## Effective Length of Receiving Magnetic Loop AntennaPart 1: Single-turn Wire Loop Antenna

How to compare the efficiency of different receiving antennas? When comparing electric receiving antennas with each other the effective length of an antenna is used as a coefficient of ratio between the voltage induced on the open-circuited terminals of the antenna and the power of the electromagnetic wave at the receiving point. And note that effective length and geometrical length of electric antennas often differ. But how to compare the efficiency of electric and magnetic receiving antennas?

### Effective length of the receiving antenna

In its physical meaning, the effective length of the receiving antenna determines the signal voltage on the open-circuited terminals of the receiving antenna depend on the intensity of the electromagnetic wave at the receiving point, expressed in terms of the strength of electric field created by this wave at this point:

(1)

where is:

UOUT – the signal voltage on the open-circuited terminals of the receiving antenna, V;
heff – the effective length of the receiving antenna, m;
E – the strength of the radiated electric field at the receiving point, V/m.

Thus, the effective length is a parameter of receiving antennas, by which their efficiency in converting radio wave energy into the voltage of the received signal is compared.

When radiating in a vacuum, an electromagnetic wave creates an electromagnetic field at the receiving point, the electric E and magnetic H components of which are related by the expression:

(2)

where is: – vacuum permittivity; – vacuum permeability.

Therefore, the effective length can characterize not only electric, but also magnetic receiving antennas, comparing them in terms of efficiency both with each other and with electric receiving antennas, regardless of their design: whether they are active (with a built-in antenna amplifier that requires power supply) or passive (no power required). After all, the purpose of any receiving antenna is to convert the energy of an electromagnetic wave into an electrical signal applied to the input of a radio receiver.

### Single-turn loop in an electromagnetic field

Let us take a single-turn wire loop as the simplest receiving magnetic antenna (see Fig.1). When the length of an electromagnetic wave, that impinges on the receiving loop antenna, is much greater than linear dimensions of it (such antenna is classified as electrically small), you can use the simplest lows of electromagnetic induction to determine the emf on the open-circuited terminals of such the loop antenna.

According to Faraday’s law of induction, the electromotive force (emf) in a wire loop, induced in it by a changing magnetic field, is equal to the rate of change of the magnetic field flux through the surface enclosed by this loop:

(3)

The flux of the magnetic field through the surface S, enclosed by the loop, is equal to the scalar product of the magnetic induction vector and the vector , that directed perpendicular to the surface, enclosed by the loop, and the modulus of which is equal to the area of this surface:

(4)

where is: – magnetic induction produced by an electromagnetic wave at the radio receiving point, T.

We will not consider here the directional properties of magnetic antennas in order to avoid exercises in trigonometry. Let the Poynting vector of an electromagnetic wave lie in the same plane as the single-turn loop under consideration, that is, the loop is located in a plane coinciding with the direction of propagation of the electromagnetic wave, in other words, it is turned in the direction of the maximum level of the received signal. In this case, the vector of the magnetic component of the electromagnetic field at any time is located in a plane perpendicular to the plane of the loop. Let the projection of the vector H onto the normal to the plane of the loop change according to the harmonic law:

(5)

where is:

Ha – amplitude of change in magnetic field strength, A/m; – signal angular frequency, rad/s.

Then the flux of the magnetic field through the surface S enclosed by the loop will change with the same regularity:

(6)

Such a magnetic field flux will induce an emf on the open-circuited terminals of a single-turn wire loop according to the expression: (7)

Thus, under the harmonically changing magnetic field strength (5) produced by an electromagnetic wave at the receiving point, the signal voltage on the open-circuited terminals of a single-turn wire loop antenna, equal in magnitude to the emf induced in the loop, will also change according to the harmonic law (7), with the same frequency and amplitude equal to:

(8)

Since the magnitude of the signal voltage on the open-circuited terminals of a receiving antenna is also the product (1) of the effective length of the antenna and the electric field strength produced by the electromagnetic wave at the receiving point, then the right parts of expressions (1) and (8) can be equated to each other:

(9)

where is:

Ea — amplitude of change in electric field strength at the radio receiving point, V/m.

Taking into account the relationship between the electric and magnetic fields at the receiving point (2), from which it follows that:

(10)

Since the vacuum permittivity and the vacuum permeability are related to the speed of light in vacuum с by the equation:

(11)

the following equation can be taken accordingly:

(12)

and the expression for the effective length of the receiving magnetic single-turn wire loop antenna will be as follows:

where is:

c = 299 792 458 m/s – speed of light in vacuum.

So, the effective length heff of a single-turn loop antenna is proportional to the area S of the surface enclosed by the loop and the signal frequency :

(13)