No need to know how to solve Maxwell equations, but a bit of theory is necessary to follow the right track. You will found here a summary of the physical rules defining the performance of the air-core and ferrite loops used for VLF reception.

This page describes first the loop antenna theory, and then contains a study of the various parameters of a loop antenna equivalent electric circuit. A SPICE model of loop antenna is then proposed.

The section References lists additional sources of information for those willing to study more in depth the loop antennas.

For those of you who are totally adverse to maths, a synthesis at the end of each section sums up the main definitions and conclusions.

The aim of any receiving antenna is to convert an electromagnetic wave into a voltage. A magnetic loop antenna is a winding of copper wire around a frame (for air-core loops) or around ferromagnetic material (for ferrite loops).

A loop antenna is actually sensitive to the magnetic field and not the electric field (it is also called a *magnetic* loop). The Faraday's law of induction (or the law of electromagnetic induction) states that the induced electromotive force *e(t)* in a loop is directly proportional to the time rate of change of magnetic flux *Φ(t)* through the loop according to the relation:

(1)

where:
*h*is the induced electromotive force, in V_{e}*Φ*is the magnetic flux accross the circuit, in webers (Wb ≡ V·s)

This equation is valid only for electrically short antennas with respect to the wavelength of interest, which is true for the VLF frequencies.

Let's define:

- the vector of the magnetic induction,
unit vector**u** - the normal vector to the frame surface,
unit vector**n** - where
*θ*is the angle between the magnetic field lines and the frame normal.

The magnetic flux is a measure of quantity of magnetism through the antenna submitted to a given magnetic induction:

(2)

For a sinusoidal magnetic field, uniform over the surface * S*, the amplitude

(3)

(4)

where:
*B*_{0}is the strength of the magnetic induction, in tesla (T ≡ Wb/m² ≡ V·s/ m²)*ω*is the angular frequency of the inductive magnetic field, in rad·s^{-1}

Equation (2) gets then:

(5)

(6)

For a loop with *N* turns, each of them of area *A*, we have , so:

(7)

and (1) is then:

(8)

Since , the RMS value of the electromotive force at the output of the antenna is:

(9)

The efficiency of an antenna is defined through the "effective height" parameter *h _{e}* according to:

(10)

where:
*V*_{rms}is the RMS value of the voltage induced at the output of the antenna, in V*h*is the effective height, in m_{e}*E*_{rms}is the RMS value of the electric field, in V/m

From and since the electric field relates to the magnetic induction through , the effective height of a loop antenna is:

(11)

where:
*h*is the effective height, in m_{e}*N*is the number of turns*A*the area of each winding, in m²*λ*is the wavelength, in m*θ*is the angle between the magnetic field lines and the frame normal

With a ferrite antenna, the magnetic induction through the antenna is increased by a factor called the relative permeability of the medium. Equation (9) becomes:

(11 bis)

where:
*µ*is the relative permeability, dimensionless quantity._{r}*µ*is specific to the medium._{r}

Quick maths will show that, taken into account the wavelengths we are interested in (15km at a frequency of 20kHz), the effective height of a loop antenna is very small compared to the signal wavelength... It is anyway possible to obtain good results with reasonable antenna designs.

The inductive magnetic field across the loop depends on the magnetic component of the electro-magnetic wave (called *H* or magnetic field strength) and on the magnetic permeability of the loop core.

The inductive magnetic field *B* relates to the magnetic field strength *H* by:

(12)

where:
*B*is the RMS value of the magnetic induction, in tesla (T ≡ V·s/m²)*µ*is the permeability of vacuum, constant of 4π10_{0}^{-7}H/m*µ*is the relative permeability, dimensionless quantity._{r}*µ*is specific to the medium._{r}*H*is the RMS value of the magnetic field strength, in A/m

For an air-core loop, we have *µ _{r}* = 1.

For a ferrite loop, the field lines are collected by the ferromagnetic properties of the core, and

The antenna will then output a RMS voltage *V*_{rms} from a given RMS magnetic field strength *H*_{rms} value according to:

(13)

A loop antenna is actually sensitive to the magnetic field and not the electric field (it is also called a

The antenna performance is influenced by the number of turns and the area of each loop. For a ferrite antenna, the permeability of the core increases the output voltage.

Also, loop antennas have a toroidal reception pattern whose amplitude is maximal in the plane of the frame and zero (in theory) in the plane normal to the frame.

- an ideal voltage generator V with a value defined by equation (13)
- the radiation resistance
*R*_{rad} - the loop inductance
*L*_{loop} - the wire inductance
*L*_{wire} - the wire resistance
*R*_{dc} - the skin effect and proximity effect resistance
*R*_{ac} - the distributed capacitance
*C*_{loop}

The equations here below are, unless otherwize stated, valid for air-core loops and ferrite-core loops. For air-core loops, *µ _{r}*=1. For ferrite antennas,

(14)

where:
*R*_{rad}is the radiation resistance, in Ω*Z*is the impedance of free space (about 377Ω)_{0}*µ*is the relative permeability of the core_{r}*N*is the number of turns*A*is the area of each winding, in m²*λ*is the wavelength, in m

Since , the equation (12) becomes:

(15)

(16)

where:
*L*_{loop}is the wiring inductance, in H*N*is the number of turns*w*is the frame side length, in m*l*is the length of the winding, in m

(16 bis)

where:
*L*_{loop}is the wiring inductance, in H*µ*is the permeability of vacuum, constant of 4π10_{0}^{-7}H/m*µ*is the relative permeability, dimensionless quantity._{r}*µ*is specific to the medium._{r}*N*is the number of turns*A*is the area of each winding, in m²*l*is the length of the winding, in m

(17)

where:
*L*_{wire}is the wire inductance, in H.*N*is the number of turns*w*is the frame side length, in m. Total wire length is 4*Nw*.*d*is the wire diameter, in m.

(18)

where:
*R*_{dc}is the wire resistance, in Ω*N*is the number of turns*w*is the frame side length, in m. Total wire length is 4*Nw*.*d*is the wire diameter, in m. Wire cross section is π*d*²/4.*ρ*is the copper resistivity, in Ω·m.*ρ*=16.78nΩ·m.

This resistance is a source of thermal white noise according to the Johnson-Nyquist formula that expresses the voltage spectral density of the thermal noise by: , in V/Hz^{½}.

*k* is the Boltzmann constant (1.38·10^{-23} J/K), *T* is the absolute temperature in kelvins, and *R* is the resistance in ohms.

The proximity effect can be more important than the skin effect, but it is much more complex to model since it is influenced by the windings geometry. It will not be calculated here.

The skin effect resistance is:

(19)

where:
*R*_{ac}is the skin effect resistance, in Ω*N*is the number of turns*w*is the frame side length, in m. Total wire length 4*Nw*.*d*is the wire diameter, in m. Wire perimeter is π*d*.*µ*is the permeability of free space (4π10_{0}^{-7}H/m)*f*is the frequency, in Hz*ρ*is the copper resistivity, in Ω·m.*ρ*=16.78nΩ·m.

(20)

where:
*C*_{loop}is the distributed capacitance of a square frame, in F*w*is the frame side length, in m*l*is the winding length, in m

This formula is also valid for ferrite antennas, by replacing *w* by the diameter of the windings.

The model parameters are:

*R*_{rad}=12.5pΩ at 10kHz, 0.125µΩ at 100kHz and 1.25mΩ at 1MHz...

This parameter can be disregarded.*L*_{loop}=5.77mH*L*_{wire}=318µH*R*_{dc}=15.6Ω*R*_{ac}=2.43Ω at 10kHz, 7.67Ω at 100kHz and 24.3Ω at 1MHz.

The skin effect has a significant contribution and adds to the ohmic resistance*R*_{dc}of the wire. Moreover, this calculation does not take into account the proximity effect that is probably more important than the skin effect since the windings are very close.*C*_{loop}=129pF

- an ideal voltage generator. The voltage is proportional to the magnetic field
- an inductance, mainly resulting from the windings inductance, but also from the inductance of the wire itself

- a radiation resistance corresponding to the "losses" resulting from the conversion of the electromagnetic energy to electric energy. Usually, it can be disregarded.
- the wire resistance and the high frequency resistance resulting from skin and proximity effects. The high frequency resistance is closely related to the antenna geometry and is usually #8801alent to or higher than the dc resistance.
- a capacity, distributed among the windings. It can be ignored for a first approximation.

The ideal voltage generator *B*_{loop} implements equation (13). A Laplace transform is used to model the frequency dependency of the voltage. *V*_{field} is a voltage generator that emulates an electric field of 1mV/m.

The schematics implements the various elements of the antenna electric model and the associated equations.

The antenna is connected to a resistive load *R*_{load}. Simulations have been done with loads of 1mΩ (close circuit), 50Ω (receiver input impedance), 10kΩ (soundcard input impedance) and 100MΩ (open circuit).

If we have a look to the open circuit plot, we have a linear response with a rising slope of 20dB per decade corresponding to the evolution with the frequency of *B*_{loop}.
From about 100kHz, the circuit starts to resonate. The resonance frequency obeys to equation (a) below:

(21)

Here, with *L*=*L*_{loop}+*L*_{wire}=6.09mH, and *C*=*C*_{loop}=129pF, we have *f*_{0}=180kHz.

Above, the impedance of the distributed capacitance *C*_{loop} gets sufficiently low to reduce the output voltage.

- a high-pass filter by (
*L*_{loop}+*L*_{wire}) and (*R*_{load}+*R*_{dc}+*R*_{ac}). - a low-pass filter by
*C*_{loop}and*R*_{load}.

By adding a capacitor *C*_{tune} to the output of the antenna, it is possible to tune it on a given frequency. Equation (19 a) can be used to calculate its capacitance.

For example, if we want to tune our antenna on NAA, 24kHz, knowing the inductance (*L*_{loop}+*L*_{wire}=6.09mH), we have *C*_{tune}+*C*_{loop}=7.22nF, that is *C*_{tune}=7.09nF.

Equation (20) gives the quality factor of the antenna:

(22)

Since

*Q* can also be determined by measuring the bandpass Δ*f* at -3dB and by using the following formula:

(23)

The measurement gives 509Hz, that is a measured Q of 47.2, in accordance with the calculated value.

A station on a close frequency (for instance DHO38 on 23.4kHz) will be strongly attenuated (by about 8dB in this case).

The real antenna quality factor will be lower since the proximity effect is not accounted for in this simulation.

With a closed circuit, the frequency response is flat. The distributed capacitance has no detrimental effect since it is shorted.

At low frequencies, the antenna acts as a resistance. At high frequencies, this resistance gets negligible against the impedance of the inductance. This impedance (*jLω*) is proportional to the frequency. Since the voltage generator is also proportional to the frequency—cf. equation (13)—, the current no longer depends on the frequency.

The transition between the resistive domain (low frequencies) and the inductive domain (high frequencies), is done according to equation (19 b). The value is slightly above 400Hz in our example.

By using an operational amplifier as a current-to-voltage converter, it is possible to have an almost null impedance at the antenna output. In this specific case, the use of a transresistance of 100kΩ will give a linear reqponse of 1V/V·m^{-1} in the whole VLF band.

By using a non-null impedance, the antenna linearity is reduced, and has finally a resonant behaviour such as the one observed with the voltage measurements.

- if one is interested in a single frequency, it is possible to tune the antenna by adding a capacitor. The antenna must then be connected to an high impedance in order to maximize the quality factor.
- if the antenna is intended to be used over a wide band, it it necessary to connect it to a short impedance and to work with its current. An operational amplifier can be used as a current-to-voltage converter. The linearity is then only limited in the low frequencies by the antenna resistance.

SID monitoring station by Lionel LOUDET is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. |

Last Update: 11 Jun 2013 |

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