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INTRODUCTIONINTRODUCTIONINTRODUCTIONINTRODUCTION BACKGROUND INFORMATION Millimeter wave sensor and communication technology may prove to be one of the key technologies of the 21st century covering a broad range of applications including high-speed telecommunication, medical diagnostics and security. The ultimate (commercial) success of these systems depends on their monolithic integration which brings about significant size and cost reduction. A crucial element in such systems is the antenna element which has to be able to send and receive signals in a desired direction in space with sufficient angle resolution. The traditional way to realize such a component is by placing a conventional antenna with a broad predefined radiation pattern in front of a parabolic reflector to achieve a high angle resolution (directivity). However, this approach demands the mechanical adjustment of the reflector to change the direction of transmission/reception: a solution not viable for integrated applications. In this project we aim for the design and realization of an electrically tunable reflector as an alternative. The reflector consists of a large number of small antenna elements each terminated by an electrically tunable capacitor (varactor). By tuning the bias voltage of the varactors, the reflection properties of the elements and thus of the whole reflector can be adjusted. Of particular importance here is the resulting antenna radiation pattern (directivity) and beam steering capabilities. THESIS OBJECTIVES As stated before, a complete monolithic integration of the antenna has several advantages and is therefore a design prerequisite. Once a basic circuit pattern has been developed and tested successfully, consistent copies can be reproduced with relative high accuracy and low manufacturing costs per unit. The most important part is therefore the realization of a successful prototype and an attempt in this direction defines the main objective of this report. Several goals have been set to reach the main objective and are listed in the following. o A preliminary study of tunable microstrip reflector arrays is required including

basic theoretical antenna concepts such as: radiation mechanism of patch antennas, reflector beam forming and array dynamic reconfiguration (by means of active element loading).

o The investigation of existing simulation techniques or eventually the development of a suitable simulation method to characterize the phase response of a single patch under illumination constitutes the next goal.

o The development of an accurate simulation model allowing the insertion of active elements (variable capacitors) in the electromagnetic analysis is also an important goal. One of the unique features of incorporating lumped circuit elements and devices into a full wave (finite element) algorithm is the capability to simulate all related electromagnetic phenomena including multimode propagation and complicated wave-device interaction. In this way, it is possible to take into

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consideration all the mutual effects between the planar radiating element and the ‘discrete’ component.

o Another goal is the investigation of the mutual coupling between adjacent microstrip patches in an array configuration. This phenomenon is expected to degrade the antenna performance in terms of side lobe levels, main beam shape, scan blindness and polarization purity.

o After that, a technique should be developed to simulate reflector arrays of reasonable size. From the obtained results, the most important antenna parameters (directivity, beam width, side lobe levels and cross-polarization levels) could be derived. The simulation technique should also allow the investigation of the beam steering capability of the antenna.

o Next, an investigation of the bias network needed to control the variable capacitors is essential. This includes a study of the physical implementation (i.e. the technological limitations imposed by the fabrication process) as well as the high-frequency parasitic effects on the antenna performance.

o Finally, the last goal consists of realizing the antenna layout (i.e. designing the masks for fabrication) using the commercial software package Cadence. The design rules which govern the patterning of the various masks needed to expose the wafer should be examined. Afterward, a series of measurements should be performed from which the theoretical performance of the antenna could be verified.

In short, the thesis provides the necessary insights, knowledge and simulation techniques required to effectively understand and successfully design a fully integrated tunable reflector array for mm-wave applications. It is worth noting that the design of the illuminating feed is not a topic that will be treated in this report, although some characteristics of the feed will be considered whenever necessary. THESIS STRUCTURE The outline of this report follows the principal steps defining the entire design procedure. The report is accordingly organized into seven chapters and two appendices. Chapter one lays the groundwork by presenting the tunable reflector array antenna. After a basic description of the design principle characterizing the theory on reflector arrays, their advantages as well as their principal drawback when compared to parabolic reflectors and conventional microstrip arrays are highlighted. Many of the topics introduced here will be revisited and expanded in later chapters. The chapter ends with a broad overview of the various applications in which reconfigurable reflector arrays play an important role. Chapter two is specifically devoted to the microstrip patch antenna and starts with an explanation of the field formation around the patch structure. Although the treatment is kept at an intuitive level, it provides useful insights in the radiation mechanism of patch antennas. Most of the considerations for millimeter wave printed antennas are discussed and a design procedure to obtain an efficient radiator is outlined. Chapter three covers the various simulation methods used to characterize a single microstrip patch under illumination. We particularly look at how the phase of the scattered fields at the patch surface can be evaluated with sufficient accuracy and within a reasonable simulation time. Furthermore, the resonant behavior of the patch is investigated by means of a geometry based on the cavity model introduced in the previous chapter. In chapter four, the varactor loaded patch is extensively studied. The influence of the varactor diode on the resonant frequency of the patch is discussed. Next, an accurate

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HFSS model is derived for the varactor diode based on its equivalent electrical circuit in reverse bias. The model is supported by several case studies revealing the behavior of impedance surfaces in the electromagnetic problem. Chapter five explains in details how the general theory of planar arrays can be applied to reflector arrays to obtain a user-specified far-field pattern. In particular, the required phase distribution at the reflector surface and the corresponding bias voltages for the variable capacitors (varactor diodes) are evaluated. The resulting antenna performance is presented by means of simulations performed on a 5x5 elements array. In this chapter, the negative effects of the bias network used to control the varactors are limited to a special case. This material is the focus of chapter six, which provides a perspective on how the control lines affects the fabrication process as well as the performance of the antenna. Chapter seven briefly covers two measurement approaches that can be used to test the actual performance of the designed reflector array. Finally, appendix A presents a brief explanation of the field equivalence principle applied on patch antennas. In appendix B, a collection of all the Matlab files used to produce (some of) the figures in this report is available.

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CHAPTER 1

THE REFLECTOR ARRAY ANTENNA The concept of reflector array antennas using microstrip radiators was originally introduced in the early 1980s [30] and has received a lot of attention during the years. Nevertheless, research in the field remains very active and is likely to continue for a long time. The basic operating principle characterizing this type of antenna and the various reasons justifying their need are the subjects of this chapter. We will particularly look in section 1.2 at its main advantages when compared with parabolic reflectors or conventional arrays. Section 1.3 discusses its major drawback as far as bandwidth performance is concerned and section 1.4 highlights several potential applications of the antenna. 1.1 1.1 1.1 1.1 ---- GGGGlobal descriptionlobal descriptionlobal descriptionlobal description As can be deduced from its name, the microstrip reflector array is a periodic arrangement of microstrip patches acting as a (flat) reflecting surface for an illuminating feed (e.g. a single antenna) positioned in front of it. The patches are printed on a dielectric substrate which is considered to be the mechanical backbone of the entire array. It provides the necessary support for all the printed components and ensures that they are properly positioned with respect to each other as well as the ground plane. A typical configuration is depicted in figure 1.1.

Figure 1.1: Reflector array antenna configuration. The feed used to

illuminate the array is most often a horn antenna.

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The main objective of the reflector array is to intercept efficiently the energy radiated by the feed and reradiate it in a user-specified direction. If the desired operation is as a receiver, the reflector array has to capture the incoming wave and focus it onto the receiving antenna. How this can be achieved will become clear from the following. The electromagnetic waves emanating from the feed hits the reflector surface and the incident energy they carried, is captured by the microstrip patches. Stated differently, the incident waves impinging on the reflector excites the patches which start resonating, provided that the frequency of the excitation signal is in the close vicinity of their individual resonant frequencies. Each patch will then act as an individual source reradiating the incident energy with a phase delay imposed by its own resonant frequency. In other words, the phase of the fields scattered by a single reflector element depends on how close its resonant frequency is to the frequency of the incident wave. It is therefore possible to adjust individually the phase shifts produced by the microstrip patches by controlling their resonant frequencies. This property can be exploited in an array configuration, together with the laws of electromagnetic interference to obtain a highly directive antenna. That means an antenna with a narrow beam pointing in a given direction.

Figure 1.2: Illustration of the reflector array geometry. An example of the

vectors and quantities to consider during the phase analysis is shown for

one of the patches (highlighted in red). The polarization of the electric field

is also shown and the position of the E-plane1 is specified. How to choose the resonant frequency of a particular element of the array (or equivalently the phase at that element) in order to prohibit radiation in a desired direction can be best understood by means of optic ray theory. Let’s assume that the

reflector array is configured to point the scattered beam in a direction given by or as

depicted in figure 1.2. The total phase of the scattered field at each array element as a

1 The E-plane is defined as the plane formed by the E-field vector and the direction of maximum radiation which is most often the broadside direction (z-axis in figure 1.2).

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result of the phase-shift produced by each resonator and the phase-shift due to propagation from the feed is:

),(),( 0 yxRkyx sφφ +⋅−= (1.a)

where R is the distance from the phase centre of the feed to the resonator whose position on the array is determined by a set of (x, y) coordinates. The free space

propagation constant is denoted by ko and φs is the phase-shift at each element. This phase-shift should be chosen so as to restitute the phase front in the pointing (scanning) angle. It should therefore compensate for the several phase delays introduced by the individual path differences, such that the fields are all in phase at any plane perpendicular to the desired direction of transmission (i.e. the desired phase front). The principle is similar to that of a conventional dish antenna, where the paraboloid shape of the reflector takes care of the phase compensation for the different rays emanating from a feed placed at its focal point. Expressed mathematically, we have:

kRrRkyx mns πφ 2][),( 00 −⋅−=r)

(1.b)

In which k denotes a positive integer (k=0, 1, 2,…) while mnRr

represents the vector

originating from the centre of the array and pointing to the mnth element. When the phases are predesigned according to this formula, the result is the situation depicted in figure 1.3. The circular arcs in figure 1.3 illustrate the wave crests from each radiator of the array at a particular time instant. Where the crests from each source overlap, the waves interfere constructively. At those overlapping points,

the phases of the fields are either equal or they differ by an integral number of 2π (2πk). The total interference pattern describes a plane wave whose phase front is as depicted in figure 1.3. In this way the reflector array collimates the spherical waves from the feed into plane waves propagating in the far-field along a well-defined direction. The vector normal to the wave front defines the direction of the outgoing wave (i.e. the main beam direction).

Figure 1.3: Illustration of the plane wave formation. The ‘spherical’ waves

radiated by each patch interfere constructively in the direction given by or

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Note that the feed is supposed to be located far enough from the reflector array surface. In that case, the incident wave emanating from the feed can be approximated by plane waves propagating along the rays extending from the feed to the reflector surface. This assumption will later on be used to analyze the patches individually (see chapter three) In the foregoing analysis, we didn’t mention how the resonant frequency of the individual elements can be adjusted or predesigned. Several techniques can be used to achieve a desired shift in the resonant frequency of the patch as can be found in the literature [3, 6]. They all consist of loading the patch with passive or active phase-shifters. Figure 1.4 illustrates several loading methods for reflector array elements whereby a planar phase front can be achieved. In the present work, an active element loading topology (using tunable capacitors in a parallel configuration) has been chosen to provide the necessary phase-shift. This is one of the key features of tunable reflector array design and will therefore be addressed in details later on (see chapter 4).

Figure 1.4: Various loading techniques, (a) identical patches with

variable-length phase delay lines, (b) variable-size dipoles or loops, (c)

variable-slots loaded patches, (d) variable-size patches, (e) variable

angular rotations and (f) varactor loaded patches (series configuration). The reflector array as described above van be viewed as a combination of a microstrip array and a parabolic reflector. It actually combines most of the best features of those two types of antennas as discussed below. 1.2 1.2 1.2 1.2 ---- General advantagesGeneral advantagesGeneral advantagesGeneral advantages The advantages of the microstrip reflector array are many-fold when compared to other types of antenna, especially parabolic reflectors and conventional arrays. In the following, the main advantages are resumed in a set of points.

o Favorable (low) profile: Because of their flat reflecting surface, microstrip reflector arrays are conformable to planar and slightly curved surfaces. They can easily be mounted or integrated into several kinds of structures like vehicles, aircrafts and/or buildings without adding a significant amount of

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mass and volume to those structures. In applications requiring deployable antennas, the flatness of the reflector greatly simplifies the folding mechanism and increases its reliability compared to folding systems used for parabolic structures.

o Scanning ability: The radiation pattern of a reflector array can be made

reconfigurable. The main beam can be designed to point either in the direction normal to the patch surface or at any large angle (up to 60°) from the broadside direction. The scan angle can even be controlled electronically, which cancels the need to mechanically adjust the direction of transmission (or reception) by physically moving the antenna. It is also possible to achieve accurate contour beam shaping by means of the proper phase synthesis technique when a large number of elements is used [2, 43].

o Feeding method: Because the array elements are spatially fed, there is no

need for complex feeding networks whose losses are often unacceptable at millimeter wave frequencies. In addition, there is sufficient flexibility in choosing the type of feed. One can choose among prime-focus feeding, offset feeding or cassegrain systems which all have their own advantages according to the application.

o High reliability: The antenna is compatible with MMIC (monolithic microwave

integrated circuit) technology. The reflector surface can be entirely fabricated with a reliable and accurate lithography process. In addition, the failure of a few elements has almost no impact on the overall antenna performance for arrays consisting of a large number of elements. It has been reported in [44] that the antenna gain will drop by about 0.5 dB if one tenth of the total elements fail to function.

o Relative good efficiency: The aperture efficiency of a reflector array is

comparable to that of a parabolic reflector. It is often defined [see 2] as the product of the spill over efficiency (related to the intercepted incident power at the aperture), the taper efficiency (due to the non-uniformity of the field amplitude distribution at the aperture), the phase efficiency (including the non-uniformity of the phase distribution at the reflector surface), the polarization efficiency (non-uniformity in the polarization of the fields at the aperture) and the blockage efficiency (caused by the feed). However, an extra term taking into consideration the radiation efficiency of the patch elements should be included for the total reflector array’s efficiency.

1.3 1.3 1.3 1.3 ---- Main disadvantageMain disadvantageMain disadvantageMain disadvantage The bandwidth of microstrip arrays is often considered to be their biggest drawback. Microstrip arrays are narrowband because of two limiting factors, namely: the poor bandwidth performance of the single element and the limitations due to the differential spatial phase delay. Both limitations are actually linked and as we shall see below, the resonating character of the patch is responsible for its narrow bandwidth. Single element limitation When a single patch is used as a stand a lone antenna (i.e. not in an array configuration), all design parameters including the width W, the length L or the value of any phase-shifting load are determined at the operating frequency. Any change in frequency requires a redesign, if the antenna performance should stay unaltered.

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However, as long as the deviations remain within a certain range (expressed in percentage of the performance at the design frequency), the antenna is said to work properly. The bandwidth of a single microstrip patch can be calculated with respect to a change in pattern shape, pattern direction, polarization, side lobe levels, VSWR or any other antenna parameter that is affected by a frequency change. In the case of a reflector, the microstrip patch is not fed with a line. The impedance bandwidth defined in terms of return loss (or VSWR) can therefore not be evaluated, since it’s a measure for the frequency range over which the patch is matched to the feed line. The quantity we shall use in this design to specify the bandwidth is the phase error (i.e. sensitivity) with respect to frequency variations. Note that the patch is dimensioned to resonate at a well defined frequency (the design frequency). As a consequence, the performances of the patch in terms of radiation will start degrading as the patch is operated at a frequency which deviates from the design frequency. If the deviation is significant enough, the patch will stop resonating and the radiation efficiency will drop considerably. This degradation in antenna performance is even more pronounced in arrays and is often attributed to a phenomenon known as differential spatial phase delay. Differential spatial phase delay limitation In array configurations, any change in the required phase at each element will strongly affect the antenna pattern and the scanning property of the array. The pointing beam will namely deviate from the desired direction. Note that, if the change in phase is linear with respect to capacitance1 variations, the desired phase shift distribution (and consequently the required progressive phase shift) at the reflector surface is conserved when the frequency changes. Any variation in frequency would only result in a uniform shift of the phase distribution for the array. This can graphically be translated into a line shifting upward or downward (according to the frequency change), but parallel to the original2 line. Figure 1.5a depicts this ideal behavior of an (imaginary) antenna with infinite bandwidth. A typical phase-capacitance curve however, has an S-shaped form expressing the resonance of the structure. The slope of this curve is a measure for the bandwidth performance of the reflector array. The steeper the curve, the larger the phase error produced by frequency variations. In other words, the bandwidth decreases as the slope of the phase characteristic increases. In figure 1.5b, the reflection phase is evaluated at normal incidence for three frequencies whereby f2 represents the design frequency. As a result of the S-shaped curve, a high sensitivity to frequency variations around the resonance and a very little dependency at the extremes are observed. A change in frequency from f2 to f3 will therefore introduce a non-uniform phase shift. The values of the loading capacitors should be chosen according to the curve corresponding to f3, in order to preserve the phase required at each element. Since those values are evaluated at the design frequency (f2), the phase of the scattered field for the elements loaded around ‘resonance’ will deviate considerably from the required values. The resulting phase

1 The same is true for any phase-shifting parameter including the dimensions of the patch (width and length) or any phase delay line attached to the patch. As we shall see in the following chapters, a (variable) capacitive loading of the patch can be used to dynamically vary the phase of the fields scattered by the patch. 2 The original line corresponds to the phase-capacitance relation obtained at the design frequency.

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error is more pronounced if the curve is falling at a sharp angle. For this reason, a small slope is desired in array design. The slope should be minimized at the center of the curve so that the phase change will not be overly sensitive to loading variations. If the curve is too steep, fabrication tolerances and load tuning may become an issue, especially at millimeter-wave frequencies.

Figure 1.5: phase-shift characteristics of a reflector array element as a

function of frequency: The ideal phase-capacitance relation (a) and a

typical microstrip patch behavior as function of the tuning capacitance (b).

The simulations have been performed in HFSS1. Another way to look at the sensitivity of the reflector array to frequency variations is by considering the difference in path between the feed and the elements of the array. As can be seen from figure 1.6, the edge elements are located farther away

from the feed than the element at the center of the array by ∆d.

Figure 1.6: Differential spatial phase delay. The phase shift at an edge

element with respect to the center element is caused by the difference in

propagation path ∆d.

1 The reader is referred to chapter three for a detailed explanation of how simulations are performed.

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The difference in path ∆d can be much bigger than the wavelength λ at the design frequency. The resulting phase delay ( dk ∆⋅− ) in radians can be many multiples of

2π and can be expressed as λ)( qnk +⋅− where n is a positive integer (n=0, 1, …) and

q a fractional number. When the frequency changes, the expression for the phase

becomes ))(( λλ ∆++⋅− qnk where ∆λ is taken positive or negative provided that the frequency decrease or increase, respectively. Since the required phase is

evaluated at the design frequency a phase error equal to ))(( λ∆+⋅− qnk is

introduced.

Note that the true time delay dk ∆⋅− is fully compensated in parabolic reflectors by the physical shape of their aperture. A microstrip patch element however, produces a

compensating phase shift ranging from zero to 2π (ideally) and is no match to the parabolic reflector as far as bandwidth performance is concerned. An important conclusion from this analysis is that the differential spatial phase delay problem can be reduced for small aperture sizes and high f/D ratios. If we aim for an aperture size of 7x7 cm2, an f/D ratio much larger than one would be necessary to

have a maximum path difference ∆d smaller than the wavelength at the operating frequency. The resulting phase shift will therefore not exceed 2π. 1.4 1.4 1.4 1.4 ---- Applications of the printed reflector array antennaApplications of the printed reflector array antennaApplications of the printed reflector array antennaApplications of the printed reflector array antenna Because of the several advantages listed above, microstrip reflector arrays have a number of attractive applications including long distance communication, high data rate telecommunication, aircraft guidance, medical diagnostics and military applications (security). A practical design includes DBS (Direct Broadcast Satellites) applications. In the receiving mode, the antenna can be mounted in building’s side walls or incorporated in rooftops. In transmitting systems, contoured beam reflector arrays can be used to provide coverage in a selected geographical region. An example has been reported in reference [45] for South America and Florida coverage. Figure 1.7 illustrates the pattern requirement and the contours of the realized antenna pattern.

Figure 1.7: South America and Florida coverage seen from 67°W geostationary orbital position (a) and illustration of the radiation pattern

contours (b). The plots are reproduced from [45]. The reflector array is also a good candidate for space applications where mass and volume are among the most important design constraints. The main areas of interest

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include deep-space exploration and Earth remote sensing. Recent improvements in those fields have led to the development of light-weighted antennas consisting of foldable membranes mounted on inflatable supporting structures. An example is depicted in figure 1.8.

Figure 1.8: Inflatable reflector arrays for space applications. The

antennas were developed at ILC Dover, Inc. for the X-band1 (a) and the

Ka-band (b). Reproduce from [46]. Another interesting and challenging application involves multiple beam reflector arrays. An example is the multi-fed reflector array for SAR (Synthetic Aperture Radar) applications in microsatellites which uses measured differences in the phase of the returned radar signal. It uses four feeds to generate four beams in the elevation plane as can be seen from the radiation pattern depicted in figure 1.9a. A similar concept presented in [47] is a reflector array for point-to-multipoint communication at 25.5 GHz (LMDS). This antenna generates three beams having the same cosequant-square pattern in elevation (see figure 1.9b) but covering each a 30° sector in azimuth. The figure shows only the central beam (H-pol) and one of the lateral beams (V-pol).

Figure 1.9: Four-beam radiation pattern of a SAR system in the elevation

plane (a) radiation pattern of a LMDS system in the azimuth plane. The

horizontal axis corresponds to the sine of the azimuth angle (b). [47] 1 The X-band is a segment of the microwave spectrum ranging from approximately 7GHz to 12 GHz. Similarly, the Ka-band extends from 26GHz to 40GHz.

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CHAPTER 2

THE MICROSTRIP PATCH The microstrip patch is a planar antenna for microwave applications. A thorough understanding of the microstrip patch will provide us with the necessary insights in the radiation mechanism of the reflector array elements. The chapter starts with a brief description of a microstrip resonator. We particularly look at how energy is efficiently coupled to and from free space depending on the field distribution underneath the patch. This leads to several constraints influencing the substrate choice as well as the patch dimensions. Those critical parameters define the typical procedure for the design of patch antennas and are the subjects of section 2.2 and section 2.3. Because our final target is the design of an array, section 2.4 discusses the design constraints imposed by the array configuration on the patch. 2.1 2.1 2.1 2.1 ---- General background and fGeneral background and fGeneral background and fGeneral background and field theoryield theoryield theoryield theory The central material for the realization of a microstrip patch antenna is called the dielectric substrate. It’s an electrically thin insulator completely covered on the back side with a metallic ground plane and partly metallized on the top by a conducting (patterned) patch. Any conductor with a high electric conductivity can be used for the ground plane and the patch. Copper is usually the preferred choice because it offers the best compromise as far as costs and electrical properties are concerned. In modern monolithic microwave integrated circuits (MMIC), aluminum (Al) is used as the prime conductor due to its compatibility with the main stream technology, although copper is also used. The form of the patch varies from basic geometrical shapes (rectangles, squares, circles…) to special configurations including sectors, rings and slots that are cut in the original profiles. The rectangular patch will further be investigated in the remaining of this section. Its basic geometry is depicted in figure 2.1. Radiation from a microstrip patch antenna can be derived from the field distribution underneath the patch or the current distribution on the conducting surfaces (patch and groundplane). This approach however, involves complicated integral equations and provides little insights in the formation of the fields and the radiation mechanism. The simple approach presented next is more understandable and is based on that given in [2]. Once the patch is energized by means of a microwave signal, charge redistribution on the lower surface of the patch, as well as on the upper side of the ground plane occurs. This charge redistribution depends on how the patch dimensions are in proportion to the wavelength of the excitation signal. As an illustration, if the frequency of the excitation wave is such that the electrical dimensions of the patch

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are in the order of half the effective wavelength1 (λeff /2), the charge redistribution is as depicted in figure 2.1.c. Negative charges are collected on one side of the patch and positive charges are repelled to the other side. A similar but inverse distribution holds for the ground plane region right beneath de patch. The established charge distribution supports the electric field distribution underneath the patch (see figure

2.1.c) and results from the current densities pJr and gJ

r. Those current densities are

controlled by an attractive mechanism, responsible for the movement of charges and characterized by attractive forces between opposite charges underneath the patch, as well as on top of the ground plane.

Figure2.1: The basic geometry of a rectangular microstrip patch. An

illustration of the fringing fields at the edges of the patch is shown.

Isometric view (a), top view (b) and cross section (c). Although the microstrip patch is essentially an open system, the picture presented above can be best understood in terms of the natural resonances of a closed rectangular cavity formed by surrounding the patch by fictitious vertical magnetic walls. As shown in figure 2.2, the magnetic walls touch the sides of the rectangular patch and are extended into the substrate until they reach the ground plane. The lowest resonances of such a cavity correspond to the formation of standing waves along its width or length. Such waves are accompanied by standing waves of electric charge and current density on the top and bottom conductors. This picture remains accurate when the frequency of the excitation (external source, etc) remains close to one of the resonant frequencies of the patch. In between those frequencies, the resulting field is a mixture of those corresponding to different resonance modes. As an illustration, two of the lowest resonant modes sustained by the length of the patch are depicted in figure 2.2. In reality however, the microstrip patch is not a closed system and the microwave energy cannot be fully contained in the region below the patch. The induced

surface currents (whose densities are denoted by pJr and gJ

r) excite different kinds

of waves which radiate the energy directly or indirectly (through the substrate) into

1 The effective wavelength is the wavelength derived from the effective dielectric constant as described in section 2.3.

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space. However, note that since the patch and the ground plane are good conductors, the radiation of electromagnetic energy should necessarily take place through the edges of the patch. Looking carefully at the fringing fields in figure 2.1.a or 2.1.b, we can distinguish two radiating slots with a uniform distribution and two secondary slots which do not contribute to the radiation because of the sinusoidal distribution of the corresponding fields. The fields on those slots have the same

magnitude but are 180° out of phase. As a consequence1, the interference of the corresponding radiated fields is destructive above the patch. The sinusoidal distribution corresponds to one half cycle variation of the electric field in the y-direction. It therefore characterizes the dominant TM010 eigen mode of the rectangular cavity formed by the patch, the ground plane and the volume in between. This dominant mode is the most important mode in antenna applications, since the TM020 mode has a broadside-null radiation pattern and the TM030 mode produces grating lobes as stated in reference [34]. This can also be deduced intuitively from the orientation of the fringing fields at the radiating edges resulting from the electric field distribution underneath the patch.

Figure2.2: Electric field distributions corresponding to two resonant

modes of the rectangular cavity, namely the TM010 and the TM020.

An HFSS simulation performed on a rectangular patch (984µm x 625µm) verifies the field theory presented in the foregoing analysis. The patch is designed to resonate at

sixty gigahertz and is deposited on top of a 150µm thick Silicon substrate (εr=11.9). The results are depicted in figure 2.3 and figure 2.4b. A plot of the electric field at the patch-dielectric interface reveals that the radiating slots are indeed formed by the edges separated by the length L of the patch. This is emphasized by the distribution of the electric field’s amplitude at the surface of the patch antenna as depicted in figure 2.3b.The uniform distribution along the width of the patch and the half sine wave dependence along the length are verified. In the middle of the patch, the electric field changes from polarity and is practically zero. The lines of surface current are shown in figure 2.4b for a patch energized by a uniform plane wave impinging normally on its surface. They are very similar to the distribution of the quadrature component of the currents corresponding to the TM010 eigen mode of a rectangular cavity [35]. This mode is characterized by longitudinal currents having nulls at the radiating edges. A sketch of this distribution is shown in figure 2.4a for a 40x60mm rectangular patch printed on a lossy dielectric substrate

(εr=4.34 and tanδ=0.02) and fed by means of a coaxial line. The quadrature component represents the imaginary parts of the surface currents. They are out of

1 This can be proven analytically by applying the ‘field equivalence principle’ on the cavity model as explained in appendix B.

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phase with the coaxial excitation currents and are highly dominant at resonance. This is indicated by the mentioned peak values on the figure.

Figure2.3: Electric field distribution at the patch-dielectric interface (a)

and an illustration of the radiating slots of a rectangular patch (b).

Figure2.4: Surface current distributions for a rectangular patch. Results

obtained from a numerical analysis and reproduced from [35]. The patch

parameters are: WxL=60mmx40mm, εr=4.34, h=0.005λ0 (a). The sketch

generated by HFSS for the 60GHz patch with the following parameters:

WxL=984.4µmx625µm, εr=11.9, h=0.03λ0 (b).

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Note that, the feeding method used to supply the excitation signal is of little concern in this discussion because it has no influence on the radiation mechanism of the patch. Provided that one remains close to the resonant frequency, once currents are excited on the patch surface, radiation can occur and the current distribution remains the same, irrespective of the feed type. However, the amplitude of the surface currents (and the amount of radiated power) is highly sensitive to the frequency. It becomes significant when the frequency of the excitation signal is close to a resonant mode of the structure. In other words, most of the supplied energy contributes to the radiation as soon as the patch is resonating. Moreover, compared to that of the excitation signal, the phase of the excited current and, therefore, the radiated wave, varies rapidly as one crosses the resonant frequency. It is exactly this property which is used to design reflector arrays based on rectangular patches, as we shall see later. To guarantee efficient radiation and make use of the rapid phase variation of the radiated signal, the patch should be designed in such a manner that one of its resonant frequencies coincides with the desired operating frequency. As stated above, this is usually the resonant frequency of the fundamental mode (TM010) which is the most suitable when a relatively broad beam, broadside to the patch surface is required. The design procedure will be given in 2.3. 2.2 2.2 2.2 2.2 ---- Choice of thChoice of thChoice of thChoice of the se se se substrateubstrateubstrateubstrate One of the first design steps consists of choosing the proper substrate. That is the substrate whose parameters fit the most with the application. There is of course a huge variety of substrate materials that are suitable for microstrip antennas. Their electrical and mechanical properties can be translated into design parameters, namely: the relative permittivity, the substrate thickness and the loss tangent. This section addresses all the considerations that should be taken into account in the choice of a particular parameter value. 2.2.1- Relative permittivity The choice of a substrate material for our design was dictated originally by the constraint of integrating the antenna in standard silicon technology. Silicon has a

relative dielectric constant (εr ) of 11.9 and is considered to be in the higher range1 of

available substrates. The relative permittivity of available substrates ranges from 1.07 to 12.9. The consequences of a substrate choice as far as antenna performance is concerned, can be resumed in a set of points:

o The higher the dielectric constant, the wider the beam width [1]. However, the increase in beam width in the E-plane is not the same as in the H-plane. The symmetry of the radiation pattern is therefore deteriorated and the cross-polar2 level is increased. This phenomenon is important, considering the fact that the scanning capability of an array is influenced by the radiation pattern of the individual radiators. Section 2.4 discusses this in more details.

o When the relative permittivity is high (εr >>1) the electric field lines concentrate mostly in the substrate [2, p817]. As a result of this, the fringing

1 The relative permittivity of available substrates ranges approximately according to:

9.121.2 ≤≤ rε 2 The reader is referred to section 5.2 for a definition and an illustration of the cross-polarization.

19

fields are reduced and the antenna radiation efficiency is reduced too. A substrate with low permittivity is therefore required for antenna applications.

o The size of the patch antenna is highly influenced by the relative permittivity

as can be deduced from the design procedure outlined in section 2.1. For a given operating frequency, the antenna size decreases as the dielectric constant increases. The size of the patch antenna is important depending on the application and is also essential in array design because it can limit the freedom on the inter-element spacing. An extremely small patch is not desired too, because it can not be efficiently loaded with active elements or even fabricated with good accuracy.

Later on in our design, a glass substrate (εr=5.5) will be chosen to circumvent the fabrication issues (see section 6.1) we encountered when trying to use a silicon substrate. It will also facilitate the integration of the active elements (varactor diodes), having high performances when fabricated in a Silicon-on-Glass transfer technology. 2.2.2- Loss tangent

The permittivity of lossy materials can be written as imre jεεε += and the loss

tangent is defined accordingly as:

=

re

imTanεεδ tan (2.a)

It‘s a measure for the dissipation of electromagnetic energy (i.e. heat) within the

dielectric material. The smaller the loss tangent (tan δ), the better the antenna efficiency based on dielectric loss. This is illustrated in figure 10 of reference [5] which is reproduced here as figure 2.5 for convenience.

Figure2.5: Antenna efficiency based on dielectric heating as function of

the substrate thickness d. On the vertical axis, η denotes the antenna efficiency as described by equation 2.g.

For εr=2.55, the efficiency is larger than 0.8 when the loss tangent (tan δ) is smaller

than 0.003 even for electrically thin substrates (h=d < 0.01λ0). In general, the smaller

20

the loss tangent, the higher the efficiency η. A similar conclusion can be drawn for other dielectric constants. For this reason, a substrate with a loss tangent as small as possible is preferred for microstrip antennas. Table 2.1 gives an idea of the performances of a glass substrate and a Silicon substrate compared to other materials often used in practice. Table 2.1: Overview of microwave substrate properties for patch antennas [6].

Material εr Tan δ Dim. Stability Chemical resistance

Temp. range (deg C)

PTFE-glass Random fiber

2.17 0.0009 fair excellent -27 to +260

PTFE-glass Woven web

2.17 0.0009 excellent excellent -27 to +260

Glass-bonded mica

7.5 0.002 excellent excellent -27 to +593

Glass 5.5 0 - excellent - Silicon 11.9 0.0004 excellent excellent -55 to +260 PTFE (Teflon) 2.1 0.0004 Poor excellent -27 to +260 2.2.3- Substrate thickness The substrate thickness is the only parameter that is useful for design since it can be varied to a certain extend. The appropriate thickness is based on the considerations listed in the remaining of this section.

o Apart from the polarization loss induced by the dielectric material, part of the microwave energy delivered to a patch antenna may be lost due to unwanted radiation into the substrate in the form of surface waves. The radiation efficiency er based on losses due to surface waves is strongly related to the thickness of the dielectric. It is defined as [5]:

swrad

radr PP

Pe

+= (2.b)

Prad denotes the radiated power and Psw represents the power coupled to surface waves. Surface waves are known to be TMn and TEn modes

1 of the substrate. The reader is referred to section 5.3 for a discussion on their creation and propagation through the substrate. They propagate above a cutoff frequency given by

4 1

nc

r

n cf

h ε⋅=

− (2.c)

where h is the thickness of the substrate, c is the velocity of light, and εr the relative permittivity. Note that under all circumstances the mode n=0 has a zero cutoff frequency and, thus, propagates into the substrate. That is the fundamental mode, often called the dominant TM0 mode. The mode TE0 does not exist in a grounded

1 The subscript n denotes the order of the mode.

21

dielectric layer. The best one can do is to prevent higher order modes (n = 1, 2, 3…) from propagating in the substrate, by choosing the thickness h of the dielectric such that the operating frequency f lies well bellow their cut-off frequency. Expressed in another way [5]:

14 −⋅<

rf

cnh

ε (2.d)

As an illustration, if the operating frequency is 60 GHz and the substrate under consideration is Silicon, we end up with the constraint:

mh µ6.378106.37819.1110604

103 6

9

8

=×=−⋅×⋅

×< − (2.e)

Substrates which satisfy the condition depicted in equation (2.d) will however allow the propagation of the zero-order mode as mentioned above. The term Psw in equation 2.b therefore refers to the power coupled into the TM0 mode. How this power depends on the substrate thickness is illustrated in figure 2.6a which shows the efficiency ‘er’ as described by (2.b). The evaluation of Prad and Psw results from an analysis based on the method of moments and carried out in [5]. A similar plot has been reported in [10] and is presented in figure 2.5b. It can be seen from figure 2.6 that the efficiency decreases as the substrate thickness increases. The TM0 mode causes the efficiency to drop below 0.75 when h is larger

than 0.025λo for high-permittivity substrates. Here, λo =c/f represents the wavelength in vacuum. More than half of the incident power is coupled to surface waves for

substrate thicknesses higher than 0.05λo. Beyond 0.05λo, the efficiency slightly starts increasing because of an improvement in the radiated power (Prad). Recall that, an increase in the substrate height results in an increase of the fringing fields which consequently improves the radiated power.

Figure 2.6: The radiation efficiency of a microstrip patch as a function of the

electrical substrate thickness h/λ0. (a) GaAs (εr = 12.8) substrate [reproduced

from 5] and (b) three different substrates [from 10].The figures can serve as

a reference.

The losses related to the propagation of the TM0 mode are inevitable but can be reduced by a judicious choice of the substrate thickness. According to reference [6,

22

p46], the power coupled to surface waves can be neglected if the substrate’s height satisfies the following condition:

02

3.0 λεπ

⋅≤r

h (2.f)

This would limit the substrate thickness to 69.2µm and 101.8µm for Silicon and Glass, respectively.

o The radiation efficiency ecd in terms of conduction and dielectric losses is also related to the substrate thickness. It is defined as [2]:

Lr

rcd RR

Re

+== η , (2.g)

where Rr denotes the radiation resistance and RL the conduction-dielectric losses. Several formulas have been reported (see [36] or [37]) for calculating the radiation resistance. An important conclusion from those expressions is that Rr weakly depends on h. The conduction-dielectric resistance RL is usually difficult to compute analytically. It is used to model the power dissipation (in the form of heat) due to currents on the metallic patch and leaky currents through the dielectric. It has been demonstrated in [5] that the radiation efficiency as described by equation 2.g increases as the substrate gets thicker (see figure 2.5). The reason for this is the corresponding decrease in field concentration for a given incident power. A less dense concentration results in less power converted into heat.

o An important limitation on the substrate thickness is dictated by the bandwidth requirements. Microstrip antennas have a very narrow bandwidth and as we know from section 1.3, a smooth variation of the phase versus capacitance curve is important in array design. Several simulations have been made to investigate the slope of the phase versus capacitance curve as function of the substrate thickness. The substrate is Silicon and the outer dimensions of the patch are the same as that of figure 2.4. The results are depicted in figure 2.7a.

Figure 2.7: Phase-capacitance characteristic (a) and phase-frequency

characteristic (b) as function of the substrate thickness h. The simulation

is carried out assuming normal incidence and for a substrate thickness

going from 50µm to 200µm. The figures illustrate the existing trade-off

between smooth variations and the achievable phase range.

23

Two behaviors can be observed from figure 2.7. The thinner the substrate, the higher

the compensating phase range. More than 330° can be reached for a thickness of 50µm over a narrow capacitance variation. A thicker substrate however, results in a slower phase variation. The corresponding smaller slope is expected to improve the bandwidth performance. A similar behavior is observed when the phase is plotted as function of frequency (see figure 2.7b). Note that the resonant frequency of the patch varies as the substrate thickness is changed, since the dimensions of the patch

are kept constant in those simulations. A substrate thickness of 150 µm appears to be a good compromise between phase range requirements and smooth variations.

o Another design quantity influenced by the substrate thickness is the directivity of the antenna. This is illustrated in figure 2.8, which depicts the directivity of a single patch as function of the dielectric thickness for two different dielectric constants [38].

Figure 2.8: Directivity as function of the substrate thickness for several

widths of the patch [reproduced from 38]. The relative permittivity is

εr=2.2 (a) and εr=9.8 (b). The figure shows that the directivity increases as the substrate gets thicker for low- as well as high-permittivity materials. Comparing the two plots (2.8a and 2.8b) confirms what was already stated in section 2.2.1: the beam width increases for higher values

of εr. As we shall see later in section 2.4, the patch should have a relative broad beam (i.e. a low directivity) when incorporated in an array configuration. 2.32.32.32.3---- PatchPatchPatchPatch design design design design The dimensions of the patch are mostly controlled by the radiation efficiency of the antenna and the operating frequency. A design procedure based on those dependencies has been developed and can be summarized as follows [2]:

1. Determine the patch width W, as outlined in section 2.3.1.

2. Derive the effective dielectric constant εr,eff corresponding to the width determined in step 1. A famous formula from literature can be used and gives:

2/1

, 1212

1

2

1−

+−

++

=W

hrreffr

εεε (2.h)

24

where h denotes the substrate thickness and εr the relative dielectric constant. The effective dielectric constant takes into account the in-homogeneity of the dielectric which partly consists of air.

3. Find the extension length resulting from the fringing fields at the edges of the patch. A very good approximation has been obtained from fitting techniques on experimental data. The results translates into the well known formula [2]:

( )

( )h

h

Wh

W

l

effr

effr

×

+−

++=∆

8.0258.0

264.03.0

412.0

,

,

ε

ε (2.i)

4. Determine the physical length of the patch, considering the extension length

∆l as explained in section 2.3.2. It is worth noting that the values obtained from this procedure are not the final design values, but provide a good starting point for further optimization. To increase the design flexibility, a Matlab program has been written based on this procedure and is included in appendix A. 2.3.1- Patch width The width of the patch influences the directivity of the antenna and consequently its beam width. This is depicted in figure 2.8. However, the antenna parameter which is highly affected by the patch width is the radiation resistance. Its value determines the radiation efficiency of the patch to a certain extend. A proper patch width should also preserve the uniformity of the electric field along the radiating slots. This is not an issue, as long as the width to length ratio remains between one and two (1<W/L<2), since undesired modes are not excited [8]. A more precise formula has been suggested in [2]:

0 2

2 1r r

vW

f ε=

+ (2.j)

This formula is highly used in practice and provides an optimum between radiation performance and power lost to high-order modes excitation. 2.3.2- Patch length The patch length has a major effect on the resonant frequency of the antenna. It should be chosen such that the patch resonates at the frequency of the incident fields. Since the lowest resonance is characterized by a half sinusoidal variation of the electric fields, the effective patch length should be in the order of half the effective wavelength of the incident wave. The effective length of the patch includes the length extension at both ends of the patch (i.e. at the radiating slots) due to fringing fields. Therefore, the physical length is given by:

lf

lLLeffr

eff ∆−=∆−= 22

12

00, µεε (2.k)

25

where µ0 =4πx10-7 Henry/m is the permeability of free space and ε0 = 8.854x10-12 farad/m the permittivity of free space. The extension length ∆l is given by (2.i) and εr,eff is the effective dielectric constant of equation 2.h. We should point out that the effective dielectric constant depends on frequency and that equation 2.h predicts with high accuracy its value at low frequencies. As the frequency increases, the value of the effective dielectric constant increases too. It saturates for high frequencies (>100GHz) at a value approaching the relative dielectric constant of the substrate. This is due to the fact that the fields concentrate more in the substrate as the operating frequency gets higher. Because we operate around 60GHz (or 94GHz), the resonant length of the patch will slightly deviate from that obtained from (2.i). The patch dimensions have been calculated according to the procedure outlined above. The obtained values are summarized in table 2.2 and table 2.3 for several dielectric constants and two different design frequencies. For the patch length, two

values are given. The values in brackets are obtained by substituting εr,eff by εr in all the formulas. The actual length of the patch should lie between those two limits. Table 2.2: Patch dimensions for a substrate thickness of 150 µm.

εr = 2.17 εr = 5.5 εr =11.9 Operating frequency L [µm] W [µm] L [µm] W [µm] L [µm] W [µm] fr = 60 Ghz 1607.2

(1543.5) 1985.8 1017.4

(936.5) 1386.7 681.7

(604.6) 984.4

fr = 94 Ghz 984.9 (932.8)

1267.5 620.2 (554.6)

885.2 408.3 (347)

628.3

Table 2.3: Patch dimensions for a substrate thickness of 250 µm.

εr = 2.17 εr = 5.5 εr =11.9 Operating frequency L [µm] W [µm] L [µm] W [µm] L [µm] W [µm] fr = 60 Ghz 1531.4

(1447.3) 1985.8 963.2

(857.3) 1386.7 632

(533.3) 984.4

fr = 94 Ghz 905.8 (841.4)

1267.5 562 (480.6)

885.2 356.4 (281.6)

628.3

2.4 2.4 2.4 2.4 ---- Array Array Array Array configurationconfigurationconfigurationconfiguration r r r requirementsequirementsequirementsequirements The designer should take those requirements into account at an early stage of the design (i.e. when designing a single microstrip patch), even though the array is assembled later on. 2.4.1- Element beam width The beam width of the patch antenna should be designed in correlation with the reflector array’s f/D ratio. This will ensure that all the array elements can support most incident angles from the feed. The energy from the illuminating feed is not captured efficiently by the outermost patches if their directivity is relatively high. This is even more pronounced if the feed is relatively close to the array surface as illustrated in figure 2.9a. The same effect applies for scanning angles as far as the radiated energy

26

is concerned. This is true for all elements of the reflector array as can be observed from figure 2.9a. Ideally, the patterns of the elements of the array should be adjusted according to the incident angles from the feed. Elements located at the edges should have a broad beam while elements in the middle should be highly directive. This would however increase the complexity of the design. A more practical solution is to make all the patterns wide, which is also favorable for the scanning angle. This can be translated into constraints on the substrate thickness and the patch width as discussed before.

Figure 2.9: The importance of the element beam width (a) and the

element reflection efficiency (b). 2.4.2- Element reflection efficiency Microstrip patches radiate best at their resonant frequency or in the close vicinity thereof. This can be best understood by realizing that the total radiated field is the superposition of three components, namely: the reradiated field, the (specular) reflected field and the scattered field. The first component is the most dominant when the frequency of the excitation signal is close to a resonant frequency of the patch. The induced surface currents are responsible for a broad beam normal to the patch surface. This is symbolically represented on figure 2.9b by the vertical green arrows. The reflected components (illustrated as blue arrows) are due to the ground plane and their energy is considered to be wasted when their direction does not coincide with the main beam direction. The same is true for the scattered fields due to any attached line and/or discontinuity. The reflection efficiency of the element depends on how much energy is wasted with respect to the reradiated energy and should be maximized for good antenna performance. Recall that the undesired components (reflected and scattered) are partly responsible for side lobes and antenna pattern degradation. Maximizing the element efficiency can be achieved by properly selecting the f/D ratio (as large as possible) and the inter-element spacing1, which lead again to constraints on the patch size and the substrate thickness. Those requirements are unfortunately opposite to those imposed by another array phenomenon known as mutual coupling.

1 The spacing between adjacent elements is strongly related to the side lobe level and the formation of grating lobes.

27

2.4.3- Mutual coupling Mutual coupling is the phenomenon by which an excited patch energizes neighboring patches in its close vicinity. This has already been addressed in section 2.2.3 and is discussed later on in more details (see section 5.3). At this stage of the design, it’s important to realize that electrically thin substrates are preferable to reduce the mutual coupling effect [3, 4]. This is due to the fact that thick substrates are prone to sustain undesired surface wave modes. A general rule of thumb requires that L/h >> 1 and W/h >>1.

28

CHAPTER 3

UNIT CELL SIMULATION MODELS The most important step in designing reflector array antennas consists of generating the data for the phase of the scattered field at the surface of a single reflectarray cell. This will be carried out using the widely used commercial program HFSS. However, even for a single patch, the analysis of the scattering problem is not straightforward. Simplifications have to be made in order to obtain meaningful results while limiting the computations time. These simplifications lead to several simulation techniques described in the remaining of this chapter, namely: the waveguide approach, the PML (Perfectly Matched Layer) approach and the cavity model. The chapter starts with a treatment of the waveguide approach. This technique usually consists of confining the analyzed region to that within a (fictitious) waveguide containing the patch and part of the substrate. Although very accurate for broadside illumination, the waveguide approach is not suitable for off-normal incidence analysis. This lead to the introduction of the PML approach based on linked boundary conditions and described in section 3.2. Finally, section 3.3 presents a third technique used to study the resonant modes of a microstrip patch antenna when treated as a rectangular cavity. 3.1 3.1 3.1 3.1 ---- The waveguide approachThe waveguide approachThe waveguide approachThe waveguide approach For the case of normal incidence, a slightly modified parallel plate waveguide can be used to simulate a plane wave impinging on the patch surface. As can be seen from figure 3.1 the side walls of the waveguide consist of two perfect electric conductors (PEC) facing each other while the front and back walls are formed by two perfect magnetic conductors (PMC). Unlike ordinary waveguides with (perfectly) conducting walls which only permit TE and TM modes, this particular choice of waveguide walls allows the propagation of TEM waves with a zero cutoff frequency. In this way, the field distribution at the aperture of the waveguide corresponds to that of a plane TEM wave normally incident upon the structure, as in the original normal scattering problem. Therefore, the reflection coefficient of the fundamental TEM mode excited in the wave guide contains the phase information needed to characterize the reflection at the surface of the patch. The fundamental TEM mode propagating in the z-direction is characterized by a y-polarized electric field, uniform in the horizontal cross-section of the waveguide. Figure 3.1b shows the corresponding field distribution at the excitation port. In the same figure, the ground plane is illustrated as a red surface and is considered to be a PEC. Note, however, that this structure also allows the propagation of conventional TE and TM modes, besides the fundamental TEM waves. However, as the former can only propagate above their corresponding cutoff frequencies, the dimensions of the

29

waveguide can be chosen judiciously to prevent those modes from propagating through the structure. The distance (d) between the PEC walls is related to the cutoff frequency of those modes according to:

oocf

nd

εµ⋅<

2, (3.a)

where n is equal to one, µo and εo are the permeability and permittivity of free space respectively, and fc is chosen to be equal to (or larger than) the highest frequency used in the simulations. The height of the waveguide is chosen with respect to the decay length ld of the first order TE and TM modes given by:

)/(1 c

dff

dl

−=

π (3.b)

Figure 3.1: A unit cell waveguide model to characterize the element

phase change. (a) isometric view; (b) port field distribution. The

characteristic impedance of the port is 376.62 Ω which is close to the

intrinsic wave impedance of free space.

For simulations at the operating frequency (60 GHz), we use a value of d (=2400µm) close to the unit cell dimension. This results in a cutoff frequency of 62.5 GHz and a

decay length of approximately 2850 µm. Note that the simulation result provides us with the scattering parameter (i.e., relation between the amplitude and phase of the incident and reflected waves) as measured from the excitation port on the top of the structure. Therefore, a de-embedding procedure is required to compensate for the height of the waveguide in order to obtain the reflection phase at the surface of the patch. This is done in HFSS by specifying a de-embedding distance as illustrated by the blue arrow in figure 3.1a.

30

Recall that the (fictitious) parallel plate wave guide can be viewed as a uniform transmission line. Therefore, the S-matrix at the excitation port is related to the S-matrix evaluated at the patch surface according to:

=

2

1

2

1

0

0

0

0

_2221

1211

_2221

1211

l

l

portexcl

l

surfacepatch e

e

SS

SS

e

e

SS

SSγ

γ

γ

γ

(3.c)

Since we are only interested in the first entry of the S-matrix, equation 3.c can be simplified to:

portexcl

surfacepatch SeS __112

__111γ= (3.d)

where γ=α+jβ is the complex propagation constant and l1 denotes the de-embedding distance. It is taken positive or negative depending on the orientation of the de-embedding (blue) arrow and its absolute value is equal to the height of the air box above the patch. A typical phase variation with respect to frequency as simulated by HFSS is depicted in figure 3.2. The simulated patch is designed to resonate at 60GHz.

Figure 3.2: Phase change versus frequency for a single reflectarray

element. The element consists of a 984µmx625µm metallic patch

deposited on a 150µm thick silicon substrate.

3.2 3.2 3.2 3.2 ---- The PMLThe PMLThe PMLThe PML approachapproachapproachapproach In the waveguide approach discussed above, the scattering phase was evaluated when both the incident and the reflected wave are assumed to propagate in a direction normal to the substrate. However, in a realistic reflector array, this is not always the case. The waves impinging on the patches located at the edges of the array are not coming from broadside but have an incident angle theta different from zero. Strictly speaking, this is true for all elements except the centered patch (see figure 2.9). The phase of the scattered field at those elements should therefore be

31

evaluated with a plane wave propagating with the right angle of incidence. The perfectly matched layer (PML) approach offers the possibility to do that. Principle In this approach, linked boundary conditions (LBC) are assigned in pairs to the walls of the unit cell terminated on top with a PML layer. By means of LBCs, the analysis of large uniform and periodic structures can be reduced to that of a single key cell. The key cell in our case is the structure that is repeated in the x- and y-directions to obtain a complete array. LBCs are defined in HFSS by means of master/slave boundaries and in contrary to the waveguide approach, they remove the conditions imposed by the PMC and PEC walls. In other words, the fields are not forced to be either tangential or orthogonal to the walls of the unit cell. The fields on the master and slave boundaries are instead linked by a user-assigned relationship, namely [17]:

slavej

master EeErr

⋅= Ψ− (3.e)

When considering an array which is designed to radiate in the (θo, φo) direction, ψ is given by:

)( vrk oo

r) ⋅=Ψ (3.f)

where ko is the wave number in free space, or)the unit vector in the (θo, φo) direction

and vrthe vector between the boundaries. This means that the boundary conditions

at the walls delimiting the unit cell are determined by the propagation direction of the scattered wave. In scanning arrays, this corresponds to the direction of the main beam in the far-field. Equation 3.f can further be simplified for a planar array whose elements are arranged in a rectangular lattice with planes of periodicity in x- and y-directions. We then obtain for each direction [7]:

=Ψ=Ψ

yoooy

xooox

dk

dk

φθφθ

sinsin

cossin (3.g),

where dx/y denotes the element spacing in the x/y- direction. In figure 3.3b, one of the master/slave boundary pairs defining the unit cell is shown. They are highlighted on the picture as cross-hatched surfaces. The left wall is the ‘slave’ and the right wall is the ‘master’. A similar relationship exists between the back wall and the front wall. In this example, the phase relation1 in HFSS between the

linked boundaries is set to (φo=180°,θo=30°) which corresponds to the specular angle. The excitation wave is a y-polarized uniform plane wave, impinging on the patch

with an angle of incidence θ=30°. Its field distribution at a distance corresponding to the zero phase position (excitation location) is revealed in figure 3.3a. Note that any desired angle of incidence can be chosen for the excitation wave.

1 The phase delay ψ can automatically be calculated in HFSS from the (φo,θo) scan angle. This angle has the same value as the angle specifying the unit vector normal to the evaluation plane (see figure 3.3b).

32

Figure 3.3: Unit cell in the PML approach. The incident wave is a

uniform plane wave (a). The outermost vertical surfaces of the air

box are assigned a periodic boundary condition (b). The evaluation

planes for a 30° scan angle (green plane) and broadside radiation (orange plane) are also shown.

Phase calculation With the LBC setup the plane wave option in HFSS can be used as excitation. However, S-parameters are not available since no port is defined. As a consequence, the phase of the scattered field is found by defining an evaluation plane located in the far-field region at a distance hevp away from the patch surface. The evaluation plane is oriented normal to the scattered ray direction to overlap with the equi-phase front of the scattered plane wave. The field calculator1 of HFSS can then be used to obtain the average phase of the scattered field at the evaluation plane according to:

∫

∫

⋅=

S

S

scatteredy

evpdss

dsEphase

)

)( _

φ , (3.h)

where S represents the surface delimited by the intersection of the evaluation plane and the vertical walls of the unit cell. Because the incident wave is modeled as a linearly y-polarized plane wave, Ey_scattered in equation 3.h denotes the y-component of the complex E-field vector. This quantity is available as an input to the field calculator in HFSS.

1 The field calculator is a tool in HFSS that enables you to perform various mathematical operations on all saved field data computed from the modelled geometry.

33

The phase1 φelt of the scattered field at the surface of the patch element can now be derived from φevp. We have:

πφφφ +−= PECevpelt (3.i)

In this equation φPEC is a reference phase evaluated using (3.h) for a PEC sheet located at the same position as the patch. φPEC is expected to be equal to

)2

( d∆⋅−λππ (3.j)

and justifies the appearance of the compensating factor π in (3.i). The second term in equation 3.j accounts for the path difference ∆d between the excitation plane and the evaluation plane. In the case of normal incidence and broadside radiation, the path difference is given by:

evphd ×=∆ 2 (3.k)

A plot of the phase (φevp) versus frequency characteristic is depicted in figure 3.4a for an incident angle of ten degrees (θ=10°,φ=0°). The corresponding reference phase φPEC is plotted in figure 3.4b.

Figure 3.4: A typical plot of the phase versus frequency characteristic as

obtained from the PML approach. The reflector is either (a) a patch

(984.4µmx625µm) printed on top of a silicon substrate or (b) a metal

plane covering the entire unit cell area. Since the field calculator does not produce the cumulative phase, a Matlab program (see appendix A) has been written to obtain smooth curves. The code provides the scattered phase based on equation 3.i. Figure 3.5a shows the results for the same patch of figure 3.4, but now under

broadside illumination (θ=φ=0°). For comparison purposes the characteristic obtained using the waveguide approach (see figure 3.2) is drawn on the same figure. Under normal incidence, the results obtained from the two methods match very well.

1 The reference is the phase of the incident field at the patch surface.

34

However, some considerable deviations around resonance can be noticed. The same

observation can be done when we compare off-normal analysis (θ ≠ 0°) to the broadside case. This is illustrated in figure 3.5b for various angles of incidence (θ=0°, 10°, 20°) while maintaining the same scan angle θ0 in all cases.

Figure 3.5: phase versus frequency characteristic.

a/ Comparison between the PML approach and the WGA. The patch is

illuminated under normal incidence

b/ An illustration of the sensitivity of the phase of the scattered field with respect to incidence angle variations.

From the foregoing discussion, we may be tempted to conclude the following: To determine (with high accuracy) the phase required at each reflector element, a simulating cell should be constructed for each desired angle of transmission. The

master/slave phase relation ψ and the orientation of the evaluation planes are namely determined by the direction of transmission (i.e. the direction of the scattered beam). This would result in an infinitely large set of simulation cells for each reflector element. All the cells in one set should be excited with a plane wave having the right angle of arrival. We should keep in mind that the value of that angle differs per reflector for a fixed feed.

Figure 3.6: phase versus frequency characteristic for several scan angles:

radiating patch (a) and reflecting PEC surface (b). The incident angle for

all simulations is set to 10 degree with respect to broadside.

35

This can be clarified by investigating the differences in the phase characteristic for various scan angles, while keeping the illumination angle constant. The results are depicted in figure 3.6. We can see that the computed average phase described by equation 3.h vary considerably with changes in the scan angle (see figure 3.6a). The corresponding reference phases (figure 3.6b) also show some variations resulting from the path difference. Although the simulations seem sensitive to the angle of the scattered wave, they all lead to the same result for the phase of the scattered field as given by equation 3.i. This is depicted in figure 3.7.

Figure 3.7: phase of the scattered field for several scan angles. Unit cell dimensions The dimensions of the unit cell should satisfy several constraints for the simulations to generate useful results. The height of the air box between the patch and the PML layer is chosen according to the following considerations:

o When putting a PML on top of the unit cell, a buffer layer (usually an air box) is required because the PML is effective against propagation modes. It should therefore lie in the far-field region.

o As stated above, the evaluation planes should not be completely or partly in contact with the reactive near field. A clearance slightly larger than the outer

boundary of the Fresnel region (2W2/λ ) is therefore needed between the patch and the observation plane.

o The observation planes shouldn’t also intersect with the substrate, imposing a minimum height of :

)tan( maxmin, θ⋅= dhairbox (3.l),

where d is the distance between the walls of the cell and θmax the maximum inclination angle of the evaluation plane with respect to the vertical z-axis.

36

To illustrate the above discussion, let’s consider a 60 GHz plane wave impinging on a

patch (W=985µm) located inside a unit cell with a λ/2 separation between its walls. For a 60° scan angle, an appropriate height for the air box is 5000µm, since:

mhairbox µλλ

2.4718)60tan(2

)10985(2 26

=+×≥−

(3.m)

The thickness of the PML and its material parameters ensure that the outgoing waves are absorbed to a certain extend1. They can be determined automatically in HFSS when the PML is created. An analytical way to determine the thickness can also be found in the literature [7, 14]. Because the electromagnetic fields are strongly attenuated in the PML, its top surface can be assigned a PEC or PMC boundary condition to proper terminate the HFSS-computational domain. There is a preference for PEC boundary conditions because they reduce the problem size. 3.3 3.3 3.3 3.3 ---- The cavity modelThe cavity modelThe cavity modelThe cavity model To validate the results obtained with the incident wave, a simulation based on the cavity model has been performed. According to the preceding approaches (WGA and PML approach), the resonant frequency of the patch is indicated by the inflection point on the sigmoïd curve representing the phase characteristic (see figure 3.2). In the cavity model however, the ‘eigenmode’ solver in HFSS is used to find the resonant frequencies (i.e. eigenfrequencies) of the geometry depicted in figure 3.8a. Note that, a user-specified external excitation (like waveports or incident waves) is not required in this approach to find the resonant frequencies of the drawn structure. For the patch, the same dimensions are used as the ones obtained from the previous analysis2. The substrate and the ground plane are truncated such that the whole structure is reduced to the volume underneath the patch.

Figure 3.8: the structure representing the cavity model.

1 The computational time increases for thicknesses enabling full absorption of outgoing waves. 2 Recall that the dimensions of the patch were chosen to obtain a resonance at 60 GHz.

37

The patch and the ground plane are assigned a perfectly electric conducting (PEC) boundary. The walls of the cavity that are separated by the width of the patch are modeled as perfectly magnetic conducting (PMC) walls, while the remaining two walls (along the length of the patch) are linked through a master/slave relationship. The validation of this model is based on the following assumptions:

−−−− The field variations in the z-direction are negligible in the region underneath

the patch for electrically small substrates (h < 0.1λo).

−−−− The electric field is nearly z-directed since the fringing fields are small for electrically small substrates.

−−−− The tangential component of the magnetic field along the vertical walls of the cavity is approximately zero because the electric current inside the patch has almost no components normal to the edge of the metallization.

In this cavity, only TM field distributions are allowed. We will be looking at the mode with the lowest order resonance frequency: the TM010. This is achieved in HFSS by setting the number of eigenmode solutions that the solver finds to one. Since the phase difference between the pair of walls in the y-direction can be controlled by the linked boundary conditions, we define it as a variable (ph_y). We then perform a parametric sweep such that (part of) a dispersion diagram can be plotted to extract the resonant frequency of the obtained cavity. The results are depicted in figure 3.9. For the TM010 mode, the distribution of the tangential component of the electric field along the side walls is expected to be as the one shown in the figure 3.8b. The field is characterized by a half cycle variation along the y-axis and a uniform distribution along the x- and z-axis. Therefore, the frequency resulting from a phase difference of

±180° corresponds to the resonant frequency of the dominant TM010 mode. We find a resonant frequency of 69.6 GHz for the drawn cavity (see figure 3.9).

Figure 3.9: the resonant frequency based on the cavity model. The vertical axis represents the real part of the first eigenmode found by the

solver of HFSS. The imaginary part which is related to the losses can be

neglected1 since the model parameters (ε,σ) are ideal.

1 The imaginary part has been checked and a value of zero was found. This confirms the fact that there are no radiation losses associated to a lossless cavity.

38

In this analysis, the fringing fields along the edges of the patch have not been taken

into consideration yet. When we extend the patch by 2∆l to account for the fringing fields at the two radiating slots, we obtain the red curve in figure 3.9. Recall that

equation 2.i can be used as an indication for the value of ∆l. For a patch with dimensions W x L = 984.4µm x 625µm printed on a 150µm thick Silicon substrate we find: ∆l = 60.7µm. The corresponding length of the extended patch is therefore 746.4µm. We observe from the figure that a length of 724.7µm yields a resonant frequency of 60 GHz for the fundamental mode, meaning that the actual length extension for this particular case deviates slightly from the value predicted by equation 2.i. This is also what we expected, since equation 2.i is only accurate at low frequencies. Moreover, the existence of fringing fields and the accuracy of the cavity model has been demonstrated. The proper dimensions of the patch to obtain a fundamental resonance at 60 GHz (or any desired operating frequency) can be determined with high accuracy.

39

CHAPTER 4

DESIGN OF VARACTOR-LOADED PATCH As announced in chapter one, an advantageous property of the reflector array antenna is its ability to achieve beam scanning by changing the phase of the wave reflected by each individual element of the array. In this section, the design of a single element of the reflector array, i.e. a microstrip patch loaded with tunable capacitors (varactor diodes) is presented. Those diodes account for a shift in the resonant frequency of the patch which, in turn changes the phase of the scattered wave. The chapter starts with a theoretical explanation of the loading mechanism. Next, an equivalent electrical circuit of the varactor is presented. This equivalent circuit is used to derive a geometrical model of the diodes in HFSS. Finally, the simulations performed in HFSS to characterize a single element are presented. 4.1 4.1 4.1 4.1 ---- Theoretical considerationsTheoretical considerationsTheoretical considerationsTheoretical considerations The resonant frequency of a patch antenna depends on nearly all design parameters. The dimensions of the patch, the thickness of the substrate and the materials used can affect the resonant frequency in different ways. Those parameters however, can not be changed once the antenna has been fabricated and are considered to be ‘passive’ phase-shifters. An ‘active’ phase-shifter is required to dynamically adjust the resonant frequency of the patch (during utilization) and can be realized by using a loading element whose impedance can be tuned electronically. There are a number of ways to load the patch antenna. One way is to mount the varactor diodes across the radiating slots of the antenna. In this way, they are in parallel with the fringing capacitances coupled to the slots. Any change in the bias voltage across the diodes will therefore tune the fringing capacitance, resulting in a

change in the electrical length extension ∆l of the patch. As a consequence, the resonant frequency can be shifted. This technique known as the parallel configuration has been used successfully in the design of several frequency agile antennas [see 22-24]. There is however one major drawback associated with this configuration. One of the terminals of the diode should be connected to the ground plane by means of a ‘via’. The realization of ‘vias’ through glass and/or thick substrates is a technological issue we weren’t able to circumvent (see chapter 6). Another way to control the resonant frequency is by loading a slot inserted in the middle of the patch (series configuration). The typical configuration is depicted in figure 4.1, in which three varactor diodes are used to bridge the gap between two halves of a patch. Because the field distribution is preserved (i.e. the fields on each half of the patch are still out of phase with respect to the fields on the other half of

40

the patch), the interference of the fringing fields at the gap is constructive. Therefore, the fields across the gap will contribute to the total radiation. Their contribution determines the radiating mode of the patch and depends on the gap width as well as the substrate thickness. We can distinguish between three modes which are qualified respectively by the dominance of the fields at the gap with respect to those at the edges: the slot mode, the hybrid mode and the patch mode [26]. When the patch operates either in the slot mode or the hybrid mode, the fringing capacitance across the gap can be varied by bridging the gap with tunable capacitors.

Figure 4.1: Parallel configuration in which the slot in the patch is loaded with varactor diodes (a) and series configuration in which the radiating

edges are loaded (b). Theoretically, if the whole gap area is covered by diodes whose capacitor values approach infinity, the loaded patch acts as a simple rectangular patch since the diodes behave as short-circuits. The resonant frequency remains unchanged. This is illustrated in figure 4.2 which shows the phase variation of a conventional patch en that of a patch loaded with three capacitors. When the capacitors have a value of 10pF, the phase characteristic almost overlap with the curve corresponding to the conventional patch.

Figure 4.2: Phase as function of frequency for several values of the

loading capacitance across the gap. The capacitors cover the entire slot aperture. For smaller capacitances the characteristic shifts to the right as

expected, meaning that the resonant frequency moves from that of a full

patch to that of a half patch.

41

Similarly, if the capacitor values approach zero, the patch can be viewed as two separate patches each with only one half of the length of the original so that the resonant frequency is doubled1. The maximum achievable resonant frequency shift is therefore set to one octave band. In practice however, it is not possible to vary the capacitor value from zero to infinity. A factor around 4 or 5 between the minimum and maximum capacitance values defines the physical limit of the reverse biased voltages. In our design, we chose not to increase the number of diodes across the gap because it would have made the slot less sensitive to the total tuning capacitance. In order to observe clear changes, the total tuning capacitance should be of the same order of magnitude as the gap capacitance. The series configuration has more advantages than the parallel configuration beside the fabrication issue mentioned above. First of all, the phase sensitivity is higher2 for the same capacitance range [18]. Secondly, the fields at the gap are much higher in magnitude than the fields at the radiating edges, provided that the patch is operated either in the slot mode or the hybrid mode. Since the fringing capacitance across the radiating slots decreases with increasing frequency, the tuning capacitors become unfeasible for millimeter wave applications. 4.24.24.24.2---- Varactor electrical circuitVaractor electrical circuitVaractor electrical circuitVaractor electrical circuit The varactors used in this design are high-performance low-loss Schottky diodes [19] developed at the Delft University of Technology (TUDelft). The diodes are fabricated on a Silicon-on-glass (SOC) substrate transfer technology and the voltage dependence of their junction capacitance is given by:

n

j

V

CVC

−

=

φ1

)( 0 (4.a)

where V denotes the bias voltage, Cj0 the zero-bias capacitance, φ (≈0.83V) the built-in potential and n equals 0.5 for a uniformly doped junction. The value of the zero-

bias capacitance is approximately 1fF/µm2. It depends on the effective area of the diodes, which is defined in the process by two or three trench plasma etch steps. For a doping profile of 1x1017 cm-3, the C-V characteristic extracted from S-parameters measurements at 60 GHz is depicted in figure 4.3. To include the varactor in the HFSS model, its equivalent circuit model (see figure 4.4) has been taken into consideration [27, 28]. It consists of a parallel connection of the junction capacitance Cj and the junction resistance Rj. The series resistance Rs and the lead inductance Ls are parasitic elements related to the contacts. They are responsible for the decrease of the Q-factor at high frequencies.

At reverse bias, Rj can be taken out of the model, since its value is larger than 1MΩ. Rs is determined by the low-ohmic contacts and can be neglected (a contact resistivity

of 10-7Ωcm2 can be realized [19]). Typical values for Ls are in the order of 10pH.

1 The coupling between the two halves of the patch through space- and surface waves is neglected. 2 It has been verified by means of an HFSS simulation.

42

Figure 4.3: C-V characteristic of the high-performance varactor

diode. The effective area is 20x5µm2 (a). Q-factor versus bias

voltage at 2 GHz [from 19].

Figure 4.4: A simplified equivalent circuit model of the varactor. The Q-factor of the diode is a critical parameter in array design. It’s namely a measure for the amount of power dissipated in the diode. A high Q-factor is therefore desirable since the dissipated energy can not be radiated. It is well known that the Q-factor is inversely proportional to the frequency and the capacitor value

(Q≈1/2πfCjRs). Because the operating frequency is high, it is of great importance to use the smallest possible tuning capacitance. A plot of the Q-factor derived1 from existing data calculated from S-parameter measurements at 2 GHz is depicted in figure 4.3b as function of the bias voltage. The measurements have been performed on a heavily-doped diode (Arsenic concentration = 1017 cm-3) with a zero bias capacitance of 5pF. 4.3 4.3 4.3 4.3 ---- HFSS HFSS HFSS HFSS varacvaracvaracvaractor modeltor modeltor modeltor model In its latest versions, HFSS offers the possibility to simulate hybrid integrated circuits consisting of distributed microwave structures and lumped elements. The recently added lumped RLC boundary conditions can model any parallel or series combination of lumped resistor (R), inductor (L) and capacitor (C). This is done by assigning a lumped element value to a well defined surface. In this way lumped elements can be included in the global electromagnetic formulation. The obtained results are based on a numerical solution of Maxwell’s equations for the entire circuit. In our design, we attribute to the actual chip surface assigned to the varactors a lumped RLC boundary condition.

1 Q(at f2) = Q(at f1) ⋅ f1/f2

43

We will first present some test cases to confirm the usefulness of the lumped element concept in this ‘global approach’. Case 1: A fifty ohm transmission line terminated by a resistor (Rload).

This simple structure is depicted in figure 4.5. The dielectric substrate is Silicon (εeff = 9.69 [2] and t = 150µm) and the operating frequency is 60 GHz. The width of the microstrip is 127.5µm.

Figure 4.5: The test structure. The microstrip transmission line has a

characteristic impedance of 50Ω and is matched to the port.

We first give to the resistor a value of 0.1mΩ. We then simulate the input impedance of the structure and we look at the imaginary part (see figure 4.6a). We observe the typical behavior as function of length for a short-circuited line. The parallel and series

resonances appear respectively at (2m+1)λeff/4 and mλeff/2 (m=0,1,2,…). As an

illustration, the second series resonance (m=1) is expected to occur at λeff/2=0.8mm.

Figure 4.6: The imaginary part of the input impedance seen at the port

for Rload=0.1mΩ (a) and the simulated impedance versus the inserted

impedance after de-embedding (b). We can also vary the value of the load resistance and de-embed the port up to the impedance surface. The real part of the input impedance reveals the presence of a numerical parasitic capacitance associated to the impedance surface. As long as the inserted impedance (Rload) remains small, the effect of the parasitic capacitance is

44

negligible and the real part of Z11 (ReZ11) faithfully reproduces Rload. When Rload gets larger, the influence of the parasitic capacitance gets bigger and ReZ11 exhibits a capacitive behavior. This effect can be compensated by placing a shunt inductance across the load resistance as illustrated in figure 4.6b. Case 2: A microstrip line loaded with a capacitor (Cload). The motivation behind this characterization model is the similarity with the configuration consisting of the patch loaded with varactors. As shown in figure 4.7 the capacitor is used to connect two pieces of a microstrip line. The lines have a characteristic impedance of fifty ohm and the substrate is glass

(εr=5.5). By means of a de-embedding procedure, we can extract the capacitance value from the admittance matrix of the two-port representing the equivalent circuit of the impedance surface. We have:

f

YimCl ⋅

=π2

11 (4.b)

We can compare the value obtained from equation 4.b to the inserted value (Cload). At 60 GHz, we find a capacitor value (Cl=160fF) which deviates from the inserted value (Cload=100fF). If we take a closer look at the imaginary part of Y11 as function of frequency (see figure 4.8a), we recognize the capacitive behavior: a straight line whose slope increases with the capacitor values. However, the slopes do not correspond to the inserted load capacitances. This means that the impedance surface does not behave like a perfect capacitor. The resonances at 113GHz, 131GHz and 147GHz for large values of the load capacitance reveal the presence of a series numerical parasitic inductance. Its value is around 5pH and can be derived from the resonant frequencies. We should note that this value increases with the size of the impedance surface. This is verified by increasing the surface in one direction while keeping the value of the capacitance constant. As can be seen from figure 4.8b, the resonant frequency shifts to the left when the length (Lvar) of the impedance surface is increased. This numerical error can be predicted and corrected when the dimensions of the surface are known and fixed.

Figure 4.7: The second test structure. The blue arrows points to the

reference lines used to evaluate the two-port matrices. On the impedance

surface, a red arrow indicates the current flow direction.

45

Figure 4.8: The imaginary part of the de-embedded equivalent

admittance for several load capacitances (a) and as function of the surface dimensions for Cload = 100fF (b).

Case 3: A resonant parallel LC-circuit. The same test structure is used as in the previous case but now, a shunt inductance (Lload=0.04nH) is also assigned to the impedance surface. In addition, we will size the

impedance surface according to the varactor chip surface: Lsurf=5µm and Wsurf=20µm. For a load capacitance of 100fF, we expect a parallel resonance at 79.6GHz. This is close to the simulated resonant frequency (fr.p=76GHz) derived from a plot of the first entry of the admittance matrix (see figure 4.9a). We can associate to this plot the equivalent circuit model depicted in figure 4.9b. The equivalent admittance G is then given by:

( )[ ]

( )[ ]1

12

2

−+−−+

=ploadloadpload

ploadload

CCLLL

CCLjG

ωωωω

[4.c]

Based on this model, we can evaluate the parasitic numerical capacitance since the parallel resonance depends on Lload, Cload and Cp

1. We find:

( )fFC

LfC load

loadsr

p 6.92

12

.

=−⋅

=π

[4.d]

The series resonance (fr.s) at 107GHz can be modeled by a parasitic inductance whose value is given by:

( ) ( ) pH

CCLf

LL

ploadloadsr

loadp 7.40

12 2.

=−+⋅

=π

[4.e]

1 The capacitance from the air gap surrounding the impedance surface can also contribute to this value.

46

Figure 4.9: The imaginary part of the de-embedded equivalent

admittance for the parallel LC-circuit (a). Equivalent circuit model of the

impedance surface (b). A series LC-circuit has also been tested by replacing the impedance surface by two sequential impedance surfaces. For L=0.1nH and C=10fF, the simulation shows a series resonance at 130GHz. The good agreement between theoretical and simulated results clearly shows the applicability of this method on planar microwave circuits. Based on those results, a model of varactors according to the equivalent circuit of figure 4.4 can be built in HFSS. Since the series inductance LS is in the same order of magnitude as the numerical parasitic inductance Lp, there is no need to define another impedance surface for it. It should be pointed out that the method is very sensitive to the environment of the discrete element since it relies on a rigorous electromagnetic simulation. In other words, the results will account for the exact location and dimensions of the impedance surface. 4.4 4.4 4.4 4.4 ---- Unit cell sUnit cell sUnit cell sUnit cell simulation resultsimulation resultsimulation resultsimulation results At this point, a geometry consisting of a single patch loaded with lumped elements – especially capacitors – can be simulated in HFSS. Several questions though need to be answered in order to properly dimension the geometry:

o What should be the size of the air gap (i.e. etched slot) in the middle of the patch?

o Which criterion will determine the size of the impedance surface representing the varactor in HFSS?

o What capacitance range can provide sufficient phase agility? Those questions are strongly related, since the answer to one of them influences the other two. The value of the capacitors depends on their actual size which in its turn affects the size of the slot etched in the patch surface. One way to deal with this reciprocal influence is to estimate the magnitude of the coupling gap capacitance. Toward this end, a microstrip transmission line with a gap discontinuity is simulated according to the method described in the previous section (refer to figure 4.7). The microstrip line has the same width as the patch and is terminated at both ends with a

47

matching port1. By modeling the gap as the equivalent circuit depicted in figure 4.10a, we can derive the values for Cg and Cp from the de-embedded Y-parameters.

The results are depicted in figure 4.10b for a gap width (Wgap) varying from 10µm to 100µm. Note that analytical formulas to approximate those capacitances can be found in the literature [6, 13]. However, their accuracy has not been proven for low permittivity substrates and S/W ratios less than 0.1.

Figure 4.10: Microstrip gap discontinuity (top view) and its lumped

equivalent circuit model (a). The equivalent capacitances Cg and Cp

extracted from a Y-parameters simulation (b). We find that the coupling gap capacitance decreases as the spacing between the two halves of the microstrip patch becomes wider. Since its average value is approximately 60fF, the load capacitance can be swept around that value. Now that we have a good estimate of the fringing capacitance across the air gap, the next logical step is to fix the width of the gap. The most suitable width is determined by the sensitivity and the range of the phase characteristic. From figure 4.11, it can be

seen that a choice of 40µm for the gap width provides a good trade-off between phase agility (phases range) and phase sensitivity (i.e. the steepness around

resonance). A realistic range of 321° can be achieved for load capacitances ranging from 25fF to 100fF. It is worth noting that the relative large thickness of the glass substrate is responsible for the limitation in the phase range. Figure 4.11b illustrates how the resonant frequency of the loaded patch moves along the suitable capacitance range. The resonant frequency corresponds to the inflection point on the sigmoid curve representing the phase characteristic. This resonating point shifts to the left as the capacitance increases. It’s also clear from the plot that the bandwidth of the element is limited, since the shift in resonant frequency is not linear with respect to the change in capacitance. The capacitance has been varied from 20fF to 100fF in 10fF steps to generate this plot. A top view of the patch geometry is depicted in figure 4.12. The polarization direction of the incident plane wave is shown, which defines the E-plane and H-plane for which the radiation patterns of figure 4.13 were drawn. The single-patch antenna has a broad beam and is therefore quite adequate as element in scanning arrays.

1 The characteristic impedance of the obtained microstrip line has been determined using the ‘LineCalc’ tool of Agilent ADS.

48

Figure 4.11: Tuning characteristics of the varactor-loaded patch. The

phase of the scattered field is plotted versus the capacitance for several

values of the gap width (a). Scattered field phase versus frequency for

various capacitances (b).

Figure 4.12: Schematic of the varactor-loaded patch.

Figure 4.13: Simulated radiation patterns of the varactor-loaded patch.

Three dimensional pattern (a), E-plane and H-plane patterns (b).

49

The radiator used in the final design, consists of a patch loaded in the middle by a slot

and two varactor diodes (see figure 4.14). The patch is deposited on a 250µm thick glass substrate. The biasing lines for an E-plane measurement are also included in the unit cell simulation setup and their effect on the phase characteristic has been investigated. The changes are negligible because the lines are perpendicular to the electric field direction (see chapter 6). As a consequence, their contribution to the specular reflected field is minimized. This contribution is further reduced as the width

of the lines gets smaller. A width of 10µm has been chosen because it corresponds to the smallest realisable trace with the available full wafer mask technology. This lower

limit can even be pushed to 2µm if a wafer stepper is used to process the wafer on which the antenna is built.

Figure 4.14: Unit cell simulation setup including the biasing lines

(a) and their effect on the phase characteristic according to the

PML approach (b). This unit cell has been reproduced to build a reflector array whose pencil beam can be scanned in the E-plane. How the capacitance values or likewise the bias voltages (for each varactor loading the single patches) are chosen to achieve this is the subject of the next chapter.

50

51

CHAPTER 5

PLANAR ARRAY ANTENNA The theory treated in this chapter is shared by all array systems and constitutes an indispensable step toward the design of our tunable reflector array. It provides the necessary insights into beam forming and the corresponding relationships between array elements. It can be shown that the far-field pattern of the antenna array can be expressed as the product of the element pattern and the array factor. We demonstrate how this pattern multiplication rule leads to the scanning ability of an array of isotropic sources. The results are subsequently used to derive the phase distribution required at the surface of the reflector array. The array terminology used in this analysis will be introduced throughout the chapter. The chapter ends with an analysis of the mutual coupling between neighbouring patches. 5.1 5.1 5.1 5.1 ---- Planar aPlanar aPlanar aPlanar array theoryrray theoryrray theoryrray theory In this analysis, we consider the coordinate system shown in figure 5.1. The array is lying in the x-y plane and the spacing between the elements is dx and dy along the x- and y-directions. The origin of the coordinate system corresponds to the centre of the

array. The spherical coordinates θ, φ and r denotes the elevation angle, the azimuth angle and the radial distance respectively.

Figure 5.1: An illustration of the planar array and the meaning of the

variables used in the analysis performed in this section.

52

5.1.1 - Array factor The array factor (AF) is actually an expression for the radiation pattern of an array consisting of isotropic sources. It’s an important factor in array design because it relates the radiation pattern to the array architecture and the relative excitation of the individual elements. The array factor of a planar array can be evaluated as the product of the array factors of two linear arrays. The two linear arrays are taken in the directions defining the plane on which the array is formed. Referring to figure 1, we can write:

yx AFAFAF ⋅= (5.a)

A derivation based on far-field superposition can be found in [2] and leads to the following expressions for AFx and AFy:

∑

∑

=

=

+⋅⋅−=

+⋅⋅−=

y

x

N

nyyny

M

mxxmx

dknjIAF

dkmjIAF

10

10

)sinsin)(1(exp

)cossin)(1(exp

βφθ

βφθ (5.b)

In equation 5.b, k0 is the wave number in free space, Im/n represents the amplitude

excitations coefficients and βx/y denotes the progressive phase difference between two successive elements along the x- and y-directions. Substituting (5.b) into (5.a), we obtain after normalization [2]:

)2/sin(

)2/sin(1

)2/sin(

)2/sin(1

y

yy

yx

xx

x

N

N

M

MAF

ΨΨ

ΨΨ

= (5.c)

Where

yyy

xxx

dk

dk

βφθβφθ

+⋅⋅⋅=Ψ+⋅⋅⋅=Ψ

sinsin

cossin

0

0 (5.d)

ψx and ψy 5.1.2 - Array pattern and beam scanning The maxima of (5.c) consisting of the main beam and the grating lobes, occur for all

combinations of (θ,φ) angles satisfying:

=±=Ψ=±=Ψ

,...2,1,0,2/

,...2,1,0,2/

nn

mm

y

x

ππ

(5.e)

The solution (θ0, φ0) following from m=n=0 corresponds to the main beam direction. It then follows from (5.e) and (5.d) that the main beam direction can be tuned by

specifying properly the values of the progressive phase differences (βx/y) between the individual radiators. We obtain:

53

⋅⋅⋅−=⋅⋅⋅−=

000

000

sinsin

cossin

φθβφθβ

yy

xx

dk

dk (5.f)

This equation can be translated into a phase –shift distribution φelt(x,y) at the surface of the array. φelt(x,y) can be viewed as the phase delay at each element with respect to a reference element1 located at the origin of the coordinate system and is given by:

000000 sinsincossin),( φθφθφ ⋅⋅⋅−⋅⋅⋅−= ykxkyxelt (5.g)

in which ),(),( yx dndmyx ⋅⋅= denotes the coordinates of the phase center of each

element. They are multiples of the spacing between the elements in the x- and y-directions. The scanning property of planar arrays is demonstrated in figure 5.2.

Figure 5.2: The three dimensional representation of the array factor [the

corresponding Matlab code is provided in appendix B].

a/ broadside radiation, dx=dy=2500µm and βx=βy=0° . b/ illustration of scanning ability, βx=0° and βy=-127.28°.

5.1.3 - Element spacing A planar array may have several lobes with intensities comparable to that of the main lobe. This can mathematically be understood by realizing that the solution to

equation 5.e is not unique. The result is a set of (θ,φ) angles that corresponds to other directions of maximum radiation and satisfying [2]:

1 In the case of an array consisting of an even number of elements, the reference is an imaginary element and the coordinates (x,y) are multiples of half the spacing between the elements.

54

±±

=

±=

±=

−

−−

x

y

yx

dm

dn

dndm

/cossin

/sinsintan

sin

/sinsinsin

cos

/cossinsin

000

0001

00010001

λφθλφθ

φ

φλφθ

φλφθθ

(5.h)

The resulting lobes appearing in the radiation pattern are referred to as grating lobes. Those lobes are unwanted in a scanning array, since the array is expected to radiate in a single - user specified - direction. To prevent grating lobes from appearing in the visible region, dx and dy have to be chosen such that the system of equations defined by (5.f) have no solution. In other words the element spacing should force the arguments of the trigonometric function (sin-1) in (5.f) to be larger than one or less than one. The resulting set of equations is difficult to solve analytically and to illustrate this discussion on grating lobes, a Matlab code (see appendix A) that yields a three dimensional view of the array factor has been developed. Figure 5.3.b shows for example the array factor of a 10x10 planar array with element

spacing larger than λ0/2 in both directions. The elements along the y-direction are all

fed in phase and the progressive phase difference along the x-direction is -203.65°. The main beam should therefore point in the (θ=45°, φ=0°) direction.

Figure 5.3: The array factor for βx=-203.65 and βy=0 [the corresponding

Matlab code is provided in appendix A].

a/ desired pattern, dx=dy=2500µm.

b/ grating lobes, dx=dy=4000µm.

Beside the main beams we also note the appearance of grating lobes at (θ=30°, φ=0°). If however the element spacing is reduced to half the free space wavelength, the grating lobes disappear as can be observed from figure 5.3.a These results agree with the conventional rule in linear array design as far as grating lobes are concerned. This rule gives an indication for the proper spacing between adjacent elements and is expressed in equation form as [15]:

55

θλ sin1

1/

+=

o

yxd (5.i)

The angle θ in the above equation denotes the maximum scan angle with respect to broadside or the maximum incident angle from the feed. 5.1.4 - Theory of the tunable reflector array The phase of the scattered field at each array element according to reflectarray theory is given by equation (1.a) and is repeated here for convenience.

),(),( 0 yxRkyx sφφ +⋅−= (5.j)

In this expression, φs(x,y) represents the phase -introduced by the varactor loading- at the element whose position in the array is defined by the coordinates (x,y). Equating (5.j) and (5.g), we can obtain the required phase of the scattered field at each element with respect to the incident field. We then have:

)]sincos(sin[),( 0000 φφθφ ⋅+⋅−= yxRkyxs (5.k)

This formula is written in some reference papers in a more generalized way, namely:

][),( 00 mns RrRkyxr) ⋅−=φ (5.l)

where 0r) is the unit vector in the direction of the main beam and mnR

r the position

vector from the center of the array to the (mnth) element. Figure 5.4 is an illustration of the required phase distribution φs(x,y) at the surface of the reflector array as calculated by equation 5.k.

Figure 5.4: The phase distribution at the surface of a 50x50 array to

obtain a collimated beam in a/ broadside direction and b/ (θ=30°, φ=90°) direction. Each square represents a cell of the array and the phase is given

in degrees.

56

As can be seen from the figure, each set of varactors (loading a patch), should have its own control lines. The phase required at each element depends on the pointing angle and also on the position of the element. It is therefore not possible to group some elements and merge their control lines (beside the common ground) if full scanning is to be achieved. If however the scanning ability is limited to two

dimensions (the φ angle is for example fixed to 90°), a plane of symmetry along the y-axis can be defined (see figure 5.4b). 5.5.5.5.2222 ---- Reflector array sReflector array sReflector array sReflector array simulation results imulation results imulation results imulation results In this section, the performance of the proposed design is presented. A 5x5 array consisting of 25 elements has been simulated in HFSS. A larger array would fall

outside the allowed model dimension range (1µm to 15x103µm). The array geometry is depicted in figure 5.5a. A uniform plane wave is incident on the array surface and a radiation boundary is enclosing the whole structure. At a radiation boundary, the electromagnetic equivalence principle is applied. The equivalent surface currents are radiated to the far field through the use of a Green function. The accuracy of the far fields is therefore dependent on the accuracy of the radiation surface currents which are related to the mesh density on the corresponding surfaces. A fine mesh provides accurate results but is limited by the computing resources. The size of the array that can be simulated is limited since it requires considerable computing time and virtual memory space (RAM capacity). In figure 5.5b, an impression of the electric field magnitude on the array surface indicates the radiating slots. It can be seen that the patches are operating close to resonance.

Figure 5.5: Geometry of the 5x5 array (a) and field distribution (b). Reflection pattern When the progressive phase difference between the elements is zero, the main beam is pointed in the broadside direction. This can easily be achieved by biasing all the varactors with the same voltage. The fields radiated by the individual elements are all in phase and their interference is constructive in the direction normal to the patch. A

57

plot of the polar three dimensional pattern, together with the radiation patterns in the E-plane and H-plane is depicted in figure 5.6 The half power beam width (HPBW) can easily be derived from figure 5.7 and

appears to be 20° . The side lobe level (computed from the E-plane data) is –13.31dB with respect to the main lobe and the directivity is 77.51 (19dB). The finite size of the array is responsible for the appearance of side lobes as well as the finite width of the main beam. This is due to the fact that contributions from a finite number of patches can not sum in exactly one direction and completely cancel out in the remaining directions.

Figure 5.6: Far-field radiation pattern of the broadside array. Co-polarization and cross-polarization The radiated field emanating from the array antenna has two mutually orthogonal components known as: the co-polarization component and the cross-polarization component. For linearly polarized antennas, the co-polarization component represents the electric field vector the antenna is intended to radiate and the cross-polarization component is the unwanted electric field component resulting from transversal currents that can exist on the patch surface (see figure 2.4). Since our antenna is y-polarized (according to the simulation setup used), the x-component of the electric field corresponds to the cross-polarization component1 and can therefore be used to derive the gain associated to cross-polarization. Figure 5.7 shows the cross- polarization characteristics in the E- and H-plane. When the array is configured to point its main beam in the broadside direction, the cross-polarization components are suppressed by 55dB or more.

1 The x-component is indeed orthogonal to both the co-polarization component and the direction of radiation.

58

Figure 5.7: Far-field radiation pattern. Co-polarized and cross-polarized

components are included. Scanning angle The scanning ability of the array has been tested for several angles of transmission. The phase distribution as derived from the written Matlab code is summarized in table 5.1. Because of the symmetry with respect to the E-plane for scan angles

whereby the azimuth angle φ is fixed to 90°, the array elements can be grouped in columns. All the elements belonging to a column can then be biased with the same voltage. The first column refers to the left-most elements of the array, while the fifth column denotes the right-most elements (see figure 5.5). An illustration of the

scanning ability is depicted in figure 5.8 for elevation angles going from -30° to 30° in 10° steps.

Figure 5.8: Scanning ability of the 5x5 array presented as a rectangular

plot.

59

Tabel 5.1: Required phase distribution and the corresponding tuning

parameters for several scanning angles. Scan angle

(θ, φ) in [deg]

Array column Required phase in [deg]

Capacitor value in [fF]

Bias voltage in [volt]

1st 180 (-180) 66.500 -1.05 2nd 270 (-90) 53.410 -2.08 3rd 0 48.711 -2.67 4th 90 38.580 -4.73

(-30, 90)

5th 180 (-180) 66.500 -1.05 1st 248.76 (-111.24) 54.700 -1.94 2nd 304.38 (-55.62) 51.555 -2.30 3rd 0 48.711 -2.67 4th 55.63 44.320 -3.40

(-20, 90)

5th 111.26 28.200 -9.60 1st 297.488 (-62.512) 51.900 -2.25 2nd 328.744 (-31.256) 50.335 -2.45 3rd 0 48.711 -2.67 4th 31.257 46.660 -2.98

(-10, 90) 5th 62.5 43.550 -3.55 (0, 90) 1st, 2nd, 3rd, 4th and 5th 0 48.711 -2.67

1st 62.5 43.550 -3.55 2nd 31.257 46.660 -2.98 3rd 0 48.711 -2.67 4th 328.744 (-31.256) 50.335 -2.45

(10, 90)

5th 297.488 (-62.512) 51.900 -2.25 1st 111.26 28.200 -9.60 2nd 55.63 44.320 -3.40 3rd 0 48.711 -2.67 4th 304.38 (-55.62) 51.555 -2.30

(20, 90)

5th 248.76 (-111.24) 54.700 -1.94 1st 180 (-180) 66.500 -1.05 2nd 90 38.580 -4.73 3rd 0 48.711 -2.67 4th 270 (-90) 53.410 -2.08

(30, 90)

5th 180 (-180) 66.500 -1.05 (1,1) (5,5) -105.42 105.46 54.3 32.4 -2 -7.1 (1,2),(2,1) (190.9) -169.1 62.5 -1.30 (1,3),(2,2),(3,1) 117.3 20 -17 (1,4),(2,3),(3,2),(4,1) 63.65 43.4 -3.60 (1,5),(2,4),(3,3),(4,2),(5,1) 0 48.711 -2.67 (2,5),(3,4),(4,3),(5,2) (296.37) -63.63 51.96 -2.24 (3,5),(4,4),(5,3) (232.73) -127.27 56.15 -1.81

(30, 45)

(4,5),(5,4) (169.1) -190.9 76 -0,61

When the array is configured to send in the θ0=30° direction one of the dominant side lobes is at -30° as illustrated in figure 5.9. This is due to the high similarity in phase distribution for an array configured to point in a direction corresponding to an

elevation angle of -30°. As can be seen from table 5.1, only the second and fourth columns require opposite phases. An increase of elements (especially in the y-

60

direction) will conceal this resemblance and consequently reduce that lobe considerably.

Figure 5.9: Polar plot of the array response when the main beam is

directed along a 30° elevation angle. Three dimensional plot (a) and E-

plane cut in dB scale (b). The tunable microstrip array suffers from scan properties that are typical of array configurations. Those issues are addressed in the remaining of this section. Scan loss The phenomenon known as scan loss is also clearly visible from the simulation results (see figure 5.8). This property of planar arrays describes the decrease in directivity for increasing scanning angles. It can be shown that the directivity of the array factor for large planar arrays can be written as [2]:

( ) yxoo DDD ⋅⋅= θπ cos (5.m),

where Dx and Dy are the directivities of linear broadside arrays in the x- and y-direction respectively. The cosine term in equation 5.m is responsible for the drop in directivity as the beam is scanned away from broadside. For an array consisting of M x N elements, we can write:

==

oyy

oxx

NdD

MdD

λλ

/2

/2 (5.n),

as a result of which, (5.m) reduces to:

( )oo NMD θπ cos⋅⋅= (5.o)

This is of course an approximation, but it states that the directivity is proportional to the total number of elements of the array.

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The directivity computed by HFSS for the 5x5 array of figure 5.5 is 77.51 (19dB). This value is very realistic and is expected to double for a 7x7 array having almost twice as many elements as the simulated array. Likewise, a 10x10 array will have a directivity of 25dB. We should point out that equation (5.m) predicts a directivity of zero when the

elevation angle θ0 (i.e. the scan angle) is ninety degree. This reduction in directivity is accentuated by the directivity observed in the radiation pattern of the individual radiators. As you move away from broadside, the field intensity diminishes (see figure

4.13). For elevation angles beyond ± 60 deg, the field intensity is down by at least 6 dB. This is even more pronounced for elements located in the middle of the array since they have almost no backscattering. The impact on the overall radiation pattern of the array is the degradation observed in the radiation pattern. Side lobe level There are several ways to quantify the side lobe level. In this report, we will refer to the level of the highest side lobe relative to the main beam. The dominant factors controlling the side lobe level of a microstrip array are [31]: amplitude and phase accuracy of the excitation, surface wave propagation, edge diffraction effects and errors related to the performance of the feed network isolation. In a reflector array however, the amplitude of the excitation can not be considered as a design parameter, provided that the array flatness is a requirement. Amplitude distributions according to the Dolph-Tschebyscheff method, the binomial approach or the Taylor design are not applicable. The consequences of the feed network isolation do not have to be considered too, but a similar effect can be attributed to the control lines of the varactors. Those biasing lines are printed on the same substrate face as the microstrip patches and any resulting discontinuity causes spurious radiation that degrades the side lobe level. It has been experimentally demonstrated that the side lobe level is limited to 15-20dB

for substrate thicknesses ranging from 0.01λ0 to 0.03λ0 [31]. In the proposed design,

the thickness of the glass substrate is 0.05λ0 and is expected to further degrade the side lobe level to a value above -15dB. The phase accuracy of the ‘excitation’ is mostly determined by mutual coupling effects and fabrication tolerances, which both affects the resonant frequency of the patch. Recall that the phase characteristic of the individual elements is extracted from a unit cell simulation. The array environment is not exactly taken into consideration and the influence of the neighbouring patches on the resonant frequency is therefore neglected. A significant error is introduced in the phase distribution at the array surface with as result a non-perfect destructive interference. For a fixed phase error, a larger array size improves the (average) side lobe level. This practical solution

is adequate due to the incredibly small size (≈1.2µm2) of the elements. It has been reported in [39] that he side lobe level with respect to the main lobe can be approximated by:

NdBL log10)( −= (5.p)

where N is the number of elements of the array. As an illustration, equation 5.p predicts a side lobe level of -13.9dB for a 5x5 elements array. On a standard wafer

62

with a diameter of 10cm and an effective area of 7.5x7.5cm2, a larger array (30x30 elements) will result in a side lobe level of -29.5dB. Edge diffraction effects are caused by surface waves reaching the outer boundaries of the array. The edges of the array act as open microstrip structures from where surface waves and guided waves1 can diffract. The diffracted waves are launched in the air and contribute to the radiation pattern by raising the side lobe level. According to reference [5], surface wave power constitutes more than 15% of the

total radiated power for electrically thick substrates (h≥0.03λ0) with high dielectric

constants (εr≥2.55). When a GaAs substrate (εr≥12.8) is used, the surface wave power accounts for almost 60% of the total power delivered to the patches. This percentage is however highly reduced in microstrip arrays consisting of a large number of elements because of destructive interference effects [32]. We can conclude from this analysis that an effective remedy for high side lobe levels is an increase in the number of individual radiators making the array. Scan blindness Another fundamental scanning property typical of phased arrays is known as scan blindness. It is caused by the resonance phenomenon that occurs when surface waves excite in synchronism the Floquet modes of the periodic structure. A detailed discussion on the subject can be found in [12] or [33]. An important result from the analysis carried in those references is the demonstrated scan angle dependence on the substrate thickness, the substrate relative permittivity and the inter-element spacing. This is summarized in figure 5.10a. It can be seen that the blindness angle is

around 60° for a λ/2 inter-element spacing and a 0.06λ0 thick substrate with relative permittivity of 5.5. Therefore, the maximum scan angle of phased arrays seldom

exceeds ±60°.

Figure 5.10: Array scan angle blindness as function of the substrate

parameters and the inter-element spacing [from 12] (a). E-plane radiation

pattern for a 60° steering angle (b). For cases 1, 2 and 3 the inter-element spacing d is 0.52λ, 0.5λ and 0.48λ respectively. For case 4, h=0.02λ and d=0.5λ.

1 The guided waves represent the normal operation of transmission lines, in which waves directed into the dielectric bounce back and forth between the upper conductor and the ground plane.

63

For large scan angles, the loaded patches suffer from scan blindness and scan loss. As an illustration, Figure 5.10b shows a plot of the radiation pattern when the array is

configured to point its main beam in the (θ=60°, φ=90°) direction. The scan blindness is also related to the performance of the individual patches. Recall that the simulated cell achieves a phase range of 320 degrees for capacitance values tuned from 25fF to 100fF. This limited phase agility will of course have an impact on the scanning ability of the array. With other words, the array will not be able to send in directions for which the corresponding phase distribution can not be realized. 5.35.35.35.3 ---- Analysis of the mutual coupling Analysis of the mutual coupling Analysis of the mutual coupling Analysis of the mutual coupling Up to now, our design has been based on the pattern multiplication rule. This rule assumes that all the patches are identical and independent of each other. In a real array however, there is an interaction between patches due to proximity and it is therefore important to investigate the validity of the pattern multiplication assumptions. This analysis will give the designer some indication of whether or not it is necessary to further minimize the mutual coupling between elements of the array. As we know from literature [4, 6], mutual coupling can have strong negative effects on antenna performance. This is especially the case in designs where thick substrates with high-permittivity are used. We start with a brief description of the transmission mechanism responsible for the coupling between patches. Based on this theory, a simulation setup quantifying that effect in a certain way is presented. 5.3.1- Causes of mutual coupling Mutual coupling between patches defines an interaction established by propagating waves. The waves originating from a patch propagate through space or are guided by the dielectric and can be partly captured by the surrounding patches. They can be classified in two main categories, namely: space waves and surface waves. The

classification is based on the elevation angle θ defining their direction of propagation. This is further illustrated in figure5.11.

Figure 5.11: Illustration of space waves and surface waves. A point current source situated at the origin of the coordinate system excites waves that propagate further in all directions [21]. Waves that are launched in the air (i.e.

2/2/ πθπ <<− ) are called space waves. The rays 1 and 2 in figure 5.bbb represent two examples of propagation direction for these kinds of waves.

64

The waves propagating downward according to )/1(sin2/ 1rεπαθπ −−=<<

are totally reflected at the ground plane and at the air-dielectric interface. If they are not well attenuated, those surface waves can travel up to the end of the substrate where they are diffracted. Each time they encounter a patch on their path, they

excite a current on it. Referring to the limiting angle α, it’s obvious that thick substrates with high permittivity sustain more surface waves.

For elevation angles beyond α, the waves are characterized by a total reflection at the ground plane and a partial reflection at the air-dielectric interface. This is illustrated in figure 5.11 by ray 4. 5.3.2- Mutual coupling simulation setups We will consider the patch printed on a Silicon substrate (see section 5.3 or table bbb of section 2.bbb). In the simulation set-up, the patch is fed by a fifty ohm transmission line excited by a wave-port. A quarter wavelength transmission line is used to match the patch to the feed line (see fig 5.12). Ports are necessary in this analysis because their S-parameters provide a measure for the degree of mutual coupling.

Figure 5.12: The microstrip patch fed with a 50 ohm transmission line.

The patch (WxL=984.4µmx633.45µm) is printed on a 150µm thick

substrate. The field distribution at the port depends on its dimensions1. The resonant frequency is affected by the feeding line, which explains the need for adjusting the length of the patch. After an optimization procedure, we find that a

length equal to 633.45µm results in a resonant frequency at 60 GHz. This is emphasized by the plots in figure 5.13 which show the return loss and the input impedance of the patch. All the data is evaluated at a reference plane corresponding to the contact point with the feed lines.

1 The dimensions of the port are chosen according to the following rule of thumb:

<<<<

hHh

WWW

port

TLineportTLine

106

105

65

Figure 5.13: Return loss (a) and input impedance of the 60GHz

rectangular patch (b). The far- field radiation pattern of the microstrip patch is depicted in figure 5.14. The shape of the pattern is very similar to the pattern obtained from a single patch illuminated by a normally incident plane wave (see figure 4.13). It is therefore considered to be accurate enough for the analysis of mutual coupling effects.

Figure 5.14: Radiation pattern of the patch fed by a transmission line. The coupling between patches depends on their position with respect to one another. In this analysis, we consider side-by-side elements. Further, we make the distinction between elements that are coupled in the x-direction (E-plane coupling) and elements that are coupled along the y-direction (H-plane coupling). E-plane coupling The corresponding simulation setup is depicted in figure 5.15, in which patches are positioned collinearly along the E-plane. We perform two different simulations to quantify the contribution of surface waves to the total coupled energy. In one of them, a PEC wall is inserted in the substrate between the two patches to prevent surface waves from propagating.

66

In both simulations, one of the ports is excited and the power collected at the other port is recorded. That amount of power can be extracted from the de-embedded S21 and S12 parameters. The results are plotted in figure 5.16 as function of the inter-element spacing dx/y. H-plane coupling A similar analysis as the one carried above is performed but in this case, the patches are aligned along the H-plane. We will refer to this arrangement as the parallel configuration. The simulation setup is depicted in figure 5.17 and the corresponding results are shown in figure 5.16. All the results are plotted on the same figure for comparison purposes. The solid lines represents the E-plane coupling and the dashed lines the H-plane coupling. Note the similarity with measured values found in the literature [4]. In this reference paper, the little contribution of surface waves to the mutual coupling effect is demonstrated by simulations based on variations of the substrate thickness.

Figure 5.15: E-plane coupling configuration and illustration of the mutual

influence.

Figure 5.16: Simulated |S21|2 values as function of the inter-element

spacing dx/y.

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Figure 5.17: H-plane coupling configuration and illustration of the mutual

influence. General remarks We can conclude from this analysis that the dominant coupling mechanism between parallel or collinear patches is through space waves. This can be explained by examining the radiation pattern of the individual radiators (see figure 5.14). The fields are stronger in the broadside direction than in the end-fire directions. Since surface waves are launched in the end-fire direction, their contribution to the mutual coupling effect is less significant than that of space waves. The analysis also shows that the coupling along the E-plane is in general more pronounced than the coupling along the H-plane. This is due to the fact that the operating frequency is the resonant frequency of the dominant TM mode. This mode propagates the best in the x-direction [16, 54]. For elements along the E-plane, the electromagnetic interaction is caused by the fundamental TM100 surface wave mode while elements along the H-plane are mainly linked by a second order (and weaker) TE surface wave mode. The spacing between adjacent elements is the only critical design parameter that can be adjusted to minimize the mutual coupling. It should be taken as large as possible. However, its value is limited at the upper end by grating lobes constraints. We will

therefore keep the value dx/y=λ/2 (i.e. 2.5 mm) for our design.

68

CHAPTER 6

ANTENNA LAYOUT AND PROCESS ISSUES In this chapter, the fabrication and testing of the reflector array antenna is discussed. The most important fabrication issue is the realization of the biasing network for the active phase shifters used to scan the main beam. How the proper bias voltage can be applied across the varactor diodes is the subject of section 6.1. Two different concepts are proposed and compared in terms of performance and most of all feasibility. The chosen concept is further investigated in section 6.2 and 6.3. 6.16.16.16.1 ---- Fabrication considerationsFabrication considerationsFabrication considerationsFabrication considerations One of the design requirements is the complete integration of the reflector array into a single integrated circuit (chip). The patches as well as the necessary passive and active circuit elements are directly realized as deposited components on the chip. The same is true for the interconnect (i.e. network of biasing lines) that is needed to control the capacitances of the varactor diodes. There are two ways to implement those control lines and in this report, they are referred to as: the “via” concept and the “line” concept. The photolithographic process used to realize those concepts determine their feasibility to a certain degree. 6.1.1- The “via” concept The main idea behind this method is to protect the bias network from the incident fields impinging on the reflector surface. Toward this end, the ground plane is used as a shield for the biasing circuitry and all the electrical connections to the back side of the wafer are formed by means of ‘vias’ through the substrate. A simplified schematic of the concept is depicted in figure 6.1. The structure consists mainly of two wafers assembled in a stack configuration. The microstrip patches and the active phase-shifters are photo-etched in the top Silicon wafer while the control lines are printed on a different wafer lying underneath. The contacts between the two wafers are established by means of solder bumps welded together in a fusion process to obtain continuous connections. The area between the bumps can be filled by an electrically insulating adhesive. Beside the fact that they ensure the electrical connection between the two wafers, the bumps have another important function regarding the thermal dissipation of the reflector. They specifically allow the generated heat to flow toward the back side of the antenna. Even though the structure is electrically tolerant to small mechanical misalignments, the use of alignment holes can result in a perfect assembly of the two wafers. This concept has already been used successfully for the realization of an integrated reflective phased array on a four inches wafer as described in [39] and [40].

69

The second wafer has a double role. First of all, it carries the driving circuits which consist of biasing lines, printed in such a manner that all the contact points are brought to the edges of the wafer. Secondly, it provides the mechanical stiffness needed to handle the antenna, given that the top wafer is actually a flexible membrane when its thickness is around 150µm.

Figure 6.1: Illustration of the antenna according to the “via” concept. The solder bumps are used in combination with metallic holes (vias) drilled through the substrate of the top wafer to supply the varactor diodes with the proper biasing voltages. However, the realization of vias is a challenging task especially through thick substrates as will be clear from the following. Note that, for an efficient radiator designed to resonate at 60GHz, a minimum substrate thickness of 150µm is required when Silicon is used as dielectric. As can be seen from figure 2.7, the sensitivity of the phase characteristic around the resonant frequency is too high to enable the realization of a phased array for substrate thicknesses below 150µm. Therefore, the steepness of the phase characteristic defines the lower limit of the substrate thickness. In practice, vias are not drilled vertically but under a 54.74° angle with respect to the horizontal plane when anisotropic wet etching is used [41]. As a result, vias has a conical shape as illustrated in figure 6.2 for a single patch on a 150µm thick substrate. The base of the cone increases as the substrates gets thicker, consequently increasing the size of the ‘via’ considerably. A cross section of the structure is depicted in figure 6.3 and reveals the connection between the two wafers for one of the radiating elements constituting the array. The metallization of the conical holes is achieved after deposition of a metal (Cu) layer by a sputtering process. For each radiator, one half of the patch is connected to the wafer containing the driving circuits by means of a solder bump and a via. Part of the ground plane is etched around the via to isolate it from the ground connection. The other half of the patch is connected to the ground plane. Note that, only one ground connection to the second wafer is required since all the patches share the same ground plane. This simplifies greatly the design of the bias network on the second wafer.

70

Figure 6.2: Illustration of the vias through the top wafer on which

patches and varactors are deposited. Isometric view (a) and top view (b).

Figure 6.3: Cross-section illustrating the biasing of the varactor diodes

loading a single patch. The dimensions of the solder bumps are taken from

[42]. We should point out that the starting material for the Silicon-on-glass process used to fabricate the varactor diodes is a standard 500µm thick Silicon substrate. As a result, the substrate should be thinned down in the region containing the array. This local narrowing is done in a first (anisotropic wet) etching step whose depth defines the distance between the ground plane and the wafer front side. Rather deep cavities (about 395µm) can be formed by this first etching step as reported in [41]. A second etching step is carried out to contact the backside of the patches. Figure 6.4 is a close-up illustrating the two etching steps process. After the realization of the vias, the area of interest (i.e. the area containing the array) can be sawed and isolated from the rest of the wafer. What then remains is a Silicon membrane (with the proper thickness) on which the radiating elements are printed and the contacts to the back side are realized. This membrane can be glued to a supporting wafer for further processing. This temporary wafer can be removed afterwards, once the membrane is fixed to the wafer containing the driving circuits.

71

Figure 6.4: Two-level bulk micromachined structures. The wafer-through

holes are created after a second (dry) etching step [reproduced from 40]. The complexity of the whole process described above involves a lot of precision and there is little certainty of obtaining a working antenna structure. Another disadvantage of the ‘via’ concept is the impossibility to drill holes in a glass substrate. Therefore, another alternative to the stack configuration has been developed and is depicted in the following section. 6.1.2- The ‘line’ concept In this configuration, the radiating elements and the control lines are located in the same plane. It is important to realize that each patch should be addressed independently of the others if the array is intended to scan in the elevation plane as well as the azimuth plane (see section 5.1.4). In other words, each patch should have its own control line. A possible way to achieve this using a single metal layer is illustrated in figure 6.5 for a 5x5 elements array.

Figure 6.5: Single-layer microstrip array configuration with control lines

printed on the same surface as the radiating elements (top view).

72

The two halves of each patch are connected to the edge of the array by means of control lines. Note that, the patches on a single row can share the same ‘ground’ line. At the edges of the array, the control lines are attached to a PCB (Printed Circuit Board) assuring the external connection to the outside world. The electrical connection between the chip (actual array) and the PCB is provided by a wire bonding technique. In this approach, the back side of the chip is attached to the PCB substrate using glue with a good thermal conductance. Next, the chip pads are individually connected to the lead frame with aluminum or gold bonding wires. The metal tracks on the PCB are 0.5mm wide and lead to connector pins on which external voltages can be applied as illustrated in figure 6.6.

Figure 6.6: Illustration of the antenna according to the ‘line’ concept (top

view). The width of the control lines on the chip depends on the exposure system used to realize the array. The alignment and exposure of the wafer can be performed either with a contact aligner or with a wafer stepper. The contact aligner uses masks with the full wafer pattern and the alignment accuracy that can be reached is about 1 µm [48]. The minimum width that can be realized with this method is 10 µm. When a wafer stepper is used, the alignment accuracy is much better (about 100 nm) and the

process allows 2µm tracks to be printed reliably. With those dimensions in mind, it is possible to investigate the effects of the control lines on the radiation pattern of the antenna. This is the subject of the next section.

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6.26.26.26.2 ---- Biasing line effectsBiasing line effectsBiasing line effectsBiasing line effects The printed biasing lines act as reflecting surfaces to the incident wave impinging on the array surface. Although their width can be made small, an HFSS simulation has been performed to find out if their influence on the reflector response is insignificant. Lines on the chip We will first consider the configuration depicted in figure 6.5. The width of the lines is chosen to be 4µm and the unit cell used to characterize the phase is depicted in figure 6.7a. The corresponding phase characteristic is plotted in figure 6.7b, together with the curve obtained from a unit cell without biasing lines. The main differences occur at resonance (around 50fF) and it seems like the insertion of the control lines does not affect the patch’s behavior. Unfortunately, a simulation of the entire array of figure 6.5 contradicts the former expectation. Figure 6.8 reveals the array patterns plotted on a dB scale for to different angles of transmission. The same information can be drawn from figure 6.9, which clearly shows the side lobe level over the entire scanning range (±50°). The polar plots of figure 6.8 also show the radiation patterns for an array of the same size but without biasing lines. There are two reasons justifying the differences observed from those plots. One is related to the phase characterization and the other is associated to the presence of the control lines.

Figure 6.7: Unit cell (a) and corresponding phase characteristic (b). Referring again to figure 6.7a, it is clear that the plotted cell does not represent a structure that can be repeated to form the entire array of figure 6.5. By using the phase characteristic derived from this cell to configure the capacitors of the array, a phase error is introduced. This error can of course be reduced if a simulation cell is built for each and every patch: a solution which is not very practical for large arrays. Figure 6.8b also reveals that the dominant side lobe points in the broadside direction. This lobe mainly consists of the specular reflection from the ground plane and the control lines (given that the array is illuminated under normal incidence). The contribution of the control lines to the dominant side lobe is clear from the figure when the radiation patterns of the two arrays (with and without biasing lines) are compared with each other. This contribution depends on several factors, namely:

74

• The effective reflecting surface defined by the number of lines and their relative width.

• The distance between the lines (which determines the degree of coupling between them)

• The orientation of the control lines with respect to the polarization direction of the electric field. Lines that are perpendicular to the electric field are less sensitive to the incident plane wave from a current point of view.

Figure 6.8: E-plane radiation patterns of the 5x5 array under normal

incidence for broadside transmission (a) and a 30° scan angle (b).

Figure 6.9: Array response in the E-plane, showing the increase in side

lobe levels caused by the control lines over the entire scanning range.

75

Note that any curvature or discontinuity in the control lines act as a radiation source. The same is true for any impedance termination of the lines. Those perturbations namely support an acceleration1 (or deceleration) of charge, which is known to be the fundamental mechanism responsible for electromagnetic radiation [2, 49]. The associated fields are radiated in all directions and contribute to the increase in side lobe levels. Lines on the PCB The effects of the lines on the PCB have also been investigated. The simulation setup used is depicted in figure 6.10. The excitation wave is a y-polarized plane wave impinging on the reflector surface under normal incidence. The conductive traces representing the control lines are 0.5mm wide and are assigned a PEC boundary

condition. The substrate material used for the 1mm thick PCB is FR-4 epoxy (εr=4.4 and tanδ=0.02), a well-known and frequently used material in the PCB industry. To ensure that the model dimensions does not fall beyond the simulation range (extending from 1µm to 10mm), a 3x3 elements array has been simulated. The array has been configured to point its main beam under a 30° elevation angle with respect to the broadside direction. The simulated radiation pattern is shown in figure 6.11a on a linear scale. Figure 6.11b shows the results for the same array without the PCB and corresponding lead frame around.

Figure 6.10: Geometry used to investigate the effects of the metal traces

on the PCB .The red arrow represents the propagation vector of the incident wave.

1 The acceleration of charge is most often caused by the external electric field (from the excitation source).

76

Figure 6.11: Radiation pattern of a 3x3 elements array with (a) and

without (b) external metal traces included in the simulation geometry. The upper parts of the two patterns are very similar. The major difference is the side lobe pointing in the broadside direction which clearly indicates the reflection from the metal traces printed on the PCB. Figure 6.11a also shows that the illuminated PCB is transparent to the incident wave. Most of the incident power goes through it, as can be deduced from the huge backside lobe. Another technological limitation is the number of available bias voltages at the output of the DAC (digital to analog converters). Note that a 7x7 elements array will require fifty outputs to properly bias the entire array. However, the array can be tested in the E-plane and the layout presented in the previous chapter can be used. It is worth mentioning that for this configuration the array response (see figure 5.8) is very similar to that obtained from an array without control lines. The differences are barely noticeable. 6666----3333 AntennaAntennaAntennaAntenna layout layout layout layout The final design step consists of generating a possible layout for the microstrip reflector array antenna. The computer program used to define the necessary masks is ‘Cadence Design System, Inc’: a very popular electronic design automation (EDA) software. Most of the design rules in terms of minimum dimensions en spacing are imposed by the Silicon-On-Glass (SOG) varactor process. Section 6.3.1 starts with the layout of the necessary lumped elements, namely the varactors and the resistors. Next, the layout of a single cell is presented, followed by a complete 7x7 elements array in section 6.3.2 6.3.1- Lumped elements layout. The process layers needed to fabricate the high-performance varactor diodes are illustrated in figure 6.12 and defined in the following.

o The ININININ layer is the first metal layer (copper). The smallest realizable trace

77

in this layer is 2µm wide and the minimum distance between two adjacent traces is 6µm. o The IIIICMCMCMCM layer is the second metal layer (copper). The smallest realizable trace in this layer is 1µm wide and the minimum distance between two adjacent traces is 1µm. o The STSTSTST, the DTDTDTDT and the KH1KH1KH1KH1 layers are used to realize the actual PN junction of the varactor diode. They define respectively the shallowest etched trench, the medium etched trench and the deepest etched trench to the buried oxide. The minimum distance with respect to the masks alignment for those layers is 0.5µm. o The COCOCOCO layer defines the front contact opening to the PN junction. Its minimum size is 1µm x 1µm and the minimum overlapping area with the ICMICMICMICM metal layer is 1µm. The minimum spacing with respect to the STSTSTST mask is 3µm. o The BNBNBNBN layer defines the back ohmic contact to the PN junction. The spacing between the BNBNBNBN mask and the STSTSTST mask should be 3µm in order to line up the two contact openings (BNBNBNBN and COCOCOCO). The minimum distance between two BNBNBNBN traces is 3µm and the overlapping area with the ININININ metal layer should be at least 2µm. o The CTCTCTCT layer is used to realize the ‘vias’ connecting the two metal layers (IN IN IN IN and ICMICMICMICM). The minimum surface area of a via is 1µm x 1µm and the minimum distance between adjacent vias is 3µm.

Figure 6.12: Illustration of the process layers needed for the integration

of varactor diodes and high-ohmic resistors in the Silicon-on-glass transfer

technology [derived from 19]. The biasing reistors are made of the Silicon epilayer. Trench etching by means of the STSTSTST, the DTDTDTDT and the KH1KH1KH1KH1 masks is also used to define those high-ohmic resistors. Note that it removes the doped Silicon in regions where the biasing resistors are printed. The resistance value is given by:

hW

LR

.ρ= (6.a)

where L, W and h are respectively the length, the width and the thickness of the deposited trace defining the resistor (see figure 6.13a). According to the SOG

process, h=0.5µm and W can be chosen to be equal to 2µm. The resistivity ρ of the epilayer is given by:

78

dnpn Nepne µµµ

ρ 1

)(

1 ≈+

= (6.b)

in which e denotes the electron charge (1.6x10-19C), n (/p) the majority (/minority) carrier concentration, µn the electron mobility and Nd the doping concentration. Recall that a high doping concentration (Nd=1x10

17cm-3) is required to obtain the highest possible Q-factor for the varactor diodes (see chapter 4). In that case the electron mobility is found to be µn=900cm

2/V.s [50] and the corresponding resistivity

of the Silicon epilayer is ρ=0.07Ω.cm. From this, the required length to realize 100kΩ resistors can be derived. We have:

mL µ1431010007.0

102001050 366

=×⋅×⋅×=−−

(6.c)

In the final design, a length of 145µm has been used. The generated layout in cadence is shown in figure 6.13c. The BNBNBNBN contact pads are highly doped and their resistance can therefore be neglected.

Figure 6.13: Simplified resistor geometry (a) and resistor layout

generated by cadence (b). 6.3.2- Reflector array layout. The generated layout for a single microstrip patch loaded with two varactor diodes is depicted in figure 6.14. The pictured unit cell is 2500µm x 2500µm. The patches are patterned in the ININININ metal layer. One half of the patch is directly connected to one terminal of the varactors (in the ININININ layer) while the other half is connected to the second terminal through CTCTCTCT vias contacting the ICMICMICMICM metal layer. In figure 6.15, the area around the varactors (inside the white circle) has been enlarged to show the high similarity with the geometry simulated in HFSS. The effective area of the varactor is 5µm x 20µm and the air gap between the two halves of the patch is 40µm. The generated layout for a 7x7 elements array is depicted in figure 6.16. The STSTSTST and KH1KH1KH1KH1 masks have been removed to increase the clarity of the picture. The size of the bond pads is 60µm x 60µm and the mask patterns are intended for a wafer stepper exposure. Consequently, the size of the entire array is 20 x 20 mm2.

79

Figure 6.14: Layout of a single varactor loaded microstrip patch.

Figure 6.15: Close-up showing in details how the air gap in the middle of

the patch is bridged (on the top) by a varactor diode.

80

Figure 6.16: Layout of a 7x7 elements array.

81

CHAPTER 7

EXPERIMENTAL MEASUREMENT OF THE ANTENNA (TESTING) To further demonstrate the effectiveness of the proposed design procedure, the reflector array antenna will have to be tested. The actual performance of the proposed antenna should be established by means of experimental measurements. The experimental results can then be compared with the theoretically predicted behavior. This section particularly focuses on the radiation pattern measurement, because the scanning capability of the array is paramount in our design. Because the operating frequency is high and the antenna is physically small, an indoor measurement can be performed. It means that the antenna properties are measured inside an electromagnetic anechoic chamber (EAC1). Two commonly used techniques known as far-field measurement and near-field measurement are discussed in section 7.1 and section 7.2, respectively. 7.17.17.17.1---- FarFarFarFar----field (compact) measurementfield (compact) measurementfield (compact) measurementfield (compact) measurement In this configuration the radiation pattern is measured with the antenna under test (AUT) in its receiving mode. A second antenna (source antenna) is used to illuminate the test antenna. It consists of a large parabolic reflector which collimates the spherical waves from a source located at its focal point into a plane wave incident on the test antenna. This plane wave is very suitable for far-fields measurements and the AUT can be located at a reasonable distance from the reflector. A simplified schematic of the test setup is depicted in figure 7.1. The antenna under test is mounted on an electromechanical system with one or more degrees of freedom, called a positioner. Two orthogonal rotational axes are required to directly measure the cut-planes of the radiation pattern. As illustrated in figure 7.1, the AUT is supposed to be located at the origin of the coordinate system. Its radiation pattern in the elevation plane and the azimuth plane can be obtained from the received signal through its motion in spherical coordinates (i.e. around the rotational axes), while keeping the source antenna stationary. The motion of the positioner (turntable) is controlled by a positioner controller located in the control room with all the necessary measurement equipment. A signal generator (synthesized source) provides the RF signal for the source antenna through one of the outputs of a directional coupler. The reflected signal is available at the second output of the directional coupler. It serves as a reference to tune the synthesized source at the correct operating frequency and to compute the complex transmission coefficient (S21 or S12) between the two antennas for each spatial

1 An EAC is a test room whose walls are covered with electromagnetic absorbing materials to minimize internal reflections of electromagnetic waves, thereby mimicking a free space environment.

82

position (elevation and azimuth). Note that the feeding source is can be fixed on the floor to prevent aperture blockage.

Figure 7.1: Illustration of an indoor test setup for compact far-field

measurement [from 51]. The received RF signal from the AUT is detected by means of a microwave receiver (generally known as a network analyzer). To make signal processing easier and less expensive, mixers are employed to down convert the high frequency (e.g. 60GHz) RF signal to a suitable intermediate frequency. Although explicitly shown in figure 7.1, the mixers are incorporated in the network analyzer. To automate the whole measurement sequence, a computer is used to drive all the instruments by means of a high-speed data bus. The computer is equipped with a suitable antenna measurement software and makes it possible to visualize the measured patterns on a monitor. 7.27.27.27.2---- NearNearNearNear----field measurementfield measurementfield measurementfield measurement In a near-field measurement, the antenna under test is measured in the transmitting mode and a second antenna is not needed. The AUT remains fixed, while a probe1 is moved to scan the field over a well-defined surface in its vicinity. For each position of the probe specified by a set of (x, y) coordinates, the tangential component of the electric (or magnetic) field is sampled. The generated (near-field) data matrix is stored and after a Fourier transformation, the far-field data is obtained. 1 The probe is a small dipole or loop antenna, depending on which type of field (electric or magnetic) one wishes to measure.

83

CONCLUSION & RECOMMENDATIONS The design of an advanced antenna technology, namely the tunable printed reflector array has been presented. Analysis of the (frequency dependent) behavior of a microstrip patch reveals a window in the frequency range over which useful resonating patch antennas can be effectively designed. It has been demonstrated and confirmed by simulation plots that the fundamental bandwidth limitation inherent in microstrip antennas is based on resonator action. It is also apparent from this analysis that a difficult challenge arises when trying to simultaneously optimize numerous performance aspects of the patch antenna including: bandwidth, phase range, directivity (beam width), side lobe levels and losses. As an illustration, it has been shown that bandwidth and radiation efficiency can be traded with one another to a certain extent. An understandable explanation (description) of the radiation mechanism by means of the transmission line model and the cavity model has been given. It is known that the microstrip patch physically radiates because of currents flowing on the patch surface and on the ground plane. In software packages (like HFSS) used to solve electromagnetic structures, those surfaces can be partitioned into elementary surfaces and the full problem space is thereby divided into smaller regions. The collection of the elementary surfaces is referred to in HFSS as the finite element mesh and their contributions through space waves and surface waves can be added up to find the total field. This complex mathematical process is used in integral equation formulations and although it provides little insights in the radiation mechanism, the calculated fields are very accurate and the results are reliable. Therefore, the use of a powerful simulation tool like HFSS based on the finite element method has been indispensable during several stages of the design. To characterize the phase of a single radiating element (loaded patch), two simulation techniques can be used in HFSS. Although very accurate for broadside illumination, the wave guide technique is not very adequate for the characterization of elements located at the edges of the array. For this reason, another method has been developed and is referred to in this report as the PML approach. It was originally used for the characterization of high-impedance electromagnetic band-gap (EBG) surfaces [9, 52]. The major differences with the wave guide approach translate into the boundary conditions delimiting the unit cell, the data extraction method and the type of excitation used. Normal incidence can be analyzed using a waveguide simulation approach with port excitation, while off-normal incidence requires fields post-processing in order to extract the necessary data from plane wave excited solutions. An accurate HFSS model based on impedance surfaces has been developed for the variable capacitors (varactors) used to control the resonant frequency of the radiating patches. A global simulation strategy can therefore be used whereby electromagnetic analysis and circuit simulation are unified and integrated seamlessly. In combination with a written Matlab code returning the required phase shift at each element of the array (for any array size), a tunable reflector array antenna has been designed. For a 5x5 elements array, the simulated antenna parameters compete very well with recent designs from the literature [18, 53]. A directivity of 77.51 (19dB), a 20° HPBW (half-power beam width), a side lobe level of -13.31dB, a cross-polarization suppression of 55dB and a ±50° scan angle has been achieved.

84

The mutual coupling between adjacent patches has been analysed. A physical insight into this fundamental array problem has been presented at hand, followed by a quantitative analysis. It turns out that the predominant coupling mechanism is via space waves and can be reduced by increasing the inter-element spacing. The implementation of the bias network needed to supply the proper voltages to the varactor diodes has been investigated. The most limitations are imposed by printed circuit technology and the corresponding manufacturing process constraints. Two implementations referred to as the ‘via’ concept and the ‘line’ concept have been presented. The ‘via’ concept is expected to result into superior antenna performances in exchange of a more complex and expensive fabrication process. A dual conclusion applies to the ‘line’ concept in which the control lines are not shielded from the incoming wave. Extraneous radiation from those control lines has been assessed and shown to badly affect the overall radiation pattern of the reflector array. In addition to the fabrication simplicity, another advantage of this method is the possibility to use an alternative substrate than Silicon by means of a post-processing transfer technology. A glass substrate for example can improve the performance of the varactor diodes in terms of their Q-factor [20]. It is known that all devices on a Silicon wafer suffer from capacitive coupling to the resistive substrate which consequently results in a dissipation of RF energy and poor quality passives. Furthermore, in recognition of the fact that the channels (outputs) of the DAC (digital to analog converter) are limited, a model which enables scanning in the E-plane has been proposed. In this model, the exposed control lines on the PCB (printed board) practically vanish completely. The resulting antenna parameters closely match the results obtained with an array without biasing traces printed on the reflector surface. The fabrication of a 30x30 elements array is possible on a standard Silicon wafer with a diameter of 10cm. The expected results can be summarized in a set of well-known antenna parameters namely: a directivity of 34.5dB, a ±50° scan angle and a side lobe level around –29.5dB. Note that the accuracy of the lithography can be translated into tolerances in the element dimensions which cause small variations in the resonant frequency. It is worth mentioning that some precautions would seem necessary, particularly at low bias levels, where the amount of power dissipated in the varactors could degrade the antenna performance [25]. The cross-polarization levels are expected to increase and the efficiency of the loaded patches with respect to the total input power can decrease considerably. This phenomenon can either be tested by measuring the current through the diodes or by plotting the cross-polarization radiation patterns of the loaded and unloaded patches. To reduce the side lobe levels, techniques involving distributions of the amplitude of the excitation currents at each patch (Taylor’s distribution, Dolph-Tschebyscheff distribution) can not be used [2]. Other techniques related to the periodical arrangement of the elements (Hexagonal arrays) can offer some great improvements in side lobe level reduction compared to the conventional quadratic grid arrangement [21]. Sub-arraying can also be used to reduce the scan blindness effect [29]. It is worth emphasizing that there is less scope for innovative designs within the limitations set by manufacturing.

85

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89

APPENDIX A: Field equivalence principle The theory presented here is based on the analysis given in [2] for aperture antennas. The microstrip patch antenna is often modeled as a resonant cavity for analysis purposes (see figure B.1). When there is sufficient knowledge on the field distributions at the apertures, the field equivalence principle can be used. According

to this theorem, the tangential components of the electric field ( tEr) on the magnetic

walls can be replaced by an equivalent magnetic current density ( sMr). A judicious

application of the field equivalence principle (whereby the volume inside the cavity is supposed to be filled with a perfect magnetic conductor) combined with image theory yields the following relation:

ts EnMrr

×−= ˆ2 (B.a)

in which n denotes the unit vector normal to the magnetic walls and pointing outwards. Because the magnetic current densities are radiating into an unbounded medium (due to image theory), the fields in the region outside the cavity can be derived from:

−=

×=

∫∫

∫∫−

−

'4

'4

ˆ

dsR

eMjH

dsR

eMajE

S

jkR

s

S

jkR

sr

rr

rr

πµω

πµωη

(B.b)

In this equation, ω represents the frequency of the radiated wave, k the wave number and η the intrinsic impedance of the medium defined as (µ/ε)1/2. As illustrated in figure B.2, the distance between the sources and the point in space where the fields are evaluated is denoted as R and the unit vector pointing in the

radial direction is ra . The surface S represents a vertical magnetic wall of the cavity.

Note that equation B.b should be derived for each magnetic wall and the total fields can be obtained by superposing the individual contributions. In this summation, the equivalent magnetic current densities on the non-radiating slots cancel each other as can be expected from (B.a)

Figure B: (1): Cavity model. The upper and lower faces are PECs while the walls

are PMCs. (2): Coordinate system used for the analysis.

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APPENDIX B: Matlab files %************************************************** ***************** % PATCH DIMENSIONS %************************************************** ***************** % THIS PROGRAM IS A MATLAB PROGRAM THAT COMPUTES THE DIMENSIONS OF THE PATCH BASED ON THE INPUT PARAMETERS. % ** INPUT PARAMETERS % 1. freq = RESONANT FREQUENCY (in GHz) % 2. epsr = DIELECTRIC CONSTANT OF THE SUBSTRA TE % 3. h = HEIGHT OF THE SUBSTRATE (in um) % ** OUTPUT PARAMETERS % 1. W = PHYSICAL WIDTH (in mm) % 2. Le = EFFECTIVE LENGTH OF PATCH (in um) % 3. L = PHYSICAL LENGTH OF PATCH (in um) function PatchDimensions(freq,epsr,h); close all ; % Check the input parameters (default: freq=60 Ghz, epsr=2.1, h=300) if (nargin < 1) freq = 60; epsr = 2.1; h = 300; end freq = freq*1e9; lambda_o=3e8/freq; %lambda in free space. lambda_d=lambda_o/sqrt(epsr); %lambda in the dielectric. W=(3e8/(2.0*freq))*(sqrt(2.0/(epsr+1.0))) * 1e6; %Width of the patch (in um) for an efficient radiat or. %W=(sqrt(h*1e-6*lambda_d)) * (log(lambda_d/(h*1e-6) ) - 1)* 1e6; %Width necessary to obtain a 50 Ohm input impedance . ereff=(epsr+1.0)/2.0 + (epsr-1)/(2.0*sqrt(1.0+12.0* h/W)) %Effective dielectric constant at low freq. dl=0.412*h*((ereff+0.3)*(W/h+0.264))/((ereff-0.258) *(W/h+0.8)); %Extended length in um due to fringing. lambda_eff=3e8/(freq*sqrt(ereff)) %Effective wavelength in the dielectric. Leff=(lambda_eff/2.0) * 1e6; %Effective length of the patch in um. %When chosen like this, the dominant mode is the TM010 L=(Leff-2.0*dl); %Physical Length of the patch in um. ko=2.0*pi/lambda_o; %Emax=sinc(h*ko/2.0/pi); %If we assume that all the field lines are in the d ielectric at high %frequencies, we have: dl2=0.412*h*((epsr+0.3)*(W/h+0.264))/((epsr-0.258)* (W/h+0.8)); %Extended length in um due to fringing. lambda_eff2=3e8/(freq*sqrt(epsr)); %Effective wavelength in the dielectric. Leff2=(lambda_eff2/2.0) * 1e6; %Effective length of the patch in um. %When chosen like this, the dominant mode is the TM010

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Le=(Leff2-2.0*dl2); %Physical Length of the patch in um. %Display output parameters disp(strvcat( 'OUTPUT PARAMETERS', '=================' )); disp(sprintf( 'PHYSICAL WIDTH OF PATCH (in um) = %4.4f' ,W)); disp(sprintf( 'PHYSICAL LENGTH OF PATCH (in um) = %4.4f' ,L)); disp(sprintf( 'EFFECTIVE LENGTH OF PATCH (in um) = %4.4f' ,Leff)); disp(sprintf( 'LENGTH EXTENSION DUE TO FRINGING(in mm) = %4.4f' ,dl)); disp(sprintf( 'PHYSICAL LENGTH2 OF PATCH (in um) = %4.4f' ,Le)); function Phase_Es_elt(M,N,Dx,Dy,F,the_o,ph_o,freq) % INPUT PARAMETERS: % % M: Number of array elements in the x-d irection % N: Number of array elements in the y-d irection % Dx: spacing in um between the elts in t he x-direction % Dy: spacing in um between the elts in t he y-direction % F: distance in meter from feed to cent er of the array % the_o: scan angle theta in degree % ph_o: scan angle phi in degree % freq: operating frequency (in GHz) % % NOTE: ONEVEN OR EVEN INTEGERS CAN BE USED FOR THE SIZE (M,N) OF % THE ARRAY. % % OUTPUT VARIABLE % % The phase required for the scattered field at each element of % the array in order to obtain a collimated b eam in the % direction (theta_o,phi_o). close all ; % Check the input parameters (default) if ( nargin < 1 ) M = 50; N = 50; Dx = 2500; Dy = 2500; F = 0.5; the_o = 0; ph_o = 90; freq = 60; end Mx = M; Ny = N; dx = Dx*1e-6; %um -> m dy = Dy*1e-6; f = F; theta_o = the_o*pi/180; %Deg -> rad phi_o = ph_o*pi/180; freq = freq*1e9; % GHz -> Hz frequency M_xcoord = ones(Mx,Ny);

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M_ycoord = ones(Mx,Ny); X_coord = (-((Mx-1)/2)*dx:dx:((Mx-1)/2)*dx)'; %vector of x-coordinates Y_coord = (-((Ny-1)/2)*dy:dy:((Ny-1)/2)*dy); %vector of y-coordinates for m = 1:Ny M_xcoord(:,m)= X_coord; end for m = 1:Mx M_ycoord(m,:)= Y_coord; end Matrix = sqrt(M_xcoord.^2 + M_ycoord.^2); Rmn = Matrix; % At this stage, Matrix or Rmn contains the distanc es from the center of % the array to each element of the array. The dista nces corresponds to % the magnitude of the position vector Rmn in eq(1) of ref['Modelling and % Design of Electronically tunable reflectarrays',S .V.Hum]. %The distance Ri from the (phase center of the) fee d to %each element on the array is: Ri = Matrix; for m = 1:Mx for n = 1:Ny Ri(m,n) = sqrt( Matrix(m,n)^2 + f^2 ); end end c = 3e8; lambda = c/freq; ko = 2*pi/lambda; % From eq1, the phase shift required at each elemen t is: Phi_mn = ko*(Ri-(M_xcoord*sin(theta_o)*cos(phi_o) + M_ycoord*sin(theta_o)*sin(phi_o))); Phi_mn_deg = Phi_mn * 180/pi; for m = 1:Mx for n = 1:Ny while (Phi_mn_deg(m,n)) >= 360 Phi_mn_deg(m,n) = Phi_mn_deg(m,n) - 360 ; end end end %display the results. %display the results. figure(1) pcolor(padarray(Rmn,[1 1], 'post' )); colorbar;

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figure(2) pcolor(padarray(Ri,[1 1], 'post' )); colorbar; figure(3) pcolor(padarray(Phi_mn_deg,[1 1], 'post' )); colorbar; disp(num2str(Phi_mn_deg));

function planar_array(M,N,Dx,Dy,the_o,ph_o,fr) % % INPUT PARAMETERS: % M: Number of array elements in the x-d irection % N: Number of array elements in the y-d irection % Dx: spacing in um between the elts in t he x-direction % Dy: spacing in um between the elts in t he y-direction % the_o: scan angle theta in degree % ph_o: scan angle phi in degree % fr: operating frequency in Ghz % % OUTPUT VARIABLES % 1. Uniform progressive phase difference i n x- and y-direction % 2. 3-D Plot of the array factor. % close all ; % %------------Check the input parameters (default)-- ---------------------- if ( nargin < 1 ) M = 10; N = 10; Dx = 2500; Dy = 2500; fr = 60; the_o = 45; ph_o = 0; end Mx = M; Ny = N; dx = Dx*1e-6; %um -> m dy = Dy*1e-6; theta_o = the_o*pi/180; %Deg -> rad phi_o = ph_o*pi/180; %Deg -> rad freq = fr*1e9; %GHz -> Hz c = 3e8; lambda = c/freq; %free space wavelength k = 2*pi/lambda; % thetaStep = 0.2; % in degrees phiStep = 0.4; theta = (0:thetaStep:180); phi = (0:phiStep:360); theta = theta*pi./180; % theta in radian phi = phi*pi./180; % phi in radian [THETA,PHI] = meshgrid(theta,phi);

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%-----------Calculation of the array factor-------- --------------------- betax = -k*dx*sin(theta_o)*cos(phi_o); betay = -k*dy*sin(theta_o)*sin(phi_o); Psix = k*dx*sin(THETA).*cos(PHI) + betax; Psiy = k*dy*sin(THETA).*sin(PHI) + betay; AFx = ((1/M)*sin((M/2)*Psix)./sin((1/2)*Psix)); %'*ones(size(phi)); AFy = ((1/N)*sin((N/2)*Psiy)./sin((1/2)*Psiy)); %'*ones(size(phi)); %AFx=sinc((Mx.*Psix./2)./pi)./sinc((Psix./2)./pi); %AFy=sinc((Ny.*Psiy./2)./pi)./sinc((Psiy./2)./pi); AF = AFx.*AFy; %----------Plot the array factor in 3-Dimension.--- --------------------- r = AF; X = r.*sin(THETA).*cos(PHI); %Transformation from spherical to rectangular. Y = r.*sin(THETA).*sin(PHI); Z = r.*cos(THETA); figure(1) h = mesh(X,Y,Z,AF); hold on % Plot the axes. corx=max(max(X)); cory=max(max(Y)); corz=max(max(Z)); corr=max([corx,cory,corz]); q1=plot3([0 corr],[0 0],[0 0], 'k' ); %trace the X-axis, 'k' set the color of %the line to black. The line passes through 2 pts w hich (x y z) coordinates %are respectively:(0 0 0) and (corr 0 0). q2=plot3([0 0],[0 corr],[0 0], 'k' ); q3=plot3([0 0],[0 0],[0 corr], 'k' ); set(q1, 'linewidth' ,1.5); set(q2, 'linewidth' ,1.5); set(q3, 'linewidth' ,1.5); t1=text(corr*1.05,0,0, 'x' , 'horizontalalignment' , 'center' ); %annotate X-axis t2=text(0,corr*1.05,0, 'y' , 'horizontalalignment' , 'center' ); t3=text(0,0,corr*1.05, 'z' , 'horizontalalignment' , 'center' ); set(t1, 'fontsize' ,14, 'fontname' , 'Times New Roman' , 'fontangle' , 'italic' ); set(t2, 'fontsize' ,14, 'fontname' , 'Times New Roman' , 'fontangle' , 'italic' ); set(t3, 'fontsize' ,14, 'fontname' , 'Times New Roman' , 'fontangle' , 'italic' ); axis off ; %remove grid and labels. axis equal ; %colorbar set(gcf, 'Color' ,[1,1,1]) view(135,20); %view(x,90) means looking from the top in the XY-plane.

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%view(90,0) means sitting on the x-axis and looking in the ZY-plane. % %--------------- display the output parameters----- ---------------------- disp([ 'progressive phase_dif in x-dir = ' ,num2str(betax*180/pi), ' degrees' ]); disp([ 'progressive phase_dif in y-dir = ' ,num2str(betay*180/pi), ' degrees' ]); %%%%%%%%%%%%%%%%%%%%%%%%%SECOND METHOD%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function Plot_phase(freq_range, ph_data, ph_ref_data, ph_WG A) %NOTE!!The function has some problem dealing with plots starting with an %increase in the phase. We deal with that by adjust ing the startvalue of %the increment variable 'x' of the 'for loop'. % Check the input parameters (default: freq_range=[ ]', ......) if (nargin < 1) freq_range = (50:0.5:70)'; ph_data = ones(1,length(freq_range))'; ph_ref_data = 0.3*ones(1,length(freq_range))'; ph_WGA = ph_data; end Freqo = freq_range; Ey_phase_av_evp = ph_data; %phase at the evaluation plane. Ey_refphase_av_evp = ph_ref_data; %reference phase at eval. plane. Ey_phase_WGA = ph_WGA; %phase of elt, evaluated using the WGA. %plot the data to see if it's the correct one. figure(1) plot(Freqo,Ey_phase_av_evp, '+blue' ); hold on plot(Freqo,Ey_refphase_av_evp, 'ogreen' ); %jumps that are less than pi should be shifted i.o. t use the function 'unwrap'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% endval = length(freq_range)-1; temp = Ey_phase_av_evp; for x=3:endval if (Ey_phase_av_evp(x)-Ey_phase_av_evp(x-1)>0) && (Ey_phase_av_evp(x)-Ey_phase_av_evp(x+1)<0) temp(x) = (temp(x-1)+(temp(x+1)-(2*pi)))/2; end end Ey_phase_av_evp = temp; %Ey_phase_av_evp(20)= -3.14; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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figure(2) plot(Freqo,Ey_phase_av_evp, '+blue' ); %plot phase at evaluation plane without a jump in t he data. Q = unwrap(unwrap(Ey_phase_av_evp)); hold on plot(Freqo,Q, 'ored' ); %compute and plot the phase produced by the element (the patch). Ey_phase = Ey_phase_av_evp-Ey_refphase_av_evp+pi; figure(3) plot(Freqo,Ey_phase, '+blue' ); Q1 = unwrap(Ey_phase); hold on plot(Freqo,Q1, 'ogreen' ); %plot the phase in degree produced by the element ( the patch). %Comparison of the PML approach with the WGA approa ch, figure(4) Ey_phase_deg = Q1*(180/pi); plot(Freqo,Ey_phase_deg, 'oblue' ); hold on plot(Freqo,Ey_phase_WGA, '+red' );