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


Kultegin Aydin


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.

iii

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

iv

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

v

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

vi

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

vii

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

viii

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


3.2 Parameters of initial coupler design . . . . . . . . . . . . . . . . . . . . . 27


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

ix

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.

x

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

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.

18

(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

19

Figure 2.12: Scanning circular array with Butler matrix feed

20

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

21

(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

23

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


S12A =S12N (1+S11N · [ESF−ELR])


S21A =S21N (1+S22N · [ESR−ELF])


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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Documents

A COMPACT WIDEBAND STRIPLINE HYBRID COUPLER