Points of Inflection
Kirt's Cogitations™ #216
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.
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