One of the key concepts in the antenna analysis and design is the isotropic radiator, which is an antenna having perfectly spherical radiation pattern. The gain of the isotropic radiator is equal to 1 (0 dBi) measured in any direction. In the majority (if not all) of the antenna books I know, one can read that no real antenna can radiate equally in all directions and the isotropic radiator is therefore hypothetical. See below two examples how the isotropic radiator is described in quite recent antenna books. 

The ARRL Antenna Book 24th Ed. (published in 2019):

“Before we can fully describe practical antennas, we must first introduce a completely theoretical antenna, the isotropic radiator.”

Rothammel’s Antenna Book, Alois Kirschke DJ0TR (published in 2019):

“It is a hypothetical antenna that is used as the basis of comparison to calculate the gain of real antennas.”

There was a scientific paper published in the 1950’s, containing a proof that you can not build a coherent isotropic radiator. But the definition of the isotropic radiator does not require it to be a coherent source. It is not relevant for establishing a reference for comparing antenna gains. Antennas having spherical radiation patterns, in which different parts of the sphere have different polarization, do exist, and they are not even complex. And this has been known for many years now! For example, a scientific paper by H. Matzner, M. Milgrom and S. Shtrikman describing an isotropic radiator was submitted for publication in 1993 and published in 1994 [1]. The article described a very simple U-shaped radiator – see below.

The radiator is in fact a quarter wave transmission line fed at one end and left open at the other end. The smaller the distance between the wires, the closer to a perfect sphere the radiation pattern is. If we design such an antenna for the frequency 14.175 MHz and set the distance between the wires to 2.5 cm, the length of both arms will be 5.275 m. Figure below shows the radiation patterns of the U-shaped antenna. The total gain is equal to 0.00 dBi +/- 0.01 dBi in any direction.

As you may remember, the quarter-wave transmission line acts like an impedance transformer and an open-circuit at its end is transformed to a short-circuit at its input (if no-loss conductors are assumed). Such an antenna requires an infinite current to achieve perfectly isotropic radiation, so, in fact, it is not very practical.

However, H. Matzner, K. McDonald and J. Henry in another scientific paper [2] published in 2008 and updated in 2013 analyzed two other isotropic radiators. 

The first one was an antenna in the form of a spherical shell of radius R = λ/4 on which a set of currents could be found that produced the same far-zone fields as the U-shaped antenna had. However, this antenna seems to be not very practical either. 

Their another proposal was an array of turnstile antennas. A single turnstile antenna is shown below.

Now, if we stack vertically two turnstile antennas, setting a distance of λ/4 between them, we will achieve almost spherical radiation pattern. The authors of [2] achieved the peak of the radiation pattern only 0.35 dB greater than the minimum. 

I created a model of such an antenna and simulated it with the NEC-5 software. I got the antenna gain ranging from -0.8 through 0.28 dBi (Figure below). Of course, the results can vary a little bit depending on the wire diameter and segmentation used in the model. The turnstile array I modeled had input impedance equal to 117 ohms when operated in resonance, and huge relative SWR=2 bandwidth equal to 11.9%, although the spherical pattern can be achieved only for the frequencies rather close to the antenna resonance frequency. As you can see, such antenna is quite practical and can be driven by a regular transmitter if only suitable matching network is used.

I also found an isotropic radiator in the exemplary antenna models distributed with the 4nec2 simulator. I mean the model called isotrop.nec which is located in the folder 4nec2\models\HFsimple. The model consists of two quad loops placed in perpendicular planes and fed with 90 degree phase shift. The side length of every quad is close to λ/4 (0.263 λ). 

The original model had the sources located in the bottom and the top wires (4 sources in total) but I found it redundant and reduced the number of sources to two. Figure below shows the antenna view and its radiation pattern. The gain of such antenna is within -0.3 .. 0.35 dBi range. Its feed point impedance is equal to 125 ohms, when in resonance and its SWR=2 relative bandwidth equals 6%. 

And finally, I would like to present an isotropic radiator I have uncovered myself purely by chance. The antenna is completely flat, as you can see below and its geometry is based on a square with 0.254 λ long sides. So, it is smaller than the previous two. Its radiation pattern is presented in the next figure below. The gain varies from -0.3 through 0.31 dBi. The antenna has a single feed point and its input impedance is 39 ohms. It is narrowband because its SWR=2 relative bandwidth is equal to 0.8 %.

Conclusion

We have enough evidence to ask the editors of the antenna books to correct their isotropic radiator definitions. They should remove the terms like "hypothetical" or "completely theoretical" from the isotropic radiator definitions.

References

[1] H. Matzner, M. Milgrom & S. Shtrikman (1994), Magnetoelectric symmetry and electromagnetic radiation, Ferroelectrics, 161:1, 213-219,  https://doi.org/10.1080/00150199408213369

[2] Matzner, H., Mcdonald, K., & Henry, J. (2008-2013), Isotropic Radiators, http://kirkmcd.princeton.edu/examples/isorad.pdf