Theory Of B Field Antenna Radiation

(Still in progress - checking the maths still)

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1. B Field Radiation From The Loop Antenna

This analysis will consider a square loop antenna, as an approximation to a circular loop, since square (or rectangular) antennas are often more convenient to fabricate, especially on a PCB substrate. Each side segment will have an equal length L and an ac current i_ac will be flowing in the loop, as shown below. The area A of the loop will therefore be  .

We first need to calculate the magnitude of the magnetic field B at a distance d from the loop, given that this current i_ac is circulating in the loop. Providing d >> L the magnetic field will be constant over a circumference defined by the distance d, in the plane of the loop. We will consider the simplest case where the field is estimated at a position perpendicular to one side of the loop.

We note that the loop’ side segments a and c are perpendicular to the direction defined by r, and therefore contribute magnetic B field components at position R. The other side segments b and d are not perpendicular, and so their magnetic contribution diminishes as . Also note that the current i_ac flowing in side a is equal and opposite to that flowing in side segment c, so that the total magnetic field at position R is equal to the difference caused by this opposition. From Biot’s law applied to a short wire length,

     ...(1)

It follows that the combined magnetic field B at point Y must be the difference

     ...(2)

Given that we will assume that the loop dimensions L are small compared to the remote point R at distance r, we will approximate . This suggests that

     ...(3)

2. Energy Density Associated With The Loop Antenna's Magnetic Field     

From electromagnetic theory we have a relationship between radiated energy density u and the corresponding electrostatic and magnetic fields E and B given by,

    ...(4)

 This implies that the magnetic energy density u_B component in the plane of the loop at a distance r caused by i_ac is,

     ...(5)

3. The Concept Of Magnetic Intensity (Power Associated With The Magnetic Component)

This energy occupies a volume in space determined by an area A and a forward distance determined by the speed of light c at a unity time dt. The magnetic intensity s_B is equal to the rate at which this energy passes through this imaginary volume in space, i.e.

     ...(6)

4. Finding The Total Magnetic Intensity Over A Closed Surface

The magnetic field strength B_f at an angular position f above or below the plane of the loop will have a cosine reduction given by

     ...(7)

It follows immediately that the magnetic intensity s_B must have a cosine-squared variation

             ...(8)

Expressed graphically as,

The total magnetic intensity S_B generated by the loop can be found by integrating s_B over the surface of a sphere with radius r. In free space, the total energy must of course remain constant for any value of R, based on conservation of energy. The magnetic intensity will reduce with distance, but the combined magnetic and electrostatic intensity integrated over any closed surface must remain constant. The surface area of a sphere is

...(9)

 If the magnetic intensity s_B was constant over the surface of this imaginary sphere, then the total magnetic energy, or power E_B attributed to this magnetic component would just be 

     ...(10)

 We note that the magnetic intensity is constant in the plane of the loop but falls of at the square of the cosine of the angle above and below this plane. If we performed the integration we would find that the actual integrated energy would become one half the value compared to a constant energy density situation. We will therefore propose the total energy associated with the magnetic field contribution from the loop to be one half the previous estimate

     ...(11)

5. Defining The Transition Zone From "Near Field" To "Far Field"

 Note: This equation must appear somewhat contradictory to the idea that the total energy remains constant, as it appears to be strongly dependant on the radius r. Bear in mind, however, that the total energy must remain constant, and we are only considering the magnetic component E_B. In the far field, one half this magnetic energy will be converted to electrostatic energy, caused by displacement currents. So where does this transition from “near field” to “far field” occur?

 Previous analysis of radiation from a standard dipole antenna suggest that two components are generated, one called a radiation-field component and the other an induction-field component. These fall off with distance r but at different rates, and are equal when . The magnetic loop generates an induction field component, which becomes associated with a radiation field component. In analogy with the dipole, we will assume the two fields become equal at the same distance. (The dipole analysis is also based on a magnetic interpretation). Substituting this value for r in equation ( 11 ) then predicts

     ...(12)

 It will now be useful to consider the loop dimensions in terms of fractional wavelength, as most loop antennae will be small compared to a wavelength. Let us define  where  represents this fractional wavelength. Substituting

     ...(13)

6. Estimating The Radiation Resistance Of The Loop Antenna

We can now observe a close relationship with the energy dissipated in a resistor when a current is flowing, i.e. . We will therefore define this “radiation resistance” R_a as

     ...(14)

Since energy must be conserved, the energy dissipated in the radiation resistance of the loop must equal the total energy flowing through the surface of this imaginary sphere. If we now substitute the values for the constants we finally derive a very simple equation for the radiation resistance of the loop antenna

     ...(15)

 We can now make some practical predictions. Let’s consider a VHF frequency of 150 MHz, which has a wavelength of 2 meters. A very large loop might be 1/10 this wavelength, i.e. 20 cm by 20 cm. A very small loop might be ten times smaller again. Lets estimate the radiation resistance for both extremes and for a loop of intermediate dimensions

Radiation Resistance For Loop Antennae at 150 MHz

 We now see the Achilles heel of the loop antenna. As the fractional wavelength reduces, the radiation resistance quickly plummets to zero due to a forth power term. In this case the ohmic losses of the loop soon dominate. 

8. Estimating The Loop Antenna Radiation Efficiency 

As an estimation guide, the wire or track segments that make up the loop will have an inductance per unit length denoted by

      ...(16)

 The reactance per unit length will be frequency dependant, i.e.

     ...(17)

Since we defined  it follows that

...(18)

 In the absence of direct measurement data, a well constructed small dimensioned coil can easily exhibit a Q factor as high as 200 at VHF, and the ohmic series loss associated with the coil, or in this case the inductive loop antenna will equal the reactance divided by this Q,

       ...(19)

 The radiation efficiency for small loop structures will therefore be determined by the ratio of radiation resistance to total loop resistance, which can be expressed in dB as

      ...(20)

 Substituting equations () and ()

     ...(21)

We can now approximately predict general loop antenna radiation efficiency, using the previous example dimensions

 Antenna Efficiency For Loop Antennae at 150 MHz Expressed in dB

 We immediately observe that the expected radiation efficiency is critically dependant on size. A magnetic B field loop antenna would generally be used when the available space for a conventional dipole antenna was absent. Even a ¼ wave helical antenna could be reduced to dimensions around  of a wavelength, so the magnetic B field antenna would probably be selected for fractional wavelength dimensions far less than this. The corresponding “loss” compared to a conventional dipole could be 29 dB or greater, depending on the size reduction needed.

 This does not rule the magnetic B field antenna out for all applications. It can be small, albeit inefficient, but many applications may accept a compromise between size convenience and radiation (or equivalent reception) efficiency.

 Strategies for improving radiation efficiency include the use of lower loss conductors (super-conductors perhaps?) or the use of ferrite material intended to increase the B field radiated from the loop.

What will happen if the loop has more than one turn? The B field will increase by an amount equal to the turns squared, but so too will the inductance and therefore the ohmic resistive loss.

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