Perhaps part of the problem is that the actual design

process is NOT about calculating the spacing and element

lengths. Rather, one CHOOSES a set of spacings and

element lengths/diameters, then models them to see whether

they give the desired pattern, gain, f/b ratio, impedance,

etc. Then you tweak one or two values and try again.

Eventually you get a feeling for which adjustments make

the biggest difference in the design, and (hopefully)

find an antenna that meets your requirements.

If you want to understand the modelling process, I'd

suggest starting with W4RNL's website - he has a whole

section on modeling, including pointers to many sources

of programs (and source code in several languages).

(

www.cebik.com/radio.html)

Another good resource is W2PV's book, "Yagi Antenna

Design". Dr. Lawson was one of the first to apply

computers to yagi modelling, and his book gives a lot

of background of how he did it. (His models don't give

exactly the same results as the current NEC-based models,

but it should give you a good idea what is involved.)

This also includes methods for calculating tapered

elements, etc.

Basically, he divided each element up into a number of

segments. He assumed a 1 amp current in the feedpoint

segment, and, using the mutual coupling between every

pair of segments (which depends on spacing, distance,

and orientation) he solved a big hairy matrix to find

the current flowing in each segment. Given that, he

could then calculate the total field at any direction

from the antenna (far field) as the vector sum of the

currents in all the segments, with the relative distances

to each (and hence phase shifts) calculated relative

to the desired point in space.

On a more simple note, I have a spreadsheet that will

scale antennas for different frequencies and element

diameters, which I wrote based on W2PV's equations.

Basically, to scale an antenna the element spacings are

simply scaled by the frequency ratio. For the element

lengths, I use his formulas to calculate the actual

reactance of each element at the original frequency,

then work backwards and find what length of the new

diameter element would give the same reactance at the

new frequency. Rather than designing my own antennas

(since I haven't found a good antenna modelling program

yet that runs on my Macintosh) I generally find a W4RNL

design or one from another source that appears to meet

my needs and scale it for the desired frequency/element

diameter. This spreadsheet isn't perfect: it doesn't

necessarily take into account capacitive coupling between

elements, for example, or metal boom effects, but I've

had pretty good luck with it so far.

The W9CF applet that I pointed to earlier includes a

brief discussion of the computational algorithm it uses.

Beyond this, you can search for information on the

Method of Moments calculations - that is the computation

core at the root of most modern modelling programs.