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UNIVERSITY OF MARIBOR
FACULTY OF ELECTRICAL ENGINEERING
AND COMPUTER SCIENCE
Matija Drgestin
ANALYSIS AND SIMULATION OF PYRAMIDAL HORN
ANTENNA
Master’s thesis
Maribor, May 2019
UNIVERSITY OF MARIBOR
FACULTY OF ELECTRICAL ENGINEERING
AND COMPUTER SCIENCE
Matija Drgestin
ANALYSIS AND SIMULATION OF PYRAMIDAL HORN
ANTENNA
Master’s thesis
Maribor, May 2019
ANALIZA IN SIMULACIJA PIRAMIDNEGA LIJAKA
Magistrsko delo
ANALYSIS AND SIMULATION OF PYRAMIDAL HORN ANTENNA
Master’s thesis
Student: Matija Drgestin
Study Programme: Master’s Study Programme of Electrical Engineering
Study Field: Telecommunications
Mentor: doc. dr. Boštjan Vlaovič, univ. dipl. inž. el.
Lector: prof. Marija Krznarić
i
Thank you!
I would like to express my deepest gratitude to my mentor Assist. Prof. Dr. Boštjan Vlaovič for his
advice, support, and guidance while writing this master’s thesis. In addition, I am very thankful
to the lector Marija Krznarić for spending time proofreading this thesis.
Furthermore, I owe special thanks and gratitude to my mother, father, brothers and my friends
for their support, encouragement and love throughout my student life, and life in general.
Matija Drgestin
ii
Analiza in simulacija piramidnega lijaka
Ključne besede: piramidni lijak, valovod, simulacija, HFSS, frekvenca
UDK: 621.396.67.029.5(043.2)
Povzetek
1. Uvod
V magistrski nalogi je opisan piramidni lijak, njegove dimenzije in parametri. Parametri antene
so podrejeni predvideni uporabi. Ob demonstraciji lastnosti razširjanja elektromagnetnega
valovanja v predavalnici in laboratoriju smo praviloma prostorsko omejeni. Ker želimo
demonstracije praviloma izvajati v daljnem polju, smo izbrali anteno manjših dimenzij. V prvem
delu naloge smo podali analitično metodo za izračun parametrov glede na željeno resonančno
frekvenco in dimenzije antene. V matematičnem modelu smo najprej izbrali željeno vrednost
dobitka antene in nato z uporabo iterativne analitične metode določili dimenzije antene.
Izračunane dimenzije so bile uporabljene pri izdelavi 3D modela antene. Sledila je simulacija 3D
modela antene s profesionalnim orodjem ANSYS HFSS z uporabo numerične metode končnih
elementov. Numerično in grafično predstavimo več parametrov antene, med drugim: dobitek,
iii
smernost, valovitost in sevalni diagram. V okviru simulacije smo preverili vpliv odstopanja
dimenzij antene na parametre antene. Predstaviti smo želeli v kolikšni meri odstopanja vplivajo
na posamezne parametre antene in na kaj je potrebno biti pozoren pri samogradnji. Kot
referenčno smo uporabili anteno z analitično izračunanimi dimenzijami, ki je napajana
neposredno preko valovoda. Pri praktični anteni smo uporabili anteno, ki je napajana preko
koaksialnega kabla in SMA konektorja. Pridobljeni rezultati so predstavljeni in komentirani.
2. Elektromagnetni valovi in valovod
V teoretičnem delu so predstavljene osnove elektromagnetne teorije, ki so potrebne za analitičen
opis piramidnega lijaka. Antena pretvarja elektromagnetno energijo v prostorski elektro-
magnetni val (sevanje) in obratno. Elektromagnetno valovanje je sestavljeno iz električnega in
magnetnega polja, ki sta pravokotna med sabo in na smer širjenja valov – potujoče transverzalno
valovanje. Za primer smo podali polvalni dipol, osnovno anteno, ki odlično predstavlja
elektromagnetno sevanje. Skladno s teorijo elektromagnetnega valovanja, ki jo podajajo
Maxwellove enačbe, sprememba električnega polja povzroča spremembo magnetnega polja in
obratno. Elektromagnetni valovi vsebujejo določene lastnosti, specifične samo za to vrsto valov.
Na svoji poti valovi pogosto naletijo na različne ovire, zato v nalogi predstavimo pojave, kot so
interferenca, odboj in lom. Nekateri od teh pojavov so ključni za dojemanje razširjanja
elektromagnetnih valov v zaprtih strukturah kot je valovod. Valovod je votla, prevodna,
praviloma kovinska, struktura, ki se uporablja za učinkovit prenos elektromagnete energije.
Uporabili smo pravokotni valovod, ki je primeren za uporabo pri višjih frekvencah in za napajanje
piramidnega lijaka. Podane so dimenzije valovoda, katere se bodo uporabile za izračun dimenzij
antene. Navedene dimenzije predstavljajo širino in višino valovoda, kjer je širina označena kot a
in znaša 28,4988 mm, b pa je označena kot višina valovoda in znaša 12,6238 mm. V tem
poglavju, smo tudi želeli izračunati t.i. mejno frekvenco, pod katero valovod ne more delovati.
Poleg tega so bile opisane komponente električnega in magnetnega polja v valovodu.
iv
Komponente polja so bile podane z enačbami ((3.7) - (3.12)), ki so razdeljene na vzdolžne in
prečne komponente. Na koncu je bila grafično predstavljena distribucija polj vzdolž valovoda.
3. Matematični model
Matematični model antene je podan analitično. Izhodišče za izračun predstavljata resonančna
frekvenca in dobitek antene. Oboje smo določili glede na željene dimenzije antene. Uporabljena
je bila iterativna metoda v kateri smo želeli zadostiti enačbi (5.1) iz poglavja 5.3, pa tudi 5.6.
Sledil je izračun dimenzije antene: višina, širina in dolžina stranic. Ob tem smo bili pozorni na
možnost fizične izdelave antene v samogradnji. V primeru referenčne antene dimenzije antene
so bile večje kot v primeru praktične antene, zato ker smo pri referenčni anteni uporabili večji
dobitek, ki je znašal 19,7 dB. Izračunane vrednosti dimenzij praktične antene (poglavje 5.6) so
podrejene demonstraciji na kratki razdalji, zato manjši dobitek antene ni problematičen.
Izračunane dimenzije antene, so bile uporabljene v procesu simulacije. Rezultati simulacije so
predstavljeni v poglavjih 5.4 in 5.8.
4. Simulacija antene
Izdelavo 3D modela antene in simulacijo smo izvedli z uporabo programskega okolja ANSYS
Electromagnetic Suite 18, ki se uporablja za modeliranje različnih prenosnih struktur, anten, RF
komponent in podobnih sklopov. Omogoča vizualizacijo elektromagnetnih polj v strukturi 3D
modelov in okolici. Pridobljeni parametri antene so dobra preverba načrtovane izvedbe antene.
Podrobneje smo preverili kako spremembe dimenzij antene vplivajo na rezultate izbranih
parametrov antene. Tako smo določili dovoljeno odstopanje dimenzij izvedene antene glede na
referenčno anteno. V poglavjih 5.3 in 5.7 je predstavljen potek simulacije in izdelava 3D modela
referenčne in praktične antene. V teh poglavjih so podrobno opisovani koraki, ki so privedli do
končnega izida, 3D modela antene. V simulaciji so uporabljene dimenzije, ki smo jih pridobili z
analitično metodo. Razlika med referenčno in praktično anteno je v načinu napajanja; pri
v
referenčni anteni je izvor elektromagnetnih valov bil valovod, med tem ko praktično anteno
napajamo preko koaksialnega kabla in konektorja SMA. Pri praktični anteni se na parametrih
pozna vpliv konektorja, ki vpliva na spremembo strukture valovoda antene.
5. Rezultati
Rezultate smo predstavili tako v tabelarični obliki kot z uporabo 2D grafov in 3D vizualizacij. Ob
simulaciji smo izbrali željeno frekvenco ter preverili smernost, dobitek, porazdelitev električnega
in magnetnega polja v anteni, razmerje stoječega valovanja in ostale parametre. Parametre
piramidnega lijaka smo predstavili tudi grafično. Simulacija je potrdila pričakovane rezultate za
oba modela antene. V obeh primerih so vrednosti pridobljenih parametrov v pričakovanih mejah,
na primer, razmerje stoječega valovanja je pod 2, večina energije se izseva. Sevalni diagram smo
predstavili v tridimenzionalnem prostoru. Iz slike je razvidno, da antena seva v predvidenem
območju. Glavni snop in stranski snopi imajo pričakovano obliko. Nekatere parametre smo
predstavili tudi dvodimenzionalno v E-ravnini in H-ravnini, saj se tako lažje odčitajo sevalni koti.
Na koncentričnih krogih je predstavljen željen parameter, na primer, dobitek, ki se praviloma
predstavlja v normirani logaritmični obliki.
6. Sklep
Osnovni namen naloge je predstavitev in opis piramidnega lijaka. V zaključku povzamemo
celotno nalogo, od raziskovalnega in teoretičnega ozadja, analitičnega izračuna, tvorbe 3D
modela, simulacije ter predstavitve in analize rezultatov. Po predstavitvi osnov teorije razširjanja
elektromagnetnega valovanja in analitičnega opisa izbrane antene, smo dimenzije antene
določili z matematičnim izračunom. Nato smo izdelali 3D model za referenčno in praktično
anteno. Rezultati simulacije za referenčno anteno so bili skladni z matematičnimi napovedmi,
kar je potrditev dobrega modela antene. Sledila je vrsta simulacij kjer smo preverili parametre
anten ter njihovo odstopanje v primeru napak pri izdelavi zaradi odstopanja dimenzij antene.
vi
Izbrane dimenzije smo spreminjali v območju od +10 mm do -10 mm, v korakih po 1 mm.
Rezultati simulacije so pokazali vplive na parametre antene, naš cilj pa je bilo preveriti potrebno
natančnost pri izdelavi tovrstne antene. Simulacije so potrdila pričakovanja in pokazale, da
nenatančnost pri izdelavi antene ne bo bistveno vplivala na delovanje v predvidenem sistemu.
vii
Analysis and simulation of pyramidal horn antenna
Key words: pyramidal horn, waveguide, simulation, HFSS, frequency
UDK: 621.396.67.029.5(043.2)
Abstract
This master's thesis describes a pyramidal horn antenna, its dimensions and parameters. The
antenna parameters are submitted to the intended use. When demonstrating the properties of
the electromagnetic wave propagation in the laboratory, we are generally spatially limited. In
order to carry out antenna demonstrations in the far field, an antenna of smaller dimensions has
been chosen. In the first part of the thesis, an analytical method for calculating the parameters
according to the desired resonant frequency and the dimensions of the antenna has been
presented. In the mathematical model, the desired gain value of the antenna was selected and
then the dimensions of the antenna, using an iterative analytical method, was determined. The
calculated dimensions were used in the design of the 3D model antenna. The simulation of the
3D model antenna was carried out by the professional tool ANSYS HFSS which uses a numerical
method with a finite number of elements. Numerically and graphically, we present several
parameters of the antenna, including gain, direction, standing wave ratio and radiation diagram.
In the simulation, the influence of the deviation of the antenna dimensions on the parameters of
viii
the antenna was examined. The objective was to present to what extent the deviations influence
the individual parameters of the antenna and what needs to be considered, while planning the
physical realization of the antenna. As a reference, we used an antenna with analytically
calculated dimensions, which is fed directly through the waveguide. In practical terms, an
antenna was fed via a coaxial cable and a SMA connector. The obtained results are presented
and commented.
ix
Table of Contents
1. INTRODUCTION ............................................................................................................................ 1
2. ELECTROMAGNETIC WAVE .......................................................................................................... 3
2.1 Properties and occurrences of electromagnetic waves ....................................................... 5
2.2 Electromagnetic spectrum ................................................................................................... 11
2.3 Maxwell’s equations ............................................................................................................ 12
2.3.1 First Maxwell’s equation ............................................................................................... 13
2.3.2 Second Maxwell’s equation .......................................................................................... 13
2.3.3 Third Maxwell’s equation ............................................................................................. 14
2.3.4 Fourth Maxwell’s equation ........................................................................................... 15
3. WAVEGUIDES .............................................................................................................................. 16
3.1 Rectangular waveguides ...................................................................................................... 18
3.2 Transverse modes ................................................................................................................ 19
3.3 Dominant mode TE10. Cut-off frequency of the WR-112 ................................................... 21
3.4 Electric and magnetic fields in the waveguide. Maxwell’s equations ............................... 25
4. ANTENNAS .................................................................................................................................. 29
4.1 Antenna parameters ............................................................................................................ 30
4.1.1 Radiation pattern. Beamwidth ..................................................................................... 31
4.1.2 Field regions .................................................................................................................. 33
4.1.3 Polarization .................................................................................................................... 35
4.1.4 Power density ................................................................................................................ 37
4.1.5 Directivity and antenna gain. Radiation intensity ....................................................... 39
x
4.2 Types of antennas ................................................................................................................ 41
4.2.1 Monopole antennas ...................................................................................................... 41
4.2.2 Dipole antennas ............................................................................................................ 42
4.2.3 Aperture antennas ........................................................................................................ 44
4.3 E-plane horn antenna .......................................................................................................... 47
4.3.1 Geometry and parameters ........................................................................................... 47
4.3.2 Aperture field distribution and far-field region. Directivity ........................................ 50
4.3.3 Optimum antenna dimensions ..................................................................................... 54
4.4 H-plane horn antenna .......................................................................................................... 55
4.4.1 Geometry and dimensions ........................................................................................... 55
4.4.2 Aperture field distribution and far-field region. Directivity ........................................ 58
4.4.3 Optimum antenna values ............................................................................................. 61
4.5 Pyramidal horn antenna ...................................................................................................... 61
4.5.1 Geometry and dimensions ........................................................................................... 62
4.5.2 Aperture field distribution and far-field region. Directivity ........................................ 63
4.5.3 Optimum antenna values ............................................................................................. 65
5. ANTENNA SIMULATION .............................................................................................................. 66
5.1 Starting the HFSS simulator ................................................................................................. 66
5.2 Mathematical calculation of the reference pyramidal horn antenna ............................... 70
5.3 HFSS design procedure of the reference pyramidal horn antenna ................................... 73
5.4 Results of the reference pyramidal horn antenna ............................................................. 79
5.5 Deviations of the reference pyramidal horn antenna ........................................................ 89
5.6 Mathematical calculation of the practical pyramidal horn antenna ............................... 100
xi
5.7 HFSS design procedure of the practical pyramidal horn antenna ................................... 102
5.8 Results of the practical pyramidal horn antenna ............................................................. 105
6. CONCLUSION ............................................................................................................................ 112
xii
Table of Figures
Figure 1: Electric and magnetic field ............................................................................................... 3
Figure 2: Manifestation of EM radiation ......................................................................................... 4
Figure 3: Manifestation of EM radiation ......................................................................................... 5
Figure 4: Reflection........................................................................................................................... 6
Figure 5: Refraction .......................................................................................................................... 7
Figure 6: Total internal reflection .................................................................................................... 8
Figure 7: Diffraction .......................................................................................................................... 9
Figure 8: Constructive interference ............................................................................................... 10
Figure 9: Destructive interference ................................................................................................. 10
Figure 10: Electromagnetic spectrum ........................................................................................... 12
Figure 11: Waveguide..................................................................................................................... 16
Figure 12: Waveguide coupling ..................................................................................................... 17
Figure 13: Wave paths in the waveguide – top view .................................................................... 17
Figure 14: Geometry of rectangular waveguide in Cartesian coordinate system ...................... 18
Figure 15: WR-112 waveguide ....................................................................................................... 19
Figure 16: Comparison between TE and TM modes in rectangular waveguide ......................... 20
Figure 17: Waveguide WR-112 dimension specifications ............................................................ 23
Figure 18: Electric and magnetic field lines within a waveguide ................................................. 26
Figure 19: Transmitting antenna ................................................................................................... 29
Figure 20: Receiving antenna ......................................................................................................... 30
Figure 21: Antenna radiation pattern ............................................................................................ 32
Figure 22: Radiation patterns ........................................................................................................ 33
Figure 23: Field regions and field distribution .............................................................................. 34
Figure 24: Types of polarization ..................................................................................................... 36
Figure 25: Graphical representation of Poynting's vector ........................................................... 38
Figure 26: Examples of monopole antennas ................................................................................. 42
xiii
Figure 27: Examples of dipole antennas ....................................................................................... 43
Figure 28: Examples of microwave antennas ............................................................................... 44
Figure 29: E-plane and H-plane antenna ....................................................................................... 46
Figure 30: Pyramidal horn antenna ............................................................................................... 46
Figure 31: Geometry of E-plane antenna ...................................................................................... 47
Figure 32: Cross-section of the E-plane horn antenna ................................................................. 48
Figure 33: Directivity as a function of aperture height ................................................................ 49
Figure 34: E-plane horn antenna patterns in E-plane and H-plane ............................................. 52
Figure 35: Table of Fresnel's integrals ........................................................................................... 54
Figure 36: Geometry of H-plane antenna ..................................................................................... 55
Figure 37: Cross-section of the H-plane horn antenna ................................................................ 56
Figure 38: Directivity as a function of aperture width .................................................................. 57
Figure 39: H-plane horn antenna patterns in E-plane and H-plane ............................................. 60
Figure 40: Geometry of pyramidal horn antenna ......................................................................... 62
Figure 41: Top view (H-plane) of pyramidal horn ......................................................................... 62
Figure 42: Side view (E-plane) of pyramidal horn ......................................................................... 63
Figure 43: Pyramidal horn antenna pattern in E-plane and H-plane ........................................... 64
Figure 44: Electronics Desktop interface ...................................................................................... 66
Figure 45: Electronics Desktop toolbar ......................................................................................... 67
Figure 46: ANSYS HFSS window with associated parts ................................................................. 67
Figure 47: Project Manager window ............................................................................................. 68
Figure 48: Properties window ........................................................................................................ 68
Figure 49: Components Library window ....................................................................................... 69
Figure 50: 3D Modeler window ..................................................................................................... 69
Figure 51: Creating a waveguide ................................................................................................... 74
Figure 52: Waveguide values ......................................................................................................... 74
Figure 53: Horn aperture values .................................................................................................... 74
Figure 54: Waveguide and rectangle ............................................................................................. 75
xiv
Figure 55: Waveguide face selection ............................................................................................. 75
Figure 56: Pyramidal horn antenna ............................................................................................... 76
Figure 57: Antenna wall thickness ................................................................................................. 77
Figure 58: Selection of perfect conductor .................................................................................... 77
Figure 59: Region around horn antenna ....................................................................................... 78
Figure 60: Excitation port values ................................................................................................... 78
Figure 61: Assignment of excitation port ...................................................................................... 79
Figure 62: Frequency sweep .......................................................................................................... 80
Figure 63: Waveguide and reference antenna dimensions ......................................................... 81
Figure 64: Results of reference antenna parameters ................................................................... 81
Figure 65: Return Loss .................................................................................................................... 83
Figure 66: VSWR ............................................................................................................................. 84
Figure 67: Directivity ...................................................................................................................... 84
Figure 68: Gain ................................................................................................................................ 85
Figure 69: Total radiated electric field .......................................................................................... 86
Figure 70: Electric field strength .................................................................................................... 87
Figure 71: Magnetic field strength ................................................................................................ 87
Figure 72: Electric field in E-plane ................................................................................................. 88
Figure 73: Electric field in H-plane ................................................................................................. 89
Figure 74: Waveguide length ......................................................................................................... 90
Figure 75: Graphical representation of simulated parameters – waveguide length .................. 91
Figure 76: Waveguide width .......................................................................................................... 92
Figure 77: Graphical representation of simulated parameters – waveguide width ................... 93
Figure 78: Waveguide height ......................................................................................................... 94
Figure 79: Graphical representation of simulated parameters – waveguide height .................. 95
Figure 80: Aperture width .............................................................................................................. 96
Figure 81: Graphical representation of simulated parameters – aperture width ...................... 97
Figure 82: Aperture height ............................................................................................................. 98
xv
Figure 83: Graphical representation of simulated parameters – aperture height ..................... 99
Figure 84: Waveguide and practical antenna dimensions ......................................................... 102
Figure 85: Antenna parameters ................................................................................................... 102
Figure 86: Pyramidal horn antenna with SMA connector .......................................................... 103
Figure 87: Lumped port ................................................................................................................ 104
Figure 88: SMA connector ............................................................................................................ 104
Figure 89: Return Loss .................................................................................................................. 106
Figure 90: VSWR ........................................................................................................................... 106
Figure 91: Directivity .................................................................................................................... 107
Figure 92: Gain .............................................................................................................................. 108
Figure 93: Total radiated electric field ........................................................................................ 108
Figure 94: Electric field in E- and H-plane ................................................................................... 109
Figure 95: Electric field strength .................................................................................................. 110
Figure 96: Magnetic field strength .............................................................................................. 110
xvi
List of abbreviations
AC Alternating Current
AM Amplitude Modulation
dB decibel
dBi decibel (isotropic)
EHF Extremely High Frequency
ELF Extremely Low Frequency
EM Electromagnetic
FM Frequency Modulation
FNBW first-null beamwidth
GHz Gigahertz
GPS Global Positioning System
HEM Hybrid Electromagnetic mode
HF High Frequency
HPBW half-power beamwidth
Hz hertz
kHz kilohertz
LF Low frequency
MF Medium frequency
MHz Megahertz
mm millimeter
PEC Perfect Electric Conductor
PTFE Polytetrafluoroethylene
SHF Super high frequency
SLF Super low frequency
SMA SubMiniature version A
xvii
TE Transverse Electric mode
TEM Transverse Electromagnetic mode
THF Tremendously High Frequency
THz Terahertz
TM Transverse Magnetic mode
UHF Ultra-high frequency
ULF Ultra-low frequency
VHF Very high frequency
VLF Very low frequency
WLAN Wireless Local Area Network
WR Waveguide Rectangular
1
1. INTRODUCTION
Communications have become an unavoidable part of human life. In the past, it was
difficult to imagine that the information can be transferred with high reliability and low latency
from one end of the world to the other and in matter of seconds. Exchange of information is
realized by different communication paths in which a certain data rate of information can be
achieved in accordance with the frequency range and the standards specifically used by the
certain communication technology. Information can be transferred by means of different
technologies, such as optical fibers and copper cables but also through waveguides and
antennas, which is known as radiocommunication. Radiocommunication falls into the area of
electrical engineering, which uses radio waves as a media to transmit and receive information.
Radio waves are electromagnetic waves arranged into a frequency range from 3 kHz to 300 GHz.
Due to various properties and occurrences of the electromagnetic waves, frequency range of
radio waves has been divided into twelve areas. Since radiocommunication development is the
cause of widespread use in a variety of areas, it is necessary to allocate frequency ranges in
exact areas to avoid interference and thus allowing a smooth operation of certain services [1,
2].
Purpose of this master's thesis is to undergo with a mathematical calculation of the pyramidal
horn antenna and to determine its dimensions, perform the process of simulating the model of
the antenna, and generate the values of the antenna parameters. Furthermore, it is necessary
to alterate values of individual antenna dimensions in the given range, aiming to determine the
deviations in relation to the original values. Deviations are important for determining the exact
antenna dimension values at which the antenna still maintains its efficiency. The idea is to carry
out an antenna calculation for the 9 GHz frequency band, with gain value above 15 dB,
preferably. After the calculation is completed, an antenna simulation will be performed in which
the individual antennas parameters such as gain, directivity, standing wave ratio, etc. will be
generated through the simulation process. The antenna dimension values will be subjected to
2
changes, which will range from -10 mm to +10 mm, in steps of 1 mm, to see how the change of
the antenna's dimensions will affect the change of the parameters. All the results will be
displayed in table and graph forms. Further in the text, this will be considered as a reference
pyramidal horn antenna.
Mentioned reference model is needed to create an antenna (later on, it is referred to as a
practical antenna) that will be used for the purposes of calculating polarization losses within
controlled conditions, in the lecture room or laboratory. Namely, the goal is to calculate the
antenna for the same frequency band but with a lower gain as the reference antenna. Lower
gain and therefore smaller antenna dimensions are needed to physically create two antennas
in order to measure polarization losses. In addition, smaller dimensions are required, as the
distance between the antennas is less than one meter.
Prior to the calculation, simulation and retrieval of antenna parameters, chapters will be
discussed in this master’s thesis that are necessary to understand the practical part of the thesis.
In essence, the electromagnetic waves, their properties and phenomena, as well as the Maxwell
equations, which are necessary to represent and understand the nature of electromagnetic
waves will be explained in following chapters. Furthermore, the waveguide will be described as
a special type of narrow, electric conduit, which is required for electromagnetic waves to
propagate towards the antenna aperture. For the purpose of this thesis, a rectangular
waveguide is selected. In addition, the thesis will contain a chapter on antennas, their
parameters and types. At the end, a conclusion regarding calculation and simulation process of
the antenna, as well as representation of obtain values will be given.
3
2. ELECTROMAGNETIC WAVE
Electromagnetic radiation represents an electromagnetic wave that consists of electric
and magnetic fields, which propagate (radiate) through free-space. An electric charge
represents an electrically charged body in space. There is always an electrical field at the points
of space around the electrical charge. The electric charge acts on all surrounding electrically
charged bodies with an attracting or repulsive force, depending on the polarity of other charged
bodies. This action of electrically charged bodies causes movement, resulting in the
manifestation of a magnetic field. Essentially, electric field is the result of changing magnetic
field, and magnetic field is the result of changing electric field. The process of mutual production
of electric and magnetic fields results in the propagation of the electromagnetic waves through
the space at a speed equal to the speed of light [3 (pp. 590-643), 4, 5, 6 (pp. 1-14)].
Figure 1: Electric and magnetic field
4
Figure 1 depicts above-mentioned fields that are mutually perpendicular to each other, as well
as they are perpendicular to the direction of wave propagation. Therefore, electromagnetic
wave is a transverse wave. Fields are illustrated in three-dimensional Cartesian coordinate
system, where coordinates are denoted with letters 𝒙, 𝒚 and 𝒛.
Figure 2 and Figure 3 provide simplified presentation how electromagnetic radiation is being
produced. A perfect example is a half-wavelength dipole antenna, which has two conducting
rods, where each rod is 𝝀/𝟒 long. The rods are connected to the AC power source (denoted as
𝒗). The voltage generator induces an electric field (denoted as 𝑬), which causes force to the
electric charge. Due to free electrons in the conducting media, an electric current (denoted as
𝒊) is generated, and it begins to flow, from the point of higher potential to the point of lower
potential. As current flows, the electric charges are formed on the rods, where one side is
positive, and the other is negative, as shown in Figure 2 [3 (pp. 590-643), 4, 5, 6 (pp. 1-14)].
Figure 2: Manifestation of EM radiation
Electric field produces magnetic field (denoted as 𝑯) and its direction is determined by the right-
hand rule. Since the half-wave dipole is connected to the alternating voltage generator, at some
point the electric current will change direction, creating a shift in electric charges, i.e. the
positive side of the rod will become negative, and negative one will become positive. By
5
changing direction of the electric current, the direction of electric and magnetic field changes
too, as shown in Figure 3. The electromagnetic field consists of two vector fields, E and H, that
are interconnected [3 (pp. 590-643), 4, 5, 6 (pp. 1-14)].
Figure 3: Manifestation of EM radiation
2.1 Properties and occurrences of electromagnetic waves
Electromagnetic waves have four important properties:
1. Unlike other (mechanical) waves that use media to propagate, electromagnetic waves
are propagated by oscillation of electric and magnetic fields. In addition,
electromagnetic wave can travel through vacuum.
2. The direction of electric and magnetic fields within electromagnetic wave is
perpendicular to one another, and both of them are perpendicular to the direction of
wave propagation, making them transverse waves.
3. In electromagnetic wave, oscillating electric and magnetic fields are in phase.
6
4. The velocity of electromagnetic waves depends only on the electric and magnetic
properties of the media in which they propagate. Electromagnetic waves travel at the
speed of light – 𝒄 =1
√ 0∙𝜇0≈ 2.998 ∙ 108 [
𝑚
𝑠]
By propagating in free-space, electromagnetic waves can encounter on certain obstacles, which
can result in various wave phenomena, such as reflection, refraction, interference, and
diffraction.
Reflection is the wave occurrence, where propagating wave is reflected from the surface. The
amount of wave, that is reflected, depends on the composite characteristics of the material or
the surface from which the wave has been reflected. Figure 4 depicts a ray of light that falls on
the smooth surface (mirror) at the angle of incidence, denoted as 𝜽𝒊. The ray of light is reflected
from the surface at the angle of reflection, denoted as 𝜽𝒓. Figure 4 shows that the angle of
incidence and the angle of reflection are equal, thus meeting the requirements for realizing the
wave reflection. Normal represents the line, which divides mentioned angles into two equal
parts. Normal is perpendicular to the surface [3 (pp. 644-678), 7, 8].
Figure 4: Reflection
7
Refraction is one of the most important wave occurrence. The electromagnetic wave refracts
as it passes from one (optically less dense) media to another (optically denser) media, due to
velocity difference of the propagated wave in different media. Change of velocity results in
change of wave propagation. Refraction can be imagined as phenomenon, where ray of light
bends when it passes from one media to another – Figure 5. Snell’s law describes refraction
sin θ1
sin θ2=
𝑣1
𝑣2=
𝑛2
𝑛1 (2.1)
where:
θ1 – angle of incidence
θ2 − angle of refraction
𝑣1, 𝑣2 – wave velocities
𝑛1 − refraction index of optically less dense media
𝑛2 − refraction index of optically denser media
Figure 5: Refraction
8
In the example, where ray of light passes from optically denser media into an optically less dense
media, a total internal reflection occurs (Figure 6). More precisely, total internal reflection
occurs, when the angle of incidence is greater than the critical angle – 𝜽𝒄. The critical angle
represents the angle, where the ray of light is no longer refracted but totally reflected, due to
passing through a denser media to a less dense media. The formula for determining the critical
angle is expressed as follows
θc = sin−1 (𝑛2
𝑛1)
Figure 6 depicts the angles of incident waves. Angles that are smaller than the critical angle
cause refraction of the ray of light at the boundary of the two media. Furthermore, incident
wave angles, that are greater than the critical angle, result in total internal reflection, where
incident wave reflects from the boundary of the two media back to the optically denser media.
Finally, incident wave angle, equal to the critical angle, causes the wave travel along the
boundary of the two media [3 (pp. 644-678), 7, 8, 9, 10].
Figure 6: Total internal reflection
9
Diffraction is a physical phenomenon that occurs when wave turns its motion behind the edge
of the barrier which the waves encounter. If the waves encounter a barrier whose dimensions
are approximate to the length of the wave, diffraction will cause the interference in the waves.
In Figure 7, it is apparent that a greater obstacle opening results in less wave spread than a
smaller obstacle opening. Essentially, diffraction describes the wave behaviour after the wave
passes through the narrow opening or bypasses a certain obstacle [3 (pp- 679-712), 7, 8].
Figure 7: Diffraction
Interference is wave occurrence that describes an interaction between two or more waves,
which occurs at the same place and time. Interference is easily represented by the example of
periodic waves. The periodic wave is a wave, which repeats itself after a certain period of time.
Elements that describe periodic waves are frequency (𝒇) or wavelength (𝝀), period (𝑻), and
amplitude (𝑨). Figure 8 depicts two sine waves that are aligned precisely, meaning that they are
in phase. The sum of their amplitudes will result in a resultant wave, which has twice the
amplitude of the initial two sine waves. This type of interference is called constructive
interference. As seen in Figure 8, crests are aligned with crests, and throughs are aligned with
throughs [3 (pp. 679-712), 11, 12].
10
Figure 8: Constructive interference
Figure 9 depicts two sine waves that are out of phase or phase-shifted. This shift causes a crest
of one sine wave to coincide with the through of another sine wave. Thus, their amplitudes are
cancelled, and the resultant wave amplitude is equal to zero. This is an example of destructive
interference [3 (pp. 679-712), 11, 12].
Figure 9: Destructive interference
11
2.2 Electromagnetic spectrum
Electromagnetic radiation can be found on different frequencies and wavelengths. Frequency
or wavelength ranges are referred to as spectrum. Electromagnetic spectrum represents the
strength of electromagnetic radiation as the function of frequency or wavelength. Spectrum
contains different frequency ranges and applications in various areas. ITU is a United Nations
regulatory body that regulates, coordinates and monitors the use of frequency domains
globally. The ITU has defined twelve frequency ranges within the radio spectrum [13, 14].
Frequency range between 3 Hz and 3 kHz (ELF, SLF, and ULF) is used in underwater
communications and mining, due to long wavelengths that can penetrate under water or even
earth. Ground dipole antenna and various types of coils and ferrite loop antennas are used in
these frequency ranges [13, 14].
Various services use frequencies between 3 kHz and 3 MHz (VLF, LF, and MF), such as
radiolocation, government and military services, long distance communications, aviation, radio
amateurism and AM radio broadcasting. Large vertical monopole antennas are used in these
ranges [13, 14].
Between the 3 MHz and 3 GHz range HF, VHF, and UHF frequencies are placed. These
frequencies are used in radio amateurism, television broadcasting, aircraft and aviation
communications, WLAN, GPS, Bluetooth, FM radio broadcasting, satellite radio, and many
more. Most commonly used antennas are Yagi-Uda dipole antenna, reflector antenna, small
monopole and helix antennas, and array antenna systems, such as reflective antenna and
collinear antenna [13, 14].
The final frequency set consists of SHF, EHF, and THF frequencies, placed between 3 GHz and 3
THz frequency range. Astronomy, microwave communications, radar, satellites, point-to-point
12
communication and X-rays are some of the applications used within this range. This frequency
range, especially super high frequency range (3 GHz – 30 GHz) is important for this thesis,
because of the small wavelength, that allows microwaves to be precisely directed into narrow
beams by horn antenna. Horn antenna is a type of aperture antenna, which is used for
frequencies above 300 MHz. It is characterized by low loss, high gain, moderate directivity, and
is relatively easy to design and build. Horn antenna will be explained in detail in the following
chapters. Frequency and wavelength ranges are shown in Figure 2.10 [13, 14].
Figure 10: Electromagnetic spectrum
2.3 Maxwell’s equations
In the late 19th century, a Scottish scientist James Clerk Maxwell developed the theory of
electromagnetic fields. Previously established laws, such as Ampère’s and Gauss’s law as well as
Faraday’s law of induction, guided him. With all gathered knowledge, he set four main equations
that describe the unified theory of the electric and magnetic fields on charges and currents, as
well as their interaction, which occurs when the fields change in time. According to those
equations, changes in the electric field cause changes in the magnetic field and vice versa [3
(pp. 623-645), 15].
13
2.3.1 First Maxwell’s equation
First Maxwell’s equation (2.2) is interpreted as Gauss’s law for electric fields. Left side of
expression describes electric field lines as open curve. These electric field lines begin on the
positive electric charge, and terminate on the negative electric charge. The right side of the
expression states that electric flux through any closed surface is equal to the sum of all the
electric charges contained within closed surface. Essentially, electric flux through a closed
surface that does not contain any electric charge is equal to zero → the source of an electric
field is an electric charge [3 (pp. 623-645), 6 (p. 193), 15 (pp. 1-38)].
∮ �⃗⃗� ∙ 𝑑𝑺 =∑𝑸𝒕𝒐𝒕𝒂𝒍
𝜺 (2.2)
where:
�⃗� – electric field strength [V
m]
𝑑 − differential
𝑆 – closed surface [m2]
𝑄𝑡𝑜𝑡𝑎𝑙 – sum of electric charges within closed surface [C]
휀 – dielectric permittivity; ε = εr ∙ ε0 = 8.854 ∙ 10−12 [F
m], εr = 1 (free-space)
2.3.2 Second Maxwell’s equation
According to this law, there are no magnetic monopoles existing in nature → magnetic field has
a source, but there are no magnetic charges. In this case, a magnetic field lines are closed, which
means that magnetic flux through a closed surface is equal to zero. There is a link between the
first and second Maxwell equation (2.3), regarding the (integral) form of equation, but they
differ in content and interpretation [3 (pp. 623-645), 6 (p. 193), 15 (pp. 43-55)].
14
∮ �⃗⃗� ∙ 𝑑𝑺 = 0 (2.3)
where:
�⃗� − magnetic field strength [A
m]
𝑆 – closed surface [m2]
2.3.3 Third Maxwell’s equation
Third Maxwell’s equation (2.4) states that any change in electric field creates a change in the
magnetic field. That was explained well in Faraday’s law of induction, which states that the line
integral of the electric field along the closed surface is equal to negative value of the magnetic
flux through the closed surface. A negative sign indicates that induced electric field opposes the
change of the magnetic flux [3 (pp. 623-645), 6 (p. 193), 15 (pp. 58-80)].
∮ �⃗⃗� ∙ 𝑑𝒍 = −𝑑
𝑑𝑡∙ ∫ �⃗⃗� ∙ 𝑑𝑺 (2.4)
where:
�⃗� – electric field strength [V
m]
𝑑𝑙 − line integral of electric field
𝑆 – closed surface [m2]
𝑑 − differential
𝑡 − time [s]
�⃗� − magnetic field strength [A
m]
15
2.3.4 Fourth Maxwell’s equation
Fourth Maxwell’s equation (2.5) is explained by Ampère's law. Ampère's law is a physical law
that states: magnetic field is generated because of electric charge movement, thus connecting
electric and magnetic occurrences. From the integral standpoint, it states that the line integral
of magnetic field along closed loop is proportional to the total electric current, penetrating open
surface enclosed by the loop. Essentially, the source of magnetic field is the electric current.
This equation is expressed as [3 (pp. 623-645), 6 (p. 193), 15 (pp. 83-108)]
∮ �⃗⃗� ∙ 𝑑𝒍 = 𝜇 ∙ ( 𝑱 + 휀 ∙𝑑
𝑑𝑡∙ ∫ �⃗⃗� ∙ 𝑑𝑺 ) (2.5)
where:
�⃗� − magnetic field strength [A
m]
𝑑𝑙 − line integral of magnetic field
𝑆 – open surface [m2]
𝜇 − magnetic permeability; μ = μr ∙ μ0 = 4π ∙ 10−7 [H
m], μr = 1 (free-space)
𝐽 − electric current density [A
m2]
휀 – dielectric permittivity; ε = εr ∙ ε0 = 8.854 ∙ 10−12 [F
m], εr = 1 (free-space)
𝑑 − differential
�⃗� – electric field strength [V
m]
16
3. WAVEGUIDES
Waveguide is constructed as hollow, metal tube usually made of copper, aluminum,
brass or even silver (Figure 11) that radiates electromagnetic waves from its conductive walls
toward the antenna aperture and ultimately into the free-space. Electromagnetic waves
propagate in a direction, which is defined by waveguide’s physical boundaries. There are
different types of waveguides based on their cross-section, such as elliptical, circular and
rectangular. Waveguide interior can be filled with dielectric, most commonly with air. Traveling
waves within a waveguide are transmitted by total internal reflection, where incident wave
impacts on a media boundary (inner walls of the waveguide) at an angle that is larger than a
critical angle (see chapter 2.1).
Figure 11: Waveguide
Since the waveguide is a type of transmission media with small losses compared to the coaxial
cable, it is ideal for wave propagation at frequencies above 3 GHz. Waveguides show lower
attenuation properties and are more capable for high power energy transmission than coaxial
cable. Waveguide uses small coupling elements, such as stubs (probes) or loops, to make the
waves easier to insert into, and extract from the waveguide. Essentially, stubs or loops are used
to generate waves into waveguide. These coupling elements come in form of a dipole (usually
half-wave dipole) antenna, with 𝝀/𝟒 in physical length for stub coupling or as short wire loop
for a loop coupling, that are embedded inside of the waveguide – Figure 12.
17
Figure 12: Waveguide coupling
Probes or loops that are embedded inside the waveguide, generate waves into it. Waves within
waveguide travel in zigzag pattern, and are successively reflected between the waveguide walls,
due to the total reflection (depicted in Figure 13). As waves travel, they hit waveguide wall at
some angle. The wave paths of these angles are larger at higher frequencies – Figure 13 a). As
the operating frequency decreases, the path between the waves becomes shorter – Figure 13
b) and 13 c). At cut-off frequency, waves “jump” up and down between the walls of the
waveguide, thus preventing the movement of energy forward – Figure 13 d) [18, 19].
Figure 13: Wave paths in the waveguide – top view
18
3.1 Rectangular waveguides
Rectangular waveguide has rectangular cross-section, which is defined by sides 𝒂 and 𝒃, where
𝒂 represents width, and 𝒃 represents height of the waveguide, where 𝑎 > 𝑏. In addition, a
hollow space within the waveguide is filled with electric permittivity, denoted as 𝜺 and magnetic
permeability, denoted as 𝝁 (Figure 14). In Cartesian coordinate system on Figure 14, there are
three dimensions: 𝒙 represents a waveguide width (denoted with letter 𝒂), 𝒚 represents a
waveguide height (denoted with 𝒃) and 𝒛 represents a direction of wave propagation.
Figure 14: Geometry of rectangular waveguide in Cartesian coordinate system
Dimensions 𝒂 and 𝒃 are used to determine the range of operating frequency of the waveguide.
WR is an abbreviation for Waveguide Rectangular, i.e. a type of standard that determines
waveguide dimensions, operating frequency range and cut-off frequency for upper and lower
operating mode. For this thesis, a WR-112 (Figure 15) has been chosen as waveguide, on which
pyramidal horn will be mounted. WR-112 waveguide is designed for frequency range between
7.05 GHz and 10 GHz, with dimensions for side 𝒂 = 28.4988 mm and for side 𝒃 = 12.6238 mm
[19, 20, 21].
19
Figure 15: WR-112 waveguide
3.2 Transverse modes
The propagation modes and the operating wavelength supported in the waveguide, depend on
its dimensions. Generally, the waveguide works best only when one mode, the so-called
dominant mode, is present. Each mode shows some special propagation properties, such as
wave (information, signal) attenuation, phase shift of electromagnetic waves, and propagation
speed. When the radiated energy of electromagnetic waves is propagated in multiple modes at
the same time, the difference in propagation speed occurs, which leads to wave distortion. In
order to eliminate this unwanted multimode effect, the cross-sectional dimensions of the
waveguide must be selected in such a way that only one mode can be transmitted at given
wavelength (frequency).
Mentioned modes are called transverse modes, and they can be divided into several groups:
transverse electric – TE, transverse magnetic – TM, transverse electromagnetic – TEM and
hybrid – HEM. In TE mode (Figure 16), there is a magnetic field component in the direction of
propagation, but not electric, since it is perpendicular (transverse) to the direction of wave
propagation (𝑬𝒛 = 0,𝑯𝒛 ≠ 0). Next, TM mode (Figure 16) contains a component of the electric
20
field in the direction of propagation (𝑬𝒛 ≠ 0), which means that magnetic component is
perpendicular to the wave propagation (𝑯𝒛 = 0). In TEM mode, both magnetic and electric field
components coincides with the direction of wave propagation, (𝑬𝒛 = 𝑯𝒛 = 0). Since in TEM
mode there are no field components in the direction of propagation, this mode is not used in
waveguide structure. TEM mode occurs in transmission lines at lower frequencies, such as
coaxial cable or parallel wire line. Lastly, hybrid mode (𝑬𝒛 ≠ 𝑯𝒛 ≠ 0) is the combination of TE
and TM modes, because in hybrid mode there are electric and magnetic field components in
the direction of the wave propagation.
Figure 16 describes electric and magnetic field lines within the waveguide. Electric field lines are
shown in blue, while magnetic field lines are shown in red colour. In TE mode, electric field is
distributed along wider side (dimension 𝒂, in 𝒙-direction) of the waveguide, in sine form, while
magnetic field is shown as uniformed loops along narrower side (dimension 𝒃, in 𝒚-direction)
of the waveguide. Wave propagation is in the 𝒛-direction, marked in green colour. Both fields
are perpendicular to each other, and while the waves propagate through the waveguide, due
to the change in the electric field, there is a change in the magnetic field and vice versa [16, 17,
19, 22].
Figure 16: Comparison between TE and TM modes in rectangular waveguide
21
3.3 Dominant mode TE10. Cut-off frequency of the WR-112
TE mode stands for transverse electric mode, meaning that there is no electric component 𝑬𝒛 in
the direction of the propagation, only magnetic – 𝑯𝒛. In rectangular waveguide design, the
dominant mode appears, when the condition 𝒂 > 𝒃 is met. Dominant mode is the mode with
the lowest cut-off frequency – 𝒇𝒄,𝒎𝒏 that can be calculated by incorporating the values of
waveguide dimensions (width and height) into the formula for cut-off frequency for any TE
mode
𝑓𝑐,𝑚𝑛 =1
2𝜋 ∙ √𝜇휀 ∙ √(
𝑚 ∙ 𝜋
𝑎)2
+ (𝑛 ∙ 𝜋
𝑏)2
[Hz] (3.1)
𝑓𝑐,𝑚𝑛 =𝑐
2𝜋 ∙ √(
𝑚 ∙ 𝜋
𝑎)2
+ (𝑛 ∙ 𝜋
𝑏)2
[Hz] (3.2)
moreover, for TE10 mode, a cut-off frequency is expressed by
𝑓𝑐,10 =𝑐
2𝑎 [Hz] (3.3)
or TE mode can be calculated by cut-off wavelength 𝜆𝑐,𝑚𝑛
𝜆𝑐,𝑚𝑛 =2
√(𝑚𝑎)
2
+ (𝑛𝑏)2 [m]
(3.4)
where:
휀 – dielectric permittivity; ε = εr ∙ ε0 = 8.854 ∙ 10−12 [F
m], εr = 1 (free-space)
22
𝜇 − magnetic permeability; μ = μr ∙ μ0 = 4π ∙ 10−7 [H
m], μr = 1 (free-space)
𝑚 − number of half-wavelength variations of the fields along dimension 𝒂
𝑎 − waveguide width, wider dimension [m]
𝑛 − number of half-wavelength variations of the fields along dimension 𝒃
𝑏 − waveguide height, narrower dimension [m]
𝑐 − speed of light; c ≈ 2, 998 ∙ 108 [m
s]
In rectangular waveguide, the TE10 is the dominant mode. This mode appears at lower cut-off
frequency; meaning that the waveguide will only propagate electromagnetic waves beyond the
cut-off frequency value. In addition, there is an upper cut-off frequency limit, where there is a
possibility of appearing of higher modes if this limit is reached [16, 20, 21, 22, 23 (pp. 237-248)].
To put things in perspective, the following example will serve to mathematically present the
value for cut-off frequency of TE10 mode for WR-112 waveguide. The WR-112 waveguide has
the following values of its dimensions (Figure 17): 𝒂 = 28.4988 mm, and 𝒃 = 12.6238 mm. Taking
into consideration that it is a TE10 mode, its index numbers (TEmn) indicate that 𝒎 = 1, and 𝒏 =
0, which means that the part of the expression containing n in the expression equals zero. By
using expression (3.2), the cut-off frequency of TE10 mode for WR-112 waveguide is
𝑓𝑐,10 =2.998 ∙ 108
2𝜋∙ √(
𝜋
0.0284988)2
= 5.25987 ∙ 109 [Hz]
and its cut-off wavelength is calculated by using (3.4)
𝜆𝑐,10 =2
√(1
0.0284988)2=
𝑐
𝑓𝑐,10= 0,0569976 [m]
23
Figure 17: Waveguide WR-112 dimension specifications
At 5.26 GHz, a WR-112 waveguide will start to propagate electromagnetic waves in TE10 mode,
the most commonly occurring mode in rectangular waveguides. In order to avoid the
occurrence of higher modes, the boundaries of cut-off frequency for a given waveguide should
be taken into consideration. Cut-off frequencies of TE01 and TE20 modes for WR-112 are given
by following expressions
𝑓𝑐,01 =𝑐
2𝑏=
2.998 ∙ 108
2 ∙ 0.0126238= 11.8744 ∙ 109 [Hz]
𝑓𝑐,20 =𝑐
𝑎=
2.998 ∙ 108
0.0284988= 10.5197 ∙ 109 [Hz] ≈ 2 ∙ 𝑓𝑐,10
which means that at 10.52 GHz and 11.87 GHz, higher modes than TE10 will appear, which will
create additional losses and attenuation of dominant mode TE10. Considering that, higher
modes will cause unwanted occurrences in the normal, working waveguide mode, it is
preferable to respect the cut-off frequencies within which the single mode can propagate [16,
17, 19, 20, 21, 22].
24
Another parameter important for wave propagation in the waveguide is guide wavelength,
denoted as 𝝀𝒈. Guide wavelength is actually a wavelength in the direction of propagation, and
it defines a distance between two in-phase points in the direction of propagation in the
waveguide (𝒛-direction). Mathematically, it is defined by cut-off frequency (wavelength) and
operating frequency – 𝒇. Guide wavelength is defined as
𝜆𝑔 =𝜆
√1 − (𝑓𝑐𝑓)2
=𝜆
√1 − (𝜆𝜆𝑐
)2
[m] (3.5)
where:
𝜆 − operating wavelength; λ =c
f [m]
𝑓𝑐 − cut-off frequency [Hz]
𝑓 − operating frequency; f =c
λ [Hz]
𝜆𝑐 − cut-off wavelength [m]
In conclusion, the waveguide transmission is only possible if the wave frequency is higher than
cut-off frequency for given operating mode. For all frequencies lower than the cut-off
frequency, the wave is attenuated exponentially, and the energy of the electromagnetic wave
would not be transmitted along the waveguide. Thus, there is an attenuation in the waveguide
that can be mathematically represented by the expression (3.6) [23 (pp. 249-260)]
𝛼 = 𝜔 ∙ √𝜇 ∙ 휀 ∙ √1 − (𝑓𝑐,𝑚𝑛
𝑓)2
(3.6)
where:
𝜔 − angular frequency; ω = 2π ∙ f [rad
s]
25
𝑓𝑐,𝑚𝑛 − cut-off frequency; expression (3.1)
𝑓 − operating frequency; f =c
λ [Hz]
휀 – dielectric permittivity; ε = εr ∙ ε0 = 8.854 ∙ 10−12 [F
m], εr = 1 (free-space)
𝜇 − magnetic permeability; μ = μr ∙ μ0 = 4π ∙ 10−7 [H
m], μr = 1 (free-space)
3.4 Electric and magnetic fields in the waveguide. Maxwell’s equations
Electric and magnetic fields, as well as Maxwell’s equations (2.2 – 2.5) were already introduced
in this thesis, while this chapter describes the field components at the aperture of the
waveguide, and explains the boundary conditions required for wave propagation within the
waveguide. Additionally, the connection between Maxwell’s equations describing
electromagnetic waves behaviour within the waveguide will be described as well.
Time-varying electric field is the direct cause of the induction of the magnetic field and vice
versa. Therefore, it is reasonable to take Maxwell’s equations into consideration in order to
describe the behaviour of the electric and magnetic fields within the waveguide, especially the
equation (2.4) described by Faraday's law of induction and equation (2.5) described by the
Ampère's law. In Figure 18, field lines of electric and magnetic fields are depicted from different
views. On cross-section, electric field lines (marked with red colour) of TE10 mode vary
sinusoidally along wider waveguide side 𝒂, while magnetic field lines (marked with blue colour)
are distributed uniformly along narrower waveguide side 𝒃, and they are perpendicular to them.
A half-wavelength variation of the electric field component (sine form), which is actually an
illustration of the electric field strength, is also shown. In addition, concentric loops of the
magnetic field can be seen from a top view of the waveguide. Lastly, side view shows, how
electric field strength rises from value zero, reaches its maximum and then falls (again) to the
value zero, and thus further along the waveguide [17].
26
Figure 18: Electric and magnetic field lines within a waveguide
In chapter 3.1 in Figure 14, the waveguide is represented in three-dimensional coordinate
system, consisting of three coordinates: 𝒙, 𝒚 and 𝒛. Considering the transmission structures,
the coordinate system is arranged so that the 𝒛-axis coincides with the longitudinal direction in
which the wave propagates, while the remaining two coordinates are in the transverse plane,
relative to the longitudinal direction. Since there are three axes in coordinate system, and two
fields that travel within a waveguide, there are total of six field components. Therefore,
magnetic field components for TE10 mode can be represented by the following expressions [17]
𝐻𝑥 = 𝑗𝛽𝑧 ∙𝑚 ∙ 𝜋
𝑎∙ (
𝜆𝑐
2𝜋)2
∙ 𝐻0 𝑠𝑖𝑛 (𝑚 ∙ 𝜋
𝑎𝑥) ∙ 𝑐𝑜𝑠 (
𝑛 ∙ 𝜋
𝑏𝑦) ∙ 𝑒−𝑗𝛽𝑧 ∙ 𝑧 (3.7)
𝐻𝑦 = 𝑗𝛽𝑧 ∙𝑛 ∙ 𝜋
𝑏∙ (
𝜆𝑐
2𝜋)2
∙ 𝐻0 𝑐𝑜𝑠 (𝑚 ∙ 𝜋
𝑎𝑥) ∙ 𝑠𝑖𝑛 (
𝑛 ∙ 𝜋
𝑏𝑦) ∙ 𝑒−𝑗𝛽𝑧 ∙ 𝑧 (3.8)
𝐻𝑧 = 𝐻0 ∙ 𝑐𝑜𝑠 (𝑚𝜋
𝑎𝑥) ∙ 𝑐𝑜𝑠 (
𝑛𝜋
𝑏𝑦) ∙ 𝑒−𝑗𝛽𝑧 ∙ 𝑧 (3.9)
while electric field components for TE10 mode are as follows
27
𝐸𝑥 = 𝜂𝑇𝐸 ∙ 𝐻𝑦 (3.10)
𝐸𝑦 = − 𝜂𝑇𝐸 ∙ 𝐻𝑥 (3.11)
𝐸𝑧 = 0 (3.12)
where:
𝛽𝑧 − propagation constant in z-direction; 𝛽𝑧 =2𝜋
𝜆 ∙ √1 − (
𝜆
𝜆𝑐)2
𝜆𝑐 − cut-off wavelength [m]
𝑚, 𝑎, 𝑛, 𝑏 → see the description of the elements after the expression (3.4)
𝜂𝑇𝐸 =𝜔∙𝜇
𝛽𝑧= 𝜂0 ∙ [1 − (
𝜆
𝜆𝑐)2
]−
1
2
− characteristic wave impedance of TE mode [Ω]
𝐻0 − amplitude constant
Since the waveguide operates in TE10 mode, the expressions (3.7 – 3.11) will be simplified,
because the element 𝒏 (TEm0) equals zero. In addition, electric field component in 𝒛-direction 𝐸𝑧
equals zero. Field components 𝑬𝒙, 𝑬𝒚, 𝑯𝒙 and 𝑯𝒚 represent transverse, while 𝑬𝒛 and 𝑯𝒛
represent longitudinal components of electromagnetic field. Each expression (3.7 – 3.12) can
be represented by one longitudinal and two transverse components. Field components must
meet boundary conditions. Boundary conditions set or restrict the boundaries of the electric
and magnetic fields within the waveguide. Boundary conditions depend on the geometry of the
waveguide itself, more precisely on its dimensions. Thus, for example, the components of the
electric field 𝑬𝒚 and 𝑬𝒛 = 0 at the value 𝒙 = 0 and 𝒙 = 𝑎, where 𝒂 is the value of the
waveguide width, while 𝑬𝒙 and 𝑬𝒛 = 0 at the values 𝒚 = 0 and 𝒚 = 𝒃, where 𝒃 is the value
of the waveguide height. Alongside waveguide geometry, conditions that should also be
respected are the wave number 𝒌 (or phase coefficient), and the cut-off wave number, 𝒌𝒄. The
wave number can be described as the number of wavelengths within the whole cycle.
28
Both expression can be written as follows [16, 23 (pp. 249-260)].
𝛽 ≡ 𝑘 = 𝜔 ∙ √𝜇 ∙ 휀 =𝜔
𝑐=
2𝜋 ∙ 𝑓
𝜆 ∙ 𝑓=
2𝜋
𝜆 [rad
m] (3.13)
where:
𝛽 − wave number, phase coefficient
𝑘𝑥 − cut-off wave number component (x-direction); 𝑘𝑥 =𝑚∙𝜋
𝑎
𝑘𝑦 − cut-off wave number component (y-direction); 𝑘𝑦 =𝑛∙𝜋
𝑏
𝑘𝑐 = √𝑘𝑥2 + 𝑘𝑦
2 → 𝑘𝑐 = 𝑘𝑥 (for TE10 mode)
29
4. ANTENNAS
Antenna (aerial) is a media or device for transmitting and receiving radio waves; more
precisely, the antenna is the mediator between waveguide and free-space. It efficiently converts
the energy of the electromagnetic wave through the transmission line, into the energy that
propagates in free-space. Antenna can act as transmitter or receiver, depending on whether it
sends or receives data. The transmitting antenna (Figure 19) converts the electromagnetic wave
from the transmission line into the electromagnetic wave in the free-space. The receiving
antenna (Figure 20) converts the electromagnetic wave from the free-space into the wave in
the transmission line. The basic function of an antenna is to adjust the wave from the free-space
with the wave within the transmission line (and vice versa), but also to direct the radiated
energy into certain parts of the space [26 (pp. 1-18), 27].
Figure 19: Transmitting antenna
Figure 19 shows a transmitting antenna, consisting of a transmitter that generates voltage 𝑽𝑨
that creates distribution of alternating current 𝑰𝑨, a transmission line (in this case, a waveguide)
30
and the antenna aperture that transmits an electromagnetic field from the waveguide into a
free-space → three-dimensional space. At the boundary between the waveguide and the
antenna, electromagnetic field lines resemble the plane wave. As the waves approach the
antenna aperture, they take shape of the concentric circles. The more they move away from
the source and the antenna itself, they are re-shaped into plane waves again [24, 25, 27].
Figure 20: Receiving antenna
In Figure 20, the opposite process takes place, where electromagnetic waves in the free-space,
which are transmitted from the transmitting antenna, spread to the receiving antenna through
the waveguide and are delivered to the receiver [24, 25, 27].
4.1 Antenna parameters
In order to describe the characteristics and performance of the antenna, it is necessary to
describe its parameters. Antenna parameters are defined in free-space, and depend on the
position of the antenna according to the Earth and surrounding objects. Some of the standard
antenna parameters include radiation pattern (diagram), antenna gain, polarization, directivity,
31
field regions, antenna efficiency, radiation intensity and power density. Some of the mentioned
parameters will be explained in this chapter [6 (pp. 301-307), 26 (pp. 1-79)].
4.1.1 Radiation pattern. Beamwidth
Antenna radiation pattern (Figure 21) represents the mathematical or graphical display of
electromagnetic radiation that characterizes the antenna, in function of three-dimensional
spatial coordinates. Essentially, the radiation diagram provides the necessary information for
the spatial distribution of electromagnetic radiation around the antenna. The radiation pattern
is most commonly referred to the distant radiation (far-field) zone where electromagnetic
radiation is represented in the form of plane waves. There are two basic types of radiation
pattern – a power diagram represents the spatial distribution of normalized radiation power,
and a field diagram represents the normalized amplitude of electric or magnetic fields. The
radiation diagram is represented in a logarithmic (decibel) or linear scale in values, relative to
the maximum radiation [6 (pp. 301-307), 26 (pp. 25-33)]
Antenna radiation diagram has two planes in the Cartesian coordinate system, which are
relevant for its presentation: horizontal and vertical plane. The angle in the horizontal plane is
the angle of azimuth, and the angle in the vertical plane is the angle of elevation. The actual
illustration of the radiation diagram is actually 3D, shown in Figure 31. In most communication
applications during transmission, all electromagnetic energy is required to be directed in the
chosen direction.
Furthermore, antenna has only one major (main) lobe and a larger number of minor (secondary)
lobes. The major lobe is defined as the lobe in the direction of maximum radiation. The minor
lobe is defined as the lobe whose direction of radiation is different from the one in which the
main lobe is directed. In a properly constructed antenna system, the minor lobe levels are
considerably lower than the major lobe level [26] (pp. 25-33).
32
Figure 21: Antenna radiation pattern
The radiation pattern (Figure 22) consists of the following parameters:
1. Direction of maximum radiation [𝑬𝒎𝒂𝒙, (𝜣𝟎, 𝝓𝟎)] – the direction in which the radiated
field has the maximum value; major lobe direction.
2. Angle of directivity (𝜣𝑫) – the angle around the direction of the maximum radiation
within which the radiated power density does not fall below half of the maximum
radiated power; HPBW → 0.707 for linear field value, 0.5 for linear power value and -3
dB for logarithmic power value.
3. Beamwidth (𝜣𝒏) – the angle between the first null points on both sides of the radiation
pattern maximum; FNBW.
4. Suppression factor of minor lobes (𝒔) – the field strength ratio between the direction of
maximum radiation and the direction of the “largest” minor lobe.
33
Figure 22: Radiation patterns
Beamwidth is one of the parameters of the radiation pattern. It indicates the angular distance
between two identical points, which are on the opposite sides of the radiation pattern
maximum. Two types of beamwidths are also mentioned, namely FNBW and HPBW. The
relationship between these two beamwidths is given by the expression 𝐻𝑃𝐵𝑊 ≈ 𝐹𝑁𝐵𝑊
2.
4.1.2 Field regions
The area around antenna consists of three field regions (Figure 23). These are the fields of
electromagnetic radiation surrounding the antenna. Given the proximity of the field, the regions
34
are divided into reactive near-field, radiating near-field (known as the Fresnel region) and a far-
field, or Fraunhofer's region. These field regions differ in the intensity of the radiated power,
since it is known that the energy decreases with the square of the distance.
Figure 23: Field regions and field distribution
The reactive near-field region surrounds the area around the antenna, where the reactive field
prevails. Although the reactive region is near the antenna, a boundary still defines the transition
between the regions. The boundary is defined as the distance from the center of the circle that
encloses a reactive near-field, and is represented by a mathematical expression
𝑅1 = 0.62 ∙ √𝐷3
𝜆 (4.1)
where:
𝑅1 − distance from the center where the antenna is located [m]
𝐷 − largest antenna dimension [m]
𝜆 − wavelength [m]
35
The next area is the radiating near-field, better known as Fresnel’s region. This region is located
between reactive and far-field region. In radiating region, the field distribution depends on the
distance from the antenna and its dimension. The boundary of the radiation near-field is
represented by a mathematical expression
𝑅2 =2 ∙ 𝐷2
𝜆 (4.2)
where:
𝑅2 − distance from the center where the antenna is located [m]
𝐷 − largest antenna dimension [m]
𝜆 − wavelength [m]
Both fields have something in common: if the antenna dimension 𝑫 is smaller than its
wavelength 𝝀 , the field regions around antenna may not exist.
Lastly, the far-field is located far from the antenna. In this region, the EM field behaves as a
plane wave, meaning that changes in the electric and magnetic fields have a uniform
distribution in the plane, which is perpendicular to the direction of propagation. If the maximum
antenna dimension – 𝑫 is considerably greater than the wavelength 𝝀 , then it can be assumed
that the distant zone begins with the distance 𝟐∙𝑫𝟐
𝝀 from the antenna. This region is called a
Fraunhofer's region [26] (pp. 25-33).
4.1.3 Polarization
From Chapter 4 in Figure 20, the fact that the electromagnetic wave in the far-field acts as a
plane wave is taken into consideration. At that distance, the electric and magnetic field vectors
are perpendicular to the direction of wave propagation, and they change their direction and
36
magnitude in time. The curve described by the top of the electric field vector is defined as
polarization (Figure 24). Polarization of EM wave defines the orientation, where maximum
power is achieved. Polarization is defined by the following parameters: axial ratio (AR), which is
the ratio of large and small ellipse axis, direction in which the electric field vector rotates;
clockwise – right and counterclockwise – left, and the direction of large axis, in relation to the
reference coordinate system.
Figure 24: Types of polarization
A polarization ellipse shape is defined by ratio between major axis 𝑶𝑨 and minor axis 𝑶𝑩
represents polarization. Knowing these parameters, one can define the axial ratio – 𝑨𝑹, whose
values are in the interval between 1 and +∞.
37
𝐴𝑅 = |𝑂𝐴
𝑂𝐵| ; [1 ≤ 𝐴𝑅 ≤ ∞] (4.3)
Figure 24 depicts the following polarizations: linear (horizontal and vertical), circular (right and
left) and elliptical (right and left). In the case of linear polarization, the amplitude of the electric
field vector is along the direction of propagation. If 𝑶𝑨 = 0, then linear polarization is
horizontal, while vertical if 𝑶𝑩 = 0. If major and minor axis are equal, then a circular
polarization occurs. The position of an electric field vector in the clockwise or counterclockwise
direction, determines the right or left circular polarization [26 (pp. 66-75), 28 (pp. 94-98)].
4.1.4 Power density
The power density parameter is closely related to the radiation pattern (see 4.1.1) and can be
described by Poynting's vector (Figure 25), which represents direction and magnitude of energy
flux of an electromagnetic field. In mathematical terms, Poynting's vector (4.4) represents a
cross product of the electric and magnetic field strengths
𝑆 = �⃗� × �⃗⃗� (4.4)
where:
𝑆 − Poynting’s vector [W
m2]
�⃗� – electric field strength [V
m]
�⃗⃗� − magnetic field strength [A
m]
Poynting’s vector direction coincides with the direction of the electromagnetic wave
propagation, while the electric and magnetic fields are perpendicular to each other (angle
between E and H is 90°; sin 𝜃 = 90° = 1).
38
Figure 25: Graphical representation of Poynting's vector
The electric and magnetic fields are seen as sine waves, and consequently, the Poynting vector
has the same shape. In the sinusoidal waveform, its average value is calculated. Thus, the
average value of the Poynting vector can be expressed as [6 (p.312), 26 (pp. 35-37)]
< 𝑆 > = |𝐸0|
2
2 ∙ 𝜇 ∙ 𝑐 (4.5)
where:
𝐸0 − electric field strength [V
m]
𝜇 − magnetic permeability; μ = μrμ0 = 4π ∙ 10−7 [H
m] , μr = 1 (air)
𝑐 − speed of light; c ≈ 2, 998 ∙ 108 [m
s]
39
4.1.5 Directivity and antenna gain. Radiation intensity
The parameter, indicating how much antenna radiation is directed in the given direction, can
be expressed numerically by the size called directivity. The directivity (denoted as 𝑫) is defined
as the ratio between the radiation intensity in the observed direction from the antenna and the
average radiation intensity that would be radiated by the isotropic radiator (see chapter 4.2).
More precisely, directivity describes how many times the radiated power of the isotropic
radiator must be greater than the radiated power of the observed antenna, so that at the same
distance the power density from the isotropic radiator would be equal to the power density of
the observed antenna. In mathematical terms, directivity can be written as [6 (pp. 325-336), 26
(pp. 41-54)]
𝐷 =𝑈(𝜃, 𝜙)
𝑈0=
4𝜋 ∙ 𝑈(𝜃, 𝜙)
𝑃𝑟𝑎𝑑 (4.6)
where:
𝑈(𝜃, 𝜙) − radiation intensity in the far-field [W/steradian]
𝑈0 − radiation intensity of isotropic radiator [W/steradian]
𝑃𝑟𝑎𝑑 − total radiated power [W]
The radiation intensity is defined as the power radiated by the antenna per unit solid angle
(steradian). Radiation intensity is the parameter of the far-field (see 4.1.2) and can be
mathematically written as [26] (pp. 37-39)
𝑈(𝜃, 𝜙) =𝑟2 ∙ |𝐸(𝜃, 𝜙)|2
2𝜂 (4.7)
where:
𝑟 − radius (distance) from the antenna
40
𝐸(𝜃, 𝜙) − electric field intensity in the far-field
𝜂 − impedance of free-space; η ≈ √𝜇
≈ 377 [Ω]
The gain of the antenna is closely related to directivity. Namely, the gain is an antenna
parameter that states how many times the total radiated power of the isotropic radiator should
be greater than the input power of the observed antenna, in order to achieve the same field
strength or equivalent power density at the same distance, and in the given direction. When
determining the gain, the spatial distribution of the radiant power and the loss of power due to
dissipation in the antenna are considered. The term describing the relationship between the
gain of the antenna and its direction is given by the following expression
𝐺 = 𝑘 ∙ 𝐷 (4.8)
where:
𝐺 − antenna gain [dimensionless]
𝑘 − efficiency factor; 𝑘 =𝑃𝑟𝑎𝑑
𝑃𝑖𝑛 [%, dimensionless]
𝐷 − antenna directivity [dimensionless]
Usually, the gain is expressed in decibels (dB) instead of being a dimensionless size. Gain in the
logarithmic scale can be calculated by the following expression
𝐺 = 10 ∙ log10(𝑘 ∙ 𝐷) [dBi] (4.9)
Expression (4.8) describes a simplified form of antenna gain, because the reference is an
isotropic radiator that is without losses. Namely, one has to bear in mind that the real world is
encountered with various losses and they need to be taken into consideration while calculating
the gain. A more complex expression for the gain is given by the expression (4.10), which is
41
described by the ratio of the radiation intensity and the total input power of the lossless
isotropic antenna [6 (pp. 370-377), 26 (pp. 61-64)]
𝐺 = 4𝜋 ∙𝑈(𝜃, 𝜙)
𝑃𝑖𝑛 (4.10)
where:
𝑈(𝜃, 𝜙) − radiation intensity in the far-field [W/steradian]
𝑃𝑖𝑛 − total input power of the lossless isotropic antenna
4.2 Types of antennas
Antennas can be classified by design, size, frequency band, shape and application in real life.
Considering the shape, the antennas can be divided into following groups: wire antennas,
aperture antennas, array antennas, microstrip (patch) antennas, and lens antennas. A brief
introduction and description of the above antennas will be given in this chapter.
A special type of the antenna is an isotropic radiator; it is an imaginary antenna that radiates
energy in all directions equally. It is used as a reference antenna by which the real antennas are
compared, and it serves as a reference for calculating antenna gain. The isotropic antenna is
presented as a point charge, in which the radiated power is uniformly distributed over the
sphere surface [13, 14, 24, 28 (p.100)].
4.2.1 Monopole antennas
Unlike the dipole antenna, there are antennas that consist of only one wire, whether it is a flat,
square, helix or round. They fall into the category of monopole antennas. Monopole antennas
(Figure 26) such as mast radiators (radio towers), whip antennas and helix antennas
42
(found in walkie-talkies) are only used to broadcast radio signals. Ground plane antenna is used
for emergency services, such as dispatchers, ambulance, firefighters and police.
Most encountered monopole antenna in practice are whip and helix antenna. These antennas
are used mostly in radio amateurism, radios and walkie-talkies, more precisely in HF, VHF and
UHF frequency bands. They are non-directional as opposed to large monopole elements, which
are used in radio broadcasting, operating between VLF and MF bands [13, 14, 26, 29].
Figure 26: Examples of monopole antennas
4.2.2 Dipole antennas
The word dipole indicates that the antenna consists of a pair of metal wires used to send or
receive the signal. Dipole antenna (Figure 27) is the most widespread type of antenna used in
the HF, VHF and UHF frequency ranges, more specifically in the application for broadcasting
television signals. In most cases commonly encountered antennas are "rabbit ears" (typical
home antenna for reception of TV signals), Yagi-Uda (known as "fish bone") and log-periodic.
43
"Rabbit ears" are characterized by a low gain of 2.5 dBi, omnidirectional (radiating in all
directions) radiation pattern, which is not suitable for antenna in practice. It is used as the basis
for the production of directional antennas. Half-wave dipole (Figure 2 and 3) is also wire
antenna. In addition, Yagi-Uda is the most recognized type of antenna encountered in practice.
It consists of a half-wave dipole with multiple passive elements (reflector or director) that
creates narrow beam (high directivity and gain), which makes Yagi-Uda a directional antenna.
The Yagi-Uda antenna directivity allows 10 dBi to 20 dBi gain. Usually it is mounted on the
rooftops for reception of TV signals. It is also used for shortwave communication at long
distance and for point-to-point communication
Lastly, log-periodic antenna is commonly mistaken for Yagi-Uda antenna, due to similar look,
since both are made of half-wave dipole antenna. However, unlike Yagi-Uda, the log-periodic
antenna has a series of half-wavelength dipoles, which gradually increase their length. Adding
a larger number of half-wave dipole elements causes an increase in bandwidth. Log-periodic
antenna is also used in television broadcasting, and other high frequency communications [13,
14, 26, 29].
Figure 27: Examples of dipole antennas
44
4.2.3 Aperture antennas
Patch and aperture antennas (Figure 28), such as parabolic reflector and horn antenna are
antennas that are used at microwave frequencies. Microstrip (patch) antennas are miniature
antennas, consisting of a thin layer of substrate, more precisely a dielectric, which is suitable for
broadcasting frequency signals. It is made by using printed circuit board technology. It is a
directional antenna, with a gain between 6 dBi and 9 dBi. The printed circuit board technology
is a simple and inexpensive process. Microstrip technology is used for filters, power dividers and
connecting leads (couplings). Because of simplicity of microstrip design, it is difficult to
overcome high power radiation, and microstrip elements are more susceptible to losses than
waveguides. Patch antennas found their application in telecommunications, such as spacecraft,
and mobile devices [13, 14, 26, 29].
Figure 28: Examples of microwave antennas
45
Parabolic antennas fall into the category of aperture or surface antennas. They use their own
surface to emit electromagnetic waves. In practice, they are known as dish satellites. They have
a curved surface, a parabolic shape, and in the center, there is an element that reflects the
waves in the free-space. This type of antenna has a high directivity and gain, up to 60 dBi, which
means that it can produce quite a narrow beam of electromagnetic waves. Therefore, it is used
in communications at large distances, such as satellite communication, radar technology, and
data exchange between two distant points [13, 14, 26, 29].
Lastly, horn antenna is one of the most prominent and simplest types of antenna in use. It is
used in the microwave frequency band and it dates back to the 1800's. Their commercial bloom
was during and after World War II. Horn antennas come in many forms, such as sectoral E-plane
and H-plane (Figure 29), pyramidal (Figure 30) and conical form. Since the title of the thesis is
oriented on the pyramidal horn antenna, a detailed description of other types of horn antennas
will not be discussed in detail; to convey the meaning and focus on this thesis is the priority.
Horn antennas are made by using a flaring process that flares end of the waveguide (neck) into
the antenna aperture (mouth) in one or two dimensions. If it is a single dimension, then the
horn antenna can fall into two categories: E-plane and H-plane sectoral antennas. If the
waveguide is flared out into direction of the electric field (𝒚-direction, see Figure 31), then the
E-plane horn is obtained. However, if the end of the waveguide has been flared out into
direction of the magnetic field (𝒙-direction, see Figure 31), then the H-plane horn is obtained.
From the perspective of the waveguide technology, expansion of the wider side – 𝒂 results in
H-plane antenna, and expansion of the narrower side – 𝒃 results in E-plane horn antenna [38].
Figures 29 and 30 consist of the following elements: 𝒂 – wider side of the waveguide (width), 𝒃
– narrower side of the waveguide (height), 𝒂𝟏, 𝒃𝟏 – dimensions of the antenna aperture after it
has been flared out.
46
Figure 29: E-plane and H-plane antenna
On the other hand, a pyramidal horn antenna (Figure 30) is flared in both dimensions, so that
its cross-section resembles a rectangle, and is often referred to in literature as the rectangular
horn antenna. To describe the pyramidal horn antenna as best as possible, it is necessary to
describe the E-plane and H-plane antennas first.
Figure 30: Pyramidal horn antenna
The main purpose of the flaring process is to increase the surface of radiated aperture, or more
precisely, to increase the directivity of the antenna. Also, flaring of the antenna aperture serves
as an adaptive element between the impedance of the waveguide 𝒁𝒘 = 50 Ω and the
impedance of the free-space 𝒁𝒇𝒔 ≈ 377 Ω. As mentioned in chapter 3, the waveguide is
usually excited with dominant TE10 mode, but the angle at which the aperture is opened must
not be too large, in order to prevent the occurrences of higher modes, and thereby reducing
the radiation efficiency [13, 14, 24, 26 (pp.719-756), 29].
47
4.3 E-plane horn antenna
The E-plane horn is a type of aperture antenna that is flared into the electric field plane. Figure
18 shows electric and magnetic field lines, making it easy to realize that the E-plane antenna is
flared in the direction of the electric field. In this chapter, a graphical representation of the E-
plane antenna will be presented, as well as its parameters and the optimal conditions for
achieving minimal attenuation and maximum directivity and gain.
4.3.1 Geometry and parameters
Figure 31 represents a geometry of E-plane antenna in coordinate system. The figure shows
antenna (and waveguide) dimensions. Values 𝒂 and 𝒃 represent a waveguide dimensions,
determined by 𝒙, 𝒚 and 𝒛 axes, as they represent a numerical value of waveguide dimensions.
In addition, 𝒙′, 𝒚’ and 𝒛’ axes represent a numerical value of antenna aperture, as well as 𝒃𝟏,
which represents a numerical value of antenna aperture in direction of 𝒚-axis. Coordinates 𝒙, 𝒚
and 𝒛 can be visualized as values before flaring process, whereas 𝒙′, 𝒚’ and 𝒛’ are values after
the flaring process is made. The origin of the coordinates 𝒙, 𝒚 and 𝒛 is located at (0, 0, 0) in the
Cartesian coordinate system [26] (pp. 719-733).
Figure 31: Geometry of E-plane antenna
48
On the other hand, Figure 32 depicts a more detailed view of E-plane horn antenna dimensions.
Dimensions 𝝆𝒆 and 𝝆𝟏 represent different lengths. Dimension 𝝆𝒆 represents a side length of
the E-plane antenna, and by using Pythagorean Theorem (see Figure 32), it is expressed as
𝜌𝑒 = √(𝜌1 )2 + (𝑏1
2)2
[m] (4.11)
where:
𝜌1 − a length from the phase center to the edge of the antenna aperture at ψ = 0 [m]
𝜌1 = 𝜌𝑒 ∙ cos 𝜓𝑒
𝑏1 − antenna aperture height [m]
𝑏1 ≅ √2𝜆 ∙ 𝜌1 → for optimum directivity
𝜓𝑒 − flare angle; 𝜓𝑒 = 𝑡𝑎𝑛−1 (𝑏12
𝜌1) → 2𝜓𝑒 = 2 𝑡𝑎𝑛−1 (
𝑏12
𝜌1) − total flare angle [°]
Figure 32: Cross-section of the E-plane horn antenna
𝒃𝟏 is the antenna dimension that represents the antenna aperture height. This dimension is
used as an approximate value when determining the optimal value of the antenna directivity.
49
Figure 33 represents a graph that describes the relationship between 𝒃𝟏 and directivity 𝑫𝑬, and
it is obvious that by increasing the 𝒃𝟏, directivity increases, and then it starts to decrease, as the
value of 𝒃𝟏 continues to increase.
Figure 33: Directivity as a function of aperture height
The flare angle is a dimension that has direct influence on antenna gain and its beamwidth. It
describes the angle of the antenna aperture, and its optimal value is between 0° and 90°. In
addition, viewed from the center of the antenna aperture (where 𝒚’ = 0), the phase angle at
any other point will not be the same as in its origin. This phase difference occurs due to the
different wave paths, which travel from the waveguide to the antenna aperture. The term
describing this difference is called the spherical phase
𝛿(𝑦′) ≅ 1
2∙ (
𝑦′2
𝜌1) (4.11a)
50
In addition, a so-called maximum phase error may occur. It occurs at 𝒚’ = 𝑏1
2, when there is a
greater difference between the length from the phase center to the center of the antenna
aperture – 𝝆𝟏 and the length from the phase center to the edge of the antenna – 𝝆𝒆. The phase
error is associated with the flare angle. If the antenna aperture is large, there is an increase in
the flare angle, and thus an increase in phase error, which affects the occurrence of wave
reflection and a decrease in antenna gain. Optimal value for E-plane horn antenna is 45°.
Maximum phase error is expressed in (4.12) [26] (pp. 719-733)
𝛿𝑀𝐴𝑋 = 2𝜋 ∙ 𝑠 (4.12)
where 𝒔 represents a peak phase error value at the E-plane antenna aperture; 𝒔|𝒃𝟏=
𝑏12
8𝜆∙𝜌1=
ρe − ρ1 . Figure 32 depicts another important dimensions of E-plane horn antenna – length of
the horn antenna from the edge of the waveguide (neck) to the very end of the antenna
(mouth). It is denoted as 𝒑𝒆. Along with the dimension 𝒃𝟏, it is the most important dimension
for the physical realization of E-plane horn. Mathematically it is expressed as
𝑝𝑒 = (𝑏1 − 𝑏) ∙ √[(𝜌𝑒
𝑏1)2
−1
4] (4.13)
4.3.2 Aperture field distribution and far-field region. Directivity
Components of the electric and magnetic field at the E-plane antenna aperture are represented
by the following expressions
𝐸′𝑦(𝑥
′, 𝑦′) ≅ 𝐸0 ∙ cos (𝜋
𝑎𝑥′) ∙ 𝑒−𝑗𝑘∙𝛿(𝑦′) (4.14)
𝐻′𝑥(𝑥
′, 𝑦′) ≅ −𝐸0
𝜂∙ cos (
𝜋
𝑎𝑥′) ∙ 𝑒−𝑗𝑘∙𝛿(𝑦′) (4.15)
51
𝐻′𝑧(𝑥
′, 𝑦′) ≅ 𝑗𝐸0 ∙ (𝜋
𝑘𝑎𝜂) ∙ sin (
𝜋
𝑎𝑥′) ∙ 𝑒−𝑗𝑘∙𝛿(𝑦′) (4.16)
while 𝑬′𝒙 , 𝑬
′𝒛 and 𝑯′
𝒚 are equal to zero – transverse field components. In addition, the field
radiated by the antenna itself in E-plane (𝝓 = 90°) can be mathematically represented by
𝐸𝜃 = −𝑗𝑋 ∙ [−𝑒𝑗𝑘∙[𝜌1∙𝑠𝑖𝑛2(
𝜃2)]
∙ (2
𝜋)2
(1 + cos 𝜃) ∙ 𝐹(𝑡1, 𝑡2)] (4.17)
where:
X =a ∙ √π ∙ k ∙ ρ1 ∙ E1 ∙ e−jkr
8r
F(t1, t2) = [𝐶(𝑡2) − 𝐶(𝑡1)] − 𝑗[(𝑆(𝑡2) − 𝑆(𝑡1)]
t1 = √k
π ∙ ρ1∙ (−
b1
2− ρ1 sin θ) t2 = √
k
π ∙ ρ1∙ (+
b1
2− ρ1 sin θ)
while radiated field in H-plane (𝝓 = 0°) can be expressed as
𝐸𝜙 = −𝑗𝑋 ∙ [cos (
𝑘 ∙ 𝑎2 ∙ sin 𝜃)
(𝑘 ∙ 𝑎2 ∙ sin 𝜃)
2
− (𝜋2)
2∙ (1 + cos 𝜃) ∙ 𝐹(𝑡3, 𝑡4)] (4.18)
where:
X =a ∙ √π ∙ k ∙ ρ1 ∙ E1 ∙ e−jkr
8r
F(t3, t4) = [𝐶(𝑡4) − 𝐶(𝑡3)] − 𝑗[(𝑆(𝑡4) − 𝑆(𝑡3)]
t3 = −b1
2 ∙ √
k
π ∙ ρ1 t4 = +
b1
2 ∙ √
k
π ∙ ρ1
52
Both planes are explicitly represented in Figure 34.
Figure 34: E-plane horn antenna patterns in E-plane and H-plane
As for the directivity of the E-plane antenna, it is necessary to find the maximum radiation
intensity and total radiated power, since it is obvious that these two parameters are actually
related to directivity, hence the expression (4.6).
Maximum radiation intensity represents a maximum radiated power of the antenna in given
direction. It can be expressed as
𝑈𝑚𝑎𝑥 = 4𝑎2 ∙ 𝜌1 ∙ |𝐸1|
2
𝜂 ∙ 𝜆 ∙ 𝜋2∙ [𝐶2(𝑞) + 𝑆2(𝑞)] (4.19)
where:
𝑎 − waveguide width [𝑚]
𝐸1 − electric field strength, const. [𝑉
𝑚]
𝜂 − impedance of the free-space [Ω]
𝐶2, 𝑆2 − Fresnel’s integrals
53
𝑞 = 𝑏1
√2𝜆 ∙ 𝜌1
Total radiated power can be calculated as
𝑃𝑟𝑎𝑑 = |𝐸1|2 ∙
𝑎 ∙ 𝑏1
4𝜂 (4.20)
Combining (4.19) and (4.20) gives the expression for directivity of E-plane horn antenna
𝐷𝐸 = 4𝜋 ∙𝑈𝑚𝑎𝑥
𝑃𝑟𝑎𝑑=
64𝑎 ∙ 𝜌1
𝑏1 ∙ 𝜆𝜋∙ [𝐶2(𝑞) + 𝑆2(𝑞)] (4.21)
Maximum directivity is achieved when 𝒃𝟏 = √2𝜋𝜌1. A more precise calculation of E-plane horn
antenna directivity is given by the following expression
𝐷𝐸 = 16𝑎 ∙ 𝑏1
𝜆2 (1 +𝜆𝑔
𝜆)
∙ [𝐶2(𝑞) + 𝑆2(𝑞)] ∙ 𝑒𝜋𝑎∙(1−
𝜆𝜆𝑔
) (4.22)
A short digression was done, due to Fresnel integral calculation process. The procedure is as
follows:
1. Let us assume that 𝒙 = 𝑏1
√2𝜆∙𝜌1. Calculate 𝑥.
2. The obtained value needs to be rounded up to a greater number (example: if 𝒙 =
0.453; round it up to 0.46).
3. Locate the number 𝒙 from Step 2 in the Fresnel integral table (Figure 35).
4. Next to the number 𝒙 from Step 3, the numbers in the columns 𝑪(𝒙) and 𝑺(𝒙) must be
rounded up to two decimal places.
54
5. In the expression 𝑪𝟐(𝒒) + 𝑺𝟐(𝒒), replace 𝒒 with number obtained from Step 4 and then
square it. The obtained value represents the solution of Fresnel's integrals [26] (pp. 719-
733).
Figure 35: Table of Fresnel's integrals
4.3.3 Optimum antenna dimensions
In this chapter, optimum values of E-plane horn antenna will be presented and described. These
values represent the dimensions of the E-plane antenna, which are required for the antenna to
achieve higher gain and outstanding directivity, with as little loss as possible. The dimensions
for achieving the optimum E-plane antenna are: 𝒃𝟏, 𝑫𝑬,𝒎𝒂𝒙, 𝜽𝑫(𝑬), 𝜽𝑫(𝑯) and 𝑨𝒆𝒇𝒇 [24].
𝑏1 ≅ √2𝜆 ∙ 𝜌1 – antenna aperture height in the E-plane direction [𝑚]
𝐷𝐸,𝑚𝑎𝑥 = 8.2∙𝑎∙𝑏1
𝜆2 − directivity of the E-plane horn antenna
55
𝜃𝐷(𝐸) = 53° ∙𝜆
𝑏1 – flare angle in the E-plane direction [°]
𝜃𝐷(𝐻) = 68° ∙𝜆
𝑎− flare angle in the H-plane direction [°]
𝐴𝑒𝑓𝑓 = 0.65 ∙ 𝑎 ∙ 𝑏1 − effective surface (area) [𝑚2]
4.4 H-plane horn antenna
Unlike E-plane horn antenna, which is flared into E-plane or in 𝒚-direction, H-plane antenna has
its aperture flared out in the direction of the 𝒙-axis or in the direction of the magnetic field.
Similar to the E-plane antenna in chapter 4.3, the geometry and dimensions of the H-plane
antenna will be described.
4.4.1 Geometry and dimensions
H-plane horn antenna is type of aperture antenna that is flared out in the direction of magnetic
field, or in H-plane or along the 𝒙-direction in the defined Cartesian coordinate system. Figure
36 depicts an antenna geometry, dimensions of the rectangular waveguide, and the antenna
itself. Dimensions 𝒂 and 𝒃 are rectangular waveguide dimensions, while 𝒂𝟏 represents a
numerical value of H-plane horn antenna aperture [26] (pp. 733-743).
Figure 36: Geometry of H-plane antenna
56
The dimensions shown in Figure 37 are similar to those of the E-plane antenna. Dimensions 𝝆𝒉
and 𝝆𝟐 represent the lengths of the antenna aperture; while 𝝍𝒉 represents the flare angle and
𝜹(𝒙′) represents the spherical angle.
Figure 37: Cross-section of the H-plane horn antenna
Dimensions 𝝆𝒉 and 𝝆𝟐 are interconnected by expression
𝜌ℎ = √(𝜌2)2 + (𝑎1
2)2
(4.23)
𝜌2 = 𝜌ℎ ∙ cos𝜓ℎ (4.24)
where:
𝜌ℎ − side length of the H-plane horn antenna [m]
𝜌2 − length form the phase center to the edge of the antenna aperture [m]
𝑎1 − antenna aperture width [m]
𝑎1 ≅ √3𝜆 ∙ 𝜌2 → for optimum directivity
𝜓ℎ − flare angle; 𝜓ℎ = tan−1 (a12
ρ2) [°, rad] → 2𝜓ℎ = 2 tan−1 (
a12
ρ2) − total flare angle
57
𝒂𝟏 is the antenna dimension representing the antenna aperture height. This dimension is used
as an approximate value when determining the optimal value of the antenna directivity. Figure
38 represents a graph that describes the relationship between 𝒂𝟏 and directivity 𝑫𝑯, and it is
apparent that by increasing the 𝒂𝟏, directivity increases, and then it starts to decrease, as the
value of 𝒂𝟏 continues to increase.
Figure 38: Directivity as a function of aperture width
Similar to the E-plane horn antenna, the spherical phase 𝜹 represents a dimension that
describes the difference in the angle in which the wave travels to the antenna aperture. It is
determined by length 𝝆𝟐 and 𝒙’, which represent a value on any part of the antenna aperture.
Spherical phase can be expressed as follows
58
𝛿(𝑥′) ≅ 1
2∙ (
𝑥′2
𝜌2) (4.25)
The spherical phase dimension is closely associated with the maximum phase error – 𝜹𝑴𝑨𝑿.
Chapter 4.3.1 explains the reason for the occurrence of the mentioned dimension. In case of an
H-plane horn antenna, the maximum phase error is manifested when 𝒙′ = 𝒂𝟏
𝟐. Along with the
spherical phase, the flare angle also affects the maximum phase error, which can be expressed
as
𝛿𝑀𝐴𝑋 = 2𝜋 ∙ 𝑡 (4.26)
where 𝒕 represents a peak phase error value at the H-plane antenna aperture; 𝒕|𝒂𝟏=
𝑎12
8𝜆∙𝜌2=
ρℎ − ρ2. Lastly, dimension 𝒑𝒉 that represents a length from the edge of the waveguide (neck)
to the antenna aperture (mouth). It is one of the important dimension of H-plane horn antenna,
along with the dimension 𝒂𝟏, as they condition the change of another antenna value. Length
𝒑𝒉 can be determined as [26] (pp. 733-743)
𝑝ℎ = (𝑎1 − 𝑎) ∙ √[(𝜌ℎ
𝑎1)2
−1
4] (4.27)
4.4.2 Aperture field distribution and far-field region. Directivity
At the H-plane antenna aperture, there are amplitude and phase distributions of the electric
and the magnetic fields. Following expressions represent those distributions
𝐸′𝑦(𝑥
′) ≅ 𝐸0 ∙ cos (𝜋
𝑎1𝑥′) ∙ 𝑒−𝑗𝑘∙𝛿(𝑥′) (4.28)
𝐻′𝑥(𝑥
′) ≅ −𝐸0
𝜂∙ cos (
𝜋
𝑎1𝑥′) ∙ 𝑒−𝑗𝑘∙𝛿(𝑥′) (4.29)
59
while 𝑬′𝒙 and 𝑯′
𝒚 are equal to zero. The part labelled with dashed lines represents the
amplitude distribution of the electric field, while the part labelled with solid lines describes the
phase distribution of the electric field along the antenna aperture. This applies to (4.14), (4.15),
(4.16), (4.29), (4.35) and (4.36) as well. On the other hand, a field distribution in the far-field
area for E-plane and for H-plane is expressed in spherical coordinate system as
𝐸𝜃 = 𝑗𝑋 × {(1 + cos 𝜃)sin 𝑌
𝑌[𝑒𝑗𝑓1 ∙ 𝐹(𝑡1, 𝑡2) + 𝑒𝑗𝑓2 ∙ 𝐹(𝑡3, 𝑡4)]} (4.30)
𝐸𝜙 = 𝑗𝑋 × {(cos 𝜃 + 1) ∙ [𝑒𝑗𝑓1 ∙ 𝐹(𝑡1, 𝑡2) + 𝑒𝑗𝑓2 ∙ 𝐹(𝑡3, 𝑡4)]} (4.31)
where:
𝑋 = 𝐸2 ∙𝑏
8∙ √
𝑘𝜌2
𝜋∙𝑒−𝑗𝑘𝑟
𝑟
𝑌 = 𝑘𝑏
2∙ sin 𝜃
𝑓1 = 𝑘′𝑥
2∙ 𝜌2
2𝑘
𝑓1 = 𝑘′′𝑥
2∙ 𝜌2
2𝑘
𝑘′𝑥 = 𝜋
𝑎1 → for E-plane
𝑘′𝑥 = 𝑘 ∙ sin 𝜃 + 𝜋
𝑎1 → for H-plane
𝑘′′𝑥 = − 𝜋
𝑎1 → for E-plane
𝑘′′𝑥 = 𝑘 ∙ sin 𝜃 −
𝜋
𝑎1 → for H-plane
t1 = √1
π ∙ k ∙ ρ2∙ (−
𝑘 ∙ 𝑎1
2− 𝑘′𝑥1,2 ∙ ρ2)
t2 = √1
π ∙ k ∙ ρ2∙ (+
𝑘 ∙ 𝑎1
2− 𝑘′𝑥1,2 ∙ ρ2)
𝑘′𝑥1,2 = 𝑘 ∙ sin 𝜃 cos𝜙 +𝜋
𝑎1
t3 = √1
π ∙ k ∙ ρ2∙ (−
𝑘 ∙ 𝑎1
2− 𝑘′𝑥3,4 ∙ ρ2)
t4 = √1
π ∙ k ∙ ρ2∙ (+
𝑘 ∙ 𝑎1
2− 𝑘′𝑥3,4 ∙ ρ2)
𝑘′𝑥3,4 = 𝑘 ∙ sin 𝜃 cos𝜙 −𝜋
𝑎1
E-plane and H-plane patterns are explicitly depicted in Figure 39.
60
Figure 39: H-plane horn antenna patterns in E-plane and H-plane
Maximum radiated intensity, radiated power, and directivity of the H-plane horn antenna are
expressed as follows
𝑈𝑚𝑎𝑥 = 𝑏2 ∙ 𝜌2 ∙ |𝐸2|
2
4 ∙ 𝜂 ∙ 𝜆∙ (𝐶′ + 𝑆′)0.5 (4.32)
𝑃𝑟𝑎𝑑 = |𝐸2|2 ∙
𝑎1 ∙ 𝑏
4 ∙ 𝜂 (4.33)
𝐷𝐻 = 4𝜋 ∙𝑈𝑚𝑎𝑥
𝑃𝑟𝑎𝑑=
4𝜋 ∙ 𝑏 ∙ 𝜌2
𝑎1 ∙ 𝜆× (𝐶′ + 𝑆′) (4.34)
where:
C′ = [C(u) − C(v)]2
S′ = [S(u) − S(v)]2
u = 1
√2∙ (
√λ∙ρ2
a1+
a1
√λ∙ρ2)
v = 1
√2∙ (
√λ∙ρ2
a1−
a1
√λ∙ρ2)
61
The maximum directivity for H-plane horn antenna is achieved when 𝒂𝟏 = √3𝜆 ∙ 𝜌1 [26] (pp.
733-743).
4.4.3 Optimum antenna values
In order to achieve the optimum performance of the H-plane horn antenna, it is necessary to
design the antenna and its dimensions as close to the optimal ones as possible. The aim is to
reduce signal attenuation, increase gain and antenna directivity. To achieve this, the following
dimensions are required: 𝒂𝟏, 𝑫𝑯,𝒎𝒂𝒙, 𝜽𝑫(𝑬), 𝜽𝑫(𝑯), and 𝑨𝒆𝒇𝒇 [24].
𝑎1 ≅ √3𝜆 ∙ 𝜌2 − antenna aperture width in the H-plane direction [𝑚]
𝐷𝐻,𝑚𝑎𝑥 = 7.8∙𝑎1∙𝑏
𝜆2 − directivity of the H-plane horn antenna
𝜃𝐷(𝐸) = 51° ∙𝜆
𝑏 – flare angle in the E-plane direction [°, 𝑟𝑎𝑑]
𝜃𝐷(𝐻) = 80° ∙𝜆
𝑎1− flare angle in the H-plane direction [°, 𝑟𝑎𝑑]
𝐴𝑒𝑓𝑓 = 0.63 ∙ 𝑎1 ∙ 𝑏 − effective surface (area) [𝑚2]
4.5 Pyramidal horn antenna
This chapter will feature a pyramidal horn antenna. Its dimensions will be mathematically and
practically described. With the aid of pyramidal horn antenna geometry, calculations of these
dimensions need to be as close as possible to the optimum values in order to achieve the best
results when simulating and designing the antenna. The optimum dimension values will result
in greater antenna gain, better directivity (minor lobe reduction), and better distribution of
radiated power density, which will contribute to a lesser power loss. After the theoretical
description and presentation of the antenna and its dimensions, a calculation will be made that
will sufficiently carry out the simulation and design procedure of the antenna in the following
chapters.
62
4.5.1 Geometry and dimensions
Pyramidal horn antenna (Figure 40) is a two-dimensional aperture antenna, i.e. it is composed
of a combination of two one-dimensional antennas; E-plane and H-plane. This means that the
pyramidal horn antenna is flared in both directions (𝒙′ and 𝒚′) and its aperture resembles a
rectangular shape; often referred to as rectangular horn antenna [26] (pp. 743-756).
Figure 40: Geometry of pyramidal horn antenna
Figure 40 depicts the dimensions 𝒂, 𝒃 mentioned and described in chapter 3.1, while 𝒂𝟏 and 𝒃𝟏
were mentioned and described in chapter 4.4.3 and chapter 4.3.3. In addition, Figure 41
contains top view of pyramidal horn antenna. Top view refers to H-plane view. Dimensions 𝝆𝒉,
𝝆𝟐 and 𝒑𝒉 were given by (4.23), (4.24) and (4.27), respectively.
Figure 41: Top view (H-plane) of pyramidal horn
63
Side view of the pyramidal horn, which refers to E-plane view, is shown in Figure 42. Displayed
dimensions 𝝆𝒆 and 𝒑𝒆 were given by (4.11) and (4.13), respectively.
Figure 42: Side view (E-plane) of pyramidal horn
4.5.2 Aperture field distribution and far-field region. Directivity
Unlike the E-plane and the H-plane horn antenna, the pyramidal horn antenna, when
distribution of the electric and magnetic fields is concerned, contains the elements of both
antennas. The terms for the field distribution at the antenna aperture are expressed as follows
𝐸′𝑦(𝑥
′, 𝑦′) = 𝐸0 ∙ cos (𝜋 ∙ 𝑥′
𝑎1) ∙ 𝑒−𝑗𝑘∙[𝛿(𝑥′)+𝛿(𝑦′)] (4.35)
𝐻′𝑥(𝑥
′, 𝑦′) = −𝐸0
𝜂∙ cos (
𝜋 ∙ 𝑥′
𝑎1) ∙ 𝑒−𝑗𝑘∙[𝛿(𝑥′)+𝛿(𝑦′)] (4.36)
where 𝜹(𝒙′) and 𝜹(𝒚′) represent spherical phases, introduced in chapter 4.3.1 and 4.4.1. In the
far-field region, electric field components are distributed as follows
64
𝐸𝜃 = |𝐸0 ∙ sin 𝜙| ∙√𝜌1 ∙ 𝜌2
𝑟∙ √𝐶′ + 𝑆′ ∙ √𝐶2(𝑞) + 𝑆2(𝑞) (4.37)
𝐸𝜙 = |𝐸0 ∙ cos 𝜙| ∙√𝜌1 ∙ 𝜌2
𝑟∙ √𝐶′ + 𝑆′ ∙ √𝐶2(𝑞) + 𝑆2(𝑞) (4.38)
where 𝝆𝟐, 𝑪’ and 𝑺’, 𝒒, 𝑪𝟐(𝒒) and 𝑺𝟐(𝒒) were described and given by (4.24), (4.32), (4.34), and
(4.19), respectively. Amplitude pattern of pyramidal horn antenna in E-plane and H-plane is
depicted in Figure 43.
Figure 43: Pyramidal horn antenna pattern in E-plane and H-plane
In order to calculate directivity of the pyramidal horn antenna, it is necessary to know
waveguide dimensions as well as the directivity of E-plane and H-plane horn antenna. The
expression for the directivity of the pyramidal antenna is given by
𝐷𝑃 = 𝜋 ∙ 𝜆2
32 ∙ 𝑎 ∙ 𝑏∙ 𝐷𝐸 ∙ 𝐷𝐻 (4.39)
65
where 𝑫𝑬 and 𝑫𝑯 were given by (4.21), (4.22), and (4.34) and for dimensions 𝒂 and 𝒃 see
chapter 3.1. The optimum dimensions of the pyramidal horn antenna will result in maximum
directivity. In addition, a phase difference between the edge and center of the antenna aperture
is required to achieve maximum directivity: for E-plane it is 90° and for H-plane it is 135°. The
optimum dimensions of the pyramidal antenna are given by the expressions for optimal values
of E-plane and H-plane horn antenna; 𝒂𝟏 and 𝒃𝟏, see chapters 4.3.3 and 4.4.3 [24, [26](pp. 743-
756)].
4.5.3 Optimum antenna values
Dimensions of the optimal pyramidal horn antenna are connected to the optimum E-plane and
H-plane horn antenna dimensions. Flaring in both planes (E- and H-) indicates that both E-plane
and H-plane antenna optimum values should be taken into consideration, when calculating and
planning a design procedure for pyramidal horn antenna. Therefore, the following dimensions
are [24]:
𝑎1 ≅ √3𝜆 ∙ 𝜌2 − antenna aperture width in the H-plane direction [𝑚]
𝑏1 ≅ √2𝜆 ∙ 𝜌1 – antenna aperture height in the E-plane direction [𝑚]
𝜃𝐷(𝐸) = 53° ∙𝜆
𝑏1 – flare angle in the E-plane direction [°]
𝜃𝐷(𝐻) = 80° ∙𝜆
𝑎1− flare angle in the H-plane direction [°]
𝐷𝑃,𝑚𝑎𝑥 = 6.3∙𝑎1∙𝑏1
𝜆2− directivity of the pyramidal horn antenna
𝐴𝑒𝑓𝑓 = 0.51 ∙ 𝑎1 ∙ 𝑏1 − effective surface (area) [𝑚2]
66
5. ANTENNA SIMULATION
Simulation of pyramidal horn antenna will be carried out in ANSYS Electromagnetic Suite
18 software. It is a software, which enables the simulation of electromagnetic fields and serves
for the modeling of arbitrary 3D devices: antennas, waveguides, RF components, filters, IoT
devices, microwave electronic devices, etc. ANSYS Electromagnetic Suite enables the design of
various types of antennas including simulation, visualization, modeling, and automation of
problems related to electromagnetism and provides quick and precise solutions to the problem.
EM Suite 18 uses High-Frequency Structure Simulator or HFSS, which is based on Finite Element
Method – problem solving numerical method in various engineering fields. HFSS offers solutions
for microwave, digital and RF applications. Its graphical interface is very intuitive and versatile.
For the purposes of this master’s thesis, software is used to simulate parameters such as
directivity, VSWR, gain, E and H fields, radiation intensity, and other antenna parameters.
5.1 Starting the HFSS simulator
To run the program, click the Microsoft Start button and select All Programs → ANSYS
Electromagnetics Suite 18 → ANSYS Electronics Desktop 2017.0. A window will appear,
representing the interface of the Electronics Desktop (Figure 44).
Figure 44: Electronics Desktop interface
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In Electronics Desktop toolbar (Figure 45), select Tools → Options → General options to set
some basic settings before launching the Project.
Figure 45: Electronics Desktop toolbar
▪ Expand HFSS-IE and select Boundary Assignment
• Check all entries
▪ Expand 3D Modeler
• Click Drawing
o Edit properties of new primitives: ☒Checked
▪ Expand Display
• Choose History Tree and check all entries, then click the OK button
On the toolbar, select Project → Insert HFSS design. The HFSS window (Figure 46) consists of
several parts:
Figure 46: ANSYS HFSS window with associated parts
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Project Manager window (Figure 47) is located on the upper left side of the HFSS program
window. It contains a design tree that shows the structure of the project.
Figure 47: Project Manager window
Properties window (Figure 48) is located on the bottom left of the HFSS program window and
allows user to change the dimensions and properties of the model.
Figure 48: Properties window
Component Library window (Figure 49) is located on the right side of the HFSS program window
and represents a library with the components of the model.
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Figure 49: Components Library window
Show Messages and Show Progress are located at the bottom right corner of the HFSS program
window. Show Messages allows user to view errors and warnings that may occur before and
during the simulation, while Show Progress shows the progress of the simulation. The last part
of the HFSS program window is the 3D Modeler window (Figure 50), which is located in the
center of the program window and contains a model display.
Figure 50: 3D Modeler window
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5.2 Mathematical calculation of the reference pyramidal horn antenna
To begin the simulation and antenna design process, it is primarily necessary to find the
appropriate dimensions. Prior to the dimension calculation, it is necessary to define the desired
gain for the antenna in order to determine the remaining dimensions, such as antenna aperture
dimensions (𝒂𝟏 and 𝒃𝟏), side length dimensions (𝝆𝒆 and 𝝆𝒉) and length of the antenna horn
from its neck to its mouth (𝒑𝒆 and 𝒑𝒉). There are total of six dimensions to be found [26](pp.
743-756).
It is known that a waveguide WR-112 is designed for X-band frequency range; between 7.05
GHz and 10 GHz. Its dimensions are 𝒂 = 28.4988 mm and 𝒃 = 12.6238 mm. The selected
frequency range for the pyramidal horn antenna is 𝒇 = 9 GHz and desired gain is 𝑮𝟎(𝒅𝑩) =
19.7 dB. For easier computation, it is better to convert gain value into non-decibel form: 𝑮𝟎 =
10𝐺(𝑑𝐵)
10 = 101.97 = 93.325. To calculate the antenna dimensions, it is necessary to select the
desired gain, within reasonable limits, and to determine if the chosen gain corresponds to the
antenna and its dimensions. One can say that the value of the gain is randomly chosen – the so
called method of attempts and failures. Knowing the operating frequency of an antenna, it is
possible to calculate its operating wavelength as 𝝀 =c
f=
2.998∙108
9∙109 = 𝟑𝟑. 𝟑𝟏𝟏�̇� mm . This allows
calibrating waveguide dimensions at the operating wavelength. Waveguide dimensions after a
calibrations are 𝒂 = 28.4988
33.3111= 0.85553 𝜆 = 28.4988 mm and 𝒃 =
12.6238
33.3111= 0.37897 𝜆 =
12.6238 mm.
Initial value, denoted as 𝝌, is an unknown part, which is needed for pyramidal horn antenna
calculation. Once the initial value is calculated, it is included in the expression (5.1) to achieve
equality on the left and right sides. The equality of the left and right side of the expression (5.1)
is required, in order to proceed further with calculation of antenna dimensions.
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(√2𝜒 −𝑏
𝜆)2
∙ (2𝜒 − 1) = [𝐺0
2𝜋∙ √
3
2𝜋∙
1
√𝜒−
𝑎
𝜆]
2
∙ [𝐺0
2
6𝜋3∙1
𝜒− 1] (5.1)
Expression (5.1) represents an initial, as well as simplified formula for pyramidal horn antenna
design procedure. It is derived from a group of expressions used in this thesis, which represents
an alternative approach for calculating pyramidal horn antenna dimensions. From that point, a
mathematical calculation of dimensions begins, which leads to physical realization of the
pyramidal horn antenna. Group of expressions used in initial formula (5.1) are:
𝐺0 ≅ 1
2∙ (
4𝜋
𝜆2∙ 𝑎1 ∙ 𝑏1)
𝑎1 ≅ √3𝜆 ∙ 𝜌2
𝑏1 ≅ √2𝜆 ∙ 𝜌1
𝑝ℎ = (𝑎1 − 𝑎) ∙ √[(𝜌ℎ
𝑎1)2
−1
4]
𝑝𝑒 = (𝑏1 − 𝑏) ∙ √[(𝜌𝑒
𝑏1)2
−1
4]
where 𝑮𝟎 represents a gain of the pyramidal horn antenna, 𝒂𝟏 and 𝒃𝟏 represent aperture width
and height, described below expressions (4.24) and (4.11), while 𝒑𝒆 and 𝒑𝒉 are given by (4.13)
and (4.27), respectively.
The first step is to calculate the initial value 𝝌.
STEP 1:
𝝌 (𝑡𝑟𝑖𝑎𝑙) =𝐺0
2𝜋 ∙ √2𝜋=
93.325
2𝜋 ∙ √2𝜋= 5.926
Incorporating value 𝝌 = 5.926 to the equation, did not match the equality of the left and right
side of the expression (5.1). It was necessary to repeat or to adjust the value of the initial value.
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In this case, it was necessary to perform at least 10 iterations before finding the initial value that
met the equality of expression (5.1). The value of the initial value that satisfied the equality on
both sides in the expression (5.1) is 𝝌 = 5.592.
The second step is to calculate the sides of the pyramidal horn antenna, 𝝆𝒆 (left and right side)
and 𝝆𝒉 (top and bottom side).
STEP 2: Calculate 𝝆𝒆 and 𝝆𝒉 by the obtained value of 𝝌.
𝜌𝑒 = 𝜒 ∙ 𝜆 = 5.592 ∙ 𝜆 = 5.592 ∙ 3.331̇ = 186.3 𝑚𝑚
𝜌ℎ =𝐺0
2
8 ∙ 𝜋3∙1
𝜒∙ 𝜆 =
93.3252
8 ∙ 𝜋3∙1
𝜒∙ 𝜆 = 6.28 ∙ 𝜆 = 6.28 ∙ 3.331̇ = 209.2 𝑚𝑚
After calculating the values of the side’s length, the next step is the calculation of the antenna
aperture dimension, 𝒂𝟏 and 𝒃𝟏.
STEP 3: Calculate 𝒂𝟏 and 𝒃𝟏 by the obtained values of 𝝆𝒆 and 𝝆𝒉.
𝑎1 = √3𝜆 ∙ 𝜌ℎ =𝐺0
2𝜋∙ √
3
2𝜋 ∙ 𝜒∙ 𝜆 =
93.325
2𝜋∙ √
3
2𝜋 ∙ 5.592∙ 𝜆 = 4.34 ∙ 𝜆 = 144.5 𝑚𝑚
𝑏1 = √2𝜆 ∙ 𝜌𝑒 = √2𝜒 ∙ 𝜆 = √2 ∙ 5.592𝜆 = 3.34 ∙ 𝜆 = 111.3 𝑚𝑚
Lastly, after the calculated dimensions of the antenna aperture, the last step is to determine
the lengths of the pyramidal horn antenna from its neck (end of the waveguide) to its mouth
(end of the antenna aperture), in both planes, 𝒑𝒆 and 𝒑𝒉.
STEP 4: Calculate 𝒑𝒆 and 𝒑𝒉 by the obtained values of 𝒂, 𝒃, 𝒂𝟏, 𝒃𝟏, 𝝆𝒆 and 𝝆𝒉.
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𝑝𝑒 = (𝑏1 − 𝑏) ∙ √[(𝜌𝑒
𝑏1)2
−1
4] = (3.34 ∙ 𝜆 − 0.37897 ∙ 𝜆)√[(
5.592𝜆
3.34𝜆)2
−1
4] = 4.73 ∙ 𝜆
= 157.6 𝑚𝑚
𝑝ℎ = (𝑎1 − 𝑎) ∙ √[(𝜌ℎ
𝑎1)2
−1
4] = (4.34 ∙ 𝜆 − 0.85553 ∙ 𝜆)√[(
6.28𝜆
4.34𝜆)2
−1
4] = 4.73 ∙ 𝜆
= 157.6 𝑚𝑚
From step 4, it is apparent that the values for 𝒑𝒆 and 𝒑𝒉 are equal, thus the condition for
physical realization of the pyramidal horn antenna is fulfilled. This completes the procedure for
determining antenna dimensions.
5.3 HFSS design procedure of the reference pyramidal horn antenna
As soon as the ANSYS HFSS program starts, it is necessary to create a new project and set the
solution type. To create a new project, select File → New in the program toolbar. After the
project has been successfully created, it is necessary to select the solution type. To set solution
type, select HFSS → Solution Type → Modal, and under Driven Options select Network Analysis,
then click the OK button. Under tab Modeler, select Units, and set default units to mm. Using
3D Modeler Materials toolbar, set vacuum as default material.
Next, draw the waveguide by selecting Draw → Rectangle, and create a shape in the main
window area. After clicking three times on the main window, the Properties window will appear
(Figure 51). Under Position type Horn_length, -a/2, -b/2. This will completely parametrize newly
created box – a waveguide. Once the dimensions have been entered, user is prompted to enter
values as follows: 157.6 mm for Horn_length, 28.4988 mm for -a/2, and 12.6238 mm for -b/2.
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Figure 51: Creating a waveguide
After all the values of the waveguide have been entered, the final layout of the entered
dimensions/values should look the same as in Figure 52. Click Apply, and then OK.
Figure 52: Waveguide values
Next, select Box1 and rename it as Horn. Select Modeler → Grid Plane → YZ. The next step is
to create a waveguide horn. Select Draw → Rectangle, and draw a shape on YZ plane. Under
Position type as follows: 0mm, -FlareA/2, -FlareB/2. Now, enter the numerical values: 144.5 mm
for FlareA, and 111.3 mm for FlareB. Final layout of entered dimensions/values should look the
same as in Figure 53. Click Apply, and then OK.
Figure 53: Horn aperture values
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Figure 54 depicts parametrized waveguide and a rectangle that need to bind together, in order
to create a pyramidal horn antenna.
Figure 54: Waveguide and rectangle
Enter Face selection mode; go to Edit → Select → Faces. Select waveguide face that is closer to
the large rectangle – Figure 55.
Figure 55: Waveguide face selection
Next, an object form selected waveguide face needs to be connected to the rectangle. In order
to do so, select Modeler → Surface → Create Object from Face. A new object will appear under
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Sheets; named Horn_ObjectFromFace1. Enter Objects selection mode; go to Edit → Select →
Objects. Then, go to Edit → Select → By Name. Select Horn_ObjectFromFace1 and Rectangle1,
click OK. While they are still being selected, go to Modeler → Surface → Connect. Selected
objects are connected. Next, click Edit → Select All, and then click Modeler → Boolean → Unite.
All of the components are combined under the object Horn – Figure 56.
Figure 56: Pyramidal horn antenna
Next, pyramidal horn shell or wall thickness (Figure 57) needs to be created. Firstly, enter Face
selection mode (Edit → Select Faces). Then, go to Edit → Select → By Name, and select Horn
object (located on left menu list). On the right list, there will be a list of Face ID’s of created Horn
object. Select all Face ID’s except its aperture, and then click OK. Go to Modeler → Surface →
Create Object from Face. Next, go to Modeler → Boolean → Unite. Lastly, determine wall
thickness of pyramidal horn. Select Modeler → Surface → Thicken Sheet. In a pop-up window,
under Thickness type wall_thickness, click OK, then under Value type 1.625. Entered number
represents a nominal value for WR-112 wall thickness.
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Figure 57: Antenna wall thickness
Next step is to assign material to the antenna. Select Edit → Select → Objects. Rename Horn
object as Horn_air (inner wall), and make sure that vacuum is selected as appropriate material.
Rename outer wall (Horn_ObjectFromFace2) to Horn_shell and under drop down menu from
Material, click Edit and select perfect conductor (Figure 58).
Figure 58: Selection of perfect conductor
Next, draw a region (Figure 59) around pyramidal horn antenna, which is going to “modify” the
air around it, and allow EM energy to propagate from the horn antenna into the surrounding
air up to its radiation boundary. Select Draw → Region. Under Padding Data select Pad all
directions similarly, and under Padding Type select Absolute offset. Type rad_dist under Value,
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and enter 8.327 mm. This will create a region, which surrounds pyramidal horn in all directions
evenly.
Figure 59: Region around horn antenna
With the created region, it is necessary to determine the radiation boundary, in order to see
how antenna will behave. To do so, select Region object, go to HFSS → Hybrid → Assign Hybrid
→ FE-BI, click OK.
Next, create excitation element or wave port (Figure 61). Select YZ plane, in order to draw a
rectangle from the back of the waveguide. Select Draw → Rectangle, select an arbitrary shape
area and click anywhere. A wave port Properties window will appear. Under Position enter:
Horn_length+waveguide_length, -a/2, -b/2, followed by Axis: X, YSize: a, and ZSize: b. Final
layout of entered dimensions/values should look the same as in Figure 60.
Figure 60: Excitation port values
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Under Attribute, change name into wave_port and click OK button. Now it is necessary to assign
excitation to the created wave port. To do so, select HFSS → Excitations → Assign → Wave
Port. Click Next, click Finish.
Figure 61: Assignment of excitation port
Save design as 3D component project. In order to do that, go to Edit → Select All to select the
antenna with created airbox and radiation boundary. Then, go to Draw → 3D Component Library
→ Create 3D Component…, and save project for future use.
5.4 Results of the reference pyramidal horn antenna
The results generated by the simulation will be presented and explained in this chapter. The
idea was to simulate the antenna parameters according to the dimensions calculated in chapter
5.2. In addition, the aim is to spot the "behaviour" of the antenna, more precisely, its parameters
in relation to the given dimensions. Specifically, the idea is to take one of the antenna dimension
and increase or decrease its value, while simultaneously monitor how its changes affects the
change of parameters such as gain, directivity, the standing wave ratio (VSWR) and return loss.
This part will be described after describing the antenna simulation procedure with the
calculated dimensions (chapter 5.2). Upon opening newly created 3D model (Draw → 3D
Component Library → Browse…), it is necessary to preform following steps:
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STEP 1: Set operating frequency
▪ Click on HFSS → Model → Create Open Region
▪ Set operating frequency to 9 GHz
▪ Set Boundary to Radiation
STEP 2: Far-field setup
▪ Click on HFSS → Radiation → Insert Far Field Setup → Infinite Sphere
▪ Set Phi: start: 0°, stop: 360°, step size 10°
▪ Set Theta: start: -180°, stop: 180°, step size 10°
STEP 3: Analyse
▪ HFSS → Analysis Setup → Add Solution Type Setup
• Set Single Frequency to 9 GHz
▪ Right-click on Setup1 → select Add Frequency Sweep
• Enter values as shown in Figure 62
Figure 62: Frequency sweep
STEP 4: Save and validate project
Click on File → Save
Click on HFSS → Validation Check…
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STEP 5: HFSS Analyser
HFSS → Analyze All
Once the 3D design is opened, a window will appear in which user can modify or confirm current
dimension values – Figure 63.
Figure 63: Waveguide and reference antenna dimensions
Next, by clicking on Radiation → Infinite Sphere1 right-click, select Compute Antenna
Parameters. A window with far-field antenna parameters will appear (Figure 64).
Figure 64: Results of reference antenna parameters
Figure 64 shows various parameters, but important ones are Peak Directivity, Peak and Realized
Gain. These parameters are described in chapter 4.1 and numerical expressions (4.6) and (4.9)
were given respectively. These values shown in Figure 64 are calculated relative to
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the isotropic radiator. Expressed in logarithmic form or in decibels, these values are
𝐷 = 10 log10(97.988) = 19.912 [dBi]
𝐺𝑃 = 10 log10(104.25) = 20.181 [dBi]
𝐺𝑅 = 10 log10(103.89) = 20.166 [dBi]
These values are approximately equal to those calculated in chapter 5.2.
The following two parameters are related to the power ratio of the incident and reflected
antenna wave. The first parameter is S11 or return loss – RL and represents the amount of power
that has been reflected from the antenna. According to the Figure 65, it can be concluded that
the antenna preforms best, i.e. resonates at 7.6869 GHz, since the value of S11 is the smallest, -
29.911 dB. Consequently, the antenna works slightly worse at 9.2222 GHz, because the loss is
higher than the loss at 7.6869 GHz, -22.4398 dB. Return loss is also known as reflection
coefficient – 𝜞, and it can be expressed as
Γ =𝑉𝑆𝑊𝑅 − 1
𝑉𝑆𝑊𝑅 + 1=
1.0660 − 1
1.0660 + 1= 0.03195 (5.2)
𝑅𝐿 = 20 log Γ = 20 log 0.03195 = −29.911 dB (5.3)
To generate S11 parameter click on Results → Create Modal Solution Data Report → Rectangular
plot. Under Geometry select Infinite Sphere1, under Category select S Parameter, and under
Function select dB. Click on New Report.
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Figure 65: Return Loss
The second parameter is a voltage standing wave ratio or VSWR (Figure 66), which represents
the measure for power that has been reflected from the antenna, due to impedance mismatch.
The VSWR is expressed as a real positive number, from 1 to ∞. For antennas, it is preferable
that the value of VSWR is low. At low VSWR values, the antenna has better performances. Ideal
value of VSWR is around 1, meaning that all the power of the incident wave is delivered to the
antenna, i.e. no power of the incident wave is reflected from the antenna.
To create a VSWR report, follow the steps on creating the S11 parameter report. Under Category,
select VSWR and click New Report.
By observing the graph of the pyramidal horn antenna shown in Figure 66, it can be concluded
that the VSWR is the lowest at 7.6869 GHz, 1.0660, which indicates that most of the power is
transmitted by the antenna.
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Figure 66: VSWR
The next three parameters are directivity, gain, and radiated electric field, described in chapter
4.1. To generate reports for directivity, gain, and radiated electric field click on Results → Create
Far Field Report → Radiation Pattern. For directivity report under Geometry select Infinite
Sphere1, under Category select Directivity, under Quantity select DirTotal, and under Function
select dB.
Figure 67: Directivity
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Figure 67 depicts value of directivity, expressed in decibel units. Its maximum value ranges
somewhere between 20 dB and 25 dB (orange colour).
For gain report under Geometry select Infinite Sphere1, under Category select Gain, under
Quantity select GainTotal, and under Function select dB. Similarly, antenna gain has the same
value as directivity, hence the expressions (4.8) and (4.9). Gain value of pyramidal horn antenna
is depicted in Figure 68.
Figure 68: Gain
For radiated electric field report under Geometry select Infinite Sphere1, under Category select
rE, under Quantity select rETotal, and under Function select dB. Figure 69 depicts antenna’s
radiated electric field. It represents the overall strength of the electric field in the given
direction.
Looking at the graphs of all three parameters, they seem to be similar, but differ in intensity,
more precisely, in decibel level. Directivity and gain are practically the same considering their
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value, while radiated electric field has the maximum decibel value between 35 dB and 40 dB
(orange colour).
Figure 69: Total radiated electric field
Additionally, the following two parameters describe the strengths of the electric (Figure 70) and
magnetic (Figure 71) fields spreading along the antenna into free-space. To generate electric
field overlay, select HFSS → Fields → Plot Fields → E → Mag_E. Click OK.
The electric field strength is expressed in volts per meter [V/m], while the intensity of the
magnetic field is expressed in ampere per meter [A/m]. In essence, both of them represent the
intensity of the electromagnetic field generated by the antenna, and as such, they propagate
into the free-space. Figure 70 shows the intensity of the electric field radiation. The strongest
radiation is in the waveguide, and it weakens as the wave travels along the waveguide toward
the antenna aperture. It is reasonable that the intensity is higher in the waveguide and it
weakens as the waves are approaching the antenna aperture, since energy of EM waves drops
in relation to the distance. To generate magnetic field overlay, select HFSS → Fields → Plot
Fields → H → Mag_H. Click OK. The intensity of the magnetic field (Figure 71) is more noticeable
than the intensity of the electric field.
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Figure 70: Electric field strength
Figure 71: Magnetic field strength
88
The last two parameters represent the electric field in the E- and H- plane. They represent
reference planes for linearly polarized microwave components, such as antennas, waveguides,
etc. E-plane (Figure 72) is the parameter that represents the direction of maximum antenna
radiation, and it contains the electric field vector. This parameter determines the polarization
of the antenna. If it is a vertically polarized antenna, then the E-plane aligns with the vertical or
elevation plane. Otherwise, if it is a horizontally polarized antenna, then the E-plane coincides
with the horizontal or azimuth plane. As for H-plane (Figure 73), if the antenna is vertically
polarized, then the H-plane coincides with the horizontal or azimuth plane, while it coincides
with vertical or elevation plane if the antenna is horizontally polarized.
To generate these reports, right-click on Results → Create Far Field Report → Radiation Pattern.
For E-plane under Geometry, select Infinite Sphere1, under Primary Sweep: Theta select All,
under Families, select Phi → Edit and select 0deg. Finally, select rE under Category, rETotal
under Quantity, and dB under Function and click on New Report.
Figure 72: Electric field in E-plane
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For H-plane under Geometry, select Infinite Sphere1, under Primary Sweep: Theta select All,
under Families, select Phi → Edit and select 90deg. Finally, select rE under Category, rETotal
under Quantity, and dB under Function and click on New Report.
Figure 73: Electric field in H-plane
This concludes design, simulation and generated results for calculated antenna dimensions
from chapter 5.2.
5.5 Deviations of the reference pyramidal horn antenna
The final step is to determine antenna dimensions deviation and to observe how deviations will
affect the antenna parameters. The aim is to select an individual dimension, change it within
the range of -10 mm to +10 mm with 1 mm step, graphically present the obtained numerical
values, and determine the boundaries of antenna dimensions in which the deviation values will
meet the criteria, in order to physically deploy the antenna with as little loss as possible.
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The first dimension taken into consideration is the waveguide length. The dimension values are
shown in Figure 74. Waveguide length marked with orange colour having numerical value of
38.1 mm, represents an original or calculated value. The parameter (marked with green colour)
results were obtained, as the value of the waveguide length changed throughout the simulation.
The change in dimension value varied between -10 mm to +10 mm from the original value.
Figure 74: Waveguide length
In addition to the numeric values shown in Figure 74, a graph was presented detailing the above
values, and indicating how the change of the waveguide length value affected the parameter
results. Figure 75 represents an antenna gain, directivity, return loss, and VSWR values in
graphical form. It is noticeable that by changing the waveguide length, there is no major
deviation from the original value in terms of parameters. Values marked with green colour
represent the best results obtained from the simulation, while those marked with black colour
represent the original ones. The numerical values shown on first graph represent the power
ratio of the pyramidal horn antenna. These numerical values can be expressed in decibel form
as follows
𝐺(𝑑𝐵𝑖) = 10 log10(104.78) = 𝟐𝟎. 𝟐𝟎 [dBi]
𝐷(𝑑𝐵𝑖) = 10 log10(99.001) = 𝟏𝟗. 𝟗𝟔 [dBi]
𝐺(𝑑𝐵𝑖) = 10 log10(103.89) = 𝟐𝟎. 𝟏𝟕 [dBi]
𝐷(𝑑𝐵𝑖) = 10 log10(97.988) = 𝟏𝟗. 𝟗𝟏 [dBi]
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Figure 75: Graphical representation of simulated parameters – waveguide length
92
It is noticeable that there are minor difference between these values, ranging between 0.03 dBi
and 0.05 dBi, which is not drastic considering antenna losses. Besides, both gain and directivity
best results were achieved at 35.1 mm waveguide length. In addition, return loss and VSWR are
practically similar, but they differ in terms of value and interpretation. The connection of the
above parameters lies in the formulas for calculating reflection coefficient and return loss (see
(5.2) and (5.3)). In order for the antenna to have small losses during transmission, it is necessary
to consider the fact that the optimal value of the VSWR parameter is below 1. In that case, no
power is reflected from the antenna, and thus there are no unwanted losses.
Next dimension taken into consideration is waveguide width, which represents a wider side of
the waveguide. The dimension values and obtained values from the simulation are depicted in
Figure 76. Waveguide width values varied between -10 mm and +10 mm from its original value
(28.4988 mm), indicated with purple colour.
Figure 76: Waveguide width
Furthermore, values presented on graphs shown in Figure 77, depict a more detailed view of
the simulated parameters values. The numerical values shown on the first graph in Figure 77
describe the power ratio of pyramidal horn antenna, while other two graphs represent values
of return loss and VSWR.
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Figure 77: Graphical representation of simulated parameters – waveguide width
94
These numerical values can be expressed in decibel form as follows
𝐺(𝑑𝐵𝑖) = 10 log10(105.59) = 𝟐𝟎. 𝟐𝟒 [dBi]
𝐷(𝑑𝐵𝑖) = 10 log10(99.713) = 𝟏𝟗. 𝟗𝟗 [dBi]
𝐺(𝑑𝐵𝑖) = 10 log10(103.89) = 𝟐𝟎. 𝟏𝟕 [dBi]
𝐷(𝑑𝐵𝑖) = 10 log10(97.988) = 𝟏𝟗. 𝟗𝟏 [dBi]
In this case, the difference between the simulated values ranges between 0.07 dBi and 0.08 dBi,
depending on the observed parameter, gain or directivity. In addition, it is noticeable that value
for gain is achieved at 37.4988 mm, while for directivity, it is achieved at 18.4988 mm.
Third dimension that was submitted for change was waveguide height, which represents the
narrower side of the waveguide. The dimension values are shown in Figure 78, where red colour
indicated waveguide’s width original value, 12.6238 mm. The change in dimension value varied
between -10 mm to +10 mm from the original value. A more detailed overview of simulated
parameters is graphically presented in Figure 79.
Figure 78: Waveguide height
The numerical values shown on the first graph in Figure 79 describe the power ratio of pyramidal
horn antenna, while other two graphs represent values of return loss and VSWR.
95
Figure 79: Graphical representation of simulated parameters – waveguide height
96
These numerical values can be expressed in decibel form as follows
𝐺(𝑑𝐵𝑖) = 10 log10(107.01) = 𝟐𝟎. 𝟐𝟗 [dBi]
𝐷(𝑑𝐵𝑖) = 10 log10(100.26) = 𝟐𝟎. 𝟎𝟏 [dBi]
𝐺(𝑑𝐵𝑖) = 10 log10(103.89) = 𝟐𝟎. 𝟏𝟕 [dBi]
𝐷(𝑑𝐵𝑖) = 10 log10(97.988) = 𝟏𝟗. 𝟗𝟏 [dBi]
There is a noticeable difference in directivity and gain values, where they vary between 0.1 dBi
and 0.12 dBi, unlike the previous results where the difference was considerably smaller. Both
gain and directivity best results were achieved at 22.6238 mm waveguide height. These
matching parameter results (at the same dimension value) are also noticeable at the waveguide
length dimension → 35.1 mm.
The fourth dimension is antenna aperture width which represents the expansion of the
waveguide in the H-plane. The dimension values and obtained values from the simulation are
depicted in Figure 80. Aperture width values varied between -10 mm and +10 mm from its
original value (144.5 mm), indicated with magenta colour.
Figure 80: Aperture width
Values from the first graph in Figure 81 represent the power ratio of the antenna, while other
two represent values of return loss and VSWR.
97
Figure 81: Graphical representation of simulated parameters – aperture width
98
These numerical values can be expressed in decibel form as follows
𝐺(𝑑𝐵𝑖) = 10 log10(103.89) = 𝟐𝟎. 𝟏𝟕 [dBi]
𝐷(𝑑𝐵𝑖) = 10 log10(98.169) = 𝟏𝟗. 𝟗𝟐 [dBi]
𝐺(𝑑𝐵𝑖) = 10 log10(103.89) = 𝟐𝟎. 𝟏𝟕 [dBi]
𝐷(𝑑𝐵𝑖) = 10 log10(97.988) = 𝟏𝟗. 𝟗𝟏 [dBi]
The best results and the original values are equal, there are no deviations, but these values were
not achieved at the same dimension value. Namely, the gain value was achieved at 144.5 mm,
while the directivity value was achieved at 141.5 mm.
The last dimension is antenna aperture height which represents the expansion of the waveguide
in the E-plane. The dimension values and obtained values form the simulation are depicted in
Figure 82. Aperture height values varied between -10 mm and +10 mm from its original value
(111.3 mm), indicated with cyan colour. A more detailed overview of simulated parameters is
graphically presented in Figure 83.
Figure 82: Aperture height
Values from the first graph in Figure 83 represent the power ratio of the pyramidal horn
antenna, while other two represent values of return loss and VSWR.
99
Figure 83: Graphical representation of simulated parameters – aperture height
100
These numerical values can be expressed in decibel form as follows
𝐺(𝑑𝐵𝑖) = 10 log10(105.50) = 𝟐𝟎. 𝟐𝟑 [dBi]
𝐷(𝑑𝐵𝑖) = 10 log10(99.473) = 𝟏𝟗. 𝟗𝟖 [dBi]
𝐺(𝑑𝐵𝑖) = 10 log10(103.89) = 𝟐𝟎. 𝟏𝟕 [dBi]
𝐷(𝑑𝐵𝑖) = 10 log10(97.988) = 𝟏𝟗. 𝟗𝟏 [dBi]
There is a slight difference in directivity and gain values, where they vary between 0.06 dBi and
0.07 dBi. Once more, gain and directivity were achieved at the same dimension value, in this
case at 107.3 mm.
5.6 Mathematical calculation of the practical pyramidal horn antenna
In this chapter, a dimensions calculation will take place. This process is identical to the one in
chapter 5.2. Here, the desired gain value is set to be 16 dBi. Non-decibel form of desired gain is
𝑮𝟎 = 10𝐺(𝑑𝐵)
10 = 101.6 = 39.811. Operating frequency, as well as operating wavelength remain
the same: 𝜆 = c
f=
2.998∙108
9∙109 = 𝟑𝟑. 𝟑𝟏𝟏�̇� mm. Waveguide dimensions remain the same as well:
𝒂 = 28.4988 mm and 𝒃 = 12.6238 mm. Initial value 𝝌 is the first step toward antenna
dimensions calculation.
STEP 1:
𝝌 (𝑡𝑟𝑖𝑎𝑙) =𝐺0
2𝜋 ∙ √2𝜋=
39.811
2𝜋 ∙ √2𝜋= 2.52775
By incorporating above value into expression (5.1), left and right side of the expression do not
match. After a close and careful adjustment of the initial value, the value satisfies the equality
of both sides is 𝝌 = 2.2974. The next step determines the values of 𝝆𝒆 and 𝝆𝒉.
101
STEP 2: Using value 𝝌, calculate 𝝆𝒆 and 𝝆𝒉.
𝜌𝑒 = 𝜒 ∙ 𝜆 = 2.2974 ∙ 𝜆 = 2.2974 ∙ 3.331̇ = 76.529 𝑚𝑚
𝜌ℎ =𝐺0
2
8𝜋3∙1
𝜒∙ 𝜆 =
39.8112
8𝜋3∙1
𝜒∙ 𝜆 = 2.7811 ∙ 𝜆 = 2.7811 ∙ 3.331̇ = 92.644 𝑚𝑚
By obtaining above values, the next step is to determine the values of the pyramidal horn
antenna aperture dimensions, 𝒂𝟏 and 𝒃𝟏.
STEP 3: Calculate aperture dimensions by using obtained values from STEP 2.
𝑎1 = √3𝜆 ∙ 𝜌ℎ =𝐺0
2𝜋∙ √
3
2𝜋𝜒∙ 𝜆 =
39.811
2𝜋∙ √
3
2𝜋 ∙ 2.2974∙ 𝜆 = 2.8885 ∙ 𝜆 = 96.22 𝑚𝑚
𝑏1 = √2𝜆 ∙ 𝜌𝑒 = √2𝜒 ∙ 𝜆 = √2 ∙ 2.2974 ∙ 𝜆 = 2.14 ∙ 𝜆 = 71.404 𝑚𝑚
Finally, the last step that will determine the lengths of the pyramidal horn antenna. If the values
from STEP 4 are equal, then the criteria for the physical realization of the pyramidal horn
antenna is achieved.
STEP 4: Determine the pyramidal horn lengths, by using the obtained values 𝒂𝟏 and 𝒃𝟏.
𝑝𝑒 = (𝑏1 − 𝑏) ∙ √[(𝜌𝑒
𝑏1)2
−1
4] = (2.14 ∙ 𝜆 − 0.37897 ∙ 𝜆)√[(
2.2974𝜆
2.14𝜆)2
−1
4] = 1.6728 ∙ 𝜆
= 55.7235 𝑚𝑚
𝑝ℎ = (𝑎1 − 𝑎) ∙ √[(𝜌ℎ
𝑎1)2
−1
4] = (2.8885 ∙ 𝜆 − 0.85553 ∙ 𝜆)√[(
2.7811𝜆
2.8885𝜆)2
−1
4] = 1.6728 ∙ 𝜆
= 55.7235 𝑚𝑚
102
5.7 HFSS design procedure of the practical pyramidal horn antenna
The process steps of HFSS design for the practical antenna are identical to the process in the
chapter 5.3. Therefore, it is not necessary to repeat these steps in this chapter. The change was
made in the antenna dimensions from chapter 5.3, over which the obtained values from a new
mathematical calculation in chapter 5.6 were applied. The new antenna dimensions values are
shown in Figure 84.
Figure 84: Waveguide and practical antenna dimensions
Antenna parameters are obtained by right-click on Radiation → Infinite Sphere1, select
Compute Antenna Parameters. A window with far-field antenna parameters will appear (Figure
85).
Figure 85: Antenna parameters
103
Generated parameters, important for the antenna are Peak Directivity, Peak Gain and Peak
Realized Gain. These values expressed in decibel from are
𝐷 = 10 log10(40.321) = 16.055 [dBi]
𝐺𝑃 = 10 log10(41.614) = 16.192 [dBi]
𝐺𝑅 = 10 log10(40.453) = 16.069 [dBi]
These gain values (approximately) correspond to the values obtained in chapter 5.6. The HFSS
model of the practical antenna (Figure 86) is slightly different from the reference antenna,
obtained in the first simulation.
Figure 86: Pyramidal horn antenna with SMA connector
104
The difference is in the excited element. First antenna was excited by the use of the wave port
that served as excitation element, which was used to “insert” electromagnetic waves (signal,
information) into the waveguide. In this case, a wave port has been replaced with a lumped
port. The lumped port (Figure 87) is a type of port used to normalize fields (to an 50 Ω
impedance) within the waveguide and match the transition between coaxial cable and the
waveguide.
Figure 87: Lumped port
In addition, by using a lumped port, only single mode field distribution can be present within
the waveguide. This eliminates the occurrence of higher modes, which reduces the possibility
of interference. In order to excite antenna with a lumped port, an SMA (Figure 88) connector
needs to be modelled first. SMA connector can be represented as coaxial cable, which consists
of inner and outer conductor (perfect electric conductor - PEC) and insulator (most commonly
made from Teflon or PTFE).
Figure 88: SMA connector
105
The mentioned SMA connector or coaxial cable has three layers; inner conductor or probe,
which serves as signal “injector”, insulator or dielectric material that separates conductors, and.
outer conductor, that serves as ground potential and it isolates signal (suppressed by
interference from the outside ) from the inner conductor.
For this case, the dimensions of these three layers are as follows
1. inner conductor (PEC) has radius = 0.635 mm and height = 12.3 mm
2. insulator (PTFE) has radius = 1.565 mm and height = 6 mm
3. outer conductor (PEC) has radius = 2.065 mm and height = 4.4 mm
In order to assign a lumped port in HFSS, click on Edit → Select → Faces. Select the insulator
part of the SMA connector. Then, right click on it, Assign Excitation → Lumped Port…, then click
Next. Under Integration Line, click New Line. Draw the integration line from one edge of the
inner conductor to the other. Click Next. In final step, click Renormalize All Modes, select Full
Port Impedance, and set it to 50 ohms. Click Finish.
5.8 Results of the practical pyramidal horn antenna
In this chapter, results generated by the simulation process will be presented and explained.
These results represent pyramidal horn antenna parameters, which are crucial for the physical
deployment of the antenna itself. Again, the necessary steps (setting up the operating
frequency, far-field setup, analyse and validation of the project) will not be examined, as they
are explained, in detail in chapter 5.4.
The first two parameters are return loss or S11 parameter and VSWR, both of them explained in
chapter 5.4. In Figure 89, it is evident, that antenna resonates at 9.3030 GHz, with the value of
-34.7860 dB. The return loss is depicted in Figure 89.
106
Figure 89: Return Loss
To generate S11 parameter, click on Results → Create Modal Solution Data Report → Rectangular
plot. Under Geometry select Infinite Sphere1, under Category select S Parameter, and under
Function select dB. Click on New Report. Second parameter is VSWR. In Figure 90, VSWR is under
the value of 2, which is preferable.
Figure 90: VSWR
Identical to the S11 parameter, VSWR also shows its best results at 9.3030 GHz, at the value of
1.0371. By using expression (5.2) and (5.3), the following values are obtained
107
Γ =𝑉𝑆𝑊𝑅 − 1
𝑉𝑆𝑊𝑅 + 1=
1.0371 − 1
1.0371 + 1= 0.01821
𝑅𝐿 = 20 log Γ = 20 log 0.01821 = −34.793 dB
To create a VSWR report, follow the steps on creating the S11 parameter report. Under Category,
select VSWR and click New Report.
Following three parameters are directivity, gain and radiated electric field. To generate reports
for directivity, gain, and radiated electric field, click on Results → Create Far Field Report →
Radiation Pattern. For directivity report under Geometry select Infinite Sphere1, under Category
select Directivity, under Quantity select DirTotal, and under Function select dB.
Figure 91: Directivity
Figure 91 represents the value of antenna directivity, expressed in decibels. Maximum value is
ranged between 25 dB and 30 dB (red colour). For gain report under Geometry select Infinite
Sphere1, under Category select Gain, under Quantity select GainTotal, and under Function select
dB.
108
Figure 92: Gain
Value of the antenna gain (Figure 92) is identical to antenna directivity, which can be confirmed
by expressions (4.8) and (4.9). Lastly, to obtain radiated electric field report under Geometry
select Infinite Sphere1, under Category select rE, under Quantity select rETotal, and under
Function select dB. It is depicted in Figure 93. This antenna parameter represents the overall
strength of the electric field in the given direction. Its maximal value (orange to red colour) is
somewhere between 35 dB and 40 dB.
Figure 93: Total radiated electric field
109
Unlike previous simulation, where antenna had larger dimensions, these three parameters,
shown as 3D plot, have slightly rough texture, more sharp side and back lobes, and the major
lobe has eminent protuberance (lump) around green area on 3D plot. This can be due to the
fact that antenna has coaxial cable “inserted” into the waveguide which can cause some signal
distortion and consequentially, a not so attractive graphical representation of the results.
Penultimate two parameters represent the electric filed in E-plane and in H-plane. These planes
are presented in two-dimensional plot. Electric fields are propagating along E-plane, and
consequentially magnetic field lines propagate along H-plane. Figure 94 depicts both planes.
Figure 94: Electric field in E- and H-plane
To generate E-plane, right-click on Results → Create Far Field Report → Radiation Pattern. For
E-plane under Geometry select Infinite Sphere1, under Primary Sweep: Theta select All, under
Families, select Phi → Edit and select 0deg. Finally, select rE under Category, rETotal under
Quantity, and dB under Function and click on New Report. Moreover, for H-plane, change Phi
value to 90deg, and click on New Report.
Lastly, two remaining parameters represent the strength or intensity of electric and magnetic
fields (Figures 95 and 96), that are propagating along the waveguide, from the probe to the
antenna aperture, and finally into the free space. To generate electric field overlay, select HFSS
110
→ Fields → Plot Fields → E → Mag_E. Click OK, while magnetic field overlay is generated by
selecting HFSS → Fields → Plot Fields → H → Mag_H.
Figure 95: Electric field strength
Figure 96: Magnetic field strength
111
This is where the practical antenna simulation process is completed. The results show that the
antenna has good performance which are confirmed through generated simulation results.
Namely, Figure 85 shows that the gain and realized gain are identical to those initially set.
Secondly, the value of the VSWR that is within the limits (under 2) indicates that the antenna
emits most of the power and that a small part of the energy reflects back to the antenna. To
support numerical values that are initially set, 3D plots shown in Figure 91-93, indicate a well-
known radiation pattern, which according to these images appears to be qualitative. It all
implies to the fact that the calculated antenna dimensions, as well as generated antenna
parameters, are an excellent indicator for the physical realization of the antenna itself.
112
6. CONCLUSION
This thesis is composed of several parts. Studies of the influence of the electric and
magnetic field have concluded that their interaction induces the creation of an electromagnetic
wave. Electromagnetic waves are dispersed freely in space. The idea is to limit their propagation
within a certain structure. We have solved this part with a metal conductive structure - a
waveguide. Namely, with the aid of Maxwell's equations and the manifestations of various wave
phenomena, it is possible to perceive the actual nature of these waves, and how they "behave"
in space. The mentioned interaction between the electric and magnetic fields is described by
the above equations. Additionally, it is important to note that waves on their path encounter
obstacles (such as waveguide walls) and are thus reflected, thereby enhancing energy
propagation. For the purpose of this thesis, a rectangular waveguide of the following
dimensions has been selected: the wider side of the waveguide 𝒂 is 28.4988 mm, while the
narrower side of the waveguide 𝒃 is 12.6328 mm. The side dimensions determine the range of
frequencies that are appropriate for the operation of this waveguide. This waveguide operates
in the range from 7 GHz to 10 GHz. The mentioned frequencies are in the range from 3 GHz to
30 GHz. This range is ideal for this master thesis for two reasons: the first one is to simulate an
antenna for the 9 GHz band. The other one refers to small wavelengths which enable the
electromagnetic waves to be directed in the narrow beam. In this way, better directivity and
gain are achieved.
To accomplish this, it is necessary to propagate the waves in the waveguide by the dominant
mode. This is achieved by determining the cut-off frequency above which it is possible to excite
the formation and spreading of the waves in the waveguide. In our case, the value of the cut-
off frequency is 5.25987 GHz. Consequently, the limit for the propagation of waves without the
occurrence of higher modes is determined and it is 10.52 GHz. Furthermore, as far as the
antenna is concerned, the pyramidal horn antenna is chosen for this thesis because it is ideal
for high frequencies and achieving good directivity and gain. Besides directivity and gain, other
113
significant antenna parameters are the standing wave ratio, return loss, radiation intensity and
others. Of course, the field regions should not be neglected, especially the far-field in which the
antenna distributes EM waves evenly. The aim is to make a calculation of the mathematical
model of the antenna which gives a confirmation for the physical realization of the antenna.
Namely, the mathematical model consists of the initial value 𝝌 and equation (5.1) for which it
was necessary to determine the desired gain of the antenna, and with the help of the waveguide
dimensions, to include those values in that equation. The problem occurred while equating the
left and right side of equation (5.1), and initial value was to be modified, to ensure that the
condition of equality was met. Once the condition was met, it was possible to calculate the
antenna dimension values. The calculation is described in chapter 5.2 (and 5.6) of this thesis,
and consists of four steps. In the last step, the values of dimensions 𝒑𝒆 and 𝒑𝒉 should be
matched, in order to satisfy the requirement for physical realization of the antenna and to start
the simulation process.
In the process of simulating an antenna, it was necessary to get acquainted with the interface
of the ANSYS Electromagnetic Suite 18. For the purpose of this simulation, a HFSS simulator was
used, which is based on the finite element method. With HFSS, it was possible to carry out a
structural analysis of the antenna. Chapter 5.1 describes the process of making a 3D antenna
model which is identical to the practical antenna model. The aim was to create a model antenna
according to the dimensions obtained from a mathematical calculation. The design procedure
began with the waveguide construction and by entering its values into the simulator. Further,
by expanding the waveguide sides, the pyramidal horn antenna was obtained, whose values
were also entered. Furthermore, it was necessary to limit the space around the antenna, by
creating a region that would strive to propagate the energy of electromagnetic waves towards
its aperture. Since two simulations were performed, one for the reference antenna that was fed
directly through the waveguide, while the other, practical antenna was fed via coaxial cable and
SMA connector.
114
When looking at the results of the generated antenna parameters, it is apparent that the
reference antenna has better results than the practical one. The reason for this is the embedded
SMA connector on the waveguide, which has affected the structure of the practical antenna.
Moreover, the results in both cases are approximate to the results obtained through the
calculation. In the example of the reference antenna, the desired gain value was 19.7 dB, while
the value obtained through the simulation was 20.166 dB. The value of the generated gain is
somewhat higher because the initial value of the mathematical model had to be altered, in
order to achieve equality of left and right side of the equation (5.1). In the example of a practical
antenna, the desired gain value was 16 dB, while the value of the simulated one was 16.069 dB.
In this case, it can be verified that by embedding SMA connector, it is possible to make good
adjustments to the waveguide and free space transition.
Furthermore, the ratio of standing waves or VSWR at the reference antenna is 1.066, while at
the practical antenna it is slightly smaller, 1.0371. In both examples, the values are optimal and
the percentage of effective radiated power is above 99%. Additionally, graphical display of gain
and directivity coincides with numerical values. They are displayed in 3D form, and the value
ranges are marked with colours representing a certain power strength, expressed in decibels.
The ultimate goal was to calculate the deviations in relation to the antenna dimensions. Namely,
the idea was to gradually change individual dimension value, in range from +10 mm to -10 mm
from the original value. Five dimensions have been considered: length, width and height of the
waveguide, as well as the width and height of the antenna aperture. The value of each change
is entered into the simulation and the generated values of the four parameters (gain, directivity,
VSWR and return loss) were recorded. The total number of simulations needed to determine
the deviation of dimensions was 100, five dimensions multiplied with 20 (range from +10 mm
to -10 mm). In this way, it was possible to gradually monitor the parameter change in relation
to the change of the antenna dimension values. Obtained results were presented in form of
tables and graphs. In addition, the goal was to determine the deviation limits in which the result
of the antenna parameters would not be compromised, while changing the values of the
115
dimensions of the antenna. It is necessary to emphasize that the modification of the antenna's
dimension does not impair its performance. Finally, each section that was covered in this thesis
led to the conclusion that it is possible to physically deploy the pyramidal horn antenna for
future measurements of polarization losses. This thesis introduces instructive things, and any
prospective reader can easily approach the radiocommunication domain, such as waveguides,
antennas, electromagnetic waves, and more.
116
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