Kirt's Cogitations™ #216
Points of Inflection
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Cog·i·ta·tion [koj-i-tey'-shun] – noun: Concerted thought or
reflection; meditation; contemplation.
Kirt [kert] – proper noun: RF Cafe webmaster.
Points of Inflection
When reading technical articles, I very often see the authors
incorrectly refer to a certain point on a curve as being the inflection point. That was the case in an article I
read today that dealt with open-loop polar modulation in EDGE amplifiers. There exists an unambiguous definition
of an inflection point, and all engineers were taught it in school. Pardon me if this seems trivial or picayune,
but the purpose of the magazine articles is to teach, so if this factoid can eliminate the error in future
articles, then it will have accomplished its objective. Here is a brief review of what an inflection point is,
and, equally important, what an inflection point is not.
inflection point is the point at which the second derivative equals zero. Accordingly, it is the point where a
curve changes from concave up to concave down. A curved region is concave up if all the points in that region lie
above a line tangent to it (in the + y-axis direction). A curved region is concave down if all the points in that
region lie below a line tangent to it (in the - y-axis direction). The small graphic that accompanies this factoid
illustrates all of these concepts (curves not plotted to scale). Another way of stating it is that directed lines
perpendicular to the tangent lines have a downward direction for downward concavity, while directed lines
perpendicular to the tangent lines have an upward direction for upward concavity. Note that not all curves have an
inflection point, while some curves have many.
A familiar inflection point, and quite likely the first one
introduced to each of us in calculus class, is the one that occurs at the axis crossing in a cubic curve: y = x3.
That is the green curve on the graph. Note that all the points on the left half of the curve lie below the tangent
line shown (or any line tangent to the curve in the left half, and therefore the left half of the curve is concave
down. To the contrary, all the points on the right half of the curve lie above the tangent line shown (or any line
tangent to the curve in the right half), and therefore the right half of the curve is concave up. The inflection
point on the y=x3 curve is at x=0, y=0. To find that point mathematically, set the second derivative, d2/dx2
(x3) = 6x, to zero and get x=0. Finally, y=03=0. Q.E.D.
The same situation is true
for the half cycle of the tangent curve (blue). The second derivative of tan(x) is d2/dx2
tan(x) = 2*sec2(x)tan(x), which is equal to zero at x=0.
Now, here is where a lot of authors go
wrong. Many of the curves we plot look like the red plot on the graph. AM/PM conversion, power versus frequency in
a bandpass filter, and component Q-factor plots are a few examples. This plot happens to be a parabola whereby y=x2.
By inspection, it can be seen that the entire curve lies below any tangent line drawn. It does not matter whether
the tangent line has a positive slope (on the left half) or a negative slope (on the right half), all points on
the curve are below it. Consequently, this curve, and those like it, have no inflection point. That this is so can
also be shown mathematically since the function y=x2 has no second derivative that can equal zero - it
always equals 2. Therefore there can exist no inflection point. Q.E.D. again.
So there you have it. Let us
resolve to never again refer to a local maximum or minimum (where the first derivative equals zero) on a curve as
an inflection point.