Proof of the relation between antenna aperture and gain

[alan123hk]

Where can I find strict mathematical proof of the relation between antenna aperture and gain which is applicable to any type of antenna ?

Aeff = Gain * (lambda^2) / (4*Pi)

Aeff - Antenna Effective Aperture
Gain - Antenna Gain
lamdda - wavelength
Pi - 3.14159

Many textbooks just show the derivation of the equation based on a simple antenna without proof that it is generally applicable to all kinds of antennas.

 

[tech99]

Where can I find strict mathematical proof of the relation between antenna aperture and gain which is applicable to any type of antenna ?

Aeff = Gain * (lambda^2) / (4*Pi)

Aeff - Antenna Effective Aperture
Gain - Antenna Gain
lamdda - wavelength
Pi - 3.14159

Many textbooks just show the derivation of the equation based on a simple antenna without proof that it is generally applicable to all kinds of antennas.

I suggest "Antennas" by John D Kraus.

 

[tech99]

A simple view is that the Effective Aperture simply tells us how much power the antenna will abstract from a wavefront having a given Power Flux Density. Such a definition does not depend on the size, shape, losses or design of the antenna. However, Effective Aperture might be similar to physical area for large antennas. For a typical dish it is about half the physical area. But for very small antennas, if we could obtain zero Ohmic losses, then the Effective Aperture would not be smaller than that of an isotropic antenna.
Even for a short dipole, the Effective Aperture is only about 10% less than for a half wave dipole.
If you refer to Kraus, you will also see that Effective Aperture comes in a few flavours. For instance, the aperture that is involved when the antenna is used as a radar reflector.

 

[alan123hk]

A simple view is that the Effective Aperture simply tells us how much power the antenna will abstract from a wavefront having a given Power Flux Density. Such a definition does not depend on the size, shape, losses or design of the antenna. However, Effective Aperture might be similar to physical area for large antennas. For a typical dish it is about half the physical area. But for very small antennas, if we could obtain zero Ohmic losses, then the Effective Aperture would not be smaller than that of an isotropic antenna.
Even for a short dipole, the Effective Aperture is only about 10% less than for a half wave dipole.
If you refer to Kraus, you will also see that Effective Aperture comes in a few flavours. For instance, the aperture that is involved when the antenna is used as a radar reflector.

Thanks for your detailed and helpful explanation.
But the core issue is that I don't understand why the Gain and Aperture are mystically interrelated by that simple and beautiful equation and it is valid for all kinds of antennas regardless of different shapes and dimensions of them. I do believe this equation have been proved to be true through strict mathematics by someone who is very talented.

 

[tech99]

Thanks for your detailed and helpful explanation.
But the core issue is that I don't understand why the Gain and Aperture are mystically interrelated by that simple and beautiful equation and it is valid for all kinds of antennas regardless of different shapes and dimensions of them. I do believe this equation have been proved to be true through strict mathematics by someone who is very talented.

The reason for the connection between Gain and Aperture is as follows.
When you say Gain, it should say Gain relative to an isotropic antenna. An isotropic antenna has a gain of 1 and an aperture area which scales up and down with wavelength^2. It is actually equal to (lambda^2)/4 pi. All the formula is saying is "how may isotropic antennas are required to give this gain?". It then finds the total aperture area by a simple multiplication.

 

[jasonRF]

The relation between the receive effective area and the transmit directive gain
## A_{eff}(\theta,\phi)=\frac{\lambda^2}{4\pi} G_d(\theta,\phi) ##
is a consequence of the reciprocity theorem of electromagnetic (or circuit) theory. Note that this is assuming the polarization of the incoming wave is matched to the polarization of the antenna, and that the antenna is connected to an impedance matched network. I just rummaged through some of the EM/antenna books on my shelf to find which ones had this.

"Field and wave electromagnetics" (2nd edition) by Cheng has a very nice derivation that boils it down to the reciprocity theorem for impedances (which he doesn't prove).

"fields and waves in communication electronics" by Ramo, Whinnery and Van Duzer (2nd edition) has a reasonable derivation but leaves some of the details to the reader so is probably not what you are looking for. It does prove the reciprocity theorem before using it.

"Antennas and Radiowave Propagation" by Collin proves reciprocity and includes impedance matching losses; it does, however, assume the antenna is fed by a coaxial line.

"Antenna theory and design" by Elliott has a nice general derivation in chapter. It may require a little more sophistication from the reader.

"Antenna Theory" (1st edition) by Balanis has a nice simple derivation that applies to only the peak gain and peak effective area, and then states without proof the result of including mismatches. Later in the chapter he then proves the reciprocity theorem and shows transmit and receive patterns for an antenna are the same.

EDIT: adding "antennas for radar and communications" by Mott, which is very good.

I'm sure a gazillion other references have this as well.

 

[mike mugumba]

Where can I find strict mathematical proof of the relation between antenna aperture and gain which is applicable to any type of antenna ?

Aeff = Gain * (lambda^2) / (4*Pi)

Aeff - Antenna Effective Aperture
Gain - Antenna Gain
lamdda - wavelength
Pi - 3.14159

Many textbooks just show the derivation of the equation based on a simple antenna without proof that it is generally applicable to all kinds of antennas.

What will be the units for gain tgere?

 

[berkeman]

What will be the units for gain tgere?

What do *you* think they will be?

 

[alan123hk]

It is not difficult to prove that the effective area of an isotropic antenna is equal to the square of the wavelength divided by 4 Pi.

 

[alan123hk]

Interestingly, although an antenna with a certain frequency and gain has a theoretical aperture, there appears to be no limit to the aperture efficiency of the antenna. In other words, there seems to be no theoretical limit to shrinking the practical antenna dimensions as long as the antennas are perfectly matched.

https://www.cambridge.org/core/book...all-antennas/10E5AD2C63ECA5073B3F9E256CEC0D5E

 

[marcusl]

The article goes on to say that electrically small antennas cannot be perfectly matched, so
“it is rather natural to say that there is a fundamental limitation applying to the size reduction of antenna dimensions.”

 

[alan123hk]

I also noticed that it was mentioned at the end of the article that due to various other constraints, it's natural to say that there are some fundamental limitations that prevent the actual size of the antenna from being infinitely small.

But I personally have different ideas. This does not appear to be a substantial theoretical limitation based on physics. It seems to be just a technical limitation for the time being. If there is a technological breakthrough in the future, humans will invent room-temperature superconductors whose resistance is very close to zero . We can then make some completely lossless inductor, capacitor and antenna impedance matches. For another example, if we invent some lossless electrolytes in the future, we can make some capacitors with small size and large capcitance, will the so-called minimum antenna size limit continue to shrink?

 

[alan123hk]

What I mean is that in the future, in addition to using zero-resistance superconductors, we may also invent some dielectric materials that are completely lossless and have a very high dielectric constant, and then we can make some small capacitors with extremely high capacitance values. All these technological breakthroughs will make it possible to continuously improve the antenna matching problem. So is it really impossible for us to infinitely reduce the size of the antenna? I personally feel that scientists and engineers currently do not seem to have a consistent conclusion on this issue.

 

[jasonRF]

There are some fundamental limits on the bandwidth of small antennas that were proved many decades ago. The wikipedia page gives the references
https://en.m.wikipedia.org/wiki/Chu–Harrington_limit
Basically, small antennas have a high Q so are narrowband.

Jason

 

[alan123hk]

Thank you for sharing this article. I'm not aware that this fundamental theoretical limit for small antennas has been discovered. In other words, even for an antenna with no loss at all, as the size of the antenna continues to shrink, the upper limit of its bandwidth continues to narrow, and an accurate upper limit can be given.
This is important because it essentially limits the effective available message traffic.

However, Wikipedia mentions some techniques that might alleviate this limitation.
"However, any antenna can be made to show a larger bandwidth than suggested by the Chu limit if there is additional resistance present to reduce the Q, and this has led to claims for antennas that have breached the limit, but none has so far been substantiated."

https://en.wikipedia.org/wiki/Chu–Harrington_limit