# Radiation impedances of wire and rod antennas.

Many people are in pursuit of the "holy grail" of electrically small, wideband, efficient antenna structures. This article attempts to explain the physical limits of electrically short wire and rod antennas, which necessarily have low radiation resistance into which it becomes increasingly more difficult to couple significant power as the length/wavelength ratio decreases. The article is short, with links into wider reading.

A generalised impedance Z = R + jX has a real part R, representing a resistance, and an imaginary part X, representing a reactance. Looked at as a circuit of components R and X, the components would be connected in series, because "impedances in series add". There is a discussion of the power flow in ac circuits containing complex impedance elsewhere . A wire or rod antenna, when driven at its operating frequency, will most often have a complex impedance, with non-zero resistance and reactance. This means that the voltage at the terminals is at some phase angle phi = arctan(X/R) to the current supplied by the feeder or transmission line connecting it to the transmitter.

The mathematical expression Z = R + jX indicates that a circuit model may be made of a resistance R in series with a reactance X, with the same current flowing through each. The power delivered by a current of complex amplitude I having some phase angle to the impedance Z is, as remarked elsewhere , II*R/2, and does not depend on the reactance X which merely alters the terminal voltage V = ZI = (R+jX)I for a given supplied drive current I. On the other hand, there is a reactive power flow to and from the reactance X twice a cycle, and this reactive power flow is represented by II*X/2 .

The energy transferred to the antenna and its environs by the reactive power flow is stored mostly in the reactive near field. The energy transferred to the antenna by the resistive power flow either (A) heats up the antenna structure (or things in the near-field region of the antenna), or else (B) it is radiated, or (C) most usually, some of each. Thus we can break down the resistive part R of the driving point impedance into the sum of a loss resistance, Rloss, which gets hot, and a radiation resistance, Rrad. As the purpose of an antenna is to radiate energy, it is therefore the radiation resistance Rrad which is most interesting. Explicitly,

The classical way to calculate the radiation resistance is to surround the antenna with a hypothetical closed surface in the far field, calculate the values of electric field and Poynting vector on this surface in terms of the antenna terminal current I, integrate the power flow per unit area, represented by the Poynting vector, all over this surface, to determine the total outward travelling power in watts, and equate this power to the quantity II*Rrad/2 as discussed above.

If one does this one finds that the radiation resistance for a thin half wave dipole is about 73 ohms, with a reactive part that depends sensitively on the dipole rod diameter, of a few tens of ohms. If one takes a fat half wave dipole, the radiation resistance can fall below 60 ohms.

Since transmission line has a real characteristic impedance, what is needed for good transfer of power from feeder to antenna is to match the radiation resistance to the feeder characteristic impedance, and tune out any residual reactance with a stub match or other matching method. So therefore we see why the coaxial line impedance of 75 ohms is often chosen as a standard, since it is close to the radiation resistance of a half-wave dipole.

Now the contribution to the radiated electric field at a point in the far field region from a current I in a small length L of antenna wire is proportional to the time rate of change of IL and, at a given frequency, we can make this contribution larger in two ways; either increase I or increase L. Since free space is a linear medium, the principle of superposition holds, and we find the total contribution to the radiated electric field strength by summing (or integrating) over all the little elemental contributions. It is accelerated charge which radiates, and the radiation contribution is proportional to the amount of charge times its acceleration. Dimensional analysis shows us that this is (Coulombs)(metres)(seconds^-2). We also see that the time rate of change of a current times an element of length is also (Coulombs/second)(metres)(seconds^-1) which is, when rearranged, just dimensionally equivalent to the accelerated charge.

For a short thin wire or rod antenna, with L much less than a quarter wavelength, I falls away linearly or uniformly to zero at the end of the rod. If the rods have plates at the ends in the form of a capacitor, then the current I may not fall to zero, since it supplies the displacement current drawn from the plates.

In any event, the far field electric-field strength will then be proportional to the length of the rods, for a given terminal current, and if we square the electric field and integrate to get the power, it is clear that the radiated power must rise (for a given terminal current) as the square of the rod lengths. Thus the radiation resistance must be proportional to L^2, and indeed, looking in the standard textbooks we are presented with the formula that

R = 80*[(pi)^2]*[(L/lambda)^2]*(a factor depending on the form of the current distribution)

The factor depending on the current distribution turns out to be [(average current along the rod)/(feed current)]^2 for short rods, which is 1/4 for a linearly-tapered current distribution falling to zero at the ends. Even if the rods are capped with plates, this factor cannot be larger than 1.

Thus we can make a table for short dipole rod antennas. L is the total length of the antenna. (for monopoles above a perfectly conducting ground, the total length and the radiation resistance are both half of the values in this table)

 L/lambda R(ohms)about 1/5 8 1/10 2.0 1/20 0.5 1/50 0.08 1/100 0.02

Thus on 160 metres, if we erect a vertical monopole antenna which is 8 metres (26.7 feet) tall, the radiation resistance will be half that for a 16 metre dipole (that is, L/lambda = 1/10). Our monopole has radiation resistance of 1 ohm. It also has a large negative reactance, depending on the rod diameter.

So, as the rod antenna is shortened, the size X of the (negative) reactance shoots up, as the antenna starts to look like a capacitor whose capacitance gets smaller as the size decreases, and so, for a given current, the terminal voltage rises sharply. It is for this reason that short antennas are very inefficient radiators; not only do they have low radiation resistance and require more current drive, but they present a serious mismatch to the resistive characteristic impedance of the feeder. Also, the higher currents required for a given radiated power dissipate more heat in the resistive loss in the skin-effect of the antenna structure, and so the efficiency drops. Even if the large capacitative reactance is tuned out successfully, the antenna then becomes inefficient, narrow-band, and prone to mistuning and mismatch problems by any alteration of the disposition of scattering objects in the near field.

### Re-radiation from near-field objects and ground

The radiation resistance measured at the antenna terminals is really an indication of the effectiveness of the antenna structure as a radiator. For a given terminal current, the total radiated power, integrated over all angles, rises as Rrad.

Thus, if the antenna is coupled to objects in the near field which re-radiate, it is possible that the radiation resistance may increase. Whether or not this is actually seen depends to some extent on the character of the coupling, or what amounts to the same thing, on the mutual impedances between antenna proper and scattering object.

If the coupled object is lossy, then the driving point resistance of the main antenna may rise without there being a corresponding increase in the total radiated power. Thus, it is quite difficult to determine how well an antenna radiates from measurements of the resistive part of its terminal impedance alone. This has in the past caused arguments among antenna engineers, particularly when small high Q antennas such as tuned loops are concerned.

The only sure way to determine whether an antenna is radiating according to its measured terminal resistance is to go into the far field and make careful field strength measurements.