Loop Antenna Design

This article describes the design, construction, and measurements performed on a 50 Ohm loop antenna for the 434 MHz band. Experiments were done to characterize the prototypes, as well to understand the impacts of the materials used. We begin with the assembly of a full wave round loop antenna, evolving the design to the 2:1 rectangle loop matched to 50 Ohm.

The journey started with the construction of many loop antenna prototypes, including two of the loop type and one folded dipole, as shown in Figure 1. Not exactly considered a loop antenna, as being better described with conventional dipole theory, the folded dipole helped to grab intuition about the others.

A full wave (one-wavelength) resonant loop antenna presents higher directivity in relation to the basic dipole, and tends to be less susceptible to electric noise or from interactions with the surroundings. This happens because the near-field nature is predominantly magnetic, where the majority of materials in the surroundings of an antenna - including human bodies - present a much more stable magnetic permeability than electric permittivity.

Measurements were performed using a Keysight N9918B FieldFox analyzer in VNA mode. Preliminary data was captured at home - in the kitchen :) - with care to position the antenna with at least one-wavelength from the surroundings.

Every measurement includes a choke at the antenna port (see Figure 6), as the way to suppress the common-mode return of the unbalanced currents - this turns out to be mandatory for the operation of balanced structures like dipoles and full wave loops. If not, the antenna behavior is totally disturbed, and you will be measuring the cable!

Loop Antenna Theory

The full wave loop antenna consists of a continuous piece of wire, mechanically mounted as a loop from the feed point. The length L of the wire has approximately one wavelength (we are going to discuss later why it is not exactly λ).  

Opposite to what is commonly thought, the current is not continuous in the full perimeter of the loop, reversing at a symmetrical cut from the feed point. This happens by the nature of the one-wavelength resonator, that when resonating at the fundamental frequency, presents a single current anti-node - for the perfectly balanced loop, the node is situated antipode to the feed.

Note: If the loop is small enough, for the current to have a single direction around the loop, the electromagnetic behavior is entirely different, and the assembly is called a small-loop antenna (for this case, you can consider the small-loop to be an inductor). The information in this article doesn't apply to small-loops.

The radiation pattern can be inferred by assuming the current distribution in the first place, or computed numerically by electromagnetic simulation software packages. The radiation pattern presented in Figure 4 was generated using the software 4NEC2.

Radiation occurs perpendicularly to the loop plane, with a directivity greater than the simple dipole, as the second will radiate in every direction around its construction axis. The directivity is approximately 1 dB higher than the dipole (we could expect 3 dB, by the simple consideration of the dipole radiating to all directions perpendicular to the construction axis, but the pattern aperture of the loop is broader than the dipole).

The near zero radiation, radially in the loop plane, is simply explained by noticing that, when looked at parallel to the plane, the opposite side of the loop always present current in anti-phase. Nulling is not perfect because the opposite currents don't overlap.

Polarization is always reverse to the feed point position. When the feed is vertical (as shown in Figure 5, with the feed at the bottom), the loop antenna will radiate with horizontal polarization, this can also be understood by looking at the current distribution.

The opposite side of the feed contains current in the same direction as in the feed, this has the effect of reinforcing the current in the horizontal direction. Looking carefully and decomposing the current vectors into their x and y components, any pair of opposite points will have the x (horizontal) component reinforced, while the y (vertical) component will null each other. The opposite scenario happens if the feed is rotated 90 degrees.

Round Loop Antenna Prototype

When prototyping, the first loop antenna assembled was of the round type (the one discussed so far in the article), aiming for grabbing a practical understanding of its behavior.

Construction followed using 2.5 mm² electrical copper wire. An SMA connector is used in the feed, also serving as the mechanical support. Firstly, a simple calculation for the wire length was performed by finding the wavelength λ for the target operational frequency F.

For λ = v × C / F, where C is the speed of light and v the velocity factor for copper wire - common dipole calculators use v between 0.94 and 0.98 - using v = 0.98 and F = 434 MHz, the wavelength got was ~0.68 m.

The electrical wire was kept with its PVC insulator, and soldered to the SMA connector. Further in the article, we are going to notice that the wire insulation changes significantly the v factor, by its dielectric properties that slow the wave propagation velocity in the vicinities of the wire surface.

For mounting the loop, a non-conductive support was prepared, keeping the antenna at least one-wavelength apart from any object. At this scale, λ is greater than the theoretical far-field distance 2 × D² / λ, as the assembled antenna diameter D is around 0.25 m. Nevertheless, the ideal measurement setup would be in an open field area.

When characterizing antennas, it is important to note that we should not be naively looking for the best return loss (deep valley on the S11 trace) in the VNA display. This happens because the VNA performs the network measurement in relation to the specified system impedance Z₀ (usually 50 Ohm).

So, for a round loop antenna, that when resonating presents a feed point impedance higher than 50 Ohm, the VNA will not show an S11 valley. By definition, the round loop antenna will be unmatched to the VNA system impedance Z₀ at resonance.

An S11 valley may appear near the resonant frequency, if - by geometrical or electrical properties - the impedance of the measured antenna approaches Z₀ (50 Ohm), leading to marking the antenna's resonance at a wrong frequency.

The proper antenna frequency can be found by displaying the S11 phase. Resonance occurs, by definition, when the electrical characteristics combine to null the reactive behavior. At this point - when looked as a black box - the circuit (in this case the antenna) becomes purely resistive, with the S11 phase crossing 0° exactly at this point.

As the VNA measurement shows, the resonance point is ~445 MHz. A direct measurement of the antenna bandwidth is not valid at this frequency, due to the unmatched Z₀ regarding the feed point impedance.

We note that the resonance frequency differs from the expected, for two main reasons: 1) the simple λ calculation considered the medium (copper wire) as a straight electromagnetic guide that did not interact with itself; 2) the dielectric effects of the wire's PVC insulator, that reduces the propagation velocity by ~5.6 % (which we will cover at the end of the article).

The resonance frequency will not be exactly as calculated, this happens from the complex interactions of the electromagnetic fields in the antenna's geometry. Either way, it is safe to assume a velocity factor that compensates for this, when designing new antennas, helping to get an operational prototype sooner.

For this round loop, the resonance occurs with a high resistive part (high in comparison to a Z₀ of 50 Ohm) making a difficult match to a standard coaxial cable. As the feed point is located at a current maxima, its impedance's resistive part is indeed the radiation resistance - assuming zero resistive loss from the wire resistance. In the next section, we explore a way of shaping the loop to achieve a lower radiation resistance.

A 50 Ohm Rectangular Loop Antenna

The radiation resistance can be modified by changes in the geometry of the loop. It is difficult to predict the impedance of a specific shape. With the help of the simulator, we arrive at an interesting case of a rectangular loop, where its height is double its width and the feed point is centered on the narrow side.

In this article, I call this antenna the 2:1 Rectangular Loop Antenna. This shape brings the advantage of very near to 50 Ohm radiation resistance, making a perfect match.

Figure 9 shows the prototype antenna with the insulation removed from the wire. The insulation was removed as an afterthought, after I observed during experiments that the resonant frequency was significantly lower than I expected.

Measurements indicate a great match to 50 Ohm and, as now the resistive part of the antenna's impedance was expected to be Z₀, we see that the phase crosses zero almost coinciding with the deep in S11.

A practical velocity factor can be back-calculated using the found resonance frequency: considering the speed of light C = 299.8, the resonance frequency F = 460 MHz and L the length of the wire (perimeter of the loop), v_2:1 (velocity factor for the 2:1 loop) is found using v_2:1 = F × L / 299.8. For this case, v_2:1 = 1.1.

Having a v_2:1 higher than unity does not mean that we are breaking the laws of physics. The interpretation is that the effective electrical length of the wire is shorter due to the specific geometry of the loop - responsible for a unique electromagnetic behavior in the near-field. The back-calculated velocity factor accounts for all these effects, collapsing them in a single design constant, including the ~95% factor usually used for bare copper wire.

Similar to the round loop, this antenna presents higher directivity than the basic dipole and carries all the advantages of the strong magnetic near-field, as presented earlier in the article. This shape of antenna is sometimes called VOHPL, standing for Vertical Oriented Horizontally Polarized Loop and, when vertical polarization is required, the loop in mounted horizontally, becoming a HOVPL antenna.

The simulated directivity is 1 dB higher than the round loop. Less power is radiated in the Z axis, this can be intuitively understood by seeing that the horizontal top and bottom segments are further apart, increasing the phase mismatch for the wave fronts traveling from top to bottom or bottom to top.

Currents in the left and right segments of the antenna are closer apart, increasing the cancelation for directions different from perpendicular to the loop plane (variations in azimuth).

2:1 Rectangular Loop Radiation Resistance

One way to parametrize the rectangular loop is by defining a design parameter K, that encodes the ratio between the height and width of the loop - height being the longer dimension. A 1:1 rectangular (square) loop has K = 1.

The 2:1 rectangular loop from this article has K = 2, meaning its height is double its width. We notice that, for increases in K, the top and bottom segments that have the majority of the radiation current decrease in length.

As the radiating elements decrease in size, the electromagnetic excitation generated by the circulating current is reduced. The shorter the segment, the lower the radiation, to the limit where, for K >> 10, the width becomes infinitesimal, and no radiation occurs.

A lower total radiated power means a lower radiation resistance for the same applied current, as P_rad = I².R_rad in which I is the total excitation current. We should expect the radiation resistance to drop more or less with the square of the loop width reduction with increased K, based on approximating the radiating elements as horizontally mounted Hertzian dipoles.

Wire Insulation And Propagation Velocity

An exposed copper wire presents a near to unity velocity factor, with common designs using values between 94% and 98%. This happens for the electric field being guided on the surface of the wire, with the currents conducted in a thin layer due to the skin-effect. The majority of the field is exposed to the air dielectric constant.

A different scenario occurs for insulated wire, where the electric field is exposed to the dielectric constant of the insulation material. The propagation velocity is slower, increasing the effective electrical length of the wire - thus reducing its resonance frequency.

The effect of the insulation material in the field velocity will be partial, as the electric field lines are not completely contained in this different electrical permittivity region, with the radial lines crossing the insulation barrier that is usually much thinner than the wavelength.

We thus expect that the influence will be partial, and the velocity factor reduced in part, not dropping entirely to the velocity factor of the insulation material. Most common electrical wires use a composite material, predominantly PVC, which has a relative permittivity ε_r ≈ 3.

If the electric field was fully immersed, the velocity factor would be v = 1 / sqrt(3) ≈ 57%. However, as we will see, the effect of the insulation will be significantly lower than this.

For the sake of experimentation, two shorted parallel-wire transmission lines were constructed, one with the wire exposed and the other with the insulation in its original form. The length of 0.165 m gives a quarter-wave resonance of 454 MHz and a half-wave resonance at twice the frequency, 898 MHz.

We expect to see the short at the end of the transmission line to be transformed to a very high impedance in the vicinities of ~450 MHz - by the quarter-wave action - and for it to become a short again exactly when the half-wave resonance is reached, near ~890 MHz.

In Figure 13, we see the phase measurement for both transmission lines. As expected, the measurement that shows the zero-phase transition at the lower frequency is for the insulated line. The line is half-wave resonant at a lower frequency because the velocity factor is smaller, meaning that the effective electrical length of the line is larger.

The insulated line, with its half-wave resonance at ~751 MHz, presents an electrical length of ~0.79 m and, when compared to the free-space length, indicates a velocity factor v_insul = F * L / 299.8 ≈ 0.82.

For the case of the bare copper line, with a half-wave resonance at ~874 MHz, the electrical length is 0.68 m, for a velocity factor v_bare = 0.96. These measurements show a relative velocity difference of ~17%, higher than experienced when testing the antennas.

The larger difference occurs because the parallel-wire transmission line presents much stronger fields, with the majority of field lines in between the conductors, augmenting the dielectric effect of the insulator. Nevertheless, the propagation velocity is not as low as the predictions for a propagating wave fully immersed in the PVC material.

Removing the insulation from the round loop shows an increase in the resonant frequency to around ~470 MHz, when compared to Figure 7 that shows the original measurement for the insulated wire. This new measurement shows a velocity factor difference of ~5.6%.

Conclusion

This article describes the working principles of full-wave loop antennas, and clearly shows the ease of construction using commonly available materials. Loop antennas present useful properties for several applications, due to its high immunity to electrical noise and compact size. The simple mechanical construction allows for the assembly of test models, enabling the back-calculation of the design parameters.

The importance of testing and measuring the design is reinforced by the demonstration of the wire insulation impact on the velocity factory. Not measured in this exercise, a higher loss from the dielectric is also expected, indicating that the construction should follow with non-insulated wire.

For the 2:1 rectangular loop, the author found an excellent result when looking for a good feed point match and, for construction with ~2.5 mm² bare copper wire, the velocity factor v = 1.1 is recommended as an initial value.

Video

References

https://physicsopenlab.org/2020/05/03/loop-antenna-for-very-low-frequency/
http://on5au.be/content/fdim/fdim5.html
https://practicalantennas.com/theory/loop/full-wave/
https://en.wikipedia.org/wiki/Velocity_factor
http://s53mv.s5tech.net/cigar/sws.html
https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Electromagnetics_II_(Ellingson)/10%3A_Antennas/10.07%3A_Directivity_and_Gain