Radio communication relies on the use of a transmit antenna to send
information "over the air" from a local position to another receiving
antenna at some remote position. The transmit antenna produces Electromagnetic
Energy comprised of an Electrostatic field traveling with a perpendicular
Magnetic field at the "speed of light" c ≡ 3 · 10^{8} m s^{-1} in a
vacuum and (very) slightly slower through an atmosphere or other medium with a
relative permittivity ε_{r}
or relative μ_{r}
permeability greater than one.

An* isotropic* antenna has a omni-directional antenna that transmits
radio signal energy, or receives radio signal energy equally in all directions.
Consequently, energy supplied to a loss-less *isotropic* antenna radiates
equally over the surface of a sphere with an surface area related to its radius ** d**
according to

* *
*A*_{sphere} = 4 · π
· **d** ^{2}

The power density ** S_{d}** at a distance

*
S*_{d} = ** P_{in}** / [ 4 · π
·

Also, the "aperture area" *A*_{aperture} (i.e.
its ability to capture a portion of radiated energy) for an isotropic antenna is
related to wavelength and commonly predicted to be

*A _{a}*

where λ
≡
^{c}/_{f}

Note
that **c** represents the "speed of light" and that **c** ≈
3 · 10^{8} m s^{-1} and * f* refers to
the frequency of the radio wave in Hz (cycles per second)

The
portion of radio wave energy "captured" by this aperture area at a
distance * d* is simply equal to the ratio of aperture area to the
surface area of the hypothetical sphere with a radius also equal to

*
P _{r}* =

i.e.
*P _{r}* /

We
can now propose a "path loss" estimation for the ratio of captured
radio wave power compared to the input or transmitted power * P_{in}*.
Expressed in decibels we have

Loss_{dB} = 20 · Log_{10}{ * d *}
- 20 · Log

(Note:
we will express loss as a **positive** quantity hence the sign reversal)

We will use this "-147.56 dB" factor to compare isotropic antennas
with other antenna types. Since loss is presented as a **positive** quantity,
antenna varieties that exhibit "gain" will have a term that is more
negative in value while this with additional power loss will have a factor that
is less negative.

Transmit and receive antennas need not be identical in design but the most common physical arrangement begins with a "dipole" configuration, developed by Heinrich Rudolf Hertz in 1886. This consists of two "quarter wavelength" conductors placed end to end with an electrical connection to the middle section.

If you would like to see derivations for path loss and radiation resistance for isotropic and dipole antennas (electrically short dipoles and half wavelength dipoles), then please click on this hyperlink Electromagnetic Equations

Also, if you would like to see a MathCAD simulation file that predicts radiation resistance for electrically short and half wave dipoles based on these Electromagnetic Equations then please click on this hyperlink Mathcad Radiation Resistance

(These hyperlinks will provide a more detailed description than my following
descriptions - and they took me several days to nut out - *oh man* am I
getting slow at maths!)

The wavelength of any electromagnetic wave is governed by the velocity of the
wave and inversely proportional to its frequency ** f**. This velocity
equals the speed of light so that the associate wavelength λ
=

*P _{in} * =

providing that electrical losses in the conducting elements and nearby materials are negligible. Given that receiving and transmitting antenna are "reciprocal" the same conversion between radio wave energy and electrical energy applies.

It
may be instructive to illustrate some of the electromagnetic radiation
properties associated with the "standard" half wave dipole for
subsequent comparison with magnetic loop antenna configurations. Dellinger's
formula (1919) expresses a relationship between two radiated magnetic field
components and instantaneous time varying AC current **I**_{ac}
flowing along the length * x* of the dipole antenna (for a half wave
dipole we have

...(4)

__ Note:__ All units are in standard SI units of meters, Amps,
seconds, Hz, and

The
first magnetic field term *H _{r} { t }* is referred to as the
"radiation field" and this term is responsible for long range radio
wave propagation. The second magnetic field term

Radio
communication requirements rely exclusively on the "far field"
radiation term *H _{r}*. We can expect that the induction field term

Let
us therefore focus on the radiation term and consider the RMS magnetic field's ** magnitude**
*
which will one half the peak value of the sinusoidal time varying field presented
in equation (4),* i.e.

...(5)

The AC current ** I_{ac}** in a dipole element has a
sinusoidal current distribution starting at

...(6)

Substituting equation (6) into equation (5) and integrating wrt * x*
(along the dipole element's length) produces

...(7)

We may want to equate the AC input power to this dipole antenna to its
resulting radiated power density * S_{d}*. The series
resistive "radiation resistance" or

*P _{in}* =

From electromagnetic theory we know that the radiated power density ** S**
(power per square meter) is related to the magnetic field strength and the
electric field strength by

...(9)

The term *R*_{εμ}
is called the "impedance of free space" with an approximate value of *R*_{εμ}
= 377 Ohms. This results in magnetic and electric field equations that are
analogous to Ohm's law for voltage, current and resistance

...(10)

Converting between magnetic and electrostatic fields can be useful but I will just use the magnetic field "companion" component for now. (This will allow direct comparison with the magnetic loop antenna predictions later on in the main content of this web chapter).

We can now substitute equations (7), (8) into equation (9) to predict the
power density ** S_{d}** produced by a dipole antenna at
a perpendicular distance

...(11)

Now lets predict the power that would be captured by an isotropic antenna with an aperture area defined in equation (1) as

* A _{a}*

Multiplying the power density in equation (11) by this aperture area now lets us relate captured power to input power applied to the dipole antenna

...(12)

We can now express this ratio as a loss in dB terms as per equation (2) i.e.

Loss_{dB} = 20
· Log_{10}{ * d *} - 20 · Log

Now recall the previous result for two isotropic antenna from equation (2)

Loss_{dB} = 20
· Log_{10}{ * d *} - 20 · Log

We can see that the path loss for a dipole to isotropic antenna is **2.15 dB**
lower than the path loss from an isotropic to isotropic antenna combination. We
conclude therefore that the dipole antenna has a relative gain of +**2.15 dB**
compared to an isotropic antenna.

This relative gain is not surprising as the dipole has some directivity.

Although the half wave dipole exhibits useful power gain over an isotropic antenna, its dimensions can be cumbersome for portable devices. As we saw previously a VHF dipole designed for operation at a frequency of f = 150 MHz would need a total length of one meter. Small portable devices such a paging receivers would obviously be incompatible with an antenna having such dimensions. In the never ending search for miniaturized electronic devices, an equally compact antenna is essential. How can this be achieved?

The
simple dipole can be reduced in length so that it is no longer resonant at the
operating frequency. When an "electrically short" dipole is used, it's
input impedance Z_{in} = R_{s} + * j* · X

__MATHCAD example__

I found an equation that predicts radiation resistance *R _{a}*
for an electrically short dipole antenna derived from Maxwell's equations. The
expression can be presented as

* R _{a}* =

where
γ represents a "fractional length ratio" of the
dipole's total length *L* compared to the theoretical half wave length ^{λ}/**2**
defined as γ
≡
*L* / ( ^{λ}/**2**
) . This formula assumes that γ
<< 1

I then found some radiation resistance data for a vertical ¼ wave antenna placed over a "near infinite" conductive ground plane from a very old radio engineering handbook

The top row refers to the radio frequency wavelength compared to the length of the antenna set to be one quarter of a wavelength. The first entry "1" implies that ¼ of the radio signal wavelength equals the electrical length of this antenna and this ratio increases as lower frequency radio waves are tested. The bottom row contains the measured radiation resistance for this ¼ wave antenna used at or below its electrical length (yes, not a very intuitive approach for data presentation)

I used the reciprocal of the top row to show "fractional length" γ
compared to a ¼ wave antenna as before. I also multiplied the radiation resistance of the
single ¼ wave antenna by __ two__ to represent a ½ wave dipole that can be
considered as two ¼ wave antennas in series.

The log-log graph scale shows a linear relationship corresponding to a power relationship

...(14b)

We see the proposed equation (blue dotted line) shows good agreement with the red measured line. Some kinks are presumable the result of measurement error.

The slightly greater than square power fall off in
radiation resistance is not unexpected as this "fudges" an
approximation to a small correction term that would be needed to incorporate the
loss of directional gain for electrically short dipoles (these tend towards
becoming omni directional). This square law relationship can be understood as a
"conversation of energy" requirement. Imagine that a ½ wave dipole
antenna is reduced in length by one half. The resulting H_{r} magnetic
field magnitude will also halve so the power density * S_{d}*
will drop by one quarter from equation (9). Since energy cannot be created or
destroyed (unless we have another "big bang") the input power to this
electrically short antenna must also be one quarter. If the input AC current is
the same, it follows that the radiation resistance must also be one quarter the
original value since

I hope this "arm waving" description will provide an *intuitive*
understanding for the slight differences between equations (14a) and (14b).

It may appear then that electrically short dipoles should
perform equally as well as an isotropic antenna and almost as well as a half
wave dipole (in terms of antenna gain). However, practical limitations exist when trying to match a low
radiation resistance to a more standard typical value of 50 Ohms used in many RF
systems. For example, the previous VHF half wave antenna could be shrunk in
length by a factor of 10 to 1 i.e. γ
= 0.1. The previous graph shows a radiation resistance of only *R _{a}*
= 0.4 Ohms. It is likely that significant power losses will occur in the
matching network required to transform this low resistance up to 50 Ohms.
Further, the electrically short antenna will exhibit a significant series
reactive component, possibly in the order of several thousand ohms of reactance
(capacitive). This would need to be tuned out with an equal and opposite
reactance. The "Q" of this ratio is Q ~ 3000 / 0.4 = 7,500.

If
the components used in the matching network had infinite Q then no power would
be lost but the antenna bandwidth would be small (-3 dB at Δ** f **=

An even further problem is associated with "de-tuning". Even if this shortened antenna could be used, it would be extremely sensitive to nearby objects that could add small additional capacitive loading and shift the antenna's resonant frequency.

It can be seen that an electrically shortened dipole may be feasible but not necessarily practical. Even at one tenth dimension, this VHF example dipole antenna would still be 0.1 meters total length and not fit into a small matchbox sized pager!

The Magnetic B Field antenna was developed to address many of the issues confronting the electrically short dipole. It is required to be physically small compared to a wavelength, have adequate radiation efficiency, provide some means to present a "reasonable" terminal impedance and be relatively insensitive to detuning effects caused by nearby objects.

The simplest loop antenna consists of a wire loop with an AC *I _{ac}*
current flowing in the loop. This current generates a magnetic field that is
responsible for radiation, in much the same way the previous dipole antenna
operated. The inductance of this loop is predictable (as I described in my
previous web chapter on inductors) and can be "tuned out" with a
capacitor placed across a gap in the loop.

This style of antenna is often used for direction finding and has a sharp "reception null" when turned broadside to the incoming magnetic field. The "ferrite rod antenna" used in small AM broadcast radios operates much the same way and will exhibit a signal null when pointed end on to a radio station.

Magnetic loop antennas also receive great interest from radio amateurs for transmission and reception at frequencies below 30 MHz and where available space required for a full dipole antenna is unavailable. I have found many excellent implementations on the web that have achieved radiation efficiencies that are comparable to a standard dipole despite useful reductions in size.

Magnetic loop antennas have some interesting directional advantages. Sometimes it is advantageous to have a directional response but in some cases it may be convenient to communicate in any direction over the surface of the earth. A vertical dipole will achieve this and produce "vertically polarized" radiation that has equal power in all directions around it. However this may be cumbersome for long dipoles. For example, a ½ wave dipole required for operation in the 80 meter amateur band would need a tower that would be at least 40 meters high! A horizontal dipole is much easier to implement, but has a "blind spot" to signals arriving at its ends. A magnetic loop antenna solves this problem. It can be placed horizontally relative to the ground and produce horizontally polarized radiation in all horizontal directions. It is also claimed that "ground effects" are less problematic for magnetic loop antennas compared to dipoles.

Some accidental reception from the electrostatic component of the incoming wave may occur due to the physical area of the conductors used in the loop antenna. This reduces the depth of the reception null as some residual energy pick up remains.

This unwanted pick-up can be avoided by placing an electrostatic shield around the loop antenna.

A metal tube can be used as shown. Note that a "gap" is needed to prevent the metal tube from acting like a shorted turn. The metal tube is grounded and the electrical connection is made to the magnetic loop either at its highest impedance point or over some fractional segment of its circumference. Although a balanced connection is preferred and offers greater rejection of stray electrostatic pick-up, and unbalanced connection is also feasible.

The circular loop is popular at low radio frequencies and may contain many turns of wire, resulting in a higher, more practical inductance. At higher VHF frequencies it may be desirable to "print" the antenna on a PCB as a square copper trace.

This configuration can be printed on an internal PCB layer and "sandwiched" between a top and bottom copper sheet acting as the previous metal tube for electrostatic shielding. In this case the "gap" can be implemented as a "slit" in the top and bottom copper shields in order to prevent these acting as a shorted turn and so interfering with the magnetic loop antenna's proper operation

This PCB magnetic loop antenna uses an internal PCB copper track as the radiating element. Top and Bottom copper sheets provide electrostatic shielding (i.e. eliminating potential "hand capacitance" detuning effects from nearby conductive objects). An insulating "gap" is need in these shields to prevent the formation of a shorted turn as this would seriously interfere with the magnetic loop antenna's operation.

The inductance of the magnetic loop is "tuned out" with a parallel capacitor as shown (placed on the top PCB layer). Two series capacitors are then included to provide an impedance transformation, and the tuning capacitor's value is reduced to compensate for their additional capacitance. (PCB Via connections are shown a yellow circles with a cross)

An optional balun may be included, allowing either a balanced electrical connection (e.g. twisted wire or 300 Ohm ribbon cable) or an unbalanced connection (e.g. 50 Ohm coax). In both cases the copper shields must be connected to a suitable adjacent ground point (otherwise they will act like plate antennas and capacitively couple electrostatic signal energy into the loop antenna)

The loop matching can also be implemented along a segment of the loop. The length of this segment can be adjusted to achieve a direct impedance match (a single resonating capacitor is still needed)

Finally, multiple turns can be implemented to achieve higher inductance as may be required for lower frequency operation. These loops can be implemented as concentric track "square spirals" and multiple internal layers can be used to stack these PCB track spirals on top of each other with via connections.

The ferrite rod magnetic loop antenna is popular for AM radios and has also been used in some (usually older) Shortwave radios.

The ferrite composition is required to exhibit low losses at the intended
operating frequency and medium relative permeability e.g. μ_{r}
≈ 50. The
actual efficiency is extremely low (i.e. less than 0.1 % compared to a dipole)
but this deficiency is more than compensated for by its convenient small size,
essential for portable radios, and the characteristic high "atmospheric
noise" associated with low radio frequencies (the actual noise floor can be
50 dB higher than thermal noise, so an antenna with -50 dB loss is not
problematic!)

The overall efficiency of a magnetic loop antenna is defined as the amount of power transmitted as radio signal energy compared to the energy transmitted by a loss-less isotropic antenna. Since antennas are reciprocal, transmit efficiency and receive efficiency are exactly equal.

The magnetic loop antenna could be made diminishingly small and retain 100 % efficiency. However this performance requires that conductor power losses are zero in addition to any associated dielectric losses. Further, the operating bandwidth will diminish eventually to zero! Clearly some practical limitations need analysis

The ohmic conductor losses are common to all magnetic loop antennas, whether
they have a circular, octagonal, hexagonal, square or triangular shape. The
total loop resistance is proportional to the total length ** l** and is affected by the conductor diameter (if circular) and the operating radio
frequency (RF)

The most common conductor used for magnetic loop antennas is copper (or silver plated copper). Copper has a typical DC resistivity of

...(15)

The
DC resistance of a length ** l** of wire
with radius

…(16)

For
example, a 40 mm length of 1 mm diameter copper wire predicts that *R _{dc}*
= 852
μΩ
.

Unfortunately the actual conductor resistance increases with AC frequency and the DC value has little bearing on the actual value. This is caused by a "skin effect" phenomenon that tends to cause the AC current to travel close to the surface of a conductor as opposed to its central mass. This reduces the "effective" area or the conductor resulting in a higher AC resistance

We
now have to consider the effective area of the (cylindrical) conductor based on
its circumference multiplied by this effective "skin depth" *δ. *

In reality the current density ** J** falls off exponentially as the
radial distance

...(17)

The
skin depth coefficient *δ*
is related to the DC resistivity * and* the operating frequency
according to the following equation

...(18)

If we use non magnetic conductors such as copper, silver or aluminium the
relative permeability μ_{r}
= 1. Equation (18) now simplifies to

...(19)

For example, the skin depth *δ
*for copper at * f* = 150 MHz will be only

The "effective area" *A _{e}* of this wire will be
defined as

...(20)

where
the wire diameter ** d** is used instead of radius

The actual AC or RF resistance of this cylindrical
wire can be predicted by substituting this effective area *A _{e}*
into the DC expression for resistance in equation (16) resulting in a new AC
resistance

...(21)

It
may be helpful to represent this AC resistance *R _{ac}* in terms of
a "fractional wavelength" γ
instead of the actual wire length

...(22)

where
the resistivity of copper is *ρ**
= *1.673
10^{-8} Ω
m^{-1 }and the fractional bandwidth γ
will be assumed to be less than one i.e. γ
<< 1 for an electrically small antenna.

It
may seem odd that the AC resistance now appears to reduce in an inverse
square-root fashion with frequency* f* but this can be
misleading as the use of this new fractional wavelength term γ
implies a shrinking length of wire as this frequency increases. If the length of
wire

Equation
(22) can now be presented for copper conductors in its simplest form
using the published resistivity *ρ
= *1.673
10^{-8} Ω
m^{-1 }for copper wire e.g.

...(23)

For
the purpose of illustrating the difference between DC resistance *R _{dc}*
and AC resistance

*R _{dc}*
= 852 μΩ

*R _{ac}*
= 40.07 m Ω

We
see that the RF resistance *R _{ac}* is 47 times higher than the DC
resistance

If we want to predict this ratio of RF to DC resistance for some other purpose, it can be expressed as

...(24)

We can interpret equation (24) as demonstrating increased "skin effect" losses as frequency increases and also that very small diameter conducting wire will tend to not exhibit significant additional skin effect losses since the skin depth will be comparable to the actual wire radius. In both cases however, increased wire diameter will always reduce ohmic loss.

**Note
1:** The variable * l* refers to the

**Note
2:** A Printed Circuit Board (PCB) magnetic loop antenna will also be affected
by "skin depth" effects. In this case, the "effective area" *A _{e}*
will become

**Note
3:** The overall surface finish also affects the RF resistance. Rough surfaces
will appear to be longer than a smooth surface as the current flow has to
accommodate small peaks and hollows. This "surface roughness" may need
to be included if optimal prediction accuracy is required.

**Note
4**: Although material parameters are often presented in non SI units e.g.
micro-ohms per cm, these conventions change with the times so I will use
standard SI units where possible for equations, e.g. resistivity is in Ohms per
meter. However it may be convenient to use units of MHz for frequency * f *
in equation (23) as this will return a value of AC resistance for

We can predict the radiation resistance Ra for a magnetic loop antenna by
first considering its open circuit voltage *E* when a time variable
magnetic flux Ø passes through its enclosed area *A*.

We know from physics that the magnetic flux Ø equals the product of a
magnetic field *B* and the loop area *A* so that

Ø = *B*
· *A *...(25)

Since B = μ
· H were μ
≡ 4 · π
· 10^{-7} we also have

Ø = μ
· *H** · A *...(26)

The open circuit voltage (EMF) induced in the loop is proportional to its area and the rate of change of magnetic flux i.e.

EMF = -∂Ø
/
∂t = -μ
· *A **· *∂*H*
/
∂ * t*
...(27)

Let us now consider an sinusoidal oscillating magnetic field with a peak
value *H _{p}* and frequency

* H* ≡ *H _{p}* · sin{
2 · π
·

Substituting equation (28) into equation (27) now reveals that

EMF = -∂Ø
/
∂t = -*2 · π
· *** f · **μ
·

The RMS magnitude *E* of this induced voltage will be

*E*
= *√2
· π
· *** f · **μ
·

were
*H _{rms}*
represents the RMS magnitude of the magnetic field

In
order for power to be extracted from this magnetic loop antenna, we require this
EMF, or some proportion of it to the presented to a load resistance. We know
that optimum power transfer occurs when the load resistance equals the source
resistance and any reactive components are cancelled. The electrically short
magnetic loop antenna will have a series inductive reactance and we will imagine
that this component is first cancelled with and equal value of capacitive
reactance. The magnetic loop antenna will then appear as a voltage source in
series with its radiation resistance *R _{a}*. To achieve maximum
power transfer the load resistance will also equal

This might seem odd - one half the received power is dissipated in the load resistance and one half in the antenna's radiation resistance. Were did this "other half" power go?

Presumably it gets radiated back into space!

Since
P = v^{2} / R it follows that the received power *P _{r}*
will be

* *
*P _{r} = *(

We
now need to relate this RMS magnetic field magnitude *H _{rms}*
to transmitted power and perhaps compare the power

**
S_{d}** = 377 · (

Substituting this result into equation (31) then predicts

* *
*P _{r} = *

where * R*_{εμ}
is called the "impedance of free space" with an approximate value of *R*_{εμ}
= 377 Ohms.

Magnetic loop antennas have a directional response that is similar to that of a dipole. Therefore it makes sense to compare the received energy from a magnetic loop antenna with the energy received by a half wave dipole.

Recalling equation (12 ) we predicted that the power *P _{r}* received
by a half wave dipole from a isotropic radiating source with an input power

...(12)

Since antennas are ** reciprocal**, the same power

In other words the receive power *P _{r}* in equation (32)
must equal the received power

OK, so let's set *P _{r}* in equation (12) equal to

...(33)

Equation (33) now demonstrates the extreme sensitivity for
the magnetic loop antenna's radiation resistance Ra with loop area and
especially with frequency. Let us consider an example square loop antenna with
the same dimensions I used previously for AC resistance *R _{ac}*
predictions, i.e.

*R _{a}
*= 3.518 · 10

Now recall the AC resistance of this length of
wire was *R _{ac}*
= 40.07 m Ω -
obviously this VHF magnetic loop antenna design will have extremely low
efficiency! The actual efficiency will be determined by the ratio of these two
series resistances, e.g.

η
≡
100 · *R _{a}* / (

From this example we predict that η = 0.0442 %

I hope the discussion so far has provided an intuitive understanding for magnetic loop antennas based on relatively straight-forward Physics without the distractions of Maxwell's equations. This intuitive approach is useful as it let's us make some reasonable predictions on how to make electrically small magnetic loop antennas with some constraints governed by efficiency relative to a half wave dipole. So far we have learnt that

The
AC resistance Rof a conductor increases with frequency
due to "skin effect" in a predictable way and dominates the
DC resistance _{ac} R_{dc} | |

This
AC resistance is proportional to the length of wire conductor
and inversely proportional to wire diameterl | |

The AC resistance does not depend on the loop geometry (shape) | |

This
AC resistance R is in series with the antenna's
"radiation resistance" _{ac}R and results in a loss
of power that can be represented as a comparative efficiency relative to a
half wave dipole_{a} | |

The efficiency of an antenna is determined by the ratio of radiation resistance to "AC (loss) resistance plus radiation resistance" | |

The radiation resistance increases with loop area so geometries that maximize area for a given conductor length will provide the highest radiation resistance and consequently the highest efficiency |

Well this reasoning is certainly easier to conceptualize than trying to muddle through a haze of differential and integral equations combined with boundary conditions! We can readily guess that a circular loop will provide the highest area for a given length of conductive wire and consequently exhibit optimum efficiency for any given frequency. An octagonal geometry will provide slightly less area and therefore reduced efficiency, followed by a hexagonal geometry, then a square geometry and finally a triangular geometry (a "folded dipole" requires a different analysis as its area approaches zero but has a radiation impedance close to 300 Ohms).

Some
examples of Area *A _{x}* to total conductor
length

*A _{circle}* =

*A _{square}* =

*
A _{triangle}* =

It
may be useful to express equations (33) and (34) in terms of fractional
wavelength defined as γ
≡
*^{l
}*/

... (35)
where * k* = 12.57, 16 or 20.78 for circle, square or
triangle geometries.

Combining these constants then produces a simple result

...(36)

Once
again. using ** l** = 40 mm at

*
R _{a}* = 28.85 μΩ
for a circular loop geometry were k = 12.57 so that efficiency
η
=
0.0719 %

*R _{a}* = 17.81 μΩ
for a square loop geometry were k = 16 so that efficiency η
=
0.0442 %

*R _{a}* = 10.56 μΩ
for a triangular loop geometry were k = 20.78 so that
efficiency η
=
0.0263 %

Since
the magnetic loop conductor loop was constant across all three examples, the AC
resistance has the same value of *R _{ac}*
= 40.07 m Ω.
We observe the circular loop to exhibit the best efficiency, followed by the
square magnetic loop, slightly better than one half the efficiency of the
circular loop. The triangular loop is, not surprisingly, even worse again !

It should be remembered that high antenna efficiency and small antenna size are mutually exclusive. The magnetic loop antenna efficiency can be improved by using wire with a higher diameter as suggested from equation (23)

...(23)

If large antennas are required, copper tubing can be used to advantage, or copper sheet or even printed circuit boards soldered together as panels in a pseudo-circle!

Let
us consider a HF example at ** f** = 30 MHz (λ
= 10 meter band). Let's consider using

Since
right angle bends are difficult we will use a circular geometry i.e. * k*
= 12.57. The AC resistance, radiation resistance and efficiency will be

*
R _{ac}* = 0.0896 Ohms

*
R _{a}* = 0.289 Ohms

η = 76.3 %

This
represents a power loss of only **1.17 dB** compared to a dipole. Given that
the magnetic loop diameter will be only 2 / π
= ** 0.637** meters compared to a half
wave dipole that requires a corresponding length of

So it appears that the magnetic loop antenna can be designed with excellent results providing that realistic dimensions are chosen. However it does have an Achilles' heal. I mentioned before that the series inductance of the loop will need to be "tuned out" with an equivalent capacitive reactance, forming a tuned circuit topology. This will probably have a very high Q due to the small resistive terms involved. As a direct consequence, the magnetic loop antenna will have a narrow operating bandwidth and may need retuning even for small frequency changes (e.g. with an amateur radio band)

I have seen many impressive approaches to this on the Internet were a number of radio amateur enthusiasts have combined geared motors to the shafts of air dielectric variable capacitors and operated these remotely.

We have actually done all the "hard theoretical work" by now so this web section should be easy. We first recall the expressions for AC resistance and radiation resistance

...(23)

...(36)

where
fractional wavelength defined as γ
≡
*^{l
}*/

This combined resistance is also in series with the loop inductance that I presented in the previous web chapter on Inductors as

...(37)

Providing
* r* > 2.5 ·

The inductive reactance *X _{s}* of this loop is

The capacitance C_{s} required to tune this reactance out is C_{s}
= 1 / [ 2 · π
· * f* ·

The "Q" of the loop will be Q = ½ · *Xs* / [ *R _{ac}*
+

* Note:* The total loop resistance is in series with the equal
value of load resistance (matched for optimum power transfer) so the expression
for operating Q contains the ½ scale term.

The -3 dB bandwidth of the loop will be BW_{-3dB} = * f* /
Q. The actual operating bandwidth may be considerably lower especially if low
VSWR performance is required for transmission.

Let us return to the previous example for ** f** = 30 MHz with

__Predictive Results__

** f** = 30 MHz (10 meter amateur radio band)

*
l *= 2 meters (total length of copper tubing)

* d* = 10 mm
(average diameter of copper pipe)

γ = 0.2 (fractional wavelength)

* **
R _{ac}* = 0.0896 Ohms (Loop AC resistance
at

*
R _{a}* = 0.289 Ohms (Loop radiation
resistance at

*R _{total}* = 0.379 Ohms (Total loop
resistance at

η = 76.3 % (Best case efficiency for resonant loop antenna with no additional power losses)

** r**
= ½ ·

L = 1.69 μH

X_{s}
= 318.6 Ohms

Q = 420.3

BW_{-3dB}
= 71.4 kHz

We can see that very high Q components will be required for tuning (and impedance matching to 50 Ohms) in order to prevent additional losses. Also the -3dB bandwidth is extremely low and retuning will be needed for even minor changes in frequency

The magnetic loop geometry affects the loop's radiation resistance and its
inductance. The AC loss resistance is * not* affected however.

...(38)

__Predictive Results__

** f** = 30 MHz (10 meter amateur radio band)

*
l *= 2 meters (total length of copper tubing)

* d* = 10 mm
(average diameter of copper pipe)

γ = 0.2 (fractional wavelength)

* **
R _{ac}* = 0.0896 Ohms (Loop AC resistance
at

*
R _{a}* = 0.178 Ohms (Loop radiation
resistance at

*R _{total}* = 0.268 Ohms (Total loop
resistance at

η = 66.5 % (Best case efficiency for resonant loop antenna with no additional power losses)

*l*
= ¼ · * l * = 0.5 meters (length
of each side of the square magnetic loop antenna)

L = 1.54 μH

X_{s}
= 289.6 Ohms

Q = 540.3

BW_{-3dB}
= 55.5 kHz

The square magnetic loop antenna exhibits slightly lower efficiency than the
circular magnetic loop antenna constructed from the same length of copper tubing
(η
=
66.5
% compared to η
=
76.3
% ) and slightly lower bandwidth (BW_{-3dB} = 55.5 kHz
compared to BW_{-3dB} = 71.4 kHz)

Does this mean that a circular loop antenna is always better? Well maybe not.
Let's imagine a confined space that has a maximum square dimension as might be
available on a balcony. The previous circular loop antenna example would have a
diameter equal to 0.6366 meters for 2 meters of copper tubing. Let's consider
what would happen if a square magnetic loop antenna could also fit this area and
have sides equal to 0.6366 meters i.e. its total conductor length * l*
could now become

**
f** = 30 MHz (10 meter amateur radio band)

**
l ****= 2.546** meters (total length
of copper tubing)

* d* = 10 mm
(average diameter of copper pipe)

γ **= 0.2546**
(fractional wavelength)

* **
R _{ac}*

*
R _{a}*

*R _{total}*

η
**=
80.4**
% (Best case efficiency for resonant loop antenna with
no additional power losses)

*l*
= ¼ · **l ****= 0.6365**
meters (length of each side of the square magnetic loop
antenna)

L**
= 2.08** μH

X_{s}**
= 392.4** Ohms

Q **=
337.1**

BW_{-3dB}**
= 89.0** kHz

Well well well. Now the square magnetic loop is "better" than the circular magnetic loop antenna if the surrounding space is applied as a constraint. It just goes to show that it is best to not jump to a conclusion prematurely!

As we have seen the magnetic loop antenna presents a rather inhospitable and daunting impedance to the relatively benign 50 Ohm resistive outside world. We therefore need to devise suitable strategies to tame the loop impedance in order for it to coexist with standard communication equipment and impedance interfaces that are typically defined at a resistive value of 50 Ohms.

There are (at least) three general strategies for impedance matching

Capacitive Impedance Transformation | |

Inductive Tap Impedance Matching | |

Coupled Inductive Loop Impedance Matching |

The capacitive impedance matching approach relies on series to parallel impedance conversion that can be presented as

R_{p} = R_{s} ( 1 + Q_{T} ^{2
}) ...(A)

X_{p} = X_{s} (1 + Q_{T} ^{-2}
) ...(B)

Q_{T} ≡ X_{s} /
R_{s} ≡ R_{p} /
X_{p} ...(C)

The variable Q_{T} represents a "transformation Q" and is
not directly related to actual operating Q although both will show similar
trends. Let us consider the last square magnetic loop example.

*
R _{s}*

We need to transform the loop impedance in parallel form to 50 Ohms

From
equation (A) with R_{s} = 50, R_{p} = 264567 we find that Q_{T}
= 72.73 (note that Q_{T} is just a parameter for this procedure)

Q_{T} = 72.73 →
X_{s} = 3636.5 (equation C) →
C_{s} = 1.459 pF at f = 30 MHz →
C_{s}1 = C_{s}2 = 2.918 pF (i.e. in series)

Also
X_{p} = 394.4
Ohms →
C_{T} = 13.45 pF at f = 30 MHz

This
C_{T} represents the total parallel tuning capacitance presented to the
loop and part of this is provided by the series capacitors C_{s}1 and C_{s}2.
From equation C we have X_{p} = R_{p} / Q_{T} = 3637.7
Ohms →
C_{p} = 1.458 pF
at f = 30 MHz

We
need to subtract this additional C_{p} from C_{T} i.e. the
corrected value of C_{T} = 11.99 pF

We have determined that a tuning capacitor will be needed with a nominal value of 12 pF for this magnetic loop antenna and two coupling capacitors each equal to 2.9 pF will be need to transform the loop impedance down to a balanced value of 50 Ohms. A balun may be included to convert balanced to unbalanced.

The capacitive impedance match is easy to implement except that all three
capacitors need to have high voltage ratings. Since we know the effective
parallel resistance of the loop, i.e. Rp = 264567 Ohms, we could reason that a
segment of the loop conductor would act like an "auto transformer"
compared to its open end. For example, a segment equal to 1/10^{th} the
total conductor length will have 1/10^{th} the voltage impressed over it
compared to the open end and so therefore have an effective parallel resistive
component equal to R_{p} / 100. In general, if we define the segment
length to be m · L where L = the length of one side and 4 · L = the total loop
length then we should have

m = 4 · L ·√ (50
/ R_{p} ) = 0.01375

Since each side had length L = 0.6366 meters we have a segment length m · L = 35.01 mm

The tuning Capacitor will be relatively unaffected (no additional reactive load added) so CT = 13.45 pF

This method can also be used to provide an unbalanced impedance match simply by shifting the ground connection to the loop center - this also allows the tuning capacitor body (ground) to be connected to the same ground point as both will be at the same RF voltage = 0 V.

Butterfly tuning capacitors are best - the main RF current does not need to pass through potentially high loss contacts.

Although this central connection is at "ground" no balance is perfect. Do not touch the ground if you are supplying even small amounts of RF power to the loop antenna as you may get a nasty RF burn.

**If you try full power you may not return from your
experimentation. **

This is probably best tried by experiment. The coupling loop may benefit from an additional tuning capacitor and could be moved relative to the mail magnetic loop to optimize the final impedance match.

The actual impedance ratio will probably be related to a ratio of the two areas.

I hope my web chapter on magnetic loop antennas has been interesting and
useful in this * Component Universe*.
These antennas can be purchased already made but can equally be designed and
built by amateur radio enthusiasts. The offer a controlled tradeoff between
size, efficiency and operating bandwidth and suit applications where space is at
a premium

In this web chapter we started with an analysis of the half wave dipole antenna. This served as reference for other antenna formats. We explored the path loss equation for an isotropic to isotropic antenna (i.e. theoretically omni-directional) and defined the "aperture area" for such an antenna.

We then compared the path loss for a isotropic transmitting antenna to a receiving dipole antenna and found a gain of about 2.15 dB, attributable to its directivity.

The concept of "radiation resistance" was also presented, along with the concept of "skin depth" in a conductor with its associated increase in power loss at higher AC / RF frequencies. This, in comparison with an antenna's radiation resistance determines its efficiency. The radiation resistance of a half wave dipole was defined at 73.13 Ohms (to 4 significant figures)

The concept of reciprocity was invoked to use the isotropic antenna as a transmitting source and the dipole as a receiving directional detector. The path loss is unchanged by this swap-around. Since the magnetic loop antenna has similar directivity to a dipole, it was used in comparison to determine the radiation resistance of the magnetic loop antenna.

Some associated concepts were included in this somewhat meandering derivational path. For example, the energy density S over the surface of a sphere was related to magnetic field magnitude H at a distance. The "impedance of free space" = 377 Ohms was also included in these conversions.

A number of magnetic loop topologies were proposed, e.g. circular, octagonal, hexagonal, square and triangular. The directivity improvement resulting from electrostatic shielding was indicated, and a hollow tube method and a PCB method was shown.

Equations were presented for antenna efficiency, impedance and tuning capacitance. Operating Q and bandwidth estimates were also provided.

Some useful generalizations "fell out" in this process I guess

The AC resistance of a loop antenna depends on the total length of the conductor, its diameter and the DC resistivity of the material and also the actual RF frequency | |

The radiation resistance increases with loop area to a forth power and is highest for a circular geometry if the total length is constant. This need not represent a real world constraint as a square loop gave better results in one example I presented! | |

Antenna efficiency is best if the radiation resistance is much grater than AC conductor loss resistance (no surprise there!) | |

The magnetic loop antenna will exhibit series inductance. The circular loop will exhibit the highest inductance | |

The magnetic loop antenna will tend to have an extremely high operating Q and therefore a narrow frequency bandwidth |

Consistent with the phrase "you can't get something for nothing" is another "you cant get 100 % antenna efficiency with minimal space and high operating bandwidth". It seems that engineering a magnetic loop antenna is, once again, the art of compromise.

Some real world examples were included to allow the reader to check the equations before attempting their own magnetic loop antenna construction projects! Also I have shown three methods that could be used to match the loop impedance to a standard 50 ohm resistive value

Phew! I have to admit this web chapter took longer than I first thought. I hope the content will be of interest and helpful. I spent some time looking on the Internet for material as a reference to my derivations but there appears to be a number of competing formulas out there. Some of them have obvious errors, as do some programs (e.g. one program assumed that inductance didn't depend on the loop geometry)

In any case I hope my web chapter on magnetic loop antennas has provided a conceptual or near intuitive appreciation of the factors that determine the performance and design constraints and trade-offs that apply to them. I have avoided abstract Maxwell's equations in favor of simpler "first year university Physics" material for magnetic fields, flux, inductance, resistance etc.

What would really be fun? Have you got any old super-conductors in your junk box? If so, these would be great for making a zero loss magnetic loop antenna. If they really have zero resistance then it should be possible to make an 80 meter magnetic loop antenna fit through the eye of a needle.

However there may be a catch - the operating Q would need to be extreme and the bandwidth would tend to zero Hz as the magnetic loop antenna's size shrank to zero loop area as well!

It might pay to consider choosing your transmitting words sparingly and speak clearly and very slowly in your deepest possible voice.

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**© Ian R Scott 2007 - 2008**