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!"