Dave wrote,

Where I = Io*cos[wt + b]. Where wt = frequency, b = phase shift along
antenna element.


wt is radians per second times seconds which is just radians, an angle. b is
time related
also. Are you thinking of perhaps kl?



By definition radiation resistance is that determined by integrating the
total radiated power over the surface of the sphere containing that
power.

Are you thinking of Rr=2Prad/|Io|^2?

[1] For a half wave dipole that converges to the value at a
current maximum. So, radiation resistance for a 1/2 wave infinitely thin
dipole is 73 ohms at the current maximum.[1]

Balanis gives (eta/4pi)Cin(2pi)=73.
Cin is .5772+ln(2pi)-Ci(2pi) which is approximately equal to 2.435.
Ci is a tabulated function.


So, the Radiation Resistance at the ends of the antenna is infinite.
[Cosine 90 degrees = 0] That means the efficiency at the ends of a 1/2
wavelength dipole is 100%. Isn't that a surprise?? [It's the same for a
dipole or a Yagi!!!!]

The Radiation Resistance at the 45 degree point from the current maximum
of a thin 1/2 wavelength dipole is 73 ohms/(cos^2(45 degrees)) = 146 Ohms.

The conclusion is that the efficiency of an antenna element varies along
it's length and can vary between maximum of infinity at the ends and
have a minimum value, that depends on the length of the antenna, at a
current maximum.

The total antenna efficiency is measured in the radiated pattern by
integrating the power density per square steradian [or square degrees]
over the full surface of a sphere divided by the power into the antenna.

So, a Yagi with 8.14 dBi (6 dBd + 2.14 dBi) gain has concentrated it's
radiated power into 15.38% of the three dimensional space defined by the
surface of a sphere. [See Note 1.] Now if the Yagi is 95 % efficient and
a dipole is 95% efficient the 6 dBd value remains constant.


Very entertaining.
73,
Tom Donaly, KA6RUH