Near and Far Fields of the Hertzian Dipole Antenna. Radiation Resistance.
In the previous lecture, we calculated the phasor `E and `H fields produced by a Hertzian dipole antenna of current `I = a_{z} I and length ÑL located at the origin of the coordinate system to be
`E(r) = a_{r}E_{r} + a_{q} E_{q} [V/m]
Where
And
In this lecture, we will carefully examine these fields and discover interesting behavior of this Hertzian dipole antenna.
Near Fields of the Hertzian Dipole Antenna
The properties of these `E and `H fields in (1)-(4) are quite different depending if we observe them electrically close or electrically far from the dipole antenna. Electrical distance here is measured by br.
As b r ®0 we can neglect the 1/( jbr)^{2} term with respect to the 1/( jbr)^{3} term in (2) and neglect the 1/( jbr)^{3} and 1/( jbr) terms with respect to the jbr)^{2} term in (3). Additionally, for both (2) and (3) we employ the series expansion
keeping only the first term b r ®0. After performing all three of these operations we find that
But
So that
From the previous lecture, we saw that Q = I/ jw so that Becomes
Following a similar process for E_{ q }in (3) we find that.
This electric field in (7) and (8) for the near fields of the Hertzian dipole antenna has exactly the same form as that for an electric dipole `p = a_{z} QÑL C-m
that you saw previously in EE 381 for static fields:
Equations (7) and (8) have exactly the same form as (9). The difference is (7) and (8) are phasors while (9) is a static field. [See Mathcad worksheet “Animated Electric Fields of the Hertzian Dipole”.] Keeping the leading term in (4) as b r ®0 gives
which is exactly the same form as the static magnetic field produced by a current element IÑL using the Biot-Savart law in magnetostatics [see equation (3) in Lecture 30], though that is not proven here. In summary, the `E and `H fields electrically close (br<< 1) to the Hertzian dipole antenna have the same form as those fields of the static problem (electric dipole, magnetic current element), but those of the antenna simply oscillate sinusoidally with time. These near fields of the Hertzian dipole antenna are consequently said to be quasi-static.
Far Fields of the Hertzian Dipole Antenna
The situation is completely different for the `E and `H fields at distances electrically far from the antenna. In the case that br >>1, then from (1) through (4):
E_{r} ? 0 (br >> 1)
These far zone fields of the antenna behave very differently than the near zone fields:
• Notice that because of the e^{- jbr} factors in , `E and `H are propagating as waves in the + a_{r} direction (away from the dipole antenna). These are called spherical waves.
•`E ^`H .
[It is intereting to observe both of these phenomenon in the Mathcad worksheet “Animated Electric Fields of the Hertzian Dipole”.]
• Both `E and `H are perpendicular to the direction of propagation ( a_{r} ) because E_{r} is vanishingly small with respect to the E_{q} term.
• E _{q} / H_{j} =h .
All of these properties sound very familiar, don’t they? These are the same characteristics of uniform plane waves (UPWs). Here, though, there is one big difference: the far fields of this Hertzian dipole antenna are proportional to 1/r. They decay in amplitude as they propagate away from the antenna. For the UPW, they didn’t decay in amplitude. The fundamental reason for this behavior is the source for a UPW is an infinite current sheet. Because of its infinite extent, the EM waves it produced didn’t decay as they propagated. Such a behavior, though, requires a source that supplies an infinite amount of power, which is not at all realistic.
Power Radiated by the Hertzian Dipole Antenna
This Hertzian dipole antenna is a much more realistic source of EM waves and it produces a finite amount of power radiated. We calculate this time average power using the Poynting vector
Substituting the far field `E and `H from (11)-(13) into (14) we Find
Or
assuming a lossless infinite space in which the antenna radiates so that h is a real number. Substituting we find
Or
This result indicates that this antenna is radiating an EM field (a wave) that is carrying time average power away from the antenna. Notice in (16) that this time average power density decays as 1/r^{2} . (The fields decay as 1/r .) We can now calculate the total radiated time average power PAV by integrating (16) over a sphere centered on the dipole antenna with a radius in the far field of the antenna such that
The integral in this expression can easily be evaluated as
Substituting this result gives
or with b = 2p /l then
Radiation Resistance and Equivalent Input
Circuit for the Hertzian Dipole Antenna
This time average power in (18) represents power that is carried away from the terminals of the antenna by the electromagnetic wave. This power will not return to the antenna. For a generator connected to the terminals of the antenna, this effect simply looks like a resistance. Even if the antenna is made from perfectly conducting wires, there is still power “lost” to radiation. In fact, we define a radiation resistance R_{r} for an antenna as a hypothetical lumped resistance that would dissipate the same amount of power as that radiated by the antenna. For a resistor,
Equating (19) with (18) we find that for a Hertzian dipole Antenna
An equivalent circuit for the input terminals to the Hertzian dipole antenna includes this radiation resistance in series with a capacitive reactance that captures the near-field terminal characteristics of the dipole antenna:
We haven’t solved for this equivalent capacitance here, but can be found in many antenna textbooks.
Example N32.1. A steel dipole antenna of length 62” and 1/8” diameter is operating at 1 MHz (an AM radio antenna, for example). Assume a Hertzian dipole antenna model.
(a) Calculate the antenna radiation resistance and Ohmic resistance.
so this antenna is electrically very short. Then using (20),
Because of the skin effect, in a wire of length L
where the surface resistance R_{s} is defined as
For this antenna of length ÑL made from steel in which s = 2×10^{6} S/m and m = m _{0} , then
Notice how small the radiation resistance is in comparison! It’s actually ten times smaller than the Ohmic resistance of the steel wire at 1 MHz. This turns out to be a universal characteristic of electrically small antennas: They are not efficient radiators of EM waves.
(b) Calculate the radiation efficiency e_{r} of this antenna. By definition, the radiation efficiency is
.
For this specific Hertzian dipole antenna
which is a very low value.
(c) Calculate the input impedance The equivalent circuit at the terminals of the antenna is
Therefore
Z_{in} = R_{ohmic} + R_{ r} - jX_{A} = 0.242 – j37.844 beta
Notice the extremely large capacitive reactance in this Z_{in} . So, not only is this antenna not an efficient radiator, but it is difficultto couple energy to it from a source! Need a matching networkto do this by resonating out the C using an L, for example, but then the antenna becomes narrow banded. Not all antennas perform this poorly! One can make dipole antennas much more efficient by making them electrically longer. Approaching l /2 in length, these antennas perform much better