# Short dipole radiation resistance - do you have experimental results?

I have seen two formulas for a short dipole radiation resistance - according to one it is about 200*(l^2/λ^2) Ω, according to the second about 800*(l^2/λ^2) Ω.
Which of these is correct, if any? (Let us consider a resonant short dipole, connected to proper inductance).
In particular, I would like to know Rrad for l = 0.1*λ.

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berkeman
Mentor
I have seen two formulas for a short dipole radiation resistance - according to one it is about 200*(l^2/λ^2) Ω, according to the second about 800*(l^2/λ^2) Ω.
Which of these is correct, if any? (Let us consider a resonant short dipole, connected to proper inductance).
In particular, I would like to know Rrad for l = 0.1*λ.
My Stutzman & Thiele antenna book derives the 2nd one of the equations you list. Where have you seen the first one derived?

berkeman
Mentor
Actually, the difference factor of 4 is suspicious. Perhaps the length of the dipole is defined differently in the two deriviations, with one using the length of an element, and the other using the length of the overall dipole...

uart
Actually, the difference factor of 4 is suspicious. Perhaps the length of the dipole is defined differently in the two deriviations, with one using the length of an element, and the other using the length of the overall dipole...
Yep, that would be it berkeman. In the "usual" formula $R = \frac{\pi}{6} Z_o (L/\lambda)^2$, the length "L" is the overall length of the dipole. So if we wanted to use the length of an individual element we would have to multiply by four.

Such a derivation would be incorrect (Lorentz gauge is not physically justified, it is only a mathematically convenient assumption of cancellation of certain quantities).

So, does anybody have experimental results regarding Rrad of a short dipole?

The difference is in what assumption is made about the current distribution on the antenna conductors.

Radiation resistance of the Hertzian dipole is 790(l/λ)^2. The Hertzian dipole assumes constant current across antenna axis. In other words current at the tips of the dipole are equal to the current everywhere else. This is obviously not possible unless it is a "top hat" type.

Another popular assumption, which is a little more realistic, is that current vary linearly with distance away from center, with zero current at tips. Radiation resistance of this turns out to be exactly 1/4 of Hertzian.

Of course this is not the actual current distribution, but an approximation that makes derivation of closed form solution feasible.

There are other assumptions that go into these formulas:
No ground plane.
Infinitely thin wire.

If you really need the radiation resistance with accuracy it is better to model it.

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How can the Larmor formula be derived?

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The Larmor formula involves radiation from a point charge. The dipole formulas previously discussed have to do with radiation from a string of infinitesimal current sources.

htg, what exactly is your goal. Are you trying to design an actual antenna?

berkeman
Mentor
The difference is in what assumption is made about the current distribution on the antenna conductors.

Another popular assumption, which is a little more realistic, is that current vary linearly with distance away from center, with zero current at tips. Radiation resistance of this turns out to be exactly 1/4 of Hertzian.
Hey, thanks for that, EMI Guy! Interesting.

It looks like the Hertzian dipole model is proper for my problem. I am trying to calculate how much power will a short dipole reradiate if an EM wave of certain intensity reaches it.

Is there anybody who has experimental results regarding Rrad of a short dipole? In particular I am interested in the case length = 0.1*lambda.

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vk6kro
You can get the input impedance of a dipole from a simulator.

I use EZNEC.

Here are some results...

Frequency 10 MHz. Wavelength 30 meters. Dipole length 3 meters. Dipole wire diameter 1 cm.

So, near enough to 0.1 wavelength long.

Input impedance at the center = 2 - J 1773
So, that is 2 ohms resistive in series with about 9 pF.

The 2 ohms would be mainly the radiation resistance as the wire is actually a piece of 1 cm (0.4 inch) pipe. Increasing the diameter of the wire to 5 cm (2 inches) reduced the input impedance to 1.9 - J1140.
That is 1.9 ohms in series with about 14 pF.

If you wanted to use this as an antenna, you would need to put an inductor of 18 μH in series with the feedpoint to cancel out the capacitance, and also drive an impedance of about 2 ohms.
Not easy to do without a lot of losses.

You can get the input impedance of a dipole from a simulator.

I use EZNEC.
Here are some results...
...near enough to 0.1 wavelength long.

Input impedance at the center = 2 - J 1773
So, that is 2 ohms resistive in series with about 9 pF.
Thanks for the results from your simulator, but:
How does the simulator calculate it? I am asking, because theoretical derivation of the widely known formulas for Rrad is based on Lorentz gauge, which has no physical justification. So it would be a little strange if these formulas agreed with experiments.

vk6kro
Short dipole radiation resistance - do you have experimental results?
You did ask for practical results.

How does the simulator calculate it?
I can't help you with that, but you could contact the guru of antenna theory, Roy Lewallan W7EL, who is the author of that simulator. He gives this email address in the simulator:
I can send it to your private mail if you like

But don't go in claiming that standard formulas are wrong. You won't get a reply if you do that.

The simulator does not give radiation resistance directly, but it can be estimated from the input impedance of a low loss antenna.

If you are interested, you could download a copy of the free demo version of this program from
http://www.eznec.com/demoinfo.htm
This program is nothing much to look at but it gives very accurate results and is easy to use.

I do know that it works on a system of segments where the antenna is split up into short segments and the center of each is regarded as a point source.

...How does the simulator calculate it?
Go to Wikipedia page for "Numerical Electromagnetics Code". You will find basic info there.

I am asking, because theoretical derivation of the widely known formulas for Rrad is based on Lorentz gauge, which has no physical justification. So it would be a little strange if these formulas agreed with experiments.
Are you trying to design an antenna here or dispute the mathematics used to derive textbook formulas? If textbook formulas are incorrect it is due to expedient simplifying assumptions, not flawed underlying physics (as far as I know).

The majority NEC's computational effort is in determining the currents along the conductors (magnitude and phase). Given this current distribution, computing the far fields is a relatively simple back-end part of the process. The textbook formulas assume a very simplified current distribution.

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berkeman
Mentor
@htg -- I can probably measure this for you after hours at work. What frequency are you interested in, and what are the physical dimensions of your antenna? What is the application?

@berkeman - what I acually want to know is the power which is reradiated by a resonant, short-circuited short dipole when a wave of given intensity P/S reaches the short dipole - say at 150 MHz, length=0.2*λ = 0.4m. So the radiation resistance that I need will perhaps correspond to uniformly distributed current in the short dipole. Perhaps you can achieve similar result by using a short dipole with disks or spheres at the outer ends, so that the extra capacitance will make the current more uniformly distributed.
If you can measure Rrad of such short dipole fed at the center, say with disks at the outer ends, it will give me a pretty good idea of what to expect.
According to what I know, the capacitance of a single disk is given by C = 8*ε0*r, where r is the radius of the disk.

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berkeman
Mentor
I don't think an electrically short reflector will be very effective. Is it supposed to be an element in a multi-element (but electrically short) array, like a Yagi?

I don't think an electrically short reflector will be very effective. Is it supposed to be an element in a multi-element (but electrically short) array, like a Yagi?
I consider using such an array. But for the beginning I want to know what to expect if I use a single resonant, short-circuited dipole.