what's the center of a log scale?

Discussion in 'Math' started by #12, Jul 9, 2012.

Nov 30, 2010
18,093
9,682
Referring to a chip called LM2941CT and a drawing that comes on the datasheet.

I'm wondering if a physical construct that responds in a logrythmic way sees the center of a 1 to 10 log graph as 5.
If you lay a tape measure from 1 to 10 on a log graph paper and divide that distance in 2, you land at about pi. That's the physical center of the graph paper. Is it relevant to the operation of a semiconductor or have I just been standing too close to what my nephew was smoking?

Pardon the geekitude but I am not educated enough to know the answer.

File size:
4.1 KB
Views:
58
2. Papabravo Expert

Feb 24, 2006
11,749
2,483
It is actually the square root of 10, which is close to pi but just a bit bigger @ 3.16227766...

The way get this is to observe that log(sqrt(10)) = 0.5 or
10 ^0.5 = sqrt(10) = 3.16227766...

Easy when you read, breath, and consume this kind of stuff in an undergraduate calculus course.

#12 likes this.

Nov 30, 2010
18,093
9,682
OK. You've located the center. Next part: does it matter to the semiconductors?
Specifically that graph that I presented.
Is the "stable region" of that chip "centered" at .5 or .316?
Page 6 on the datasheet.

File size:
314.8 KB
Views:
45
4. Papabravo Expert

Feb 24, 2006
11,749
2,483
The left hand side is centered at 0.316...
Hard to say about the wider part; in particular it doesn't look symmetrical, but I could be wrong.

Nov 30, 2010
18,093
9,682
That's OK. You answered my primary question.
What's the math for the center of any 2 points on a log graph (assuming I'd rather use a calculator instead of a vernier caliper)?

6. Papabravo Expert

Feb 24, 2006
11,749
2,483
If x an y are any two points on a log scale then the mid point m is:
Code ( (Unknown Language)):
1.
2. log(m) = (log(x) + log(y)) / 2
3. 2 * log(m) = log(x) + log(y)
4. 2 * log(m) = log(x*y)
5. log(m^2) = log(x*y)
6. m^2 = x*y
7. m = √(x*y)
8.
The midpoint on a log scale has another name and that is the geometric mean
Her is the wiki:

http://en.wikipedia.org/wiki/Geometric_mean

#12 likes this.

Nov 30, 2010
18,093
9,682
You have finished everything I wanted. Thank you.

8. WBahn Moderator

Mar 31, 2012
23,201
6,993
Why would a semiconductor either know or care about how something is drawn on a graph?

What I think you are really getting at is the notion of what does the "center" of a region mean? That's one of those questions that seems so simple on the surface but can get really complicated really quickly as soon as you are talking about anything that is anything but purely linear in its characteristics.

In this case, if you are asking what resistance would park you at the 'center' of the stable region, I would have to ask how you define center of the stable region? The definition would probably involve something like having equal phase margins on both sides, or some such. I would not be at all surpised to find that different, but equally reasonable, definitions of the center of the stable region produced different answers.

Nov 30, 2010
18,093
9,682
"Does it matter to the semiconductors" or, "Why would a semiconductor care about which graph paper you used" is an important half of a 2 part question. Nobody actually answered that and I don't know the answer.

The answer, "geometric mean" tells me that accumulated errors in the components used would have to be 316% (in this case) in either direction from the geometric mean before wandering out of the acceptable limits. From this point of view, it makes sense to me. More sense than 100% allowable tolerance error from 5 to 10 and 500% allowable tolerance error from 1 to 5.

Nobody in their right mind would expect 100% in accumulated component tolerance error but this math is just a tool. Some other circuit might find this approch useful.

I thought the datasheet might define the center of the stable region. Apparently I was wrong, but I don't know more than the datasheet does. Tossing this question into the public forum allows many people to consider it, and some of them might know the answer. That's why I posted the question.

10. WBahn Moderator

Mar 31, 2012
23,201
6,993
I suspect that, for most considerations on something like this, geometric mean will be the most useful metric.

What you would really like to have is a contour map within the region of stability where they give a feel of how stable things are at various points in that region. But that would be more data for them to take and, worse, it brings up that whole issue of what is the appropriate metric of stability to use. It's tempting to say, "Well, they should use whatever metric they used to determine the edges of the region." But what we don't know is whether each point on the edge was determined by the same issue. There may be several different factors at play and some are the critical parameter at different regions. In fact, I wouldn't be a bit surprised if this isn't exactly what is happening in the transition from the narrow region at the left to the broader region to the right.