WIRELESS POWER TRANSFER: EFFICIENCY, FAR FIELD, DIRECTIVITY, AND PHASED ARRAY ANTENNAS by Abigail Jubilee Kragt Finnell

A Thesis Submitted to the Faculty of Purdue University In Partial Fulfillment of the Requirements for the degree of

Master of Science

Department of Electrical and Computer Engineering Indianapolis, Indiana August 2021 THE PURDUE UNIVERSITY GRADUATE SCHOOL STATEMENT OF COMMITTEE APPROVAL

Dr. Peter Schubert, Chair Department of Electrical and Computer Engineering

Dr. Maher Rizkalla Department of Electrical and Computer Engineering

Dr. Lauren Christopher Department of Electrical and Computer Engineering

Approved by: Dr. Brian King

2 TABLE OF CONTENTS

LIST OF TABLES ...... 5

LIST OF FIGURES ...... 6

ABSTRACT ...... 7

1 INTRODUCTION ...... 8

2 FUNDAMENTALS ...... 10

2.1 Propagation of Signals and Power ...... 10

2.1.1 The Friis Equation ...... 11

2.1.2 The Goubau Equation ...... 11

2.1.3 The Far Field Equation ...... 12

2.2 Wireless Power Beaming ...... 14

2.3 Gain and Directivity ...... 15

2.3.1 Example: Transmit Antenna Trade-off Study ...... 17

2.4 Sidelobe Levels ...... 18

2.5 Non-Traditional Phased Array Antenna Architecture ...... 21

2.6 Other Components of Wireless Power Transfer ...... 22

3 FAR FIELD DISTANCE STUDY ...... 25

3.1 Far Field Model ...... 26

3.2 MATLAB Model ...... 27

3.3 Laboratory Work at IUPUI ...... 33

4 FREE SPACE TRANSMISSION EFFICIENCY STUDY ...... 40

4.1 Efficiency Equations ...... 41

Assuming a Constant D(φ) ...... 42

Assuming D(φ) is Parabolic on a Logarithmic Scale ...... 43

Using Numeric Integration ...... 44

4.2 Analysis of a Uniform PAA ...... 45

4.3 Past WPT Experiments ...... 47

4.3.1 The Microwave Powered Helicopter ...... 47

4.3.2 The JPL Experiment ...... 49

3 4.3.3 The Goldstone Experiment ...... 50

4.3.4 Equation Comparison ...... 51

5 DISCUSSIONS AND ANALYSIS ...... 53

6 CONCLUSIONS AND FUTURE WORK ...... 55

6.1 Future Work ...... 57

REFERENCES ...... 59

A MATLAB Code ...... 62

B Laboratory Data ...... 73

4 LIST OF TABLES

B.1 Lab Data: 1.45 m Distance ...... 73

B.2 Lab Data: 1.2 m Distance ...... 77

B.3 Lab Data: 0.84 m Distance ...... 81

5 LIST OF FIGURES

2.1 Transmit Antenna Costs [ 14 ] ...... 18

2.2 Grating Lobes for 1.5 λ Spacing, Zero Element Phase Shift ...... 20

2.3 Linear Phased Array Antenna, θ = π/4 ...... 23

3.1 Far Field Square Rectenna Size vs. Distance ...... 28

3.2 Comparison of Singleton and 2x2 Array Phase at 125 mm ...... 29

3.3 Comparison of Singleton and 2x2 Array Phase at 154 mm ...... 29

3.4 Comparison of Singleton and 2x2 Array Phase at 600 mm ...... 30

3.5 Comparison of Singleton and 2x2 Array Phase at 1000 mm ...... 30

3.6 4x4 Antenna Array: 125 mm ...... 31

3.7 4x4 Antenna Array: 160 mm ...... 31

3.8 4x4 Antenna Array: 200 mm ...... 31

3.9 4x4 Antenna Array: 350 mm ...... 31

3.10 4x4 Antenna Array: 850 mm ...... 32

3.11 4x4 Antenna Array: 2 m ...... 32

3.12 Far Field Circular Rectenna Size vs. Distance ...... 33

3.13 Comparison of Traditional and Modeled Far Field for PAAs ...... 34

3.14 Laboratory Setup at IUPUI ...... 35

3.15 Laboratory Setup ...... 36

3.16 Comparison of Laboratory Results with MATLAB Data ...... 37

3.17 Far Field Model Comparison ...... 39

4.1 Power Beaming Block Diagram ...... 40

4.2 Layout Used for Efficiency Calculations ...... 41

4.3 Maximum Directivity vs. Lambda Spacing ...... 46

4.4 Maximum Directivity vs. Number of Antennas ...... 47

4.5 Microwave Powered Helicopter Experiment [ 31 ] ...... 48

4.6 The JPL Experiment [ 33 ] ...... 50

4.7 Goldstone Experiment [ 35 ] ...... 51

4.8 Free Space Efficiency Equation Comparison ...... 52

6 ABSTRACT

This thesis is an examination of one of the main technologies to be developed on the path to Space Solar Power (SSP): Wireless Power Transfer (WPT), specifically power beam- ing. While SSP has been the main motivation for this body of work, other applications of power beaming include ground-to-ground energy transfer, ground to low-flying satellite wireless power transfer, mother-daughter satellite configurations, and even ground-to-car or ground-to-flying-car power transfer. More broadly, Wireless Power Transfer falls under the category of radio and microwave signals; with that in mind, some of the topics contained within can even be applied to 5G or other RF applications. The main components of WPT are signal transmission, propagation, and reception. This thesis focuses on the transmission and propagation of wireless power signals, including beamforming with Phased Array An- tennas (PAAs) and evaluations of transmission and propagation efficiency. Signals used to transmit power long distances must be extremely directive in order to deliver the power at an acceptable efficiency and to prevent excess power from interfering with other RF technology. Phased array antennas offer one method of increasing the directivity of a transmitted beam through off-axis cancellation from the multi-antenna source. Besides beamforming, another focus of this work is on the equations used to describe the efficiency and far field distance of transmitting antennas. Most previously used equations, including the Friis equation and the Goubau equation, are formed by examining singleton antennas, and do not account for the unique properties of antenna arrays. Updated equations and evaluation methods are presented both for the far field and the efficiency of phased array antennas. Experimental results corroborate the far field model and efficiency equation presented, and the implications of these results regarding space solar power and other applications are discussed. The results of this thesis are important to the applications of WPT previously mentioned, and can also be used as a starting point for further WPT and SSP research, especially when looking at the foundations of PAA technology.

7 1. INTRODUCTION

The number of papers regarding power beaming has increased significantly even as recently as in the year 2020. The reasons for this are numerous. Power beaming is seen as a field with increased potential at a time when transmitting and receiving antenna technologies are beginning to mature and WPT demonstrations are becoming more common. Additionally, international interest in SSP has increased; Japan, China, and the UK have all invested in SSP research. While the first inquiries into SSP were conducted in the 1970s, revived interest into this technology is unsurprising as alternate energy sources continue to be in high demand, and SSP has the unique capability of constant power, night and day, almost year-round. That being said, there are still many areas of SSP to be explored; much work on this subject is necessary before space solar power will be ready for wide-scale use. As mentioned above, much of the exploration into SSP (and consequently WPT) started in the early 1970s with interest from NASA. At the time, worldwide tensions about oil and energy garnered increased attention to renewable energy resources, and the idea of space solar power—while not original at the time—was considered as a potentially promising tech- nology. Some of the first experiments into power beaming were conducted in 1975 and developed with the help of William C. Brown at Raytheon, NASA, and JPL. In one ex- periment at Raytheon, an end-to-end efficiency of 54.18% was achieved at a distance of1.7 m. In another experiment at the JPL Goldstone facility later that year, Brown transferred 270 W over 1.54 km with a record-setting rectenna efficiency of over 80%. While this was a huge achievement, the large distance paired with the relatively small transmitting and receiving antenna resulted in a path loss of approximately 89%. As these experiments show, the main components and largest challenges of wireless power transfer have all been present from the beginning: incredibly high directivities are required to increase transmission effi- ciencies over long distances; safety, control, and careful evaluation are needed in all steps of the process; careful rectenna configurations are required for maximum energy harvesting; and beam steering is necessary for careful transmission. In addition to this, other compo- nents of WPT become relevant at high power densities: low sidelobe levels are required to

8 prevent the power level accessible to bystanders and external equipment from causing harm or interference, as well as pin-point accuracy and complete beam control. The following work details and develops aspects of these key WPT components, specifi- cally power beaming via phased array antennas or unconventional antenna configurations re- garding directivity, SLL reduction, and free space path efficiency. Although the overall thrust of the work is towards Space Solar Power, there are many applications in which power beam- ing would be beneficial, and much of this work can be expanded to other RF applications. Additionally, smaller-scale applications that include power beaming will allow for increased funding and demonstrations of WPT, forming stepping stones to SSP. These applications may include ground-to-ground WPT, ground-to-low-orbit-satellite WPT, ground-to-car or ground-to-flying-car WPT, and others. Any situation in which the transfer of power would be helpful, but the implementation of cables would be unreasonable or impossible, power beaming can be a solution. Other peripheral applications can include any radio or RF communications, especially 5G, which relies heavily on PAAs for signal steering. Although this type of application can be seen as largely different from WPT, as the goal is to transmit signals embedded with information rather than power, some principles (including efficiency estimations and far-field specifications) can be seen to overlap.

The remainder of this thesis will be organized as follows. Chapter 2 will form a complete introduction to wireless power transfer, including all of the assumptions and information

that form the background for the remaining sections. Chapter 3 will detail all of the models, experiments, and results discussing far field analysis including experimental designs and

results from the lab at IUPUI. Chapter 4 will contain all of the models and results discussing the free space transmission efficiency, including a comparison of equations with regard topast

wireless power transfer experiments. Chapter 5 will include the discussion and analysis of experimental results, including the potential impact this work has in the realm of SSP and

power beaming in general. Finally, Chapter 6 will include conclusions and recommendations for future work.

9 2. FUNDAMENTALS

The fundamentals of wireless power transfer are extremely important to the discussions in this work. This chapter will discuss WPT fundamentals, including widely used equations such as the Friis and Goubau equations, the far-field equation and background behind the associated division into near-field and far-field distances, the history behind the current re- search in WPT, and other WPT subjects. Beyond allowing for a comprehensive background for the subject at hand, this section should serve as a starting point and overview for students who are interested in studying wireless power transfer. Although there are many sources for students to use, resources such as textbooks often lag behind the state of the art. This is especially true in the area of WPT, which is currently experiencing rapid development. Additionally, textbooks can be too specific and detailed for a big picture view, and resources easily found online are too vague to give an accurate background of current RF technol- ogy with enough detail to provide the stepping stones to further research. This chapter is an attempt to bridge that gap by providing an overview with enough specific background information to allow for further study, as well.

2.1 Propagation of Signals and Power

When initially delving in to the topic of antennas and RF technology, some of the first equations one might encounter are the Friis equation, the Goubau equation, and the far field equation. These equations give an overview of the technology at hand: the Friis and Goubau equations give an estimation of propagation efficiency, and the far field equation, as titled, provides a baseline distance required for a system to operate in the far field. Unfortunately, without care, these equations can be misused by applying them in situations that are not applicable and were not intended upon their formation. This is especially true with the Friis and Goubou equations, which were formulated with singleton antennas in mind. It is also true in that many methods of antenna evaluation have been designed for signal transmission, in which the power delivered only matters in terms of appropriate signal-to-noise ratios, and not in applications of power delivery itself.

10 2.1.1 The Friis Equation

The Friis equation was first introduced by Harald T. Friis in May of 1946 in a paper that

has been cited over 900 times [1 ]. It was given as a formula for the transmission of RF power in free space, as follows: Pr ArAt = 2 2 (2.1) Pt d λ

Where Pt is the power delivered to the transmitting antenna, Pr is the power recovered from the receiving antenna, Ar is the effective area of the receiving antenna, At is the effective area of the transmitting antenna, d is the distance between the antennas, and λ is the wavelength. One of the biggest challenges for this equation is the evaluation of the effective area of the antenna. In the original document, equations are given for effective areas of many different antenna types, including dipole antennas, isotropic antennas, parabolic reflectors, and horns. This is necessary because, while dependent on the physical area, the effective area is actually defined as the area at which the incident power per unit area multiplied by the effective area is the total power. This definition, while relatively straightforward, can be difficult to easily measure, especially for antenna arrays and antennas with very large antenna gains. These limitations make this equation less useful for cases involving large scale wireless power transfer or unusual antenna geometry.

2.1.2 The Goubau Equation

Another commonly used efficiency equation is the Goubau equation, as shown:

2 η = 1 − e−τ (2.2)

s AtAr τ = (2.3) dλ

This equation was first developed by Georg Goubau and Felix Schwering, and has been widely used: “On the Guided Propagation of Electromagnetic Wave Beams” has been cited

over 250 times [2 ]. This equation is based on the examination and evaluation of reiterative

11 wave beams, in which the resulting power distributions in the Fresnel zone (as explained

in Section 2.1.3 ) repeat themselves without expansion of energy through the use of phase

transformers to guide the beam [3 ]. In another publication, Goubau’s work is explicitly

dependent on the Fresnel-Kirchhoff theory that excludes super-gain antennas [4 ]. These overlooked requirements result in an equation that can easily be misinterpreted to produce very large efficiencies. The requirement to exclude super-gain antennas andthe requirement for phase transformations, in the form of dielectric lenses, are often ignored; the

resulting efficiencies are overstated. For this reason and reasons mentioned in Section 2.1.1 , the free space efficiency will be studied later in this thesis.

2.1.3 The Far Field Equation

The far field is defined as the region in which the directivity pattern of the propagated beam is a function only of the angle, not of the distance from the transmitter. In other words, it is the region in which the transmitter can be viewed as a point source producing a spherical wave or a plane wave. Other regions of note are the near field and the Fresnel region. The near field is theregion before a coherent wave is formed; it includes the effects of imperfections of the transmitting antenna and evanescent waves. The Fresnel region is in between the near field and the far field; the effects of evanescent waves are no longer present, but the propagated beam

does not yet act as a point source [5 ], [6 ]. The Fresnel region, unlike the near field, can sometimes be used to advantage in wireless power transfer, either by design or by necessity. Because the antenna dimensions of the currently used SSP concept are very large to ensure transmission efficiency, the link would be considered to be in the Fresnel region, notthe far field region. The Fresnel region can also facilitate transmission as a plane wavefrom phased array antennas, rather than as a spherical wave. As a note: the Fresnel zone has a different definition than the Fresnel region. The Fresnel zone is an ellipsoidal regionin space surrounding both the transmitting and receiving antennas, and is defined in order to examine how obstructions near the antennas will affect transmission.

12 In order to capture the idea of the far field without intense analysis determining if a given antenna setup fits the above definition, the far field equation is used. This equation, while widely applied, is rarely explained. The full derivation is included here for easier access by future students, because I have not been able to find it anywhere else. The typical equation used to determine the far field is derived through the phase difference caused by the difference in the distance from one edge of the transmitting antenna tothe receiving point (Redge) and the distance between the transmitting antenna center and the π receiving point (Rc nt r). The allowable phase difference typically used is . The phase e e 8 difference can be converted into a physical distance; the corresponding physical distanceis the wavelength divided by sixteen. So, the distances from one edge and the center of the λ transmitter to the observation point must be different by no more than : 16

λ |R − R | ≤ (2.4) edge center 16

For an observation point directly in front of the transmitter, the difference between the two distances can also be written in terms of their geometrical components, with D being the diameter of the transmitting antenna, using the Pythagorean theorem:

q λ D2/4 + R2 − R ≤ (2.5) edge edge 16

This equation can be simplified:

D2 R λ λ2 + R2 ≤ R2 + edge + (2.6) 4 edge edge 8 162

D2 R λ λ2 ≤ edge + (2.7) 4 8 162

Considering the distance will be much larger than the wavelength, this simplifies as:

D2 R λ ≤ edge (2.8) 4 8

13 When solving for Redge and labeling it R, this becomes

2D2 R ≥ (2.9) λ

Which completes the derivation. It is important to note that this is the far field equation for electromagnetically long antennas; that is, antennas which are larger in diameter than the wavelength they emit. For electromagnetically short antennas, such as patch antennas, the far field distance is typically considered tobe 2λ. This differentiation is the basis for the examination of the far field specifically for phased array antennas, which will be discussed at a later point.

2.2 Wireless Power Beaming

With the acknowledgement that the analysis and origins of this work are rooted in RF technology used for communications and/or short distance wireless power charging, and thus many of the results contained within could therefore be retroactively used in those fields, the remainder of this work will be primarily focused on wireless power beaming, especially with regards to phased array antennas. For clarity, although the term “Wireless Power Transfer” can also refer to short distance wireless power charging, in the context of this paper, it refers to long distance (longer than a few wavelengths) power beaming. The topic of WPT has increased in popularity as a subject of research significantly in the past decade. The main objectives of power beaming, as opposed to other technologies using propagated microwaves, is reliable and cost-effective power transfer, rather than information transfer, among other objectives such as control, simplicity, efficiency, and feasibility. The use of phased array antennas for WPT is significantly different for transmitting and receiving antennas. For receiving antennas, phased array antennas are used to collect RF energy; for simplicity and robustness, many receiving antennas (named rectennas) are designed to collect incoming RF waves on an individual or sub-array basis, which allows for conversion to DC power on a large scale without the added complication of directional focusing.

14 On the other hand, the directional and beam-forming aspect of the transmitting phased array antenna is one of the most important aspects of long-distance WPT. The efficiency of a WPT system is based in part on the capability of the phased array antenna to deliver the most power possible to the receiving antenna surface. For space solar power, this requires large arrays, a narrow beam, and considerable control.

2.3 Gain and Directivity

One of the most important metrics when considering a transmitting system is the gain or directivity of the antenna array. The directivity is a measurement that can be thought of as the amount of power propagated in any given direction. It is formally defined as the radiant intensity (U(θ, φ), measured in Watts per steradian) divided by the average power (the total power output of the antenna array divided by 4π steradians):

U(θ, φ) D(θ, φ) = (2.10) P/4π This equation gives a ratio value for each angle of propagation; however, the directivity of an antenna or antenna array is often given as the maximum directivity of the entire antenna pattern, listed in dB. Gain, then, is the directivity multiplied by the efficiency of the transmitting antenna, η:

G(θ, φ) = ηD(θ, φ) (2.11)

Typical directivities of patch antennas are around 5-7 dB, whereas an isotropic antenna would have a directivity of 1 (or 0 dB); in that case, all directions receive equal radiation. Many antennas, including horn antennas, parabolic antennas, and others, have com- monly used equations that estimate the directivity. Often, the main way to increase the directivity of an antenna is to increase the size. For patch antennas, antenna gain is typ- ically increased by the formation of an antenna array, with specific tapering methods and

antenna arrangements being an area of considerable interest [7 ]–[10 ].

15 Another metric used to evaluate antenna arrays is the beam form. The beam form, like the directivity and gain, gives an idea of how much energy is propagated in each direction, however, it does not follow the same definition. The most commonly used way to find the beam form of an antenna array is through an array factor. In this case, the beam form E(θ, φ) is a product of the antenna gain D(θ, φ) and the array factor A(θ, φ):

E(θ, φ) = D(θ, φ)A(θ, φ) (2.12)

Where the array factor is dependent on the antenna configuration. For example, if the array is a linear, uniformly spaced array with N antenna elements, the array factor is:

N X jφn A(θ, φ) = ane (2.13) n=1

Where an is the amplitude of the nth antenna element and φn is the phase [11 ]. The phase can be described in terms of element location:

φn = kdn cos θ + δn (2.14)

Where k is the wave number (1/λ), dn is the nth element spacing, and δn is any additional phase shift from a phase shifter or other components. Although this is a fine way to calculate the sidelobe level reduction or to get a general idea of the directivity pattern, the beam form should not be confused with directivity. Because it is a multiplication of a ratio and a scaling factor, it no longer follows that the integral of the total directivity (in ratio form) is constant (4π), as would be the case if the directivity followed the original defining equation: the directivity, when taken as an integral overthe whole transmitted sphere, should be the total power divided by the average of the total power, i.e., 4π. This disconnect between the typically used method of describing a phased array antenna and the directivity equation will be discussed again later. Another effect often disregarded by directivity calculations is mutual coupling. Because antennas in phased arrays are within the near field of each other, each antenna’s radiation pattern will be affected by the presence of other antennas. This causes a minor decrease

16 in total gain that should be considered when developing a practical array but is commonly disregarded in most phased array antenna discussions. For this reason, in this work, the mutual coupling between antennas will be considered to be negligible, although it is an area

of research that has considerable interest and should be considered in the future [12 ], [13 ]. The previously mentioned equations for different types of antennas are a basic wayto estimate the gain of a given antenna. For a more detailed evaluation of the antenna gain, RF solvers and finite element analysis can be used to find the directivity of a given antenna configuration. This is especially helpful to include the effects of mutual coupling inaphased array. In practice, the directivity of an array can be found experimentally by recording the power received at a given distance while rotating the transmitting antenna to give a full view of the antenna power distribution. Unfortunately, this method of finding the gain is often impractical; the area of testing must be completely isolated from any external RF signals, and the size of antennas in question are often prohibitively large. A mix of small-scale testing and antenna modeling is often preferred.

2.3.1 Example: Transmit Antenna Trade-off Study

To emphasize the usefulness of the directivity as a metric for system design, an example of directivity evaluation is presented, including work done for the company Van Wyn and

results included in WISEE papers in 2019 and 2020 [14 ]–[17 ]. This study was a cost analysis for the transmitting antenna of a Sitallite Stratospheric Platform (a sitting satellite). In this example, the size of the transmitter was considered the primary variable; with a larger transmitter comes higher initial cost but also higher directivity, efficiency, and long- term electricity costs. Also considered in this study was the size of the receiving antenna. A larger receiving antenna would require a larger system overall to compensate for weight, and require more energy, but would also encompass more area to receive energy and would therefore be more efficient.

Figure 2.1 shows a cost analysis of the transmitting setup. As the transmitter becomes larger, the directivity of the transmitter increases, which allows more of the energy beamed

17 Figure 2.1. Transmit Antenna Costs [14 ] by the transmitter to be projected across the rectenna surface and thus increases efficiency. Because the efficiency increases, the amount of power required to be fed into the transmitting antenna decreases, as indicated by the falling costs of the associated generator. Also consid- ered in this analysis was the cost of the diesel required to run the generators; this application was to be implemented in a remote location. The total lowest cost is indicated by the red diamond; the increasing transmitter cost and decreasing generator and diesel costs provide a clear minimum on the curve of the total cost.

2.4 Sidelobe Levels

In any given antenna distribution pattern, there tends to be a singular “main lobe”, which has the highest directivity and is pointed towards the intended beam direction, and lower-directivity “sidelobes”. These sidelobes, while containing much less energy than the main lobe, are an important consideration in large scale wireless power transmission due to the high power levels involved.

18 Sidelobe levels are one of the main showstoppers of SSP for the time being. If sidelobe levels are not appropriately contained, the excess energy could cause significant problems for RF signal communications outside of the transmission area. Maximum incident RF energy for bystanders in areas adjacent to the receiver is also a serious consideration, but is less likely to be an issue. There are many methods previously explored in the subject of sidelobe level reduction, and additionally, many methods of forming RF transmission beams include the sidelobe level

as a limiting parameter [18 ], [19 ] One method of sidelobe level reduction is tapering of the phased array. In this method, the antenna elements forming the array are supplied with different power levels; typically, the antenna elements in the center are supplied with higher power levels than elements on the edge. This allows for the resulting beam to have a higher power distribution in the intended beam direction and less power directed elsewhere. Commonly used tapers are a Gaussian taper, a step-wise taper, and a Dolph-Chebychev taper, although there are many others. Another method of sidelobe level reduction is the placement of the antennas within the phased array. There are many antenna configurations, including in a line, in a circle, square placement, triangular placement, and others. The distance between antennas can be changed as well, although the spacing is generally dependent on the grating lobes. Grating lobes are features of a phased array antenna distribution that can appear when

the intended beam direction and the antenna element distribution cause spatial aliasing [20 ]. These lobes tend to be much higher than sidelobes, and are the result of coherence of the

beam in undesirable directions. As mentioned in Section 2.3 , the array factor and resulting antenna directivity pattern are a function of the amplitudes and phases of the individual elements. For example, when considering a uniformly spaced linear array, propagating in the direction θ, the individual antenna element phases can be calculated (similarly to Equation

2.14 ) as follows:

dn2π φn = sin θ (2.15) λ

19 Where λ is the wavelength and dn is the nth element spacing. This results in an array with the direction of maximum propagation being as follows:

λφ ! θ = sin−1 (2.16) d2π As this gives the direction in terms of a sine wave, there could be multiple different solutions if the antenna spacing and primary angle of propagation are not properly considered (the individual antenna element phases can be considered as φ + 2πm where m is a whole number). These different solutions are the grating lobes.

Figure 2.2. Grating Lobes for 1.5 λ Spacing, Zero Element Phase Shift

As an example, if a linear antenna array with an antenna spacing of 1.5λ is propagating straight forward, then the phase delay of each antenna is zero, and an alternate solution to

Equation 2.16 when m is 1 is 41.8°. This is shown in Figure 2.2 ; four antennas marked with blue triangles are shown with their associated radiation patterns. The coherence of the beam in the broadside direction is shown with the horizontal line; clearly, the zero-phase-difference antenna array propagates in that direction. The grating lobes are also present, shown as the

20 line slanted 41.8°to the right; although this is not the goal of this phase configuration, the phase becomes coherent in that direction anyway. With sidelobes and grating lobes in mind, the antenna placement and directions of propa- gation must be evaluated for directivity and associated efficiency, sidelobe levels, and possible grating lobes as well. All in all, the large number of variables involved and variations al- lowed make sidelobe level reduction one of the most complicated and interesting problems on the pathway to space solar power. Currently, the highest SLL reduction reported is -

120dB [21 ]. This configuration is dependent on an extremely large phased array setupwith a Dolph-Chebychev taper and minimal antenna failures.

2.5 Non-Traditional Phased Array Antenna Architecture

One of the main concerns of wireless power beaming is the cost of the total system. This cost includes not only the cost of the system components, but the cost of system transportation and setup, especially in the case of space solar power. With this in mind, systems with fewer components or lighter components can be seen as advantageous. Two potential methods for obtaining high results (as described above in terms of high directivity and low sidelobe levels) with fewer components are heterogeneous arrays, which use multiple different types of antennas in an attempt to increase directivity, and sparse arrays, which have selectively less antenna elements in different locations around the array. The idea of a heterogeneous array was conceived as a solution to issues presented by preliminary results of antenna array power distributions for very low sidelobe levels, as de-

scribed by Schubert in 2016 [21 ]. In this paper, an extremely low sidelobe level (-120dB) is the result of an extremely large phased antenna array with a Dolph-Chebychev taper. Because of the large dimensions of the array and the specificity of the taper, the elements at the center of the array require power levels that are several orders of magnitude larger than the elements at the edges. To attempt to alleviate the issues this causes with power distribution in a large array, antenna elements with a higher natural directivity were con- sidered for the central elements of the array, and elements with lower directivities for the edges. Unfortunately, initial results in the examination of this method were not favorable.

21 Because the elements had different radiation patterns from one another, they did notactas a cohesive phased array, and produced distribution patterns with lower directivity patterns than either element in a homogeneous array. Sparse arrays, on the other hand, are a widely examined method to reduce antenna

mass, volume, and costs [22 ], [23 ]. There are many different methods for implementing sparse arrays. Some methods involve removing antennas from the array randomly; others involve specific densities based on geometry or distance from the antenna center. Although the results of these arrays are more promising than heterogeneous arrays, they still produce less directive antenna patterns with higher sidelobe levels. Because of this, sparse arrays may be more practical for smaller scale applications in which sidelobe levels are not of such high importance. One method adjacent to sparse arrays is the idea of using unpowered antenna elements. Although this method does not help reduce the number of antennas used, it may reduce the cost of the supporting electronic equipment with less severe results than sparse antennas themselves. In models of this method, arrays with selected antennas remaining unpowered cause less disruption from full antenna results than arrays with the elements removed alto- gether. This may be because of mutual coupling effects; unpowered antennas provide the same electronic environment for their powered peers as powered antennas do, which, as in the case of the heterogeneous array versus the homogeneous array, could allow for higher beam coherence of the antenna as a whole.

2.6 Other Components of Wireless Power Transfer

There are many other components of Wireless Power Transfer that are discussed in detail in current publications. Some of these components will be briefly discussed here, including beam steering, link communications, and retrodirective antennas. One of the main advantages of Phased Array Antennas is their beam steering capability.

As discussed in Section 2.4 and shown in Figure 2.3 , the direction of the propagated wave is controlled by the phase delivered to individual antenna elements. This allows for steering even in situations where physical maneuvering of the antenna itself is not feasible, for ex-

22 ample, in kilometer-wide solar arrays. The technology of phase shifting itself is one area of interest not covered in this thesis, although there are many efforts to increase accuracy and

improve PAA control systems [24 ]–[26 ].

Figure 2.3. Linear Phased Array Antenna, θ = π/4

The communication between the transmitting and receiving antennas is very important, especially for risk reduction in links that have especially high directivities or power densities. In all space solar power configurations, there must be a way to ensure stable and fast commu- nication between the ground and the transmitter in case of emergencies. The frequency and power of this communication must be considered so as to not interfere with the transmitting link or vice versa.

One method of both communication and beam steering is retrodirective arrays [27 ]. In this method, the phase of the incoming beam to the receiving antenna is conjugated and used to send a pilot beam back to the exact location of the transmitting array. If the pilot beam is absent, the transmitter de-phases the antenna array, acting as an isotropic source and thereby reducing the amount of power sent in any one direction to prevent potential harm.

23 This method is the prevalent form of beam steering currently considered for space solar power, but there are still many questions to be answered about its specific implementation.

24 3. FAR FIELD DISTANCE STUDY

This chapter will discuss work examining the far field, specifically in regard to phased array antennas. A new model for the far field is presented, along with modeling done in MATLAB and experimental results, with the goal of understanding the transmission of an antenna array.

The far field, as discussed in Chapter 2 , is an important concept for ensuring coherence of phase across a receiving array. Although the impact of this phase difference depends on the configuration of the receiving antenna itself, it is an important consideration totake into account when looking at the transmission efficiency. Lack of phase coherence can cause decreases in efficiency due to the cancellation of power as it is collected by the receiving antenna. With this in mind, a fresh look at the far field of phased array antennas is required. While each individual element is electrically small, and so the far field for a single element would be 2λ, it does not make sense to adapt this as the far field for a phased array antenna. It also does not make sense to adapt the entire size of the array as the size to be used in a far field calculation, because the phase result at each point is not only determined fromthe distance from one side versus the other; it is also determined from the phase of the individual elements. For this reason, the far field distance of phased array antennas is discussed. One application for an examination of the far field distance is to reduce the necessary

distance required for antenna testing [28 ]. If a phased array antenna is used, as in the following discussion and in 5G applications, the traditional far field equation can be unnec- essarily limiting and examining the reasons behind it can produce smaller testing distances and consequently lower costs. Another reason to examine the far field is to glean more information about the power density at any given point between the transmitter and receiver. The maximum power density in a transmission setup is an important parameter to be aware of in order to ensure safety measures are followed.

25 3.1 Far Field Model

As discussed in Section 2.1.1 , the far field is neatly described in terms of the phase difference at a receiving point for electronically large antennas andas 2λ for electrically small antennas. This definition is somewhat lacking in terms of phased array antennas; the entire array is electrically large, but the individual elements are electrically small. Additionally, one of the main benefits of phased array antennas is the electronic steering implemented by adjusting the phase of individual elements. Because of this, the far field definition is lacking; since the phase can be changed from one edge of the antenna array to the other, allowing for the phase along a receiving plane to be manipulated, it no longer makes sense to define the far field in that way. The far field itself, separate from its typically used equation, is defined as the distanceat which all variation in directivity is a function of azimuth and elevation, not distance. The near-field is the region at which strong inductive or capacitive effects exist. Neither ofthese definitions allow for an examination of mid-range phased array antennas, at which power could be transferred but before the phase pattern acts as if it is from a point source. Instead, for this evaluation, this transition zone is examined as a function not only of distance, but of receiver size as well. As opposed to looking at the far field as caused by the phase difference due to the transmitting antenna size at a singular receiving point, the receiving antenna field is considered as the area of coherent phase over a plane produced by the resulting beam of a transmitting antenna. The limit for the coherence of phase will be π/2 radians, or λ/4. In other words, at a given distance from the transmitter, all points that have a resulting phase within π/2 radians of each other will be considered to be in the receiving antenna field. This value was chosen to ensure minimal interference of phase at the receiving antenna to maximize power received. This is a divergence from the traditional far field model; however, because this coherence of phase is needed to ensure efficiency of collection at the rectenna rather than to ensure each point at the far field has a coherent phase, it is more acceptable.

26 3.2 MATLAB Model

This evaluation of the receiving plane was modeled in MATLAB. Select code from MAT-

LAB is shown in Appendix A . The goal of the MATLAB model was to be able to provide an antenna array setup and a distance and evaluate all possible points that resulted in a phase that would allow coherence across a rectenna. There are many different starting points possible for this analysis; although any antenna configuration and individual antenna beam pattern would be allowed, for simplicity and coherence with lab work discussed later in this chapter, a patch antenna beam pattern was used along with a square, uniform array. The resulting beam pattern along a receiving plane was calculated by summing the resulting

electromagnetic field of each antenna, as described by Shinohara [11 ]. For ease of modeling, the transmitting antenna was assumed to be a uniform antenna array, with a square arrangement of patch antennas with 0.8λ spacing. The points in the far field were found by determining which points arranged in a square were within π/2 radians of phase with each other. The rectenna could be any shape; for this test, a square was chosen to match the shape of the transmitting antenna and because a square rectenna is easy to design and visualize. Additionally, the frequency was chosen to be 2.4 GHz to match the frequency of the laboratory experiments and because 2.4 GHz and 5.8 GHz are the most used frequencies for wireless power transfer analysis due to the atmospheric losses at those frequencies. A series of different antenna sizes were tested for maximum far field rectenna

size across various distances, as shown in Figure 3.1 .

Initially, the far field rectenna sizes shown in Figure 3.1 can seem somewhat chaotic, but they are, in fact, completely dependent on the size and shape of the antenna arrays

in question. Figures 3.2 through 3.5 , for example, compare the phase plane produced by a singular antenna vs. a 2x2 antenna array for various distances, labeled as 1 through 4 in red

on Figure 3.1 . In these figures, the placement of the antennas is marked by a black asterisk, and a square surrounding all possible points on a rectenna at that distance is marked in black. The color map of the figures is a color wheel, so that there isn’t a large differencein color for the phases 0 and 2π.

27 Figure 3.1. Far Field Square Rectenna Size vs. Distance

Figure 3.2 shows the comparison of phase at 125 mm: one wavelength away from the antenna. In reality, even an electrically small antenna would have a far field distance of at least 2λ; however, the relationship between antenna placement and phase is easier to see at this distance for a 2x2 antenna, so these figures are used for discussion purposes. The difference between the singular antenna and the 2x2 antenna phase pattern isquite clear. The singular antenna element produces a perfect phase pattern, whereas the phase produced by the 2x2 antenna element is actually more coherent across the plane at this close

distance. Similarly, in Figure 3.3 , the single antenna element produces a regular phase; the rectenna size has increased a slight but regular amount. On the other hand, the size of the 2x2 rectenna has decreased considerably, because at this distance, rather than allowing for more coherence, the phases of the 2x2 antenna cancel each other out and produce a smaller

possible rectenna. At a distance of 600 mm, as in Figure 3.4 , the rectenna sizes are almost the same, although the effects of the antenna array are still visible (especially across the diagonals; there is a difference in antenna spacing when viewing the antenna array across √ that axis, 0.8λ × 2 vs. 0.8λ).

28 Figure 3.2. Comparison of Singleton and 2x2 Array Phase at 125 mm

Figure 3.3. Comparison of Singleton and 2x2 Array Phase at 154 mm

While the effects of the antenna placement are easiest to see for a 2x2 antenna, thetrends

continue for larger sizes. Figures 3.6 through 3.11 show the results of the same analysis for

a 4x4 antenna array, shown at the distances marked in green in Figure 3.1 , as well as the

circular rectenna results, which are shown in total in Figure 3.12 . Similarly to the 2x2 antenna, the resulting maximum square rectenna for a 4x4 antenna increases and decreases in size, depending on the coherence of the antenna array at that point.

29 Figure 3.4. Comparison of Singleton and 2x2 Array Phase at 600 mm

Figure 3.5. Comparison of Singleton and 2x2 Array Phase at 1000 mm

Another feature of note is that at some points, especially at smaller sizes when the physical arrangement of the antenna is more prominent, the square rectenna size is larger, whereas at other points, especially when the effects of the arrangement have faded, the circular rectenna is larger. Either way, it remains higher than the maximum size of a rectenna for a single transmitting antenna for a considerable distance.

30 Figure 3.6. 4x4 Antenna Array: 125 mm Figure 3.7. 4x4 Antenna Array: 160 mm

Figure 3.8. 4x4 Antenna Array: 200 mm Figure 3.9. 4x4 Antenna Array: 350 mm

Another trend to note is that there is a point at which the mid-range effects examined above taper off and the maximum rectenna size for a given setup begins to trendtoward the result of the singleton antenna. This point could be seen as the beginning of the PAA’s far field; the resulting field begins to act only as a function of azimuth and elevation, notof distance. This far field point is compared with the traditional far field equation inFigure

3.13 . For this study, the traditional far field distance is linear with transmitting array

31 Figure 3.10. 4x4 Antenna Array: 850 mm Figure 3.11. 4x4 Antenna Array: 2 m area; the transmitting arrays are square, and the far field equation is proportional to the largest diameter squared. The points at which the array-based rectenna sizes trend toward the singleton rectenna size, found as percent reductions in the difference between the two where the initial difference is the point at which the trend first starts and there arenomore discontinuities, are linear as well, with an R2 value of 0.997 and 0.999 for an 80% reduction and a 70% reduction, respectively. This result further emphasizes the differences between singleton antennas and phased array antennas; the far field of a phased array antenna canbe significantly closer than the traditional far field equation would expect. While this specific relationship could change with the arrangement of the phased array antenna in question, it is still an important result to keep in mind. As shown in the above discussion, the MATLAB work provides a clear example of the lim- itations of the existing antenna distance models. Although the traditional model is useful in many applications, it does not account for the complexity that phased array antennas bring to the table. The result of this analysis would indicate that receiving antennas could poten- tially be larger than they are currently, allowing for more area of collection and potentially, higher efficiencies.

32 Figure 3.12. Far Field Circular Rectenna Size vs. Distance

Additionally, the power at a given distance for mid-range transmission may vary dramat-

ically within short distances; as the phase analysis in Figures 3.1 and 3.12 show, the rectenna phase coherence could change, causing differences in power. For example, a car that is being charged by wireless power beaming may need to adjust its position by mere meters in order to charge more efficiently. Examining phased array antennas through this lens could allow for greater design confidence, and could reduce power variability.

3.3 Laboratory Work at IUPUI

In addition to the theoretical work on and far field of antenna arrays, laboratory experi- ments were conducted to corroborate results.

Experimental Setup

The general setup of the RF laboratory at IUPUI, as shown in Figure 3.15 , included a transmitting antenna, a receiving antenna (Furious FPV Two Slices Patch Antenna 2.4GHz

33 Figure 3.13. Comparison of Traditional and Modeled Far Field for PAAs

RHCP), and the associated testing equipment; VNA RF signal generator (TPI Synthesizer, Model No. TPI-1001-B, Serial No. 0184), coaxial cables and connectors, oscilloscope (LeCroy Wavepro 7300A 3 GHz Oscilloscope), power amplifiers (WiFi Signal Booster 3000mW 2.4GHz 35dBm), and isolating foam, among others. To ensure accurate readings, the peripheral equipment in the RF lab was as out of the way as possible; in particular, anything including metal or RF waves was out of the line of sight from the transmitter to the receiver. Additionally, all metal equipment was separated from the testing equipment with isolating foam, if possible. All WiFi devices in the lab were placed on airplane mode to reduce interference. Any material that could potentially interfere with the RF transmission was out of the way, including personnel. Measures were taken to prevent potential harm, including staying out of the way of the transmitted beam and ensuring that persons with pacemakers or other similar health equipment stayed well away from any excess radiation. The VNA was connected to the laptop through a USB. The power amplifiers were con- nected in series with the RF transmission and connected to the power source. Wooden

34 Figure 3.14. Laboratory Setup at IUPUI frames were be used to set up the transmitting and receiving antennas, as shown in Figure

3.15 . The oscilloscope was used to take phase measurements. The TX waveform was set as the reference and the phase from zero-to-peak of the received waveform was measured. Although this did not measure the phase difference between the source and the receiver, it did measure the phase difference of the receiving antenna in different locations withthe same reference point (the phase measured directly from the VNA). The oscilloscope used is capable of measuring signals up to 3 GHz. The signal measured is 2.4 GHz, so while a regular measurement is possible, a better measurement was found using the FFT function of the oscilloscope, which displays the result based on multiple cycles of periodic signals. The purpose of this experiment was to determine the phase difference across a receiving antenna plane at specific distances in order to determine the maximum possible sizeofa receiving array at that point. Fields were tested at 84 cm, 120 cm, and 145 cm distances, limited by the space allowed in the lab. For each distance, measurements were taken across the array, starting at the center and moving outwards in increments of approximately λ/2 until the phase measured more than π/2 radians from the center measurement. The point

35 (a) Transmitting Setup (b) Receiving Setup

Figure 3.15. Laboratory Setup at which the phase difference became prohibitively large, that is, larger than π/2, was then determined to be the edge of the maximum possible rectenna at that distance. The lab experiment was adjusted with time spent in the laboratory; some of the pre- liminary results indicated the beam direction was not broadside, but slightly off axis. To prevent this, the transmitting antenna was changed from an adjustable four-antenna setup that required coaxial cables to each individual element to a design that only required one connection, because any sharp curve in the coaxial cable could cause a change in the phase and direct the beam off-axis, and having four different connections exacerbated this issue. The coaxial cable curve to the individual transmitting elements was not feasible to eliminate or correct for with the equipment on hand. With that correction made, the off-center nature was reduced considerably, but is still slightly present. One possible explanation for this could be the curvature of the coaxial cables used to connect the single source to each of the built-in antennas; because this PAA unit was a single piece, it was not possible to measure each of the individual phases or amplitudes. Another source of this off-center result could be due to a twist in the receiving plane, which was hard to mitigate without the use of materials that would have also disrupted the beam.

36 The results of the transmission experiment were compared with the MATLAB simulations of the same setup to determine if the MATLAB analysis matches real world data.

Experimental Results

The resulting phase differences across the receiving field generally agree with the farfield

discussions in Sections 3.1 and 3.2 . All final data from the experiment is copied in Appendix

B . An example of the far field phases is shown in Figure 3.16a . In this figure, the phase at each measured location is shown with the color bar to the right. A red circle the same size as a circular rectenna previously measured by MATLAB is overlaid on the data points for reference; as previously mentioned, there was an offset from center in all laboratory results, so the red circle is actually centered at (−0.13, 0) (m). Each data point outlined in black is within π/2 of the minimum point on the graph, which happens to be at (−0.26, −0.065)

(m). This figure is compared with the MATLAB results in Figure 3.16b . Although the MATLAB data is clearly much more precise, the trend of a flat circular area of similar phase surrounded by points of different phase holds true.

(a) Laboratory Results: 120 cm (b) MATLAB Equivalent Results

Figure 3.16. Comparison of Laboratory Results with MATLAB Data

37 Since the oscilloscope only measures the relative phase for each data point, not the total

phase, the phase difference has been considered from the center point. Figure 3.16 shows that the phase produced by the 2x2 transmitting antenna is coherent approximately as

expected, but the expected Far Field from the MATLAB model in Section 3.2 , as shown in red, does encompass some points with a phase difference slightly higher than π/2 radians, and excludes others with slightly less; the measured largest far field is not quite circular. This discrepancy could be explained by any number of potential sources of error, including stray EM waves from external sources or reflection of the source from the metal flooring, errors in measurement, or receiving antenna plane stability, as the cardboard used to host the rectenna began to bend with use. Another source of error is that the cardboard hosting the rectenna did not quite encompass the whole field; the testing of points near the bottom

of the array, as shown in Figure 3.16 , was cut off. Adjusting the height of the receiving array creates additional uncertainty, so these data points were excluded. As many of these sources of error as possible were considered while planning, including layering the isolating foam and repeating measurements. Methods to improve accuracy are included in the Future

Work section in Chapter 6 . The results of the laboratory experiments generally agree with the model previously

explored, as shown in Figure 3.17 . The 2x2 Circular Model line comes from the MATLAB data explained above, and the lab data is as described in the previous paragraphs. The traditional far field equation line comes from the maximum size that a receiving antenna

could be while still including the transmitter in its far field, as per Equation 2.9 . The error bars shown for the laboratory data stem from the sources of error listed above. The repeated trials to find the phase produced a standard deviation of around 0.06 radians, and the potential drift of the antenna’s phase over a testing period at most amounted to 0.09 radians. One of the biggest sources of error was the curvature of the cardboard; while this was measured and adjusted for, it could still contribute to potential error.

As one can see from Figure 3.17 , the receiving antenna size in the far field is much closer to the model discussed than the traditional far field equation. One reason for this may be because the traditional far field equation mandates a phase difference of nomore than λ/16, whereas the model discussed relaxes this standard to λ/4. However, because

38 Figure 3.17. Far Field Model Comparison this phase difference is no longer phase difference at a singular point from two different spots on a transmitting antenna but rather the phase difference between two different points on a receiving antenna, that could, additionally, be configured to disregard the phase of the incoming beam, this relaxation is seen as appropriate in phased array configurations.

As mentioned in Section 3.1 , the receiving antenna resulting from typical far field analysis discussions may be smaller than would allow for maximum efficiency. One drawback of this analysis is that only PAAs of modest size are considered. In the case of space solar power, and many other WPT applications, the size of PAAs are very large. The size of the PAA examined in the lab was around 16 cm x 16 cm; although this is larger in diameter than the wavelength used (12.5 cm) it is still much smaller than the PAAs required for SSP, which could be on the scale of kilometers. While the antenna sizes examined could be applicable for some WPT cases, for example, wireless power transfer to electric or flying cars, more analysis should be done on large scale PAAs.

39 4. FREE SPACE TRANSMISSION EFFICIENCY STUDY

This chapter will examine the free space efficiency of phased array antenna systems. Asmen-

tioned in Section 2.1 , the previously used equations are not entirely applicable to wireless power beaming; the purpose of this study is to remedy this and provide a useful, compre- hensive efficiency analysis starting point. There have been comparisons of the typically used efficiency equations in the past; however, the most common method of evaluation beyondthe

equations mentioned is numerical analysis of the system in question [29 ]. This study goes beyond that to create easy to understand, easy to use equations that predict the efficiency of a system efficiently and realistically.

Figure 4.1. Power Beaming Block Diagram

The overall efficiency of WPT systems has many components, related to the blocksin

Figure 4.1 ; there is the efficiency of the DC-to-RF system, the efficiency of the transmit- ting antennas, the free space transmission efficiency, the efficiency of the receiving antennas, and the efficiency of the RF-to-DC system. This chapter specifically examines free space efficiency: the power available for capture at the receiving antenna divided bythepower transmitted across the surface of the transmitting antenna. This definition of the free space efficiency and the definition of directivity are used to provide a simple, comprehensive method that can be used for any type of WPT system and is not based on specific antenna configu- rations or previous efficiency approximations.

40 4.1 Efficiency Equations

The following equation formulation will be based on the directivity of the transmitting

antenna and the geometry of the transmission setup, as shown in Figure 4.2 . The directivity

is defined as in Equation 2.10 : the radiant intensity divided by the average power. The angle Φ indicating the area of reception can be found using the dimensions of the transmission:

d/2! Φ = tan−1 (4.1) R

Figure 4.2. Layout Used for Efficiency Calculations

where d is the diameter of the receiving antenna and R is the distance between receiving and transmitting antennas. Although an azimuth angle of this kind would typically be represented by θ, in the case of power beaming, θ is used for the steering direction of the beam, so Φ is chosen for clarity. The total power can be calculated as the radiant intensity over all angles: Z 2π Z π Ptotal = U(θ, φ) sin θ dθ dφ (4.2) 0 0

And so the power at the receiving antenna can be calculated as

Z 2π Z Φ D(θ, φ) PR = Ptotal sin θ dθ dφ (4.3) 0 0 4π

41 If we assume radial symmetry, this equation simplifies to:

Φ Ptotal Z PR = D(φ) sin φ dφ (4.4) 2 0

Assuming a Constant D(φ)

If D(φ) is constant, integral is quite simple:

Φ Ptotal Z PR = D sin φ dφ (4.5) 2 0

PtotalD Φ PR = [− cos φ] (4.6) 2 0

PtotalD PR = [1 − cos Φ] (4.7) 2

" !!# PtotalD −1 Di/2 PR = 1 − cos tan (4.8) 2 R

  PtotalD R PR = 1 − q  (4.9) 2 2 2 R + Di /4 This derivation gives a simple, easy to use equation, but has limited accuracy; care must be taken to only apply this equation across receiving antennas in which the constant D(φ) assumption is reasonable. This condition is what prevents efficiencies higher than unity; only a couple dB decrease in directivity can be allowed from center to edge, which limits the size of the receiving area. Another necessary condition of this equation is that the receiving antenna be in the far field of the transmitter as a whole. Typically used antenna elements in aPAAwouldno doubt be in the far field of the receiver individually, but the receiver must also be farenough away for the transmitting antenna to form a single coherent beam.

42 Assuming D(φ) is Parabolic on a Logarithmic Scale

Typically, antenna directivity is displayed on a dB scale, and is shown to have a roughly parabolic curve. If this were a perfect assumption, the curve would be in the form

2 D(φ) = De−φ /β (4.10)

where D is the maximum directivity and β is some shaping constant. This is equivalent to a Gaussian distribution with µ = 0 and the standard deviation found from the maximum directivity and half-power beam width. Additionally, this can be found by approximating the dB curve with a second-order Taylor series. If b is one half of the half power beam width (HPBW) found from a pre-existing directivity pattern of the transmitting antenna, approximated from known configurations or calculated with AWR or other antenna analysis tools, then β can be found:

1 2 D(b) = D = De−b /β (4.11) 2

− ln(2) = −b2/β (4.12)

β = b2/ ln(2) (4.13)

This allows for the directivity to be written:

2 2 D(φ) = De−φ ln(2)/b (4.14)

The total power can then be found as:

Z Φ Ptotal −φ2ln(2)/b2 PR = De sin φ dφ (4.15) 2 0

43 The sin φ component can be approximated by φ on a small enough scale, so this equation simplifies as:

Z Φ Ptotal −φ2ln(2)/b2 PR = De φ dφ (4.16) 2 0

With a u-substitution of u = −φ2 ln(2)/b2, this becomes

2 Z −Φ2 ln(2)/b2 −PtotalDb u PR = e du (4.17) 4 ln(2) 0

2 PtotalDb h −φ2 ln(2)/b2 i PR = 1 − e (4.18) 4 ln(2)

Again, this equation should only be used in cases when the assumption of a Gaussian distribution and approximation of sin φ as φ hold true. Both φ and b should be in units of radians.

Using Numeric Integration

The integral form of the efficiency can be found numerically as well. This method, while not allowing for a simple, concise equation, does allow for a better approximation of efficiency for any pattern of directivity that is not easily approximated by the previous sections, and

has been used before [30 ]. It is important to note that the total efficiency (if taking the angle from zero to180 degrees) should be unity. Any pattern of directivity used should be scaled accordingly; if the integral from zero to 180 is not one, the integral over the desired area should be divided by this “total” efficiency. It is very helpful in system design to have a baseline equation that can estimate the expected gain. The following section discusses some of the relationships between antenna array designs and directivity, and the formation of a design equation for a uniform antenna array that can be used in conjunction with the efficiency equations presented.

44 4.2 Analysis of a Uniform PAA

As mentioned above, it was desired to have an easy-to-use equation for the directivity of a phased array antenna in order to predict the efficiency of a given transmission setup. Although there are many factors that can change antenna efficiency, a simple PAA setup was chosen for this evaluation. This allows for an estimation of directivity and efficiency for an antenna array that is simple to arrange and easy to replicate. The design chosen was a square array of uniformly spaced, uniformly powered patch antennas, with uniform phase. This design was modeled in MATLAB and AWR to find the directivity for differing transmit antenna sizes and antenna spacings.

The MATLAB code for this evaluation is found in Appendix A . The results found from MATLAB and AWR were very similar, besides a shift in directivity in AWR that resulted in all antenna arrangements with the same number of elements producing the same amount of directivity, regardless of the change in half power beam width. This is an indication that the AWR phased array wizard uses the antenna factor to find the gain, which as discussed

in Section 2.3 , is incorrect. AWR has been contacted about this issue. Their response, in part, is as follows:

[...the Phased Array Wizard] is considered to be a part of our Visual Systems Simulator (VSS), not Microwave Office (MWO). That means that it is partof a high-level behavioral approach to a complete system simulation rather than a precise/complete solution. At the practical level, this means that our “gain” is measured as signal power gain since it uses generic VSS measurements and it is not customized to phased array definitions. We take into account the array factor and element radiation patterns to calculate the signal gain at the output of the array. Since phased arrays are used in VSS as part of larger communications systems, we need to be able to track the signal power when they are present. [...] you would likely want to use AXIEM to do the actual design of your phased array.

In short, there is agreement about the fact that the antenna factor is used, but it is determined to be acceptable for that tool because it is primarily used for large scale com- munication system analysis; for more exacting power analysis, a different tool should be used. This is one example of the fact that most antenna tools are used for communications,

45 not power transfer; it is important to evaluate tools that are used to ensure accuracy for alternate applications. For this reason, the resulting MATLAB directivity is used to evaluate the directivity for different antenna configurations. The directivity was determined for a number ofdif- ferent configurations, changing both the spacing and the number of antenna elements; still maintaining a square array. The resulting directivities were found to increase linearly with respect to number of antenna elements per side. The relationship between the directivity

and the antenna spacing is a phenomenological one, and is shown in Figures 4.3 and 4.4 .

Figure 4.3. Maximum Directivity vs. Lambda Spacing

In general, the directivity increases linearly with the number of antenna elements, and the slope of that linear relationship also increases linearly with the lambda spacing. One possible reason for the difference between the shape of the curve at small lambda spacing and at large lambda spacing could be grating lobes. Because grating lobes start to appear around 1λ, the shape of the resulting directivity pattern changes; this could result in a differently shaped relationship than at closer spacings. As a result, this directivity in tandem with the efficiency equations above gives a baseline equation for the efficiency of a square uniformly spaced, uniform phase antenna. Although this is not a configuration likely to be used in practical WPT applications, it shows thatthis

46 Figure 4.4. Maximum Directivity vs. Number of Antennas type of relationship is possible to attain, and further relationships with different antenna arrangements, tapers, or directions could be an area of future work.

4.3 Past WPT Experiments

Most experiments of the size required to actually test the efficiency equations detailed above are too large for simple testing. Previous WPT experiments can be examined to give valuable insight into real-world applications, especially in cases that are much larger than feasible in a laboratory. Unfortunately, many of the details of past experiments are hard to find or simply unavailable. Several experiments are described here in detail to

form a comparison between the Friis (2.1 ), Goubau (2.2 ), and “Common Sense” (4.9 ,4.18 ) efficiency equations, and all sources of uncertainty regarding the details of the experiments are explained.

4.3.1 The Microwave Powered Helicopter

One of the first microwave power transfer experiments was conducted by William C.

Brown in 1964 [31 ]. The purpose of this experiment was to determine the feasibility of a

47 microwave powered helicopter; without the need to come back down to earth to refuel, a helicopter platform could stay up in the air indefinitely.

Figure 4.5. Microwave Powered Helicopter Experiment [31 ]

In this experiment, a microwave-powered helicopter was flown on a tether system, as

shown in Figure 4.5 , at a height of 50 feet for 10 hours. The transmitting antenna in the setup consisted of a trapezoidal feed horn and a 10-foot diameter ellipsoidal reflector. The 4ft2 rectenna for this project was one of the first ever, developed to receive the incoming energy with a weight lower than traditional receiving antennas at the time, and also without some of the associated cooling issues. The system was powered by a 5kW magnetron, and the frequency used was 2450 MHz. The reported DC output of the rectenna was 270 watts. The rest of the numbers used in the following analysis are educated guesses, as more accurate numbers are unavailable. The rectenna efficiency is not given. At a later point, William C. Brown boastsan 80% efficient rectenna system; this is used as an approximation in this case aswell.The transmitting antenna efficiency is not given, but the horn antenna used is quite similar to an antenna used in a different experiment, with a reported efficiency of around 68%.

48 The other important parameter that was unrecorded is the directivity of the transmitting system. In this case, the directivity was estimated through a commonly used equation to find the gain of horn and parabolic antennas:

4πA G = eA (4.19) λ2

Where A is the area of the mouth of the antenna and eA is the aperture efficiency.

In this case, since only the directivity and not the gain is of interest, eA was presumed to be subsumed in the “antenna efficiency” parameter. While the largest diameter ofthe transmitting antenna is recorded, the smaller diameter is not; an 8ft diameter is estimated from an examination of the photo of the transmission setup. The resulting directivity is around 37dB, which is an appropriate number for an antenna of this size.

4.3.2 The JPL Experiment

Another well-known wireless power transfer experiment was conducted by NASA in col-

laboration with the Jet Propulsion Laboratory (JPL) in May of 1975 [32 ]. In this experiment, the rectenna developed by William C. Brown was tested and verified for quality assurance, producing the highest efficiency on record.

This experiment was conducted with a horn antenna as shown in Figure 4.6 transmitting at a frequency of 2.45GHz towards a receiving array at a distance of 1.702m. The transmitter had a diameter of 57cm. The receiving antenna consisted of an array of half-wave dipoles placed in a triangular lattice. The largest number of antennas across the receiving array, based on system diagrams included in the report, was 12, and the spacing between them was assumed to be between 0.75λ and 1λ. The reported transmitting, receiving, and link efficiencies are 68.3%, 80.8%, and 54.2% respectively. The directivity in this example was found through diagrams in the technical memorandum for the relative power density of a dual-mode horn antenna based on position. The link distance in this experiment is really quite small in comparison to the transmitting and receiving antenna sizes. In the report from JPL, the free space power loss is considered

49 Figure 4.6. The JPL Experiment [33 ] to be negligible. This result is corroborated by all equations considered; as a result, this experiment is only included for completeness.

4.3.3 The Goldstone Experiment

One of the most famous long-distance wireless power transfer experiments was the Gold-

stone experiment, also conducted as a collaboration between JPL and NASA [34 ]. In this experiment, a beam was sent almost a mile over the Mojave Desert, lighting up a series of

bulbs with the transmitted power, as shown in Figure 4.7 . The link distance was 1.54km, and the frequency was 2388 MHz. The transmitting antenna was a parabolic antenna with a diameter of 26m. The largest receiver dimension was 7.242m, but the shape was not regular; the entire area was 24.5 m2, including 17 sub- arrays that were each 1.162 m by 1.207 m. The recorded path loss was 81.5%, and the Receiving antenna efficiency was recorded to be 81.5%. The directivity of the antenna was again calculated from antenna pattern

50 Figure 4.7. Goldstone Experiment [35 ] diagrams given in the technical document, which are in close agreement with a calculation

of the antenna directivity as a parabolic antenna similar to the calculations in Section 4.3.1 .

4.3.4 Equation Comparison

The data given for the experiments above are used to compare the Friis, Goubau, and

“Common Sense” equations, as shown in Figure 4.8 . As previously mentioned, the efficiencies for the JPL experiment all indicate that the free space loss is negligible, as reported. For the other experiments, the “Common Sense” efficiency equation both when considering the directivity as constant and when considering it as a Gaussian curve, come closer to the recorded efficiency than either the Friis equation or the Gaubou equation do. While this comparison shows promising results for the common sense equation, another take-away from this section: there is a dearth of quality information regarding long-distance

51 Figure 4.8. Free Space Efficiency Equation Comparison wireless power transfer experiments. Many experiments were considered and not included in this section, because there was not enough information about the transmitting antenna, receiving antenna, resulting efficiencies, or all three. During the course of research inthis subject, a call for data was requested of 167 WPT practitioners, but no new sources were

identified [36 ]. Not only must there be more long distance WPT experiments, but the results must be carefully recorded and shared to increase the wide-scale knowledge of this important subject across the board.

52 5. DISCUSSIONS AND ANALYSIS

The results presented in this thesis are relevant in a wide variety of applications.

As mentioned in Chapter 3 , the result that the far field is unnecessarily restrictive for phased array antennas could result in smaller testing areas for the PAAs used in 5G ap- plications. Also, the examination of the phase distributions produced by PAAs could have potential applications for mid-distance wireless power beaming. It also provides a stepping point for the examination of the power distribution surrounding a phased array antenna, which could have applications in many areas. Another potential application for this work, as discussed in the introduction, is Space Solar Power. Although there has been much discussion about this technology since it was

first introduced in the late 1970s, the basic setup is not much changed[37 ]–[41 ]. In the most common design, a 1 km diameter transmitting antenna with a geosynchronous equatorial orbit (GEO) at around 35 thousand km transmits power to an elongated receiving antenna

with a collection area with a 10 km diameter [37 ]. The free space efficiency from the economic analysis of the SSP concept was considered to be almost negligible; the ionospheric and atmospheric transfer efficiency was considered to

be 90% at worst [40 ]. In a different document written by William C. Brown on the efficiency

of the rectenna, the free space efficiency of the beam was considered to be over 90%[41 ]. In actuality, when considering a transmission of that distance to a rectenna of that size, the transmission efficiency is not negligible. In order to achieve an efficiency of 90%, a directivity of around 89 dB must be achieved

(as implicated by the analysis of Chapter 4 . If a 60dB directivity was used instead, the efficiency would only be around 0.122%. When considering the size of the transmitting antenna, the requirement of an 89 dB beam is a high order. If considering the transmitting antenna to be a square, uniformly powered phased array antenna with a 0.8λ spacing at 5.8 GHz, then the directivity would be only

around 47.63 dB (from the analysis presented in Section 4.2 ). For an 89 dB beam, the size of the transmitting antenna would need to be around 950% larger.

53 As this analysis shows, the free space efficiency is not a negligible parameter when con- sidering space solar power. Although directivities of this magnitude aren’t impossible with the transmitter size given, it is important to recognize this as a limiting parameter; the technology is not there yet, and cannot be assumed to be there for system designs going forward. This area requires work, and should not be ignored. Along the same vein, this work also impacts many other technologies, including any tech- nology requiring long distance wireless power transfer; examples include mother-daughter satellite configurations, high altitude stratospheric pseudo-satellites (HAPS), and power transfer to remote areas including disaster recovery and Forward Operating Bases (FOBs). If care is not taken to ensure accurate analysis of the efficiencies involved, the resulting designs will be estimated as smaller than actually required, and progress of wireless power transfer will be stalled. Accurate design equations are necessary for progressing this technology, and should be shared as much as possible.

54 6. CONCLUSIONS AND FUTURE WORK

As discussed in the previous chapters of this work, the differences between phased array an- tennas and singleton antennas and the differences between power transfer and signal transfer should not be neglected.

In Chapter 2 , the fundamentals of wireless power transfer were discussed. In many ways, the differences between WPT and radio-wave based signals are small; both relyon fundamental properties of energy transfer and phase based signals. However, assuming that there are no differences, especially in the case of the far field and efficiency equations, isan oversimplification that can lead to uninformed design.

As discussed in Chapter 3 , the far field of an antenna is based almost entirely onthe phase at given receiving points. In the traditionally used equation, the far field is determined to be the distance at which the phase difference caused by different distances between the edges of the transmitter and the receiving point became negligible. This can be an important metric, because it ensures that other governing equations for WPT still apply.

However, as examined in Chapter 3 , the phase plane of a phased array antenna acts differently. It does not make sense to label the far field as the point where each individual antenna’s far field begins, because the phased array has not yet formed a coherent beam. It also does not make sense to label the far field as the point at which the phase from each individual element is within a certain phase difference, because that would create a definition that is unnecessarily restrictive, especially since the nature of a phased array antenna is to change the phase of individual elements based on design needs. Instead, as discussed, a far field model can be produced that encompasses the unique nature of a phased array antenna; the resulting phase at each point can be determined over a receiving plane to determine the maximum possible receiving antenna size at that distance. Examining the far field in this way has the potential to produce more flexible transmitting and receiving antenna designs, and provides a clearer picture of the resulting phased array antenna beam. Also discussed is the evaluation of the free space efficiency of wireless power transfer.

As discussed in Chapter 2 , previously used equations such as the Friis equation and the

55 Goubau equation have fundamental limitations that, while not vitally important in most applications, make their use in WPT applications limited. The Friis equation is defined through the effective area of an antenna. In the sameway that the size of a phased array antenna does not effectively convey information about the far field the same way the size of a singleton antenna does, the size of a phased array antenna does not convey information about the effective area the same way a singleton antenna does. This causes a fundamental limitation when applying the Friis equation to WPT. The Goubau equation, similarly, was designed through the evaluation of reiterative wave beams. Although it can be applicable in some situations, it does not make sense for most applications of long-distance wireless power transfer, because there is no mechanism for the reiteration of the phase profile. It also does not include super-gain antennas; thisisa potential limitation, as there is no theoretical limit to the gain of a phased array antenna. With these limitations of efficiency equations in mind, the equation formulation inChap-

ter 4 is presented. The efficiency of any WPT setup can be evaluated with this method. Although the most accurate evaluations would require additional information, including the directivity of the transmitting antenna, the simplified equations can give a general idea for the efficiency of a layout, as well. These equations could help the design of long-distance wire- less power transfer by providing realistic, easy-to-use equations that estimate the efficiency of a given system. One important note from this work is the result of a comparison of the beam profiles from

AWR and individual analysis. As mentioned in Section 4.2 , the “Phased Array Wizard” in the AWR tool uses an antenna factor to find the beam form and labels it the gain. Although this is perfectly acceptable as a method to find the general profile for directivity for agiven array, it is not perfect; it does not actually determine the gain as can be used per the

definition as in Chapter 4 . This should serve as a reminder that while most microwave tools used for system analysis should be able to be converted from use on signal transfer systems to power transfer systems, not all parameters are the same or have the same importance. While the general approach of antenna engineers seems to be rather lax, it’s important to remember that all equations, even the Friis and Goubau equations which together have

56 been cited over a thousand times, are formed through a series of assumptions which may or may not hold when the application is far enough removed from the source.

6.1 Future Work

Given the complex nature of wireless power transfer and phased array antennas in par- ticular, there are many more studies that could be conducted in this area in the future. Additional experiments regarding long-distance wireless power transfer are absolutely

necessary. As mentioned in Chapter 4 , there is a lack of WPT experiments in general with enough information available to draw reasonable conclusions. Additionally, experiments that

can mitigate sources of error as mentioned in Chapter 3 , such as potential reflections of the transmitted beam and instability in receiving antenna equipment, should be pursued. This could include testing within an anechoic chamber, or potentially testing in locations of future use, such as the tops of buildings or out-of-doors. One large potential area of study is the expansion of the phased array antenna models presented in this work to include other directions, layouts, and tapering. All of the phased array antennas evaluated in this work were designed to transmit power straight forward (broadside direction). Additionally, most of the antennas were designed in a square forma- tion, with regular spacing (usually 0.8λ). Also, all of the arrays considered were uniform. One of the largest benefits of phased array antennas, aside from their directional capabil- ities, is the benefit of increased gain with only changes to layout or tapering. For example,

the evaluation of the directivity of phased array antennas, as provided in Section 4.2 , could be completed with a Dolph-Chebychev taper and a triangular spacing, which would increase the directivity considerably for the same number of antenna elements. Many studies about the best possible configurations for phased array antennas to form the incredibly narrow beams required for SSP are already underway; inclusion of those methods for this type of analysis could be helpful as well. Another hugely important area of study is the mutual coupling between antenna elements and their effect on the antenna directivity and phase. Although the mutual coupling is

57 ignored in many cases, it could have a significant impact on the overall directivity ofan antenna system. One possible method of mutual coupling analysis is as follows. An element would be studied in multiple different electronic environments; its electrical field would be determined as it is by itself as well as in various antenna configurations. The difference in the resulting electrical field for that singular element would be the result of mutual coupling. Thisdif- ference in resulting field could then be applied to an entire array through the arrayfactor and resulting beam form, and that beam form could be compared to the array without the mutual coupling factor and to the array profile as determined through finite element analysis, which would include the effects of mutual coupling as well. Other considerations for mutual coupling could include the effect that mutual coupling has on sparse arrays, and if the effect of unpowered elements rather than removed elements

is, in fact, the result of mutual coupling, as is hypothesized in Section 2.5 . Mutual coupling is possibly one of the most important areas of future study, as it directly relates to many of the issues discussed in this thesis. All in all, this area of research has many avenues to pursue, and will require much work before the goal of reliable, clean, constant Space Solar Power is possible.

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61 A. MATLAB Code

Attached is some of the MATLAB code used at various points in this work. The main file is as follows.

1 %THIS IS TO PLOT THE PHASE AT A GIVEN DISTANCE ACROSS AN ENTIRE FIELD 2 %Functions used in this code: pondProp3D , farFieldSizeCirc , farFieldSizeSq 3 %AJ Finnell , 2021 4 5 %constants 6 c = 299792458; %m/s 7 freq = 2.4*10^9; %Hz 8 lambda = c/freq; %m 9 spacing = 800*c/freq; %space between antennas: 0.8* lambda (in mm) 10 11 %calculation of time required to encompass an entire wave , in ps 12 maxTime = ceil(10^12/freq) + 10; 13 14 %recieving field setup 15 sizeField = 90; %90x90 tested grid 16 sqSize = 14; %size of each edge; aprox 1/9th lambda spacing 17 hand = figure('position ' ,[100,100,600,570]); %handle for resulting MATLAB figure 18 edgeForPlot = 0:14:14*(sizeField-1); %x and y vector for plotting 19 edgeForPlot = edgeForPlot - mean(edgeForPlot); 20

62 21 exes = 1805:5:2500; %distances to be tested 22 uptoAnt = 6; %largest antenna size to be tested 23 resSec = zeros([length(exes),uptoAnt]) '; 24 25 for txSize = 1:uptoAnt 26 27 %antenna location setup 28 k = 1; 29 clear antennas 30 for i = 1: txSize 31 for j = 1: txSize 32 antennas(k,:) = [0,i*spacing,j*spacing]; %x,y,z coordinates for antenna 33 k = k+1; 34 end 35 end 36 antennas(:,2) = antennas(:,2) - mean(antennas(:,2)); % centered in grid 37 antennas(:,3) = antennas(:,3) - mean(antennas(:,3)); 38 39 %antenna phase shift setup 40 shift = zeros(size(antennas(:,3))); 41 42 %antenna amplitude setup 43 amps = ones([1,length(antennas)]); %uniform amplitudes 44 45 secRsizePAA = zeros(size(exes)); 46 47 %TEST LOOP

63 48 for h = 1:length(exes) %distance in mm 49 50 xDist = exes(h); 51 %clear previous loop info 52 resField = zeros(sizeField); 53 resFieldReal = zeros(sizeField); 54 clf(hand ,'reset'); 55 clear T; 56 57 58 %phase across field 59 for i = 1:sizeField 60 for j = 1:sizeField 61 clear C; %C is the aggregate sine wave of all antennas formed by the loop 62 C = zeros([1,maxTime]); 63 for time = 1:ceil(10^12/freq) %10^12/ freq is the anticipated wavelength in ps 64 for k = 1:length(antennas(:,1)) 65 C(time) = C(time) + pondProp3Dexp( antennas(k,1),antennas(k,2), antennas(k,3),freq,amps(k),shift(k ),time+10000000,xDist,edgeForPlot( j),edgeForPlot(i)); 66 end 67 end 68 rPart = real(C); 69 C = imag(C); 70 t = 1;

64 71 while(abs(C(t)-max(C)) > 0.0000001) 72 t = t+1; %find when the max is: phase shifted to that pt 73 end 74 resField(j,i) = t*freq*2*pi/10^12; %time shift to radians 75 t = 1; 76 while(abs(rPart(t)-max(rPart)) > 0.0000001) 77 t = t+1; %find when the max is: phase shifted to that pt 78 end 79 resFieldReal(j,i) = t*freq*2*pi/10^12; %time shift to radians 80 end 81 end 82 83 %far field points across field 84 T = farFieldSizeCirc(edgeForPlot,resField); 85 rad = max(T(length(T),1),T(length(T),2)); 86 secRsizePAA(hmm) = pi*(rad)^2; 87 88 %to plot circle around FF 89 ang = linspace(0,360)*pi/180; 90 xf = rad.*cos(ang); 91 yf = rad.*sin(ang); 92 zf = 4.5*ones(size(ang)); 93 94 %plot results and save figure as jpg 95 %plot phases

65 96 surface(edgeForPlot,edgeForPlot,resField,'edgecolor ', 'none') 97 colormap(rainbowMap(100)) 98 caxis([0,2*pi]); 99 hold on 100 %plot surrounding circle 101 plot3(xf,yf,zf,'r','LineWidth ' ,3); 102 %plot antennas 103 plot3(antennas(:,2),antennas(:,3),2*pi*amps,'*',' MarkerSize ',10,'MarkerEdgeColor ','k') 104 view (2) %sets 2-D view 105 axis([min(edgeForPlot),max(edgeForPlot),min( edgeForPlot),max(edgeForPlot)]) 106 axis equal 107 xlabel("X Distance (mm)") 108 ylabel("Y Distance (mm)") 109 title(sprintf('Phase Plot at a Z Distance of %d mm', xDist)) 110 saveas(hand,sprintf('FF_sq_for%dAat%d.jpg',txSize , xDist)); %save as jpg 111 112 end 113 114 resSec(txSize,:) = secRsizePAA '; 115 116 end

The function farFieldSizeCirc is as follows (farFieldSizeSq is the same, but with a series of points in a square checked instead of a circle).

1 function [rectenna] = farFieldSizeCirc(edges,phases)

66 2 %This function receives a plane that includes all phases , location of each point given by edges (in mm). The rectenna is produced by searching for the first phase outside of bounds in circles checked through interpolated phases. 3 %AJ Finnell , 2021 4 5 %first get center phase 6 x = 0; 7 y = 0; 8 n = 1; 9 10 %find center 11 i = 1; 12 while(edges(i) < x) 13 i = i+1; 14 end 15 j = 1; 16 while(edges(j) < y) 17 j = j+1; 18 end 19 %interpolation 20 lt = phases(i-1,j); 21 rt = phases(i,j); 22 t = (rt - lt)/2 + lt; 23 lb = phases(i-1,j-1); 24 rb = phases(i,j-1); 25 b = (rb - lb)/2 + lb; 26 ph = (t-b)/2 + b;

67 27 28 originPhase = ph; 29 30 rectenna(n,:) = [x,y,ph-originPhase+pi]; 31 n = n+1; 32 33 34 %then get the phase for each surrounding ring 35 test = 1; 36 for k = 1:length(edges) 37 38 radius = k*14; 39 leng = 2*pi*radius; 40 numbertest = ceil(leng/14); 41 angles = (0:360/numbertest:360)*pi/180; 42 x = radius.*cos(angles); 43 y = radius.*sin(angles); 44 45 fin = n; 46 47 for m = 1:numbertest 48 49 i = 1; 50 while(edges(i) < x(m)) 51 i = i+1; 52 if i > length(edges) 53 test = 0; 54 sftl = sprintf("Search Field Too Large") 55 break

68 56 end 57 end 58 59 j = 1; 60 while(edges(j) < y(m)) 61 j = j+1; 62 if j > length(edges) 63 test = 0; 64 sftl = sprintf("Search Field Too Large") 65 break 66 end 67 end 68 69 if test == 0 70 break 71 end 72 73 %interpolation 74 lt = phases(i-1,j); 75 rt = phases(i,j); 76 if lt-rt > pi/2 77 lt = lt - 2*pi; 78 elseif rt-lt > pi/2 79 rt = rt - 2*pi; 80 end 81 t = (rt - lt)*(x(m) - edges(i-1))/(edges(i) - edges(i -1)) + lt; 82 lb = phases(i-1,j-1); 83 rb = phases(i,j-1);

69 84 if lb-rb > pi/2 85 lb = lb - 2*pi; 86 elseif rb-lb > pi/2 87 rb = rb - 2*pi; 88 end 89 b = (rb - lb)*(x(m) - edges(i-1))/(edges(i) - edges(i -1)) + lb; 90 if b - t > pi/2 91 b = b - 2*pi; 92 elseif t - b > pi/2 93 t = t - 2*pi; 94 end 95 ph = (t-b)*(y(m) - edges(j-1))/(edges(j) - edges(j-1) ) + b; 96 ph = ph - originPhase+pi; 97 ph = mod(ph,2*pi); 98 99 rectenna(n,:) = [x(m),y(m),ph]; 100 n = n+1; 101 end 102 103 %making sure rectenna is in FF 104 if (max(rectenna(:,3)) - min(rectenna(:,3)) > pi/2) 105 rectenna(fin:n-1,:) = []; 106 n = length(rectenna) + 1; 107 test = 0; 108 break 109 end 110

70 111 end 112 113 end

The function pondProp3Dexp is as follows. The 1D and 2D versions were conceptualized through the ripples propagating in a still pond after stones are dropped in; thus the name “pondProp”.

1 function [propVals] = pondProp3Dexp(xLoc, yLoc, zLoc, freq, amp, phase, time, x0, y0, z0) 2 %This function takes the input values of one patch antenna source and outputs the propagation , using angleLookup for the patch antenna distribution. 3 % xLoc: x value of the source origin (mm) 4 % yLoc: y value of the source origin (mm) 5 % zLoc: z value of the source origin (mm) 6 % freq: frequency of the source (hZ) 7 % amp: amplitude of the source 8 % phase: phase shift of the source (rad) 9 % time: time that the source has propagated (ps) 10 % x0: initial x-axis value (mm) 11 % y0: initial y-axis value (mm) 12 % z0: initial z-axis value (mm) 13 %AJ Finnell , 2021 14 15 c = 299792458; %speed of light (m/s) 16 17 %FOR A POINT 18 dist = sqrt((y0-yLoc)^2+(x0-xLoc)^2+(z0-zLoc)^2); %distance from antenna (x,y,z) to point in question (x0,y0,z0)

71 19 if (dist/(1000*c) > time/10^12) %if it hasn 't reached that point yet , it's zero 20 propVals = 0; 21 else 22 %find angles for angleLookup 23 if abs(z0) < 0.0000000001 24 phi = 0; 25 else 26 phi = atan(y0/z0)*180/pi; 27 end 28 theta = atan(sqrt(y0^2+z0^2)/x0)*180/pi; 29 mag = angleLookup(phi,theta); 30 31 %propagation calculation 32 propVals = mag*(1000*amp/dist^2)*exp(sqrt(-1)*((2*pi*freq )*(time/10^12-dist/(1000*c))-phase)); %if it has reached that point , it's at this amplitude in the cycle , divided by r^2. 33 end 34 35 end

72 B. Laboratory Data

Below is the lab data referenced in Chapter 3 . All of the X and Y distances are measured from the center of the transmitting array. The trial data are all measured as a time difference from the same reference point, in nanoseconds.

Table B.1. Lab Data: 1.45 m Distance

X (m) 0 0 0 0 0 0 0 0 0 Y (m) 0 0 0 -0.065 -0.13 -0.195 -0.26 0.065 0.13 Trial 1 57 57 62 66 78 115 167 55 82 Trial 2 62 55 67 64 84 122 171 53 75 Trial 3 64 63 60 63 75 120 170 56 81 Trial 4 62 70 68 61 83 122 167 59 68 Trial 5 54 62 59 66 86 121 165 62 71 Trial 6 61 54 50 60 78 120 169 57 69 Trial 7 58 56 56 64 80 112 173 59 70 Trial 8 61 53 59 63 81 120 159 63 75 Trial 9 56 56 59 59 86 121 168 61 68 Trial 10 60 52 56 69 85 118 165 59 74

X (m) 0 0 0 0 0 0 -0.065 -0.065 -0.065 Y (m) 0.195 0.26 0.325 0 0 0 0 -0.065 -0.13 Trial 1 107 158 193 48 57 49 38 55 73 Trial 2 107 146 198 48 48 48 36 48 75 Trial 3 102 148 194 49 46 46 46 45 72 Trial 4 110 145 197 47 44 49 44 59 68 Trial 5 99 140 194 52 47 50 34 52 62 Trial 6 113 137 190 51 50 50 39 56 68 Trial 7 108 139 190 47 44 38 35 50 64 Trial 8 111 129 187 52 50 49 46 51 72 Trial 9 101 134 189 46 44 49 41 48 68

73 Table B.1. Lab Data: 1.45 m Distance (Continued)

Trial 10 100 131 194 51 50 47 46 48 68

X (m) -0.065 -0.065 -0.065 -0.065 -0.065 -0.065 -0.065 0 0 Y (m) -0.195 -0.26 0.065 0.13 0.195 0.26 0.325 0 0 Trial 1 113 157 47 68 91 131 190 37 36 Trial 2 120 147 50 62 96 126 195 43 36 Trial 3 117 161 44 65 85 123 191 37 42 Trial 4 109 153 50 68 92 120 193 37 45 Trial 5 111 160 49 62 99 133 191 41 40 Trial 6 110 151 53 60 90 124 192 38 41 Trial 7 117 143 45 66 94 134 188 36 36 Trial 8 114 153 46 67 88 127 187 41 36 Trial 9 113 152 46 65 87 134 193 45 41 Trial 10 113 155 42 59 92 129 191 40 41

X (m) 0 -0.13 -0.13 -0.13 -0.13 -0.13 -0.13 -0.13 -0.13 Y (m) 0 0 -0.065 -0.13 -0.195 -0.26 0.065 0.13 0.195 Trial 1 40 14 12 38 91 134 38 61 97 Trial 2 45 19 23 38 82 128 29 61 92 Trial 3 42 14 20 36 92 131 32 58 83 Trial 4 40 17 26 47 82 130 27 61 84 Trial 5 46 23 20 38 86 139 34 52 88 Trial 6 38 21 36 43 77 130 34 63 95 Trial 7 38 17 18 41 87 137 36 53 90 Trial 8 40 19 25 49 80 130 40 50 85 Trial 9 34 26 19 40 86 132 39 61 97 Trial 10 31 15 25 44 79 139 30 53 89

X (m) -0.13 -0.13 0 0 0 -0.195 -0.195 -0.195 -0.195 Y (m) 0.26 0.325 0 0 0 0 -0.065 -0.13 -0.195

74 Table B.1. Lab Data: 1.45 m Distance (Continued)

Trial 1 150 188 50 47 51 23 32 42 75 Trial 2 141 192 45 45 41 28 41 38 80 Trial 3 151 202 47 47 48 26 26 41 80 Trial 4 146 195 48 44 44 28 27 35 76 Trial 5 137 188 45 43 47 34 34 34 68 Trial 6 143 200 46 39 43 28 31 41 79 Trial 7 134 197 52 51 52 25 28 36 73 Trial 8 146 193 51 45 48 29 35 42 85 Trial 9 145 194 46 45 50 29 34 38 76 Trial 10 150 186 49 47 49 28 31 39 86

X (m) -0.195 -0.195 -0.195 -0.195 -0.195 -0.195 0 0 0 Y (m) -0.26 0.065 0.13 0.195 0.26 0.325 0 0 0 Trial 1 142 37 66 104 152 203 37 38 45 Trial 2 140 47 58 104 153 201 46 43 41 Trial 3 141 37 59 107 152 211 48 39 38 Trial 4 140 39 64 108 148 203 39 50 43 Trial 5 145 38 65 108 155 203 42 50 45 Trial 6 136 38 60 102 148 202 40 45 42 Trial 7 140 39 61 105 154 201 40 45 44 Trial 8 139 45 63 104 154 198 40 44 46 Trial 9 142 39 52 103 152 202 43 42 46 Trial 10 137 42 58 100 147 200 46 49 35

X (m) -0.26 -0.26 -0.26 -0.26 -0.26 -0.26 -0.26 0 0 Y (m) 0 -0.065 -0.13 -0.195 0.065 0.13 0.195 0 0 Trial 1 112 126 125 169 117 138 138 38 44 Trial 2 107 122 132 165 112 137 174 49 47 Trial 3 117 125 138 171 114 133 164 37 51

75 Table B.1. Lab Data: 1.45 m Distance (Continued)

Trial 4 114 121 125 159 125 139 169 45 49 Trial 5 114 128 131 164 121 145 166 40 40 Trial 6 109 127 132 173 116 140 167 38 50 Trial 7 110 126 140 170 120 143 163 39 43 Trial 8 114 127 133 160 111 135 157 39 51 Trial 9 114 127 132 167 111 123 171 41 39 Trial 10 114 119 132 162 119 141 164 40 53

X (m) 0 -0.325 -0.325 -0.325 -0.325 -0.325 -0.39 0.065 0.065 Y (m) 0 0 -0.065 -0.13 0.065 0.13 0 0 -0.065 Trial 1 50 136 136 163 143 161 197 119 128 Trial 2 46 141 145 164 151 164 205 119 136 Trial 3 39 142 135 168 147 162 200 116 135 Trial 4 43 138 140 154 153 156 203 123 131 Trial 5 48 133 136 155 138 153 203 122 136 Trial 6 48 148 143 167 146 159 208 119 131 Trial 7 43 131 138 167 154 161 202 112 128 Trial 8 45 132 135 162 151 156 201 117 133 Trial 9 41 145 138 158 145 157 207 117 127 Trial 10 41 137 142 159 142 157 199 120 140

X (m) 0.065 0.065 0.065 0.065 0.065 0 0 0 0.13 Y (m) -0.13 -0.195 0.065 0.13 0.195 0 0 0 0 Trial 1 138 168 126 147 168 34 36 42 150 Trial 2 140 161 123 142 171 43 45 45 157 Trial 3 133 166 128 143 168 42 46 45 152 Trial 4 134 154 122 145 176 42 37 42 159 Trial 5 142 163 129 157 172 46 42 45 163 Trial 6 132 167 127 142 169 50 50 43 148

76 Table B.1. Lab Data: 1.45 m Distance (Continued)

Trial 7 136 160 139 145 174 43 42 33 154 Trial 8 132 153 123 145 169 42 45 31 146 Trial 9 142 161 125 146 172 44 44 33 161 Trial 10 135 165 127 137 168 41 36 41 147

X (m) 0 Y (m) 0 Trial 1 42 Trial 2 34 Trial 3 50 Trial 4 54 Trial 5 39 Trial 6 37 Trial 7 48 Trial 8 53 Trial 9 47 Trial 10 50

Table B.2. Lab Data: 1.2 m Distance

X (m) 0 0 0 0 0 0 0 0 0 Y (m) 0 -0.065 -0.13 -0.195 -0.26 0.065 0.13 0.195 0.26 Trial 1 11 -4 4 14 46 26 42 85 144 Trial 2 1 -11 3 16 49 22 41 89 148 Trial 3 7 -10 4 17 59 26 43 87 147 Trial 4 -2 -2 3 21 37 30 39 90 141 Trial 5 4 0 3 28 36 28 40 87 148 Trial 6 5 -1 4 26 43 26 44 93 140 Trial 7 2 -12 1 16 50 27 36 84 145

77 Table B.2. Lab Data: 1.2 m Distance (Continued)

Trial 8 3 -1 3 28 50 25 40 86 147 Trial 9 3 -3 4 14 53 26 40 88 150 Trial 10 1 -8 3 19 52 31 40 89 145

X (m) -0.065 -0.065 -0.065 -0.065 -0.065 0 -0.065 -0.065 -0.065 Y (m) 0 -0.065 -0.13 -0.195 -0.26 0 0.065 0.13 0.195 Trial 1 -14 -18 -16 4 33 -2 1 26 79 Trial 2 -14 -23 -10 0 34 -9 4 25 73 Trial 3 -14 -12 -12 -5 31 -4 1 27 75 Trial 4 -12 -24 -11 0 40 -1 -3 27 73 Trial 5 -6 -24 -8 3 35 -2 4 38 72 Trial 6 -13 -22 -11 -3 35 -8 3 30 78 Trial 7 -8 -19 -9 -4 42 -7 5 29 75 Trial 8 -16 -20 -16 0 37 -7 -1 25 81 Trial 9 -7 -12 -14 6 38 -5 1 31 78 Trial 10 -13 -14 -8 9 38 0 -3 31 77

X (m) -0.065 -0.13 -0.13 -0.13 -0.13 -0.13 -0.13 -0.13 -0.13 Y (m) 0.26 0 -0.065 -0.13 -0.195 -0.26 0.065 0.13 0.195 Trial 1 108 -62 -63 -51 -13 25 -34 -4 4 Trial 2 108 -57 -58 -51 -28 4 -33 -1 47 Trial 3 113 -58 -60 -53 -36 24 -32 -7 38 Trial 4 110 -59 -64 -51 -28 10 -34 -10 43 Trial 5 107 -59 -64 -48 -32 11 -34 -3 38 Trial 6 115 -57 -67 -50 -28 13 -33 -4 41 Trial 7 113 -56 -66 -52 -28 6 -35 4 49 Trial 8 113 -60 -57 -51 -33 6 -36 -2 49 Trial 9 110 -59 -58 -58 -25 12 -40 1 48 Trial 10 113 -59 -64 -50 -28 14 -38 -1 49

78 Table B.2. Lab Data: 1.2 m Distance (Continued)

X (m) -0.13 0 -0.195 -0.195 -0.195 -0.195 -0.195 -0.195 -0.195 Y (m) 0.26 0 0 -0.065 -0.13 -0.195 -0.26 0.065 0.13 Trial 1 108 -8 -50 -65 -48 -21 8 -30 8 Trial 2 115 5 -45 -53 -48 -19 17 -30 6 Trial 3 108 12 -55 -64 -46 -19 28 -23 4 Trial 4 110 4 -44 -49 -47 -22 28 -24 4 Trial 5 113 11 -46 -54 -51 -19 27 -30 5 Trial 6 110 -1 -50 -59 -58 -21 16 -26 3 Trial 7 112 -3 -57 -65 -48 -24 19 -27 4 Trial 8 117 3 -53 -56 -54 -18 23 -32 1 Trial 9 112 4 -48 -64 -47 -18 21 -31 5 Trial 10 111 -3 -48 -55 -46 -22 23 -19 5

X (m) -0.195 -0.195 -0.26 -0.26 -0.26 -0.26 -0.26 0 -0.26 Y (m) 0.195 0.26 0 -0.065 -0.13 -0.195 -0.26 0 0.065 Trial 1 43 105 24 13 19 36 64 6 38 Trial 2 42 111 25 13 16 41 65 1 41 Trial 3 46 111 18 12 26 46 66 -1 43 Trial 4 51 101 21 18 28 49 66 1 41 Trial 5 48 104 20 10 26 41 64 -9 38 Trial 6 54 106 23 18 25 44 63 -5 32 Trial 7 45 109 25 15 31 42 68 -4 35 Trial 8 45 103 19 15 26 42 67 -1 38 Trial 9 45 102 23 11 16 47 62 -7 37 Trial 10 47 113 21 22 23 46 67 -4 34

X (m) -0.26 -0.26 -0.325 -0.325 -0.325 -0.325 -0.325 -0.325 -0.325 Y (m) 0.13 0.195 0 -0.65 -0.13 -0.195 -0.26 0.065 0.13 Trial 1 58 108 58 50 8 54 95 75 93

79 Table B.2. Lab Data: 1.2 m Distance (Continued)

Trial 2 69 111 52 47 46 66 97 70 95 Trial 3 65 112 59 46 51 60 88 64 91 Trial 4 64 104 54 45 51 66 86 74 89 Trial 5 61 112 63 42 55 62 88 73 88 Trial 6 66 110 62 40 54 69 88 69 98 Trial 7 60 102 62 51 45 68 93 66 98 Trial 8 66 102 56 51 43 62 93 67 96 Trial 9 69 107 56 50 47 60 91 78 95 Trial 10 69 102 55 45 52 62 85 68 97

X (m) -0.325 -0.39 -0.39 -0.39 -0.39 -0.39 -0.39 -0.455 0 Y (m) 0.195 0 -0.065 -0.13 -0.195 -0.26 0.065 0 0 Trial 1 135 89 90 70 94 127 124 136 -14 Trial 2 139 96 87 74 83 114 121 129 -9 Trial 3 135 90 92 75 95 110 126 139 -15 Trial 4 141 94 86 73 85 130 121 134 -10 Trial 5 136 88 89 80 86 119 124 136 -9 Trial 6 136 98 85 81 94 121 128 142 0 Trial 7 138 89 81 71 98 124 114 145 -9 Trial 8 127 93 91 72 99 119 115 143 -8 Trial 9 135 93 85 77 95 120 119 136 -3 Trial 10 141 93 88 73 94 109 125 139 1

X (m) 0.065 0.065 0.065 0.065 0.065 0.065 0.065 0 0.13 Y (m) 0 -0.065 -0.13 -0.195 -0.26 0.065 0.13 0 0 Trial 1 61 61 53 62 82 84 103 -6 79 Trial 2 66 50 48 60 82 75 104 -12 86 Trial 3 60 56 58 56 78 79 103 -10 94 Trial 4 57 52 54 65 80 76 109 -14 89

80 Table B.2. Lab Data: 1.2 m Distance (Continued)

Trial 5 62 46 54 55 92 81 100 -4 83 Trial 6 61 52 59 62 84 83 102 -12 89 Trial 7 63 55 54 60 84 75 104 -10 85 Trial 8 67 47 51 60 88 75 99 -7 86 Trial 9 68 60 49 65 89 76 103 -2 87 Trial 10 66 55 60 62 83 76 107 -16 89

X (m) 0.13 0.13 0.13 0.13 0.13 0.195 Y (m) -0.065 -0.13 -0.195 -0.26 0.065 0 Trial 1 79 82 77 107 111 130 Trial 2 79 86 71 107 115 129 Trial 3 82 74 73 110 108 126 Trial 4 78 78 81 112 115 129 Trial 5 82 74 81 118 116 128 Trial 6 89 80 78 112 113 122 Trial 7 82 83 70 114 116 131 Trial 8 85 83 81 121 112 123 Trial 9 83 81 84 120 114 134 Trial 10 82 77 76 117 117 132

Table B.3. Lab Data: 0.84 m Distance

X (m) 0 0 0 0 0 0 0 0 -0.065 Y (m) 0 -0.065 -0.13 -0.195 -0.26 0.065 0.13 0.195 0 Trial 1 90 88 83 108 158 112 151 211 76 Trial 2 92 89 79 120 161 101 156 209 73 Trial 3 85 85 92 113 172 113 145 210 75 Trial 4 94 87 84 107 167 112 146 206 82 Trial 5 91 93 84 112 158 105 152 210 86

81 Table B.3. Lab Data: 0.84 m Distance (Continued)

Trial 6 91 93 77 114 170 111 152 202 87 Trial 7 88 88 90 109 173 109 149 212 78 Trial 8 88 89 86 110 165 104 146 210 79 Trial 9 95 98 92 113 163 109 153 213 81 Trial 10 89 83 90 118 164 113 146 204 80

X (m) -0.065 -0.065 -0.065 -0.065 -0.065 -0.065 -0.065 0 -0.13 Y (m) -0.065 -0.13 -0.195 -0.26 0.065 0.13 0.195 0 0 Trial 1 79 82 108 159 112 143 214 80 61 Trial 2 91 80 107 159 104 142 206 75 55 Trial 3 91 81 109 160 113 151 209 88 64 Trial 4 86 84 103 156 108 141 208 89 65 Trial 5 85 79 106 164 105 140 214 93 67 Trial 6 81 82 105 167 108 147 212 77 61 Trial 7 84 82 110 156 112 151 221 79 62 Trial 8 83 85 106 165 106 143 212 84 55 Trial 9 81 80 112 162 117 146 212 83 57 Trial 10 90 82 103 158 108 138 205 84 58

X (m) -0.13 -0.13 -0.13 -0.13 -0.13 -0.13 -0.13 -0.13 -0.195 Y (m) -0.065 -0.13 -0.195 -0.26 0.065 0.13 0.195 0.26 0 Trial 1 56 52 83 139 88 137 179 278 87 Trial 2 46 56 80 144 94 130 172 277 74 Trial 3 51 55 101 141 93 133 173 286 74 Trial 4 55 58 87 139 86 125 170 288 87 Trial 5 55 61 83 136 91 127 174 283 80 Trial 6 52 56 91 145 89 131 178 282 79 Trial 7 56 56 92 138 90 133 184 288 76 Trial 8 55 53 91 137 94 126 175 290 80

82 Table B.3. Lab Data: 0.84 m Distance (Continued)

Trial 9 54 51 88 138 90 131 177 281 82 Trial 10 50 49 85 145 88 132 186 277 71

X (m) -0.195 -0.195 -0.195 -0.195 -0.195 -0.195 -0.195 0 -0.26 Y (m) -0.065 -0.13 -0.195 -0.26 0.065 0.13 0.195 0 0 Trial 1 74 77 107 174 108 141 214 83 169 Trial 2 72 70 109 181 106 146 208 71 168 Trial 3 69 71 103 172 110 143 215 81 168 Trial 4 70 70 108 177 105 137 215 80 168 Trial 5 69 69 98 173 108 145 219 81 172 Trial 6 68 79 97 174 102 149 211 86 180 Trial 7 60 80 100 173 102 142 216 90 179 Trial 8 65 72 98 175 107 145 212 80 173 Trial 9 66 76 105 170 106 148 214 82 175 Trial 10 60 78 95 177 103 147 221 82 170

X (m) -0.26 -0.26 -0.26 -0.26 -0.325 0 0.065 0.065 0.065 Y (m) -0.065 -0.13 -0.195 0.065 0 0 0 -0.065 -0.13 Trial 1 171 172 206 195 252 89 145 141 140 Trial 2 164 170 190 193 265 85 149 136 137 Trial 3 162 163 190 196 259 82 148 129 134 Trial 4 173 163 192 199 260 86 141 132 135 Trial 5 167 169 199 194 256 80 148 129 133 Trial 6 167 167 201 198 255 84 144 140 141 Trial 7 163 169 199 195 257 95 147 143 137 Trial 8 165 169 192 192 261 86 148 136 130 Trial 9 168 173 196 188 259 84 148 139 140 Trial 10 164 174 195 198 264 85 152 130 130

X (m) 0.065 0.065 0.065 0.065 0 0.13 0.13 0.13 0.13

83 Table B.3. Lab Data: 0.84 m Distance (Continued)

Y (m) -0.195 -0.26 0.065 0.13 0 0 -0.065 -0.13 -0.195 Trial 1 140 194 163 216 89 183 170 163 182 Trial 2 142 188 172 216 82 186 178 156 172 Trial 3 145 179 176 207 80 183 174 152 183 Trial 4 143 173 184 213 82 186 168 159 172 Trial 5 146 183 171 210 83 187 170 161 181 Trial 6 146 187 166 208 75 182 170 160 178 Trial 7 149 190 176 211 80 181 171 161 176 Trial 8 148 184 169 210 76 184 170 159 175 Trial 9 148 187 172 215 78 184 167 153 175 Trial 10 143 187 167 215 77 187 169 161 176

X (m) 0.13 0.13 0 0.195 Y (m) -0.26 0.065 0 0 Trial 1 211 193 88 232 Trial 2 198 199 82 222 Trial 3 213 196 85 226 Trial 4 200 189 84 222 Trial 5 202 195 83 224 Trial 6 208 190 87 222 Trial 7 206 191 84 222 Trial 8 205 193 82 216 Trial 9 211 197 81 223 Trial 10 210 194 82 223

84