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^{Radiation pattern of the loop antenna combined with sense antenna}

The null is now opposite to the maximum, while they where at 90° before adding the sense element. When the null direction has been determined, the NDB direction is the exact opposite.

^{Radiation pattern of the loop antenna combined with sense antenna}

In the figure above, the red curve denotes a negative polarity (a 180° phase shift). Both the loop and sense antennas have a power peak of 0dB, they are equal. The null is now opposite to the maximum, and the maximum is +3dB, meaning twice the power of the loop alone. It is easy to determine the null direction. When the null direction has been determined, the NDB direction is the exact opposite.

That's correct. For the antenna pattern shown in the question, the angle between direction of nulls and peaks is 90°. When sensing a null (or a peak) there are two possible and opposite directions for the beacon. This ambiguity has to be removed. This could be done by triangulation, with a second reading of the angle of arrival after the receiver has moved a bit, but in the modern ADF antenna this is not required, a sense antenna is added to the loop antenna to change the radiation pattern (i.e. the sensitivity vs. the bearing):

The reason we prefer to sense the null direction is the signal received is fading at the larger rate near the null than it is increasing in intensity near the peak. The change being more pronounced near the null, sensing the null is easier and gives a better accuracy.

Electromagnetic waves simply

Contrary to what is often thought, the classical high-level principles behind EM waves, which explain how the loop antenna can work, are quite easy to summarize.

The very common representation of an EM wave:

^{Electric and magnetic components of an EM wave. Source}

A wave propagates in 3D space and its electric and magnetic fields are not dissociable, they are a single electromagnetic field which energy is carried by photons (though according to Richard Feynman, EM sources produce not physical particles or waves, but wave-like “probability amplitudes” that propagate at c in space. QED! Well... let's stick to good ol' photons!).

A NDB transmitter is an electric generator, generating an alternating current in a basic vertical conductor which length is tweaked to make it resonant at the frequency used (it's a dipole antenna). Three laws are then governing what happens:

• Ampère-Maxwell's law says 1/ the vertical current creates a circular magnetic field around the antenna (Ampère's initial idea). The field intensity follows the variation of the current, 2/ the same happens with an electric field instead of a current (Maxwell addition).

• Faraday's law of induction says a varying magnetic field creates an electric field. This field opposes the magnetic flux which created it according to Lenz's law. Lenz's law is a kind of reaction principle.

So AC current creates a variable magnetic field. The energy of the magnetic field comes from the current in the antenna.

As the magnetic field is variable, using the two laws above alternately, we see when a magnetic field varies a variable electric field is generated. When an electric field varies a variable magnetic field is generated.

The energy of the electric field comes from the magnetic field, as postulated in Lenz's law: The electric field opposes the magnetic field changes, therefore it is a force, aka the back electromotive force (back-emf).

What's great is an actual current (electrons) is required only to generate the initial magnetic field, the subsequent magnetic fields are due to electric fields, not depending on a conductor and/or electrons to expand:

^{Electric and magnetic fields generated from a point of a vertical antenna in a given direction}

From left to right: A current is created in the transmitter antenna. If we focus on any point of the antenna, the local current creates a magnetic field around the point. This field induces now vertical loops of electric field, each loop in turn creates a new magnetic field, and so on.

The two fields, normal to each other, are the two inseparable aspects of the electromagnetic field. They exist at the same time, and the repeating process leads to their propagation as wave. As fields have no mass, they can travel at the speed of light.

This process occurs in 3D space, we see concentric shells of electric field radiating from the whole length of the antenna, and between these shells the magnetic field linking the shells:

^{Still image of electric and magnetic fields. Source (animation).}

The electric field at the top and bottom of the antenna is missing, we can predict a "cone of silence" at the vertical of some radio aids using vertical antennas.

  The wave crosses conductors while expanding. A conductor acts as a reception antenna. Depending on the conductor ability to couple more or less strongly with the electric and/or magnetic fields, a given portion of the energy from the fields is converted back into current, by virtue of the same reversible laws.

The ADF receiver uses a small-loop antenna (the circumference is small compared to the wavelength), which principle is to sense the magnetic field of the wave, the antenna is actually a coil.

Let's look at an everyday life example: Hearing a sound without seeing the source doesn't prevent us to determine its direction. Unless the sound is coming from straight ahead or straight behind, one ear receives the sound first, the same for any reflection, our brain does the rest. Moreover our ear pinna is not symmetrical, this also allows to separate a front sound from a back sound at the same distance, or one low from one high. This localization is irrespective of the mean intensity of the sounds.

Likewise the loop antenna reacts to the combined result of instantaneous amplitudes at all points of the antenna. According to the law of induction, two points of the loop receiving the signal with a phase difference due to the difference of distance from the transmitter (left side below) will be at a different potential, creating a current in the conductor between these points:

• Note the antenna is actually much smaller than the wavelength and the phase difference is very small. But the loop antenna is a coil, it is easy to have multiple turns of wire, each turn "collecting" an additional amount of the magnetic field. However as the increasing impedance cancels partly this gain, a reasonable compromise has to be found.

"LW/SW receiver" with a ferrite coil antenna (a coil which inductance is reinforced by a ferrite rod) must be oriented in the direction of the station to maximize reception.

The gain is provided in decibels, which is the logarithm of the ratio between the intensity in the direction considered, and the intensity in the direction of the maximum. By principle the maximum is 0dB (100%) and other values are negative (less than 100%).

There is a problem with the loop antenna: It has a symmetrical radiation pattern, when a null is found, there are still two possible opposite directions for the transmitter. The solution is to use an additional element, the sense antenna, which is an antenna with the same sensitivity regardless of the bearing. It is said to be omnidirectional and is usually coupling with the electric field.

A wave creates some current in each antenna. In the loop it is proportional to the magnetic field difference at the two sides of the loop. As the sign of the difference is inverted when the direction of the signal is changed by 180°, we should be able to detect whether the source is on the front or on the rear with the loop alone. But the magnetic field amplitude is actually a sinusoid and the sign of the current is also inverted after half a period. This current alone cannot indicate the signal direction.

The sense antenna is omnidirectional, meaning neither the sign nor the "mean" magnitude of the current change with the source incident angle. In addition, the instantaneous changes in the loop and in the sense antenna are perfectly in sync since they are created by the same wave (some trivial correction of the phase may be required, depending on the design).

Thus if we subtract the two currents, the instantaneous change (the $\small \sin 2 \pi f$ factor) is cancelled from the result, and the result is sensitive to the signal bearing. The resulting pattern has a heart shape (cardioid):

Back to the audio field, this antenna system is very similar to the sound engineering technique called MS stereo using one mono (mid) and one stereo (sides) microphones, though this technique is valid for only a 180° bearing range.

^Source

That's correct. For the antenna pattern shown in the question, the angle between direction of nulls and peaks is 90°. When sensing a null (or a peak) there are two possible and opposite directions for the beacon. This ambiguity has to be removed. This could be done by triangulation, with a second reading of the angle of arrival after the receiver has moved a bit, but in the modern ADF antenna this is not required, a sense antenna is added to the loop antenna to change the radiation pattern (i.e. the sensitivity vs. the transmitter relative bearing):

The reason we prefer to sense the null direction is the signal received is fading at the larger rate near the null than it is increasing in intensity near the peak. The change being sheer near the null, sensing the null is easier and gives a better accuracy.

Details follow. I added a section about electromagnetic waves details at the end of this answer for those who are interested.

The ADF receiver uses a small-loop antenna (the circumference is small compared to the wavelength), which principle is to sense the magnetic field of the wave, the antenna is actually a coil with a single turn, ensuring it is unsensitive to the electric field.

The loop antenna reacts to the combined result of instantaneous magnetic field at all points of the antenna. According to the law of induction, two points of the loop receiving the signal with a phase difference due to the difference of distance from the transmitter (left side below) will be at a different potential, creating a current in the conductor between these points:

The gain is provided in decibels, which is a handy way to express a ratio, here the ratio between the intensity in the direction considered, and the maximum intensity found when varying the direction. A decibel value is the logarithm of the ratio, a logarithmic scale being more usable than a linear one when the values vary by several orders of magnitude. By principle the maximum is 0dB (100%, a ratio of 1/1) and other values are negative (less than 100%, a ratio smaller than 1/1).

There is a problem with the loop antenna: It has a symmetrical radiation pattern, when a null is found, there are still two possible opposite bearings for the transmitter. The solution is to use an additional antenna, the sense antenna, which is an antenna with the same sensitivity regardless of the bearing. It is said to be omnidirectional and is usually coupling with the electric field.

A wave creates some current in each antenna which instantaneous amplitude is $A \sin 2 \pi f t$, where $A$ is the maximum amplitude.

• In the loop $A$ depends on the antenna gain, thus varies with the transmitter bearing.

• In the sense antenna $A$ is constant, regardless of the transmitter bearing. This $A$ in the sense antenna is made equal to the $A$ in the loop antenna by a trivial adjustment.

Now the polarity of the signals:

• In the loop the sign is positive when the bearing is in [270..(0)..90] and negative in [90..(180)..270].

• In the sense antenna the sign is always positive.

The polarity in each antenna indeed depends on how we measure the signal and can be inverted, but the important point is the polarity is inverted in the loop and constant for the sense antenna. Whatever they the actual signs, we get the same result if we sum the two signals when the loop is aligned on the transmitter bearing and $A$ is maximum:

• On one side $A \sin 2 \pi f t + A \sin 2 \pi f t = 2 A \sin 2 \pi f t$

• On the other side $-A \sin 2 \pi f t + A \sin 2 \pi f t = 0$

The 180° ambiguity has been removed. For other bearings, when the loop is not aligned with the transmitter, $A$ for the loop won't have the maximum value and the result will be between 0 and $2A$, a heart shape curve (cardioid):

^Source

Electromagnetic waves simply

The very common representation of an EM wave:

^{Electric and magnetic components of an EM wave. Source}

A radio wave, synonymous for an electromagnetic field, propagates in 3D space and its electric and magnetic fields are not dissociable, they are the two sides of the same coin. The classical high-level principles behind EM waves, explaining how the loop antenna can work, are quite easy to summarize, at least in the far field, the area where they actually propagate as we know:

• When a current flows in a wire, a magnetic field exists around the wire as well as an electric field. If these fields vary they creates in turn another varying magnetic/electric pair, and so on. The wave propagation is just the result of electric and magnetic fields expanding from the wire.

• The condition is the first magnetic field must vary, which implies the current in the wire has to be variable, it cannot be a constant DC.

Said otherwise feeding a wire with AC automatically creates a wave. Fortunately if the wire is not tweaked to resonate at the AC frequency, the wave energy is tiny. We can use electric devices on 50/60Hz without creating significant waves because a resonant antenna for this frequency has to be 6 000km long.

Electric current in an open circuit

A NDB transmitter is an AC current generator feeding, e.g., a basic vertical dipole which length is tweaked to make it resonant, like a piano string is tweaked to produce a specific sound. You may wonder how electrons can circulate in an open circuit. The truth is they don't exactly circulate, they oscillate around a fixed position:

• It's well known currents are created by the drift of electrons, but what is actually the velocity of this drift? Surprisingly it is very low: In a DC flashlight it takes about 24 hours for electrons to travel the 10 cm wire connecting the battery to the LED. It's exactly like water traveling from the pumping station to the kitchen.

• An antenna can be seen as a capacitor, mutual capacitance effects exist between the two elements of the antenna. Electrical loads can move on a small distance without leaving the conductor, increasing a bit the local loads density, said otherwise the wire can store some energy at different locations.

• If the antenna is fed with AC, electrical loads are moved back to their initial location after half a cycle, and are pushed in the opposite direction during the next half cycle, their mean position is constant. Electrons oscillate in the open circuit without leaving it, still an AC current is created.

Wave propagation

When a current is created in an antenna, the propagation of the wave can be described by the laws of electromagnetism:

• Ampère-Maxwell's law says 1/ a sinusoidal current creates a magnetic field around the antenna (Ampère's initial idea). The field intensity follows the variation of the current, 2/ the same happens with an electric field instead of a current (Maxwell addition).

• Faraday's law of induction says a varying magnetic field creates an electric field. This field opposes the magnetic flux which created it according to Lenz's law. Lenz's law is to electric field what inertia is to mass (and therefore links both fields like mass is linked to inertia).

When a magnetic field varies a variable electric field is generated and when an electric field varies a variable magnetic field is generated. This leads to an infinite process:

Source: Wikipedia.

The energy of the electric field comes from the magnetic field, as postulated in Lenz's law: The electric field opposes the magnetic field changes, therefore it is a force, aka the back electromotive force (back-emf).

The wire (antenna) is only required to accelerate electrons and generate the initial magnetic field. The subsequent expanding magnetic fields are strictly due to electric fields, electrons and conductors play no role:

The two fields, normal to each other, are the two inseparable aspects of the electromagnetic field. They exist at the same time, and the repeating process leads to their propagation as wave. As fields have no mass, they can travel at the speed of light.

The wave crosses conductors while expanding. A conductor acts as a reception antenna. Depending on the conductor ability to couple more or less strongly with the electric and/or magnetic fields, a given portion of the energy from the fields is converted back into current, by virtue of the same reversible laws.

That's correct. For the antenna pattern shown in the question, the angle between direction of nulls and peaks is 90°. When sensing a null (or a peak) there are two possible and opposite directions for the beacon. This ambiguity has to be removed. This could be done by triangulation, with a second reading of the angle of arrival after the receiver has moved a bit, but in the modern ADF antenna this is not required, a sense antenna is added to the loop antenna to change the radiation pattern (i.e. the sensitivity vs. the azimuth):

There is a problem with the loop antenna: It has a symmetrical radiation pattern, when a null is found, there are still two possible opposite directions for the transmitter. The solution is to use an additional element, the sense antenna, which is an antenna with the same sensitivity regardless of the azimuth. It is said to be omnidirectional and is usually usually coupling with the electric field. The two antennas can be combined in such a way the resulting pattern has a heart shape (cardioid):

One lobe of the loop antenna (here in blue) adds with the sense antenna, the other is subtracted from the sense antenna, creating a gain dissymmetry.

The ambiguity isn't possible anymore. The null is now opposite to the maximum, while they where at 90° before adding the sense element. When the null direction has been determined, the NDB direction is the exact opposite.

Back to the audio field, this antenna system is very similar to the sound engineering technique called MS stereo using one mono (mid) and one stereo (sides) microphones, though this technique is valid for only a 180° azimuth range.

That's correct. For the antenna pattern shown in the question, the angle between direction of nulls and peaks is 90°. When sensing a null (or a peak) there are two possible and opposite directions for the beacon. This ambiguity has to be removed. This could be done by triangulation, with a second reading of the angle of arrival after the receiver has moved a bit, but in the modern ADF antenna this is not required, a sense antenna is added to the loop antenna to change the radiation pattern (i.e. the sensitivity vs. the bearing):

There is a problem with the loop antenna: It has a symmetrical radiation pattern, when a null is found, there are still two possible opposite directions for the transmitter. The solution is to use an additional element, the sense antenna, which is an antenna with the same sensitivity regardless of the bearing. It is said to be omnidirectional and is usually coupling with the electric field.

A wave creates some current in each antenna. In the loop it is proportional to the magnetic field difference at the two sides of the loop. As the sign of the difference is inverted when the direction of the signal is changed by 180°, we should be able to detect whether the source is on the front or on the rear with the loop alone. But the magnetic field amplitude is actually a sinusoid and the sign of the current is also inverted after half a period. This current alone cannot indicate the signal direction.

The sense antenna is omnidirectional, meaning neither the sign nor the "mean" magnitude of the current change with the source incident angle. In addition, the instantaneous changes in the loop and in the sense antenna are perfectly in sync since they are created by the same wave (some trivial correction of the phase may be required, depending on the design).

Thus if we subtract the two currents, the instantaneous change (the $\small \sin 2 \pi f$ factor) is cancelled from the result, and the result is sensitive to the signal bearing. The resulting pattern has a heart shape (cardioid):

The null is now opposite to the maximum, while they where at 90° before adding the sense element. When the null direction has been determined, the NDB direction is the exact opposite.

Back to the audio field, this antenna system is very similar to the sound engineering technique called MS stereo using one mono (mid) and one stereo (sides) microphones, though this technique is valid for only a 180° bearing range.