Theoretical evaluation, analysis and design of surface-mounted waveguide (SMW) components for on-substrate integrated microwave applications

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by

Jan Schorer

B.Eng., University of Applied Science Ravensburg Weingarten, 2010 M.Eng., University of Applied Science Ravensburg Weingarten, 2012

A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of

DOCTOR OF PHILOSOPHY

in the Department of Electrical and Computer Engineering

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Theoretical Evaluation, Analysis and Design of Surface-Mounted Waveguide (SMW) Components for On-Substrate Integrated Microwave Applications

by

Jan Schorer

B.Eng., University of Applied Science Ravensburg Weingarten, 2010 M.Eng., University of Applied Science Ravensburg Weingarten, 2012

Supervisory Committee

Dr. Jens Bornemann, Supervisor

(Department of Electrical and Computer Engineering)

Dr. Poman So, Departmental Member

Dr. Afzal Suleman, Outside Member (Department of Mechanical Engineering)

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Supervisory Committee

Dr. Jens Bornemann, Supervisor

Dr. Poman So, Departmental Member

Dr. Afzal Suleman, Outside Member (Department of Mechanical Engineering)

ABSTRACT

This dissertation presents the research on a novel combination of well proven con-cepts for passive electromagnetic wave-guiding components. The goal of this work is to overcome and minimize losses occurring in frequency-selective structures. The work aims to contribute to an improvement in the application of conventional and Substrate Integrated Waveguide (SIW). It is proposed to mount conventional waveg-uide structures on the surface of printed circuit boards containing substrate integrated waveguides. The crossover technology is referred to as Surface Mounted Waveguide (SMW). Theoretical investigations are performed, proving the validity and superiority of the proposed structure focusing on the elimination of losses, while maintaining low space consumption and printed circuit board technology compatible manufacturing processes. Additionally, a mode matching technique is developed and successfully ap-plied to prototype such components. The validation of this method reveals superior computational speed when compared to commercial available electromagnetic field solvers. The proposed structures are validated by measurements of several proto-types, including coupled SMW resonator filters, combined SMW and SIW resonator filters, a SMW triple-layer diplexer and single individual SMW resonator filters. The experimental verification shows good agreement between theory and measurements.

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Moreover, the comparison to other technologies proves the superiority of the proposed structures.

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List of Tables

Table 2.1 Comparison of waveguide technologies . . . 20 Table 4.1 Comparison of quality factors of single resonator filters . . . 65 Table 4.2 Comparison of quality factors of single resonators of SMW filter 73 Table 4.3 Comparison of quality factors of individual resonators of extended

SMW filter . . . 75 Table 4.4 Comparison of quality factors of five resonator filters . . . 77 Table 4.5 Quality factors of single individual resonator SMW filter . . . . 94 Table 4.6 Comparison of different SMW designs . . . 97

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List of Figures

Figure 2.1 Schematic rectangular waveguide . . . 13

Figure 2.2 Schematic rectangular waveguide modes; left TE10, right TM11 15 Figure 2.3 Schematic rectangular waveguide resonator . . . 17

Figure 2.4 Schematic substrate integrated waveguide . . . 18

Figure 2.5 Spacing of substrate integrated waveguide via holes . . . 19

Figure 2.6 Mode matching sketch . . . 21

Figure 2.7 Double step discontinuity . . . 25

Figure 2.8 E-plane T-junction . . . 26

Figure 2.9 4-port junction . . . 29

Figure 2.10 Schematic of S parameters for a passband filter . . . 34

Figure 2.11 Equivalent circuit model for waveguide filter . . . 35

Figure 2.12 Extracted pole with phase shifters . . . 36

Figure 2.13 Impedance inverter for waveguide resonator coupling . . . 37

Figure 3.1 Cascaded S matrices . . . 40

Figure 3.2 Flow chart of filter design routine . . . 41

Figure 3.3 S matrix sub-block iris . . . 42

Figure 3.4 S matrix sub-block of E-plane iris . . . 43

Figure 3.5 S matrix sub-block of E-plane iris with resonator . . . 43

Figure 3.6 S matrix sub-block double of E-plane iris . . . 44

Figure 3.7 S matrix sub-block of E-plane single SMW resonator . . . 45

Figure 3.8 Scattering parameters comparison for five resonator SIW filter 47 Figure 3.9 Via hole conversion circular to square . . . 48

Figure 3.10 Scattering parameters comparison for SIW to CWG transition and resonator . . . 49

Figure 3.11 SIW to CWG transition with resonator (dimensions in [mm]) 49 Figure 3.12 Different model handling of SIW to CWG transition . . . 50 Figure 3.13 Scattering parameters comparison for five resonator SMW filter 50

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Figure 4.1 SMW assembly stack . . . 55

Figure 4.2 Schematic measurement setup . . . 56

Figure 4.3 Schematic of error compensation during de-embedding process 57 Figure 4.4 Flow graph and picture of through calibration standard . . . . 58

Figure 4.5 Flow graph and picture of reflect calibration standard . . . 59

Figure 4.6 Flow graph and picture of line calibration standard . . . 59

Figure 4.7 Single CWG resonator setup (dimensions in [mm]) . . . 61

Figure 4.8 Q factor of dielectrically filled CWG vs. permittivity (including loss tangent) . . . 62

Figure 4.9 Single SIW resonator setup (dimensions in [mm]) . . . 63

Figure 4.10 Single SMW resonator setup (dimensions in [mm]) . . . 64

Figure 4.11 Losses of single SMW resonator . . . 65

Figure 4.12 CWG five resonator prototype, h=1.5 (dimension in [mm]) . . 66

Figure 4.13 SIW five resonator prototype (dimension in [mm]) . . . 66

Figure 4.14 SIW five resonator prototype scattering parameters . . . 67

Figure 4.15 SMW five resonator prototype (dimension in [mm]) . . . 68

Figure 4.16 Field lines for different SMW coupling methods . . . 69

Figure 4.17 SMW five resonator design employing MS coupling . . . 70

Figure 4.18 S parameter SMW five resonator design, iris vs. MS coupling . 71 Figure 4.19 Tolerance analysis of five resonator SMW prototype . . . 72

Figure 4.20 Picture of SMW prototype . . . 72

Figure 4.21 SMW five resonator prototype; simulation and measurement . 73 Figure 4.22 Wide-band transition detailed view (dimension in [mm]) . . . 74

Figure 4.23 Extended SMW filter with extended transitions (dimension in [mm]) . . . 75

Figure 4.24 Extended SMW simulation . . . 76

Figure 4.25 SMW with SIW resonator setup (dimensions in [mm]) . . . . 78

Figure 4.26 SMW with SIW resonator prototype . . . 78

Figure 4.27 SMW with SIW resonators scattering parameters . . . 79

Figure 4.28 Study for SMW diplexer input coupling . . . 81

Figure 4.29 Study for SMW diplexer output coupling . . . 83

Figure 4.30 SMW diplexer model structure (dimensions in [mm]) . . . 85

Figure 4.31 Picture of the manufactured diplexer prototype . . . 86

Figure 4.32 SMW diplexer scattering parameters . . . 87

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Figure 4.34 Model of distributed single resonator filter . . . 89 Figure 4.35 Tolerance analysis of single resonator SMW model . . . 91 Figure 4.36 Picture of manufactured individual single resonator prototype 92 Figure 4.37 SMW individual resonator scattering parameters . . . 93 Figure 4.38 Footprint of single individual SMW filter variations . . . 94 Figure 4.39 SMW individual resonator example scattering parameters . . 95

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List of Symbols

α attenuation constant β phase constant ψ phase shifter matrix D diagonal matrix I unit matrix

LE,H matrix of T E, T M coupling integrals

M coupling matrix

S generalized scattering matrix

sK generalized scattering matrix for imp. value calculation

Sxy partial scattering matrix, port y to port x

Z impedance matrix

T generalized transfer matrix δ loss tangent

δ0,x Kronecker delta function

permittivity

0 permittivity constant 8.854 × 10−12F/m

r relative permittivity

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ηc characteristic intrinsic impedance γ propagation constant ˆ n normal vector λg0 guided wavelength µ permeability

µ0 permeability constant 4π × 10−7V s/(Am)

µr relative permeability

ω normalized angular frequency ρ skin depth

σ conductivity [S/m]

k_x.y.z complex wavenumber

Z complex impedance ~

B magnetic flux density ~

D electric flux density ~

E electric field ~ez,x,y unit vector

~

H magnetic field ~

J current density ~

JS surface current density

A waveguide discontinuity cross section area a waveguide width

aeq equivalent width of dielectric filled waveguide

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aSIW width of substrate integrated waveguide

b waveguide height

Bhe,mn backward traveling wave

bw filter bandwidth C lossless capacitance c0 speed of light in vacuum

Che,mn cross section function of waveguide discontinuity

d diameter of substrate integrated waveguide via holes dB decibel

e waveguide resonator length f frequency

fc cutoff frequency

Fhe,mn forward traveling wave

fkx corner frequencies of pass-band for filter design

fl,h corner frequencies of stop-band for filter design

hsub height substrate

j imaginary unit

K impedance inverter value k0 free-space wavenumber

kc cutoff wavenumber

m index for waveguide mode N degree / order of filter n index for waveguide mode

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p pitch of substrate integrated waveguide via holes Q quality factor R propagation region Rs surface losses ri ripple of filter in [dB] rl return loss in [dB]

T E transverse electric wave in waveguide T M transverse magnetic wave in waveguide vc phase velocity

Vh,e vector potential of electromagnetic field

X complex impedance

x axis/component of cartesian coordinate system y axis/component of cartesian coordinate system

Yhe,mn admittance of waveguide mode

z axis/component of cartesian coordinate system

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Acronyms

Q quality factor.

CAD computer aided design. CNC computer numerical control. CWG conventional waveguide. DUT device under test.

EM electromagnetic.

ESR equivalent series resistor. FEM finite element method. MMT mode matching technique. MS micro-strip.

PCB printed circuit board. SIC substrate integrated circuits. SIW substrate integrated waveguide. SMW surface mounted waveguide. SoS system on a substrate.

TRL thru / reflect / line. VNA vector network analyzer.

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ACKNOWLEDGEMENTS

I would like to express my sincere gratitude to my supervisor Prof. Dr.-Ing. Jens Bornemann for the inspiration, his continuing support, encouragement and patience. I cannot think of a better mentor for my time spent on this work.

I would like to thank my family: my parents and my sisters for all the support throughout the past years I spent researching for this thesis and the help in manag-ing my life between two continents in general.

Last but not least I would like to thank my colleagues and friends Lisa, Zamzam, Mahbubeh, Farzaneh, Manuel and James for all the good moments and inspirational discussions.

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Introduction

1.1 Surface mounted waveguide

The world is becoming more and more connected and digitalized in every aspect. Emerging concepts like Big Data [1] or the demand for wireless connectivity in every corner on the planet via satellites [2] are only two examples of the ongoing devel-opment. In the process to handle the always increasing data volumes, which are expected to increase at an even higher growth rate in the future [3], the industry pro-viding the technology for this development is demanding components accommodating these higher data rates. Simultaneously, they are pushing for smaller, highly integra-tive system solutions [4] to accommodate new concepts such as, e.g., the internet of things or the enhancement of public mobile communications standards (5G).

As state of the art transmission technology, conventional waveguide (CWG) [5] has been and is still used in high frequency applications. Applications such as, e.g., com-munication links, land and space based, or radar technology are still heavily relying on this well proven technology. However, this technology does not necessarily meet the needs for those newly emerging applications as described above, being sometimes too bulky for integration and generally too expensive for high volume production. Other transmission line technologies like printed circuit board (PCB) structures, e.g., micro-strip (MS), coplanar waveguide or slot line [6] meet the demand for a cost effi-cient production and a high degree of integration at a first glance. Looking closer at their performances and comparing them when used in applications typically employ-ing CWG technology, their performance falls significantly short. They neither reach the required quality factor (Q ) when applied in a filter setup, nor do they provide

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enough electromagnetic (EM) shielding to substitute the CWG technology in a satis-factory way. In the early 2000s, a new waveguide concept was proposed: the substrate integrated waveguide (SIW) [7]. It is the attempt to combine the advantages of CWG technology with those of PCB technology. Since then, much research effort has been invested in this new developing and promising technology concept. SIW technology has presented itself as a versatile, highly integrative, low cost solution for signal trans-mission in the GHz range. Although many CWG structures and concepts have been successfully translated towards SIW technology, e.g. couplers, filter structures, power dividers, there are still some performance issues left over to deal with which the recent development has not yet overcome. One main issue, preventing the technology from being deployed on a large scale, are the losses occurring in the dielectric of the PCB substrate.

This work focuses on a combination of the well proven CWG technology and the developing SIW technology. The combination is made by mounting CWG like struc-tures on top of a SIW. One new aspect is an extension of the SIW technology away from pure, simplified and almost two dimensional H-plane designs, towards E-plane designs extending the structure fully into a third dimension. This extension is verified with a new mode matching technique (MMT) calculation routine. The purpose of this setup is to enhance current SIW filter structures, mainly passive bandpass filters. Several different filter prototypes will be investigated with the focus of realizing an enhancement of the currently achievable Q factors. Therefore, one main goal is to overcome or alleviate a major bottleneck of the SIW technology: losses [8]. Another new benefit of these surface mounted structures is the suitability and scalability for high volume production. This fits into the ongoing movement towards the system on a substrate (SoS) and substrate integrated circuits (SIC) approach [4]. Here different technologies, active as well as passive components, are combined on one substrate and optimized towards an economic production process. Compared to emerging ad-ditive manufacturing concepts like 3 − D printing [9], the technology researched here relies on and employs well known standardized processes and can be applied without any further improvements of the fabrication techniques used. The surface mounted cavities can easily be manufactured in a deep drawing process, with thin walls and an efficient use of material. Furthermore, those cavities can be integrated in a pick and place, reflow soldering assembly lane. Such lanes are state of the art for a PCB production process. This newly proposed structure represents a practical and appli-cable enhancement of SIW technology, and will be referred to as surface mounted

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waveguide (SMW) in this work.

1.2 Outline

This thesis starts with an introduction that gives a brief overview where the proposed work is embedded and the main intention, benefits, respectively, the author sees in conducting this research. It also includes an overview of the current state of the art technology, and the research achievements of others, working in the same or in closely related areas. Also the distinct contributions originating from this research are highlighted, and achievements are backed up with a list of research results. The second chapter focuses on an in depth summary of the theoretical background stating the framework for the conducted research. It starts from Maxwells Equations, focusing on wave propagation with respect to losses in the different technologies, namely CWG and SIW. Also the calculation method used to theoretically describe the SMW, namely MMT, is explained, followed by the theories required and used to synthesize filters.

In the third chapter, the MMT is combined with filter synthesis. It also presents a calculation routine established to efficiently design SMW filter prototypes.

The fourth chapter focuses on the investigation of the prototypes. It starts with an investigation on single resonators realized in the different technologies. A first prototype, a directly coupled five resonator SMW filter is presented, verifying the desired and proposed advantages of SMW by evaluating theoretical, simulated and measured data across the different technologies. The second prototype combines SIW and SMW resonators for the purpose of a better selectivity. The third prototype is a three layer diplexer design enhancing the transition from SIW to SMW into two layers and efficiently utilizing the available PCB surface area. The final fourth prototype employs individual single SMW cavities. All prototypes are simulated and measured. Furthermore, there are additional designs, which either present simulated preliminary studies leading up to the prototype design, or are simulated variations of the actual realized prototype designs.

The conclusion in Chapter 5 gives a summary of the obtained results, followed by a section outlining possible further investigations and directions of the presented technology with a focus on easy manufacturing and application.

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1.3 Contributions of others

This section gives an overview of the current standing of research in the areas relevant to the proposed SMW structure. CWG structures have been studied in an extensive manner, a good summary of the fundamentals can be found in [5]. Focusing on filters realized in CWG technology, [10] gives a good review of the state of the art as well as a profound overview of the basics of microwave filter design in conjunction with outlining the limitations and capabilities of CWG structures. Looking at equivalent circuit theory as an aid in microwave filter design, [11] is a good reference. Focusing on the losses of CWG and the connected Q factor, investigations towards the behav-ior of CWG resonators are made in [12], reporting unloaded Q factors in the range of multiple ten thousands.

Shifting the focus to surface mounted structures, there are design proposals integrat-ing resonators in planar PCB circuits. Reference [13] proposes a cavity filter fed by a standard MS transmission line. The coupling to the cavity is achieved via a slot. In [14] another MS to waveguide converter, namely a MS-to-waveguide transducer, is employed to achieve coupling to a coupled resonator structure. A surface mounted filter is suggested in [15]. This filter uses additional cavity coupling sections to couple the signal from a MS transmission line into the coupled cavities. All three proposed structures use MS transmission lines to couple the signal. Therefore, they fall short in performance, employing conventional transmission lines [6], not providing a fully EM shielded environment. Also there is no quantification of the Q factor reported in these publications.

The SIW technology was proposed around 2000. One of the first publications is for instance [7]. Since then, many CWG structures, mainly in the H plane, have been successfully transferred to SIW technology. Also structures involving the E plane are reported in literature: a surface mounted stub filter [16], couplers [17],[18], power dividers [19] and a surface mounted horn antenna [20]. In line of the SoS approach [4], first active surface mounted components have been integrated in SIW technology like for example a low noise amplifier [21].

In terms of fundamental electrical characteristics of the SIW technology, research on the losses has been conducted in, e.g., [22]. In [23] SIW resonators are investi-gated, reporting unloaded Q factors in the lower multiple hundreds. Further research towards achievable unloaded Q factors is carried out in [24], investigating the dimen-sions of SIW resonators in reference to the Q factor. To reduce the losses, another

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recent attempt is made in [25] by extruding the substrate in the SIW resonators or in the entire SIW [26]. However, the height of the substrate restricts the achievable Q factors for these cutout designs mentioned last.

When calculating waveguide structures, the MMT [27], [28] as well as the BI-RME method [29] are used. While the BI-RME method is mainly proposed together with SIW technology, the MMT is used and has been extensively studied in conjunction with CWG, c.f. [28], [30]. In addition, the MMT is adapted to SIW technology in [31] and [32]. Also it has been used to successfully calculate and optimize SIW components like for instance a diplexer presented in [33].

1.4 Contributions originating from this work

Contributions originating from this work are:

Adaption of MMT for the combination of SIW and CWG.

The MMT is developed for a transition from SIW to CWG by splitting up the struc-ture in individual parts (see [34]). In detail, different irises, resonators and T-junctions are cascaded. The method and its theoretical foundation are described in Section 2.3.

Development of filter design routine to prototype SMW filters.

The MMT is combined with a Chebyshev filter synthesis and an optimization tech-nique employing impedance/admittance inverters (see Section 2.4) to fully automati-cally prototype the dimensions of SMW filters. The routine is build up in single blocks being easily adjustable to different SMW structures (see [35]). The entire routine is described in Chapter 3.

Study on possible transition between SMW and CWG.

In addition to the magnetic E-plane iris coupling, a different coupling method utiliz-ing a micro strip transmission line is investigated (see Section 4.3 and [36]).

Study on losses in single SIW, CWG and SMW resonators.

A comparison based on numerical simulations amongst single resonators in CWG, SIW and SMW is carried out. Also a parametric study of a single SMW resonator is presented (see [36] and [37]). The results are presented in Section 4.2.

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Study on the losses occurring while combining SMW and CWG .

Furthermore, the effect on the Q factor is observed when mounting several coupled SMW resonators on a PCB fed via a SIW. The results are obtained using the eigen-mode solver of a numerical electromagnetic software suite. In conclusion, the results are compared to a pure SIW and a pure CWG filter comprised of the same number of coupled resonators (see Section 4.3 and [36]).

Five resonator SIW and SMW filter prototypes.

A first prototype SMW filter is built and measured (see [37]) prooving the validity and superiority in terms of the Q factor when compared to a SIW filter (see Section 4.3 and [36]).

Five resonator SMW filter prototype with additional SIW resonators. The SMW structures’ stop-band performance is enhanced by employing SIW res-onators in the PCB layer, showing significantly better selectivity of the filter (see Section 4.4 and [38]).

Three layer diplexer prototype.

The transition from SIW to CWG is extended into a second CWG layer and utilized to prototype a diplexer for K band (see Section 4.5 and [38]) showing the expandabil-ity and flexibilexpandabil-ity of the concept.

A pass-band filter prototype with single individual SMW resonators. Single SMW air cavities are employed to gain more flexibility in the frequency re-sponse with a simplified structural design. The prototype exhibits less sensitivity to manufacturing tolerances and better compliance with thermal expansion restrictions during the reflow-oven soldering process (see Section 4.6 and [39]).

1.5 Publications related to this work

• J. Schorer, J. Bornemann, and U. Rosenberg, “Comparison of surface mounted high quality filters for combination of substrate integrated and waveguide tech-nology,” Proc. Asia-Pacific Microwave Conf., pp. 929–931, Nov. 2014 [36]

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• J. Schorer and J. Bornemann, “Mode-matching analysis for double-layered sub-strate integrated and rectangular waveguide filter technology,” Int. Conf. Com-putational Electromagnetics (ICCEM), pp. 158–160, Feb. 2015 [35]

• J. Schorer, Z. Kordiboroujeni, F. Taringou, L. Locke, and J. Bornemann, “De-sign of substrate integrated waveguide components,” Proc. 8th Global Symp. Millimeter-Waves, pp. 1–3, May 2015 [40]

• J. Schorer and J. Bornemann, “A mode-matching technique for the analysis of waveguide-on-substrate components,” Proc. IEEE MTT-S Int. Conf. Numerical Electromagnetic Multiphysics Modeling Optimization (NEMO), pp. 1–3, Aug. 2015 [34]

• J. Schorer, J. Bornemann, and U. Rosenberg, “Design of a surface mounted waveguide filter in substrate integrated waveguide technology,” Proc. 45th Eu-ropean Microwave Conf., pp. 757–760, Sep. 2015 [37]

• J. Schorer, J. Bornemann, and U. Rosenberg, “Mode-matching design of sub-strate mounted waveguide (SMW) components,” accepted for IEEE Trans. Mi-crow. Theory Techn., pp. 1–8, June 2016 [38]

• J. Schorer, J. Bornemann, and U. Rosenberg, “Design of low loss substrate mounted waveguide (SMW) filter employing individual resonators,” accepted for IEEE MTT-S Int. Microwave Symp. Dig., pp. 1–3, May 2016 [39]

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Chapter 2 Theory

2.1 Basic theoretical foundation

This work focuses on a contribution towards improving passive microwave circuitry by minimizing losses occurring when transmitting electromagnetic waves in different media. Therefore, this theory chapter starts with a brief overview of the necessary theoretical background. It first develops the general wave equation based on Maxwell’s equations followed by looking at the propagation mechanism for plane waves. In a further step, the main causes for propagation losses are identified, followed by showing the relation between losses and the quality factors as well as the definition and purpose of the quality factor when qualifying and quantifying the characteristics of passive filters. The information contained in this Section (2.1) is gathered from several textbooks: [41], [42] and [11].

2.1.1 Electromagnetic waves

Maxwell’s equations in differential form can be written as [42]:

∇ · ~D = ρ, (2.1) ∇ · ~B = 0, (2.2) ∇ × ~H = ~J + ∂ ~D ∂t , (2.3) ∇ × ~E = −∂ ~B ∂t. (2.4)

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Observing Equation 2.3 and 2.4 and substituting ~D = ~E and ~H = 1_µB, one can~ obtain the wave equation in a differential form, representing the electrical field ~E and the magnetic flux density ~B:

∇2_{− µ}∂2 ∂t2 − µσ ∂ ∂t ~ E = 0, (2.5) ∇2_{− µ}∂2 ∂t2 − µσ ∂ ∂t ~ B = 0. (2.6)

Assuming a homogeneous propagation media (∇ = 0 and ∇µ = 0) with σ rep-resenting the conductivity and under the assumption of linearity, one can transfer this equation into the frequency domain. For Equation 2.5 one obtains the following representation:

∇2_{E + ω}_~ 2_{µ ~}_{E − jωµσ ~}_{E = 0,} _(2.7)

also called the homogeneous vector Helmholtz equation.

Setting the propagation direction along the z-axis one can state: ∂

∂x = ∂

∂y = 0, (2.8)

for the x − y plane, leading to a simplification of Equation 2.7: ∂E_x

∂z2 + ω

2_µE

x− jωµσEx = 0, (2.9)

and the solution:

Ex(z, t) = Ex,fe−αzcos (ωt − βz + ∠f) + Ex,beαzcos (ωt + βz + ∠b). (2.10)

Ex(z, t) is a superposition of two traveling waves in ±z direction identified with the

subscripts f for the forward and b for the backward traveling wave. The terms α and β are factors determining the propagation of the wave and can be obtained from Equation 2.9:

p−ω2_{µ + jωµσ = α + jβ = γ} _(2.11)

The magnetic field component H_y can be obtained by taking Equation 2.4 and trans-ferring it into the frequency domain:

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By using ~B = µ ~H this leads to: H_y(z) = γ jωµ Ex,fe −γz_{− E} x,be γz_{= H} y,fe −γz + H_y,beγz. (2.13) This equation can be expressed in the same way as Equation 2.10:

Hy(z, t) = Hy,fe−αzcos (ωt − βz + ∠f) + Hy,beαzcos (ωt + βz + ∠b). (2.14)

2.1.2 Wave propagation and losses

Observing a uniform plane wave propagating in a lossy medium in z-direction along a wave-guiding structure, one can denote the electric field E(z, t) and the magnetic field H(z, t) as shown in Equation 2.10 and 2.14. Please note that these equations are a general case, only valid for the propagation of an uniform plane wave and do not hold for the propagation in a CWG nor a SIW. The exact propagation mechanism in such structures will be shown later in Section 2.2. Focusing on the occurring losses, they are represented by the terms α, the attenuation constant and β, the phase constant. Combining them one can state the complex propagation constant γ as shown in Equation 2.11. Both constants (α and β) are dependent on the permittivity r( = 0r), the permeability µ (µ = µrµ0) and the conductivity σ of the propagation

medium. They can be expressed in the following way [42]:

α = ω √ µ √ 2 s r 1 + σ ω 2 − 1, (2.15) β = ω √ µ √ 2 s r 1 + σ ω 2 + 1. (2.16)

α represents the real part of the complex propagation constant and is therefore the cause for the attenuation of the wave amplitude while traveling in a lossy media. β affects the phase of the propagating wave and is therefore a cause for dispersion. Both parameters are also influenced by the material’s conductivity σ. Since the conductivity appears in the numerator of either α and β, a small conductivity leads to lower losses in the propagation medium.

Another measure connected to the losses is the characteristic impedance ηc of the

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amplitudes of the electric and magnetic fields: ηc = E_x,f H_y,f = s jωµ0µr σ + jω0r . (2.17)

The term also connects the permittivity , conductivity σ and the permeability µ. Looking at their relation, one can verify the assumptions made for the lossy case based on the observation of α and β. It becomes obvious that as small as possi-ble values for the permittivity and conductivity are leading to a high characteristic impedance. For the free space propagation case, the conductivity is σ ≤ 3 × 10−15 and the characteristic permittivity is at r = 1, leading to a characteristic impedance

of ηc= 377Ω.

Often the losses are denoted as a factor called the loss tangent tan(δ) (see Equation 2.20). This factor can be derived based on the equivalent series resistor (ESR) de-rived from the principle of equivalent circuit theory (presented for instance [43]); a mathematical representation is given in Equation 2.18.

ESR = σdielec

ω2_C, (2.18)

with C being the lossless capacitance of the dielectric. Observing the ESR in an alternating current/ field, hence introducing a complex plane, the loss tangent can be derived as follows. tan(δ) = ESR | XC | , (2.19) where XC = 1 ωC. (2.20)

To identify another source of losses, one has to shift the focus away from the prop-agation medium and the conjoined propprop-agation losses towards the conductors. A comprehensive study is presented in [44]. The conductors or conductive walls are guiding the wave along its propagation path. The wave, more specificly the magnetic field of the wave, induces time varying surface currents JS(t) onto the surrounding

material:

ˆ

n × ~H(t) = ~JS(t) (2.21)

These currents encounter the specific resistance or conductivity σconductor of the

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conductivity, the smaller are the losses.

The third class of losses are radiation losses. These are typically occurring when the wave-guiding structure is not closed entirely and energy is leaking out of the structure. CWG and SIW can be considered as closed structures. Therefore, radiation losses do not contribute to the total amount of losses. The minor leakage which might occur in SIW structures is included in the dielectric losses.

2.1.3 Quality factor Q

A measure for classification and qualification of resonating structures is the so-called quality factor Q (see 2.22). This dimensionless number describes the losses occurring in resonating structures and is typically applied in, e.g., filter technology. A general definition (see [45]) of this number can be given as:

Q(ω) = ωmax. energy stored

losses (2.22)

Hence a resonator obtains a higher quality factor when the occurring losses are eliminated or minimized for a fixed amount of stored energy, typically limited by its dimensions. The following equation combines the different types of losses described above towards the total Q factor:

1 Qtot = 1 Qdielectric + 1 Qconductor + 1 Qradiation (2.23) indicating the variables to tackle when trying to minimize the total amount of losses. The combination and weighting of the different loss types is accomplished by treating them as a circuit consisting of parallel elements, according to a concept borrowed from linear circuit theory. By inverting the losses, the smallest Q factor obtains the dominant influence. All three different losses are weighted equally.

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2.2 Waveguide technology

The two applied wave-guiding technologies for this work are CWG technology and SIW technology. The following section develops the electromagnetic field representa-tion for both technologies. It lays the basic theoretical foundarepresenta-tion for the intended application and combination of both. The overview is kept brief, since this set of formulas is well established: for more details the reader is referred to classic electro-dynamics books like [43], [44] or [5].

2.2.1 Conventional waveguide technology

Figure 2.1: Schematic rectangular waveguide

Starting with the wave equation developed in Section 2.1.1, Equation 2.5 and 2.6 in the frequency domain:

∇ × ~H = jω ~E, (2.24)

∇ × ~E = −jωµ ~H, (2.25)

can be split up into six cartesian components namely: E_x, E_y, E_z and H_x, H_y, H_z leading to the two independent equations representing the longitudinal (with the z-axis as propagation direction, see Figure 2.1) field components. This is achieved by combining Equation 2.24 and 2.25 with the Laplacian operator (Equation 2.26)

(∆t+ k2− kz2) " E_z H_z # = 0, (2.26)

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with k = ω2_{µ and k}

z = β − jα leading to the following Helmholtz Equations (2.27

and 2.28). ∂2 ∂x2 + ∂2 ∂y2 + k 2_{− k}2 z E_z = 0, (2.27) ∂2 ∂x2 + ∂2 ∂y2 + k 2_{− k}2 z H_z = 0, (2.28)

Further, utilizing the Bernoulli separation for cartesian coordinates, one can state for T M waves:

E_z = E₀f (x)g(y)e±j(β−jα), (2.29) and introduce the two differential equations:

− k2 x = 1 f d2_f d2_x, (2.30) − k2 y = 1 g d2_g d2_y, (2.31)

which then need to fulfill the separation equation (2.32).

k_x2+ k_y2+ k2_z = k2 = ω2µ. (2.32) With the help of the two trigonometric equations:

f (x) = C1sinkxx + C2coskxx, (2.33)

g(y) = C3sinkyy + C4coskyy (2.34)

one can either solve for H_z = 0 (Dirichlet conditions) with the boundary conditions E_z = 0 for x = 0, x = a, y = 0, y = b (with a and b representing the physical dimensions of the rectangular waveguide, c.f. Figure 2.1), obtaining the T Mmn modes

of the waveguide: E_z = E₀sinmπx a sin nπy b e ±jk_zz_, _(2.35)

or for E_z = 0 (Neumann conditions) with the boundary condition ∂Hz

∂n = 0, obtaining T Emn modes: H_z = H₀cosmπx a cos nπy b e ±jk_zz_. _(2.36)

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Figure 2.2: Schematic rectangular waveguide modes; left TE10, right TM11

T M11 mode, are displayed. The schematic shows the field lines of the E and the H

field.

The propagation characteristics of the different modes are determined by k_z, the complex wavenumber: k_z;m,n = r ω√µ −mπ a 2 −nπ b 2 . (2.37)

To obtain a propagating wave in the waveguide, k_z has to become real. This lead to the cutoff wave number kc which can be determined from Equation 2.37:

kc;m,n= r mπ a 2 +nπ b 2 , (2.38)

simplifying to the term:

fcT E10 =

c

2a, (2.39)

for the fundamental T E10 mode cutoff frequency. In a lossless waveguide, below this

cutoff frequency, k_z is imaginary and the modes are evanescent, meaning they are no longer propagating and die off exponentially in z-direction with the attenuation of e−αz where α can be determined as:

α = r mπ a 2 + nπ b 2 − ω√µ. (2.40)

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resis-tance losses (conductor losses) Rs and propagation losses. Hence the attenuation

constant α is different for every T E and T M mode:

αT Emn = 2Rs bη0 q 1 − kc2/k02 1 + b a kc2 k02 + b a ρ 2 − kc2 k02 m2_{ab + n}2_a2 m2_b2_{+ n}2_a2 , (2.41) αT M mn = 2Rs bη0 q 1 − kc2/k02 m2b3+ n2a3 m2_b2_{a + n}2_a3, (2.42)

with the resistance Rs [42] encountered by the surface currents Js(t) ( see Equation

2.21): Rs= r ωµ 2σconductor , (2.43)

and the skin depth ρ (c.f. [5]):

ρ = √ 1

πf µσconductor

. (2.44)

Waveguide resonator

Dealing with frequency selective passive wave-guiding structures, resonators are of fundamental importance. They are basically confined waveguide pieces causing a certain wavelength, corresponding to their dimensions, to resonate, c.f. Figure 2.3. The inherent resonant frequency can be calculated using Equation 2.45 [42]

fm,n,l = c0 2√µrr s m a 2 +n b 2 + l e 2 . (2.45)

Observing the fundamental T E101 resonance, the formula simplifies. It is possible to

connect the surface losses Rs to this air filled resonator structure by [42]

Qconductor = (kae)2bη 2π2_R s 1 a3_{(2b + e) + (2b + a)e}3, (2.46)

clearly showing that the Q can be increased by expanding the height of the resonator (b dimension in the denominator of Equation 2.46). Assuming a dielectric filled resonator, the total Q can be determined from the ratio of the conductor losses and

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Figure 2.3: Schematic rectangular waveguide resonator

the volume, i.e. dielectric losses as outlined in Equation 2.23. 1 Qresonator = 1 Qvolume + 1 Qconductor . (2.47)

This leads to the critical equation for this work when observing SMW resonators [42]: 1 Qresonator = tanδ + (kae) 2_bη 2π2_R s 1 l2_a3_{(2b + e) + (2b + a)e}3 . (2.48)

2.2.2 Substrate integrated waveguide technology

SIW is a relatively new developing technology. It transfers the CWG principle onto PCBs, offering a new additional transmission line technology for SICs. A SIW, a schematic is shown in Figure 2.4, consists of a rectangular structure realized in PCB technology. The metallic coating of the substrate acts as top and bottom walls of the waveguide. The waveguide’s side walls are realized by metallized via holes. Here the via holes are realized in a circular shape. While there are also elliptical shapes and rectangular shapes reported in literature, the circular shape is the contour suiting best an easy manufacturing process. A study of the via hole spacing is carried out in [46], and the result of this study is summarized in Figure 2.5. The figure plots the relation between the diameter d and the pitch p of the via holes, normalized to the cut off wavenumber. It identifies several cases constraining the SIW, namely structural

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Figure 2.4: Schematic substrate integrated waveguide

stability (over perforation), radiation leakage into the surrounding substrate and the band gap. A ratio of 0.5 ≤ d/p ≤ 0.8 has been proven to provide a good compromise between structural stability and leakage of the wave. Adhering to this ratio, one can neglect leakage or radiation losses occurring in SIW. To observe the electric and magnetic field of a wave propagating in a SIW, the same set of formulas as for CWG (developed in Section 2.2.1) can be used. The SIW is treated as a dielectric loaded waveguide. Several formulas interrelating the SIW width aSIW to the width aeq of an

equivalent CWG have been published for example in [47], [48] and [49]; the simplest one is [47]:

aSIW = aeq+

d2

0.95p. (2.49)

Another more recent one is [49]:

aSIW = aeq+ p 0.766e0.4482d/p− 1.176e−1.214d/p . (2.50)

There was no implication of enhanced accuracy when modeling resonating structures and frequency selective transitions with newer formulas like Equation 2.50. The height b is represented by the substrate thickness hsub. Applying these equations, it

is possible to calculate the cutoff frequency for a SIW: fcSIW T E10 =

1 √

µ2aeq

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Figure 2.5: Spacing of substrate integrated waveguide via holes

Due to the non continuous sidewalls of the SIW, a surface current flow along the sidewalls in longitudinal direction along the z axis is not possible. Hence a SIW is not capable of supporting T M modes or other T E modes than T Em0 modes.

Losses occurring in a SIW can be classified into three different categories:

conductor losses are caused by the characteristic conductivity σ of the metallized via holes and the top and bottom coating of the substrate.

dielectric losses result from the substrate’s permittivity r and the related loss

tangent δ of the substrate.

radiation losses due to the non continuos sidewalls cause possible leakage of the wave into the surrounding substrate.

Trying to minimize the overall losses, the main parameters to adjust are the height of the substrate hsuband the via hole parameters d and p. The height hsubdetermines the

ratio of the conductor and dielectric losses. Increasing the height lowers the influence of the conductor losses on the overall loss, since the dielectric losses are outweighing the conductor losses.

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by choosing a proper spacing of the via holes (c.f. Figure 2.5). This results in the dielectric losses being the main cause for loss in a SIW. By selecting a proper substrate for the desired operating frequency, meaning a low characteristic permittivity and loss tangent δ, the dielectric losses can be reduced.

2.2.3 Overview and comparison of waveguide technology

Table 2.1 qualifies the advantages and disadvantages for the two waveguide tech-nologies from above. Additionally, another frequently used PCB transmission line technology, namely MS, is added to justify the extra manufacturing effort necessary when producing SIW, SMW components. The Q factor values show that SIW repre-sents a reasonable compromise between CWG and MS technologies.

CWG SIW MS

manufac. tech. metal profiles PCB PCB

space consumption high height of PCB one layer of PCB losses conduct. conduct. dielec. conduct. dielec. radiation Q factor Q > 103 ₁₀2 _{< Q < 10}3 _{Q < 10}2

Table 2.1: Comparison of waveguide technologies

2.3 Mode matching

MMT is an extensively studied computational calculation method applied mainly in the area of conventional waveguide [28]. It is based on a modal analysis of disconti-nuities in a propagation medium. It can be efficiently applied when the wave-guiding structures have fixed boundaries, as it is the case for both technologies, CWG and SIW, surveilled here. One of the advantages of MMT is the accuracy the method provides. In view of this work, MMT is used to calculate interconnections between CWG, SIW and SMW. The structures modeled are: a plain waveguide of arbitrary length, double step discontinuities with different propagation media, a three port T-junctions extending into the E plane and a four port intersection, also extending into the E plane. This section first gives an overview of the basic principle of a full T Emn and T Mmn mode analysis (More fundamental information, can be found in

[28]). This basic principle is then extended to describe the models for the structural discontinuities used to calculate SMW components.

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2.3.1 Theory

The modal analysis of a waveguide is based on the vector potentials [50] which can be obtained from the electromagnetic field of the waveguide (Equation 2.24 and 2.25). Looking at the respective components in propagation direction, this leads to:

~ E = −1 jω0 ∇ × ∇ × (Vez~ez) + ∇ × (Vhz~ez), (2.52) ~ H = −1 jωµ0 ∇ × ∇ × (Vhz~ez) + ∇ × (Vez~ez). (2.53)

These vector potentials V are calculated for different regions of the structural waveg-uide discontinuity as outlined in Figure 2.6. The obtained vector potentials can be

Figure 2.6: Mode matching sketch expressed as [28]: V_hzR = M X m=0 N X n=0 q ZR h,mnC R h,mn(x, y) F_h,mnR e−jkRzh,mnz_{+ B}R h,mne jkR zh,mnz , (2.54) V_ezR = M X m=1 N X n=1 q YR e,mnC R e,mn(x, y) F_e,mnR e−jkRze,mnz_{+ B}R e,mne jkR ze,mnz . (2.55) R is an index describing the region the vector potential can be ascribed to. In this example (Figure 2.6) Region I and II. Z (Equation 2.56) and Y (Equation 2.57) are mode specific impedances and admittances, respectively [28]

Z_hmnR = ωµ0 kR zhmn = 1 YR hmn , (2.56)

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Y_emnR = ω0 kR zemn = 1 ZR emn , (2.57)

with the wavenumber:

k_zhmnR = k_zemnR = r k2 0 − mπ aR 2 −nπ bR 2 . (2.58)

C are eigenfunctions of the waveguide modes cross section. They are further employed to set the different regions, vector potentials in a relation depending on the x and y components of the fields. They also contain a power normalization [28]. Examples for such eigenfunctions and the corresponding power normalization for a rectangular waveguide are given in Equations 2.76 to 2.79. FR _{and B}R _{are the forward and}

backward traveling wave amplitudes of the observed waveguide modes.

Adhering to the boundary conditions in a waveguide at the discontinuity (z = 0) for both regions R = I, II and with the cross section areas AI and AII.

E_x,yI |z=0,AI = E

II

x,y|z=0,AII, (2.59)

E_x,yII|z=0,AI−AII = 0, (2.60)

H_x,yI |z=0,AII = H

II

x,y|z=0,AII, (2.61)

Equations 2.59 to 2.61 ensure the continuity of the field. This allows for a matching of the transverse components Ex, Ey, Hx, Hy of the two different vector potentials Vh,ezI

and VII

h,ez and leads to the following M matrices (Equation 2.62 to 2.65). The M

matrices are calculated by the coupling integrals over the common cross section area A of both regions using Equations 2.59 to 2.61. They can be expressed as:

Mhh= Z A (∇CI_h,i× ~ez)(∇CIIh,k× ~ez)dA, (2.62) Mhe = Z A

(∇CI_h,i× ~ez)(∇CIIe,k)dA, (2.63)

Meh = Z A (−∇CI_e,i)(∇CII_h,k× ~ez)dA, (2.64) Mee= Z A

(∇CI_e,i)((∇CII_e,k)dA, (2.65) where mode indices m, n from region I are represented as i and mode indices p, q from region II as k. To simplify and organize the calculation, the modes are ordered

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according to their cutoff frequency. This reduces the double sums in Equation 2.54 and Equation 2.55. Interrelating the two different regions based on energy conservation and the wave amplitudes FRand BR, one can establish the following equation system [28]:

FI+ BI = LE(FII − BII), (2.66)

LH(FI− BI) = FII + BII, (2.67)

where the impedances Z, admittances Y and the M matrices are contained in LE:

LE =   D(qYI h,mn)MhhD( q ZII h,pq) D( q YI h,mn)MheD( q ZII e,pq) D(qYI e,mn)MehD( q ZII h,pq) D( q YI e,mn)MeeD( q ZII e,pq)  . (2.68)

LE contains four sub matrices, mapping the two sets of modes (T E and T M ) to

each other. D represents a diagonal matrix containing the square root of either the admittance or the impedance of the modes. LH can be calculated from 2.61 as

LH = LTE due to the reciprocity of the discontinuity.

As a last step, the generalized scattering matrix is obtained utilizing LE and LH:

S11 = (LELH + I)−1(LELH − I), (2.69)

S21 = LH[I − (LELH + I)−1(LELH − I)], (2.70)

S12= 2(LELH + I)−1LE, (2.71)

S22= I − 2LH(LELH + I)−1LE. (2.72)

This generalized scattering matrix contains sub matrices for the different modes de-rived from their coupling. Neglecting losses to verify the consistency of the calculation method, the generalized scattering matrix has to satisfy the following condition:

S = ST = S−1. (2.73)

Another aspect to take into consideration is the convergence of the MMT method. To achieve a sufficient accuracy, the so called upper cutoff frequency, the cutoff frequency of the highest order mode included in the calculation, is a critical factor. Several rules of thumb interrelating this frequency either to the T E10 mode or to the ratio of the

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in line of this work observes the amount of energy captured in the considered modes, for the lossless case, and derives the number of modes accordingly.

2.3.2 Modeled structures

Plain waveguide [28]

The most simple and easiest structure to model using the mode based analysis is a plain waveguide. Due to the lack of discontinuities, a plain waveguide of finite length with a perfect absorber at its end will not cause any reflections to the EM field. Therefore, the S11 and S22 components of the generalized scattering matrix will be

zero: S = " 0 Dw Dw 0 # , (2.74) with: Dw = " e−jkzh,mnl ₀ 0 e−jkze,mnl # . (2.75)

However, the waveguide will support propagating and even evanescent modes until they are attenuated completely. Dw is a diagonal matrix since coupling between T M

and T E is not possible.

Double step discontinuity [28]

The second modeled structure is a discontinuity in the waveguide. The discontinuity is of rectangular shape in the x - y cross section. The presented analysis can be applied to a single change in width (x), height (y) or to a change in both directions as presented in Figure 2.7. The calculation of this discontinuity follows the general procedure described above. The specific C terms for this structure are [28]:

C_h,mnI = _q 2 aI_bI₍mπ aI )2+ ( nπ bI)2 cos(mπ_aI x) p1 + δ0,m cos(nπ_bIy) p1 + δ0,n , (2.76) C_e,mnI = _q 2 aI_bI₍mπ aI )2+ ( nπ bI)2 sin(mπ aI x) sin( nπ bI y), (2.77)

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Figure 2.7: Double step discontinuity C_h,pqII = _q 2 aII_bII₍qπ aII)2+ ( pπ bII)2 cos(_aqπII(x − ( aI_−aII 2 ))) p1 + δ0,q cos(_bpπII(y − ( bI_−bII 2 ))) p1 + δ0,p , (2.78) C_e,pqII = _q 2 aII_bII₍qπ aII)2+ ( pπ bII)2 sin(qπ aII(x − ( aI− aII 2 ))) sin( pπ bII(y − ( bI− bII 2 ))), (2.79) where δ0,i is the Kronecker delta, leading to the coupling integrals represented in

Equations 2.62 to 2.65. The coupling integral 2.63 vanishes for this structure. This routine can be implemented in a function to be used in a filter design routine (see Chapter 3).

T-junction [30]

To model a connection from the SIW to the SMW, this work uses a T-junction in the E-plane. This T-junction is also modeled using a full T E-T M analysis. As outlined in Figure 2.8, one needs to extend the general MMT routine by an additional region RIII. The vector potential of this region can be found analogous to the vector potential of the other regions. For region RIII_{, the z coordinate has to be substituted with the y}

coordinate to represent the correct propagation direction of the wave incident at the boundary of region three and four. The vector potentials for RIII _are:

V_hyIII = G X g=0 U X u=0 q ZIII h,guC III h,gu(x, z)

F_h,guIIIe−jkIIIzh,guy_{+ B}III

h,gue jkIII

zh,guy

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Figure 2.8: E-plane T-junction V_eyIII = G X g=1 U X u=1 q YIII

e,guCe,guIII(x, z)

F_e,guIIIe−jkIIIze,guy _{+ B}III

e,guejk

III ze,guy

. (2.81)

Using the three different vector potentials from RI, RII and RIII, it is possible to determine a superposition for region RIV (see Figure 2.8). This superposition is established as: V_hzIV = − M X m=0 N X n=0 q ZI h,mn(∇C I h,mn× ~ez) sin(k_zh,mnI (z − bIII)) sin(kI zh,mnbIII) F_h,mnI + B_h,mnI + M X m=1 N X n=1 q ZI e,mn(∇C I e,mn)

j sin(k_ze,mnI (z − bIII)) sin(kI ze,mnbIII) F_e,mnI + B_e,mnI + P X p=0 Q X q=0 q ZII h,pq(∇C II h,pq× ~ez) sin(kII zh,pqz) sin(kII zh,pqbIII) F_h,pqII + B_h,pqII − P X p=1 Q X q=1 q ZII e,pq(∇Ce,pqII ) j sin(kII ze,pqz) sin(kII ze,pqbIII) F_e,pqII + B_e,pqII − G X g=0 U X u=0 q ZIII h,gu(∇C III h,gu× ~ey) sin(kIII zh,gu(y − bI)) sin(kIII zh,gubI)

F_h,guIII + B_h,guIII +

G X g=1 U X u=1 q ZIII e,gu(∇C III e,gu) j sin(kIII ze,gu(y − bI)) sin(kIII ze,gubI)

F_e,guIII + B_e,guIII , (2.82)

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V_ezIV = −j M X m=0 N X n=0 q YI h,mn(∇C I h,mn) cos(kI zh,mn(z − bIII)) sin(kI zh,mnbIII) F_h,mnI + B_h,mnI − j M X m=1 N X n=1 q YI e,mn(∇C I e,mn× ~ez) cos(kI ze,mn(z − bIII)) sin(kI ze,mnbIII) F_e,mnI + B_e,mnI + j P X p=0 Q X q=0 q YII h,pq(∇C II h,pq) cos(k_zh,pqII z) sin(kII zh,pqbIII) F_h,pqII + B_h,pqII + j P X p=1 Q X q=1 q YII e,pq(∇C II e,pq × ~ez) cos(k_ze,pqII z) sin(kII ze,pqbIII) F_e,pqII + B_e,pqII − j G X g=0 U X u=0 q YIII h,gu(∇C III h,gu)

cos(k_zh,guIII (y − bI)) sin(kIII

zh,gubI)

F_h,guIII + B_h,guIII −

j G X g=1 U X u=1 q YIII e,gu(∇C III e,gu× ~ey) cos(kIII ze,gu(y − bI)) sin(kIII ze,gubI)

F_e,guIII + B_e,guIII .

(2.83)

Knowing all four vector potentials, it is possible to set up an equation system similar to the one presented in Equations 2.69 - 2.72 to obtain the generalized scattering matrix. Therefore, the vector potentials are matched for the following locations: z = 0 for VI

h,e and Vh,eIV:

FI− BI _{= −D}I T(F

I_{+ B}I_{) + D}II

S (F

II _{+ B}II_{) + L}I−III_(FIII _{+ B}III_), _(2.84)

z = bIII _{for V}II

h,e and Vh,eIV:

FII − BII _{= −D}I S(F

I_{+ B}I_{) + D}II

T (F

II _{+ B}II_{) + L}II−III_(FIII _{+ B}III_), _(2.85)

y = bI, bII for V_h,eIII and V_h,eIV:

FIII − BIII = LIII−I(FI+ BI) + LIII−II(FII+ BII) + DIII_S (FIII + BIII). (2.86) Here diagonal matrix DR_T, represents the reflected incident wave and DR_S the short-circuited wave: DI,II_T =     j tan(kR h,zbIII) 0 0 j tan(kR e,zbIII)     , (2.87)

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DI,II_S =     j sin(kR h,zbIII) 0 0 j sin(kR e,zbIII)     , (2.88)

and DIII_T due to the interchanged coordinates:

DIII_T = "

j cot(k_h,yIIIbI) 0 0 j cot(kIII_e,ybI)

#

. (2.89)

The LR−R matrices are coupling matrices, like those presented in Equation 2.68. Rearranged in matrix notation, Equations 2.84 to 2.86 are:

   BI FII FIII   =    −(I − DI T) −D II S −L I−III DI_S (I − DII_T ) −LII−III

−LIII−I _−LIII−II _{(I − D}III

S )    −1    −(I + DI_T) DII_S LI−III −DI_S (I + DII_T ) LII−III

LIII−I LIII−II (I + DIII_S )

      FI BII BIII   , (2.90)

and transformed into the form of the generalized scattering matrix:    BI FII FIII   =    S11 S12 S13 S21 S22 S23 S31 S32 S33       FI BII BIII   . (2.91)

The entire routine can be implemented in a function, tailoring it to a modular appli-cation in a filter design routine (see Chapter 3).

Four port junction

For an efficient use of the available substrate’s surface area, another structure is required, utilizing the top and bottom plane. The fundamental waveguide configu-ration at heart of a transition to two independent SMW branches can be seen as a four port junction. Such an intersection extends into the E plane, therefore a full T E − T M analysis has to be performed. A schematic of this junction type is pre-sented in Figure 2.9. The method used to model this structure in MMT is extended from the T-junction model presented in [30]. The four port junction is divided into

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Figure 2.9: 4-port junction

five regions. Regions one to four are determined by the physical dimensions of the connected ports. The vector potentials for the regions can be derived similar to those of the T-Junction. As already seen when modeling the T-junction, the longitudinal wave propagation direction is changing in RIII and RIV to the coordinate y and −y, respectively. The vector potentials for RIII are given in Equations 2.80 and 2.81 and the ones for RIV _are:

V_hyIV = L X l=0 F X f =0 q ZIV h,lfC IV h,lf(x, z) F_h,lfIV e−jkzh,lfIV y_{+ B}IV h,lfe jkIV zh,lfy , (2.92) and V_eyIV = L X l=1 F X f =1 q YIV e,lfC IV e,lf(x, z) F_e,lfIVe−jkze,lfIV y _{+ B}IV e,lfe jkIV ze,lfy . (2.93)

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In the next step the superposition of all four regions RI _{to R}IV _{is established, yielding}

the vector potential of region RV:

V_hzIV = − M X m=0 N X n=0 q ZI h,mn(∇C I h,mn× ~ez) sin(kI zh,mn(z − bIII)) sin(kI zh,mnbIII) F_h,mnI + B_h,mnI + M X m=1 N X n=1 q ZI e,mn(∇C I e,mn) j sin(kI ze,mn(z − bIII)) sin(kI ze,mnbIII) F_e,mnI + B_e,mnI + P X p=0 Q X q=0 q ZII h,pq(∇C II h,pq× ~ez) sin(kII_zh,pqz) sin(kII zh,pqbIII) F_h,pqII + B_h,pqII − P X p=1 Q X q=1 q ZII e,pq(∇C II e,pq) j sin(kII_ze,pqz) sin(kII ze,pqbIII) F_e,pqII + B_e,pqII − G X g=0 U X u=0 q ZIII h,gu(∇C III h,gu× ~ey)

sin(k_zh,guIII (y − bI)) sin(kIII

zh,gubI)

F_h,guIII + B_h,guIII +

G X g=1 U X u=1 q ZIII e,gu(∇C III e,gu) j sin(kIII ze,gu(y − bI)) sin(kIII ze,gubI)

F_e,guIII + B_e,guIII +

L X l=0 F X f =0 q ZIV h,lf(∇C IV h,lf × ~ey) sin(kIV_zh,lfy) sin(kIV zh,lfbI) F_h,lfIV + B_h,lfIV − L X l=1 F X f =1 q ZIV e,lf(∇C IV e,lf) j sin(kIV ze,lfy) sin(kIV ze,lfbI) F_e,lfIV + BIV_e,lf , (2.94)

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and V_ezIV = −j M X m=0 N X n=0 q YI h,mn(∇C I h,mn) cos(kI zh,mn(z − bIII)) sin(kI zh,mnbIII) F_h,mnI + B_h,mnI − j M X m=1 N X n=1 q YI e,mn(∇Ce,mnI × ~ez) cos(kI ze,mn(z − bIII)) sin(kI ze,mnbIII) F_e,mnI + B_e,mnI + j P X p=0 Q X q=0 q YII h,pq(∇C II h,pq) cos(kII zh,pqz) sin(kII zh,pqbIII) F_h,pqII + B_h,pqII + j P X p=1 Q X q=1 q YII e,pq(∇C II e,pq × ~ez) cos(kII ze,pqz) sin(kII ze,pqbIII) F_e,pqII + B_e,pqII − j G X g=0 U X u=0 q YIII h,gu(∇C III h,gu) cos(kIII zh,gu(y − bI)) sin(kIII zh,gubI)

F_h,guIII + B_h,guIII −

j G X g=1 U X u=1 q YIII e,gu(∇C III e,gu× ~ey)

cos(kIII_ze,gu(y − bI)) sin(kIII

ze,gubI)

F_e,guIII + BIII_e,gu +

j L X l=0 F X f =0 q YIV h,lf(∇C IV h,lf) cos(kIV zh,lfy) sin(kIV zh,lfbI) F_h,lfIV + B_h,lfIV + j L X l=1 F X f =1 q YIV e,lf(∇C IV e,lf × ~ey) cos(kIV ze,lfy) sin(kIV ze,lfbI) F_e,lfIV + B_e,lfIV . (2.95)

In order to obtain the general scattering matrix for this structure, one has to establish an equation system. These equations can be determined through matching of the four vector potentials RI _{- R}V I _{to the superposition vector potential R}V _{at the respective}

boundaries: z = 0 for VI

h,e and Vh,eV :

FI − BI = −DI_T(FI+ BI) + DII_S (FII + BII)+ LI−III(FIII + BIII) − LI−IV(FIV + BIV),

(2.96)

z = bIII, bIV for V_h,eII and V_h,eV : FII − BII _{= −D}I

S(F

I_{+ B}I_{) + D}II

T (F

II _{+ B}II₎₊

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y = bI_{, b}II _{for V}III

h,e and Vh,eV :

FIII − BIII _{= −L}III−I_(FI_{+ B}I_{) + L}III−II_(FII_{+ B}II₎₊

DIII_S (FIII + BIII) − DIV_T (FIV + BIV),

(2.98)

y = 0 for VIV

h,e and Vh,eV :

FIII − BIII = −LIII−I(FI+ BI) + LIII−II(FII+ BII)+

DIII_T (FIII + BIII) − DIV_S (FIV + BIV), (2.99) where DR_T represents the reflected incident wave and DR_S the short-circuited wave. For region I and II those matrices are given in Equations 2.87 and 2.88. For region III and IV they can be established as:

DIII,IV_T = "

j cot(k_h,yIII,IVbI,II) 0

0 j cot(kIII,IV_e,y bI,II) # , (2.100) DIII,IV_S =     j

sin(kIII,IV_h,y bI,II₎ 0

0 j

sin(ke,yIII,IVbI,II)

   

. (2.101)

The LR−R matrices represent the coupling (see Equations 2.68). Defining region I as input, one can rearrange the equation system (Equation 2.96 to 2.99):

      BI FII FIII FIV       =       (I − DI_T) DII_S −LI−III LI−IV DI_S (I − DII_T ) LII−III −LII−IV

LIII−I −LIII−II _{(I + D}III

T ) −D

IV S

LIV −I −LIV −II _DIII

S (I − D IV T )       −1       (I + DI_T) −DII S L I−III _−LI−IV −DI S (I + D II T ) −L II−III _LII−IV

−LIII−I _LIII−II _{(I − D}III

T ) +D

IV S

−LIV −I LIV −II −DIII_S (I + DIV_T )

            FI BII BIII BIV       , (2.102)

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and solve for the generalized S parameters:       BI FII FIII FIV       =       S11 S12 S13 S14 S21 S22 S23 S24 S31 S32 S33 S34 S41 S42 S43 S44             FI BII BIII BIV       . (2.103)

2.4 Filter synthesis

This section outlines the theoretical design and synthesis of filters. The surface mounted waveguide prototypes investigated in line of this work can either be classified as coupled resonator filters or single individual resonator filters. The two synthesis techniques suiting these filter types are the Chebyshev and the extracted pole syn-thesis. First the theory for both techniques is summarized. In the next step, both synthesis methods are interrelated to the mode matching technique described in Sec-tion 2.3.

2.4.1 Chebyshev filter design

The Chebyshev filter synthesis method [11] uses a prototype low pass filter for the synthesis. In general there are two different types of Chebyshev filter: Type I and II. The first one can be characterized by less stop band ripple, the latter one exhibits less ripple in the passband. Compared to the Butterworth method, the Chebyshev method is superior due to steeper skirts at the corner frequencies [11]. Taking the calculation effort into account, it is also superior to elliptic function synthesis, while still maintaining accurate results. A schematic of a passband filter is displayed in Figure 2.10. The synthesis procedure starts with the specification of the return loss ripple ri (S11 parameter in the pass band), represented as a normalized value in [dB]:

ri = −10 log (1 − (10  − rl 20dB   )2) (2.104)

In the next step, the filter bandwidth bw and corresponding corner frequencies fk1

and fk2 are defined. Knowing the waveguide cutoff frequency fcof the T E10 mode for

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Figure 2.10: Schematic of S parameters for a passband filter ([42]): λg0 = 1 2( vc pf2 k1− fc2 + vc pf2 k2− fc2 ). (2.105)

Applying the guided wavelength, the defined design goal attenuation al,h of the S21

parameter and the frequencies fl,h, one obtains the degree N , number of resonators,

of the filter: N = max              arcosh     1 r 10 ri 10dB − 1 r 10 ah 10dB − 1     arcosh(ωh) , arcosh     1 r 10 ri 10dB − 1 r 10 al 10dB − 1     arcosh(ωl)              , (2.106)

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where ωh and ωl are normalized angular frequencies for the upper and lower stop

band corner. Knowing the degree of the filter, one can transfer the canonical form of the obtained synthesis results into a ladder network topology [11]. It accommodates coupled, as well as single individual resonator filters. The different coefficients, e.g., capacitances and inductances of the single elements of the filter (c.f. Figure 2.11, g0...gn+1) can be calculated. In the next step, the waveguide resonators are modeled

Figure 2.11: Equivalent circuit model for waveguide filter

as equivalent circuit resonators. To match the different resonating circuits, impedance inverters are introduced. These impedance inverters are realized as discontinuities in the waveguide structure represented as K inverters in the equivalent circuit model (see Figure 2.11) [11].

2.4.2 Extracted pole filter design

Filters with single distributed cavities, the second filter type investigated in this work, are introducing transmission zeros in the S21 parameter [51]. These transmission

zeros are typically sensitive to the design. Using the extracted pole filter synthesis [51], one can control these transmission zeros by defining them during the synthesis process. Starting with a filter transfer function, obtained from a filter synthesis, e.g. Chebyshev, elliptic integrals and the definition of the ABCD transfer Matrix T

T = 1 jF " A1 + jA2 B C A1− jA2 # , (2.107)

Theoretical evaluation, analysis and design of surface-mounted waveguide (SMW) components for on-substrate integrated microwave applications

Contents

List of Tables

List of Figures

List of Symbols

Acronyms

Introduction

1.1

Surface mounted waveguide

1.2

Outline

1.3

Contributions of others

1.4

Contributions originating from this work

1.5

Publications related to this work

Chapter 2

Theory

2.1

Basic theoretical foundation

2.1.1

Electromagnetic waves

2.1.2

Wave propagation and losses

2.1.3

Quality factor Q

2.2

Waveguide technology

2.2.1

Conventional waveguide technology

2.2.2

Substrate integrated waveguide technology

2.2.3

Overview and comparison of waveguide technology

2.3

Mode matching

2.3.1

Theory

2.3.2

Modeled structures

2.4

Filter synthesis

2.4.1

Chebyshev filter design

2.4.2

Extracted pole filter design