Analysis of Circular Polarization of Cylindrically Bent Microstrip Antennas

When circularly polarized (CP) microstrip antennas are bent, the polarization becomes elliptical. We present a simple model that describes the phenomenon. The two linear modes present in a CP patch are modeled separately and added together to produce CP. Bending distorts the almost-spherical equiphase surface of a linearly polarized patch, which leads to phase imbalance in the far-field of a CP patch. The model predicts both the frequency shifting of the axial ratio band as well as the narrowing of the axial ratio beam. Uncontrolled bending is a problem associated especially with flexible textile antennas, and wearable antennas should therefore be designed somewhat conformal.


Introduction
Microstrip antenna, or the patch antenna, is a common choice for wearable antenna topology.The ground plane isolates the antenna from the human body, and as a result, the impedance and radiation characteristics are not significantly affected by the user.In addition, the ground plane reduces radiation into tissue.The low profile makes the antenna easy to integrate into clothing.
For an antenna to be truly wearable, it must be flexible, and its performance must not change significantly because of deformation.Design rules for conformal antennas can to some extent be used in wearable antenna design, but since the shape is subject to change, the antenna designer must ensure operation in many different use cases.Usually this means that the antenna should be designed to be broadband.
Circular polarization (CP) is commonly used in wearable off-body systems.For example, the GPS satellite positioning system uses CP.When a body-worn antenna is moving, its orientation is subject to change, resulting in polarization mismatch loss if linear polarization is used in the system.However, circular polarization requires careful shaping of the antenna, and is easily distorted when the wearable antenna conforms to the user's body.
Cylindrically bent linearly polarized (LP) patches have been thoroughly analyzed in the past.The input impedance of a cylindrically bent LP patch was derived in [1,2], and the equations for the radiated fields were also presented in [1].The equations for the input impedance in [1,2] do not predict any resonant frequency change for a thin patch because of the approximations used.However, the frequency shift has been reported in many measurements [3][4][5] as well as simulations [6].To our knowledge, bent CP patch antennas have only been analyzed on a conical surface [7], but not cylindrical.
In this study we analyze the behavior of a bent CP patch.The radiated fields of the CP antenna are modeled by adding together the fields of two simulated LP patches, one in vertical and one in horizontal polarization.We assume that the circular polarization results from two orthogonal modes excited in a 90-degree phase shift.We then bend the two LP antennas that produce the two modes and model the bent CP antenna by adding the fields of the bent LP antennas.
Proper phasing in the farfield is the key to achieve circular polarization.The equiphase surface of an LP antenna is nearly spherical in the main lobe, and when the fields of two LP antennas are added together, good phasing (good CP) can be retained in a wide angular region.Bending distorts the equiphase surface, and as a result, the axial ratio in the sum field deteriorates.
By modeling the two modes with separate LP patches we aim to dissect the situation into easily understood parts.The model gives insight into the phenomena associated with bent CP patches: change in input impedance, frequency shift of the axial ratio band, and distortion of the axial ratio beam.
This paper is structured as follows: Section 2 introduces the simulation models.The simulation results for the LP patches are briefly given in Section 3, and in Section 4 they are used to model the CP patch.Section 4.4 comments on the validity of the model, briefly comparing the results with measurements.The results are used to give guidelines for wearable antenna design in Section 5, and 6 concludes the work.

Model
Probe-fed LP rectangular patches were simulated using the finite integration technique implemented in the CST software [8].The patches were 60 mm by 74 mm rectangles, on a 3 mm thick substrate with dielectric constant 2.3, and on a 148 mm square ground plane.The feed probe was inset 18 mm from the long edge of the patch and centered along that edge.The resonant frequencies of the two patches were almost equal, with the resonant frequency of the Eplane bent patch tuned slightly higher to achieve circular polarization.Figure 1 illustrates the structure.
These LP patches were bent in the E-and H-planes, conforming to a cylinder.The bending radii (30 mm to 300 mm, or 0.15 to 1.6 wavelengths) correspond to the curvatures found in the arms of children and adults and the torso.Bending in the E-plane (yz-plane) involves bending the y-directional current flow lines.The bending directions are illustrated in Figure 1.
The radiation patterns of the bent LP patches were simulated, and from the results, the phase centers were calculated.Phase center is the imaginary point from which the radiated spherical wavefront seems to emanate.A unique phase center does not generally exist, but an approximate location can be calculated using the field phase information in the main beam region.
Circular polarization was modeled as a phased sum of two orthogonal LP modes.The phase shifting was adjusted to achieve a 90-degree phase difference in the farfields of the flat antennas.The input impedance of the CP patch is the sum of the input impedances of the LP modes, and the farfield is the sum of the far fields of the LP patches, with the same phase shifting for both the impedance and the fields.The phasing was optimised for the flat antenna, and the same phasing was then used for all the different bending radii and all frequencies to simulate a phase-shifting network.

Results for LP Patches
The results for the impedance, detuning, far fields, and semi phase centers of the linearly polarized patches are summarized in this section.increases the input impedance, whereas H-plane bending decreases it.Figure 2 shows the simulated impedances and resonant frequencies.The simulated effects of bending the LP antennas are listed in Table 1.
Previous measurements have shown that E-plane bending increases the resonant frequency of an LP patch antenna [3,4].The theoretical examination in [1] does not predict the detuning because of the approximations used.According to [1], the input impedance will decrease in H-plane bending and increase in E-plane bending (this was shown for substrate thickness 1.59 mm and dielectric constant 2.32).
The change in impedance magnitude as well as the detuning was repeated in our simulations.Detuning was minor when the E-plane bending radius was very large (100 mm or more, which corresponds to 0.5 wavelengths) and, between radii of 30 and 80 mm a decrease of 1 mm in the radius introduced about 0.9 MHz detuning.Table 2 lists the simulated resonant frequencies of the linearly polarized patch bent in the E-plane.

Far Field and Semi Phase Centers.
The phase center of an antenna is the imaginary point from which the spherical wavefront seems to emanate.Such a point is not guaranteed to exist, but an approximate location can be calculated, for example, in the main beam region.Moreover, separate semi phase centers can be calculated in the principal planes, considering only the field phase in that plane: in the E-plane, the circular (two-dimensional) wavefront seems to emanate from the point P E .The H-plane semi phase center is denoted by P H .If the points P E and P H coincide, the wavefront is spherical; if not, the phase front is astigmatic, as illustrated in Figure 3.
For the purposes of this study, we calculated the E-and H-plane semi phase centers using the phase information of the pattern in a 60-degree aperture cone around the main lobe.This is approximately the 3dB beam.
The phase center of a flat LP patch lies approximately in the plane of the patch, at the center.The semi phase centers coincide, and the wavefront is approximately spherical.
Bending moves the semi phase centers apart from each other and makes the phase front astigmatic.Figure 4 shows that especially in the E-plane bending the semi phase center locations depend heavily on the bending radius.Figure 5 illustrates the locations of the two semi phase centers in two bending cases.If the antenna is bent in the H-plane, both semi phase centers move behind the antenna, P H slightly more than the P E .Bending in the E-plane, on the other hand, moves P E backward and P H to the front.Thus the equiphase surface is badly distorted, resembling a donut (flat in the H-plane) rather than a sphere.This is illustrated in Figure 6.Within the frequency range of interest the semi phase center locations for each bending radius are quite stable, moving 5 mm at the maximum over the band.Note that the interesting frequency band is narrow, since we are mainly interested in the band where the axial ratio of the modeled CP patch is small.
The semi phase centers also move in the xy-plane (in the plane of the patch), but this movement is much less pronounced than the movement along the z-axis (forward and backward).Gain, cross-polarization level, and efficiency are not significantly affected by bending.

The CP Patch according to the Model
The results for the bent linearly polarized patch antenna will now be used to explain the behavior of a bent circularly polarized antenna.To generate circular polarization, the two linear modes are added together with such a phasing that a 90-degree phase shift is achieved at the target frequency.The same phase shifting is used in the input impedance.
The changes in the input impedance due to bending are discussed first.Then the simulated changes in the far field phase are used to explain the axial ratio in the main lobe (zaxis, θ = 0).We will see that the axial ratio band shifts in frequency.Finally, the angular region of good axial ratio is examined with the help of the semi phase centers P E and P H . Using the semi phase centers instead of the full far field phase information allows us to squeeze the information into one number (semi phase center location) and, hence, to explain the situation intuitively.

Impedance and Matching.
The impedance of a CP patch was modeled as a phased sum of the impedances of the LP patches.Plotted on the Smith chart (Figure 7), the reflection coefficient shows a loop commonly seen in the case of CP antennas.In the reflection coefficient of the bent CP patch (modeled as the sum of the impedances of two bent LP patches), the loop is seen to shift counterclockwise.
In the return loss of the CP patch (inset of Figure 7) we see how the wide notch with two minima turns into two separate notches when the antenna is bent.This might lead to the conclusion that circular polarization is lost, but actually we will still find an axial ratio minimum between the two notches.It is noted that the frequency difference between the pits in the sum (Figure 7) is smaller than the difference of the pits of two modes modeled separately (Figure 2).

Axial Ratio versus Frequency Using Far Field Phase.
To produce a perfect circular polarization, an antenna must radiate two LP modes of equal amplitude in a 90-degree phase shift.Table 3 lists the limits to produce circular polarization with an axial ratio less than 3 dB.From the table it is evident that phase shifting is more critical than amplitude balance.
From the simulation data, we have extracted the field phase differences of the bent LP patches at θ = 0 (z-axis).One of the LP antennas is now bent in its E-plane and the other in its H-plane.The phase differences at the original center frequency are tabulated in Table 4.
When the antenna is bent, the phase difference between the fields at the center frequency grows too large.To reduce the phase difference, we must introduce extra phase difference between the impedances.A larger impedance phase  difference can be found at a higher frequency.We will now observe a good axial ratio at this higher frequency, provided that the amplitude difference is small enough (see Table 3).

Axial Ratio in the Main Beam Region Using Semi Phase
Centers.Even at the frequency where we have a good axial ratio on the z-axis, the shape of the angular region with AR < 3 dB ("AR beam") is distorted.For the flat antenna, the AR beam is of the same shape as the main lobe, but bending makes the AR beam narrower.Consider a CP antenna with two radiating current modes: mode no. 1 producing vertical polarization and mode no. 2 horizontal, as illustrated in Figure 8.The main lobe points toward the z-axis.The horizontal plane is the H-plane for mode no. 1 but the E-plane for no.2, and vice versa for the vertical plane.When mode no. 2 is set to lead by 90 • , the resulting polarization is right-handed circular.We will assume a 90 • phase difference on the z-axis.Now if this antenna is bent, the semi phase centers of the modes #1 and #2 will move apart, especially along the z-axis, as was seen in Figure 4.This results in additional phase shift, which is listed in Table 5.Note that the table only lists values for one bending radius, and as seen from Figure 4, the semi phase center location depends heavily on the bending radius.The amplitudes of the modes do not change much, unless detuning causes a significant mismatch loss.
Bending deforms the angular region of good axial ratio: when one mode produces an equiphase surface nearly spherical and the other of a donut shape, it is clear that the phase difference between the fields will vary even in the main beam region.Sufficiently far away from the z-axis we can use the semi phase centers to give information about the field phase difference.In the horizontal bending (the first two columns in Table 5), the semi phase centers of the modes #1 and #2 are 18 mm apart from each other, which corresponds to a phase difference of 32 • , as compared to the case where the semi phase centers coincide.We assumed that the field phase difference at θ = 0 (z-axis) is exactly 90 • , but away from the z-axis we now have a phase difference of 90 • − 22 • = 68 • in the vertical plane and 90 • + 10 • = 100 • in the horizontal plane.This means that circular polarization cannot exist in the vertical plane, and the AR beam is deformed.If the antenna was bent in the vertical plane the situation would be the opposite: no circular polarization in the horizontal plane.

International Journal of Antennas and Propagation
Figure 9 illustrates the axial ratio of a flat and a bent antenna at the same frequency.We see good axial ratio in the φ = 0 plane (horizontal plane in the previous example) even with the bending radius 80 mm.Bending more makes the axial ratio beam very narrow.
The deformation of the AR beam occurs in addition to the frequency shift of the good axial ratio.

Comments on the Model.
The model is suited to describe a dual-feed CP antenna with a 90-degree hybrid splitter between the two ports.We can then assume a good isolation between the ports, and the modes can be treated independently.
The shift of the axial ratio band has been reported, for example, in [9] as well as in our measurements and simulations.In our measurements we validated the model by measuring the input impedance and radiation pattern of a single-feed detached-corner CP patch antenna [10] bent in four planes spaced 45 • .A shift of the axial ratio band towards lower frequencies has also been reported, but this phenomenon cannot be explained using this model.However, this very simple model already predicts the shift in impedance and axial ratio and sheds light on the reasons why this happens.

Implications to Wearable Antenna Design
During the research it became clear that wearable antennas should be designed somewhat conformal: for example, if the antenna is to be placed on the sleeve of a jacket, it should be designed on a cylindrical surface.A small deviation in the cylinder radius is less harmful than a change from flat to cylindrical shape.This is seen especially in Figure 4: slight bending changes the equiphase surface from spherical to astigmatic.
Particularly CP antennas should be designed as conformal as possible, close to their expected bending radius, and bending those antennas should be limited to relatively large radii only (e.g., minimum 100 mm at 1.5 GHz).CP antennas should therefore be placed on the back rather than on the sleeve of a coat.
Bending makes the main lobe wider and hence lowers the gain.This phenomenon is not harmful in wearable antennas, where all-round coverage is often sought for.The crosspolarization in linearly polarized antennas does not change significantly.We can say that linearly polarized antennas are, in general, more robust with regard to bending than circularly polarized antennas.
If possible, bending any patch antenna, be it circularly, linearly, or dual polarized, should be limited to the H-plane only.Bending in the E-plane results in a resonant frequency shift which can lead to a strong mismatch loss.This is the most severe effect that bending has on microstrip antennas.

Conclusion
A simplified model of a bent CP patch antenna has been described.In the model, the two orthogonal polarization components that together form the circular polarization were treated separately.In simulations, one component was bent in its E-plane and the other in its H-plane.Circular polarization was then modeled by adding these modes together with a proper phase shifting.
The model predicts the frequency shift of the axial ratio band and explains why the angular region with good axial ratio is deformed.Also the change in the input impedance was predicted.
Finally, some guidelines for wearable antenna design and placement were given.

3. 1 .Figure 1 :
Figure 1: Simulated LP patch: flat, bent in H-plane, and in E-plane.Dot marks the feed probe position.H-plane is the xz-plane and Eplane the yz-plane.The main lobe is in the θ = 0 (z-axis) direction.

Figure 2 :
Figure 2: Simulated input reflection coefficient of the bent LP patch.(a): bending in the H-plane, (b): E-plane bending.Insets show return loss in dB.

Figure 3 :
Figure 3: How to calculate the semi phase centers from the equiphase surface.(a): The astigmatic (not spherical) wave front.(b): The circular wavefront in the H-plane (xz-plane) emanates from P H . (c):In the E-plane (yz-plane), P E is the center of the circular wavefront.P H and P E do not coincide.

Figure 4 :Figure 5 :
Figure 4: Semi phase center z-coordinates.(a): H-plane bending, (b): E-plane bending.Horizontal lines show the location of semi phase centers as calculated for a very large bending radius.

Figure 6 :Figure 7 :
Figure 6: Equiphase surfaces.(a): The spherical equiphase surface of the flat patch antenna.Semi phase centers coincide.(b): The astigmatic equiphase surface of the E-plane bent patch antenna.P H (in the xz-plane) is in front of the antenna, but P E (in the yz-plane) is behind it.The astigmatism is exaggerated for clarity.

Figure 9 :
Figure 9: The 3dB axial ratio area (shaded) in the main beam.Radial coordinate is θ = 0 • • • 80 • , polar φ.Main lobe direction is θ = 0.The figures are best understood as a map of the polar region of the spherical coordinate system.(a): almost flat antenna, shown both 3D and in (θ, φ) coordinates.(b): 80 mm and 60 mm radii.

Table 1 :
Simulated effects of bending LP patch antennas."+" indicates a strong effect.

Table 2 :
Detuning caused by bending in the E-plane.

Table 3 :
Limits for circular polarization with axial ratio <3 dB.

Table 4 :
Simulated field phase difference near θ = 0, observed at a constant frequency.

Table 5 :
Semi phase center z-coordinates when the antenna is bent on a 80 mm radius.The semi phase centers lie approximately at z = 0 when the antenna is flat.Distance between semi phase centers is also expressed as phase shifting (degrees) at 1515 MHz.