The Magnetic B Field or loop antenna is relatively
inefficient as we saw in the previous analysis. The radiation resistance is
small compared to the loops ohmic loss. It is therefore necessary to transfer as
much energy as possible into this ohmic loss in order to maximize radiation
efficiency, and through antenna reciprocity, the antenna's reception capability.

How then can we estimate the loop antenna's ohmic resistance? The loop is essentially a single turn solenoid or "coil". It may be square, rectangular or circular, but square geometries are the most practical for fabrication, and other than this, there seems to be little reason to adopt any particular geometry over another. Printed Circuit Board (PCB) antennae are likely to be most cost effective, especially in miniature receiver or transmitter applications.

We
begin with some useful formula for estimating the resistance of wires at radio
frequencies and then inductance of wires in various geometries

We begin with the published resistivity of copper

…(1)

so
that the total resistance of a length ** l**
of copper wire with radius

…(2)

If we consider a 40 mm length of 1 mm diameter copper wire we find that .

The
resistance of a conductor tends to increase with frequency as the current
flowing through it tends to crowd towards the outer surface. This sets up an
exponential radial distribution in current density ** J** given by

…(3)

where
refers to the distance from the
outer surface to some internal position and
is defined as a ** “skin
depth”,** which identifies where the majority of conduction
occurs.

…(4)

or
alternatively, since
, and for copper wire where

…(5)

If
we assume that
, then the ** skin depth**
for copper wire will be
. This depth will usually be far less than the wire diameter, but
does define the average depth
current flows over (which can be shown by integrating equation (3). The
effective area

Given
this interpretation

…(6)

Providing
** f** is sufficiently high so that skin depth issues dominate, the ac
resistance for a length

…(7)

Using
the previous example 40 mm length of 1mm diameter copper wire with a DC
resistance of 852
we find that this increases to
at a frequency of 150 MHz, i.e.
almost 50 times higher at VHF!

We will now estimate the inductance
of several loop geometries,

In
this example, 40 mm of 1 mm diameter copper wire will have an inductance of *21.51
nH.*

Let
us now imagine this 40 mm length of 1 mm diameter wire bent into a square shape
so that
in this equation. The new
inductance will be ** 18.17 nH**. A slight reduction has occurred,
presumably because opposing side segments have magnetic fields in opposition.

Now
let’s imagine the same length of wire bent into a circle. In order to retain a
circumference of 40 mm its radius will need to be 6.366 mm. The predicted
inductance will therefore be ** 20.99 nH**. We note that this is
slightly higher than the square loop, as each opposing magnetic side segment is
maximally distant from each other. The total inductance is still less than the
original straight wire example.

We
will concentrate on square loop antennae, as these are usually easier to
fabricate than circular versions. We recall that

The
inductive reactance component of the loop will therefore be

…(11)

Again,
a square with a side segment length of 10 mm made from 1 mm diameter copper wire
will have an inductive reactance of 17.13 Ohms at ** f** = 150 MHz.

We
recall from equation (7) that
where ** l** represented
the total wire length, so that the modified expression with l representing the
side segment length must be,

…(12)

The
ac resistance at ** f** = 150 MHz will be 40.08
, which will
be in series with the radiation loss component
. From the previous chapter this was predicted to be
where
represents the fractional length of
one side compared to a wavelength. The radiation resistance at a given frequency
is therefore

…(13)

Continuing with the example, the
radiation resistance of a 10 mm square loop will be 29.2
at ** f** = 150 MHz. This is significantly less than the ohmic loss, as
would be expected for an electrically small loop antenna.

The equivalent loop antenna reactance will therefore be

The maximum achievable “Q” of
the loop can be defined as the inductive reactance divided by the ohmic losses
associated with its conductor,

Recalling
the ohmic loss resulting from the skin effect mechanism

…(14)

And
the loop inductive reactance

…(15)

We predict the maximum achievable Q
of the loop to be

…(16)

This suggests that the maximum achievable Q for a small square loop antenna with 10 mm side segments made from 1 mm wire to be 408.6. In practice, other losses will be dominant, such as dielectric loss and effective series resistance associated with resonating capacitors. A more practical total Q prediction will be in the order of 200 ~ 300.

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to** Magnetic B Field Antenna