A COMPACT WIDEBAND STRIPLINE HYBRID COUPLER
Transcript of A COMPACT WIDEBAND STRIPLINE HYBRID COUPLER
The Pennsylvania State University
The Graduate School
College of Engineering
A COMPACT WIDEBAND STRIPLINE HYBRID COUPLER
A Thesis in
Electrical Engineering
by
Ryan Campbell
© 2019 Ryan Campbell
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Master of Science
December 2019
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.
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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.
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Table of Contents
List of Figures viii
List of Tables ix
Acknowledgments x
Chapter 1Introduction 1
Chapter 2Background 32.1 Beamforming and Antenna Arrays . . . . . . . . . . . . . . . . . . . . . 32.2 Linear Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1 Beam Scanning . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Planar Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Circular Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5 Butler Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5.1 Design Methodology . . . . . . . . . . . . . . . . . . . . . . . . 132.5.1.1 Design Example . . . . . . . . . . . . . . . . . . . . . 15
2.5.2 Butler Matrix Excitation for Circular Arrays . . . . . . . . . . . . 182.6 Hybrid Coupler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Chapter 3Wideband Hybrid Coupler 233.1 Design Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2 Initial Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3 Fabrication and Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3.1 Design Parameters and Simulation . . . . . . . . . . . . . . . . . 283.3.2 Design for Fabrication . . . . . . . . . . . . . . . . . . . . . . . 31
3.4 TRL Calibration Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.4.1 Twelve and Eight Term Error Models . . . . . . . . . . . . . . . 33
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3.4.2 Acquiring error terms . . . . . . . . . . . . . . . . . . . . . . . . 383.4.3 TRL Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.5 TRL Kit Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.6 Coupler Redesign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Chapter 4Future Work 47
Chapter 5Conclusion 51
AppendixAsymmetric Stripline 521 Stripline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
1.1 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Bibliography 55
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List of Figures
2.1 Progressive phasing of a linear array’s elements result in a beam in the θ
direction [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Two Element Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Pattern multiplication in two element array of Hertzian dipoles . . . . . . 7
2.4 Changing excitation phase difference of a two element array of Hertziandipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.5 Radiation pattern for a 25 element array with different scan angles, θ . . . 9
2.6 Planar Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.7 Circular Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.8 Example 4x4 Butler Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.9 Determining progressive phase shifts of outputs of a Butler Matrix . . . . 14
2.10 16X16 Butler Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.11 Pattern of circular array with increasing number of modal contributions . 19
2.12 Scanning circular array with Butler matrix feed . . . . . . . . . . . . . . 20
2.13 Hybrid coupler symbol with labeled ports . . . . . . . . . . . . . . . . . 21
2.14 Ideal S-matrix of 90-degree hybrid coupler . . . . . . . . . . . . . . . . . 21
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2.15 Microstrip Implementation of Hybrid Coupler . . . . . . . . . . . . . . . 21
2.16 Microstrip Implementation of Hybrid Coupler . . . . . . . . . . . . . . . 22
3.1 Narrowband 8×8 Butler matrix at 4GHz . . . . . . . . . . . . . . . . . . 23
3.2 Tandem stripline ultra-wideband Butler matrix . . . . . . . . . . . . . . . 24
3.3 Ultra-wideband Butler matrix magnitude 3.3(a) and interport phase differ-ences 3.3(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4 Coupler Design Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.5 Ku/K band coupler simulation results . . . . . . . . . . . . . . . . . . . . 29
3.6 S Band coupler simulation results . . . . . . . . . . . . . . . . . . . . . . 30
3.7 Coupler parameters including microstrip-to-stripline transitions . . . . . . 31
3.8 First fabricated coupler iteration . . . . . . . . . . . . . . . . . . . . . . 32
3.9 Second Fabricated Coupler Iteration . . . . . . . . . . . . . . . . . . . . 33
3.10 Illustration of Short-Open-Load-Thru Calibration . . . . . . . . . . . . . 34
3.11 Forward error model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.12 Reverse error model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.13 Eight term error model . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.14 Circuit representations of the four SOLT calibration standards . . . . . . . 38
3.15 Circuit diagram of thru standard showing reference plane . . . . . . . . . 39
3.16 Circuit diagram of line standard showing reference plane . . . . . . . . . 39
3.17 Open circuit capacitance model . . . . . . . . . . . . . . . . . . . . . . . 40
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3.18 Short circuit inductance model . . . . . . . . . . . . . . . . . . . . . . . 41
3.19 Initial TRL kit design with multiple views . . . . . . . . . . . . . . . . . 42
3.20 Fabricated TRL kit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.21 Updated Coupler Design . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.22 Final coupler design magnitude response comparison (simulations aredashed lines) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.23 Final coupler design output port phase difference comparison . . . . . . . 46
4.1 Phase shifter mock-up design (left) beside the coupler design (right) . . . 47
4.2 Butler matrix using mock-up phase shifter design . . . . . . . . . . . . . 48
4.3 Butler matrix circuit simulation using ideal phase shifters . . . . . . . . . 49
4.4 Input reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.5 Transmission from port 1 excitation . . . . . . . . . . . . . . . . . . . . 50
4.6 Progressive phase shifts from port 1 excitation . . . . . . . . . . . . . . . 50
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List of Tables
2.1 Number of phase shifters per row for the type of hybrid used. . . . . . . 13
2.2 Progressive phase shifts of 16X16 Butler Matrix . . . . . . . . . . . . . . 16
3.1 Coupler Design Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Parameters of initial coupler design . . . . . . . . . . . . . . . . . . . . . 27
3.3 Coupler Design Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 Error Term Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.5 TRL Kit Design parameters . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.6 TRL Kit Design parameters After scikit-rf Verification . . . . . . . . . . 43
3.7 Updated Coupler Parameters . . . . . . . . . . . . . . . . . . . . . . . . 44
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Acknowledgments
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
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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.
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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
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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,
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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
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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
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(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
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(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
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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
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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.
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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
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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
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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
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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
15
ψ1 = −11.25° ψ5 = −56.25° ψ9 = −33.75° ψ13 = −78.75°ψ2 = 168.75° ψ6 = 123.75° ψ10 = 146.25° ψ14 = 101.25°ψ3 = −101.25° ψ7 = −146.25° ψ11 = −123.75° ψ15 = −168.75°ψ4 = −78.75° ψ8 = 33.75° ψ12 = 56.25° ψ16 = 11.25°
Table 2.2: Progressive phase shifts of 16X16 Butler Matrix
shifters can be determined using 2.7 and 2.8. The results as well as the full 16X16 Butler
Matrix are shown in Figure 2.10.
16
Figu
re2.
10:1
6X16
But
lerM
atri
x
17
2.5.2 Butler Matrix Excitation for Circular Arrays
If the excitation function of a circular array of 2N +1 element is taken, as in [10], to be
F(φ) =N
∑m=−N
Cme jmφ (2.9)
then each of the terms of the sum are referred to as a phase mode. The current of each of
these modes is then Ime jmφ , and the constant Cn is, from [15],
2πK jnInJn
(2πaλ
)(2.10)
These orthogonal current modes have the same form as the outputs of N ×N Butler
matrices [8]. What this means is a Butler matrix can be used to excite the current modes
needed to develop a pencil beam from a circular array. However, this is only true if the
appropriate phasing is applied to the inputs of the Butler matrix. This is done by applying a
fixed phase shift to each of the inputs of the matrix. If in addition to this fixed phase shift
a linear phase progression is established at the input of the form 0φ ,1φ , ...,Nφ , then the
beam will be scanned to the φ direction. Figure 2.11 illustrates how a beam is formed by
combining higher and higher order modes. This system was described theoretically and
experimentally by Sheleg [15]. A diagram of this system is shown in Figure 2.12.
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(a) Modes -1 to 1 (b) Modes -2 to 2
(c) Modes -4 to 4 (d) Modes -6 to 6
(e) Modes -8 to 8 (f) Modes -15 to 15
Figure 2.11: Pattern of circular array with increasing number of modal contributions
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Figure 2.12: Scanning circular array with Butler matrix feed
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2.6 Hybrid Coupler
Figure 2.13: Hybrid coupler symbol withlabeled ports
[S] =−1√
2
0 j 1 0j 0 0 11 0 0 j0 1 j 0
Figure 2.14: Ideal S-matrix of 90-degreehybrid coupler
A hybrid coupler, also called a 3dB hybrid is a specific type of coupled line coupler
with 4 ports: An input, an isolated port, and two output ports. Each of the output ports
gives half of the input power and are 90° (or 180°) offset from each other. This device is
ubiquitous in microwave engineering. Some applications include splitting an input into
real and imaginary (or in-phase and quadrature) components for signal processing, and
their use in beamforming networks. Figure 2.13 shows the circuit symbol for a hybrid
coupler, and Figure 2.15 shows a microstrip implementation of a hybrid coupler called
a branchline hybrid coupler. The downside of this implementation is the very narrow
operational bandwidth as shown in Figure 2.16. There are a huge number of hybrid
Figure 2.15: Microstrip Implementation of Hybrid Coupler
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(a) Magnitude
(b) Phase
Figure 2.16: Microstrip Implementation of Hybrid Coupler
coupler implementations on all kinds of substrates and with a vast assortment of frequency
ranges [2–6]. Because of their wide use, there seems to constantly be research on improving
any number of parameters of the hybrid coupler, especially size and bandwidth.
22
Chapter 3
Wideband Hybrid Coupler
As mentioned in Section 2.5, one of the primary building of a Butler matrix is the hybrid
coupler. Previous research in our lab had focused on the development of a narrowband 8×8
Butler matrix to be used as part of a larger, 64×64 matrix. This implementation, shown
in Figure 3.1 used the branchline hybrid coupler implementation described in Section
2.6, along with fixed length transmission line phase delays to provide the necessary phase
shifts. The two biggest shortcomings of this design are the narrow bandwidth and the
Figure 3.1: Narrowband 8×8 Butler matrix at 4GHz
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size of the footprint. Having a narrow bandwidth isn’t necessarily a bad thing, but for our
application the desire was for a much wider operational bandwidth. There are however few
downsides of having a smaller footprint. This design was 195 mm×104.6 mm. Another
student used existing techniques to create a 2− 18 GHz design [16]. This design drew
heavily from works like [5, 6], utilizing a tandem offset segmented stripline coupler and
Schiffman-style phase shifters [17–19]. With this, they were able to achieve ultra-wideband
performance (the 4×4 implementation and some of its results are shown in Figures 3.2
and 3.3, respectively), but there was still interest in shrinking the overall size of the matrix.
Figure 3.2: Tandem stripline ultra-wideband Butler matrix
The hybrid coupler in that design was 26.4 mm×9.5 mm, and given that the bulk of the
Butler matrix is the hybrid coupler (and the crossover in the middle, which is made of
two cascaded hybrid couplers), this research focused on the development of a compact
wideband 90° hybrid coupler.
When designing a hybrid coupler, the key metrics are: the appropriate phase difference
24
(a)
(b)
Figure 3.3: Ultra-wideband Butler matrix magnitude 3.3(a) and interport phase differences3.3(b)
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
Parameter ValueW1 0.8 mmW2 2.0 mmWc 0.4 mmW50 0.1975 mmL50 0.5 mmLt 2.5 mmh1 10 milh2 2 mil
Table 3.2: Parameters of initial coupler design
GHz as indicated by Figures 3.5. Regarding the key metrics outlined in the introduction
of this section, Figure 3.5(a) shows an equal power split between ports 2 and 4 with a
maximum S11 of −13.2 dB and isolation of at least 13.2 dB at port 3. Figure 3.5(a) shows
a flat phase response across the indicated band with a maximum phase error of 5.23° or
27
+5.8%/−1.2%.
3.3 Fabrication and Testing
Following the successful simulation of the design in Section 3.2, it needed to be verified
through fabrication. Considering the small size, the initial design would have been difficult
to fabricate and test. For this reason, the design was scaled up to operate in S band, verified
through simulation, then fabricated.
3.3.1 Design Parameters and Simulation
The design parameters for the S band coupler are shown in Table 3.3. These parameters
yielded an identical 28.6% bandwidth centered at 3.5 GHz. Figure 3.6(a) shows this
operational band as well as a maximum S11 of −15.9 dB and isolation of at least 14.7 dB.
Figure 3.6(b) shows a flat phase response across the indicated band with a maximum phase
error of 1.39° or +3.5%/−1.7%.
Parameter ValueW1 4 mmW2 10 mmWc 2 mmW50 0.87 mmL50 0.5 mmLt W2 +LOLo 5 mmh1 50 milh2 10 mil
Table 3.3: Coupler Design Parameters
28
(a) Magnitude
(b) Output phase difference
Figure 3.5: Ku/K band coupler simulation results
29
(a) Magnitude
(b) Output phase difference
Figure 3.6: S Band coupler simulation results
30
3.3.2 Design for Fabrication
To fabricate the structure, the top and bottom layers were milled out on an LPKF S103
milling machine, and the center layer was cut on the milling machine, then chemically
etched (as it was too thin to allow for milling). The feed method was a microstrip to stripline
transition. Figure 3.7 shows the additional parameters associated with this transition. The
Figure 3.7: Coupler parameters including microstrip-to-stripline transitions
design for fabrication went through several iterations. Figure 3.8 shows the first iteration.
The additional parameters are: l f eed = 15 mm, and w f eed = 1.85 mm. In this iteration,
31
there were only four screws on the corner of the structure to join the three layers. This
minimal number of screws was not enough to ensure the three layers were completely
laminated resulting in a non-functional device. After the first design failed, more screws
Figure 3.8: First fabricated coupler iteration
were added to improve the lamination between layers. This redesign is shown in Figure 3.9.
The first iteration also made it apparent that to be able to test the fabricated device would
require the construction of a Thru-Reflect-Line (TRL) calibration kit. The theory of TRL
calibration and the process for designing the kit for this device are explained in the next
section.
32
Figure 3.9: Second Fabricated Coupler Iteration
3.4 TRL Calibration Theory
Calibration is the removal of (or accounting for) systemic errors in a measurement. Ef-
fectively, calibration moves what is referred to as the reference plane to a certain location
in the measurement chain depending on the type of calibration used. An example of
the reference plane being moved to the ends of the coaxial measurement cables after a
Short-Open-Load-Thru (SOLT) calibration is shown in Figure 3.10.
3.4.1 Twelve and Eight Term Error Models
The theory behind VNA calibration is derived from a signal flow graph analysis of the
analyzer. Figures 3.11 and 3.12 show the signal flow graphs for forward and reverse
measurements of a 2-port device with embedded error terms. This is referred to as the
twelve term error model. Table 3.4 provides definitions for each of these error terms. Using
signal flow graph analysis techniques, we arrive at the following expressions for measured
33
Figure 3.10: Illustration of Short-Open-Load-Thru Calibration
S-parameters [20]:
a1M
b1M
b2M1 S21A
S12A
S11A S22AEDF
ETF
ELF
ERF
ESF
a1A b2A
b1A a2A
EXF
DUT
Figure 3.11: Forward error model
Table 3.4: Error Term Definitions
Measurement Tracking Response Mismatch LeakageInput Reflection ERF ESF EDFForward Transmission ETF ELF EXFReverse Transmission ETR ELR EXROutput Reflection ERR ESR EDF
34
b′1M
b′2M
a′2M
1
S21
S12
S11 S22
ERR
ELR ESR EDR
ETR
a′1A b′2A
b′1A a′2A
EXR
DUT
Figure 3.12: Reverse error model
S11M =b1M
a1M= EDF+
ERF(
S11A +S21ELF·S12A(1−S22A·ELF)
)[1−ESF ·
(S11A +
S21ELF·S12A(1−S22A·ELF)
)]
S22M =b′2Ma′2M
= EDR+ERR
(S22A +
S21ELR·S12A(1−S11A·ELR)
)[1−ESR ·
(S22A +
S21ELR·S12A(1−S11A·ELR)
)]
S12M =b′1Ma′2M
=(S12A ·ETR)
(1−S11A ·ELR) · (1−S22A ·ESR)−ESR ·S21AS12A ·ELR+EXR
S21M =b2M
a1M=
(S21A ·ETF)(1−S11A ·ESF) · (1−S22A ·ELF)−ESF ·S21AS12A ·ELF
+EXF
These four equations can be solved for the actual S-parameters, resulting in the following
equations:
S11A =S11N (1+S22N ·ESR)−ELF ·S21NS12N
(1+S11N ·ESF)(1+S22N ·ESR)−ELF ·ELF ·S21NS12N
35
S22A =S22N (1+S11N ·ESF)−ELR ·S21NS12N
(1+S11N ·ESF)(1+S22N ·ESR)−ELF ·ELF ·S21NS12N
S12A =S12N (1+S11N · [ESF−ELR])
(1+S11N ·ESF)(1+S22N ·ESR)−ELF ·ELF ·S21NS12N
S21A =S21N (1+S22N · [ESR−ELF])
(1+S11N ·ESF)(1+S22N ·ESR)−ELF ·ELF ·S21NS12N
where
S11N =S11M−EDF
ERF,S21N =
S21M−EXFET F
S12N =S12M−ETR
ET R,S22N =
S22M−EDRERR
If instead measurements are made at all four test receivers, two for incident waves and
two for scattered waves, the result is referred to as the eight term error model. Figure
3.13 shows the signal flow graph for this model, but it is easier to analyze the model using
cascaded T-parameters [20].
a1M b2M
b1M a2M
S21A
S12A
S11A S22A
a1A b2A
b1A a2A
E10 E32
E00 E11 E22 E33
E01 E23
DUT
Figure 3.13: Eight term error model
36
T-parameters are defined as:
b1
a1
=
T11 T12
T21 T22
a2
b2
(3.1)
where a1,a2,b1,b2 represent the normalized waves. If the measurement chain is considered
to be the port 1 error box cascaded with the DUT cascaded with the port 2 error box, then
the T-parameters are
TMeasured = TPort 1TActualTPort 2 (3.2)
where
TMeasured =
(S21MS12M−S11MS22M)S21M
S11MS21M
−S22MS21M
1S21M
TPort 1 =
1E10E32
(E10E01−E00E11) E00
−E11 1
TActual =
1E10E32
(S21AS12A−S11AS22A)S21A
S11AS21A
−S22AS21A
1S21A
TPort 2 =
1E10E32
(E32E23−E33E22)
−E33 1
The actual T-parameters (for which the S-parameters can be found easily through a trans-
37
formation) can then be found by
TActual = T−1Port 1TMeasuredT−1
Port 2 (3.3)
3.4.2 Acquiring error terms
The error terms for the Twelve-term error model are most commonly acquired by performing
an SOLT calibration. This type of calibration uses the four standards shown in Figure 3.14.
According to [20], only ten error terms are needed because the crosstalk between two ports
is usually lower than the noise floor of the VNA, however many VNAs include the option
to measure this error term. The ten error terms are taken from ten measurements: short,
open, and load for each of the two ports, and the thru standard in both directions. Because
(a) Short (b) Open (c) Load (d) Thru
Figure 3.14: Circuit representations of the four SOLT calibration standards
the measurement we were trying to make required the use of a TRL kit instead of an SOLT,
that will be the focus of this section. As explained in [20], the eight-term error model
actually only has seven independent values. Because the eight-term model is frequently
converted to the twelve term error model, two additional error terms, ΓF and ΓR must be
measured. It turns out that these reflection coefficients can be measured during the thru
38
standard measurement.
3.4.3 TRL Standards
Figure 3.15: Circuit diagram of thru standard showing reference plane
The thru standard, also called a zero-length thru is essentially the two transmission lines
being used in the measurement placed flush against each other. In the case of microstrip
measurements, for example, this is manifested as just a length of transmission line. By
definition, this standard has 0 reflection and 0 phase (hence zero-length). This has the effect
of setting the reference plane at the center of the thru standard, as illustrated in Figure 3.15
Figure 3.16: Circuit diagram of line standard showing reference plane
The line standard is the same two transmission lines as used in the thru standard with
an additional quarter-wavelength section of transmission line between them. This quarter
wavelength is typically taken to be a quarter wavelength at the geometric mean (√
flow fhigh)
39
of the frequency range being measured. Between 20° and 160° is taken as a rule of thumb
for the electrical length of the line standard. If the frequency range would make the electrical
length go beyond this range, additional line standards should be used. Figure 3.16 shows a
circuit diagram of the line standard.
Finally, the reflect standard is any standard that provides some equal reflection at both
ports. Often this is realized as a short or open circuit. Both of these have limitations. An
open circuit will always have some stray capacitance. This stray capacitance is modeled as
a third order polynomial, represented as four capacitors in parallel, as shown in Figure 3.17
and summarized in Equation 3.4.
C( f ) =C0 +C1 f +C2 f 2 +C3 f 3 (3.4)
Similarly, a short circuit has associated inductance Equation 3.5 and Figure 3.18, though
Figure 3.17: Open circuit capacitance model
with a much less pronounced effect than the open circuit capacitance.
L( f ) = L0 +L1 f +L2 f 2 +L3 f 3 (3.5)
40
Figure 3.18: Short circuit inductance model
3.5 TRL Kit Design
A full TRL kit was designed using the steps outlined for each of the standards above. The
parameters were dictated by the 3− 4 GHz design from Section 3.2. Table 3.5 details
each of these design parameters, and Figure 3.19 illustrates each of these parameters. It
should be noted that a chamfer was added after discovering the 50Ω width of the microstrip
was too wide for the connectors used. After fabricating this TRL kit, and performing a
Table 3.5: TRL Kit Design parameters
Parameter Thru [mm] Line [mm] Reflect [mm]wfeed 1.85 1.85 1.85w50 0.85 0.85 0.85lmicrostrip 15 15 15lstripline 10 10 10lλ/4 — 8.73 8.73h1 50 mil 50 mil 50 milh2 10 mil 10 mil 10 mil
TRL calibration, the line standard was measured. This showed the kit was unsatisfactory
41
(a) Thru (b) Reflect
(c) Line
Figure 3.19: Initial TRL kit design with multiple views
for several reasons. The S21 showed magnitude above 1 which is not physically possible.
Additionally, the phase should be linear across the frequency range of the measurement,
and should be 90° at the geometric center of the frequency range. For the fabricated kit,
42
neither of these were true. The failure of this kit showed the need for a means of validating
prospective TRL kits. To do this, scikit-rf, a python library, was used to apply a calibration
from .sNp files from prospective TRL kit simulations. The .sNp files of the line and thru
standards were then both “measured” in scikit-rf and verified to have the desired behavior.
A new TRL kit was developed using this procedure which exhibited the proper behavior.
The new parameters for this kit are given in Table 3.6. Figure 3.20 shows each of these
Table 3.6: TRL Kit Design parameters After scikit-rf Verification
Parameter Thru [mm] Line [mm] Reflect [mm]wfeed 1.85 1.85 1.85w50 0.85 0.85 0.85lfeed 6.5 6.5 6.5lstripline 5 5 5lλ/4 — 8.73 8.73h1 50 mil 50 mil 50 milh2 10 mil 10 mil 10 mil
standards fabricated. These values were then used to update the original coupler design.
(a) Thru (b) Reflect
(c) Line
Figure 3.20: Fabricated TRL kit
43
3.6 Coupler Redesign
With the new values taken from the redesign of the TRL kit, the coupler parameters were
as summarized in Table 3.7. Figure 3.21 shows the CAD model of this design as well as
the fabricated version. The results of these updates were vastly improved results over
Table 3.7: Updated Coupler Parameters
Parameter ValueW1 0.8 mmW2 2.0 mmWc 0.4 mmW50 0.1975 mmL50 0.5 mmLt 2.5 mmh1 10 milh2 2 mil
(a) CAD model (b) Fabricated coupler
Figure 3.21: Updated Coupler Design
the original measurements and better agreement with the expected characteristics. Figure
3.22 shows the magnitude response, and Figure 3.23 shows the phase. The differences
44
between simulation and measurement are likely the result of the difficulty with making this
measurement as well as imperfections in the fabricated TRL kit and coupler.
45
Figure 3.22: Final coupler design magnitude response comparison (simulations are dashedlines)
Figure 3.23: Final coupler design output port phase difference comparison
46
Chapter 4
Future Work
With the success in developing this compact wideband hybrid coupler, other work utilizing
a similar topology can be explored. In particular, if a phase shifter could be developed
using this offset coupling technique, it could be incorporated into a Butler matrix design.
This would further aid in creating a compact Butler matrix structure. Figure 4.1 shows an
example of what this kind of phase shifter might look like. This phase shifter design has a
Figure 4.1: Phase shifter mock-up design (left) beside the coupler design (right)
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
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