The Poynting Vector, Power Transmission
and the CFA
By David Jefferies
D.Jefferies@surrey.ac.uk
Text Reprint by Permission
(Additional graphics by antenneX)

The Poynting Vector and Power Flux Density
The units of electric field strength E are volts/metre, and the units of magnetic field strength H are amps/metre. Thus the product (E times H) has units (volts amps)/(square metre) or watts per square metre. A quantity having these units is called a "power flux density". We pose the question, "under what conditions does a product of an electric field strength, a magnetic field strength and an area represent a real power flow?"

The traditional answer to this question is that the Poynting vector S = E x H is the vector product of the E vector and the H vector; it has dimensions watts/(sq metre) and is in a direction at right angles to the plane containing the E and H vectors, and can be interpreted as a local power flow at right angles to the EH plane, proportional to the area of the element of surface times the E field strength times the H field strength times sin(the angle between E and H).

This interpretation is very suspect. A little thought shows we can set up a permanent magnet to create a static H field, and a pair of charged stationary plates to set up a static E field not everywhere in the same direction as the magnetic field. There is no movement. There is no power supplied to maintain the charges or the magnetic field. There is no power dissipated anywhere in the scenario. Yet there is a local elemental Poynting vector, which, if we interpret it as a local power flow, has no basis in the Physics of the situation.

The solution to this conundrum is to consider the integrated Poynting vector only over an entire closed surface in 3-d space. If the integral over the closed surface is not zero, then we can be sure that there is a flow of power from the inside to the outside of the surface, or from the outside to the inside.

In the case of our static charges and our magnet, such an integral over an entire closed surface works out to zero whatever shape the closed surface is.

Now we consider the case where a battery (d.c.) supplies a pair of wires connected to a resistor. There is power drawn from the battery; it is conveyed by the wires ("waveguide??") to the resistor and dissipated there. There is a magnetic field generated around each of the wires by the current flowing in them, and there is an electric field between the wires because of the voltage drop across the resistor. However, this is a static problem, as was our first example of the magnet and the charged plates.

We perform our integral of the Poynting vector in various ways. If the surface surrounds the battery only, with the wires passing through the surface, there will be an outward directed total to the integral, this total being equal to the power supplied by the battery. If the surface surrounds the entire circuit, the integral will be zero. If the surface surrounds just the resistor, with the wires coming in through the surface. the integral will give an inward directed power flow just equal to the power dissipated in the resistor. If the surface cuts the wires twice, but does not include either the battery or the resistor, the integral is again zero as the power flow in equals the power flow out.

Those keen readers with numerical NEC modelling code may like to assume a geometry for this scenario and check that these statements are born out by the simulations. If you do this, email me (D.Jefferies) and I'll post a credit.

The CFA ("Crossed-Field Antenna")
The CFA has been proposed many times, by various people, as a method of reducing the size of radiating antenna structures while still radiating very significant amounts of power. The idea is to set up, independently, oscillating electric and magnetic fields (perhaps using capacitor plates for the E field and a coil for the H field) at right angles to each other (or nearly so) and thereby produce an outward directed Poynting vector which it is claimed represents real radiating power in the far field.

From the arguments above, we are naturally suspicious of this procedure as we have seen a counter example. The acid test is to integrate the quantity E x H (the mathematical vector product of the E and H field strengths) over an entire surface which surrounds the supposed radiating CFA and all its generators and power supplies . Only if this gives a significant radiating power may we assume that the structure works as advertised.

There is currently a proposal to build a CFA transmitter on the Isle of Man to cover NW Europe or some significant proportion thereof. It will be very interesting to see if this works. Low frequencies are proposed, to give long range ground wave communications; the normal antenna structures would be very large and the Island's planners would not be happy. For this reason, a small CFA (if it works) would be advantageous.

Having read the descriptions above, what do you think will happen when they try this out? Of course, as with all science the acid test is a properly constructed and implemented experiment, not speculations of a theoretical nature, especially on a web site.

Therefore it is nice to read in the magazine "Radio Today" for October 1999 (vol 17 number 10) on pages 11-13 a report of experimental measurements on practical CFAs which have been built in Egypt. One interesting result is that the quoted measured field strength at a distance of 76.9 kilometres from a CFA at 1.161 MHz and a stated power of 30 kW is 14 mV/metre. If we assume that the 30 KW power is all radiated and is distributed uniformly across a half-sphere at this radius, we calculate an r.m.s. electric field strength of 17.5 mV/metre. It would appear, therefore, that this antenna is radiating nearly all its stated power.

This antenna is stated to be located at Tanta, near Cairo, and to be installed on the roof of the transmitter. The height of the antenna is said to be 8.2 metres which is 3.2% of the wavelength (258.4 metres) at 1.161 MHz. The construction consists of a circular disk capacitor plate above a ground plane, used to convey displacement current dD/dt in a vertical direction, which presumably generates horizontal loops of magnetic field H according to the fourth Maxwell equation curl(H) = dD/dt + J. Here, unlike the assumptions in most antenna theory books, it is the displacement current which gives rise to the magnetic field, rather than conduction current on the antenna elements. (There is a question about currents on the outside of the feed, assumed to be coaxial cable). Combined with this magnetic field is an electric field generated by an electrode above the disk, roughly in the form of an inverted cone, starting on the cone surface and ending on the ground plane. The displacement current is arranged to be in phase with the electric field, which means that the electric field generating the displacement current is in phase quadrature with the electric field from the conical electrode.

Now this is a bit naive, because the electrode (D-plate) that generates the displacement current also generates a strong electric field of its own. The electrode (E-plate) that generates the electric field (that is, the inverted cone electrode) also generates a displacement current of its own. The magnetic field from this second displacement current is in phase quadrature with the primary magnetic field from the D-plate. If we add the magnetic field generated by this additional displacement current to the D-plate-generated magnetic field, and we similarly add both the electric field contributions, then we go out about an additional antenna diameter from the structure and integrate (E x H) over a surface, taking into account the phase relationships, it is not clear at all what the outcome will be.

Normally, for a capacitative load on the end of a transmission line, the power reflection coefficient is unity and the voltage at the terminals is in phase quadrature with the current. It is not clear from the descriptions given, what the driving point impedances are of the two capacitative elements as they are in close proximity and fed in phase quadrature. Neither is it clear how the radiated power is transferred from the feeds.

The crux of the experimental data (as reported) is the (unstated) efficiency of the antenna. We have accepted that the field measurements indicate that 30 kW of power is radiated, but we are not told the input power level. It is well known that a short antenna has a high radiation reactance compared to its radiation resistance. This radiation reactance needs to be tuned out, to get power transfer from the generator to the radiated field. That makes the antenna narrow bandwidth or high Q. However, the Q can be reduced and the bandwidth increased by making the antenna less efficient; that is by adding significant dissipative loss resistance in series with the radiation resistance. The power delivered by the generator is therefore largely supplied to the dissipative loss resistance, and only a small fraction of it is radiated. It is this 30kW of radiated power which is claimed for this antenna.

In a small antenna, at these power levels, the dissipated power would normally (see next section) make the structure hot. It might even melt the conductors. The property of the CFA (as described) that strikes one most strongly, is that it has a large surface area of conductor to carry the currents. Thus it can be very inefficient electrically without getting too hot, and be a classical inefficient electrically small antenna having corresponding wide bandwidth, and still function as described. However, the mechanisms describing the radiation process are suspect.

It would be very interesting to be told the supplied input power of this 30kW-radiating CFA.

Nevertheless, if this arrangement works as the description says that it does, and also with sensible efficiency, it would be very interesting to see if the NEC antenna modelling code for numerical simulation predicts its performance accurately. If not, then there are deficiencies in all the currently-used software for antenna modelling.

The bandwidth and efficiency of electrically small antennas
There is a well-thought-of paper by R.C Hansen, called "Fundamental limitations in antennas", and published in the IEEE proceedings in February 1981, volume 69 number 2. The essential result, for our purposes, is that the theoretical minimum value of the Q-factor for an antenna which is very much smaller than a wavelength is given by (approximately) Qo = ([lambda]/[2 pi r])^3, if the antenna and its supporting transmission line(s) and generator(s) are all contained within a sphere of radius r, providing the antenna is 100% efficient. If the Q factor is spoiled by resistive loss, so that the power efficiency becomes (eta), then the minimum Q factor for the radiating structure becomes (eta)Qo.

In the case of the CFA at Tanta described above, if we assume that the antenna and all its associated drive circuitry is contained within a sphere of radius 4.1 metres (we are told the height is 8.2 metres) then the value of Qo is around 1000. But we are also told that the measured Q factor is at most the inverse of the fractional bandwidth, (27 KHz at 2:1 VSWR in 1,161 kHz) which puts it at 43 at the maximum. Thus the efficiency is at most 43/1000 or 4.3%. If this were the case for this antenna structure, and we believe the field measurements which put the radiated power at 30kW, then the input power would be of the order of 30/(0.043) kW or 700 kW. It is difficult to believe that an antenna of this size would not melt, given a dissipated power of over half-a-megawatt in the structure.

Thus, according to traditional antenna sources, this antenna probably cannot work in the way we are led to believe. It is a classic case of "theory vs experiment" and the usual scientific solution to such a paradox is to believe the experiment and re-evaluate the theory.

Therefore, independent tests of the technology are desirable. If anyone else can get this kind of performance from such a small structure they should tell us. Meanwhile, there seems to be much reported evidence of experiments which fail to reproduce these results. This evidence should not be dismissed.

See Author's Biography -30-