theoretical analysis and simulation of microwave...

UPTEC F 17008

Examensarbete 30 hpJanuari 2017

Theoretical analysis and simulation of microwave-generation from a coaxial vircator

Martin Hägg

Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box 536 751 21 Uppsala Telefon: 018 – 471 30 03 Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student

Abstract

Theoretical analysis and simulation ofmicrowave-generation from a coaxial vircator

Martin Hägg

High-power microwave, HPM, systems can be used as non-lethal weaponswith the ability to destroy or disturb electronics, by damaging internal circuits andinducing high currents. Today microwave sources are being developed with peakpowers exceeding 1 GW, one of these devices is the vircator, a narrowband sourcewhich is unique to the HPM community. In order to understand and developmicrowave sources like the vircator it is necessary to have computer models, assimulations gives an invaluable understanding of the mechanisms involved duringoperation, saving time and development costs.This thesis presents the results from a theoretical analysis and a simulation study usinga well known electromagnetic particle-in-cell code, Computer Simulation TechnologyParticle Studio. The results are then compared to measured data from a HPM system,the Bofors HPM Blackout. The results show that CST PS can be used to design andstudy the coaxial vircator with good results.

ISSN: 1401-5757, UPTEC F 17008Examinator: Tomas NybergÄmnesgranskare: Dragos DancilaHandledare: Patrik Hermansson

iii

Sammanfattning

Mikrovågsteknik har länge varit ett viktigt område för militärt bruk, då enav de mest väsentliga militära tillämpningarna, radarn använder sig utavmikrovågor för att kunna detektera farkoster och robotar. Efter attmikrovågor började användas i militära applikationer har en del forskninggått till att försöka skapa vapen som använder sig utav mikrovågor. Ettvapen som utnyttjar mikrovågor kan ha många fördelar jämfört medtraditionella vapen. Det skulle kunna användas mot obemannade farkoster,missiler och kommersiell elektronik samtidigt som vapnet inte skadarmänniskor.

Genom att generera mikrovågs pulser med hög effekt, i gigawattområdet,kan man förstöra eller störa elektronik på avstånd. Dagens mikrovågsvapenanvänder sig utav elektrisk energi och omvandlar denna energi tillelektromagnetisk strålning med hjälp av en pulsgenerator och en strålkälla.Lagringsenheten för denna energi kan vara något så vanligt som ett 9-voltsbatteri och pulskällan består oftast av ett kondensatorpaket som laddas urunder kort tid och skapar en puls med hög spänning. Mikrovågornagenereras sedan i strålkällan för att därefter riktas mot målet med hjälp envågledare och en antenn. När mikrovågorna träffar målet kan de förstöraelektriska komponenter eller störa elektronik och sensorer så att exempelvisen robot missar sitt mål.

Elektromagnetiska vågor uppstår då laddningar accelererar eller bromsasin, då delar av deras rörelseenergi omvandlas till elektromagnetiskstrålning. Vanligtvis genereras mikrovågor med hög effekt i vakuumtuberdär elektroner accelereras och bromsas in under kontrollerade former ochtillåts därmed skapa mikrovågor. Det finns flera typer av strålkällor för attgenerera mikrovågor, bland annat i hemmet så producerar mikrovågsugnenmikrovågor med hjälp av en magnetron. På BAE Systems Bofors ABanvänds en virkator, virtuell katod oscillator som mikrovågskälla i derasicke-dödande strålvapen, Bofors HPM Blackout.

För att kunna designa och optimera denna strålkällan som används är detnödvändigt med datorsimuleringar och teoretiska beräkningar. I dettaprojekt används ett datorprogram, CST Particle Studio i syfte att modelleraden aktuella strålkällan, virkatorn. Programmet använder en metod somkallas för PIC, Particle-In-Cell. Denna metod används för att kunna följa deladdade partiklarnas bana genom de elektromagnetiska fälten, i vårt fallelektroner som rör sig inuti strålkällan. Resultatet från datorsimuleringarnajämförs sedan med teoretiska beräkningar och mätdata från det aktuellasystemet för att avgöra om programmet är lämpligt för design ochoptimering av virkatorn.

v

AcknowledgementsFirst of all I would like to thank my supervisor Patrik Hermansson andmy colleagues at BAE, Mats Jansson and Denny Åberg who have helpedme with guidance and numerous questions throughout the process of thismaster thesis. I would also like to thank my supervisor Dragos Dancila atUppsala University.

vii

Contents

Acknowledgements v

1 Introduction 31.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Project Description . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Theory 52.1 Characteristics of High-Power Microwaves . . . . . . . . . . 52.2 Microwave Sources . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Vircator Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.1 Vircator Operating Principles . . . . . . . . . . . . . . 82.3.2 Field Emission . . . . . . . . . . . . . . . . . . . . . . 102.3.3 Space Charge Limited Current . . . . . . . . . . . . . . 11

Child-Langmuir Law . . . . . . . . . . . . . . . . . . . 12Langmuir-Blodgett law . . . . . . . . . . . . . . . . . 14

2.3.4 Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . 152.3.5 Frequency Characteristics . . . . . . . . . . . . . . . . 152.3.6 Mode Characteristics . . . . . . . . . . . . . . . . . . . 16

2.4 The High-Power Microwave System . . . . . . . . . . . . . . 172.4.1 Marx Generator . . . . . . . . . . . . . . . . . . . . . . 182.4.2 Microwave Source . . . . . . . . . . . . . . . . . . . . 192.4.3 Waveguide and Antenna . . . . . . . . . . . . . . . . 19

2.5 PIC-Solver and CST PS . . . . . . . . . . . . . . . . . . . . . . 202.5.1 Finite Integration Technique . . . . . . . . . . . . . . . 20

3 Method 213.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.1 Test Structure . . . . . . . . . . . . . . . . . . . . . . . 233.2.2 Excitation of the Vircator . . . . . . . . . . . . . . . . . 243.2.3 Particle Source . . . . . . . . . . . . . . . . . . . . . . 263.2.4 Waveguide Ports . . . . . . . . . . . . . . . . . . . . . 26

3.3 Monitor and Probes . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Results 294.1 Current and Particle Properties . . . . . . . . . . . . . . . . . 29

4.1.1 Anode Current . . . . . . . . . . . . . . . . . . . . . . . 314.2 Anode-Cathode Voltage . . . . . . . . . . . . . . . . . . . . . . 314.3 Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.4 Virtual Cathode Characteristics . . . . . . . . . . . . . . . . . 33

4.4.1 Phase Space of the Particles . . . . . . . . . . . . . . . 354.5 Output Characteristics . . . . . . . . . . . . . . . . . . . . . . 36

viii

4.5.1 Time-Domain . . . . . . . . . . . . . . . . . . . . . . . 364.5.2 Frequency-Domain . . . . . . . . . . . . . . . . . . . . 364.5.3 Output Power . . . . . . . . . . . . . . . . . . . . . . . 38

5 Discussion 41

6 Conclusions 43

Bibliography 45

ix

List of Figures

2.1 Wideband vs. narrowband power spectrum. . . . . . . . . . 72.2 Three types of vircator, a) axial vircator, b) coaxial vircator

and c) reflex triode. . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Vircator regions. . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Phase space for a axial vircator, from a CST simulation. . . . 92.5 Energy level scheme for field emission for a metal. Picture

from Wikipedia, [12]. . . . . . . . . . . . . . . . . . . . . . . . . 112.6 Planar diode in vacuum, each conducting plate is approxi-

mated as infinitely long in the vertical direction. . . . . . . . 122.7 The electric field for the TM01 mode in a circular waveguides

cross section (left) and a longitudinal cut (right). . . . . . . . 172.8 The electric field for the vertical polarization of the TE11 mode

in a circular waveguides cross section (left) and a longitudinalcut (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.9 Photo of the Bofors HPM Blackout system with a compactMarx generator. . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.10 Block diagram over a HPM weapon system. . . . . . . . . . . 182.11 Schematic figure over the Marx generator. . . . . . . . . . . . 19

3.1 A overview of the experimental HPM system together withthe HV-supply. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 HPM system inside the anechoic chamber. . . . . . . . . . . . 223.3 Side-view of the inward-emitting coaxial vircator structure. . 223.4 Free field D-dot probes. . . . . . . . . . . . . . . . . . . . . . . 233.5 Coaxial vircator test structure in CST. . . . . . . . . . . . . . 243.6 Anode foil model in CST. . . . . . . . . . . . . . . . . . . . . . 243.7 Discrete voltage ports inside the vircator structure. . . . . . . 253.8 Excitation signal used in the simulations. . . . . . . . . . . . 253.9 Particle source on the inside of the cathode cup. . . . . . . . 263.10 A view of the two waveguide ports. . . . . . . . . . . . . . . 27

4.1 Plot showing the simulated total current through the vircator. 304.2 Experimental total current through the vircator. . . . . . . . 304.3 Number of macro-particles emitted. . . . . . . . . . . . . . . . 314.4 Simulated current at the anode. . . . . . . . . . . . . . . . . . . 314.5 Simulated AK-voltage. . . . . . . . . . . . . . . . . . . . . . . 324.6 Experimental AK-voltage. . . . . . . . . . . . . . . . . . . . . 324.7 Simulated impedance. . . . . . . . . . . . . . . . . . . . . . . 334.8 Experimental impedance. . . . . . . . . . . . . . . . . . . . . 334.9 The virtual cathode evolution between T= 22 ns to T= 43 ns.

At T = 22 ns we can seen that the discrete ports affects theparticles distribution, causing a cross-shaped inner distribution. 34

4.10 The virtual cathode evolution between T= 45 ns to T= 90 ns. 34

x

4.11 Side-view of the particles inside the vircator at T = 32 ns. . . 354.12 Plot showing the x-component of the E-field at T = 70 ns. . . 354.13 Phase space of the particles at T = 70 ns. . . . . . . . . . . . . 364.14 Simulated output signal in time domain. . . . . . . . . . . . . 364.15 Simulated output signal in frequency domain. . . . . . . . . 374.16 Frequency spectrum of the output signal for the experimental

system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.17 View of the phase-depending E-field at the main frequency. . 384.18 Simulated output power in time domain. The purple curve

shows the TM01-mode and the green curve shows the twopolarizations of the TE11 mode and the blue curve shows thetotal output power for all (16) modes. . . . . . . . . . . . . . 38

4.19 Experimental output power, for the TE11 mode. . . . . . . . 39

xi

List of Abbreviations

AK Anode-CathodeCL Child-LangmuirCST Computer Simulation TechnologyDC Direct CurrentEM ElectroMagneticFDTD Finite-Difference Time-DomainFFT Fast Fourier TransformFIT Finite Integration TechniqueHPM High-Power MicrowaveHV High VoltageIC Internal CircuitMILO Magnetic Insulated Line OscillatorPBA Perfect Boundary ApproximationPEC Perfect Electrical ConductorPIC Particle-In-CellPRF Pulse Repetition FrequencyPS Particle Studiorf radio frequencySCL Space-Charge LimitedTE Transverse ElectricTM Transverse MagneticTWT Travelling-Wave TubeUAV Unmanned Aerial VehicleUHV Ultra-High VacuumVC Virtual Cathodevircator VIRtual CAthode oscillaTOR

xiii

Physical Constants

Electron charge e = 1.602 176 620× 10−19 CElectron mass m0 = 9.109 383 56× 10−31 kgPermittivity of Free Space �0 = 8.854 187 82× 10−12 F m−1Planck constant h = 6.626 070 04× 10−34 J sSpeed of Light c0 = 2.997 924 58× 108 m s−1

xv

List of Symbols

D distance mE electric field V m−1

I current AJ current density A m−2

Φ work function JT time sV voltage VW width mα transparency %� permittivity F m−1

f frequency Hzγ Lorentz factorm mass kgω angular frequency rad s−1

φ electric potential Vr radial position mρ charge density C m−1

v speed m s−1

3

Chapter 1

Introduction

BAE Systems is one of the worlds largest defence contractors whichdevelops high technology defence systems for the future. The group hasmore than 84,500 employees around the world and they are represented inSweden by their companies Weapon Systems Bofors and BAE SystemsHägglunds located in Karlskoga, Örnsköldsvik, Stockholm and Linköping.This thesis project was performed at BAE Systems Bofors AB in Karlskoga.

1.1 Background

On today’s modern battlefield electromagnetic effects are becoming a moreand more important field of study. This is mainly due to that the modernsociety and warfare relies heavily on electronics, which means that anelectromagnetic weapon could have devastating effects. A weapon of thiskind could shut down telecommunications and power supplies withoutdestroying buildings, roads and not hurting humans.

The concept of using electromagnetic weapons originates from when theU.S. military observed the effects from electromagnetic shock waves createdby nuclear detonations in the 1940’s. Also the British studied how theycould use HPM, high-power microwaves as a weapon during radar studiesin World War II. Under the last decade extensive research has been made inHPM weapons, currently both anti-electronics and non-lethalanti-personnel weapons are being deployed today [1],[2].

1.2 Project Description

The goal of this project is to participate and contribute to the developmentof a non-lethal directed energy weapon using high-power microwaves toknock out electronics in different target objects.

The main focus is to study the behaviour of the charged particles and thetime varying electromagnetic field inside the radiation source by usingnumerical methods, in this case the software CST Particle studio. The maintask is to investigate if it is possible to simulate the microwave source, thevircator, and get reasonable results that corresponds to experimental resultsmeasured in the lab. The project was an interaction of numericalsimulations and lab work and some analytical derivations.

4 Chapter 1. Introduction

The results of this study can be used to evaluate if CST PS can be used tooptimize the design of the source and study how different geometries andconfigurations used in the microwave source will affect the performance. Agood computational model can help the development of this system andachieve a better understanding about the physics involved.

1.3 Previous Work

At BAE Systems Bofors in Karlskoga there has been research anddevelopment of non-lethal direct energy weapons for over a decade,research around this weapon types are important in order to be able todevelop protection against these kinds of threats. During this perioddifferent kinds of system configurations and microwaves devices has beentested and evaluated. Researchers at BAE has over the years published acouple of papers on their research, such as Frequency Dependence of theAnode-Cathode Gap Spacing in a Coaxial Vircator System and A Compact HighPower Microwave System - IEEE Xplore [3], [4].

5

Chapter 2

Theory

2.1 Characteristics of High-Power Microwaves

The general definition of HPM, High Power Microwaves according to [1] isdevices that exceed 100 MW in peak power and operate on frequenciesbetween 1 to 300 GHz. Conventional microwave sources ranges fromcommunication devices which operates in the lower power region up tolinear colliders which have the highest output power. Recently solid statedevices have begun to take over the conventional microwave sources in thelower power region [5].

Radio frequency energy is created by acceleration and de-acceleration ofcharges in oscillating circuits. Microwaves with a peak power over 100 MWare generated in vacuum electron devices and the devices use the kineticenergy of the electrons, sometimes under the influence of controlling electricand magnetic fields. Microwave vacuum tubes include devices like theMagnetron, TWT (Travelling wave tube), Klystron and MILO (MagneticInsulated Linear Oscillator). Other sources include the Gyrotron, theVircator (Virtual Cathode Oscillator) and the beam plasma generator [6].

The outcome of an HPM attack can either be a hard kill or a soft kill. A hardkill means that the internal components of the device will not function aftera manual restart or after a period of time. By our experience the destructionof the device is due to the semiconductors devices like transistors and IC’s,Integrated Circuits, are put in "forbidden" modes, causing the device todestroy itself. A hard kill that destroys internal circuits is what the militaryoften desires. The soft kill can have different types of scenarios, it candisturb the target while it is exposed to the HPM beam, resulting in e.g. thatthe incoming missile misses its target or that a UAV loses its control signaland crashes. Another scenario is that the target will need a manual restart tofunction again or it will be functioning only after a given time.

All electronics can work as antennas even if it is a bad antenna and there aretwo ways to couple the electromagnetic energy into the device, front doorcoupling and back door coupling. Front door coupling is when theelectromagnetic pulse is coupled into the device through a componentdesigned to receive or transmit signals e.g. sensors, antennas and receivers.Back door coupling is when the electro-magnetic pulse couples into thedevice on frequencies it was not designed to communicate on and it is oftenmore complex. The EM-wave can couple through cables and openings intothe device and expose components. To protect a device from these incoming

6 Chapter 2. Theory

EM pulses the device needs to be fully contained inside a metal structure i.e.a Faraday cage but this will make the device unable to communicate. Evenoptoelectronics are susceptible to EM-waves [7], [8]. The electric fieldstrength required for a severe PC-crash that requires a manual restart isaround 4.5 kV/m, when the optimal coupling angle is known but also lowerfields can cause malfunction [9].

2.2 Microwave Sources

It is common to distinguish between two different kinds of HPM-sources,narrow band and wide band sources. The wide band source sends out asignal with a large bandwidth compared to the narrowband as seen infigure 2.1. There are several advantages and disadvantages with each kindof microwave source. It is easier for a wideband source to couple the energyinto the device, because it will cover more frequencies and since it is notpossible to know in advance which the optimal coupling frequency is, thewideband source will have a better chance to couple the energy into thedevice. A big disadvantage is that the energy density in a wideband pulsewill be far less compared to a narrowband pulse. This could mean that evenif we manage to couple the EM-pulse into the device, we will not have thedesired effect because the energy absorbed by the device is to small to affectthe device. In order to achieve thermal stacking, which is when the timebetween pulses is so short that the heat does not have time to diffuse, weneed that the PRF, Pulse-repetition frequency to be atleast 100 kHz [10].

A narrowband source on the other hand will concentrate the energy in asmall band of frequencies and if it is tuned into the right frequency for thetarget object, the energy will be sufficient to knock out the electronics.However if the device operates on a different frequency it will remainunaffected. A narrowband HPM source that has the ability to changefrequency during operation is considered to be the ultimate solution, if wecould make a frequency sweep during a burst we can find the couplingfrequency and transmit enough energy into the target object to knock it out.Most of the HPM systems today for weapon research are narrowbandsources and a system with the ability to change frequency during operationare highly sought after.

2.3. Vircator Theory 7

FIGURE 2.1: Wideband vs. narrowband power spectrum.

2.3 Vircator Theory

This thesis will focus on the vircator and mainly the coaxial vircator, whichis used as the microwave source in the HPM system at BAE systems.Therefore it is important to know a bit about the physics and the workingprinciples of the vircator.

Vircators are one of the most interesting and popular HPM sources today,mainly because they are simple to build and can be made compact. Theyrequire no external magnetic field and they have a low impedance whichpermits high power operation on low voltage. These devices are capable ofoutput powers of a magnitude of GW in the 1-10 GHz frequency range butthere are also a few drawbacks with the vircators, mainly their lowefficiency which is well below 10%. Another downside is that they oftenhave a problem with gap closure, which causes the current in the vircator toincrease, which will lead to frequency chirps upward over time whichaffects the resonance and lowers the output power. Despite thesedrawbacks, vircators are the most promising devices today and they arevery popular in all countries with an HPM-program.

Virtual cathode oscillators are a type of sources which generates theirradiation when a electron beam is injected inside a waveguide or a cavityand the injected electron beam current surpasses the local space chargelimiting current, resulting in the formation of a virtual cathode. This virtualcathode will then oscillate at approximately the injected beam plasmafrequency and there will also be reflex electron oscillations. Both of thesetwo things will contribute to the generation of microwaves. Sources of thistype includes the axial vircator, coaxial vircator, reflex triode and thefeedback vircator, seen in figure 2.2 [10].

8 Chapter 2. Theory

FIGURE 2.2: Three types of vircator, a) axial vircator, b)coaxial vircator and c) reflex triode.

2.3.1 Vircator Operating Principles

The standard vircator, the axial vircator, can be characterized as a planardiode in vacuum with a anode that is transparent to electrons to someextent. First, the cathode is pulsed with a negative voltage pulse, −V0,leading to field emission of the electrons, which will accelerate towards theanode, which is on a positive potential compared to the cathode, in this casegrounded. The electrons have their highest velocity when they arrive at theanode. Since the anode is semi-transparent to the electrons, some of theelectrons will pass through it, with the energy E = eV0 and they will thenstart to slow down. At a distance of about the cathode-anode distance theelectrons kinetic energy will be equal to their potential energy and this willresult in that the electrons stop and they create a negative potential well−V0, the space charge which has accumulated here will exceed the localspace charge limit and it will slow down other incoming electrons. Thevirtual cathode is formed and consists of a cloud of electrons and thevircator can be split into two regions shown in figure 2.3, region 1 and 2. Inregion 1 the vircator can be seen as a simple diode and in the axial vircatorwe can view it as a planar diode with the current density J1. In region 2 thevirtual cathode is formed at the about the AK-gap distance where thecurrent density will be J2 = α · J1, where α is the anodes electrontransparency [7].


FIGURE 2.3: Vircator regions.

Figure 2.4 shows the phase space characteristics for the particles in a axialvircator, here we can see the anode position where the momentum for theelectrons reaches it is maximum. The position of the virtual cathode isincluded in the plot, here we also see escaping electrons, that will lower theefficiency of the device.

FIGURE 2.4: Phase space for a axial vircator, from a CSTsimulation.

The cathode is usually covered in velvet or carbon fiber that acts as aexplosive emitter and the anode is often made of a semitransparentstructure that allows the electrons to pass through the anode. Commonlanode materials are stainless steel, copper and aluminium. Lately

10 Chapter 2. Theory

researchers have had good results using pyrolytic graphite in both theemitter and the anode [11].

When the virtual cathode is formed two things will happen, first the virtualcathode will start to move closer to the anode until the electron kineticenergy is sufficient to pass through the virtual cathode and the virtualcathode is dissolved. Then the electron beam can start to propagate again,this leads to an oscillating cloud of electrons and this phenomena happensat approximately the beam plasma frequency, ωB .

The other thing that will occur is that a part of the electrons will passthrough the virtual cathode and some will be reflected, the electrons whichpasses through the VK are escaping while the reflected electrons will bepropagating back towards the diode where they will be reflected by thecathode potential, this is called reflexing. The oscillation of the virtualcathode and the reflexing will generate microwaves which for the most partare at different frequencies but they can be tuned to the same frequency forincreased performance [7], [10]. One of the main reasons vircators have alow efficiency is due to oscillating electron cloud and the reflexing electronswill oscillate at different frequencies and therefore create destructiveinterference which lowers the power of the radiated electromagnetic wave.

2.3.2 Field Emission

The reason behind the intense electron beam generation is field emission.This phenomenon is explained by the emission of electrons from a metal,semiconductor or dielectric under the influence of a strong electric field.This effect is purely quantum-mechanical with no classical analogue and theexplanation behind this is quantum tunnelling of electrons. The theory offield emission from metals was introduced by Fowler and Nordheim in1928. Their study resulted in the Fowler-Nordheim formula for currentdensity, for emission from metal to vacuum:

J =e3

8πht2(y)

E

Φexp

−8π√

2mΦ32

3heEθ(y)

, (2.1)where h is Planck’s constant, m is the electron mass, Φ is the work functionof the metal, t(y) and θ(y) are functions of the variable

y =e√eE

Φ(2.2)

The reason behind this tunnelling is that the electron wave function doesnot vanish at the classical turning point but decays exponentially into thebarrier. Then there will be a limited probability that the electron will befound outside of the potential barrier.

The tunnelling process for a metal at low temperature can be illustrated byfigure 2.5 below.


FIGURE 2.5: Energy level scheme for field emission for ametal. Picture from Wikipedia, [12].

A bit of metal can be viewed as a potential box where all the electron liesbelow the Fermi level, which in return lies below the vacuum level by acertain energy. The vacuum level represents the potential energy of anelectron at rest outside the metal without an applied electric field. Thedistance between the metal’s Fermi level and the vacuum level is called thework function, Φ. When a strong field is applied, the potential outside themetal will be deformed along the line AB according to 2.5. Then a triangularbarrier is formed where the electrons can tunnel and most of the emissionwill happen close to the Fermi level.

When applying an external electric field to the potential barrier, Fowler andNordheim found that the width of the potential barrier decreased, whichresults in that the electrons can tunnel more easily through the barrier. Theelectric field required for electrons to tunnel through a potential barrier with1 nm width is around 4 · 107 V/cm and this is for a flat surface but often thesurface is not flat at a microscopic level so that the field is enhanced andthus requires a lower E-field for the emission to occur [13], [14].

2.3.3 Space Charge Limited Current

The electrons emitted from the cathode will create a current, the magnitudeof this current is related to the applied potential and the anode-cathodedistance. This current that flows between two conductors was described byChild-Langmuir in the early 20th century. Later Langmuir and Blodgettdiscovered that current flowing between two conductors under allnon-relativistic regimes varies with V 3/20 , where the initial electron speed iszero and V0 is the applied voltage between the two plates. These discoveriesresulted in the Child-Langmuir law, which describes current flow in a1-dimensional steady state planar diode and later the Langmuir-Blodgettlaw which describes the SCL current in a coaxial diode. These laws can thenbe generalized to the relativistic regime using simple methods [15].


Child-Langmuir Law

For the 1-dimensional planar axial diode we can describe the currentflowing between two conducting plates the using CL law. Starting with twoconducting plates in vacuum with a distance D between the two andapplying a voltage V0 to the anode according to the figure 2.6.

FIGURE 2.6: Planar diode in vacuum, each conducting plateis approximated as infinitely long in the vertical direction.

The current density, J , in the region between the conducting plates candescribed by:

J(z) = ρ(z)v(z) = −JSCL, (2.3)

where ρ(z) is the charge density and v(z) is the electrons speed. We can findthe velocity v(z) using conservation of energy. The RHS of 2.4 is the kineticenergy of a electron at the anode and the LHS shows the potential energy ofa electron at the cathode.

1

2mv2(z) = eϕ(z) (2.4)

Here m is the electrons mass, e is the charge of the electron and ϕ(z) is theelectric potential. ϕ(z) must follow Poisson’s equation:

∇2ϕ(z) = −ρ(z)�0

(2.5)

Equation 2.3 and 2.4 can then be rewritten as:

ρ(z) = −JSCLv(z)

(2.6)

v(z) =

(2ϕ(z)e

m

) 12

(2.7)


Insert 2.6 and 2.7 into equation 2.5:

∇2ϕ(z) = JSCL√2ϕ(z)e

m

1

�0(2.8)

We rewrite equation 2.8 as:

∂2ϕ(z)

∂z2=JSCL�0

(2ϕ(z)e

m

)− 12

=JSCL�0

(m

2eϕ(z)

) 12

(2.9)

We get a second order nonlinear differential equation which we can solve byrewriting 2.9:

∇2ϕ(z) = Kϕ(z)−12 , (2.10)

using:

K =JSCL�0

√m

2e(2.11)

Letϕ(z) = azb (2.12)

Differentiate twice

∇2ϕ(z) = Kϕ(z)−12 = ab(b− 1)zb−2 (2.13)

Insert 2.12 into 2.13

K(azb)−12 =

K√a

= ab(b− 1)zb−2 (2.14)

We equate powers:

b− 2 = − b2⇒ b = 4

3(2.15)

⇒ ϕ(z) = az43 (2.16)

The potential at the anode is ϕ(D) = V0 and now we can get a

⇒ V0 = aD34 ⇒ a = V0

D43

(2.17)

Then we get the expression for ϕ:

⇒ ϕ(z) = V0D

43

z43 (2.18)

By using equation 2.18 we can write ρz and vz using equation 2.5 and 2.7

ρz = −�0∇2ϕ(z) = −�0∂2

∂z2V0

D43

z4/3 =�0V0

D43

4

9

1

z23

(2.19)

vz =

(2ϕ(z)e

m

) 12

=

√2e

m

(V0z

43

D43

) 12

=

√2eV0m

z23

D23

(2.20)


Using 2.3 together with 2.19 and 2.20 we get the space-charge-limitedcurrent for a 1D planar diode the so called Child-Langmuir law.

JSCL = −

�0V0D

43

4

9

1

z23

·

√2eV0m

z23

D23

= 49

�0V320

D2

√2e

m(2.21)

In the relativistic case the kinetic energy for the electrons can be written as

Ek,rel = m0c2(γ0 − 1) = e(V0 + ϕ) (2.22)

where γ0 is the Lorentz factor given by 2.23

γ0 =

√1− v

2

c2= 1 +

e(V0 + ϕ)

m0c2(2.23)

The relativistic solution of the Child-Langmuir law is complicated but it canbe approximated to 2.24 with less than 1% error by [15].

JSCL,rel =2�0m0c

3

eD2(γ

230 − 1)

32

(√

3− 1)γ−0.3920 + 1(2.24)

Langmuir-Blodgett law

The current density for a coaxial diode is described by theLangmuir-Blodgett law, which is derived in similar way as the planar diodebut in cylindrical coordinates and the motion of the electrons is assumed tobe in the radial direction.

For the non-relativistic case the current density is described by equation2.25 [7].

Jc =8�0π

9

√2e

m0

V320

2πr2β2(2.25)

Where β is a function represented by the infinite series 2.26.

β = lnr

rc− 2

5ln2

r

rc+

11

120ln3

r

rc− 47

3300ln4 u+ ... (2.26)

Where r is the radial position and rc is the cathode radius. Zhang et al. [15]gives a approximate solution for the Langumir-Blodgett law in therelativistic regime for the steady-state 1-dimensional coaxial diode:

Jc,rel =�0m0c

3

er2aA(rrc

)

(γ23 − 1)

32

(√32 H(u)− 1)γ

−0.3920 + 1

, (2.27)

whereA(u) = 1− u−1 − u−1 lnu (2.28)

H(u) = 2 + 0.2617 lnu+ 0.0091 ln2 u− 0.0014 ln3 u, u ∈ [ 1500

, 500] (2.29)


This relation gives a maximal relative error of 5% when the voltage rangesfrom 0.5 kV to 10 GV. The analytical current for our coaxial vircator can thenbe calculated using 2.30 [7].

Ic = Jc,rel2πraW (2.30)

where ra is the anode radius and W is the width of the emitter.

2.3.4 Bremsstrahlung

Most HPM devices produces its radiation from free electrons moving invacuum, converting their kinetic energy to rf energy. Since an electron withconstant velocity in vacuum does not radiate, the electrons should eitheraccelerate or de-accelerate in order to produce EM waves in vacuum. It iscommon to classify the interaction between the electron beam and themicrowave radiation to three types; Cherenkov or Smith-Purcell radiationwhich are slow waves that propagates with a phase velocity slower than thespeed of light, transition radiation and bremsstrahlung. For a vircator theproduction of microwaves are due to bremsstrahlung.

We cannot only consider the radiation from a single particle, a so calledspontaneous radiation. In HPM devices, such as a vircator, the number ofparticles are huge. It is therefore more convenient to talk about coherentradiation where all the electron are bunched together and radiates with thesame phase. Because of that the radiated power scales to N2 compared to Nin the spontaneous radiation, where N is the number of particles [10].

2.3.5 Frequency Characteristics

When the virtual cathode is formed, two things will happen, first the virtualcathode position will oscillate back and forth at the beams plasmafrequency, ωb. Secondly a part of the electrons will be transmitted throughthe virtual cathode while some electrons are reflected back towards theanode which eventually will be reflected again towards the virtual cathodeby the cathodes potential. This is called reflexing and these two processesoften occur at different frequencies [10].

The oscillation of the virtual cathode happens around the beams relativisticplasma frequency, for a thin annular beam inside a coaxial structure we got:

fp =1

2π

(nbe

2

�0mγ0

) 12

= 8.98× 103[nb(cm

−3)

γ0

] 12

Hz (2.31)

Where nb is the electron density of the beam at the anode. We can express2.31 in more practical units as:

fp(GHz) = 4.10

[J(kA/cm2)

βγ0

] 12

(2.32)


Whereβ = (1− 1

γ20)12 (2.33)

Between the real and virtual cathode a potential well is formed whichcauses the trapped electrons to reflex in the well. These electrons will bebunched together and radiate at the reflexing frequency 2.34, where d is theAK-distance: [10].

fr =1

4T=

1(4∫ d0 dz/vz

) (2.34)In non-relativistic more practical units, 2.34 becomes:

fr(GHz) = 2.5β

d(cm)(2.35)

It is difficult to develop a analytical expression for the coaxial vircator sincethe process in the vircator are highly nonlinear and time-dependent and it istherefore very hard to find an accurate analytical solution. There exists1-dimensional analytical models which give estimates for the diode current,the space charge limited current, oscillation frequency and the position ofthe virtual cathode. But the electron trajectories is not 1-dimensional so a1-dimensional model may not give accurate estimates and it is thereforenecessary to use a 2-dimensional analytical expression, that treats theelectrons with relativistic-fluid Maxwell’s equations [16].

For the coaxial vircator, Xing et al. have made a 2-dimensional steady stateanalysis for the plasma frequencies of the coaxial vircator [16]. Theapproximate solution in the relativistic regime is:

f =4.77 · 107

(rc − ra)ln

[γ0

√rarc

+

(γ20rarc− 1

) 12

](2.36)

In the non-relativistic case the solution can be reduced to 2.37, which is agood approximation for the main frequency when the applied voltage isbelow 500 kV.

f = 9.44 · 104 (ra/rc)14

ra − rc√V0 (2.37)

2.3.6 Mode Characteristics

In a coaxial vircator the anode and the cathode are cylinders on the sameaxis, the electrons are oscillating in the radial direction creating a doughnutshaped virtual cathode. Previously, researchers have only considered theelectron beam interacting with the TM01 mode, which seems natural sincethe radiation is cylindrically symmetric and thereby cancels the radialcomponent of the electric field along the whole central axis. The TM01 modeis illustrated in figure 2.7. However, more recent publications proposes thatthe TE11 has a strong contribution the overall EM power since the TE11mode is the fundamental mode in a circular wave-guide and has a strongradial component which will enhance the EM-power in the mode. We alsohave to consider that in a experimental device we cannot expect to have

2.4. The High-Power Microwave System 17

perfect symmetrical emission and geometry which then will increase thegrowth of the TE11 mode. Figure 2.8 shows the vertical polarization of theelectric field in the TE11 mode.

FIGURE 2.7: The electric field for the TM01 mode in a circularwaveguides cross section (left) and a longitudinal cut (right).

FIGURE 2.8: The electric field for the vertical polarization ofthe TE11 mode in a circular waveguides cross section (left)

and a longitudinal cut (right).

For a microwave weapon it is preferable to have the EM-radiation in theTE11 mode, since the radiation in the TM01 is maximum off axis and it istherefore not suitable for irradiating objects. It has been shown that even forperfect coaxial symmetry the EM-field gain for both of these modes hasnearly the same amplitude and for the most experimental systems one canexpect the TE11 mode to grow and dominate [17].

Since the TE11 mode is preferred for radiating objects, researchers hasfurther enhanced this mode by sectioning the emitter into two circularsections of 90 degrees to favour the TE11 mode and being able to choose thepolarization [7].

2.4 The High-Power Microwave System

The whole HPM system is shown in figure 2.9.


FIGURE 2.9: Photo of the Bofors HPM Blackout system witha compact Marx generator.

It consists of a Marx generator which creates the voltage pulses to themicrowave source. The vircator is shown in the middle together with thewaveguide and the antenna. A block diagram over the system is shown infigure 2.10. The system also includes a vacuum pump, which allows thesystem to operate under UHV-conditions.

FIGURE 2.10: Block diagram over a HPM weapon system.

2.4.1 Marx Generator

A Marx generator is used to create very high voltage pulses, in the order ofhundreds of kV, it can create this kind of pulses with a power supply with asignificant lower voltage. These kind of generators are often used in highenergy-physics and to simulate the effects of lightning. It was invented bythe German electrical engineer Erwin Otto Marx in 1924.

2.4. The High-Power Microwave System 19

FIGURE 2.11: Schematic figure over the Marx generator.

The Marx generator consists of several stages of capacitors that are chargedin parallel through resistors or with inductors to eliminate the resistivelosses and decrease the charging time. The generator is charged by ahigh-voltage DC-generator.

The spark gaps act as open switches when the capacitors are charging andthe load, in our case the vircator is isolated by the last spark gap. When thecapacitors are fully charged the spark gap between each capacitor in figure2.11 is closed which results in that all of the capacitors are now connected inseries. This will result in that the voltage for each capacitor will be added tothe output according to n · V , where n is number of capacitors and V is thecapacitor voltage. The capacitor circuit then discharges through the loadand the charging resistors. For example, a charging voltage of 10 kV and 13stages of capacitors will result in a output voltage of 130 kV. After a time thespark gaps stops conducting and the HV-voltage supply can start to chargethe capacitors again.

2.4.2 Microwave Source

The microwave source used in this system is a coaxial vircator capable ofGW-level output levels in the L- to S-band. Our vircator is enclosed inside astainless steel vacuum vessel, as it requires vacuum to be operated. Thecathode consists of a material with velvet-like properties and the anode ismade of a highly conductive material which is semi-transparent for theelectrons. The microwave source is constructed so that repetitive operationis possible [3].

2.4.3 Waveguide and Antenna

The waveguide has a cylindrical cross-section and a inner radius of 10 cm.Both the antenna and the waveguide is constructed in stainless steel and theantenna window is made of transparent polymer. The antenna is a conicalhorn antenna with a gain of about 17 dBi, which is also pumped to vacuumin order to withstand the high electric fields [3].


2.5 PIC-Solver and CST PS

Simulations and modelling are important tools to be able to understand anddesign HPM-devices. In order to both simulate the movements of chargedparticles and the time-varying electromagnetic fields, a much used methodis PIC, Particle-in-cell simulation, which is frequently used for simulatingplasma.

First a mesh-grid is applied over the simulation volume and super-particlesare placed inside the cells, in our case electrons. Since real systems consist ofextremely large amounts of particles it is convenient to represent theparticles as macro-particles, in PIC simulations often called super-particles.One of these super-particles can represents millions of real particles e.g.electrons, it is allowed to rescale the particles since the Lorentz force onlydepends on the charge to mass ratio so these "computational particles" willbehave exactly as a real particle. Without these super-particles thecomputing time would be significantly longer.

The iteration cycle is as follows:

1. Assign charge on grid points.

2. Compute the electromagnetic fields according to Maxwell’s equations.

3. Evaluate the Lorentz force and the charged particle movement on aexplicit time-domain scheme where the time step is controlled by themesh grid size.

4. The movement of the charged particles leads to a change in currentdensity which serves as the source term in Ampére’s law. Allowingcurrent and charge conservation.

Where many of the PIC-simulation programs use FDTD, finite differencetime domain, CST PS uses FIT, finite integration technique in order todiscretizie Maxwell’s equations.

2.5.1 Finite Integration Technique

Finite integration technique is a discretization scheme for Maxwell’sequation. This method uses the integral form of Maxwell equations to apreassigned grid.

The grid is located on a dual orthogonal grid, a e.g Cartesian grid, togetherwith an explicit time integration scheme such as leap-frog, which leads toless time-consuming and less memory using efficient algorithms.

To accurately discretize the spatial domain CST uses a hexahedral gridtogether with a perfect description of rounded solids also called perfectboundary approximation, PBA. This method cover all the applications ofelectromagnetic’s from static up to high frequency [18].

21

Chapter 3

Method

This chapter will describe the model used for simulation and theexperimental setup.

3.1 Experimental Setup

The experimental setup consist of a Marx generator, an inward-emittingcoaxial vircator, a cylindrical wave-guide and a conical horn antenna. Apicture of the HPM system is shown in figure 3.1.

FIGURE 3.1: A overview of the experimental HPM systemtogether with the HV-supply.

All the experiments are performed inside a anechoic chamber, as shown infigure 3.2. The main purpose of the anechoic chamber is to absorb themicrowaves and serve as a measuring chamber [4].

22 Chapter 3. Method

FIGURE 3.2: HPM system inside the anechoic chamber.

A schematic picture of the inward emitting coaxial vircator used in thesystem is shown below in 3.3. The anode structure is made of an highlyconductive material with a geometrical transparency of about 80 %, whichthe electrons can pass through. The emitter is circular symmetric andconsists of a material with velvet-like properties. The Marx generator ischarged to the desired voltage using a high voltage DC-supply, thereafterthe vircator is pulsed negatively resulting in production of microwaves.

FIGURE 3.3: Side-view of the inward-emitting coaxial virca-tor structure.

The radiated electric field is measured using two D-dot probes in the farfield region, one of the probes measures the vertical component and theother probe measures the horizontal component of the electric field as seenin 3.4. The probes are connected to a oscilloscope, Tektronix TDS7404B, ourmeasured data is then analysed in a MATLAB script and the frequency isobtained using fast Fourier transform. The probe configuration in the usedmeasurement setup only allows for measurement of the TE11-mode. The

3.2. Simulation Setup 23

voltages and currents related to the vircator are measured using integratedcurrent and voltage sensors connected to an Tektronix TDS7104 oscilloscope.

FIGURE 3.4: Free field D-dot probes.

3.2 Simulation Setup

The simulations are performed using a program called CST Studio suitewith the module Particle Studio, see section 2.5 [19]. A test structure of thevircator is built inside the program, with some simplifications. Later fromthe results we can extract important parameters such as frequency, outputpower and particle movement etc.

3.2.1 Test Structure

The geometry of the test structure is shown below in figure 3.5, allcomponents are made of PEC, perfect electric conductor and the wholecavity is filled with vacuum. The geometrical structure of the vircator issimplified compared to the real vircator but all the important dimensionsare the same. The dimensions have also been distorted by hand due toconfidentiality.


FIGURE 3.5: Coaxial vircator test structure in CST.

Our anode is modelled as a infinitely thin cylinder with a sheettransparency of 80 % which is the same as the geometrical transparency ofthe anode in the lab system, the anode material is also set here as PEC. CSTPS also allows either a fixed transparency or a energy dependenttransparency, which can be preloaded using a ASCII-file. Theenergy-dependent transparency of our used anode model is complicated tofind an expression for, therefore we have a fixed transparency. The anodefoil in CST can be seen in figure 3.6.

FIGURE 3.6: Anode foil model in CST.

3.2.2 Excitation of the Vircator

In the experimental system the cathode cup is pulsed negatively using acoaxial cable. Where in our computer model the voltage signal to thecathode is applied using four discrete voltage ports. These ports are

3.2. Simulation Setup 25

symmetrically placed between the outer cavity wall and the cathode baseaccording to 3.7. In the program the voltage ports are modelled as thinwires, where more symmetrically placed ports results in less disturbance onthe EM modes, in our case four ports was chosen. The discrete voltage is setto the same value used in the lab for the real system which is 468 kV.

One thing to consider is that the discrete voltage ports in CST acts as idealvoltage source, which has no internal resistance. This means that the voltageports will always give the desired voltage no matter what current is drawn.This is not the case in our lab system, where we have a impedance in ourload, the vircator which will change during operation and also our Marxgenerator has a limited amount of stored electrical energy compared to theideal case. We must take this into consideration when we later compare thesimulated results to the empirical results.

FIGURE 3.7: Discrete voltage ports inside the vircator struc-ture.

The voltage ports are excited using a excitation signal, which in CST canhave different shapes such as a smooth step, square wave or any arbitrarycurve loaded using an ASCII-file. In our simulation we use a smooth stepshown in 3.8, which resembles the first half of our voltage pulse in ourexperimental system. The rise-time for our smooth step is, TRise = 15 nsand Ton = 250 ns, which is in the same region as the real pulse.

FIGURE 3.8: Excitation signal used in the simulations.


3.2.3 Particle Source

The particles i.e. electrons are emitted through a circular source, the emitteron the inside of the cathode cup, which covers the whole innercircumference. The emission model used in CST has velvet-like properties.In our simulations we use the explosive emission model in order to mimicthe experimental system, the emission model has the following parameters:

• EThreshold = 2 MV/m

• tEmission,risetime = 5 ns

Where a too short emission rise-time could create unwanted harmonics inour simulations. The number of emission points are adjusted to the meshsize, every mesh cell contains a emission point, shown in 3.9. The electronswill be emitted with zero initial velocity and they are emitted with noangular distribution.

FIGURE 3.9: Particle source on the inside of the cathode cup.

3.2.4 Waveguide Ports

Waveguide ports are introduced to extract and absorb EM-waves. There aretwo ports in the structure, the first waveguide port is place at the far back ofthe structure, its purpose is to absorb the modes. The second waveguideport is placed at the waveguide output in order to record the output signal,the ports are visualized in figure 3.10.

3.3. Monitor and Probes 27

FIGURE 3.10: A view of the two waveguide ports.

We only consider the first 16 modes, since the higher propagation modescontribution are very small compared to the early order modes. Howeverwe are mainly concerned with the dominant TE11 and TM01 modes. Themode calculation frequency is set to the estimated oscillation frequency.

3.3 Monitor and Probes

In order to record the requested results different probes and monitors mustbe defined inside the program. Field monitors are defined, such as E-field,H-field and E-energy in both time- and frequency domain that solves thefield on staggered grid points in the whole computational volume. Alsovoltage and current monitors are placed in the structure the record the inputvoltage, AK-gap voltage and the vircator currents. PIC position monitor isdefined that will track the charged particles dynamics inside the volume,also PIC phase space monitors is included.

29

Chapter 4

Results

This chapter will compare the results from simulations with experimentalmeasurements from the lab together with analytical calculations for certainparameters. The goal is to see if the program gives satisfactory resultscompared to measurements and also achieve a deeper understanding of thephysics involved during the vircator’s operation.

4.1 Current and Particle Properties

Figure 4.1 shows the simulated total current through the vircator during apulse. This current can be calculated in two ways; by adding the totalemission current from the emitter together with the collision current at theemitter which has opposite sign resulting in the yellow curve in 4.1, or byadding the currents which flows through the discrete ports which is thepurple curve, where both of these curves stabilizes at aroundITotal,Simulated = 26kA.

We can also get a analytical expression for the total current flowing in thevircator, by calculating the current just outside the emitter using equation2.30 and 2.27. Assuming the same current density in the AK-gap region andjust changing the area to just outside the emitter we getITotal,Analytical = 21.3 kA.

30 Chapter 4. Results

FIGURE 4.1: Plot showing the simulated total currentthrough the vircator.

The experimental current through the system is shown in 4.2, it is measuredat the Marx generator. This current is a bit lower, ITotal,Experimental = 17 kAand the shape of the current curve is deviating due to different shapes in theexcitation signal, as discussed in 3.

FIGURE 4.2: Experimental total current through the vircator.

Figure 4.3 shows the time evolution of the numbers of macro-particlesemitted from the emitter.

4.2. Anode-Cathode Voltage 31

FIGURE 4.3: Number of macro-particles emitted.

4.1.1 Anode Current

We can calculate the current at the anode using equations 2.30 and 2.27, weget IAnalytical = 15.9 kA. Figure 4.4 shows the simulated anode current inCST, ISimulated ≈ 16 kA.

FIGURE 4.4: Simulated current at the anode.

4.2 Anode-Cathode Voltage

Figure 4.5 shows the simulated AK-voltage and the correspondingexperimental voltage can be seen in 4.6. They both have a max-amplitudearound 470 kA, the shape of the curve is the result of the limited energystored in the Marx.


FIGURE 4.5: Simulated AK-voltage.

FIGURE 4.6: Experimental AK-voltage.

4.3 Impedance

In figure 4.7 and 4.8 we can see the impedance change during operation. Wehave a typical diode behaviour, in the beginning of the pulse we have a highimpedance but after some time the current can easily flow between cathodeand anode resulting in a lower impedance, around ZSimulated =18 Ω for thesimulated and between ZExperimental ≈15-20 Ω for the experimental system.

4.4. Virtual Cathode Characteristics 33

FIGURE 4.7: Simulated impedance.

FIGURE 4.8: Experimental impedance.

4.4 Virtual Cathode Characteristics

Figure 4.9 and 4.10 shows the time evolution of the virtual cathode at atransverse cross-section at the emitter during a pulse. The color ramp showsthe particles energy, blue is the lowest energy and red is the highest. Theelectrons are emitted from the outer ring and accelerating inward towardsthe transparent anode, where they get their maximum velocity resulting inthe red ring, the electrons that passes through the anode then de-acceleratecreating the virtual cathode where they have zero velocity, resulting in theinner blue ring as seen below.


FIGURE 4.9: The virtual cathode evolution between T= 22 nsto T= 43 ns. At T = 22 ns we can seen that the discrete ports af-fects the particles distribution, causing a cross-shaped inner

distribution.

FIGURE 4.10: The virtual cathode evolution between T= 45ns to T= 90 ns.

We can see that the shape of the virtual cathode, the inner ring of electronswith low energy, changes quite a lot with time. At the beginning there is aperfect symmetry which will result in the production of TM01 modes. Afteraround T = 40 ns, a non-symmetrical virtual-cathode starts to appear, thiswill cause the output to be in TE11-modes since it is the fundamental modein a cylindrical waveguide.

This breakdown of the symmetrical virtual cathode could be explained byan accumulation of excessive charge in the region between the cathode andthe virtual cathode, which changes the electron beam dynamics as the timeprogresses leading to breakdown [20]. This phenomena can also be seen ifwe study the output power over time.

Figure 4.11 shows a side-view of the of the particles in the vircator at T =32ns, the color scheme shows the velocity of the electrons. Here we can seethat some of the "escaping" electrons in the waveguide are acceleratedtowards the frontal wave-port by the electric-field, the "lost" acceleratingparticles extracts EM-energy from the output signal which lowers theoutput power.

4.4. Virtual Cathode Characteristics 35

FIGURE 4.11: Side-view of the particles inside the vircator atT = 32 ns.

The position of the virtual cathode can be obtained from figure 4.12. Theplot shows the x-component of the E-field at the emitter at 70 ns, the changein sign of Ex shows the position of the virtual cathode,DV C−Anode ≈ 1.37 ·DAK .

FIGURE 4.12: Plot showing the x-component of the E-fieldat T = 70 ns.

4.4.1 Phase Space of the Particles

Figure 4.13 shows the phase-space of the particles in the vircator at T = 20ns. The y-axis shows the normalized momentum and the y-position is onthe x-axis. Since it is not possible at the time to plot the phase space incylindrical coordinates in CST, the plot can be a bit tricky to understandsince it is a cylindrical geometry plotted in Cartesian coordinates.


FIGURE 4.13: Phase space of the particles at T = 70 ns.

4.5 Output Characteristics

4.5.1 Time-Domain

The simulated output signal in time-domain for all 16 modes is shown infigure 4.14, the y-axis shows the square root of peak power.

FIGURE 4.14: Simulated output signal in time domain.

4.5.2 Frequency-Domain

Figure 4.15 shows the FFT of the output signal in CST, where the mainfrequency at fSimulated = fsim Hz = fanal Hz.

4.5. Output Characteristics 37

FIGURE 4.15: Simulated output signal in frequency domain.

We have a analytical expression for the frequency from 2, equation 2.37, thisgives a theoretical frequency, fAnalytical = fanal Hz.

The frequency for the experimental system is shown in figure 4.16, from thiswe can extract the main frequency which is fExperimental = fexp Hz= 0.91 · fanal Hz.

FIGURE 4.16: Frequency spectrum of the output signal forthe experimental system.

Furthermore in figure 4.17 we see the electric field in a cut-plane normal tothe x-axis at the main frequency, where the E-field is normalized to 1 W.Inside the waveguide it is possible to see the frequency of the outgoingwave, the escaping electrons will be accelerated towards the waveport bythis field. These accelerated electrons will extract energy from the outgoingEM-wave which lowers the efficiency of the vircator.


FIGURE 4.17: View of the phase-depending E-field at themain frequency.

4.5.3 Output Power

The electric field is proportional to power of the EM-wave, therefore it issufficient to only present output power. The power output curve from CSTis shown in figure 4.18, almost all output power is in either the TM01 or theTE11 mode. We should only consider the first 65 ns when we look at thesimulated curve since the excitation signal is different compared to reality asaforementioned, the power output for the TE11 mode is around P = Psim W.

FIGURE 4.18: Simulated output power in time domain. Thepurple curve shows the TM01-mode and the green curveshows the two polarizations of the TE11 mode and the blue

curve shows the total output power for all (16) modes.

4.5. Output Characteristics 39

Below in 4.19 the output power for the experimental system is shown with amaximum around Pexp = 0.8 · Psim W.

FIGURE 4.19: Experimental output power, for the TE11mode.

41

Chapter 5

Discussion

Computer simulations are frequently used for research regarding HPMdevices and many papers have been published on this subject. CST’ssoftware is especially adapted for modelling particle sources and particleaccelerators. Therefore, even before we started this study we expected to getreasonable results.

When setting up the vircator model in CST some simplifications have to bemade, for practical reasons. Firstly, the geometry of the model is "cleaned"and by doing this we reduced the time to build the geometry and themeshing is simplified. Also small changes in geometry are of littleimportance for the results. In the simulation model all components areperfect electric conductors, which should affect the end results considerably.The anode foil model we are using has a fixed transparency, the reasonbehind this is that it is very difficult to get an estimate of theenergy-dependent transparency for the our used vircators anode design.

If we look at the total current through the vircator we notice that thesimulated, analytical and experimental results differ somewhat but they arestill in the same region. The reason for the difference between analytical andsimulated total current is that the analytical solution, only considers theelectron motion in 1-dimension where the CST model on the other handallows the electrons to move in all dimensions, thereby allowing a differentcurrent density compared to the 1-dimensional model. The experimentalcurrent is the lowest, an explanation for this is that in the experimentalsystem we do not have PEC, which results in resistive losses lowering thecurrent.

The current at the anode cannot be measured in the lab, therefore we onlyhave the simulated and analytical results, they are close to each other andthe variation could be due to reasons mentioned in the paragraph above.

The simulations gives a good understanding of the process inside thevircator and it is interesting to study the particles 3-dimensional plots whichshows the formation of the virtual cathode. These 3-dimensional plots is agood match to what we expected from the theory, one minor negativeaspect is that we cannot plot the phase space of the particles in cylindricalcoordinates which makes our current phase-space plot complicated toevaluate and understand. The formation of the virtual cathode happens atabout twice the AK-gap distance which is about what we should expectfrom theory.

42 Chapter 5. Discussion

The frequency behaviour of the device is investigated in 4.5. There we cansee that the simulated and analytical main frequency is essentially the same,this is because the frequency is only dependent on the voltage and theAK-gap. Our experimental measurements shows a lower frequency, thereason behind this lower frequency could be that the applied voltage in theexperimental system is only at 468 kV for a very short time and thendecrease, resulting in a lower average voltage during the pulse andtherefore a lower frequency.

When we compare the output power we need to take into consideration thatthe voltage pulses between simulation and the lab system are quite differentand in the simulations we have an ideal voltage source compared to reality,hence we should only look at the beginning of the pulse in 4.18. Whendoing so we see that the simulated output power is around 20% higher thanthe experimental which we should expect since both the diode voltage andthe current is lower in the experimental plots. We also see that during thestart-up process inside the vircator we have perfect symmetry in theemission which results in only TM01 modes, this we cannot always expectfrom a experimental device. After a while there is some interferenceintroduced resulting in growth of the TE11 mode, which is what weexpected from theory.

There are some limitations regarding this research, one major point is thatwe cannot simulate the pulsed power supply, the Marx generator in theprogram. At this moment there is no way to include a dynamic source inCST, which would make the simulations come closer to reality. Also wecould use a excitation signal closer to the experimental signal, for increasedaccuracy. Throughout the simulations no time was invested into adaptingthe emission model, the default settings was used since it gave reasonableresults. During this study no uncertainty analysis for the experimental datawas performed because the main focus was on simulations and theory,therefore no measurement errors for the collected lab data is presented andaccounted for.

A possible follow-up research would be to try to optimize the design of thevircator with regards to efficiency or mode output, since we now know thatthis program works satisfactory for modelling this type of vircator. CST PShas built in optimization algorithms, where we can vary parameters such asAK-gap and voltage etcetera in order to find the best geometricalparameters which gives the highest efficiency.

43

Chapter 6

Conclusions

A coaxial vircator structure used in a non-lethal microwave weapon hasbeen analytically and numerically investigated using a particle-in-cell code.The main focus on this study was to evaluate if CST PS can be used forstudying and designing the coaxial vircator. The simulations was also doneto achieve a better understanding of the physical processes involved in theoperation of the coaxial vircator.

CST PS have produced similar results to the analytical and experimentalresults. Under the same input parameters CST produces current andfrequency characteristics in the same region as the experimental system alsothe output power was similar to the power produced in the lab system. Thedominant mode produced was the TE11 mode when a disturbance wasintroduced, this results is in line of what other researches has found.Deviations between simulations and empirical results can be explained bythat the program uses ideal voltage sources and the conductors are perfectlyconducting compared to reality. Also there could be a slight mismatch forthe emission model and the anode transparency compared to our empiricalmodel which could contribute to differences.

CST PS code gives acceptable results and the vircator model used could befurther improved for increased accurately. The conclusion is that CST PScan be used for studying and designing the coaxial vircator.

45

Bibliography

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AcknowledgementsIntroductionBackgroundProject DescriptionPrevious Work

TheoryCharacteristics of High-Power MicrowavesMicrowave SourcesVircator TheoryVircator Operating PrinciplesField EmissionSpace Charge Limited CurrentChild-Langmuir LawLangmuir-Blodgett law

BremsstrahlungFrequency CharacteristicsMode Characteristics

The High-Power Microwave SystemMarx GeneratorMicrowave SourceWaveguide and Antenna

PIC-Solver and CST PSFinite Integration Technique

MethodExperimental SetupSimulation SetupTest StructureExcitation of the VircatorParticle SourceWaveguide Ports

Monitor and Probes

ResultsCurrent and Particle PropertiesAnode Current

Anode-Cathode VoltageImpedanceVirtual Cathode CharacteristicsPhase Space of the Particles

Output CharacteristicsTime-DomainFrequency-DomainOutput Power

DiscussionConclusionsBibliography

theoretical analysis and simulation of microwave...

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