage as it is produced by an amplifier (apparent).
4. The two voltages are summed or subtracted (obvious, KVL). For example, in an inverting amplifier the input voltage and the output voltage are summed (travelling along the loop you will see consecutively + -, + - or - +, - +); in a non-inverting amplifier they are subtracted (travelling along the loop you will see + -, - + or - +, + -).
5. The total voltage (V₁ + V₂) applied across the impedance is different from the initial V₁ (more than obvious).
6. The current (V₁ + V₂)/R is different from the initial one because the voltage is different (obvious, Ohm's law).
7. As a final result, the circuit impedance seen from the side of the input source is different (modified) as the applied voltage is the same but the current is different (obvious, Ohm's law)...

It would be extremely interesting and funny for me if someone can prove that some of the humble assertions above are original research... Circuit dreamer (talk, contribs, email) 11:04, 31 July 2010 (UTC)

About the name

It is preferable to insert some words (in the lede or in a separate section) about the history of Miller theorem. But I am not clear about the origin of the name. Is the theorem proposed by Miller or the theorem is named after Miller? And who is Miller here - John Milton Miller, the inventor of Miller effect, or other person? Please, help.

Showing the relation with Miller effect

We have to reveal the connection between Miller theorem and Miller effect by answering the question "Is the Miller effect a special case of the Miller theorem, or is the Miller theorem a generalization of the Miller effect?" (see Miller effect talk). IMO both the assertions are right but as the Miller theorem is proposed later maybe it is more reliable to consider the theorem as a generalization of the Miller effect.

Is there a feedback?

Miller's arrangement does not obligatory implies feedback. If the input voltage source V₁ is almost ideal (with zero internal resistance), there is no feedback as it fixes the circuit input voltage and V₂ cannot change it through the impedance Z. If the input source is a real voltage source with some internal resistance or a resistor is connected between the source and the amp's input or the input source is just a current source, there is a feedback.

Needed assistance

There is a need at least of two pictures representing the two Miller theorem arrangements (the main and the dual one). They may be redrawn from this source. Also, the first aspect of Miller theorem and its dual as tools for creating equivalent circuits may be enlarged. Circuit dreamer (talk, contribs, email) 11:04, 31 July 2010 (UTC)

Intriguing web discussions

Miller's theorem and virtual ground

"There are many misinterpretations of such a simple theorem, including textbooks by Razavi, Sedra, this forum, etc..." It sounds confusing, especially if it is true...

"Miller's concept is actually based on the substitution of an impedance Zµ with two series ones, Zµin and Zµo, in such a way that their total impedance equals Zµ, and the potential at their interconnection equal zero, which is a virtual ground..." An interesting assertion (Zµin i= Zµin + Zµo); it turnes out the actual impedance is divided into two series connected impedances with virtually grounded intermediate point? For now, I can't realize what it means...

I WANT TO GET A DEEP UNDERSTANDING OF Miller's theorem

"And I think the system must be linear so we could use it..."
"Linearity is not a compulsion... the only requirement is a capacitor between input and output and some amount of change between input and output..." Interesting... K can be not a constant? Circuit dreamer (talk, contribs, email) 16:40, 1 August 2010 (UTC)

What is Miller's theorem?

"Instead of one impedance, which connectes two non-grounded nodes, Miller's Theorem allows this impedance to be broken down into two parallel impedances..." - very figuratively spoken. Circuit dreamer (talk, contribs, email) 21:14, 1 August 2010 (UTC)

This page is unnecessary

A mention of the term Miller Theorem could be added to the Miller Effect page, making this page redundant. If this page remains it needs to be truncated to just a short paragraph. It is very seriously over-worked. Zen-in (talk) 16:08, 2 August 2010 (UTC)

I'm given to agreeing, given that the author seems quite open about his use of Wikipedia to explore new treatments of electronics. Might be worth pinging some relevant WikiProjects to see what they think. Chris Cunningham (user:thumperward: not at work) - talk 09:31, 23 August 2010 (UTC)

I oppose. Miller Effect is (at least in german literature) only the effect that a feedback capaticity is largly with voltage amplification looking from input. Miller Theorem is more general way in analysing feedback circuits. --Biezl (talk) 17:55, 9 September 2010 (UTC)

About the lede

I have been waiting with the hope that some reliable wikipedians will bring Zen-in to his senses and will educate him about the topic. Alas, as usual, I have to do this donkey work...

A few days ago Zen-in replaced the lede with the scanty sentence "The Miller Theorem describes the effect of negative feedback through an impedance in a circuit that has gain". There are two thoughts in this sentence and both the thoughts are inaccurate. Both negative feedback and gain are not obligatory in this arrangement. Miller theorem is absolutely valid also in the case when there is no negative feedback and in the case when the circuit attenuates.

Let's now consider, sentence by sentence, the lede that you have removed:

"Miller theorem applies to the process of creating equivalent circuits" shows the purpose of the theorem.

"This general circuit theorem claims that a floating impedance element supplied by two connected in series voltage sources may be split into two grounded elements with corresponding impedances" is a short description of the Miller theorem arrangement.

"There is also a dual Miller theorem about impedance supplied by two connected in parallel current sources" is a short description of the dual Miller theorem arrangement.

"The two versions are based on the two Kirchhoff's circuit laws" does not need an explanation.

"Miller theorems are not only pure mathematical expressions. These arrangements explain important circuit phenomena about modifying impedance (Miller effect, virtual ground, bootstrapping, negative impedance, etc.) and help designing and understanding various popular circuits (feedback amplifiers, resistive and time-dependent converters, negative impedance converters, etc.)" mentions extremely useful applications of the two theorems.

"They are useful in the area of circuit analysis especially for analyzing circuits with feedback and certain transistor amplifiers at high frequencies" shows some classic Miller theorem applications.

"There is a close relationship between Miller theorem and Miller effect: the theorem may be considered as a generalization of the effect and the effect may be thought as a special case of the theorem" reveals the relationship between Miller theorem and Miller effect.

Zen-in, I have the feeling that you do not distinguish Miller effect from Miller theorem. Miller effect is just a special case of Miller theorem that is something more general.

Please, read the article and try to realize the simple truths behind the phenomenon. Circuit dreamer (talk, contribs, email) 15:30, 30 September 2010 (UTC)

Yes I will read it but seriously I think all that is needed here are just a few sentences so I will continue my edits and selective pruning, etc. Please do not apply my user name to your edits with the wiki formatting as it has the appearance that the edits were authored by me. If you continue to post insults and other defamatory material it will be necessary for me to cite you for uncivil conduct. The same applies if you continue to apply my signature to edits you have done. There are more constructive things you can do. Zen-in (talk) 06:14, 1 October 2010 (UTC)

Don't you think that it is strange, to put it mildly, first to remove the entire lede and then to read the article (shoot first, ask questions later)?!? Circuit dreamer (talk, contribs, email) 18:02, 1 October 2010 (UTC)

References

Why is circuit dreamer allowed to this stuff above?

Get a website! 82.169.255.79 (talk) 19:08, 1 May 2011 (UTC)

Because he's an egomaniac that has to let the world now how brilliant and gifted he thinks he is. He truly believes he has a gift when it comes to explaining, but he rambles incoherently.

He has systematically ruined many electronics pages on Wikipedia as a result. And why can he still post; 'circuit dreamer' is the same person as 'light current' (a banned user)!

At least some people have been reverting other pages back to their more normal forms without his terribly confusing "descriptions" and sea-of-ink diagrams. — Preceding unsigned comment added by 130.65.251.51 (talk) 22:57, 20 April 2015 (UTC)

Needs Another Draft or Two

Aside from the controversies related to original research and the concepts presented here, this article needs to go through at least one more draft. In particular, two compositional issues need to be addressed.

The first issue involves sentence structure. It is often convoluted and abstruse. The arrangement of subject, object, verbs, and adjectives--especially compound adjectives--makes this article difficult reading. Let one example stand for them all. The second sentence in the lede says this:

"The general circuit theorem asserts that a floating impedance element supplied by two connected in series voltage sources may be split into two grounded elements with corresponding impedances" (emphasis added).

I take it that "connected in series" is a compound adjective that modifies "voltage sources." As a compound, it should be hyphenated: "connected-in-series." In English, though adjectives are often placed before the noun they modify, in this case it's very confusing. This would be much clearer:

"two voltage sources connected in series."

In this revision, "connected" takes its proper place as a verb, the past participle of "to connect." "In" returns to its role as a preposition, and "series" is now a simple adjective. The phrase can be further improved by eliminating a redundancy: "connected." To say, "connected in series" is like saying "past history." So, here are both of my suggestions combined:

"two voltage sources in series . . ."

There are other issues with that sentence, especially the reference to the "general circuit theorem." Is this Miller's "general circuit theorem"? It seems like it, because the following sentence begins, "There is also a dual Miller theorem . . . " If "general circuit theorem" is Miller's, it should read, "Miller's general circuit theorem . . ."

The second issue has to do with odd or distracting word choice. Here is but one example:

"These arrangements explain important circuit phenomena about modifying impedance (Miller effect, virtual ground, bootstrapping, negative impedance, etc.) and help designing and understanding various popular circuits (feedback amplifiers, resistive and time-dependent converters, negative impedance converters, etc.)."

"Popular"? This word suggests a sort of democratic value judgment. I know that sounds absurd, and it absolutely is. That's why the word is so out of place. I suspect the author meant to write something like this:

". . . help designing and understanding various commonplace circuits . . . "

Now, having tossed out examples of my two issues, I fully expect the author would have seen and corrected these and other compositional problems all on his or her own if only the article had been run through the word processor a few more times. It's not too late to do so now. Prof. Todd Carney / Southern Oregon University 09:20, 6 July 2011 (UTC) — Preceding unsigned comment added by Tcarney57 (talk • contribs)

The Miller Theorem as presented is elementary circuit analysis applied to a simple Miller feedback circuit

As I have been familiar with Miller feedback since the 1950's, when I built a hi-fi preamp using it for the tone controls using a high gain video pentode for each one, Miller feedback is the effect illustrated by the second (non-inverting amplifier) and third (inverting amplifier) figures, where the circuit gain is determined by the two resistances (or impedances) in the circuit. These are the most common ways to achieve controlled gains using op amps, including integrators and other analog operations. Things like Miller's theorem as presented here is a topic in circuit analysis. Miller feedback is not a Wikipedia topic, but could be a very short one, and would make the points of most interest to designers and engineers. -motorfingers- (talk) 16:12, 4 May 2014 (UTC)

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