Antennas Fed in Phase Quadrature

Introduction
Readers of articles in antenneX will by now be familiar with the class of compact antenna which have two independent feeds, phased in quadrature (that is, with 90 degrees or pi/2 radians phase shift between the feeds) linked by some kind of reactive structure. These antennas include the EH antenna and the well-worn CFA concept, and loop antenna variants as well as the more familiar capacitative structures.

In this short article, it is shown that these antennas can absorb power from one of their feeds, and deliver an equal amount of power (apart from losses, and a very little radiation) to the other. In this way, they may appear to be accepting power efficiently, at least from one of the feeds, and accordingly seem to have low Q and high efficiencies. Of course, this power is going nowhere other than being absorbed by a generator, transmitter, or its associated internal impedance.

Circuit theory
Let us start by looking at the very simplified circuit diagram in Figure 1.

The diagram in Figure 1 shows two generators, in phase quadrature, each with internal impedance Zo, feeding opposite ends of a two port circuit which is entirely reactive. For simplicity in the subsequent mathematics, we have represented this reactive antenna structure as being a series component only, with imaginary impedance X ohms. X can be positive (inductive) or negative (capacitative), so the development here encompasses both small capacitative antennas and small loop antennas. If we were to include the shunt reactance components on each port as well, the mathematics would become very much more complicated, almost intractable, but the essence of the argument remains unchanged.

The quadrature nature of the generators is taken care of by making them of equal amplitude V, but whereas one is V the other is taken to be jV. In complex notation, a phase shift of 90 degrees is equivalent to multiplication by j. Of course, it makes no difference where the origin of time is assumed, and so all we need is that the ratio of these generator voltages is j. This subsumes the case where one generator is run at +45 degrees, and the other at –45 degrees.

And this voltage drives current through the series combination of the reactance jX and each of the internal generator impedances Zo, in total an impedance of

Dividing the voltage by this total impedance produces a value for the current I in the loop of

And so the input impedance may be calculated looking into port 1; a few lines of algebra reveals that this input impedance is

A similar calculation also reveals that the input impedance looking into port 2 is

For more realistic and complicated networks, the mathematics proceeds similarly but the results are not quite as simple to interpret.

In exactly the same way that it is possible to charge a car battery, by supplying current against its terminal voltage, it is possible also to supply energy to an a.c. voltage source by appropriate phasing of the terminal current. In these two expressions above, the real part is proportional to X and represents power delivered in this manner to the generator. (If the internal generator impedance Zo takes a complex value, part of the power delivered to the source is dissipated in the Zo component.)

Interpretation
The impedances Z1 and Z2 represent the loads presented to the two arms of the generator(s) taking into account the presence of the other arm. Zo and X are taken to be real numbers; X can take positive and negative values, depending on whether it is inductive or capacitative. The power flow direction between the two quadrature generators depends on the sign of X.

We see that the effect of driving in phase quadrature on the two ports simultaneously is to produce a real impedance component, +/- X/2 depending on the phasing of the generators. What this means is that power is extracted from one of the generators and delivered to the other. There is also a transformation of the real internal impedance of the generators into a reactive component.

If we interpret the power flows as representing the Q of the antenna structure, it is not difficult to see that the antenna appears to be absorbing power from one generator and delivering it to the other. Monitoring the behaviour of the antenna by making supply side measurements therefore gives us no information as to whether any of this power is actually being radiated.

Real-life considerations
Of course, in real life scenarios there is likely to be only one transmitter, whose output is split into two paths through a so-called phasing network. In addition, the transmitter is likely to have poorly-defined properties as far as its output impedance is concerned. This is especially so if it consists of an efficient class C stage with an associated tank circuit and matching network. Thus, Zo is not likely to be a nicely behaved 50 or 75 ohm resistive component, but may be reactive and highly non-linear, depending on the instantaneous value of the output voltage. Also, the two Zo components may not take the same value, and may be complex so that additional power is dumped into them. The power balance, whereby equal powers are delivered and absorbed, probably will not be exact. Therefore, the transmitter appears to be delivering net power, via the antenna structure, to its own internal impedance. In the non-linear case, it will be very difficult to interpret the instantaneous power flow in the structure, and the Q factor may not be at all well defined. Thus, the bandwidth over which the antenna appears to radiate effectively (in so far as it does at all) may be larger than anticipated by the CHU limits for antennas of this size. I do not recall any discussion of the CHU limits for non-linear antenna structures.

Conclusions
It is suggested that these considerations go a long way to explaining the many and various problems people have had with matching to multiply-fed antennas. Taken together with the spurious radiation from unbalanced feeds and adjacent coupled structures, the antenna may be expected to work in some installations better than in others, and to provide sufficient encouragement to the keen experimenter to encourage persistence. We are gradually arriving at a better grasp of this class of antenna structure, which appears not to work as claimed but nevertheless has sufficient complexities of behaviour that it is not simple to see why this might be.

The advice to antenna experimenters is, please avoid multiply-fed antenna structures. If you must experiment with this class of antenna, then you have to characterise the performance by making careful measurements of the radiated far-field strengths as suggested by Alan Boswell, rather than by trying to deduce efficiency from measurements of antenna Q or from other supply-side investigations. –30-

Dr. David J. Jefferies
School of Electronics and Physical Sciences
University of Surrey
Guildford GU2 7XH
Surrey, England
D.Jefferies email
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