Hertzian Dipole Antenna.
In the previous lecture, we discussed the fundamental sourcefield relationship that can be used to calculate the `E and `H
fields produced by sinusoidal steady state line currents. In this
process, we first compute the phasor vector magnetic potential
where R = |r – r| , then compute the phasor magnetic field as
Using Ampere’s law, we determine the phasor electric field
What might be surprising to you is that one of the most difficult
challenges an antenna designer faces is determining the current
distribution `I (`r) on a particular antenna. This is often a
complicated problem and usually the only accurate solution is to
use computational electromagnetics methods. There is a famous class of straight wire antenna that has an accurate approximate solution for the form of the current on the antenna. This class of antenna is known as a thin-wire dipole antenna:
It is called a “dipole” antenna because there are two wires or
poles comprising the antenna. The form of the current on the wire is known quite accurately for thin wires (a <
It is infinitesimally short with a uniform current distribution.
Because
then for sinusoidal steady state this means that I = jbQ. The ends of this short antenna are capacitively loaded by small metallic spheres or disks to create a current that is approximately
uniform along its length in Without the capacitive loading the current would be approximately triangularly shaped with a maximum value at the center of the antenna and linearly decreasing to zero at both ends, as illustrated in `A for a Hertzian Dipole Antenna
Following the procedure described at the beginning of page 1,
we’ll now determine the `E and `H fields produced (i.e., “radiated”) by the Hertzian dipole antenna. This three-step process begins with the calculation of `A from (1)
The current `I(`r) is that for the Hertzian dipole antenna, which we will assume is located at the center of a spherical coordinate system:
Substituting for this assumed uniform current into (1) gives
Because this antenna is infinitesimally short then R is
approximately constant over the entire range integration.
Consequently, the integral in (4) can be performed trivially
leading to
(Notice that R has now become r.)
So we now have completed the first step and have obtained an
analytical expression for `A. The next step is the calculation of `H . Because of the spherical symmetry of the dipole antenna, we’ll use the spherical coordinate system to simply the mathematics.
We can convert `A in (5) to spherical vector components quite easily by using the relationship
az= ar, cosq –aq sin q
Giving
Where `H for a Hertzian Dipole Antenna
Now that we have successfully determined `A produced by the
Hertzian dipole antenna, we can now determine the `E and `H
fields, beginning with `H . As we saw in (2)
Because of the ¶ symmetry, ¶/¶¶ ®0 which, using the
determinant form of curl in the spherical coordinate system,
gives
such that
• Using
• Using
Consequently, using we find `H to be
Factoring out ( jb)2 = -b2 from both terms in (14) we arrive at the final form of the magnetic field produced (or “radiated”) by the Hertzian dipole antenna E for a Hertzian Dipole Antenna
Now that H has been determined, we can solve for the E field
produced by this Hertzian dipole antenna according to (3). Using
the determinant form of curl in the spherical coordinate system
Such that
Substituting for H from (15) into (16), and simplifying, gives Summary
So, this concludes the calculation of the `E and `H fields
produced (or “radiated”) by the Hertzian dipole antenna. There
is a wealth of information contained in these field solutions for
`E in (17)-(20) and `H in (15) that we will carefully pick through in the next lecture.
