This article describes a method to match short antennas to standard transmission lines. It's mainly intended for radio amateurs working on 137 kHz and 500 kHz bands, because when dealing with long waves, matching the antenna is always a problem. Traditional matchboxes designed for short waves do not have enough capacitance and inductance to work on low frequencies, and large commercially available variable capacitors and inductors are expensive and hard to find. Furthermore, installing an appropriate antenna, let's say a quarter wavelength tower, is out of reach for the common amateur. So, all long wave ham antennas are always too short and are a nightmare to match to a 50Ω line.

What one usually does is trying to run a wire as long as possible as high as possible. Any short antenna can be matched with this method, such as towers with or without capacitive hat, T antennas, inverted-L antennas, random wires, and so on, as long as they are small compared to the wavelength; shorter than a quarter wavelength in any case. Let's imagine one wire going vertically for 8 meters and than horizontally for 45, and take this antenna as an example, as shown in the figure below.

This is already a big antenna for a backyard, but really small compared to the 2200 m (600 m) wavelength of the 137 kHz (500 kHz) band.

The impedance of an antenna can be simulated with a NEC based program by "drawing" and approximating its structure and four our example the results are as follows:

Simulationfrequency |
Antennaimpedance |
Equivalentcapacitance |
Radiationresistance |

137 kHz | (0.7 – j3900) Ω | 298 pF | 0.01 Ω |

500 kHz | (1.4 – j960) Ω | 330 pF | 0.12 Ω |

The impedance has a very low resistance in series with a very high capacitive reactance which is typical of short antennas and the results look plausible.

If the antenna cannot be simulated, one can still build an antenna as high and as long as possible and then measure the static capacitance. Since we are far below the first resonance, the antenna will look like a capacitor, and with some luck we can measure its static capacitance with a capacitance meter. Please remark that unfortunately some capacitance meters use a too high frequency and gives inconsistent results: in order to be accurate, the frequency should be as low as possible, but definitively not higher than the design frequency (the antenna has to be short at the measuring frequency). Connecting a capacitance meter to the antenna taken as example gives 295 pF; not far from the simulated values and the static capacitance is a really good starting point.

Another method to measure the static capacitance of the antenna is to connect
an inductor of known value in parallel to its feed point and to measure the
resonance frequency.
Again, this frequency has to be as low as possible to make sure the antenna
is still "short" and a large inductor must be used.
Once the capacitance has been determined, the capacitive reactance can
be easely calculated with the formula X_{c} = 1/(2πfC).

The two above described measuring methods only determine the capacitive reactance but not the resistance. It's pretty hard to measure values as low as a few ohms while having kiloOhms of reactance in series, but knowing the exact real part of the impedance is not really important, since this will have little effect on the design of the match, as we will see later. Assuming just one or two Ohms is accurate enough.

So, the antenna looks like a big capacitor with a small resistor in series,
which we would like to transform in purely resistive impedance Z_{c},
let's say 50 Ω with a matching network.
If we plot the antenna impedance __Z___{a} on the complex plane,
we find a point very close to the vertical axis, much lower than the
horizontal axis. Z_{c} is of course located on the horizontal axis on
the right, as one can see in the figure below:

The goal is to reach Z_{c} starting from __Z___{a} as
shown by the dashed line. No component can achieve this in one step and we
will match the impedance in two steps.

We have several choices: if we connect a series capacitor C_{s} to
the antenna, a parallel inductor L_{p} or a step down transformer N:1
(more turns on the transmitter side), __Z___{a} will be
transformed in a higher impedance (we are moving away from the origin) and
this situation must be avoided because a high impedance means a high voltage,
much higher than the already high voltage typical of a short antenna.
Let's imagine to connect a 1 kW transmitter on our antenna at
137 kHz: with an impedance around 3900 Ω the voltage on the
feed point will be 2 kV!
This is already high enough to make some trouble and we definitively don't
want to deal with higher voltages in the matching network.

If we connect a parallel capacitor C_{p} or a step up transformer 1:N
(more turns on the antenna side), __Z___{a} will be transformed in
a lower impedance (we are moving towards the origin) and this situation must
be avoided because this reduces both the reactance and the resistance.
The resistance of a short antenna is already very low (in the Ohm range),
reducing it will dramatically increase the losses in the matching network and
this ahs to be avoided as well.

So, the only choice is to connect a series inductor L_{s} to the
antenna in order to reduce ("resonate away") the huge capacitive
reactance of the antenna without reducing the resistive part of the impedance.
In the figure this means moving vertically in the up direction.
It's good to remark that the best choice is also consistent with common sense,
as one would most probably think in matching a capacitive load with a series
inductor.

Now that we determined that the first element of the matching network is a
series inductor L_{s} connected to the antenna, we have to determine
the second element that will finish the match.

As it can be seen in the figure above, here we have three choices.
The first idea is to resonate away completely the capacitive reactance with
L_{s} and find the intermediate impedance __Z___{i2} which
lays in the real axis and than use a step down transformer N:1 (more turns on
the transmitter side) to reach Z_{c}.
This solution could work but requires a transformer which takes longer to
build, adds some difficulties to the design and requires more copper and has
therefore higher losses.

The second possibility is to over-compensate the capacitive reactance of the
antenna with a slightly larger series inductor L_{s} reaching point
__Z___{i3} which is slightly inductive and lays on the constant
admittance circle passing through Z_{c}.
Using a parallel capacitor C_{p} we can reach Z_{c}.
This solution can work as well and good capacitors have very little losses,
but the capacitance required is very large.
In order to match the antenna of our example at 137 kHz some 200 nF
are required: not easy to find and not easy to adjust as variable capacitors
are usually in the few 100 pF range.

The third and preferred option is to slightly under-compensate the capacitive
reactance of the antenna with a slightly smaller series inductor L_{s}
reaching point __Z___{i1} which is slightly capacitive and lays on
the constant admittance circle passing through Z_{c}.
Using a parallel inductor L_{p} we can reach Z_{c}.
This solution has several advantages: first it requires only a small parallel
inductor; it has the shortest path between __Z___{a} and
Z_{c} (meaning low losses) and is easier to build.

So, after having analyzer the different options we have determined that the impedance follows the path shown in the plot below:

With a series inductor L_{s} connected at the antenna port and a
parallel inductor L_{p} connecter at the transmitter port.
Our matching network looks like the following diagram:

Now, to determine the value of the two inductors, one can use the two equations above and with our antenna we get:

Frequency |
Antennaimpedance |
Seriesreactance |
Parallelreactance |
Seriesinductor |
Parallelinductor |

137 kHz | (0.7–j3900) Ω | 3894 Ω | 6.0 Ω | 4.5 mH | 6.9 μH |

500 kHz | (1.4–j960) Ω | 952 Ω | 8.5 Ω | 300 μH | 2.7 μH |

In order to simplify this operation the following calculator can be handy:

Just enter the working frequency f, the transmitter impedance Z_{c}
(usually 50 Ω), the antenna impedance __Z___{a} and hit
the button to calculate the required series and parallel inductances
L_{s} and L_{p}.

To calculate the required number of turns, an empirical formula that works well is described in the ARRL Handbook. Transformed for metric units is as follows:

Where d is the diameter in mm, l is the length in mm and N is the number of turns. The calculated inductance L is in μH. When designing a large coil, it's more practical to use the turns spacing s in mm than the total length. s is defined as l/N.

In order to simplify coil design, the following calculator can be handy:

Just enter the core diameter d, turns spacing s, the desired inductance L and hit the "Calculate N" button to computer the number of turns N and the overall coil length l. As verification, one can enter the number of turns N and hit the "Calculate L" button to compute the inductance L and the overall coil length l, based on the same diameter d and spacing s.

The values we have just calculated are approximate since are based on
simulations, on low precision measurements, on arbitrary assumptions and on a
simplified theoretical model.
Therefore real inductors require some adjustment and must be variable.
Changing the transmit frequency inside the band requires retuning as well.
Experience shows that only L_{s} needs adjustment, L_{p} is
less sensitive to frequency changes and once tuned can stay fixed for the
whole band.

There are several ways to make a variable inductance and the easiest is to change the number of turns. One could wind a coil with more turns than needed and than make connections to intermediate taps, as shown in the figure below.

There is always debate about shorting unused turns or not (the dashed line in the figure). Shorting them results in higher losses while leaving them open may result in high induced voltage and arcing. As we will discuss later, in an air core solenoid the coupling factor between the turns is low and doesn't work well as autotransformer; on the other hand if the unused coil section happens to be resonant at the working frequency high voltage may result. This is quite unlikely since the working frequency is low and fairly fixed; a resonance requires significant stray capacitance which is unlikely with a small coil section. It's therefore wiser to leave unused turns open reducing losses unless arcing takes place leaving no other choice than shorting them. If the unused turns have to be shorted it's better not to short them all but to leave a small gap to let the field "escape", as shown in the figure below:

Another way to have a variable inductor without involving too much mechanics is to build a variometer as summarized below:

The coil is split in half and a smaller coil is inserted inside the first one and connected in series. By turning the small coil inside it's possible to increase the total inductance (when both coils are aligned in phase) or to decrease it (when both coils are aligned with opposite phase). A variometer has higher losses because the wire is longer, but on the other hand it's easy to build and has no sliding contact.

The final solution is to use a tapped coil for L_{p} and a variometer
for L_{s}.
L_{p} is small and is not adjusted frequently, usually on once when
the antenna is first tuned, so a tapped coil is acceptable.
L_{s} needs to be adjusted every time the working frequency is moved,
even a few hundred Hz, making the variometer solution more practical because
of its continuously adjustable value.

A variometer is easy to build with readily available materials. Two plastic pipes of different diameters, insulated copper wire and a plastic rod for the shaft are the main components. The figure below shows the cross section and the way the coils are connected together.

Whit the formula (or the calculator) we discussed before, on determines the required number of turns on the external (stator) coil. This coil is split in two to let same space for the shaft to cross the core. A second coil, small enough to turn inside, is built and connected to the shaft and forms the rotor. All inductors are connected in series. Since the rotor only needs to move 180°, there is no need for sliding contact since flexible wire will do nicely the job.

It's hard to determine exactly the range of variation of the inductance of a
variometer, since this strongly depends on the coupling factor between stator
and rotor coil, but as a rule of thumb, the experience shows that the total
inductance will be something in-between
L_{stat}±L_{rot} and
L_{stat}±2L_{rot} and is a fair starting point.
The best swing is obtained when all the coils are symmetrical.
If the variation happens to be too small, one can still add a second layer
of turns on the rotor.

A question still remains: do we really need two separate inductors?
Well, theoretically yes, the two inductors need to be separate and not
coupled at all.
On the other hand, if we talk about air core solenoids, the answer may be yes
under certain conditions.
In an air core solenoid, the coupling between turns is fairly low and further
decreases as the distance between the turns increases.
It's therefore possible to wind on the same core both L_{s} and
L_{p}, adjacent to each other.
The whole matching network looks than as an autotransformer, but it's not (or
it's a very bad one), as explained later.
L_{p} can be adjusted by selecting a suitable tap on the main coil
and L_{s} can be adjusted by turning the variometer.
This works only if the coupling factor is low: meaning that there is no
magnetic core inside the coil, the turns are on only one layer in proximity
of L_{p} (coupling factor is much higher for two windings one inside
the other)the variometer inside L_{s} is far from L_{p}, and
L_{p} is small compared to L_{s}.

The final antenna tuner looks like the above diagram. This really looks like an autotransformer, but it's not really one. If it was an autotransformer it could not cancel the reactive component of the impedance. The antenna could not be matched, since it will just transform down the complex impedance of the antenna into another smaller complex impedance, something like (0.01–j50) Ω, but to match the antenna it's necessary to remove the reactance and to transform the resistance to 50 Ω. Just adjusting one parameter is not enough: the matching network must transform both.

On the other hand, if we imagine that part of the coil works like an inductance, cancelling out the reactive part of the antenna impedance, the rest of the coil cannot be an autotransformer either, because once the reactance has been resonated away, the impedance is real and very low. It would require a step-down transformer (more turns on the transmitter side) instead of a step-up transformer, which is how the antenna tuner looks like.

In order to check the coupling between the turns and the autotransformer effect, the setup shown in the figure below has been built:

The coil has a diameter of 200 mm, a length of 280 mm, 100 turns
and a tap on the 4^{th} turn.
The transformer turn ratio is therefore 1:25.
A voltage U_{p} of 1 V_{pp} has been applied on the 4
turns and the voltage across the whole 100 turns U_{s} has been
measured.
Several measurements were taken at different frequencies and show that for
this the voltage U_{s} is around 4 V_{pp} from a few kHz
until about 400 kHz.
The transformer ratio is therefore 1:4, very far from the expected 1:25,
meaning that only the first few turns are coupled and that the rest of the
coil is just an inductor.
In this case, if the frequency is increased to 463 kHz we have a
resonance and U_{s} goes up to 55 V_{pp}, much more than
what a 1:25 step-up transformer would do.
Further increasing the frequency lowers U_{s} down again to
4 V_{pp} until other resonances occurs.

It's best to think of this antenna tuner as two separate inductors, one in
parallel with the transmitter and the other in series with the antenna, but
because of the coupling between the two, the required values will be slightly
different, especially for the small L_{p}, but this it's not a big
issue, since, it's just a matter of finding the correct tap on the coil.

Now that we have determined how our antenna tuner will be and how many turn we'll have to wind, let's spend a few words about the construction. First of all, we should try to design a tuner with minimal losses, but this is not an easy task. The wire used to wind the coils will be responsible for the majority of the losses and choosing a large diameter wire is a good idea. "Litz" wire is definitively the best option, but this is almost impossible to find in large diameters. The second preferred option is enameled copper wire, which is easier to find but expensive. The third option is to use PVC insulated electrical installation wire. This wire is relatively cheap, easy to find and available in large diameters. There is one detail one should take care of: the copper must look "red" and not be tinned. Tinned wire has very high losses at radio frequency and should be avoided. PVC insulation is not the best option because of its poor tg(δ), but the frequency is low and can be accepted; Teflon isolation would be a better option.

Concerning the core, the best core is no core at all, but this won't be possible for such a big coil. Ceramic cores are very, but very hard to find in large sizes, so the third option is to use PVC pipes. These pipes are available in many large diameters, are cheap and are easy to work. They are a bit lossy, but again, the frequency is low and can be accepted for amateur use.

In short antennas, the resistive part of the impedance is highly dominated by losses. Trying to reduce losses is always a good idea. For example one could use a few (or more) buried radials to increase ground conductivity near the antenna (where the current is higher). Another idea is to use more than one aerial conductor running parallel at some distance apart: this will split the current and reduce the losses.

Using this antenna tuner is quite easy: since the topology of the matching network has only two parameters to adjust, there is no risk to find a "wrong" and lossy match.

If one happens to have an antenna analyzer that works at the desired
frequency, this is by far the best option to adjust the tuner.
If not, just connect the antenna and the transmitter (with a suitable SWR
meter) and select low power operation.
Adjust the variometer first is to find the position where the SWR is the
lowest, then move the tap between L_{s} and L_{p} to find the
best possible match (don't forget to switch the transmitter off before moving
the tap).
There is nothing more to do as this will usually allow match as good as 1.2:1
if there are enough taps.
If the match is not good enough, try to repeat these two operations again
maybe adding some more intermediate taps to the coil.
If frequency is changed, usually only the variometer needs to be adjusted
again and the tap can stay fixed for the whole band.

Please remark that the majority of SWR meters designed for short-waves operations do not work for long waves. One should check before with a 50 Ω dummy load if the instrument gives reasonable results at the desired frequency.

The above described matching networks have been built and tested.
The following pictures show a variometer designed for the 137 kHz band.
The main coil can be adjusted between 3.75 and 4.63 mH and matches the
antenna used as example at 137.3 kHz with an inductance of 4.39 mH.
We can therefore calculate the capacitance of the antenna in 306 pF.
In order to achieve the required inductance with less wire, some additional
turns have been added inside the main coil (on the opposite side of
L_{p}).
The rotor of the variometer is also a double layer coil.
The loaded Q of this matching unit (loaded with the antenna) is about 60.

The second variometer is designed for the 500 kHz band and can be adjusted from 214 to 598 μH by moving the variometer and selecting one of four taps on the upper side of the coil. It matches the same antenna at 500 kHz with 288 μH and we can calculate the capacitance at this new frequency in 352 pF. The loaded Q of this antenna tuner is about 80.

A simple way of matching an electrically short antenna to a standard 50 Ω transmission line has been explained. Both theoretical and practical aspects have been addressed. The goal is to help radio amateurs in tuning their (short) antenna with the hope that more and more hams will be interested in medium and long wave frequency bands.

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