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Analytical Design and Optimization of a WR-3 Synthesized using Direct Coupled Cavities

Project in Applied Physics, Uppsala University, January 2017

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

Supervisor: Dragos Dancila Department of Engineering Sciences, Solid State , Uppsala University

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

1 Contents

1 Introduction 4 1.1 Introduction to ...... 4 1.2 Project Description ...... 5

2 Theory 6 2.1 & Cavities ...... 6 2.1.1 Waveguides ...... 6 2.1.2 Resonance Cavities ...... 8 2.2 Filter Theory ...... 9 2.2.1 Scattering Parameters ...... 9 2.2.2 Chebyshev by the Insertion Loss Method ...... 10 2.2.3 Chebyshev Low-Pass Prototype ...... 12 2.2.4 LP Prototype to Bandpass Transformation ...... 13 2.2.5 Waveguide Resonator Filters ...... 14 2.3 ...... 15

3 Method 17 3.1 Design Specifications ...... 17 3.2 Analytical Filter Design ...... 17 3.3 Extraction of Iris Opening Widths ...... 19 3.4 Filter Simulation & Tuning in HFSS ...... 20 3.5 Diplexer Simulation & Tuning in HFSS ...... 22

4 Results and Discussion 24 4.1 Filter Design ...... 24 4.1.1 Initial Design Performed with Analytical Calculations and CMS ...... 24 4.1.2 Optimized Design (HFSS) ...... 25 4.2 Diplexer Design ...... 27

5 Summary and Conclusions 31

2 List of abbreviations

AC -

BP - Bandpass

BS - Band-stop

CMS - 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 - Frequency

RL - Return Loss / Reflection Loss

SHF - Super High Frequency

TE - Transverse Electric

TEM - Transverse Electromagnetic

TM - Transverse Magnetic

UHF -

3 1 Introduction

1.1 Introduction to Microwave Technology have widespread use in many fields of technology such as telecommunication [1], military and 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 of high resolution detection of ships and airplanes [1, 4, 3]. Furthermore, many materials exhibit resonance phenomena due to interaction with EM waves in the microwave region, which also makes microwaves useful for molecular/nuclear spectroscopy for material analysis [3].

The frequency interval of electromagnetic (EM) radiation which corresponds to microwaves ranges from ca 300 MHz to 300 GHz [4]. The microwave region can be divided into the subsets Ultra High Frequency (UHF), Super High Frequency (SHF) and Extremely High Frequency (EHF), as shown in table 1. Table 1: Microwave frequency bands

Frequency band Frequency (GHz) (m) UHF 0.3 - 3 1 - 0.1 SHF 3 - 30 0.1 - 0.01 EHF 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 certain frequency bands for microwave systems to operate within, while simultaneously rejecting unwanted . 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 microwave filters, the most common ones including the use of , or waveguides. For high power 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 with high fabrication cost, which may make other devices such as striplines or microstrips more feasible for those frequencies. An important application of microwave filters is the case where several devices operating on different frequency bands are required to be received on a common and sub- sequently sent on along to different outputs. One way to prevent interference between the different devices is to project the from each device onto the antenna using a multiplexer. A multiplexer is constructed by combining several microwave filters (one for each frequency band) with a power distribution network. The multi-band coming from the shared antenna is inserted into the multiplexer, which splits the multi-band signal and separates it into several single-band channels. In each channel, a microwave filter allows transmission of a single frequency band, while rejecting the

4 others, thus preventing cross-talk between the different channels. A multiplexer with only two separate frequency 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 inversely proportional to the operating frequency. Thus, higher operating frequencies allow for more compact systems, which makes possible Micro-Electro-Mechanical Systems (MEMS)[6] such as lab-on-a-chip technology, as well as facilitate the need for smaller and lighter 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 frequency ranges, such as extremely high frequency microwaves and sub-millimeter waves.

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

The goal of the project presented in this paper is to demonstrate the design and optimization of a diplexer operating within the frequency range of 220-325 GHz. This frequency range corresponds to 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 by connecting 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 found in [1, 5, 8, 9, 10].

5 2 Theory

In order to design a coupled resonator waveguide diplexer, some theoretical background study has been required. Literature has been studied to understand the nature of waveguides, and how these can be used 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 with terminated ends function as electromagnetic resonance cavities, making them well suited for applica- tions in signal (e.g. microwave filter technology). EM filters are an essential component of coupled 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 Chebyshev type bandpass filters is described in sections 2.2.2 - 2.2.4, which can be realized by electromagnetically coupled waveguide resonance cavities [1, 12, 13]. Finally, section 2.3 presents the general purpose and configuration 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 certain path. These structures are useful in high frequency AC electronics, where the frequency is high enough that 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 can consist of one continuous piece of conducting material or of several conductors put together. Waveg- uides can either be empty, or be filled with some medium. For electromagnetic waves, the modes of the waves allowed to propagate through the waveguide are dependent on the dimensions of 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; meaning that the electric field has at least one component in the transverse direction relative to the direction of propagation), TM (transverse magnetic) and TEM (transverse electromagnetic). TEM waves are supported by transmission lines and waveguides consisting of more than one conductor, e.g. a parallel plate waveguide. Rectangular waveguides, consisting of only one conductor, only allow for TE and TM waves to propagate through them [1]. −→ −→ Assuming that the general static electric field E and magnetic field H propagating in thez ˆ- direction, divided into its transverse and longitudinal components, can be written as

−→ −→  −jβz E (x, y, z) = e (x, y) +ze ˆ z(x, y) e (1) and −→ −→  −jβz H(x, y, z) = h (x, y) +zh ˆ z(x, y) e , (2) where β is the wave , 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

6 Figure 1: A rectangular waveguide of width a, height b and infinite length. (Taken with permission from www.commons.wikimedia.org/wiki/User:Zykure, CC BY-SA 3.0) can then be reduced to the wave equation

 ∂ ∂  + + k2 h (x, y) = 0, (3) ∂x2 ∂y2 c z

−jβz p 2 2 where Hz(x, y, z) = hz(x, y)e and kc = k − β 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 separation of variables) gives an expression for the cutoff wave number of the waveguide as

r mπ 2 nπ 2 k = + , (4) c a b

where a is the larger one of the two dimensions not in the direction of propagation and b is the smaller one (see figure 1). The integers m and n refer to the Tmn mode of the electric wave. The corresponding cutoff frequency can be written as 1 f = √ k , (5) cmn 2π µ c

√1 where µ is the plane wave speed of 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 of the modes for a rectangular waveguide geometry [1].

7 2.1.2 Resonance Cavities By closing off a section of a rectangular waveguide, a cavity is created in which the electromagnetic waves can store energy in the form of standing waves. Due to the addition of two more walls with normal vector in the direction of the propagation of the wave, there is now a possibility of waves being reflected and propagating in the negativez ˆ-direction, giving rise to the standing waves at some resonance frequencies. Equation (1), which describes the electric field in the waveguide, becomes with the new geometry

−→ E (x, y, z) = −→e (x, y)A+e−jβmnz + A−ejβmnz, (6)

where A+ and A− are the of waves travelling in +ˆz-direction and −zˆ-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 of the waveguide was found, but with the additional boundary conditions set by the new walls inserted into the Maxwell equations. It then follows that the resonance wave number for a wave mode TEmnl in a waveguide cavity with dimensions b < a < d (see figure 2) can be written as [1]

r mπ 2 nπ 2 lπ 2 k = + + , (7) mnl a b d

with the corresponding resonance frequencies being

1 f = √ k . (8) mnl 2π µ mnl

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

r 1 1 π 2 π 2 f = √ k = √ + . (9) 101 2π µ 101 2π µ a d

8 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. or interference from other signals) from a signal of interest. Fundamental characteristics of filters include low-pass (LP), high-pass (HP), bandpass (BP) and band-stop (BS) behaviors. In the field of RF or , a filter is a 2- network used for manipulating the frequency response of a RF or microwave system. Here, the term port or wave-port simply refers to the means by which a microwave signal enters or exits a system (for example, a waveguide or a network of coupled resonance 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 filter using coupled resonator waveguide technology.

2.2.1 Scattering Parameters The frequency response of a general multiport network is represented by so called scattering matrices S = [Sij]. Sij is known as a scattering parameter, which is defined as the ratio of the outgoing/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 of th a section of , with waves ai and bi entering/exiting the system at the i port. If a signal is excited at port 1, then S21 can be thought of as the transmission amplitude and S11 as the reflection amplitude of the system. Scattering parameters that can be mathematically described by rational 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 of the incident and reflected waves, respectively, at ports 1, 2, ··· , n, then the relationship between the incident and reflected contributions in the network is described by equation (10) [11].       b1 S11 S12 ··· S1n a1  b2  S21 S22 ··· S2n  a2    =     (10)  .   . . .. .   .   .   . . . .   .  bn Sn1 Sn2 ··· Snn an

2.2.2 Design by the Insertion Loss Method The insertion loss method is a well established method for filter design. With this approach, the frequency response of a filter is specified by its insertion loss IL, defined in equation (11) [1].

IL = 10 log PLR. (11) 2 It can be noted that the power loss ratio PLR is simply the reciprocal of |S21| , which can be written as

M(ω2) P = 1 + , (12) LR N(ω2) where M, N are polynomials of ω2. As such, IL is equivalent to through the filter. When designing a nth order Chebyshev filter, the power loss ratio is specified as

2 2 ω  PLR = 1 + k Tn , (13) ωc

10 where T ( ω ) is a Chebyshev polynomial of the nth order and ω is the cut-off frequency of the n ωc c filter. The frequency response, produced by this type of filter (see figure 4 for an example), will exhibit a passband ripple amplitude of 1 + k2, but will have a sharper passband edge and steeper slope when compared to a maximally flat (or Butterworth type) filter. For reference, some Chebyshev polynomials are listed in table 2.

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

n Tn(x) 1 x 2 2x2 − 1 3 4x3 − 3x 4 8x4 − 8x2 + 1

At ω far away from ωc the insertion loss becomes approximately

k2 2ω 2n IL ≈ 10 log , (14) 4 ωc 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 a bandwidth of 10 GHz and a passband ripple of 3 dB.

11 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-pass filter design which is normalized to have a cutoff frequency of ωc = 1 (or sometimes fc = 1 Hz) and input impedance of 1Ω. A general filter of any type (low-pass, high-pass, band-pass or band-stop) can then be constructed from the prototype by making the appropriate frequency scaling and impedance transformation. 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 as tabulated values for Chebyshev filters with different passband ripple, can be found in several textbooks on microwave engineering and filter design [1, 9, 5]. For reference, table 3 lists some element values for 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 impedance g0 = 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 1.0 2 0.6648 0.5445 1.2210 3 0.8516 1.1032 0.8516 1.0 4 0.9314 1.2920 1.5775 0.7628 1.2210 5 0.9714 1.3721 1.8014 1.3721 0.9714 1.0 6 0.9940 1.4131 1.8933 1.5506 1.7253 0.8141 1.2210 7 1.0080 1.4368 1.9398 1.6220 1.9398 1.4368 1.0080 1.0 8 1.0171 1.4518 1.9667 1.6574 2.0237 1.6107 1.7726 0.8330 1.2210 9 1.0235 1.4619 1.9837 1.6778 2.0649 1.6778 1.9837 1.4619 1.0235 1.0

12 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 and with normalized element values gi, connected to source/load impedances of 1 . Transformation from the LP prototype to a desired bandpass filter is achieved by replacing the LP filter 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 domain as described in equations (15) - (16) [1]. 1  ω ω  ω → ω0 = − 0 (15) ∆ ω0 ω ω − ω ∆ = 2 1 (16) ω0

ω1 and ω2 are the angular frequencies of the lower and higher passband edges, corresponding to the desired frequency response, ∆ is the desired bandwidth and the center frequency, ω , is defined as √ 0 the geometric mean ω1ω2.

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

j  ω ω0  jXk = − Lk, (17) ∆ ω0 ω

j  ω ω0  jBk = − Ck, (18) ∆ ω0 ω

where Lk and Ck are the series and shunt of the LP prototype filter. The 0 0 resulting component values, Lk and Ck, of the LC-circuit replacing the series inductors of the LP prototype are found by combining equations (15)-(18) and are thus given by equations (19) and (20),

0 Lk Lk = (19) ∆ω0

0 ∆ Ck = . (20) ω0Lk Similarly, the component values of the LC-circuits replacing the shunt capacitors of the LP proto- type are given by equations (21) and (22),

0 ∆ Lk = (21) ω0Ck

0 Ck Ck = (22) ∆ω0 Figure 6 depicts a circuit schematic of the bandpass filter resulting from transforming the LP prototype from figure 5.

13 Figure 6: Bandpass filter schematic.

2.2.5 Waveguide Resonator Filters A Chebyshev-like filter response can be achieved by electromagnetically coupling several resonance cavities together. The coupling is achieved by making an opening (iris) in the shared wall between the resonance cavities, allowing the electromagnetic field to propagate through the opening and resonate in the neighbouring cavity as well. The strength of the coupling (i.e. the level of power transfer from one resonating cavity to another) directly depend on the dimensions of the iris. The smaller the iris is, the weaker the coupling, and vice versa. If the iris is too large (overcoupled), there will be no resonance in the cavity and the wave will just propagate through without reflecting. If the iris is too small (undercoupled), all wave intensity will be reflected on the wall and there will be no transfer of energy through the iris to the neighbouring cavity. The order of the Chebyshev response is given by the number of resonators that are coupled together to make the filter. A higher order filter will therefore be a larger physical structure consisting of more cavities than a lower order filter, which might 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.

  m11 m12 ··· m1n m21 m22 ··· m2n M =   (23)  . . .. .   . . . .  mn1 mn2 ··· mnn

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

14 self-coupling of the resonators. These self-coupling elements are related to the resonance frequency of the corresponding cavity in relation to the frequency f0 at the center of the passband. If all resonators were tuned to the center frequency, the corresponding self-coupling elements in the diagonal would all be zero [13]. The introduction of resonator apertures will affect the analytically calculated resonance frequencies of the cavities, as calculated in equation (8), and thus shift the frequency response of the filter. To get the desired frequency response, the iris sizes must be tuned to the right values. When properly tuned, each cavity will contribute one reflection zero (a dip in the reflection amplitude S11), which corresponds 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 will be close enough to form a continuous passband. The bandwidth, ripple and center position of the passband 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 from separate channels into a single signal, and project it onto a common port (or vice versa). They are generally synthesized by combining a set of channel filters with a power-distribution network (see figure 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 two channels occupying the shared port.

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

15 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 8 describes, qualitatively, how a typical frequency response could look, where |S21| and |S31| are plotted against some frequency span, which has been normalized in the figure. Note that, in general, the two channel 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 with a shared cut-off frequency.

Figure 8: Frequency response of a diplexer.

16 3 Method

This section presents the design and simulation of a WR-3 waveguide diplexer, composed of coupled resonator waveguide filters. Filters are designed through analytical calculations, according to the design specifications outlined in section 3.1. Calculations are presented in section 3.2. After the physical dimensions of the filters are extracted, simulation and optimization of the filter designs are performed in HFSS [14], which is a computer aided design (CAD) software program, based on finite element 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 consists of two separate channel filters, connected to a common port. The diplexer is synthesized by initially designing and optimizing each channel filter independently and subsequently assembling the diplexer by 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 resonators and 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 channel B respectively, will occupy two disjoint frequency bands in the WR-3 frequency range of 220 - 325 GHz. WR-3 is a standard for rectangular waveguides, which specifies the transverse inner dimensions of the waveguide as a = 863.6 µm and b = 431.8 µm, where a, by convention, is usually considered to be the width of the waveguide and b is considered to be the height. The center frequencies of channels A and B will be 265 GHz and 300 GHz respectively. Both channels should have a bandwidth of 15 GHz and passband ripple of LAr = 0.0432 dB, corresponding to a return loss of -20 dB in the passband.

3.2 Analytical Filter Design This section outlines the analytical design process of the diplexer channel filters, following the method described in [8].

17 The filters consist of five iris coupled waveguide resonator cavities (see figure 10). Such a filter is known to be able to achieve a N th order Chebyshev-like response, where N is equal to the number of resonators [8]. The frequency response of the filter is determined, mainly, by the lengths, li, of the cavities and the sizes, di of the iris openings. The dimensions of the waveguide resonator filters can be analytically calculated by first designing a low-pass prototype filter with normalized input impedance and cutoff frequency. Appropriate transformations are then performed to transform the low-pass prototype filter to a bandpass filter with the desired frequency response. The waveguide resonator 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 series of shunt inductors between two transmission lines [1]. The coupling irises are then modeled as shunt susceptances Bi (see equations (24) - (26)) [8].

1 − ωR g1 B1 = p (24) ωR/g1 1  ω2 √ Bk = 1 − gkgk−1 (25) ω gkgk−1

1 − ωR gN−1 BN = p (26) ωR/gN−1

2 p 2 2 LAr/10 π R = 2k + 1 − 4k (1 + k ) ≈ 0.9802, where k = 10 − 1, and ω = 2 (β2 − β1)/β0.

q 2 2 √ The of a waveguide mode is given by βj = kj − kc , where kj = 2πfj µ is the free-space 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 s BW 1 ki,i+1 = , (27) f0 gigi+1

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

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

φ λ l = i g0 , (28) i π 2

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

1 −1 2 −1 2  φi = π − tan + tan (29) 2 Bi+1 Bi Now remains only to extract the physical size of the iris openings, which is presented in the next section.

3.3 Extraction of Iris Opening Widths Consider a circuit composed of two coupled resonators with resonant frequency f. It can be shown that, when the symmetry plane of the circuit is replaced with a , the coupling coefficient is altered, resulting in a lower resonant frequency fe [9]. Similarly, replacing the symmetry plane with an open circuit, increases the resonant frequency to fm. A relationship between the coupling coefficient and the shifted resonant frequencies is given by equation (30) [9].

2 2 fe − fm kij = 2 2 (30) fe + fm

In the case of two waveguide resonators coupled by an iris opening, the short circuit is equivalent to letting a wall with a perfect electric conductor (PEC) boundary condition span across the width of the 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 resonant frequencies are then calculated by the software for each case, while incrementally increasing the width of the iris opening. The coupling coefficient is extracted by inserting the resonance frequencies into equation (30) and the result is plotted as a function of the coupling iris width. The appropriate iris opening widths di for the filters can then be graphically read (or interpolated) from figure 11, assuming the analytically calculated coupling coefficients from equation (27).

19 Figure 11: Coupling coefficient vs iris opening, extracted through HFSS simulations and calculated with equation (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 the physical 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 37 µm. A simulated electromagnetic signal is fed into and out of the filter via two waveguide extensions which 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 feeding a frequency-swept signal into port 1 and solving for the outgoing signal at each port, as a result of the wave at port 1.

20 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, the iris opening widths, the length of the waveguide extensions and the waveguide thickness of each filter are all optimized to generate the desired frequency response in terms of bandwidth, center frequency and 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 to allow electromagnetic excitations to travel between neighbouring resonators, and how the resonance phenomenon gives rise to standing waves in each resonator, with maximal amplitude in the center of the cavity.

21 Figure 13: 3D plot of the electric field magnitude inside the channel A filter for a transmitted signal at 265 GHz, 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 (see figure 9). Since each channel filter was designed with five resonance cavities, this design means that two resonance cavities has to be shared between both diplexer channels, and thus the original filter design is perturbed to some extent. HFSS simulations of this design, however, resulted in the channel filters 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 been designed or tuned with this particular topology in mind. Initially, attempts were made to recover the specified frequency response by tuning the coupling apertures and resonators (as was done with each individual filter). However, since this proved to be both very difficult and time consuming, the choice was 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 are simply 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.

22 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, which allowed the original filter design to remain unchanged. As expected, HFSS simulations of this design yielded far superior results, although slightly detuning the filters. Further optimization of the entire diplexer was therefore necessary to achieve the desired passband return loss of -20 dB. Optimization was performed in HFSS, in the same way as for each individual filter, using a combination of the software’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 excited into the common port. As can be seen in the figure, the 265 GHz signal is only permitted to propagate through 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.

23 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 opening widths, di, resonator phase lengths φi and the physical resonator lengths, li from the analytical design of the channel filters, A and B, respectively. Coupling coefficients were extracted both analytically, using equation (27) and via simulations with CMS [17]. The ones obtained from CMS ended up being used for extracting the physical iris opening widths for the initial models as these were deemed more accurate, with the exception of the coupling 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 637 2 1.3721 6.9024 3.60 3.37 266.9 2.8967 708 3 1.8014 9.4886 3.60 3.37 266.9 2.9339 717 4 1.3721 9.4886 4.90 4.59 303.3 2.8967 708 5 0.9714 6.9024 5.74 - 333.7 2.6108 639 6 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 534 2 1.3721 9.3481 3.18 2.89 251 2.9587 587 3 1.8014 12.796 3.18 2.89 251 2.9865 592 4 1.3721 12.796 4.33 3.86 282 2.9587 587 5 0.9714 9.3481 5.07 - 316 2.6955 535 6 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 obvious that neither of the filters satisfy the demands (as specified in section 3.1) in terms of bandwidth, return loss, bandpass ripple or center frequency. The dimensions of the filters, therefore, need to be optimized in order to obtain the sought response. This is, of course, not surprising, since the analytical design is by no means a perfect description of a waveguide coupled resonator filter, but should be viewed as a first approximation.

24 (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. The results 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 and the 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 was adjusted 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 entire passband 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 81 d1 = d4 303 266 -37 d2 = d3 267 245 -22 l1 = l5 637 680 43 l2 = l4 708 746 38 l3 717 752 35

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

25 Table 7: Optimized filter dimensions of channel B filter.

[µm] Design Optim ∆ d0 = d5 316 369 53 d1 = d4 282 232 -50 d2 = d3 251 208 -43 l1 = l5 534 564 30 l2 = l4 587 617 30 l3 592 622 30 for the channel A filter and 13.8 GHz for the channel B filter, the bandwidths are somewhat narrower than originally specified. Center frequencies are ≈265.6 GHz resp. ≈300.7 GHz, which is very close to the goal center frequencies.

As for the return loss, the two filters differ both qualitatively and quantitatively. For the channel A filter, a return loss of ≤ -20 dB could not be achieved uniformly in the entire passband. The choice was therefore made to prioritize a greater negative return loss in the center of the passband at the expense of a less negative return loss near the passband edges. For the channel B filter, a satisfactory return 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 filter interact with one another on some level, changes of the physical dimensions of single resonators or coupling apertures affect the sensitivity of all other parts of the whole filter. As a result, when tuning each resonator and coupling aperture, it is extremely difficult to predict the immediate effect on the frequency response as the result of any single adjustment.

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

Figure 17: Frequency response of channel filters after optimization. S21 represents insertion loss or transmitted signal 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 filters without any optimization and before the wedge in the T-junction was introduced. Both channel A and B passbands have relatively small ripples but both have a significant insertion loss throughout most of the band, giving a much higher return loss than the ≤ -20 dB goal value. The passband center frequency of channel B has been shifted toward higher frequencies when assembling the diplexer and is 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 the two filters and the T-junction when put together.

27 Figure 18: Frequency response of the unoptimized diplexer design from figure 14 before a wedge was introduced into 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 the diplexer is shown in table 8. The wedge is positioned 863.6 µm from the irises of channel A and B that 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 of both channels, and through further optimization the passband of channel B was shifted to have the desired center frequency of ca 300 GHz. However, as can be seen in the plot, both channels still experience some significant passband ripple. Attempts at optimizing the complete diplexer structure was unsuccessful at achieving the minimally acceptable passband ripple within the time frame of the project.

Difficulty in achieveing an optimal frequency response is likely, in part, due to interaction between the 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 tools based on gradient and pattern search methods failed to converge to satisfy conditions for the desired frequency response. Manual tuning of the channel filter dimensions resulted in, at best, marginal improvement at the expense of significantly narrower passbands.

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

Section 3.1 specifies a desired maximum passband ripple of 0.0432 dB. In order to achieve this, the reflection loss of the diplexer channels is required to be ≤ −20 dB in the passbands. The reflection loss of the tuned diplexer is shown in figure 20. As is clear, neither of the channels were able to fulfil the requirement at any point in the passband. For channel A, the local reflection loss maxima in the 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 A frequency response than vice versa. It is unclear why this is the case.

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

Table 8: Optimized diplexer dimensions

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

30 5 Summary and Conclusions

This report presents the design and optimization of a WR-3 diplexer, synthesized using iris coupled waveguide resonators. The diplexer channel filters feature a 5th order Chebyshev response and are initially designed using the insertion loss method and later modeled as waveguide filters in HFSS. The physical 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 loss of ≤ −20 dB. The bandwidth of the passbands was decreased in the optimization process in order to improve the return loss however, due to difficulties in tuning the diplexer channels, it was not possible to 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 diplexer would 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-domain analysis, 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 coupling matrix 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 until a bandpass return loss of -20 dB is achieved, since a closer proximity of the resonance peaks in the frequency 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 likely need to change the overall design of the diplexer in some way. It is reasonable to assume that a higher filter 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 peaks into the frequency response, which would allow closer proximity of the peaks in addition to a wide bandwidth.

The realizability of the design could also be further investigated. For example, simulations can be performed 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 methods would affect the performance of the device is important when assessing the viability of this design.

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