Building Antennas Using "Extrapolation"

by
N4UJW

**So what is "extrapolation"?
Where do I buy it? How do I use it in antenna work?**

Does it come in
different lengths, diameters? What tools are required?

I'm
lost....tell me more!

**Actually, extrapolation is a method
using math to arrive at an "unknown" or a set of "unknowns" when you
have a set of "knowns" to work with and is a logical method of arriving at
the "unknowns" you are looking for. So what does all this
mean......**

**First, we must know the definition of
extrapolation....**

Here are some simple definitions of
extrapolation:

**1. (Mathematics) Estimating a function
at a point which is larger than (or smaller than) all the points at which
the value of the function is known.**

**2.
Mathematics:**** To estimate (a value
of a variable outside a known range) from values within a known range by
assuming that the estimated value follows
logically from the known values.**

Get out the calculator and learn how "ratios" can be used to
Extrapolate.....or not!

You probably "extrapolate" many
times during your day around the house or on your job without knowing
it.

**Here is an example....you want to
make 1 cup of coffee but all you have is a 10 cup coffee maker and
you certainly don't want 9 cups just sitting there getting stale or
wasted.**

**You "know" that in order to make 10
cups of coffee, you normally would use, as an example, 5 scoops of coffee
for 10 cups of brew! But....you only want 1 cup, not
10!**

**So how do you "extrapolate" how much
coffee to take out of the can to make 1 cup of coffee using the same
strength as if you had made 10 cups?**

**Here's how using
extrapolation: (in the form of ratios)**

**You take the "knowns" and then arrive
at a final ratio of water to
coffee for that 1 cup of coffee you so desperately need to get the
body going!**

**It is "known" that 5 scoops of coffee,
plus 10 cups of water = 10 cups of coffee when brewed to your taste.
(Assuming this is the way you normally make it)**

**Using math extrapolation we get a ratio between scoops required and the final pot of
coffee:**

**10 cups / 5 scoops = 2 (a ratio
of 10 cups of water to 5 scoops of coffee) for 10 cups of
brewed coffee.**

**We only want 1 cup as the final result
so we extrapolate the ratio of coffee to
water using:**

**5 scoops / 10 cups of water =
.5 (that's decimal 5 in case it's
not clear on your browser)**

**So for 1 cup of coffee you would use 1
cup of water and .5 scoops (1/2 scoop) of
coffee!**

**Now to prove the strength is the same
using math, as if you are making 10 cups, just multiply .5 X 10 = 5 scoops of coffee!**

**The whole idea here
is that you use ratios when comparing dissimilar
things.**

Another simple example using extrapolation:

You
want to make some instant pudding from an 8oz package of mix
you bought at the store.

When you opened the mix, you accidentally
spilled half of the mix on the floor. Don't worry, the ants will be happy
to clean up the mess! Now you only have 4 oz or half of the mix left to
make the pudding.

You read the directions on the package and it says
you need 2 quarts of milk plus one 8 oz package of mix to make 4 cups of
pudding...but wait...you only have 1/2 of that 8oz package left to
use...so how much pudding will it make and how much milk will you need to
make the pudding as thick as it normally would be?

Now working with the
"knowns" that you have which came with the
instructions:

**"Knowns".... 8oz of mix plus 2 quarts
of milk = 4 cups of pudding and you only have 4oz of mix! **

So what
is the "unknown"...how much milk you need to
use in order to use all of the mix?

**Using the extrapolation technique to get a ratio to work
with:**

8oz mix / 2 cups milk = 4 (a 4 to 1 ratio)

So you need a 4 to
1 ratio of mix to milk to get the same strength pudding.

You simply use
4oz of mix plus the ratio of 4 to 1 milk to get the same
mixture.

**2 quarts / 4 = .5 or 1/2 as much milk
as you would have needed for the full 8oz of mix!**

Yum,
Yum!

**Using these two examples about is a
very simple method that many of us use without really thinking about the
fact that we are using some simple math to find an "unknown" using
"knowns". It is sort of a "logical" excercise for our brains that we
use during many days and with many things in life. **

This technique can be applied to working with ham radio
antennas in many cases....read on.

**How to Use Extrapolation
when working with antennas!**

**Using this method described
above when working with many ham radio antennas can be a fantastic
aid if you don't have "formulas" to work with
when you are attempting to use antenna plans designed for one
particular band on another band or frequency.**

**Now let's get to some examples using
antennas rather than staying in the kitchen and getting
FAT!:**

**Just pretend for a moment that you have
plans for a 10 meter "xyz" antenna and you want to build one of the same
"type" antennas for use on the 17 meter band BUT...there are no formulas mentioned in the original
plans. You are in the dark...or are you?**

The 10 meter
"xyz" antenna is designed for 28.400mhz and has a total
length of 20 feet!

28.400mhz and 20 feet are your "knowns" for the
"xyz" antenna.

**The "unknowns" you are looking for is
to use the same antenna "extrapolated" to the 17 meter band and you want
to know the length...the
"Unknown".**

**Using some simple math and your
"knowns" and "unknowns" we have:**

"Known" 10 meter design frequency =
28.400mhz

"Known" Total length = 20 feet

"Known" new design
frequency = 18.130mhz (in the 17 meter band)

"Unknown" total
length of new 18.130mhz antenna

**The math:
(Remember, we need to arrive at a ratio between
28.400mhz and 18.130mhz)**

28.400 / 18.130mhz = 1.5664644 (round that off to 1.56) << There is your ratio in red!

So the ratio in physical size would be 1.56
to 1, meaning the 18.130mhz antenna will be 1.56 times LONGER than the
28.400mhz "xyz" antenna. (Remember your going down in frequency so
the antenna will be longer and/or larger).

Now using this ratio of 1.56
to 1 and using our "knowns" and "unknowns" we get:

__20 feet__ for
28.400mhz __X 1.56__ (the ratio) = 31.20 feet for the new 17 meter
"xyz" antenna!

**Using "real life
antennas" for extrapolation**

Now lets use a real antenna and a
standard antenna formula and not that "xyz" antenna that does not
exist and also use the "extrapolated" method in this example as a
comparison to see if this "method" really works:

**Using a dipole as an example we
use the standard formula of 468 / freqmhz = total length in
feet.**

**At 28.400mhz the dipole length would be
468 / 28.4 = 16.47 feet total length.**

**BUT...you
want to use the design on 18.130mhz...what would the length
be***.....(remember, you no longer have the formula to work with in
this example* and you need to find the ratio between the 28.400mhz frequency and the
18.130mhz frequency...all you have is the frequency
and length of the original in the plans......

The math: ( we did not round off in this
example)

**28.400 / 18.130 = 1.5664644 ratio (so
the 17 meter antenna is 1.56+ longer than the 10 meter antenna length in
the plans.**

**Original 28.400mhz total length =
16.478873 feet**

**We now have a "known" of 16.478873 feet
for the dipole, and by using 1.56 as the ratio when we did the math above,
we can now us it to find the length of the 18.130mhz frequency
antenna!**

**16.478873 X 1.5664644 = 25.81 feet ( the new 17 meter antenna extrapolated length rounded
off).**

**Now lets check the math results
compared with using the standard formula. (Remember, we did not have
formulas to work with in the original "antenna plans" for the 10
meter antenna but now we do!**

**468 / 18.130 = 25.81 feet (using the formula)**

**You will notice the
"extrapolated" length is the same as is you had the formula to work
with!!!!!**

**Another
example:**

You are looking at the plans for a 20
meter vertical antenna but want to use the antenna re-designed for 2
meters on 146mhz! The plans state that the vertical length is say....16.47
feet long and was designed for 14.200mhz...the author of the 20 meter
vertical forgot to say what formula he used in
designing the antenna.

**You procede using the same techniques
as before:**

14.200 / 146.00mhz = .0972 ratio between the physical length
of the 20 meter version and the 2 meter version. Remember that the length
of the 20 meter version is much longer than the 2 meter
version.

**Using the math ratio we
get:**

16.47 X .0972 = 1.6 (this is still in feet) 1.6 feet X 12 to
get inches would give us 19.2 inches!

So everything else being equal,
the new 2 meter antenna "extrapolated" to a 2 meter length would give us
the 20 meter version "scaled" to 2 meters and should work fine.

**Now
dust off that calculator and play with this method. Practice,
practice.**

*"It is important to
note that when using this method to arrive at the "unknown" length of that
antenna, that differences in construction materials, frequency, size
of elements or wire, the surroundings, height above ground and many other
variables associated with the "nature" of various antennas will make
this method 'not perfect' by any means, but it should get you in the ball park...you now have to find your section,
row and seat for the game. *

In other words, you may still have to
experiment with the "new" extrapolated antenna to get it to perform well
while remembering that in the beginning, you had no idea how to even get
into the ball park without the "ticket" (the formula)....that's the fun
part of antenna experimenting!* *

*Now go into the kitchen and spill some pudding mix! You deserve a
break! *

What's that......no
milk?.....just make some coffee....it should be ready by the time you get
back from the grocery store...I'll take 2 cups, the other (insert
number of cups here) are for you...do the math.........!

73
Don!"