Roy Lewallen wrote:
Pete Bertini wrote:
As an academic exercise, for example using a tapered
line to match an OCD with a known FP impedance of 200 ohms
down to a 1:1 balun at the 50 ohm coax line transition point.
What is the recommended minimum length of cable needed to
perform a four to one characteristic impedance change in the
open wire for relating to the wavelength?

Pete

It's not practical to make a 50 ohm open wire line. The wires would have
to be extremely close together.

But as to "recommended" minimum length, it depends on who does the
recommending and what your criteria are. The length of line necessary to
transition from one impedance to another depends on at least three
things: the amount of transformation required, the type of taper (e.g.,
linear, exponential, Klopfenstein), and how close the impedance match
has to be (usually specified as input reflection coefficient). There's a
brief treatment in Johnson, _Antenna Engineering Handbook_.

In Brown, Sharpe, Hughes, and Post, _Lines, Waves, and Antennas_, it's
stated that the tapered section must be at least a quarter wavelength
long, and that if less than that, "it will fail in its matching function".

In Skilling, _Electric Transmission Lines_, ". . . a 2 to 1 change of
characteristic impedance many be accomplished by using a tapered section
that is 1 wavelength long. A good impedance match results at this
frequency and at all higher frequencies. Results are fairly good if the
tapered section is as little as 1/2 wavelength long, particularly if the
impedance ratio required is less than 2 to 1. It is not worth while to
use a tapered section much shorter than 1/2 wavelength, for a taper that
is 1/4 wavelength or shorter will give nearly as much reflection as will
the abrupt change of impedance at a mismatch. Those frequencies for
which the length of the taper is a multiple of a half-wavelength are
transmitted with the least reflection. This may be a guide in choosing
the length of the taper." The author goes on to state that linear and
exponential tapers give practically the same results, and summarizes
with a recommendation that a tapered section be between 1/2 and 1
wavelength long at the lowest frequency of use.

Roy Lewallen, W7EL

I've seen essentially the same ... bigger than 1/2 (or 5/8) wavelength
works well. BUT... that's for a "good match".. if you're willing to
tolerate a "not so good match" I think you can get much shorter.

There's a lot of antenna designs that use this sort of scheme to match,
say the 100 ohm impedance of a helix to 50 ohms, over a distance on the
order of 1/4 or 1/5 wavelength. Whether it's truly a tapered
transmission line or a semi-distributed lumped L and C is probably a
good question. Certainly, one could use a string of lumped L/C networks
to make a synthetic transmission line that has mechanical length much
shorter than 1/2 wavelength. So, when does a series of lumped LCs
become a transmission line?

You also see tapered transitions in microstripline all the time, and
they're often shorter than 1/2 wavelength. But maybe that's more in the
nature of a "distributed matching network" rather than a pure
transmission line.

There's a goodly amount of empiricism in both antennas and stripline
design. Especially if you're talking about an balanced pair with
separation smaller than the diameter/width of the conductor, it's
probably not a "simple" transmission line.