# Tricky little op amp.

#### MrAl

Joined Jun 17, 2014
8,555
There's no need to guess -- just do that math.

[in reference to: A>>1+R1/R2]

If it weren't dimensionless, then the units would be there. If it lacks dimensions, then it is dimensionless (seems rather obvious when put that way, doesn't it?).

This, of course, is only if the units are properly tracked in the first place. If someone is routinely sloppy with units, then it's not surprising that they would never have confidence whether a number in an equation is dimensionless or not because they are used to seeing dimensioned quantities presented as pure numbers and then just tacking on whatever units they think the answer should have regardless of the units that their work actually results in.

That's how we slam hundred million dollar space probes into planets or let airliners filled with passengers run out of fuel mid-flight.
Hello again,

Well we know that it 'should' be dimensionless for two reasons (and not because the units are not there):
1. 'A' is usually understood to be dimensionless.
2. R1/R2 is one resistance divided by another resistance or Ohms/Ohms which is dimensionless.

But in doing that we are taking the units from the context not from any stated or unstated units.
Now maybe it does not always work (track units as you said) but if we say:
R2>1

then since we have units of Ohms on the left we assume units of Ohms on the right. I've never seen anyone demand that we state the units for every single statement, although i agree that would be nice if they did. That would require writing:
"R2>1 with R2 in Ohms and '1' in Ohms."
Yeah sure if we had symbols it would be nice too:
"R2>1W"
where "W" here is used to represent "Ohms".
or even:
"R2>1R"
which is also typical.

Yeah your last line shows how problems can arise because of poor communication. That one time was a doozy too
I thought someone was just joking around when they told me for the first time that the thing crashed into Mars because of a units mixup. I still couldnt believe it i had to look it up to be sure. NASA=Not Always Sure About

#### WBahn

Joined Mar 31, 2012
26,398
Hello again,

Well we know that it 'should' be dimensionless for two reasons (and not because the units are not there):
1. 'A' is usually understood to be dimensionless.
2. R1/R2 is one resistance divided by another resistance or Ohms/Ohms which is dimensionless.
But in doing that we are taking the units from the context not from any stated or unstated units.
'A' is not just "usually understood". It is, by definition (for this circuit), the ratio of an output voltage to an input voltage and is therefore dimensionless.

Now maybe it does not always work (track units as you said) but if we say:
R2>1

then since we have units of Ohms on the left we assume units of Ohms on the right.
No, we do NOT have units of ohms on the left, we have the dimension of electrical resistance. Ohms are just one possible specific unit of measure of resistance. And we should never just "assume" that it has the specific units that we would just like to have it have. The fact that you've wasted a bunch of time trying to assert that such a claim has meaning and failed in every attempt to do so is strong evidence of this.

Just as if we were working a problem involving distances and someone claimed to end up with

D1 > 1

What is the units on the left? Meters? Feet? Light-years? Angstroms?

I have asked this several times and you refuse to answer it. Could it be because you CAN'T answer it in any way that makes sense because it is inherently meaningless?

The left has DIMENSIONS of length. Thus the right has to have DIMENSIONS of length. There are many, many units of length, but they are all commensurable (can be compared, added, and subtracted).

For instance, if I tell you that Person 1 is 90 tall, is that person really tall or really short?

If there is a requirement that Person 1 (h1) must be taller than Person 2 (h2), namely

h1 > h2

and if I tell you that h1 = 60 and h2 = 100, is the condition met or not?

What if I tell you that h1 = 60 inches and h2 = 100 cm? Is the condition met or not?

Gee, units kinda matter, don't they?

I've never seen anyone demand that we state the units for every single statement, although i agree that would be nice if they did. That would require writing:
"R2>1 with R2 in Ohms and '1' in Ohms."
It would require no such thing. Go look at any of the hundreds of posts that I have made where I have worked a problem through from beginning to end. They have all been worked tracking units consistently at each step of the process, whether that step involved symbolic quantities, specific values, or a mixture of the two.

Symbolic quantities carry their dimensions. Numbers must have explicit units. The very fact that you end up in a situation where you think you would need to do that is another red flag that you have made a mistake somewhere and ended up with a result that is meaningless -- no amount of patching is going to fix that.

If I conclude that I have a requirement that

h1 > h2

I do not have to say what units are used for either h1 or h2. It is perfectly reasonable to replace these symbolic quantities with specific quantities, such as

60 in > 100 cm

That is a perfectly reasonable and valid expression. I can very easily compare 60 inches to 100 centimeters - and I do NOT do it by just comparing the number '60' to the number '100'. I do it by comparing the quantity '60 inches' to the quantity '100 cm'. I can convert them to the same units if I need to, but in this case just a passing familiarity with the relative scale of the two units is sufficient for me to answer the question of whether the inequality is satisfied.

Yeah sure if we had symbols it would be nice too:
"R2>1W"
where "W" here is used to represent "Ohms".
or even:
"R2>1R"
which is also typical.
And here again you make the same mistake that led you down a rabbit hole before. The constraint R2 > 1 was MEANINGLESS to begin with!. The 1 on the right was dimensionless. It did NOT have dimensions of resistance and most certainly did NOT have units of ohms. YOU chose to ARBITRARILY and without ANY justification CHANGE the right hand side by tacking on a unit that did not belong there because you refused to even consider the reality that the fact that it was dimensionally unsouond was a red flag screaming out that it was meaningless. So you arbitrarily assigned a random meaning to it and ended up with a meaningless constraint that was shown repeatedly to be neither necessary nor sufficient with specific counter examples and you insisted on then trying to further patch your conclusion with yet further conclusions based on assigning random and arbitrary meanings that did nothing but produce yet more constraints that were meaningless and shown to be so.

#### MrAl

Joined Jun 17, 2014
8,555
'A' is not just "usually understood". It is, by definition (for this circuit), the ratio of an output voltage to an input voltage and is therefore dimensionless.

No, we do NOT have units of ohms on the left, we have the dimension of electrical resistance. Ohms are just one possible specific unit of measure of resistance. And we should never just "assume" that it has the specific units that we would just like to have it have. The fact that you've wasted a bunch of time trying to assert that such a claim has meaning and failed in every attempt to do so is strong evidence of this.

Just as if we were working a problem involving distances and someone claimed to end up with

D1 > 1

What is the units on the left? Meters? Feet? Light-years? Angstroms?

I have asked this several times and you refuse to answer it. Could it be because you CAN'T answer it in any way that makes sense because it is inherently meaningless?

The left has DIMENSIONS of length. Thus the right has to have DIMENSIONS of length. There are many, many units of length, but they are all commensurable (can be compared, added, and subtracted).

For instance, if I tell you that Person 1 is 90 tall, is that person really tall or really short?

If there is a requirement that Person 1 (h1) must be taller than Person 2 (h2), namely

h1 > h2

and if I tell you that h1 = 60 and h2 = 100, is the condition met or not?

What if I tell you that h1 = 60 inches and h2 = 100 cm? Is the condition met or not?

Gee, units kinda matter, don't they?

It would require no such thing. Symbolic quantities carry their dimensions. Numbers must have explicit units. The very fact that you end up in a situation where you think you would need to do that is another red flag that you have made a mistake somewhere and ended up with a result that is meaningless -- no amount of patching is going to fix that.

If I conclude that I have a requirement that

h1 > h2

I do not have to say what units are used for either h1 or h2. It is perfectly reasonable to replace these symbolic quantities with specific quantities, such as

60 in > 100 cm

That is a perfectly reasonable and valid expression. I can very easily compare 60 inches to 100 centimeters - and I do NOT do it by just comparing the number '60' to the number '100'. I do it by comparing the quantity '60 inches' to the quantity '100 cm'. I can convert them to the same units if I need to, but in this case just a passing familiarity with the relative scale of the two units is sufficient for me to answer the question of whether the inequality is satisfied.

And here again you make the same mistake that led you down a rabbit hole before. The constraint R2 > 1 was MEANINGLESS to begin with!. The 1 on the right was dimensionless. It did NOT have dimensions of resistance and most certainly did NOT have units of ohms. YOU chose to ARBITRARILY and without ANY justification CHANGE the right hand side by tacking on a unit that did not belong there because you refused to even consider the reality that the fact that it was dimensionally unsouond was a red flag screaming out that it was meaningless. So you arbitrarily assigned a random meaning to it and ended up with a meaningless constraint that was shown repeatedly to be neither necessary nor sufficient with specific counter examples and you insisted on then trying to further patch your conclusion with yet further conclusions based on assigning random and arbitrary meanings that did nothing but produce yet more constraints that were meaningless and shown to be so.

Hello,

It seems to me that you are making an unnecessarily big deal out of this.
Have you read any physics books lately?
If you dont know how to judge the units from the context i can not do anything about that.

"D1>1"
has no context, therefore we dont know what units D1 (or the 1) is.

However, compare with:
"D1=1N4148"
Now what are the units, "Diodes" ?
No of course not, but at least we know we are dealing with a diode.

R1
on the other hand when we are talking circuits, is always Ohms unless stated otherwise.

Of course "R1>1 Ohm" is better than "R1>1" but the context tells us that if it is not stated then it is more than likely Ohms.
If we wrote:
"R1>10megohms"
or:
"R1>2.2k"
then we know that the units are a multiple of ohms respectively.

There is also the case where all the resistors are given in K Ohms. Then we could interpret:
"R1>2"
to mean:
"R1>2k"

Of course the second is better, but we dont just ignore the context either in the first.

It's entirely up to you how you want to do it. But you'll find that many authors assume some units or write in unit free form.
A good example is when dealing with transformer equations. Often the author states the units ONE TIME near the beginning or else writes in a unit free form where the units dont matter because you can choose your application requirement: V=L*W*H.
Now are you going to argue that V could be in meters and L,W,H could be in inches if we dont specify all the units? I dont think so

#### WBahn

Joined Mar 31, 2012
26,398
Hello,

It seems to me that you are making an unnecessarily big deal out of this.
Have you read any physics books lately?
Actually, physics (and even more so chemistry) texts are much better about properly tracking units than most texts.

Sadly, math texts and engineering texts are among the worst.

The reason is actually pretty straightforward -- people learn their early math from textbooks written by mathematicians who write math textbooks. Mathematicians, by and large, work in an abstract space and don't think in terms of physical measurement very often. They also work in a world where if they mess up the units, they get a wrong answer, instead of in a world where if they mess up the units things can break and people can die.

As a consequence, students learn bad habits from the bad example set forth by their K-12 math classes (and I was no different). So now let's look at the bulk of college texts in engineering -- unfortunately a large fraction of them are written by people that have little to no real-world engineering experience. So to them messing up the units still just means having to add one more item to the textbook's errata sheet. The physical sciences aren't nearly as bad because many of the textbook authors there have at least had a lot of physical research experience that has taught them the real-world penalties of being sloppy with units.

I can't even count the number of problems I have found that were worked by the authors (or their grad students) and were wrong SOLELY as a result of failing to track their units and, instead, just tacking on the units they wanted to have onto the end result.

I watched someone die a pretty horrible death (and was literally within three feet of being that person instead) because someone couldn't be bothered to track their units through their work and, instead, just tacked the units they wanted the answer to have onto the end. The result, as the accident investigation revealed going through his work, is that he converted from inches to centimeters by multiplying the number at a certain line of his work by 2.54. The only problem was that, at that point in the work, that number was not a diameter, but an area and thus needed to be converted from square inches to square centimeters by multiplying by 2.54². But, like you and so many others, he couldn't be bothered to properly track his units and was content to just be lazy and assume what the result meant. It cost him his life and almost cost me mine.

If you dont know how to judge the units from the context i can not do anything about that.

"D1>1"
has no context, therefore we dont know what units D1 (or the 1) is.

However, compare with:
"D1=1N4148"
Now what are the units, "Diodes" ?
No of course not, but at least we know we are dealing with a diode.
I told you very explicitly that the example at hand was dealing with distances. By now claiming that D1 could somehow be a particular diode you are just being absurd in your efforts to dodge a question that you can't even begin to answer.

R1
on the other hand when we are talking circuits, is always Ohms unless stated otherwise.

Of course "R1>1 Ohm" is better than "R1>1" but the context tells us that if it is not stated then it is more than likely Ohms.
And yet, because of this false belief of yours, you repeatedly drew conclusions that were demonstrably wrong! You made a mistake and ended up with a fundamentally meaningless expression. When you got A >> R1+R2, it was MEANINGLESS. But you insisted on then assuming that it means that the numerical value of the gain has to be greater than the numerical component of the value of the sum of the resistances when they happen to be expressed in ohms, which was shown was completely WRONG. That is neither a necessary nor a sufficient condition. It is meaningless.

When you later claimed that R2>>1, that 1 did not have units of resistance at all. Not ohms or any other unit of resistance. It never did. By arbitrarily tacking on the units that you wanted it to have you changed the result and, in the process, ended up with yet another incorrect conclusion that was shown to be completely wrong.

How many times are you going to have to draw blatantly wrong conclusions before the notion that, perhaps, the result you are patching up with your assumptions is wrong to begin with?

It's entirely up to you how you want to do it. But you'll find that many authors assume some units or write in unit free form.
A good example is when dealing with transformer equations. Often the author states the units ONE TIME near the beginning or else writes in a unit free form where the units dont matter because you can choose your application requirement: V=L*W*H.
Now are you going to argue that V could be in meters and L,W,H could be in inches if we dont specify all the units? I dont think so
For an equation like this the author didn't need to state units ANY time at all. The symbols carry their dimensions.

Assuming that this is an equation for the volume of a rectangular solid, then V has dimensions of volume (which is commensurable with the cube of dimensions of length) while L, W, and H all have dimensions of length. Multiply L, W, and H together and guess what, you get the cube of dimensions of length -- the same as the V on the other side of the equation.

Are you REALLY claiming that that equation only applies to a particular set of units that has to be spelled out ahead of time? That YOU can't use that equation to find the volume of a box that is 1 foot long, 1 meter wide, and 1 yard tall and express the result in cubic inches?

I most certainly am NOT going to argue that V could be in meters, any more than I would argue that the volume of my gas tank could be in inches. I sure hope you don't think that V could EVER be in meters.

Let's set L = 1 foot, W = 1 meter, H = 1 yard. That means that V = 1 ft·m·yd. While not usual, like acre-foot it is a perfectly valid unit of volume (and which just happens to be slightly over 17 kin² or just shy of 279 liters or 0.279 m^3.

If, given L and W, you wanted to know what H needs to be in order to get a particular volume, then you have

H = V/(L·W)

Pick some volume: 811 microacre-foot.
Pick some L: 0.621 millimiles
Pick some W: 10 giga-angstroms

Do I really have to provide you with a different equation that only works for those specific units.

Last edited:

#### WBahn

Joined Mar 31, 2012
26,398
Hello again,

Well we know that it 'should' be dimensionless for two reasons (and not because the units are not there):
1. 'A' is usually understood to be dimensionless.
2. R1/R2 is one resistance divided by another resistance or Ohms/Ohms which is dimensionless.

But in doing that we are taking the units from the context not from any stated or unstated units.
Now maybe it does not always work (track units as you said) but if we say:
R2>1

then since we have units of Ohms on the left we assume units of Ohms on the right.
Let me try a different tack.

Hopefully you will agree that there are many physical quantities for which there have been many, many very different units of measure and those units can differ by several orders of magnitude. In most cases that is because the need to measure that quantity developed independently in different parts of the world at different times in history and, as a consequence, there was no effort to agree on a single unit of measure for that quantity. Pick pretty much any quantity that has been being measured for more than a couple hundred years and you likely have a shining example -- the more common the usage, the greater the number and variety of units. Units of distance are perhaps the extreme with common measures ranging from the angstrom to the light-year -- a difference in scale of 26 orders of magnitude.

We just happen to have a single common measure for electrical resistance not because there is anything fundamental about an ohm, but merely because it developed within a relatively small community of people in sufficiently adequate communication with each other that they settled on a single unit for their purposes and as the need to measure electrical resistance became more common it was only natural that people new to the concept adopted the single existing unit of measure in use. It's pure historical coincidence.

Now, imagine that people had started measuring electrical resistance just a century or two before in several different places and they weren't in communication with each other. Is it not reasonable to expect that many very different measures of resistance would have resulted. Some might have measured electrical current in electrons per hour while others might have done so using a unit of charge that was equal to that of Avogadro's number of electrons per a unit of time that is one one-millionth of the length of a day. Similarly, the units of voltage, which is energy per unit charge, might have been newton-meters per coulomb in one place, foot-pounds per electron in another, and any number of other commensurate units in other places. In each of these places they would have likely used their local units for voltage and current to define a unit of resistance and, for simplicity, let's assume that they also gave it a name in honor of someone. So we might have ohms, johnsons, smeigles, bobs, and many more. Just as common units of distance span a couple dozen orders of magnitude because the scales of distance used by different communities range from the atomic to the astronomical, so too do the scales at which people work with quantities of energy, voltage, charge, and time. So it would not be at all surprising to have ended up in common units of resistance that likewise different by many, many orders of magnitude.

Now, consider that this exact same problem was given to ten different people from ten different places that used ten different units of resistance that ranged from something on the order of a microohm to something on the order of a gigaohm (i.e., fifteen orders of magnitude) -- Let's say that 10^6 smeigles = 1 ohm and 10^9 ohms = 1 bob. What if all ten of them used the exact same line of reasoning that you did and came up with a result of

A >> R1 + R2 or R2 >> 1

and then each of them made the exact same claim that you did in order to justify that this makes sense and that the context means that the 1 has units of their preferred unit of resistance. So one of them would be saying that R2 must be much larger than one smeigle (or 1 microohm) while another would be saying that R2 must be much larger than one bob (or 1 gigaohm), while you would be insisting that R2 must be larger than one ohm.

Which of them is right? Every last one of them would have just as valid a claim as you do. Look at the problem -- there is absolutely NOTHING in that problem that puts any preference of one unit of resistance over another. NOTHING. NOT ONE THING. They would have imposed an interpretation using a smeigle or a bob purely because they wanted to, just as you imposed a unit of ohms because that's what you wanted to -- and each of you would have been equally wrong and the wrongness of their conclusions would have been as equally demonstrable as the wrongness of yours was and, very possibly, they would have continued to insist that they were somehow assuming the "right" units and everyone else was mistaken (just like you are doing).

You yourself provided a perfect case in point when you said:

on the other hand when we are talking circuits, is always Ohms unless stated otherwise.
...
There is also the case where all the resistors are given in K Ohms. Then we could interpret:
"R1>2"
to mean:
"R1>2k"
Again, look at the original problem. There are NO units stated. So your claim is that this somehow means that R1 and R2 just MUST be in ohms and that

A >> R1 + R2 and R2 >> 1

therefore mean that the gain must be much larger than the numerical sum of the two resistors when expressed in ohms and the same for R2.

But what if that same diagram had had a note saying that the resistors were to be expressed in kilohms or milliohms, or megohms. Would you then claim that in each case the interpretation of those expressions somehow magically changes by a span of nine orders of magnitude? Even though the equations were based on an analysis of the exact same circuit using only symbolic variables for each and every quantity?

Or does it possibly make more sense that the conclusion that A must simply be much greater than R1+R2 is fundamentally faulty and that it should prompt a further examination of the results in order to recognize that it isn't the quantity of (R1+R2)/A that has to get small compared to 1 (which is just a number, not a resistance -- look carefully at where that 1 came from in that equation), but that it must get small compared to R2, meaning that A must be much greater than (1 + R2/R1). And notice how it makes no difference whether we are using ohms, megohms, microohms, smeigles, or bobs -- we come to the exact same conclusion regardless of which units of resistance we want to work with.

NOTE: For those interested, there is a lot of confusion on the proper spelling of MΩ, kΩ, and μΩ. There seem to be four common ways, (e.g., mega ohm, mega-ohm, megaohm, and megohm). I found sites that claimed that each of these is the correct way in turn. However, NIST can be considered an authoratative source on topics like this (in that they are party to the Conventions that establish these definitions and take great pains to be consistent with the them).

https://www.nist.gov/pml/nist-guide-si-chapter-9-rules-and-style-conventions-spelling-unit-names

I've tried to be compliant with them, but it is easy to mess it up since, especially for resistance, we have two special cases (MΩ and kΩ) where the the final vowel of the prefix is omitted while the rest (GΩ, mΩ, μΩ and the rest) the final vowel is retained even though it often results in a double-o that is not pronounced as a double-o, but rather as the distinct prefix and unit. Although it should be noted that the NIST document does NOT state that megohm and kilohm are correct, but merely cite a reference that acknowledges that they (along with hectare) are the three commonly used exceptions to the rule that "When the name of a unit containing a prefix is spelled out, no space or hyphen is used between the prefix and unit name."

The referenced document is IEEE/ASTM SI 10-2002, IEEE/ASTM Standard for Use of the International System of Units (SI): The Modern Metric System. Unfortunately, I couldn't find a copy online and IEEE wants \$74 to download it. No thanks.

So as of right now I am of the opinion that the use of megaohm and kiloohm are acceptable (and possibly even preferred -- the IEEE document would probably clarify which) but that megohm and kilohm are at least sufficiently established in common use as to be defacto acceptable (and possibly preferred).