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The thesis of Ryan Campbell was reviewed and approved∗ by the following:
Gregory Huff
Professor of Electrical Engineering
Thesis Advisor
Timothy Kane
Professor of Electrical Engineering
Kultegin Aydin
Professor of Electrical Engineering
Head of the Department of Electrical Engineering
∗Signatures are on file in the Graduate School.
ii
Abstract
A new compact design for a hybrid coupler using asymmetric stripline is introduced.The design process for two different frequency ranges and simulation results for thosefrequency ranges is presented. Following the design process and simulations, the couplerwas fabricated and tested. This required the development of a Thru-Reflect-Line (TRL)calibration kit. The process for the design and verification of this kit is also presented.Finally, the calibrated results for the coupler are given. Future work is discussed includingthe design of a phase shifter using a similar design process to the coupler. It is thenshown how the coupler presented by this work and the hypothetical phase shifter might beincorporated into a larger Butler matrix design.
I would like to extend my gratitude first to my two committee members, Dr. GregoryHuff, who was kind enough to extend an invitation to remain a part of his research groupfollowing his move, and Dr. Timothy Kane, whose candor is always appreciated.
I would also like to thank the department staff for making the transition as well as themaster’s process painless.
My close friends deserve thanks as well for welcoming me with open arms and pro-viding assistance whenever it was needed.
Finally, I would like to express my sincerest thanks to my mother. She supports mein all that I do and for that I am eternally grateful.
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Chapter 1
Introduction
Hybrid couplers are a ubiquitous device in microwave engineering and have been realized
in a myriad of transmission line topologies [2–6], etc. Their ubiquity stems from their
useful properties, namely the ability to split power equally among output ports, and provide
a constant phase difference between these ports. This provides utility in fields such as
signal processing (converting incoming signals in to in-phase and quadrature components),
radar, beamforming, communications, test and measurement, and just about every other
field which interacts with microwave domain [7]. As such, there has been much research
and development of these devices since their introduction. Of particular interest is the
miniaturization of these devices. Miniaturization enables the circuits which use these
devices to be smaller themselves which is frequently desirable.
This thesis investigates a novel hybrid coupler design in stripline, a type of transmission
line structure. The exact structure is asymmetric stripline, meaning there are two conductors
embedded in dielectric separated by a third piece of dielectric. This device was of interest
due to recent research in the lab on beamforming using Butler matrices. As will be discussed
in Section 2.5, Butler matrices are, at their most basic, comprised of two components: phase
shifters and hybrid couplers. As will be further explained, hybrid couplers tend to take up
1
the bulk of the board space of Butler matrices thus finding a compact coupler design would
enable the Butler matrix overall to be more compact.
2
Chapter 2
Background
This chapter provides background on topics starting at a high level and becoming more
granular. This order follows the development of the motivation for the study of the device
investigated in this thesis.
2.1 Beamforming and Antenna Arrays
Beamforming refers to the use of an antenna array with its constituent elements excited
in such a way as to develop main beam with greater directivity than those elements have
individually. This technique finds application in RADAR, direction finding, communication
networks, etc [8]. Beamsteering is the process by which the main beam is pointed in a
specific direction. Beamsteering can be basically divided in to two categories: Mechanical
and Electronic. Mechanical beamsteering is the physical movement of the antenna array to
make it face in a direction of interest. Electronic beamsteering is when adjustments to the
RF chain are adjusted to produce a beam in a direction of interest. This could be through
the use of phase shifters, differing amplitudes of power applied to each antenna, etc. An
illustration of how phasing array elements can produce a beam in a specific direction is
3
given in Figure 2.1. In this figure, a progressive phase delay at each of the antenna elements
Figure 2.1: Progressive phasing of a linear array’s elements result in a beam in the θ
direction [1]
produces a plane wave travelling in the θ direction.
Electronic beamsteering affords several advantages over mechanical beamsteering. For
example, electronic beamsteering is much faster than physical beamsteering and it reduces
the mechanical complexity of the system, which generally translates to a reduction in
physical side and a reduction in failure rate citation. This isn’t to say electronic beamsteering
isn’t without its downsides. For one, the reduction in mechanical complexity is juxtaposed
with the complexity of the circuitry involved, e.g. the phase shifters, the antenna design,
4
etc. Additionally, the techniques used in electronic beamsteering can introduce significant
grating lobes, particularly at wide scan angles. To understand this more, Section 2.2
explains the behavior of linear arrays and how and why grating lobes can appear.
2.2 Linear Arrays
A common starting point for analyzing linear phased arrays is to consider the simplest
linear array: two identical antenna elements separated by some distance, d and assuming
no mutual coupling between them [9]. This setup is shown Figure 2.2. Given these
Figure 2.2: Two Element Array
assumptions, the far field can be expressed as the sum of electric fields of each of the two
antennas. Further assuming these antennas to be infinitesimal Hertzian dipoles whose far
5
field electric field expression is known, results in 2.1.
Et(r1,r2,θ1,θ2) = E1(r1,θ1)+E2(r2,θ2)
= aθ jηkI0l4π
e− j[kr1−(β/2)]
r1cosθ1 +
ekr2+[(β/2)]
r2cosθ2
(2.1)
Furthermore, since there was the assumption of being in the far-field, the following ap-
proximations can be made: r1 ≈ r2 ≈ r for r terms outside of complex exponentials (i.e.
magnitude variations) and θ1 ≈ θ2 ≈ θ , r1 ≈ r− d/2cosθ , r2 ≈ r+ d/2cosθ for terms
inside complex exponentials (i.e. phase variations). With these approximations and using
Euler’s Formula, 2.1 can be reduced to:
Et(r,θ) = aθ jηkI0le− jkr
4πrcosθ
2cos
[12(kd cosθ +β )
](2.2)
Putting the equation in this form shows the far field approximation of a two element array
of infinitesimal Hertzian dipoles is equal to the element radiation pattern multiplied by
some factor. This factor (in s in 2.2) is known as the array factor. This approximation can
be extended to any configuration of identical antenna elements. Specifically, the far field
antenna pattern for an array of identical elements is equal to the element pattern multiplied
by the array factor which is dependent only on the geometry of the array [9]. This pattern
multiplication is shown in Figure 2.3
6
(a) Element pattern (b) Array factor
(c) Full pattern
Figure 2.3: Pattern multiplication in two element array of Hertzian dipoles
2.2.1 Beam Scanning
The β term in 2.2 is the progressive phasing between the elements of the array. Varying
this value allows the beam to be steered to a certain direction. While this effect isn’t very
apparent in a two element array, the effect of the phase differences between elements can
still be seen. Figure 2.4 shows the patterns for β = [π/4,π/2,2π/3,−π/2]. If the number
of elements is increased, the result is what is referred to as a linear array. The array factor
7
(a) β = π/4 (b) β = π/2
(c) β = 2π/3 (d) β =−π/2
Figure 2.4: Changing excitation phase difference of a two element array of Hertzian dipoles
for this configuration is [9]:
AF =N
∑n=1
e j(n−1)ψ (2.3)
where ψ = kd cosθ +β
8
N is the number of elements in the array. Several approximations can be made to reduce
this to:
(AF)n ≈
[sin(N
2 ψ)
N2 ψ
](2.4)
Setting ψ = 0 yields a maximum in the theta direction. Solving for β in this case shows
if a beam is desired in the θ direction, then β =−kd cosθ . Figure 2.5 shows the field for
several scan angles (θ ) in this case.
(a) θ = 0 (b) θ = π/6
(c) θ = π/3 (d) θ = π/2
Figure 2.5: Radiation pattern for a 25 element array with different scan angles, θ
9
2.3 Planar Arrays
Section 2.2 described linear arrays and how changing the difference in excitation phase can
enable the beam to be scanned in the θ direction. Expanding the array in to two dimensions
allows the beam to be scanned in both θ and φ . The definition of the various parameters
which constitute the planar array are illustrated in Figure 2.6. Like with the linear array, the
Figure 2.6: Planar Array
array factor for the planar array can be simplified to:
AFn(θ ,φ) =
1M
sin(M
2 ψx)
sin(
ψx2
) 1N
sin(N
2 ψy)
sin(ψy
2
) (2.5)
10
where
ψx = kdx sinθ cosφ +βx
ψy = kdy sinθ sinφ +βy
Again, if it is desired to steer the main beam to (θ ,φ), set ψx = ψy = 0 and solve for (βx,βy)
to find the appropriate phasing for the elements.
2.4 Circular Arrays
Moving beyond the rectilinear domain allows for the discussion of circular arrays. Circular
arrays can be thought of as basically a linear array wrapped around a cylinder of radius a.
The rest of the design parameters of circular arrays are illustrated in Figure 2.7. Circular
arrays are attractive because they allow for 360 degree beam scanning with a constant
sidelobe level. This is in comparison to linear and planar arrays, as exampled in Figure
2.5. Many techniques exist to generate proper excitation of the elements of a circular
array [10], but the one focused on here is a Butler Matrix (with some other components,
but the workhorse is the Butler matrix) as described in Section 2.5.2.
11
Figure 2.7: Circular Array
2.5 Butler Matrix
A Butler Matrix, first described in [11], is an NxN network with N input ports and N output
ports. Through a series of couplers and phase shifters, it is able to provide a discrete set
of progressive phase shifts determined by which of the input ports is excited. Having
this set of these progressive phase shifts is attractive from a beam forming perspective
because it provides a method of exciting an array to provide a discrete number of main
beam directions in a very simple (from a hardware perspective) device. As an illustration
of the beamforming property, consider the 4x4 matrix in Figure 2.8. In this figure, exciting
port 1 results in beam 1, etc. Additionally, Butler matrices provide a means of calculating
the Fast Fourier Transform (FFT) [12]. This property is useful in circular beamforming, as
12
Figure 2.8: Example 4x4 Butler Matrix
described in Section 2.5.2.
2.5.1 Design Methodology
Previous works have systematized and simplified the design procedures for Butler Matrices
[13, 14]. The most common configuration of a Butler Matrix is one with N = 2n ports.
Given this configuration, the number of hybrid couplers required to implement it is Nn2 ,
spread evenly across n rows. These can be either 90° or 180° hybrids, but the choice
determines how many phase shifters are required per row, detailed in Table 2.1 (k is the
index of the row with k = 1 being the row closest to the output ports. The total number of
phase shifters required is (n−1).
90° hybrids N/2180° hybrids N
2−2k−1
Table 2.1: Number of phase shifters per row for the type of hybrid used.
To determine the value of a row of phase shifters, it is helpful to create a diagram like
13
Figure 2.9: Determining progressive phase shifts of outputs of a Butler Matrix
Figure 2.9. Moody explains the value of progressive phase shifts between each of the
outputs is given by
ψn =±2p−1
N180° (2.6)
Here, p is the index of the output beam and the sign of the phase difference depends on
whether the beam being considered is to the left (+) or to the right (−) of broadside. Moody
additionally explains it is only necessary to determine ψ1 and then use the relationship of
pairs indicated in Figure 2.1. In words, this figure shows pairs of inputs should add to some
fraction of 180°. The pairs are endpoints of increasing powers of two. Adjacent pairs will
add to 180°, endpoints of pairs of four will add to 90°, and so on with the angle decreasing
by half with each row. When using this pair relation, sign is not considered for the sum of
pairs, but when finished determining the absolute value of each ψ , the sign alternates, e.g.
if ψ1 is −45°, then ψ2 will be 135°.
After the values of ψn have been found, the values of the phase shifters follow. Each
row of phase shifters follows a row of hybrid couplers. For the first row, the phase shifters
14
are placed at the endpoints of groups of four (e.g. on lines 1, 4, 5, 8, 9, 12, etc). The values
of these phase shifters are given by
φn = 90°−|ψn| (2.7)
where ψn is the the value of the progressive phase shift corresponding to the line under
consideration (this makes more sense with the example at the end of this section). Phase
shifters on every row besides the first are placed in groups of k at the endpoints of increasing
powers of two, where k is the index of the row (e.g. for row two, phase shifters are placed
on lines 1, 2, 7, 8, 9, 10, 15, 16, etc.). The values for these phase shifters is given
by
φn = 90°−2(k−1)|ψn| (2.8)
2.5.1.1 Design Example
What follows is an example of a 16X16 Butler Matrix using the design methodology
outlined in the section above. First, the value of ψ1 must be determined. Using 2.6
ψ1 =2(1)−1
16180° = 11.25°
The sign of the ψ1 can be arbitrary, but each subsequent ψ must alternate in sign. The
remaining ψ values are found using the pairs shown in Figure 2.9.
Now that all the progressive phase shifts are known, the values for each of the phase
between the thru and coupled ports (90°), the even power split between the thru and coupled
ports, a high level of isolation in the isolated port, and the bandwidth.
25
3.1 Design Parameters
The primary element of the coupler is the overlapping circle structure. A smaller circle
is cut from a larger circle set off-center. This semi-annulus is then split in half, and one
half is raised by the height of the center substrate. This is then duplicated and joined to
result in what’s shown in Figure 3.4. Table 3.1 provides further explanation of each of the
parameters shown in this figure.
Figure 3.4: Coupler Design Parameters
26
Parameter DefinitionW1 Radius of the inner circle cut from the larger circleW2 Radius of the larger circleWc "Coupling width" - Width of the first coupling sectionW50 50 Ω line widthL50 Length of 50 Ω lineLt Length of transition from 50 Ω line to WcLO How far beyond the major coupler structure the transition line extendsh1 Outer substrate height (not pictured)h2 Middle substrate height (not pictured)
Table 3.1: Coupler Design Parameters
3.2 Initial Design
The initial design was focused between the Ku and K bands. The parameters for this
initial design are given in Table 3.2. This design resulted in a 28.6% bandwidth at 17.5
huge benefit, at least in terms of Butler matrix design: it changes layers of the asymmetric
stripline. This eliminates the need for a crossover so frequently required in planar Butler
47
matrices. Figure 4.2 shows what this kind of Butler matrix might look like. This design
could be compacted further, but even in this state it is 24 mm×24 mm. In the event a phase
Figure 4.2: Butler matrix using mock-up phase shifter design
shifter like this could not be developed, the following simulation shows the viability of this
coupler design in a Butler matrix. It is a simple circuit simulation using ideal phase shifts.
Figure 4.3 shows the setup for this simulation and Figures 4.5-4.6 show the magnitude
48
response and progressive phase shifts.
Figure 4.3: Butler matrix circuit simulation using ideal phase shifters
Figure 4.4: Input reflection
49
Figure 4.5: Transmission from port 1 excitation
Figure 4.6: Progressive phase shifts from port 1 excitation
50
Chapter 5
Conclusion
This work presented the a new compact stripline hybrid coupler. In particular, two designs
were presented for two different frequency ranges with their results. These results showed
identical fractional bandwidths (28.6%) indicating the ease of scalability of this design.
These bandwidths were defined as having reflection coefficients below−10 dB and isolation
above 10 dB. Additionally, the phase difference between the output ports had little < 5%
error, or offset from 90°. The larger of these two designs was fabricated and a TRL kit
developed for measurement. This necessitated a redesign of both the TRL kit and the
coupler and these redesigns had performance similar to simulations.
This new coupler design could be incorporated, along with a similarly designed phase
shifter, into a Butler matrix design. The benefit is the small footprint of this coupler which
would serve to make the overall Butler matrix more compact.
51
Chapter
Asymmetric Stripline
1 Stripline
Stripline is a transmission line structure comprised of two large ground planes separated
by some thickness of a dielectric material. The actual transmission line is a strip (or
multiple strips) of copper embedded in this dielectric. Figure .1 shows a cross section of
this geometry.
Figure .1: Basic stripline cross section
52
1.1 Characteristics
The typical mode of operation of stripline is Transverse Electromagnetic or TEM waves.
From this, [21] and [22] show, as summarized by [7], that the characteristic impedance of
stripline can be found in terms of its effective width.
Z0 =30π√
εr
bWe +0.441b
(.1)
We
b=
Wb−
0 for W
b > 0.35
(0.35−W/b)2 for Wb < 0.35
(.2)
This however applies only to this symmetrical case. For asymmetrical stripline (Figure
.2), the following equations apply [23]
Z0 =
60√
εrln[
4bπK1(W, t)
]for W
b < 0.35
94.15W/b1− t
b+ K2(b,t)
π
1√εr
for Wb > 0.35
(.3)
where
K1(w, t) =W2
[1+
tπ +W
(1+ ln
4πWt
)+0.255
(t
W
2)]
(.4)
K2(b, t) =2
1− tb
ln[
11− t
b+1]−[
11− t
b−1]
ln
[1(
1− tb
)2 −1
](.5)
53
Figure .2: Asymmetric Stripline
54
Bibliography
[1] COMMONS, W. (2016), “Phased Array Animation with Arrow,” File: Phased arrayanimation with arrow 10frames 371x400px 100ms.gif.URL https://upload.wikimedia.org/wikipedia/commons/4/4a/Phased_array_animation_with_arrow_10frames_371x400px_100ms.gif
[2] Lubing Sun, Y. Zhang, Z. Qian, D. Guan, and Xingjian Zhong (2016)“A compact broadband hybrid ridged SIW and GCPW coupler,” in 2016IEEE MTT-S International Microwave Workshop Series on AdvancedMaterials and Processes for RF and THz Applications (IMWS-AMP),pp. 1–3.
[3] Arriola, W. A., J. Y. Lee, and I. S. Kim (2011) “Wideband 3 dBBranch Line Coupler Based onλ/4Open Circuited Coupled Lines,”IEEE Microwave and Wireless Components Letters, 21(9), pp.486–488.
[4] Hantula, P., N. Jaiyen, and R. Tongta (2018) “A 3-dB QuadratureCoupler Using Broadside Striplines for FM Power Amplifiers,” in2018 IEEE International Workshop on Electromagnetics:Applicationsand Student Innovation Competition (iWEM), pp. 1–2.
[5] Anselmi, M., M. Pingue, A. Manna, R. Flamini, and L. Cosmi (2014)“Design and Realization of 3 dB hybrid stripline coupler in 0.5 -18.0 GHz,” 2014 44th European Microwave Conference.
[6] Jain, A. and T. Mittal (2016) “Design and Simulation ofUltra-Wideband 3 dB Hybrid Tandem Coupler,” in 2016 InternationalConference on Micro-Electronics and Telecommunication Engineering(ICMETE), pp. 476–481.
[7] Pozar, D. M. (2012) Microwave engineering, John Wiley & Sons.
[9] Balanis, C. A. (2016) Antenna theory, 4 ed., John Wiley & Sons,Inc.
[10] Davies, D. (1983) “Circular Arrays,” in Handbook of AntennaDesign (E. Rudge, K. Milne, A. Olver, and P. Knight, eds.),vol. 2, chap. 12, Peter Peregrinus Ltd., London UK.
[11] Butler, J. and R. Lowe (1961) “Beam-Forming Matrix SimplifiesDesign of Electronically Scanned Antennas,” Electronic Design, 9,pp. 170–173.
[12] Nester, W. (1968) “The fast Fourier transform and the Butlermatrix,” IEEE Transactions on Antennas and Propagation, 16(3), pp.360–360.
[13] Moody, H. (1964) “The systematic design of the Butler matrix,”IEEE Transactions on Antennas and Propagation, 12(6), pp.786–788.
[14] MacNamara, T. (1987) “Simplified design procedures forButler matrices incorporating 90 hybrids or 180 hybrids,” IEEProceedings H - Microwaves, Antennas and Propagation, 134(1), pp.50–54.
[15] Sheleg, B. (1968) “A matrix-fed circular array for continuousscanning,” Proceedings of the IEEE, 56(11), pp. 2016–2027.
[16] Singh, I. (2019) Design of Ultra-Wideband Butler Matrices,Master’s thesis, Texas A&M University.
[17] Schiffman, B. M. (1958) “A New Class of Broad-Band Microwave90-Degree Phase Shifters,” IRE Transactions on Microwave Theoryand Techniques, 6(2), pp. 232–237.
[18] –-–-–- (1966) “Multisection Microwave Phase-Shift Network(Correspondence),” IEEE Transactions on Microwave Theory andTechniques, 14(4), pp. 209–209.
[19] Quirarte, J. L. R. and J. P. Starski (1993) “Novel Schiffmanphase shifters,” IEEE Transactions on Microwave Theory andTechniques, 41(1), pp. 9–14.
[20] Dunsmore, J. P. (2012) Handbook of microwave componentmeasurements, John Wiley & Sons Inc.
56
[21] Howe, H. H. (1982) Stripline circuit design, Artech House.
[22] Bahl, I. and D. Trivedi (1977) “A Designer’s Guide to MicrostripLine,” Microwaves, 16, pp. 174–176, 178, 180, 182.
[23] Johnson, H. W. and M. Graham (1993) High speed digital design:a book of black magic, PTR Prentice Hall.
[24] Engen, G. F. and C. A. Hoer (1979) “Thru-Reflect-Line: AnImproved Technique for Calibrating the Dual Six-Port AutomaticNetwork Analyzer,” IEEE Transactions on Microwave Theory andTechniques, 27(12), pp. 987–993.