How much power can one draw from a short dipole antenna?

Let us consider a short dipole resonant receiving antenna (i.e. inductance-loaded to make it resonant). It is exposed to an EM field of given intensity. How much power can one draw from it? Is the load resistance in such a case given by
R = SQRT(L/C) ?
I have seen the Friis equation, but I find it quite unbelievable that the received power does not depend upon the size of the receiving antenna.

Hello,

When you can make a lossless match, the effective area (Aeff) of a straight radiator is < 0.25lambda:

Aeff = 1.5*(lambda)^2/(4pi).

The gain of such a short radiator is 1.5 (perpendicular to the radiator).

output power = Aeff*(power flux density).

Power flux density is in W/m2

Power flux density = (Ep)^2/377

Ep = amplitude of E-field (assuming plane free propagating wave).

The problem with short dipoles is their low radiation resistance and high capacitive component. So you need a very good inductance to avoid that all your power is lost in the inductor. Because of large (capacitive component)/Rrad ratio, bandwidth is very small.

Using thick conductors (or even strips), reduces the capacitive component (as capacitance increases). This will result in reduction of the required inductance.

In real world, antennas with size << 0.25 lambda have poor efficiency.

Larger antennas can have gain due to directivity. The more gain you want, the larger (in size w.r.t. wavelength) the antenna must be.

It is NOT TRUE that effective area is independent of the size of the antenna - for a loop antenna I can easily prove it using Maxwell's equations. But I do not know what to do in the case of a dipole antenna.
What you wrote is GENERALLY ACCEPTED BUT FALSE.

Before shouting, please read carefully.

We were discussing small dipoles (<<0.25 lambda), no loss and perfectly matched (as your question relates to maximum power extracted).

For the inductive loop, the result is the same, when no loss, perfectly matched, and size < 0.25 lambda, the effective area is only dependent on wavelength.

For short dipole, Prad = 0.5I^2Rr where Rr=20pi^2(l/lambda)^2

For a loop antenna you can calculate the voltage from the Faraday's law, the current can be computed e.g. from the fact that the wave impedance of a resonant circuit is SQRT(L/C) or from the condition that the maximum flux of magnetic induction due to current induced in a single period is equal to the maximum flux due to the incoming wave.
There is an obvious dependence upon the size of the loop.
So do not repeat any more what you wrote. Think it over - what you wrote is generally accepted but false.

Hello,

I hope some other people with experience in designing small loop and dipole antennas will comment also.

If you are willing to read some other views, do a search on fundamental limits antennas wheeler.

Your reasoning based on sqrt(L/C) maybe valid for the traveling wave case, but in a small dipole and loop antenna, standing wave behavior in combination with the radiation loss dictates the input impedance of a small loop or dipole. Both can be seen a a short piece of transmission line.

When you really do the math (with some approximations), you will come to the result as shown by testing test.

I did the calculations for the case of interest to me (150 MHz, 2 cm loop) and the result is that when I draw 2W of power from the loop, the radiation loss is 4 microwatts. So it is negligible.
It is obvious both intuitively and after doing calculations that the size of an antenna is a fundamental factor influencing how much power we can draw from it.

Yes, the size of the antenna plays a keyrole as Prad depends upon the length of the dipole antenna.

I've always been sure it does. Do you have a quantitative expression for the received power given intensity of the incoming wave and parameters of the dipole antenna?

Well out of the transmitted power, 50% is dissipated and 50% is usable power delivered to load. However, polarization mismatch may count. You need to consider Friis equation for this.

Hello,

For the Friis formula, you need know the antenna parameters (gain and wavelength, or effective aperture (energy based) ). This he don't know for the electrically small dipole case, so he first has to solve that issue.

Regarding the loop case:

As htg stated, you can calculate the EMF based on the magnetic field (for electrically small loops). But with the EMF only, you don't know the power that can be extracted.

Now we place a loop in a plane wave field, with best orientation for maximum output voltage. This loop has zero resistive losses (see it as a cryogenic case). Based on the geometry you can calculate the inductance and as the frequency is known, you can calculate the reactance.

Now we add a capacitor in series to cancel the inductive component. What will be the current when I short circuit the loop, or what resistor do I have to connect to this LC circuit to derive maxim power from it (loop has zero resistance)?

This current is not infinite as the current through the loop generates a secondary far field. The far field can be calculated based on the current and the magnetic moment (product of current and area enclosed by the electrically small loop).

The energy radiated by this field (see testing test's posting) causes a so-called radiation resistance. Even a lossless coil in free space experiences a loss (due to radiation). Both for the loop and the short dipole, radiation resistance can be calculated from simple formulas, so you don't have to do the full math yourself. When you know Rs, you can calculate the theoretically maximum available power (for zero Ohms loop resistance).

By using reciprocity theorem for antennas, it is sometimes easier to see things from the transmitting case (were gain for RX is same as gain for TX).

Both a small loop and small dipole have maximum directivity of 1.5 (free space case) . From the directivity and wavelength you can calculate the effective area for the receive case (lossless case). In case of loss (what is normally the case with small loops and dipoles), you have less gain, hence less effective area. Finding the loss can be done by calculation, Q-factor measurement, etc.

The situation becomes difficult when the field isn't plane or when conducting objects are close by. The reradiated field from the loop induces current in these objects and these currents do generate a field as well. This results in a new overall radiation pattern, hence other directivity and gain. For some simple geometry you can do the calculation by hand, but nowadays a simulator can ease the work.

A special case is when the exciter (that generates the incident field) is in the near field of the receiving loop. Then the impedance of the exciters antenna will change significantly. Compare this with 2 inductivity (or electrically) coupled LC circuits (as used in wireless electric power transfer, RFID and RF Electronic Article Surveillance).

Friis equation is generally accepted, but false. The received power depends upon the size of the antenna.

Added after 4 minutes:

It is also generally accepted, but false, that the effective area does not depend upon the size of the antenna.

Hello htg,

What are your final goals? Your replies will result in silence from people that really could help you. This may also be valid for other requests from you.

1. The Friis formula has antenna size in it, as there is a relation between size and gain.

2. Only in the lossless case, under certain conditions, effective area (power/energy based) in free space is independent on size (See Wheeler's work). Under most practical circumstances, effective area is related to size. Antennas can have a smaller or larger effective area then their physical area.

A half wave matched dipole has a small physical area while the effective area is about 0.125(lambda)^2. Arrays with good side lobe suppression have smaller Ae (with respect to size).

I explicitely agree to WimRFP's comment on attitude when asking for help or contributing to a discussion at edaboard, apart from
the question who's right or wrong.

Ballanis Antenna Theory has a detailed paragraph on effective size and area of antennas. He claims, that an aperture
antenna's (e.g. horn, waveguide) effective area can't be larger than it's physical area, but he also derives a constant
effective area for small lossless dipoles.

I have read that a small dipole has constant gain independently of its size.
So you say I should believe Friis formula? What is the lower limit on the distance between antennas?

Hello,

For a small dipole (or small loop) the directivity is independently of size (D = 1.5). When you are able to make a lossless match to such a dipole (so radiation efficiency 100%), the Directivity equals the Gain.

In practice when the size << 0.5lambda, your matching network will introduce significant loss, so your actual gain will be less.

Regarding the Friis formula, this one is only valid under free space propagation and both antennas must be in their far field zone.

As a very conservative value, you can use

rff = 2(size)^2/lambda or 0.5lambda, [m]

Take the largest value and do this for each antenna, the zones may not overlap.

When you convert the antenna gain (not Directivity) to the effective area, you can directly calculate the output power based on the plain wave field intensity (W/m^2).

Please not that this is only valid for plain wave field, and no other objects close to the dipole antenna. The reason for this is that objects close to the dipole modify the radiation pattern, hence the gain and radiation resistance ( Re(Zin) ).

When the source's antenne is in the transition or reactive zone of the receiving antenna, strange things will occur.

If for a small loop the effective area depends only upon wavelength, then it contradicts Maxwell's equations. I've always thought they were true.

Hello,

The statement regarding effective area and loop size for small losless loopantennas does not violate maxwell's equations. You need to apply them in full, not just the faraday induction law.

In addition to the reference of FvM, you may also check: "Electromagnetic Waves & Antennas", S. J. Orfanidis, chapter 14. this discusses the gain and effective area of small dipoles also.

"Electrically Small, Superdirective, and Superconducting Antennas", R.C. Hansen. This book also discusses bandwidth and matching issues for electrically small antennas from a more practical point of view.

I checked Orfanidis (it is actually chapter 15 that discusses effective area), but he does not derive the essential statement.
I suspect that it is not true. You do not seem to have any explanation either.