DJ3EI's blog

Summary

Isotropic radiators can or can not exist, depending on your definition of that term.

Traditionally, the term has been defined to refer to a fictitious antenna that transmits linear polarization in all directions and also equal power in all directions. Such an antenna cannot exist. We outline the traditional argument that shows that any antenna that transmits linear polarization in all directions has one direction into which it transmits no power.

However, if we also allow circular and elliptical polarization, it is possible to construct an antenna that transmits some power into all directions, with the understanding that power in any one direction is averaged over one period of the HF frequency involved. This is what this paper argues.

Extending the present line of thought, one could define a notion such as “weak isotropic radiator”. This is an antenna that transmits the same power into every direction, with the fine print that elliptical polarization is now allowed and power is averaged over one period of the HF frequency involved.

The author happens to speculate arbitrarily good approximations of such a “weak isotropic radiator” could be constructed. However, for the present paper, this speculation is out of scope.

The iso-what???

The “isotropic radiator” is a theoretical construct. It represents a worst case, as far as directional antennas are concerned. Directional antennas are supposed to concentrate transmitted energy into a wanted target direction. By contrast, the definition of the isotropic radiator is: It is an antenna that transmits equal amounts of power in every direction. It is the least directional antenna conceivable.

Also, the isotropic radiator is traditionally considered lossless.

Isotropic radiators aren’t actually built, but they are used as a theoretical construct to gauge other, more directional antennas. The power a directional antenna sends to its target direction when fed with a certain amount of transmitter power is compared with the power an isotropic radiator sends to that (or any other) direction, when both are fed with the same transmitter power. Express the power ratio in Decibel, and you have the definition of dBi.

This is all quite familiar to most radio amateurs.

It is impossible for theoretical reasons

It is also well-known among radio amateurs interested in antenna theory that an isotropic radiator is strictly a theoretical construct. It cannot be built in practice.

The traditional proof goes like this: Consider a huge sphere with the antenna at its center. The sphere is so large that it is safely in the antenna’s far field. (If you are a mathematician and want to be precise about this, use limits.) For every point P of the sphere, define a vector at P: Its direction is given by the electric field direction of the wave at P as it is coming from the central antenna, and the length of that vector is proportional to the electric field strength at that point. As the sphere is in the far field, the vector’s direction is tangential to the sphere’s surface at P.

(Alternatively, you could also rig up the same argument that is about to follow with the magnetic field instead of the electric. It’s your choice.)

Now, there is a mathematical theorem affectionately called the “hedgehog theorem” or, alternatively, the “harry ball theorem”. It states that any continuous vector field tangent to a sphere’s surface becomes zero in at least one point of the sphere.

By that theorem, there is one point where our wave vector is of length zero. In other words, there’s no field at that point. So no power is transmitted from our antenna towards that point’s direction.

As we have found one direction into which the antenna does not transmit at all, whatever antenna we have at the center of the sphere certainly isn’t an isotropic radiator.

By implication, that means no (finite-sized) antenna can be an isotropic radiator.

Or is it?

This argument invites a bit of scrutiny, for it disregards time.

There is a thing called “circular polarization”. It is somewhat (though not entirely) similar to what would be transmitted by a dipole that rotates very quickly. More precisely, circular polarization can be understood as a wave with a polarization plane that rotates quite quickly, namely once per period of the HF frequency, while the electric and magnetic field strength at the same place remain constant.

Now let us place an antenna producing circular polarization into the center of our giant sphere. The theorem then tells us that there must be a point on the sphere receiving no power at any given moment. But over time, during one period of the radiation, that point may well move.

So, if circular polarization is allowed (or its wilder cousin, elliptic polarization, is), it is at least conceivable we may be able to build an antenna that approximately transmits the same power to every direction when averaged over one radiation period. If we agree to call such an antenna a “weak isotropic radiator”, it is at least feasible arbitrarily good approximations of weak isotropic radiators might exist. However, this is only speculation. I do not argue this case here. Later edit: If you want to go down this rabbit hole, I suggest the academic paper Haim Matzner, Kirk T. McDonald: Isotropic Radiators.

An example

But I give a pertinent example. It shows the hedgehog theorem argument can be worked around with circular or elliptical polarization.

Scope

Before we start, let me repeat: I’m not concerned with producing a good approximation of a weak isotropic radiator. I just discuss the traditional argument that uses the hedgehog theorem. What I’m after is: To demonstrate an antenna that transmits some power to all directions. I’m not concerned about power levels transmitted to different directions being equal, just about all of them not vanishing.

A concrete example antenna

I look at the common cross dipole that is fed with two signals of the same frequency that are π/4 out of phase. (π/4 in radians is the same as 90° in degrees, if you prefer degrees.) This is a simple antenna that can be used to generate circular polarization.

Details for those not familiar with this antenna: It consists of two straight half-wave dipoles that are perpendicular to each other. The dipole centers are quite close, but without actually touching. The dipoles can be fed independently of each other.

It transmits into all directions

Claim: When the two dipoles are fed with two π/4 out of phase signals, there is no direction in the far field of this antenna towards which the antenna does not radiate, assuming radiated power is averaged over time, e.g., over one period of the HF being fed into the antenna.

The main technical argument needed for this claim: Consider the superposition of two linearly polarized waves of the same frequency, traveling in the same direction, so the waves are π/4 out of phase. If both of these “component waves” come in with the same power and polarizations are perpendicular, their superposition can also be viewed as a circular polarized wave. It is really the same thing. In mathematical terms, the two out of phase waves can be viewed as linearly independent vectors in some suitable vector space. That they are linearly independent is easy to see if the waves’ polarizations point into different directions. But is also true if polarizations are parallel or anti-parallel. Linear independence implies: The total wave can only carry no power whatsoever (when integrated over one HF period) if both of the two component waves are zero.

In passing: This argument just given does not depend on the phase being precisely π/4. Any phase difference will produce the linear independence needed, with the exception of integral multiples of π/2.

That out of the way, let us now come back to our cross dipole. Let us consider all different directions as they extend from the cross dipole into its far field.

There are four special directions into which only one dipole radiates: These are precisely the directions into which the four dipole conductors point. If the conductor of one dipole points into one such direction, that dipole contributes nothing to the radiation towards that direction. But the other dipole certainly does. So the total antenna does transmit into those four directions. So these four directions are dealt with.

Any other direction, both dipoles transmit some power towards that direction. Consider a point P in that direction, at a certain distance out in the far field.

Both dipoles are basically equally far away from P.

One could spend some thought to find out whether the waves hitting P coming from the two dipoles are still precisely π/4 out of phase, or else only approximately π/4. Actually analyzing this in detail is not needed. It is enough to show that the phase difference will stay close enough to π/4 so it remains safely away from both 0 and π/2.

So, by our previous argument, the total field at P will not vanish, as both waves don’t.

In other words, any P sees a transmitted signal.

q.e.d.

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