Analytical Design and Optimization of a WR-3 Waveguide Diplexer

Synthesized using Direct Coupled Resonator Cavities

Project in Applied Physics, Uppsala University, January 2017

Authors: Markus Back, Rickard ViikMaster Programme in Engineering Physics, Uppsala University

Supervisor: Dragos DancilaDepartment of Engineering Sciences, Solid State Electronics, Uppsala University

Abstract

A WR-3 coupled waveguide resonator diplexer is designed, analytically, using the insertionloss method, and subsequently simulated and optimized in HFSS. The design features ten iriscoupled resonator cavities, assembled with a power divider in a T-junction topology. The diplexerchannel filters yield a 5th order Chebyshev type frequency response, centered around 265 GHzand 300 GHz, respectivley. The resulting diplexer channels have bandwidths of 13 GHz and 11.6GHz, respectivley, and a maximum passband return loss of -6.35 dB for channel A and -6.95 dBfor channel B. A usable diplexer should have a passband return loss of at most -20 dB. Furtheroptimization or, alternatively, changing the features of the design is needed to reduce the passbandripple to a level for which the diplexer is usable in practice. Possible improvements can be madeby making the channel passbands narrower or, alternatively, increasing the number of resonatorsin the channel filters.

1

Contents

1 Introduction 41.1 Introduction to Microwave Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Project Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Theory 62.1 Waveguides & Resonance Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.2 Resonance Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Filter Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.1 Scattering Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.2 Chebyshev Filter Design by the Insertion Loss Method . . . . . . . . . . . . . . 102.2.3 Chebyshev Low-Pass Prototype . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.4 LP Prototype to Bandpass Transformation . . . . . . . . . . . . . . . . . . . . 132.2.5 Waveguide Resonator Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Diplexers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Method 173.1 Design Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Analytical Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 Extraction of Iris Opening Widths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4 Filter Simulation & Tuning in HFSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.5 Diplexer Simulation & Tuning in HFSS . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Results and Discussion 244.1 Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.1.1 Initial Design Performed with Analytical Calculations and CMS . . . . . . . . 244.1.2 Optimized Design (HFSS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2 Diplexer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5 Summary and Conclusions 31

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List of abbreviations

AC - Alternating Current

BP - Bandpass

BS - Band-stop

CMS - Coupling Matrix Synthesis

EHF - Extremely High Frequency

EM - Electromagnetic

HFSS - High Frequency Structure Simulator

HP - High-Pass

IL - Insertion Loss

LP - Low-Pass

MEMS - Micro-Electro-Mechanical Systems

PEC - Perfect Electric Conductor

PMC - Perfect Magnetic Conductor

RF - Radio Frequency

RL - Return Loss / Reflection Loss

SHF - Super High Frequency

TE - Transverse Electric

TEM - Transverse Electromagnetic

TM - Transverse Magnetic

UHF - Ultra High Frequency

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1 Introduction

1.1 Introduction to Microwave Technology

Microwaves have widespread use in many fields of technology such as telecommunication [1], militaryand law enforcement systems [2], astronomic remote sensing [3], heating [1], medical technology [4],etc. The use of microwaves became important for military use during World War II for the purpose ofhigh resolution radar detection of ships and airplanes [1, 4, 3]. Furthermore, many materials exhibitresonance phenomena due to interaction with EM waves in the microwave region, which also makesmicrowaves useful for molecular/nuclear spectroscopy for material analysis [3].

The frequency interval of electromagnetic (EM) radiation which corresponds to microwaves rangesfrom ca 300 MHz to 300 GHz [4]. The microwave region can be divided into the subsets Ultra HighFrequency (UHF), Super High Frequency (SHF) and Extremely High Frequency (EHF), as shown intable 1.

Table 1: Microwave frequency bands

Frequency band Frequency (GHz) Wavelength (m)

UHF 0.3 - 3 1 - 0.1SHF 3 - 30 0.1 - 0.01EHF 30 - 300 0.01 - 0.001

Furthermore, the frequency region of 300 GHz - 3000 GHz, contiguous to EHF, is known as sub-millimeter waves or THz waves.

All of the aforementioned technological application areas of microwaves employ the use of mi-crowave filters in some way or another. The use of filters is motivated by the need to select certainfrequency bands for microwave systems to operate within, while simultaneously rejecting unwantedfrequencies. Much of the theory of microwave filters was developed in the 1950’s by Matthaei, Young,Jones and others, which resulted in an extensive handbook containing standard methods for filter de-sign which is still relevant today [5]. There are several methods of design and synthesis of microwavefilters, the most common ones including the use of microstrips, striplines or waveguides. For highpower and low loss devices in the microwave region, the preferred method is to use waveguides [1].For lower frequencies, the low loss property of waveguides comes at the prize of bulky structures withhigh fabrication cost, which may make other devices such as striplines or microstrips more feasible forthose frequencies.

An important application of microwave filters is the case where several communications devicesoperating on different frequency bands are required to be received on a common antenna and sub-sequently sent on along to different outputs. One way to prevent interference between the differentdevices is to project the signals from each device onto the antenna using a multiplexer. A multiplexeris constructed by combining several microwave filters (one for each frequency band) with a powerdistribution network. The multi-band signal coming from the shared antenna is inserted into themultiplexer, which splits the multi-band signal and separates it into several single-band channels. Ineach channel, a microwave filter allows transmission of a single frequency band, while rejecting the

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others, thus preventing cross-talk between the different channels. A multiplexer with only two separatefrequency band channels is called a diplexer.

1.2 Project Description

An important reason for the suitability of high frequency microwaves or THz waves for certain appli-cations, as opposed to lower frequency waves, is that the size of microwave systems are often inverselyproportional to the operating frequency. Thus, higher operating frequencies allow for more compactsystems, which makes possible Micro-Electro-Mechanical Systems (MEMS)[6] such as lab-on-a-chiptechnology, as well as facilitate the need for smaller and lighter communication devices in e.g. air-planes, ships and spacecrafts where limitations to weight and size are of high importance. Furthermore,as technology progresses, more and more of the microwave spectrum is used up by existing technology.This pushes research into development of technological solutions, able to operate in higher frequencyranges, such as extremely high frequency microwaves and sub-millimeter waves.

The larger context of this project is a collaborative effort (MEMS THz Systems (SSF)) betweenUppsala University (UU), Royal Institute of Technology (KTH) and the Swedish Defence ResearchAgency (FOI) to develop proof-of-concept prototypes of MEMS operating in the THz frequency bandof electromagnetic waves [7].

The goal of the project presented in this paper is to demonstrate the design and optimization ofa diplexer operating within the frequency range of 220-325 GHz. This frequency range correspondsto EHF to THz microwaves, making hollow rectangular waveguides suitable as a basis for the design.The channel filters of the diplexer can thus be realized by a series of connected waveguide cavities(resonators), coupled to one another by small apertures (irises). The diplexer is then constructed byconnecting each channel filter to a waveguide T-junction. The recommended dimensions for rectan-gular waveguides operating in this frequency range is specified by the WR-3 standard.

The chosen methodology for the design of the demonstrated diplexer is based on methods foundin [1, 5, 8, 9, 10].

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2 Theory

In order to design a coupled resonator waveguide diplexer, some theoretical background study has beenrequired. Literature has been studied to understand the nature of waveguides, and how these can beused to allow transmission of EM (electromagnetic) signals through different types of RF (Radio Fre-quency) and microwave systems [1]. Furthermore, section 2.1.2 discusses how waveguide sections withterminated ends function as electromagnetic resonance cavities, making them well suited for applica-tions in signal modulation (e.g. microwave filter technology). EM filters are an essential component ofcoupled resonator waveguide diplexers. Section 2.2.1 briefly discusses so called scattering parameters[11], which describe the frequency response of such filters. A useful method for designing Chebyshevtype bandpass filters is described in sections 2.2.2 - 2.2.4, which can be realized by electromagneticallycoupled waveguide resonance cavities [1, 12, 13]. Finally, section 2.3 presents the general purpose andconfiguration of a microwave diplexer [10].

2.1 Waveguides & Resonance Cavities

2.1.1 Waveguides

Waveguides are conducting structures that allow waves to propagate through them along a certainpath. These structures are useful in high frequency AC electronics, where the frequency is high enoughthat the propagating wave nature of the alternating electromagnetic fields cannot be ignored. Differ-ent types of waveguides allow for different types of waves to propagate through them and they canconsist of one continuous piece of conducting material or of several conductors put together. Waveg-uides can either be empty, or be filled with some dielectric medium. For electromagnetic waves, themodes of the waves allowed to propagate through the waveguide are dependent on the dimensionsof the waveguide and of the composition of the conductors that makes up the waveguide. The dif-ferent possible modes of propagation for electromagnetic waves are TE (transverse electric; meaningthat the electric field has at least one component in the transverse direction relative to the directionof propagation), TM (transverse magnetic) and TEM (transverse electromagnetic). TEM waves aresupported by transmission lines and waveguides consisting of more than one conductor, e.g. a parallelplate waveguide. Rectangular waveguides, consisting of only one conductor, only allow for TE andTM waves to propagate through them [1].

Assuming that the general static electric field−→E and magnetic field

−→H propagating in the z-

direction, divided into its transverse and longitudinal components, can be written as

−→E (x, y, z) =

[−→e (x, y) + zez(x, y)]e−jβz (1)

and −→H (x, y, z) =

[−→h (x, y) + zhz(x, y)

]e−jβz, (2)

where β is the wave propagation constant, the TE wave mode will, according to the Maxwell equations,depend only on Hz (the longitudinal component of the magnetic field) [1] . The Maxwell equations

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Figure 1: A rectangular waveguide of width a, height b and infinite length. (Taken with permission fromwww.commons.wikimedia.org/wiki/User:Zykure, CC BY-SA 3.0)

can then be reduced to the wave equation

(∂

∂x2+

∂

∂y2+ k2c

)hz(x, y) = 0, (3)

where Hz(x, y, z) = hz(x, y)e−jβz and kc =√k2 − β2 is the cutoff wave number, i.e. the wave

number corresponding to the lowest frequency wave that is able to propagate through the waveguide.In a rectangular waveguide geometry, the solution to equation (3) (e.g. using the method of separationof variables) gives an expression for the cutoff wave number of the waveguide as

kc =

√(mπa

)2+(nπb

)2, (4)

where a is the larger one of the two dimensions not in the direction of propagation and b is thesmaller one (see figure 1). The integers m and n refer to the Tmn mode of the electric wave. Thecorresponding cutoff frequency can be written as

fcmn =1

2π√µεkc, (5)

where 1√µε is the plane wave speed of light through the medium in the waveguide. The lowest value

for the cutoff frequency is fc10 which corresponds to the mode T10. This also is the most dominant ofthe modes for a rectangular waveguide geometry [1].

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2.1.2 Resonance Cavities

By closing off a section of a rectangular waveguide, a cavity is created in which the electromagneticwaves can store energy in the form of standing waves. Due to the addition of two more walls withnormal vector in the direction of the propagation of the wave, there is now a possibility of wavesbeing reflected and propagating in the negative z-direction, giving rise to the standing waves at someresonance frequencies. Equation (1), which describes the electric field in the waveguide, becomes withthe new geometry

−→E (x, y, z) = −→e (x, y)

(A+e−jβmnz +A−ejβmnz

), (6)

where A+ and A− are the amplitudes of waves travelling in +z-direction and −z-direction respec-tively and −→e (x, y) is the transverse components of the field [1].

The resonance frequencies of the cavity can be found analogously to how the cutoff frequency ofthe waveguide was found, but with the additional boundary conditions set by the new walls insertedinto the Maxwell equations. It then follows that the resonance wave number for a wave mode TEmnlin a waveguide cavity with dimensions b < a < d (see figure 2) can be written as [1]

kmnl =

√(mπa

)2+(nπb

)2+( lπd

)2, (7)

with the corresponding resonance frequencies being

fmnl =1

2π√µεkmnl. (8)

The most dominant mode, corresponding to the lowest order resonance frequency of the cavity, isthen the TE101 mode with frequency

f101 =1

2π√µεk101 =

1

2π√µε

√(πa

)2+(πd

)2. (9)

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Figure 2: A rectangular waveguide cavity

2.2 Filter Theory

In general, a filter is a signal processing device used to eliminate certain undesired frequency contents(e.g. noise or interference from other signals) from a signal of interest. Fundamental characteristics offilters include low-pass (LP), high-pass (HP), bandpass (BP) and band-stop (BS) behaviors. In thefield of RF or microwave engineering, a filter is a 2-port network used for manipulating the frequencyresponse of a RF or microwave system. Here, the term port or wave-port simply refers to the means bywhich a microwave signal enters or exits a system (for example, a waveguide or a network of coupledresonance cavities), i.e. it can be understood as a signal input/output.

The following sections outlines some basic theory on how to design a microwave bandpass filterusing coupled resonator waveguide technology.

2.2.1 Scattering Parameters

The frequency response of a general multiport network is represented by so called scattering matricesS = [Sij ]. Sij is known as a scattering parameter, which is defined as the amplitude ratio of theoutgoing/reflected wave at the ith port and an incident wave at the jth port.

An example is presented in figure 3, which features a schematic of a 2-port system, consisting ofa section of transmission line, with waves ai and bi entering/exiting the system at the ith port. If asignal is excited at port 1, then S21 can be thought of as the transmission amplitude and S11 as thereflection amplitude of the system. Scattering parameters that can be mathematically described byrational expressions of polynomials are thus equivalent to transfer functions.

9

Figure 3: Schematic of a 2-port system, composed of a section of transmission line.

Consider an n-port system and let a1, a2, · · · , an and b1, b2, · · · , bn represent the amplitudes ofthe incident and reflected waves, respectively, at ports 1, 2, · · · , n, then the relationship between theincident and reflected contributions in the network is described by equation (10) [11].

b1b2...bn

=

S11 S12 · · · S1nS21 S22 · · · S2n

......

. . ....

Sn1 Sn2 · · · Snn

a1a2...an

(10)

2.2.2 Chebyshev Filter Design by the Insertion Loss Method

The insertion loss method is a well established method for filter design. With this approach, thefrequency response of a filter is specified by its insertion loss IL, defined in equation (11) [1].

IL = 10 logPLR. (11)

It can be noted that the power loss ratio PLR is simply the reciprocal of |S21|2, which can bewritten as

PLR = 1 +M(ω2)

N(ω2), (12)

where M, N are polynomials of ω2. As such, IL is equivalent to power transmission through thefilter. When designing a nth order Chebyshev filter, the power loss ratio is specified as

PLR = 1 + k2T 2n

( ωωc

), (13)

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where Tn( ωωc) is a Chebyshev polynomial of the nth order and ωc is the cut-off frequency of the

filter. The frequency response, produced by this type of filter (see figure 4 for an example), will exhibita passband ripple amplitude of 1 + k2, but will have a sharper passband edge and steeper slope whencompared to a maximally flat (or Butterworth type) filter. For reference, some Chebyshev polynomialsare listed in table 2.

Table 2: Chebyshev polynomials of order n = 1− 4.

n Tn(x)

1 x2 2x2 − 13 4x3 − 3x4 8x4 − 8x2 + 1

At ω far away from ωc the insertion loss becomes approximately

IL ≈ 10 log(k2

4

(2ω

ωc

)2n), (14)

from which the required filter order n can be determined, by specifying a desired ripple and inser-tion loss at some ω far away from ωc [8].

Figure 4: Frequency response example of a 5th order Chebyshev BP filter, centered around 50 GHz with abandwidth of 10 GHz and a passband ripple of 3 dB.

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2.2.3 Chebyshev Low-Pass Prototype

A standard procedure for synthesizing filters is to begin with a low-pass prototype, i.e. a low-passfilter design which is normalized to have a cutoff frequency of ωc = 1 (or sometimes fc = 1 Hz) andinput impedance of 1Ω. A general filter of any type (low-pass, high-pass, band-pass or band-stop) canthen be constructed from the prototype by making the appropriate frequency scaling and impedancetransformation.

The LP prototype is modeled as a lumped element circuit consisting of discrete electrical compo-nents, as depicted in figure 5.

Figure 5: Low-pass filter prototype, N = 3.

Formulas for calculating the prototype element values g0, · · · , gN + 1 (see figure 5), as well astabulated values for Chebyshev filters with different passband ripple, can be found in several textbookson microwave engineering and filter design [1, 9, 5]. For reference, table 3 lists some element valuesfor Chebyshev LP-prototype filters with a passband ripple of 0.0432 dB.

Table 3: Element values for N th order Chebyshev LP prototype filter with normalized input impedanceg0 = 1Ω, cutoff frequency Ωc = 1 Hz and passband ripple LAr = 0.0432 dB [9].

N g1 g2 g3 g4 g5 g6 g7 g8 g9 g10

1 0.2000 1.02 0.6648 0.5445 1.22103 0.8516 1.1032 0.8516 1.04 0.9314 1.2920 1.5775 0.7628 1.22105 0.9714 1.3721 1.8014 1.3721 0.9714 1.06 0.9940 1.4131 1.8933 1.5506 1.7253 0.8141 1.22107 1.0080 1.4368 1.9398 1.6220 1.9398 1.4368 1.0080 1.08 1.0171 1.4518 1.9667 1.6574 2.0237 1.6107 1.7726 0.8330 1.22109 1.0235 1.4619 1.9837 1.6778 2.0649 1.6778 1.9837 1.4619 1.0235 1.0

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2.2.4 LP Prototype to Bandpass Transformation

As can be seen in figure 5, the LP prototype circuit is modeled as a sequence of series inductors andshunt capacitors with normalized element values gi, connected to source/load impedances of 1 Ohm.Transformation from the LP prototype to a desired bandpass filter is achieved by replacing the LPfilter circuit components with shunt susceptances Bk and series reactances Xk, modeled as LC-circuits(see figure 6). The values of the new circuit components can be found by scaling the frequency domainas described in equations (15) - (16) [1].

ω → ω′ =1

∆

( ωω0− ω0

ω

)(15)

∆ =ω2 − ω1

ω0(16)

ω1 and ω2 are the angular frequencies of the lower and higher passband edges, corresponding tothe desired frequency response, ∆ is the desired bandwidth and the center frequency, ω0, is defined asthe geometric mean

√ω1ω2.

The series reactances and shunt susceptances are given by equations (17) and (18) [1],

jXk =j

∆

( ωω0− ω0

ω

)Lk, (17)

jBk =j

∆

( ωω0− ω0

ω

)Ck, (18)

where Lk and Ck are the series inductance and shunt capacitance of the LP prototype filter. Theresulting component values, L′k and C ′k, of the LC-circuit replacing the series inductors of the LPprototype are found by combining equations (15)-(18) and are thus given by equations (19) and (20),

L′k =Lk

∆ω0(19)

C ′k =∆

ω0Lk. (20)

Similarly, the component values of the LC-circuits replacing the shunt capacitors of the LP proto-type are given by equations (21) and (22),

L′k =∆

ω0Ck(21)

C ′k =Ck

∆ω0(22)

Figure 6 depicts a circuit schematic of the bandpass filter resulting from transforming the LPprototype from figure 5.

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Figure 6: Bandpass filter schematic.

2.2.5 Waveguide Resonator Filters

A Chebyshev-like filter response can be achieved by electromagnetically coupling several resonancecavities together. The coupling is achieved by making an opening (iris) in the shared wall between theresonance cavities, allowing the electromagnetic field to propagate through the opening and resonatein the neighbouring cavity as well. The strength of the coupling (i.e. the level of power transfer fromone resonating cavity to another) directly depend on the dimensions of the iris. The smaller the irisis, the weaker the coupling, and vice versa. If the iris is too large (overcoupled), there will be noresonance in the cavity and the wave will just propagate through without reflecting. If the iris is toosmall (undercoupled), all wave intensity will be reflected on the wall and there will be no transferof energy through the iris to the neighbouring cavity. The order of the Chebyshev response is givenby the number of resonators that are coupled together to make the filter. A higher order filter willtherefore be a larger physical structure consisting of more cavities than a lower order filter, whichmight limit the highest available order if there are limits on the physical dimensions.

The strength of the coupling between the resonator cavities can be expressed in the form of a cou-pling matrix M (equation (23)), which contains all mutual coupling coefficients between the resonators.

M =

m11 m12 · · · m1n

m21 m22 · · · m2n...

.... . .

...mn1 mn2 · · · mnn

(23)

The matrix element mij is the strength of the coupling between cavity i and cavity j, and n isthe number of cavities in the filter. Since there is no directional dependence on the strength of thecoupling, the matrix will be symmetric, i.e. mij = mji. The diagonal elements mii represent the

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self-coupling of the resonators. These self-coupling elements are related to the resonance frequency ofthe corresponding cavity in relation to the frequency f0 at the center of the passband. If all resonatorswere tuned to the center frequency, the corresponding self-coupling elements in the diagonal would allbe zero [13].

The introduction of resonator apertures will affect the analytically calculated resonance frequenciesof the cavities, as calculated in equation (8), and thus shift the frequency response of the filter. Toget the desired frequency response, the iris sizes must be tuned to the right values. When properlytuned, each cavity will contribute one reflection zero (a dip in the reflection amplitude S11), whichcorresponds to a peak in the signal transmission amplitude at that frequency through the filter [12].If the resonator dimensions and their mutual couplings are well tuned, the transmission peaks willbe close enough to form a continuous passband. The bandwidth, ripple and center position of thepassband depend directly on the dimensions of the cavities and the coupling matrix (iris dimensions).

2.3 Diplexers

Diplexers are devices used in RF and microwave systems to combine two disjoint-band signals fromseparate channels into a single signal, and project it onto a common port (or vice versa). They aregenerally synthesized by combining a set of channel filters with a power-distribution network (seefigure 7) [10]. The components of the diplexer are connected by transmission lines, such as waveg-uides. Each filter is designed to allow its corresponding channel’s signal to pass through unaffected,while simultaneously rejecting the other channel’s signal. This prevents interference between the twochannels occupying the shared port.

Figure 7: Schematic of a diplexer configuration. Signals of frequencies f1 and f2 enter the system and are splitby the power divider. Each channel filter then selects its corresponding signal to pass through to the output.

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The frequency response of a diplexer is described in terms of its scattering parameters, S21 and S31,which respresent the amplitudes of the transmitted signals of channels A and B, respectively. Figure 8describes, qualitatively, how a typical frequency response could look, where |S21| and |S31| are plottedagainst some frequency span, which has been normalized in the figure. Note that, in general, the twochannel passbands of a diplexer are neither necessarily non-contiguous or have a narrow bandwidth,but could depending on the desired function, for example, be realized by a HP- and a LP-filter witha shared cut-off frequency.

Figure 8: Frequency response of a diplexer.

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3 Method

This section presents the design and simulation of a WR-3 waveguide diplexer, composed of coupledresonator waveguide filters. Filters are designed through analytical calculations, according to thedesign specifications outlined in section 3.1. Calculations are presented in section 3.2. After thephysical dimensions of the filters are extracted, simulation and optimization of the filter designs areperformed in HFSS [14], which is a computer aided design (CAD) software program, based on finiteelement methods. A 3D model of the diplexer is later assembled in HFSS from the optimized filters,and optimization is further performed on the whole structure (section 3.5).

3.1 Design Specifications

The diplexer, presented in this report, is a 3-port coupled waveguide resonator network. It consistsof two separate channel filters, connected to a common port. The diplexer is synthesized by initiallydesigning and optimizing each channel filter independently and subsequently assembling the diplexerby connecting the filters in a T-junction topology (see figure 9).

Figure 9: Topological schematic of a T-junction coupled resonator diplexer. Black squares represent resonatorsand lines represent couplings.

In terms of topology and channel bands, the diplexer is intended to follow the design of T. Skaik in[15]. The two channels of the diplexer, which from here on will be referred to as channel A and channelB respectively, will occupy two disjoint frequency bands in the WR-3 frequency range of 220 - 325GHz. WR-3 is a standard for rectangular waveguides, which specifies the transverse inner dimensionsof the waveguide as a = 863.6 µm and b = 431.8 µm, where a, by convention, is usually consideredto be the width of the waveguide and b is considered to be the height. The center frequencies ofchannels A and B will be 265 GHz and 300 GHz respectively. Both channels should have a bandwidthof 15 GHz and passband ripple of LAr = 0.0432 dB, corresponding to a return loss of -20 dB in thepassband.

3.2 Analytical Filter Design

This section outlines the analytical design process of the diplexer channel filters, following the methoddescribed in [8].

17

The filters consist of five iris coupled waveguide resonator cavities (see figure 10). Such a filter isknown to be able to achieve a N th order Chebyshev-like response, where N is equal to the numberof resonators [8]. The frequency response of the filter is determined, mainly, by the lengths, li, ofthe cavities and the sizes, di of the iris openings. The dimensions of the waveguide resonator filterscan be analytically calculated by first designing a low-pass prototype filter with normalized inputimpedance and cutoff frequency. Appropriate transformations are then performed to transform thelow-pass prototype filter to a bandpass filter with the desired frequency response. The waveguideresonator dimensions are then calculated by relating them to the bandpass filter component values.

Figure 10: Schematic of the top view of a coupled resonator waveguide filter.

For the analytical design, it has been assumed that a waveguide filter can be viewed as a seriesof shunt inductors between two transmission lines [1]. The coupling irises are then modeled as shuntsusceptances Bi (see equations (24) - (26)) [8].

B1 =1− ωR

g1√ωR/g1

(24)

Bk =1

ω

(1− ω2

gkgk−1

)√gkgk−1 (25)

BN =1− ωR

gN−1√ωR/gN−1

(26)

R = 2k2 + 1−√

4k2(1 + k2) ≈ 0.9802, where k = 10LAr/10 − 1, and ω = π2 (β2 − β1)/β0.

The dispersion relation of a waveguide mode is given by βj =√k2j − k2c , where kj = 2πfj

√µε is

the free-space wavenumber of the mode and kc = π/w0 is the cutoff wavenumber [1].

The coupling coefficients ki,i+1, which are later used to extract the physical size of the iris open-ings, are calculated as shown in equation (27),

18

ki,i+1 =BW

f0

√1

gigi+1, (27)

where BW is the bandwidth of the filter and f0 is the center frequency of the passband. Theelement values gi that were used are defined for a LP prototype filter with a 5th order Chebyshevresponse and a passband ripple of 0.0432 dB (see table 3).

The physical length of each resonator is related to its electrical length φ as shown in equation (28),

li =φiπ

λg02, (28)

where λg0 = 2π/β0. The electrical length is, in turn, given by equation (29) [8].

φi = π − 1

2

(tan−1

2

Bi+1+ tan−1

2

Bi

)(29)

Now remains only to extract the physical size of the iris openings, which is presented in the nextsection.

3.3 Extraction of Iris Opening Widths

Consider a circuit composed of two coupled resonators with resonant frequency f. It can be shownthat, when the symmetry plane of the circuit is replaced with a short circuit, the coupling coefficient isaltered, resulting in a lower resonant frequency fe [9]. Similarly, replacing the symmetry plane with anopen circuit, increases the resonant frequency to fm. A relationship between the coupling coefficientand the shifted resonant frequencies is given by equation (30) [9].

kij =f2e − f2mf2e + f2m

(30)

In the case of two waveguide resonators coupled by an iris opening, the short circuit is equivalentto letting a wall with a perfect electric conductor (PEC) boundary condition span across the width ofthe iris and, similarly, the open circuit is equivalent to a perfect magnetic conductor (PMC) wall.

In order to extract the appropriate widths of the iris openings for the waveguide filters, equation(30) is used together with HFSS simulations. Two waveguide resonators coupled by an iris is simulated,with a PEC and PMC wall, respectively, spanning the width of the iris opening. The resulting resonantfrequencies are then calculated by the software for each case, while incrementally increasing the widthof the iris opening. The coupling coefficient is extracted by inserting the resonance frequencies intoequation (30) and the result is plotted as a function of the coupling iris width. The appropriate irisopening widths di for the filters can then be graphically read (or interpolated) from figure 11, assumingthe analytically calculated coupling coefficients from equation (27).

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Figure 11: Coupling coefficient vs iris opening, extracted through HFSS simulations and calculated withequation (30).

3.4 Filter Simulation & Tuning in HFSS

Figure 12 shows the 3D model, generated in HFSS, of the channel A filter after optimization of thephysical filter dimensions. The filter was simulated as an air filled perfect electric conductor (PEC),composed of five iris coupled resonator cavities. The waveguide walls and irises have a thickness of 37µm. A simulated electromagnetic signal is fed into and out of the filter via two waveguide extensionswhich act as wave ports. The filters frequency response is given by the S-parameters, S11 and S21,which represent the reflection/insertion loss of the device. The S-parameters are generated by feedinga frequency-swept signal into port 1 and solving for the outgoing signal at each port, as a result ofthe wave excitation at port 1.

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Figure 12: HFSS model of the channel A filter.

Starting with the analytical design values shown in tables 4 and 5, the lengths of the cavities, theiris opening widths, the length of the waveguide extensions and the waveguide thickness of each filterare all optimized to generate the desired frequency response in terms of bandwidth, center frequencyand reflection loss. Tuning is performed in HFSS partly by iteration, using the software default Quasi-Newton optimization algorithm [16], and partly by manual fine tuning.

Figure 13 shows a 3D plot of the electric field magnitude of a simulated electromagnetic wave,propagating through the channel A filter. The wave is simulated with a frequency of 265 GHz,which lies in the center of the filter’s passband. The plot shows how the coupling apertures act toallow electromagnetic excitations to travel between neighbouring resonators, and how the resonancephenomenon gives rise to standing waves in each resonator, with maximal amplitude in the center ofthe cavity.

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Figure 13: 3D plot of the electric field magnitude inside the channel A filter for a transmitted signal at 265GHz, generated from HFSS simulation.

3.5 Diplexer Simulation & Tuning in HFSS

The first attempt at synthesizing the diplexer from the optimized channel filters was performed fol-lowing the topological design featured in [15], i.e. in a T-topology with eight resonance cavities (seefigure 9). Since each channel filter was designed with five resonance cavities, this design means thattwo resonance cavities has to be shared between both diplexer channels, and thus the original filterdesign is perturbed to some extent. HFSS simulations of this design, however, resulted in the channelfilters being severely mistuned, to the extent of being completely unusable. This should, perhaps,not have been entirely surprising, when one considers that the individual channel filters had not beendesigned or tuned with this particular topology in mind. Initially, attempts were made to recover thespecified frequency response by tuning the coupling apertures and resonators (as was done with eachindividual filter). However, since this proved to be both very difficult and time consuming, the choicewas made to abandon this design completely, in favor of a different one.

A different and simpler approach was tried, in which the input ports of both channel filters aresimply connected to a WR-3 waveguide T-junction, which then serves as a power divider. In addition,a wedge was introduced into the center of the junction, as this has been demonstrated to generally im-prove the insertion loss of a power divider [10]. The HFSS model of this design is presented in figure 14.

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Figure 14: Top view of the HFSS model chosen for the diplexer.

The new design consists of ten resonators and no cavities are shared between the channels, whichallowed the original filter design to remain unchanged. As expected, HFSS simulations of this designyielded far superior results, although slightly detuning the filters. Further optimization of the entirediplexer was therefore necessary to achieve the desired passband return loss of -20 dB. Optimizationwas performed in HFSS, in the same way as for each individual filter, using a combination of thesoftware’s built-in optimization tools and manual fine tuning.

Figures 15a and 15b, respectively, show magnitude surface plots of the electromagnetic field prop-agating in the complete structure as microwave signals of (a) 265 GHz and (b) 300 GHz are excitedinto the common port. As can be seen in the figure, the 265 GHz signal is only permitted to propagatethrough channel A, and the 300 GHz signal can only propagate through channel B.

(a) 265 GHz input signal. (b) 300 GHz input signal.

Figure 15: Surface plots of the electromagnetic field propagating in the diplexer, for different input signals.

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4 Results and Discussion

4.1 Filter Design

4.1.1 Initial Design Performed with Analytical Calculations and CMS

Tables 4 and 5 show the element values gi, susceptances Bi, coupling coefficients, ki,i+1, iris openingwidths, di, resonator phase lengths φi and the physical resonator lengths, li from the analytical designof the channel filters, A and B, respectively.

Coupling coefficients were extracted both analytically, using equation (27) and via simulationswith CMS [17]. The ones obtained from CMS ended up being used for extracting the physical irisopening widths for the initial models as these were deemed more accurate, with the exception of thecoupling of the ports to the first/last resonators (these could not be generated with CMS).

Table 4: Analytical design parameters for channel A filter.

i gi Bi ki,i+1 [%] di [µm] φi [rad] li [µm]eq. 27 CMS

0 1 - 5.74 - 333.7 - -1 0.9714 2.0238 4.90 4.59 303.3 2.6108 6372 1.3721 6.9024 3.60 3.37 266.9 2.8967 7083 1.8014 9.4886 3.60 3.37 266.9 2.9339 7174 1.3721 9.4886 4.90 4.59 303.3 2.8967 7085 0.9714 6.9024 5.74 - 333.7 2.6108 6396 1 2.0524 - - - - -

Table 5: Analytical design parameters for channel B filter.

i gi Bi ki,i+1 [%] di [µm] φi [rad] li [µm]eq. 27 CMS

0 1 - 5.07 - 316 - -1 0.9714 2.4658 4.33 3.86 282 2.6955 5342 1.3721 9.3481 3.18 2.89 251 2.9587 5873 1.8014 12.796 3.18 2.89 251 2.9865 5924 1.3721 12.796 4.33 3.86 282 2.9587 5875 0.9714 9.3481 5.07 - 316 2.6955 5356 1 2.4977 - - - - -

The frequency response of each filter, as simulated using the analytical design is shown in figure 16.The filters were simulated with a, comparatively, narrow waveguide thickness of 1 µm. It is obviousthat neither of the filters satisfy the demands (as specified in section 3.1) in terms of bandwidth, returnloss, bandpass ripple or center frequency. The dimensions of the filters, therefore, need to be optimizedin order to obtain the sought response. This is, of course, not surprising, since the analytical designis by no means a perfect description of a waveguide coupled resonator filter, but should be viewed asa first approximation.

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(a) Channel A frequency response. (b) Channel B frequency response.

Figure 16: Frequency response of channel filters before optimization. S21 represents insertion loss or trans-mitted signal and S11 represents return loss or reflected signal.

4.1.2 Optimized Design (HFSS)

Tables 6 and 7 list a comparison of the physical filter dimensions, before and after optimization. Theresults are qualitatively similar for both filters, in that the coupling of the ports to the first/last res-onators (d0, d5) was increased, the coupling between adjacent resonators (d1-d4) was decreased andthe lengths of all resonators (l1-l5) were increased.

In order to achieve a passband return loss as close as possible to -20 dB, the bandwidth wasadjusted to be slightly narrower than the design specification. It was found, during optimization,that increasing the waveguide thickness achieved a narrower bandwidth, but also shifted the entirepassband to higher frequencies. The resonator lengths were increased to compensate for this.

Table 6: Optimized filter dimensions of channel A filter.

[µm] Design Optim ∆

d0 = d5 334 415 81d1 = d4 303 266 -37d2 = d3 267 245 -22l1 = l5 637 680 43l2 = l4 708 746 38l3 717 752 35

Figure 17 shows the frequency response of the optimized filters. Clearly, the optimized filters are asignificant improvement upon the analytical design, in terms of all specified demands. At 12.55 GHz

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Table 7: Optimized filter dimensions of channel B filter.

[µm] Design Optim ∆

d0 = d5 316 369 53d1 = d4 282 232 -50d2 = d3 251 208 -43l1 = l5 534 564 30l2 = l4 587 617 30l3 592 622 30

for the channel A filter and 13.8 GHz for the channel B filter, the bandwidths are somewhat narrowerthan originally specified. Center frequencies are ≈265.6 GHz resp. ≈300.7 GHz, which is very closeto the goal center frequencies.

As for the return loss, the two filters differ both qualitatively and quantitatively. For the channelA filter, a return loss of ≤ -20 dB could not be achieved uniformly in the entire passband. The choicewas therefore made to prioritize a greater negative return loss in the center of the passband at theexpense of a less negative return loss near the passband edges. For the channel B filter, a satisfactoryreturn loss was much more consistently achieved over the entire passband.

Difficulty in optimizing the filter response is due to several causes. Since all parts of the filterinteract with one another on some level, changes of the physical dimensions of single resonators orcoupling apertures affect the sensitivity of all other parts of the whole filter. As a result, when tuningeach resonator and coupling aperture, it is extremely difficult to predict the immediate effect on thefrequency response as the result of any single adjustment.

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(a) Channel A frequency response. (b) Channel B frequency response.

Figure 17: Frequency response of channel filters after optimization. S21 represents insertion loss or transmittedsignal and S11 represents return loss or reflected signal.

4.2 Diplexer Design

Figure 18 shows the frequency response for the diplexer assembled from the two individual filterswithout any optimization and before the wedge in the T-junction was introduced. Both channel Aand B passbands have relatively small ripples but both have a significant insertion loss throughoutmost of the band, giving a much higher return loss than the ≤ -20 dB goal value. The passband centerfrequency of channel B has been shifted toward higher frequencies when assembling the diplexer andis now located at approximately 303 GHz. Compared to the individual channel filters (see figure 17)it is clear that the diplexer needs further optimization due to the EM-field interaction between thetwo filters and the T-junction when put together.

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Figure 18: Frequency response of the unoptimized diplexer design from figure 14 before a wedge was introducedinto the T-junction.

Figure 19 presents the frequency response of the diplexer after insertion of a wedge into the T-junction (see figure 14) and some optimization has taken place. The optimized dimensions of thediplexer is shown in table 8. The wedge is positioned 863.6 µm from the irises of channel A and Bthat are adjacent to the T-junction. The thickness of the wedge 65 µm and the length is 240 µm.Insertion of the wedge resulted in slightly improving the insertion loss throughout the passbands ofboth channels, and through further optimization the passband of channel B was shifted to have thedesired center frequency of ca 300 GHz. However, as can be seen in the plot, both channels stillexperience some significant passband ripple. Attempts at optimizing the complete diplexer structurewas unsuccessful at achieving the minimally acceptable passband ripple within the time frame of theproject.

Difficulty in achieveing an optimal frequency response is likely, in part, due to interaction betweenthe channel filters, when assembled into the complete diplexer structure, resulting in increased sensi-tivity of the channel filters to changes in the physical dimensions. HFSS built in optimization toolsbased on gradient and pattern search methods failed to converge to satisfy conditions for the desiredfrequency response. Manual tuning of the channel filter dimensions resulted in, at best, marginalimprovement at the expense of significantly narrower passbands.

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Figure 19: Frequency response of the tuned diplexer design from figure 14 after a wedge was introduced intothe T-junction.

Section 3.1 specifies a desired maximum passband ripple of 0.0432 dB. In order to achieve this, thereflection loss of the diplexer channels is required to be ≤ −20 dB in the passbands. The reflectionloss of the tuned diplexer is shown in figure 20. As is clear, neither of the channels were able tofulfil the requirement at any point in the passband. For channel A, the local reflection loss maxima inthe passband range from ≈ −10.5 dB to ≈ −6.4 dB, and for channel B from ≈ −11.8 dB to ≈ −7.0 dB.

Most efforts were put into tuning channel A, as it proved much harder to affect than channel B.Changes in the physical dimensions of channel B filter had a much greater effect on the channel Afrequency response than vice versa. It is unclear why this is the case.

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Figure 20: Reflection loss of the tuned diplexer design from figure 14 after a wedge was introduced into theT-junction.

Table 8: Optimized diplexer dimensions

[µm] Channel A Channel B

d0 = d5 438 374d1 = d4 283 234d2 = d3 257 228l1 = l5 674 579l2 = l4 740 628l3 750 631

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5 Summary and Conclusions

This report presents the design and optimization of a WR-3 diplexer, synthesized using iris coupledwaveguide resonators. The diplexer channel filters feature a 5th order Chebyshev response and areinitially designed using the insertion loss method and later modeled as waveguide filters in HFSS. Thephysical dimensions of the filters were optimized in HFSS to achieve the specified frequency response,and the filter models were later assembled into a complete diplexer.

The diplexer channels were specified to have passband center frequencies at 265 GHz and 300 GHz,15 GHz bandwidths and a passband ripple of ≤ 0.0432 dB, corresponding to a passband return lossof ≤ −20 dB. The bandwidth of the passbands was decreased in the optimization process in order toimprove the return loss however, due to difficulties in tuning the diplexer channels, it was not possibleto meet the specified demand of ≤ −20 dB return loss within the time frame of the project. As such,the quality of the diplexer cannot be considered high enough for practical use.

It is unclear, at this point, whether further optimization of the physical dimensions of the diplexerwould be sufficient to achieve the originally specified frequency response, using this particular design,although it is conceivable that investigation into other methods of optimization such as time-domainanalysis, as described in detail in [12], might yield better results. Furthermore, T. Skaik has demon-strated a different analytical design method for coupled resonator diplexer synthesis using couplingmatrix optimization [10], which may be a viable alternative to the method used in this project.

The substantial difficulty in optimization indicates that this design is insufficient to simultane-ously fulfil the demand for both a wide bandwidth of 15 GHz and a high return loss of ≤ −20 dB.Recommendations for further improvement include further decreasing the channel bandwidths untila bandpass return loss of -20 dB is achieved, since a closer proximity of the resonance peaks in thefrequency response tends to result in a greater return loss. This would make the diplexer usable,however, at the expense of bandwidth. If a wide bandwidth is considered essential, one would likelyneed to change the overall design of the diplexer in some way. It is reasonable to assume that a higherfilter order is necessary to achieve a great enough return loss while maintaining a wide bandwidth.Increasing the number of resonators in each filter results in introducing additional resonance peaksinto the frequency response, which would allow closer proximity of the peaks in addition to a widebandwidth.

The realizability of the design could also be further investigated. For example, simulations can beperformed to investigate the effects of using different realistic signal feeding mechanisms, such as e.g.coaxial probes. It is also important to take into account possible limitations of fabrication methods.Exploring how different manufacturing materials and the accuracy of available fabrication methodswould affect the performance of the device is important when assessing the viability of this design.

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