Diplomarbeit
Investigation of high power limitation of waveguide elements at FLASH
Untersuchung zur Leistungsbegrenzung von Wellenhohlleiterelementen bei FLASH
Sebastian Göller
Institut für Experimentalphysik Universität Hamburg September 2008 1. Gutachter:
Prof. Dr. Jörg Rossbach
2. Gutachter:
Dr. Stefan Choroba
Kurzfassung
Diese Diplomarbeit beschäftigt sich mit der Leistungsbegrenzung von Wellenhohlleiter- elementen bei FLASH. Die untersuchten (rechteckigen) Hohlleiter sind vom WR650 (R14) Standard [165,1 mm × 82,55 mm] und werden bei 1,3 GHz betrieben. Die theoretische Begrenzung, die durch den elektrischen Durchschlag (spark) bestimmt ist, liegt bei 58 MW. Es zeigt sich jedoch, dass die praktische Begrenzung in der Gröÿenordnung von 5 MW liegt. Die vorliegende Arbeit untersucht verschiedene As- pekte der Leistungslimitierung, wie Ein uss der Form/Geometrie der Hohlleiter, Ein- uss der Ober äche, Ein uss von HOMs (Higher Order Modes) und den Ein uss der Gasfüllung.
Der Ein uss von HOMs und der Ober äche werden theoretisch behandelt. Die Be- grenzung aufgrund der Geometrie wird mit Hilfe des Simulationsprogramms Micro Wave Studio (MWS) betrachtet. Zur Untersuchung der Eigenschaften der Luft im Hohlleiter wurden verschiedene Messungen mit einem speziellen Aufbau gemacht. Hierbei wurde der Ein uss der Luftfeuchtigkeit, Temperatur und des Druckes untersucht. Zudem wur- den Versuche mit Sticksto unternommen. Diese Messungen stellen den Hauptteil der Arbeit dar.
Die Arbeit zeigt ein Modell zur Bestimmung der (praktisch übertragenen) Leistung, aufgrund der genannten Eekte. Auÿerdem geben die Ergebnisse der Messungen zum Ein uss der Gasfüllung Hinweise auf technische Verbesserungsmaÿnahmen zur Hochleistungsübertragung von Hohlleiterverteilungen. Abstract
This diploma thesis investigates the power limitation of waveguide elements at FLASH. The investigated (rectangular) waveguide elements are of the WR650 (R14) standard [165.1 mm × 82.55 mm], driven at 1.3 GHz. The theoretical limitation, which is given by the electrical breakdown (spark) is in the order of 58 MW. Practical experiences at FLASH show that the power capability is in the order of 5 MW. This investigation considers several aspects of power limitation, like shape/form of the waveguide, surface eects, in uence of HOMs (Higher Order Modes) and the in uence of the gas lling.
The in uence of HOMs and the surface eects are described theoretical. The limi- tation due to the geometry is described by simulations with the program Micro Wave Studio (MWS). To investigate the eect of the air properties to the limitation, several measurements with a special setup have been done. Investigated properties are humid- ity, temperature and pressure. Experiments with nitrogen have also been done. These measurements are the main part of this thesis.
This thesis shows a model to determine the (practical transmitted) power, due to the named eects. Furthermore the results of the measurements with dierent gas llings are given hints to increases the power capability of high power RF transmission lines. Contents
1 Introduction 1 1.1 Outline ...... 1 1.2 FLASH ...... 2
2 Theory of waveguides 4 2.1 Transporting RF in a rectangular waveguide ...... 4 2.2 Transmitted power in waveguides ...... 6 2.3 Average power for pulsed mode ...... 8 2.4 Standing waves (De nition of the VSWR) ...... 10
3 Theory of breakdown 12 3.1 Field emission (Fowler-Nordheim theory) ...... 12 3.2 Breakdown in gases ...... 14 3.3 Electrical strength of gases ...... 15 3.4 SF6 ...... 17 3.5 Humidity ...... 19 3.6 Pressure/Temperature ...... 20
4 Practical power capability 22 4.1 Gas lling ...... 22 4.2 Form of the waveguides ...... 22 4.3 Surface eects ...... 26 4.4 Higher order modes ...... 27 4.5 Calculation of limitation ...... 27 4.5.1 In uence of the VSWR to the power capability ...... 28 4.5.2 In uence of HOMs to the power capability ...... 29
5 Simulation with Micro Wave Studio 31 5.1 E-bend and H-bend ...... 32 5.1.1 E-bend with groove ...... 33 5.2 Bellows ...... 35 5.3 One-stub tuner ...... 38 5.3.1 Measurement device ...... 41 5.4 HOM simulation ...... 41
6 Measurement assembly 43 6.1 Measurement procedure ...... 47 6.2 Taking spark pictures ...... 48 6.3 Observing time of the spark ...... 48 7 Measurement results 50 7.1 Data analysis ...... 50 7.2 Results for the `8 sparks` measurements ...... 51 7.2.1 Humidity measurement ...... 52 7.2.2 Temperature measurement ...... 54 7.2.3 Pressure measurement ...... 55 7.2.4 Nitrogen measurement ...... 56 7.2.5 Synthetic air measurement ...... 58 7.3 Results for the `34 sparks` measurements ...... 59 7.4 Spark timing ...... 63 7.5 Pulse length and repetition rate ...... 63 7.6 Systematic errors ...... 64
8 Summary and Conclusion 65 8.1 A model of power limitation ...... 65 8.2 Summary of the measurement results ...... 66
A Surface investigations i
B Tables for the HOM simulation of the E-bend ii
C Labview program for taking spark pictures iii
D Spark pictures iv 1 INTRODUCTION
1 Introduction
1.1 Outline This diploma thesis investigates the high power capability of waveguide elements at FLASH (F reier Elektronen Laser H amburg ).
FLASH is a high gain FEL (Free Electron Laser) in the VUV (Vacuum Ultra Violet) and soft X-ray wavelength regime, set up at DESY, Hamburg. The produced radiation can be used for many scienti c investigations ranging from physics, chemistry and biol- ogy to material sciences, geophysics and medical diagnostics.
To accelerate the electron beam, superconducting niobium cavities have been devel- oped and produced by the TESLA (Tera Electronvolt Superconducting Linear Acceler- ator) Technology Collaboration working at a frequency of 1.3 GHz.
The high power RF (Radio Frequency) for accelerating the electrons is transported by a waveguide distribution consisting of several waveguide elements from the klystrons to the cavities. Four klystrons are required to deliver power to the RF gun at the beginning and to six cryo modules, which comprises the (nine cell) cavities. Figure 1 shows the distribution part, which is at the end of the distribution line in the accelerator tunnel. This distribution part is connected to a cryo module, which comprises eight cavities. These cavities are normally used with a eld gradient in the order of 20 MV/m. The power transported to one cavity can be up to 300 kW.
Figure 1: Example for a waveguide distribution at FLASH for one accelerator module (8 cavities). red: load; grey: circulator
The rectangular waveguide elements used at FLASH are of the WR650 standard. The cross section of this standard is a = 6.5 inch (165.1mm)× b = 3.25 inch (82.55mm).
Experience at FLASH shows that the practical power capability of the WR650 waveguides are well below the theoretical limit. Instead of reaching the theoretical limit of 58 MW with air inside the waveguide, the real practical limit is near the 5 MW level which is required in many parts of the waveguide distribution of FLASH.
This treatise contains two theory parts concerning power transmission with waveg- uides (2) and electrical breakdown in gases (3). These parts are followed by a chapter (4) about practical power capability, which shows the eects for limitation. To give an idea about the theoretical behaviour of waveguide elements, several simulations with a
1 1.2 FLASH 1 INTRODUCTION simulation program (Micro Wave Studio) have been done. The results of these simula- tion are presented in chapter 5. The following chapter (6) is about the developed and used measurement setup. Chapter 7 explains the results for the measurements about dierent gas conditions. The last chapter (8) summarizes the results and cognitions.
1.2 FLASH The Free Electron Laser in Hamburg has been emerged as the result of the TTF (Tesla Test Facility). For a few years it has been delivering coherent radiation with short wavelengths and a high brilliance (number of photons per time unit, wavelength interval, aperture angle and cross section). Since October 2007 the design beam energy of 1 GeV and design wavelength of 6.5 nm have been available. FLASH can produce very short photon pulses, which is why it is very useful for many chemistry/biological experiments. Figure 2 shows the schematic build-up of FLASH (from [FLASH]).
Figure 2: Schematic representation of FLASH (details in text)
In the electron source (RF gun) electrons are emitted of the cathode by irradiation with a laser (photo eect) and then accelerated up to 5 MeV. Six acceleration modules (each comprises 8 cavities) and two bunch compressors create electron bunches with a beam energy up to 1 GeV and a bunch length of a few µm (<300 µm). The charge of the bunches is in the order of 1 nC, what aords surge currents up to 2.5 kA. Behind the accelerator section follows an energy collimator, where the band width of the energy can be reduced. In the undulator section nally the radiation pulse in the soft X-ray and VUV regime is created. In the undulator the electrons follow a slalom trajectory and start to radiate spontaneously.
The lasing process is initiated by this spontaneous undulator radiation, which in- teracts with the `slaloming` electrons and starts to modulate the density of the bunch. As a consequence, the radiation power is ampli ed, which in turn increases the lon- gitudinal density modulation. In total it follows an exponential rising of the irradi- ated power. This so called SASE (Self Ampli ed Spontaneous Emission)-eect requires a high charge density of the bunches, low emittance and a small energy width (see [Schneidmiller/Saldin/Yurkov]).
2 1.2 FLASH 1 INTRODUCTION
FLASH is with its quality of the bunches and the reached wavelength, the leading Free Electron Laser worldwide.
The next generations of free electron lasers are already planned and goes to construc- tion [for example: LCLS at SLAC (USA) or SCSS at RIKEN-Institute (Japan)]. One of these machines is the XFEL, which will be set up at DESY in Hamburg. The goals for the XFEL are an electron beam energy up to 20 GeV and reaching wavelengths in the region of 0.1 nm to 6.5 nm. This requires a total length of 3.4 km. Photon pulses of 100 fs will be available for this European Community project. FLASH is the pilot project for the XFEL, where many machine parts have been tested to make the XFEL possible.
3 2 THEORY OF WAVEGUIDES
2 Theory of waveguides
2.1 Transporting RF in a rectangular waveguide Transporting high power RF (Radio Frequency) is done by waveguides. In this thesis the investigated waveguides are rectangular with conducting walls. The convention is to lay a coordinate system with x- and y-axis in the cross section plane (x||a (width); y||b (height)). Normally it is a = 2· b. The z-axis points at propagation direction. If there is no other annotation, the waveguide will be lled with air [r = µr = 1].
Figure 3: Convention for a rectangular waveguide: x-direction is || to a, y-direction is || to b
The solution for the `waveguide problem` is given (as usual) by solving the Maxwell- equation with this special boundary conditions (width a, height b). This solution is given by (1) and (2) for the TEnm-mode. TE means transversal electric eld, so there is no E- eld in propagation direction (Ez = 0). The H- eld has a z-component. The TEnm-mode can also be called Hnm-mode. The indexes n,m show the number of burls (n in x-direction, m in y-direction).
Ez = 0
nπx mπy E = −E cos( )sin( )sin(ωt − βz) x 0 a b nπx mπy E = E sin( )cos( )sin(ωt − βz) (1) y 0 a b mπy nπx H = −H cos( )cos( )cos(ωt − βz) z 0 b a mπy H = H cos( )sin(ωt − βz) x 0 b nπx mπy H = H cos( )sin( )sin(ωt − βz) (2) y 0 a b
4 2.1 Transporting RF in a rectangular waveguide 2 THEORY OF WAVEGUIDES
The eld strength amplitude E0 and H0 are the maximum electric/magnetic eld values.
The mode used at FLASH and for the measurements is the TE10 (n = 1 and m = 0), also called fundamental mode. So there is only an E- eld parallel to the y-direction (one burl), while the H- eld is rotating around the E- eld (see gure 4).
Figure 4: y-component of the electric eld and eld density for the TE10-mode in a rectangular waveguide
The phase constant describes the repetition of the eld pattern in a distance of 2π β β and is given by the following equation:
ω2 m2 n2 β2 = − π2( + ) (3) c2 b2 a2 The critical ω 2 2 shows that there is a critical value of angular frequency βc = ( c ) − β given by:
m2 n2 ω2 = π2c2( + ) (4) c b2 a2 or as wavelength:
2 λc = (5) q m2 n2 b2 + a2 Below (above) this frequency (wavelength) no propagation takes place. By substitu- tion of (5) in (3), the guide wavelength results: r λ 2 λg = λ 1 − ( ) (6) λc
5 2.2 Transmitted power in waveguides 2 THEORY OF WAVEGUIDES
The critical (also called cut o) wavelength for the TE10-mode in the WR650 stan- dard (a = 165.1 mm, b = 82.55 mm) is λc = 330.2 mm. The waveguide wavelength λg results for 1.3 GHz (∼= 230.76 mm) from (6) to 322.62 mm. q The factor 1 − ( λ )2 is an important parameter for the waveguide. λc
E0 The eld impedance ZF = in the waveguide can be calculated by the impedance H0 in free space Z0: r λ 2 Z0 = ZF 1 − ( ) (7) λc
q µ0 The value of the impedance is for vacuum and air (r = µr = 1): Z0 = = 377Ω. 0 Another interesting aspect is the phase velocity and the group velocity of electro- magnetic waves in waveguides. The phase velocity vph is given by: c c c vph = = = > c (8) β p ω 2 nπ 2 mπ 2 q λ ( c ) − ( a ) − ( b ) 1 − ( )2 λc
The phase velocity for the WR650 standard and 1.3 GHz is vph = 1.4 c! The group velocity vg can be calculated as follows: dω c2 c2 vg = = β = < c (9) dβ ω vph The group velocity of course is smaller than c (it is 0.71 c for WR650). This behaviour of the group velocity is called dispersion (in free space there is no dispersion).
2.2 Transmitted power in waveguides
In the following the TE10(H10) mode is used to calculate the transmitted power in a waveguide. The elds can be calculated by setting n = 1 and m = 0 in (1) and (2). Here the coupling between E0 and H0 is used to eliminate H0 with the help of the eld E0 E0 impedance ZF = → H0 = . H0 ZF 0 E0 sin( πx )sin(ωt − βz) ZF a ~ πx ~ (10) E = E0sin( a )sin(ωt − βz) H = 0 0 − E0 cos( πx )cos(ωt − βz) ZF a The transmitted power by an electromagnetic wave can be calculated with the help of the Poynting vector. This is the de nition of the Poynting vector:
S~ = E~ × H~ This vector describes an energy ux density of a wave through the cross section of a space element[W/m2].
6 2.2 Transmitted power in waveguides 2 THEORY OF WAVEGUIDES
The Poynting vector S~ points at propagation direction of the wave. By using (10) for the electric and the magnetic eld, the Poynting vector looks as follows:
E2 0 sin( πx )cos( πx )sin(ωt − βz)cos(ωt − βz) ZF a a ~ 0 (11) S = E2 0 sin2( πx )sin2(ωt − βz) ZF a The x-component is an oscillation of the energy in x-direction, which does not account for the power. The transmitted energy of the wave is described by the z-component. It is necessary for practical transmission to build a time average of the Poynting vector S~ (see [Nolting]). This could be done by averaging the time dependent parts sin(ωt − βz) · cos(ωt − βz) respectively sin2(ωt − βz) over one period. The x-fraction becomes 0 (does not account to the energy transport), while the sin2(ωt − βz) of the z-component (for ) becomes 1 : z = 0 2
T = 2π 1 Z ω 1 sin2(ωt)dt = T 0 2 The result for the time averaged Poynting Vector S~¯ of a planar wave is :
0 ~ S¯ = 0 (12) E2 1 0 sin2( πx ) 2 ZF a
Figure 5: Geometry of a plane wave. The Poynting vector points at transmission direction
To come from the time averaged Poynting Vector S~¯ to the transmitted power P, a eld integral of the averaged Poynting vector over the cross section, where the wave travels through, has to be calculated (for a more detailed calculation see [Meinke]).
7 2.3 Average power for pulsed mode 2 THEORY OF WAVEGUIDES
Due to the fact, that only the z-direction for energy transport is interesting, the integral becomes one dimensional:
Z Z a Z b 1 E2 πx P = S~¯dA~ = 0 sin2( )dxdy Area x=0 y=0 2 ZF a 1 E2 Z a πx 1 E2 = 0 b sin2( )dx = 0 ab (13) 2 ZF x=0 a 4 ZF This is the (theoretical) power going through a waveguide with the cross section a×b (see also [Harvey] and [Lorrain/Corson]).
To calculate the power for a waveguide the impedance ZF given by (6) has to be set in (13). The formula with calculated constants for the WR650 standard
(λc = 330.2 mm) and a wavelength of λ = 322.62 mm is:
q λ 1 − ( )2 2 1 λc 2 −5 m A 2 P = E0 ab = 0.65 · 10 E0 (14) 4 Z0 V
The calculated power by setting E0 to the breakdown voltage of air (30 kV/cm = 3 MV/m) is for a traveling wave in a straight waveguide 58.5 MW.
It should be mentioned, that in this calculation no consideration has been done to things like electrical conductance, surface and gas conditions. So 58.5 MW is the the- oretical power, a straight waveguide would transmit in a perfect world, without any disturbing eects.
The main result of these considerations is that the transmitted power of a planar wave is proportional to the square of the maximal electric eld E0 and to the cross section a × b, normal to the wave propagation.
E0 is as the maximal electric eld amplitude the most interesting parameter for power limitation.
2.3 Average power for pulsed mode Many applications do not run in the so called CW-(continuous wave) mode, but in a pulsed mode. Then the peak power is only for a de ned pulse length present. Such a microwave pulse can be approximated by a rectangular shape. The generation of these pulses is done by a modulator. Figure 6 shows a typical pulse for FLASH. FLASH is running with a repetition rate of 5 Hz and a total RF pulse length of 1.4 ms.
8 2.3 Average power for pulsed mode 2 THEORY OF WAVEGUIDES
Figure 6: Rectangular microwave pulse with a pulse length of 1.4 ms
The average power depends on the repetition rate fr and the integral about the peak power of the pulse tp:
Z tp Pave = fr Ppeakdt (15)
where the peak power (Ppeak) is de ned by (13). With a rectangular pulse length tp, (15) becomes to: Pave = frtpPpeak. The factor frtp is also called duty cycle and is for FLASH in the order of 1 %.
The average power is the main constitutive factor for the temperature of the waveg- uide, which can limit the power capability (see chapter 3.6)
9 2.4 Standing waves (De nition of the VSWR) 2 THEORY OF WAVEGUIDES
2.4 Standing waves (De nition of the VSWR) A standing wave in a microwave transmission line can be realized by a short circuit at the end of the line. The short circuit is a normal conductive plate, which terminates the line and re ects the wave. At the short circuit the electric eld wave reaches its zero position, so the re ected wave also starts with the zero position. The re ected wave interferes with the forward wave parts.
The wave nodes become constant in space. Between the nodes the eld oscillates among its maxima and minima. With a short circuit is the wave totally re ected, thus the electric eld becomes two times larger. So the breakdown level has to divide by two. The power capability of a waveguide with a short circuit is reduced to a quarter of the value without short circuit ( E0 2 P ). ( 2 ) → 4 In a normal distribution line only a small wave part is re ected. This can be illus- trated by a barrier, which re ects a part of the wave, but also lets some forward wave parts pass (see gure 7). These re ected parts of the wave disturb the forward wave parts by building up a standing wave.
Figure 7: Schematic of an interference between forward and re ected wave parts [blue (dashed): forward wave, green (dash - dotted): re ected wave, black: sum of re ected and forward wave]
10 2.4 Standing waves (De nition of the VSWR) 2 THEORY OF WAVEGUIDES
The VSWR (Voltage Standing Wave Ratio) is used to describe the re ected part of the wave in a transmission line. The VSWR describes the ratio between the maximum and minimum voltage (Vmax, Vmin) induced by the wave. These values are given by the forward traveling wave Vf and the returning (re ected) wave Vr: V V + V VSWR = max = f r (16) Vmin Vf − Vr Instead of using the voltage induced by the wave, the electric eld can also be used as equal description for the SWR: SWR = Emax . Emin The smallest VSWR is 1, because there is only one forward traveling wave in the waveguide and no re ected parts (VSWR = Vf +0 ). In a common transmission line re- Vf −0 ected wave parts always occur, so a VSWR of exact 1 is only theoretically possible.
1.1 - 2 are common VSWR values for waveguide transmission lines.
11 3 THEORY OF BREAKDOWN
3 Theory of breakdown
The breakdown of the electric eld (called spark) in a waveguide is the limit for the power capability and is discussed in this chapter. It is a process with a stochastical component like all processes of particle collision. The stochastical behaviour can be in uenced by many parameters. That is why it is important to hold as much as possible of the parameters constant for experiments.
3.1 Field emission (Fowler-Nordheim theory) In gases always certain free electrons occur. The electrons may arise by radiation with UV light (photo eect), by eld emission (in a high electric eld) or just by thermal uctuations. The eld emission is explained here and it is described, why it has no in uence to the breakdown in a waveguide.
Field emission explains the eect of emitting electrons from a metal with the help of high electric elds. The electrons have to tunnel (quantum mechanical) through the potential wall of the metal. The electric eld increases the tunnel probability, which can be described by the Fowler-Nordheim equation (see [Forbes]).
2 − b J(E) = aE e E (17) J(E) is the emission current density, which depends on the electric eld E on the surface. a and b are constants, which consist of natural constants and the local work function Φ for the electrons : √ 3 e 1 4 2me 3/2 a = and b = Φe 8πh Φ 3e~
−19 −31 e is the electron charge (1.62 · 10 C), me is the electron mass (9.1 · 10 kg) and is the Planck constant ( −34 Js [ h ]). h 6.62 · 10 ~ = 2π The basic Fowler-Nordheim equation can describe the eld emission for high electric elds. In principle this equation is calculated for the ground state of the electrons. But (17) can also be used for room temperature due to the fact, that the Fermi distribution for electrons is nearly similar for T = 0 K and T = 293 K. The electric eld has to be in the order of 1 GV/m, to get a appreciable current density with (17). Many experiments and practical applications with large metal surfaces (∼ cm2) show that the principle dependency between current density and electric eld arises already at eld values in the order of 1 MV/m (for details see [Habermann]). The so called enhanced eld emission arises due to small particles and surface defects like scratches. Many investigations about the physical mechanism of the geometrical un- evenness and dielectric particles on the surface have been done. The most enhancing factor is assigned to micro spikes in the metallic surface.
12 3.1 Field emission (Fowler-Nordheim theory) 3 THEORY OF BREAKDOWN
External particles, which lay on the surface are not so important (see [Habermann]). The electric eld E is enhanced at such a micro spike by a factor β. So in equation (17) the electric eld has to be replaced by βE, where E is still the macroscopic eld:
2 − b J(E) = a(βE) e βE (18)
With the help of the β-factor, it is possible to calculate the electric eld for many spike geometries. The most appearing spike geometry for metallic surfaces is a cylinder (height h, ra- dius r) with a hemisphere (radius r), which gives following β (see sketch on the left):
h βcylinder = r + 2
Another cylindric spike to describe the surface is a truncated cone (height h, radius r) with an aperture angle of α, which gives a β of :
h βcone = r + 3 · cosα A standard industrially manufactured metallic waveguide surface ( h ) has a of r < 1 β 2 (see appendix A). So called `hair structures` with h on the surface are responsible r > 1 for eld enhancement by micro spikes. Figure 8 shows the current density of (18) with a β of 50 (relativly rough surface).
Figure 8: Logarithmic plot of (18) for a eld enhancement factor β of 50
13 3.2 Breakdown in gases 3 THEORY OF BREAKDOWN
The gure shows a strong rise at 10 MV/m that results in a signi cant current den- sity above 25 MV/m.
For electric elds in the order of 1 MV/m (or less) β has to be in the order of 1000 to get a signi cant current density. It is impossible to reach such values with the surface of the aluminum waveguides. The micro spikes of this surface have a geometrical β in the order of 2 (see chapter 4.3). So (enhanced) eld emission is not an eect, which in uences breakdown in the discussed waveguides. The in uence given by the surface is discussed in chapter 4.3.
3.2 Breakdown in gases Microwave breakdown in gases is induced by a rapid (exponential) growth in time of the free electron density in the waveguide (`avalanche eect`). The freed electrons follow the electric eld and build up a spark channel. The gas which is normally an insulator becomes conductive. The basic physics involved in the microwave-induced breakdown process can be described with the continuity equation for the free electron density:
∂n = ν n − ν n + ∇(D∇n) (19) ∂t i a
where n is the electron density, D is the diusion constant and νi and νa are the ionization and attachment frequencies. D and νa primarily only depend on the gas pres- sure p, while νi in addition also depends strongly on the magnitude of the electric eld β E. This dependence is often approximated as νi ∝ E , where β is a parameter that depends on the gas (from [Anderson]).
For the geometry of two parallel plates (waveguide) and a homogeneous RF electric eld, the continuity equation can be simpli ed to
dn D (20) = νin − νan − 2 n ≡ νnetn dt Le
where the diusion length Le, is directly related to the distance between the plates Le . is the `netto` ionization frequency due to the competition between L = π νnet(p, Le,E) ionization and the two loss mechanisms (see [Anderson]). Breakdown occurs when the production rate by primary ionization of the electrons becomes greater than or equal to the loss of electrons by diusion to the surrounding walls and attachment to neutral gas molecules.
The solution for (20) grows exponentially in time for the breakdown condition:
νnet(p, Le,E) ≥ 0. The breakdown electric eld Eb given by νnet(p, Le,E) = 0 → Eb = Eb(p, Le) increases for high pressures and for low pressures leading to the characteristic U-shaped form of the so called (empirical) Paschen curve.
14 3.3 Electrical strength of gases 3 THEORY OF BREAKDOWN
A Paschen curve is a measured curve showing the breakdown voltage U as a function of the product of gas pressure p and sparking distance s for a particular gas. Figure 9 shows such Paschen curves for air, nitrogen and SF6. Nitrogen and air have for 1 bar and a sparking distance of 10 mm the same breakdown eld of kV , while SF6 Eb = 30 cm has with nearly kV a breakdown value three times higher. Eb = 100 cm
Figure 9: Paschen curves for air (at normal conditions), nitrogen and SF6
The free path length for small pressures of the electrons become large compared to the distance of the plates. So the electrons cannot ionize a gas atom and will be absorbed by the waveguide walls, where they might release a second electron by the photo eect. This requires adequate electron energy, so for decreasing pressure the breakdown voltage increases. For large pressures the free path length of the electrons becomes smaller. The electrons have to accelerate within a smaller distance to reach the ionization energy. Thus the breakdown voltage increases.
3.3 Electrical strength of gases The actual dielectric strength essentially depends on the chemical characteristics of the gas. The electro negativity of the gas is important for the breakdown threshold. The electro negativity describes the ability to attract electrons towards itself in a co- valent bond. Elements like uorine and oxygen have the highest electro negativity. A molecule can attract more or less electrons besides the chemical bonding.
15 3.3 Electrical strength of gases 3 THEORY OF BREAKDOWN
In a water molecule for example the oxygen atom can attract the electrons from the hydrogen atoms, thus a dipole is created. In normal oxygen molecules, both atoms at- tract the electrons in the same way. So water can be called `more electronegative` than oxygen.
Here are some values for some atoms given by the Pauling scale (from [Mortimer]):
uorine: 4.0; oxygen: 3.5; sulfur: 2.5; hydrogen: 2.1
The electro negativity is de ned about the dierence between bonding energies within a molecule (for detailed de nition see [Mortimer]).
Fluorine is the atom with the highest electro negativity and acts as reference for the other elements. The electro negativity dierence between sulfur and uorine is 1.5, while the dierence in a bonding between oxygen and hydrogen is 1.4. This shows that SF6 is more electronegative than water.
In electronegative gases (like SF6) negative ions are established by attachment of electrons with neutral molecules. Thus free charge carriers are dissipated. So the at- tachment rate for SF6 is much higher than for air or other gases. This is the reason for the large breakdown threshold of SF6. In opposite the noble gases have small breakdown thresholds (although the ionisation voltage of the noble gases is larger), because they are not able to attach electrons. The ionisation voltage for most gases is in the same order (see table 1). So the main parameter for a high breakdown level of a gas is a high electro negativity, which reduces the free electrons in the gas.
Besides the electro negativity a minor in uence comes to the breakdown voltage from the gas density and the ionisation voltage. High density and high ionisation voltage in- creases the threshold. The low ionisation voltage and (relative) low density of oxygen are responsible for the (compared to SF6) low breakdown value, although oxygen is an electronegative gas.
There is normal air in all parts of the waveguide distribution at FLASH, except for the area directly behind the klystron. The normal air consists of 78 % nitrogen, 21 % oxygen and 1 % noble gases (also 0.04 % CO2). The breakdown threshold of 30 kV/cm is a mean value of the values for nitrogen and oxygen. Due to the fact that air contains oxygen, air can be counted to the (weak) electronegative gases.
16 3.4 SF6 3 THEORY OF BREAKDOWN
gas density [ kg ] ionisation voltage [ ] breakdown threshold [ kV ] m3 V cm
air 1.21 - 32 N2 1.17 15.8 33 O2 1.33 12.8 29 SF6 6.15 15.9 90 H2 0.08 15.4 19 He 0.17 24.5 10 Ne 0.84 21.6 2.9 Ar 1.66 15.8 6.5
Table 1: Breakdown thresholds for dierent gases for DC applications (values for 20 ◦C and 1013 mbar) (from [Kind/Kärner])
3.4 SF6 Behind every klystron at FLASH a circulator with a load is installed, to protect the klystron from re ected power. These area is very sensitive to sparks, because of the high power. That is why these areas are lled with SF6 by a small overpressure (typical 0.2 - 0.4 bar).
SF6 is a non toxic, achromatic gas, which is used in many high voltage applications. Beneath its high breakdown value of 90 kV/cm, it has good erasing properties. SF6 is chemically inert. Figure 10 shows the chemical structure of SF6.
Figure 10: SF6 molecule
17 3.4 SF6 3 THEORY OF BREAKDOWN
The sulfur atom (in the middle) bonds to each of the six uorine atoms. Because of its chemical dullness, SF6 is not reactive with air, but it displaces the air. This might become dangerous, if the SF6 displaces the oxygen concentration in a closed room.
If a spark occurs, the bonds between the sulfur and the uorine might break. Then it is possible that toxic uorides arise. These toxic SF6 decomposition products are results of the reaction with water, oxygen or metals like aluminum (from the waveguide walls). Some of these reactions with oxygen and water are listed below.
with oxygen with water
SF6 → SF4 + 2F SF4 + H2O → SOF2 + 2HF 2SF6 + O2 → 2SOF2 + 8F SOF2 + H2O → SO2 + 2HF 2SF6 + O2 → 2SOF4 + 4F SOF4 + H2O → SO2F2 + 2HF
Table 2: Possible reaction of SF6, induced by a spark (from [Kahle])
The possible decomposition products are one reason, why there are many attempts to reduce the SF6 in high power application. One possibility is mixing the SF6 with nitrogen (respectively air) or oxygen. For a mixture of 25 % nitrogen and 75 % SF6, the breakdown threshold is still in the order of 60 kV/cm (from [Kahle]).
It might occur a breakdown in SF6 even under the threshold value. Inhomogeneous electric elds at borders/ edges of the surface are responsible for this behaviour (for more information see [Hess]).
18 3.5 Humidity 3 THEORY OF BREAKDOWN
3.5 Humidity The breakdown level for gases changes with the degree of humidity. Water vapor has an electronegative eect, so free electrons can be attached. A bigger concentration of water molecules (absolute humidity) per volume increases the breakdown value. In a homogeneous eld this eect is not so large, but in an inhomogeneous eld the increase of the breakdown level is remarkable (see [Köhrmann]).
Figure 11 shows the static breakdown eld of air for normal air pressure (1 atm = 760 Torr ≈ 1 bar) and various partial water pressures for two plates with distance d.
Figure 11: Breakdown threshold for dierent humidities for a pressure of 760 Torr according to [Köhrmann] (for details, see text)
The curve A is for a partial pressure of water pw of 0 Torr (dry air). Curve B is for pw = 10 Torr ˆ= 13.3 mbar and curve C is also for pw = 13.3 mbar, but it is a another measurement made by Schumann (see [Köhrmann]).
In practice it is not easy to use this eect of the higher threshold by increasing the humidity because of rusting materials and other technical problems with humidity.
19 3.6 Pressure/Temperature 3 THEORY OF BREAKDOWN
3.6 Pressure/Temperature The power capability should increase for an ideal gas with a square for the pressure, due to the fact that the breakdown eld increases linear with the pressure ( rst approxima- tion of Paschen curve behaviour for the pressure area about 1 bar). So the ratio of two power capabilities at dierent pressures (p1 > p2) should increase in the following way: P ower p1 p1 = ( )2 (21) P owerp2 p2 In practical the exponent is not two, but gets values like 3/2 or 4/3 (from [Rajtsin]).
The physical magnitude in uencing the breakdown in the Paschen curve is the gas density. The variation of the pressure is the practical implementation to vary the gas density.
The temperature T is (beside the pressure p) also directly related to the gas density N by the ideal gas law: V N p = · k · T (22) V B −23 where kB = 1.38 · 10 J/K is the Boltzmann constant and the gas density is given by N (numbers of particle in the volume ). V N V
So the gas density N decreases in case of increasing temperature (and for constant V T pressure): N 1 . → V ∝ T The breakdown threshold of the electric eld falls with the temperature of the gas (for constant pressure). So with this behaviour the power capability of a heated waveguide should decrease for higher temperatures by the power of 2 ( 1 ). P ∝ T 2 In practice is the exponent not 2, but also in the order of 4/3 like for the pressure (see also [Rajtsin]).
20 3.6 Pressure/Temperature 3 THEORY OF BREAKDOWN
The function 2·300 2 describes the Paschen behaviour for the relative power ( T [K]+300 ) capability normalised to 300 K (27 ◦C). Figure 12 shows this decrease of the relative power capability for increasing temperatures. At a temperature of 360 K (87 ◦C) the power capability has decreased to a value of 0.8.
Figure 12: Plot for the decrease of the relative power capability (normalised to 300 K)
The limitation by the temperature is an interesting point for waveguide applications with a large average power, which heats up the waveguide.
21 4 PRACTICAL POWER CAPABILITY
4 Practical power capability
Several eects can be considered as a reason for power limitation:
• Properties of the gas inside the waveguide (humidity, temperature, pressure)
• Shape or form of the waveguides (straight, bends, bellows)
• Surface eects
• HOM (Higher order modes) generated by the Klystron
4.1 Gas lling The limit for the power capability is given by the breakdown threshold of the gas. Several magnitudes like temperature, pressure, humidity of the gas aect this aspect. The breakdown voltage of a gas is described by Paschens law. Instead of using air some parts of the waveguide distribution at FLASH are lled with SF6. The eect of the limitation by the gas is described in detail in chapter 3.
4.2 Form of the waveguides Microwave transmission lines are seldom able to proceed in a straight direction. So waveguide elements could have several forms/shapes, each for a dierent function. The form of a waveguide can have a big in uence to the power capability. The problem is the eld enhancement, which depends on the geometry. The eld enhancement has been calculated for six dierent elements with the help of numerical simulations (details in chapter 5).
Six dierent elements have been simulated:
• 1000 mm straight For understanding the power limitation in waveguide elements with dierent shapes, the straight acts as reference. The used straight waveguides at FLASH are man- ufactured by Spinner. The ideal straight waveguide has a theoretical power capa- bility of 58 MW.
• H-bend (radius: 150 mm) and E-bend (radius: 100 mm) The dierence between E-bend and H-bend is the bending plane. The E-bend is bended round the `long` (x-direction) side, while the H-bend is bended round the `small` (y-direction) side. The plane parallel to the x-direction is called `E-plane`
due to the fact that for the TE10, this plane con nes the electric eld. The plane parallel to the y-direction is accordingly called `H-plane`.
22 4.2 Form of the waveguides 4 PRACTICAL POWER CAPABILITY
The used E-bends normally have a groove at one end to reduce the VSWR. The groove reduces the re ected wave parts, thus the VSWR is decreased for the trans- mission line.
Figure 13: left: E-bend with radius 100 mm [side view]; right: H-bend with radius 150 mm [top view]
• 300 mm more exible bellow and 300 mm less exible bellow Bellows are exible waveguide elements, which compensate mechanical stress in a waveguide distribution. Two bellows with dierent grades of exibility have been simulated. The main dierence between both bellows is the geometry. The more exible bellow has acute edges between its ripples, while the less exible bellow comprises of a sequence of `rounded ripples` (for details see chapter 5). Picture 14 shows the two bellows manufactured by MegaIndustries.
Figure 14: left: more exible bellow; right: less exible bellow
23 4.2 Form of the waveguides 4 PRACTICAL POWER CAPABILITY
• One-stub tuner A one stub-tuner is a waveguide device with a (adjustable) post. A one-stub tuner can be used for several things, like changing the VSWR in a transmission line or tuning phases. This device is also useful to create sparks for investigation at a localised point. A waveguide with a xed post has been used for the measurements and has been simulated with Micro Wave Studios (results in chapter 5.3).
Figure 15: One-stub tuner with an adjustable post
A waveguide distribution line can have only the power capability of its weakest el- ement. Waveguide elements with many square edges have a bigger enhancement than those with a nearly at geometry.
Table 3 shows a few waveguide producers, who produce the WR650 standard. The table shows every information given by the producers and gives an idea about the power capability for the mentioned waveguide elements.
The mentioned power capabilities are relative low due to the fact, that the companies issue a guarantee to their products. Especially the given CW- (continuous wave) power values are very small. This table gives also a hint about the big dierence between practical and theoretical power capability.
24 Firm Address Component Material VSWR (peak) Power [MW] Losses [dB/m] Remarks
Spinner straight Al <1.02 - - (Germany) www.Spinner.de E-bend Al <1.06 - - over all H-bend Al <1.06 - - frequency range AINFO straight Al/Cu <1.05 27.8 0.2(max) (China) www.ainfoinc.com E-bend (Miter 90◦) Al <1.15 13.9 0.55 (max) temperature range: H-bend (Miter 90◦) Al <1.15 13.9 0.55 (max) -40 ◦C - 70 ◦C MegaIndustries straight - - min. Freq: 19.6 0.0073 (USA/ME) www.megaind.com - - max. Freq: 27.8 0.0073 bellow plated brass - 11 0.032 average power: 150 kW Microtech www.microtech- bellow - 1.06 10.7 0.033 CW-power: 20 kW (Scotland) inc.com Credowan www.credowan. bellow - 1.1 12.5 0.01- temperature range: (England) co.uk 0.02 -50 ◦C - 180 ◦C E-bend Al/ 1.05 - - - ATM www.atm Cu-brass 1.08 (USA / New York) microwave.com H-bend Al/ 1.05- - - Cu-brass 1.08 Odd Tvedt & Co www.oddtvedt.no bellow - - - - (Norway) Penn Engineering www.penn bellow Al/brass - 10.7 - CW-power: 20 kW (USA / California) engineering.com
Table 3: Waveguide producers list 4.3 Surface eects 4 PRACTICAL POWER CAPABILITY
4.3 Surface eects Limitation by the surface is given by physical magnitudes like roughness and conduc- tance of the surface. The ohmic losses of the surface associate with the skin depth λd of the electric eld. The skin depth λd describes how far an electric eld in ltrates a conductor till it is reduced by a factor of e−1:
r 1 λd = (23) πfµ0µrσ
where f is the frequency of the electric eld, µrµ0 is the (relative) permeability and 6 σ is the conductivity. The skin depth of aluminum (σAl = 37.7 · 10 S/m) at 1.3 GHz is 2.6 µm.
Unless the surface is smooth to the order of the skin depth, the currents in the surface traverse longer paths. This increases ohmic losses and leads to attenuation.
Figure 16 shows the surface pro le for a straight aluminum waveguide measured with a (Mitutoyo SJ-301) pro lometer. The roughness of the surface is in the order of 5 µm. The maximum surface peaks are about 10 µm with a length of ∼ 50 µm. So the ratio of peak height to length ( h ) is 1/5. This shows that the surface is very smooth and electric l eld emission is strongly suppressed (see chapter 3.1).
Figure 16: Surface pro le of a 200 mm straight waveguide; vertical: 5 µm/scale division, horizontal: 100 µm/scale division (total 4mm)
The surface of an E-bend is rougher than the surface of the straight waveguide, because of the bending surface. Measurements show that the surface eld enhancement factor h (from (3.1)) is β = r + 2 in the order of 2 (3 for the E-bend) for the analysed waveguides (details concerning the surface measurements in appendix A).
26 4.4 Higher order modes 4 PRACTICAL POWER CAPABILITY
4.4 Higher order modes
Higher order modes are wave modes TEnm with integer multiple frequencies of the fun- damental mode (1.3 GHz → 2.6 GHz, 3.9 GHz,... ). Higher order modes are excited by resonance eects in the klystron. HOMs can build up standing wave parts, which reach an electric eld above the breakdown threshold. Depending on the mode frequency, more wave types are established. According to the cut o frequency (5), the number of wave types increases for higher modes. The modes of 2.6 GHz have for example seven possible propagable wave types (TE10, TE01, TE20, TE11, TM11, TE21, TM21). The number increases up to 14 for the modes of 3.9 GHz and it is already 24 for 5.2 GHz.
To reduce this disturbing eect, a HOM lter is required, which damps the higher modes.
The theoretical power reduction by HOMs is described in chapter 4.5.2. Some nu- merical simulations with MWS concerning HOMs can be found in chapter 5.4.
4.5 Calculation of limitation
The theoretically maximum level of the transported power Ptheo through a waveguide is (as shown) 58 MW. The following formula gives an idea, which factors have to be added to the power formula (13), to get the practical transported power. Some recip- rocal factors have been added to give a rst explanation for the reduction of the power capability.
1 1 VSWR + 1 1 VSWR + 1 2 2 2 (24) P = E0 ab 2 2 ( ) = Ptheo 2 2 ( ) 4ZF f β 2 · VSWR f β 2 · VSWR The additional factor β (surface factor) is added to give consideration to industrial manufacturing concerning the surface. Chapter 4.3 mentions, that a eld enhancement in the order of 2 (3 for the E-bend) happens at the surface.
The ( eld enhancement) factor f is established to consider the shape (geometry) of the waveguide. In a straight waveguide is the eld enhancement f = 1, but in other formed waveguide elements (bend, bellow,etc.) is f 6= 1. The amplitude of the enhance- ment depends on the geometry. The factor is the ratio of the maximum electric eld of the element and the straight waveguide (f = Eelement ). Estraight Both factors (f and β) have to be squared to get their in uence for the power capa- bility, because they are related to the electric eld.
The factor VSWR+1 2 is described in the next section. ( 2·VSWR )
27 4.5 Calculation of limitation 4 PRACTICAL POWER CAPABILITY
4.5.1 In uence of the VSWR to the power capability
The factor VSWR+1 2 in (24) considers the fact of re ected wave parts, which disturb ( 2·VSWR ) the propagation. It is the ratio between the forward power Pfor and the theoretically maximum power Pm in the transmission line running to the match.
The maximum electric eld Em is the sum of forward and re ected elds and can be described with the help of the VSWR in the following way:
2·VSWR Em = Efor VSWR+1
The electric eld of the forward component Efor is the amplitude that would exist in a perfect matched line. So the maximum eld Em is equal to the forward eld Efor for a VSWR of 1. Em becomes larger than Efor for VSWR values bigger than 1 (not perfect matched line).
By squaring and inverting this expression, it describes the theoretical reduction of power in a transmission line by the VSWR (for details see [Sucher/Fox]).
→ Pfor = ( VSWR+1 )2. Pm 2·VSWR
The degradation ratio Pfor between forward power P and the theoretical Pm for (maximum) power running to the match Pm decreases to a lower limit of 1/4.
This lower limit occurs because the square of the maximum electric eld can be increased by only a factor of four relative to the forward wave at large values of the VSWR:
VSWR+1 2 VSWR→∞ ( 2·VSWR ) → 1/4 This also shows that a waveguide with a short circuit have only a quarter of the power capability, than a waveguide without short circuit. The degradation of the power capability with the VSWR is shown in gure 17.
28 4.5 Calculation of limitation 4 PRACTICAL POWER CAPABILITY
Figure 17: Degradation of breakdown power by the VSWR (according to [Gilden/Gould])
4.5.2 In uence of HOMs to the power capability HOMs are generated in the klystron. In a disadvantageous case these HOMs might reach the breakdown level and lead to a spark. The power of the HOMs is distributed in an indeterminable way, so it is di cult to give a good solution for this problem (see also [Gilden/Gould]). The worst case is selected for the following consideration, in which all power is transmitted by only one wave type of the HOM.
To indicate the relative power in the harmonic mode a factor H is de ned. H is the ratio between the breakdown power of the harmonic (Phighermode) to the breakdown power of the main mode (Pfundamentalmode) for a given waveguide application.
H = Phighermode Pfundamentalmode
The factor H describes, how much power the mode maximal can transmit relative to the fundamental mode. The mode can for example transmit only 80 % of the power for the fundamental mode for H = 0.8 and then reaches the breakdown threshold.
The HOM limits the power capability for values smaller than 1. If the value of H is larger than 1, the limitation will be given by the fundamental mode (T10). The HOM is then not interesting for limitation.
29 4.5 Calculation of limitation 4 PRACTICAL POWER CAPABILITY
Small values of H close to 0 describe the `bad` HOMs, while values close to 1 (or larger) are not disturbing.
To show the disturbing eect of the HOMs with small ratios H, it is useful to de ne the following factor, where the in uence of the HOMs with the VSWR of a transmission line is correlated : (1 − H) · VSWR.
The following formula describes the reduction of the relative power capability of HOMs.
1 R = (25) 1 + (1 − H) · VSWR The function is normalised to 1. So there is no disturbance for H = 1.
Figure 18 shows the fractional reduction of power with increasing VSWR for dierent values of H. The smallest H values give the lowest power capability. So it is useful to reach large power capability levels for the HOMs in a transmission line with large VSWR, to avoid problems with HOMs.
Figure 18: Degradation of breakdown power by a harmonic for dierent values of H
In chapter 5.4 some power ratios H have been calculated for an E-bend.
30 5 SIMULATION WITH MICRO WAVE STUDIO
5 Simulation with Micro Wave Studio
Micro Wave Studio (MWS) is a simulation program, developed by CST, which gives the option to simulate RF waves running through a special geometry de ned by the user.
The solving method used in Micro Wave Studio, called transient solver, simulates a RF pulse of a prede ned frequency interval. The mean frequency of this interval is 1.3 GHz. This RF pulse starts at one speci ed port and moves through the geometry (designed by the user) to other speci ed ports (`transient`). All simulated elements have been simulated with two ports (input and output) for the travelling wave simulation and one port (only input) for the standing wave simulation.
A short circuit plate consisting of (10 mm) aluminum, which re ects the wave has been installed for the standing wave simulation. The boundary conditions for the metal part of the simulated waveguides have been set to aluminum as electric conductor for the walls (in sketches blue colored). The boundary conditions for the `air` parts have been set to vacuum (there is almost no dierence to air with an r = 1.00059). Trials to use a PEC (Perfect Electric Conductor) instead of the aluminum with a con- ductivity of 3.72 · 107 S/m show no dierence in the results.
Figure 19: Field distribution in a 1000 mm straight waveguide [TE10-mode] for a RF power of 1 W; maximum eld amplitude is 278 V/m (red)
MWS uses a mesh to approximate the geometry developed by the user. In this case the hexahedral mesh was used. By setting the mesh properties it is possible to re ne the mesh for a better approximation. This increases the calculation time by the power of 4. Two parameters (`lines per wavelength` and `lower mesh limit`) have been used to re ne the mesh. Both parameters are set to 10 by MWS default value. To get a good
31 5.1 E-bend and H-bend 5 SIMULATION WITH MICRO WAVE STUDIO approximation for the dierent geometries (especially for the bellows with its edges and curves) these two parameters have to be increased.
By starting the transient solver MWS calculates every eld value at each place of the geometry.
The interesting value is the maximum electric eld for each element, which has been calculated by MWS. The RF pulses running through the elements are normalised to 1 W. By calculating the amplitude factor A the maximum transported power for each element can be determined.
The amplitude factor A is the ratio between the breakdown voltage in air Eb = Eb 3MV/m (30 kV/cm) and the maximum simulated electric eld E1W for 1 W (A = ). E1W For getting the maximum transported power Pmax till reaching breakdown limit, the square of the amplitude has to be divided by 2 A2 . → Pmax[W ] = 2 The amplitude factor has to be divided by two, to consider the fact, that not each point of the cross section reaches the maximum electric eld. This is in principle the averaged eld integral of the Poynting vector over the cross section.
The simulated eld E1W for the straight waveguide (1000 mm) is 278 V/m, the resulting power is Pmax = 58 MW. This eld value can be calculated by setting 0.5 W in (14).
The simulated eld E1W is 556 V/m for the straight waveguide with short circuit plate and the resulting power is 1 3MV/m 2 . So as expected the Pmax = 2 ( 556V/m ) = 14.5MW calculated electric eld is two times larger, while the power capability is only a quarter of the value without short circuit (see chapter 2.4).
5.1 E-bend and H-bend The H-bend has been simulated with a bending radius of 150 mm, while the E-bend has a radius of 100 mm. The results are given together with the results for the straight in table 4. The dierence in the power capability is about 50 %. The H-bend behaves in principle like a straight waveguide (concerning the capability), while the E-bend has less than half of this capability. This dierence can be explained by the eld enhancement, which happens, by `bending the electric eld around the corner of the E-bend` (see gure 20). The electric eld in the H-bend is `just` shifted in the propagation plane. The enhancement factor for (24) is in the order of f = 1.5 for the E-bend and f = 1.02 for the H-bend. The eld is not exactly doubled for the E-bend with short circuit. This might be a problem with the simulation by rotating the electric eld in the bend. So the breakdown power for the E-bend with short circuit is more than a quarter of the value for the E-bend without short circuit.
32 5.1 E-bend and H-bend 5 SIMULATION WITH MICRO WAVE STUDIO
component electric eld Emax [V/m] power [MW] VSWR (at 1.3 GHz) straight 278 58 1 with short circuit 556 14.5 - H-bend 282 56.6 1.03 with short circuit 564 14.3 - E-bend 414 26 1.13 with short circuit 809 6.9 -
Table 4: MWS results for the straight and the bends
Figure 20: Image of the (absolute) eld evaluation of an E-bend. At the inner bending side the eld becomes maximal (red)
5.1.1 E-bend with groove A second simulation with a groove at the end of the bending part has been done for the E-bend ( gure 21). This groove has a depth of 4 mm and a width of 10 mm. The edges between the bend and the groove have been simulated with dierent groove radius in the range of 0 to 0.6 mm. This simulation should show the in uence of the groove to
33 5.1 E-bend and H-bend 5 SIMULATION WITH MICRO WAVE STUDIO the power capability and how small dierences in the bend geometry of edges change the power capability.
groove radius [mm] electric eld Emax [V/m] power [MW] VSWR (at 1.3 GHz) 0.1 1136 3.49 1.11 0.2 971 4.77 1.11 0.3 900 5.56 1.11 0.4 819 6.71 1.11 0.5 785 7.30 1.11 0.6 749 8.02 1.11
Table 5: Dependence of the groove radius to the power
So even small changes in the bending of edges can increase the capability very much. Figure 21 shows the behaviour of the electric eld at the groove. The wave is coming from the right.
Figure 21: Side view (y-z plane) of an E-bend with groove (aluminium parts are blue). Groove radius is 0.3 mm
The largest eld value arises at the edge of the inner groove (small red eld), while the eld is comparable small in the groove.
Figure 22 is a plot of the values of table 5. The breakdown power increases linear with the groove radius.
34 5.2 Bellows 5 SIMULATION WITH MICRO WAVE STUDIO
Figure 22: Plot of the dependence between power and groove radius
5.2 Bellows Two kinds of bellows with dierent geometries concerning the ripples have been simu- lated. One bellow has a sinus-like geometry for its ripples with a radius of r = 2.38 mm, at both planes (for x- and y-direction). This bellow is called the `less exible` bellow ( gure 23).
Figure 23: Bellow geometry for the less exible bellow [the x-z-plane looks similar]
35 5.2 Bellows 5 SIMULATION WITH MICRO WAVE STUDIO
The other (`more exible`) bellow has a parable like geometry with a radius r = 2.5 mm for its ripples and a corresponding height of r2 = 6.25 mm.
Figure 24: Bellow geometry for the more exible bellow [the x-z-plane looks similar]
The smallest distance between the ripples is b − 2 · 0.37 · r2 = 74.75 mm ( gure 24). The smallest distance is analogous for the less exible bellow b−2mm−2·r = 75.75 mm.
The advantage of the less exible bellow is its rounded shape without any edges, which gives a higher capability and better VSWR, than for the more exible bellow. This disadvantage of the more exible bellow, is the price for the larger grade of exi- bility.
The results for both bellows (with short circuit) are given in table 6. The electric eld for the more exible bellow with short circuit does not reach the doubled value (581 V/m), because the geometry disturbs the interference of forward and re ected wave. This eect is not so large for the less exible bellow. The enhancement factor for formula (24) is in the order of f = 2 for the bellows.
The mesh properties for the bellow simulation have to be set to very high values (3.000.000 mesh cells) to get a ne mesh, which approximates the edges and curves in an acceptable way. This increases the calculation time to several hours (up to 17 hours with 1.8 GHz Intel Pentium 4, 1 GB RAM).
Figure 25 shows the eld distribution for the exible bellow. The maximum eld arises at the deepest part of the ripple. This is the rounded (acute) part with the smallest distance between top and bottom of the waveguide.
36 5.2 Bellows 5 SIMULATION WITH MICRO WAVE STUDIO
That is why the eld is enhanced at this part. At both sides a waveguide adapter (WR650) with a length of 165 mm (half wavelength) has been placed.
component electric eld Emax [V/m] power [MW] VSWR (at 1.3 GHz) less exible bellow 576 13.5 1.005 with short circuit 1044 4.1 - more exible bellow 581 13.3 1.04 with short circuit 879 5.8 -
Table 6: MWS results for the bellows
Figure 25: Top view for the eld distribution of the more exible bellow
37 5.3 One-stub tuner 5 SIMULATION WITH MICRO WAVE STUDIO
5.3 One-stub tuner The one-stub tuner has been simulated, because it is similar to the used measurement device. Figure 26 shows such kind of device with a length of 350 mm, where the cylinder is positioned at 240 mm (from point of origin).
Figure 26: One-stub tuner with a cylinder length of 32 mm
The radius of the cylinder for the simulation is always 12 mm. The simulation has been done for dierent cylinder lengths from 4 mm to 72 mm. The results for the maximum electric eld, breakdown power and VSWR are shown in table 7.
length of electric eld Emax [V/m] power [MW] VSWR cylinder [mm] (at 1.3 GHz) 4 647 10.7 1.01 8 883 5.7 1.03 14 1041 4.1 1.11 18 1198 3.1 1.2 22 1378 2.4 1.34 27 1591 1.8 1.64 32 1784 1.4 2.15 42 1962 1.2 4.8 52 1741 1.5 15.2 62 1391 2.3 64 72 1225 3 585
Table 7: Results for the one-stub tuner
The impedance of the device can be approximated by a capacitance for small lengths of the cylinder. When the depth of penetration is between ≈ 0.7 · b - 0.9 · b, it starts to act like a short circuit (from [Collin]). The impedance is inductive then.
38 5.3 One-stub tuner 5 SIMULATION WITH MICRO WAVE STUDIO
Figure 27: Circuit diagram for a post in a waveguide left : small cylinder length - capacitive right: cylinder is totally deployed - inductive down: cylinder in middle position - LC series network
In the middle position of the post (≈ 40 mm), the problem behaves (for 1.3 GHz) like a LC series network. It becomes resonant and the electric eld reaches its maximum value. Figure 27 shows the circuit diagrams for the dierent cylinder lengths.
Figure 28: Power behaviour of the one-stub tuner
Figure 28 shows, how the power decreases in the capacitive part, till it reaches the middle and the resonance condition. Then the power increases again in the inductive area. The maximum eld for the resonance condition (length = 42 mm) arises not in the end point of the sphere but 3 mm beside.
39 5.3 One-stub tuner 5 SIMULATION WITH MICRO WAVE STUDIO
Figure 29 shows a side view (y-z plane) of the device. The power source is on the left side. The maximum eld pattern is at the surface of the sphere 3 mm to the right from the middle point.
Figure 29: Side view of the one-stub tuner (cylinder length = 42 mm), wave comes from the right side
It is useful to have a look on the change of the phase to visualise the behaviour described above concerning the development from a capacitance about a series network to an inductance. The dierence between a capacitance and an inductance is a phase shift of 90◦. Figure 30 shows the relative phase changes depending on the cylinder length for the forward wave (S21 - parameter). The phase shift is nearly 90◦.
Figure 30: Relatives phase of the forward wave in the one-stub tuner
40 5.4 HOM simulation 5 SIMULATION WITH MICRO WAVE STUDIO
5.3.1 Measurement device A one-stub tuner like waveguide with a xed post has been used for the measurements. The diameter of the post is 10 mm and the height is 40 mm. The sparking level is dras- tically reduced with this device. The MWS calculation results are 200 kW as sparking power level and a standing wave ratio of 2.75. Figure 31 shows the decrease of the electric eld from its maximum point at the post middle to the bottom of the waveguide. The fast decrease of the eld also shows, that sparks might occur at higher power levels. The breakdown value (3 MV/m) is only close to the post reached. 200 kW is only the lower border for sparking, where the spark probability becomes larger than zero.
Figure 31: Electric eld in the waveguide with post (d = 10 mm, h= 40 mm) with a calculated sparking level of 200 kW.
5.4 HOM simulation To get an idea about the dimension of the higher order modes, a simulation with the E-bend for all propagable modes for the frequencies 2.6 GHz, 3.9 GHz and 5.2 GHz has been done.
41 5.4 HOM simulation 5 SIMULATION WITH MICRO WAVE STUDIO
The E-bend has been chosen, because it is a relative sensitive element concerning HOMs.
An interesting parameter is the VSWR for each mode. The most disturbing mode is that one, which builds up the largest standing wave. Table 8 shows the VSWR and power ratio H for the seven propagable wave types of the 2.6 GHz modes.
The ratio H describes the power ratio between the maximal transmitted power by the HOM and the maximal transmitted power by the fundamental mode (see also chap- ter 4.5.2).
The power ratio of the HOMs is normally less than 1, because the maximum electric
eld of the HOMs is bigger, than the eld for the fundamental mode (T10). Thus the HOMs determine the maximal transmitted power in the E-bend. If the ratio H is bigger than 1, the mode will not be interesting, because the limitation is given then by the fundamental mode (see chapter 4.5.2).
wave type cut o frequency VSWR at 2.6 GHz power ratio H [GHz]
TE10 0.908 1.04 1 TE01 1.817 1.04 1.58 TE20 1.817 1.02 0.93 TE11 2.032 1.03 0.87 TM11 2.032 1.07 0.84 TE21 2.57 2.5 0.18 TM21 2.57 3.6 0.08
Table 8: Results for the 7 propagable wave types of the 2.6 GHz modes
The relative bad VSWR (and power ratio) of the last two wave types can be explained with a simulation problem. The critical (cut o) frequency for these modes is 2.57 GHz. So it is very close to the limit of 2.6 GHz. These wave types behave in principle like the other types of the 2.6 GHz mode and should have similar VSWR values and power ratios.
The tables for the 3.9 GHz and the 5.2 GHz modes can be found in the appendix B. The behaviour of these modes is similar. The VSWR increases a little bit for higher cut o frequencies.
If the power ratio H for a mode becomes too small in a waveguide distribution, a HOM lter will be required.
42 6 MEASUREMENT ASSEMBLY
6 Measurement assembly
The main part of the measurement setup is a waveguide with a post (diameter: 10 mm, height: 40 mm). In this setup sparks are generated.
The calculated sparking level is 200 kW and the VSWR is 2.75 (see chapter 5.3.1). The measured VSWR value for all measurements is in the order of 2.98. There is a small dierence between calculated and measured value, because the whole waveguide system has an in uence to the VSWR. This has not been taken into account in the calculation. All measurements have been done at klystron 6 at FLASH, which is a setup for test- ing waveguide components and also the reserve for klystron 5.
Figure 32 shows a schematic side view of the waveguide system of klystron 6. The circulator area is lled with SF6. The input of the dry air system (output for nitrogen) is behind the gas window. So the whole area behind the window up to the terminal load is lled with air/nitrogen under the investigated parameters. A HOM lter has been required, because measurements in the past have shown, that the HOMs generated by the klystron are responsible for sparking in the waveguide system.
Figure 32: Schematic side view of Klystron 6
43 6 MEASUREMENT ASSEMBLY
The klystron has been driven with 11 kV at the modulator, which corresponds to 128 kV at the klystron cathode. All measurements have been done with a pulse length of 1 ms and a repetition rate of 5 Hz (if there is no other notation).
A H-bend with a spyhole is at the end of the waveguide system. Behind the H-bend is a directional coupler, which has been used to measure the power (forward and re ected). The waveguide with the post follows then. Figure 33 shows a side view of the whole measurement setup.
Figure 33: Side view of the measurement setup (in the middle: the waveguide with the post)
A camera has been placed at the spyhole of the H-bend for getting images of the sparks. The waveguide with the post is wired with a heating coil and heat insulated by a foil for the temperature measurements. Beside the H-bend is the (60 dB) directional coupler for measuring the power (forward and re ected). The dry air output is followed by a waveguide with a sound detector, which switches o the RF after a spark occurred. Finally there is the input for the nitrogen/ synthetic air and the termination load.
44 6 MEASUREMENT ASSEMBLY
Figure 34 shows a picture of the post in the waveguide. The spyhole of the H-bend is visible behind the post.
Figure 34: Picture of the used post; line of vision is towards the H-bend
To realize dierent gas llings in the waveguide, a gas distribution line for nitrogen and synthetic air has been build up. It is possible with this gas distribution, to ll nitro- gen/synthetic air with a constant pressure in the testing area. It is also possible to work with a continuous speci ed ow, because a ow meter is available for this distribution.
To controll the nitrogen content in the waveguide volume, an oxygen sensor has been used. The nitrogen measurements have been started, when the oxygen content has reached less than 1%.
45 6 MEASUREMENT ASSEMBLY
A compressor with a drying system (called `dry air machine`) has been used for ex- periments with (dry) air. The machine has been modi ed with a mixing system, which mixes water with dry air. It is possible with this system to handle every humidity in between 0.5 % to 60 % in the waveguide. A ow meter at the compressor allows to control the gas ow in the waveguide. The humidity is measured after the air ows through the waveguide.
Figure 35: The mixing system for the humidity measurements, installed on the dry air machine
In gure 35 one can see the mixing system installed at the compressor. The glass tube in the center is lled with water, which is bubbled up by air coming from the compressor. This `wet` air goes to the mixing tube (metal cylinder at the right side). Dry air comes from the drying system and also ows to the mixer. The humidity can be regulated by adjusting more or less pressure to the `wet air line` and the `dry air line`. The two regulators are also visible. The dry air pressure regulator is the valve in the center. In gure 35 the used humidity sensor and the ow meter are also visible (left side).
46 6.1 Measurement procedure 6 MEASUREMENT ASSEMBLY
6.1 Measurement procedure To take account for the stochastical behaviour of sparks a special procedure has been developed. This procedure is an agreement between getting statistics and using time e ciently for measurements. For getting a spark the sweep function of the signal gen- erator has been used. With increasing power the spark probability increases. The used sweep makes steps of 0.1 dBm every 2 s and starts with an adjusted start value (oset). A sound detector switches o the RF, when it sparks. Then a break of 5 minutes follows to re ll the waveguide and get the same gas conditions as before. After 5 minutes the RF and the sweep is started again to get the next spark.
Figure 36: Used sweep function of the signal generator
The start value was 380 kW up to 08.07.2008. At this date the start value was changed to 90 kW. The value has been changed according to gas mixture demands concerning nitrogen. (see chapter 7.2.4)
Due to the fact, that the sparking power has big deviation/changes (up to 50 %) from day to day, it can not be handled as an absolute value. So it is useful to de ne a reference value. This reference value is for practical reasons the (daily) sparking level of dry air (humidity: 1 %, pressure: 1 bar, temperature: 28-30 ◦C, gas ow: 1500 l/h). These are the so called standard conditions for dry air.
47 6.2 Taking spark pictures 6 MEASUREMENT ASSEMBLY
6.2 Taking spark pictures Behind the spyhole a camera for taking spark pictures has been placed. Two dierent cameras have been used during the measurement. The rst camera is a standard web camera (from Logitech). The second camera is an industrial digital camera produced by Basler (`Basler camera`).
The cameras are controlled by a Labview program, which triggers the camera and saves the taken pictures to hard disk. The program compares the (mean) light intensity acquired by the camera with an adjustable value. If the light intensity exceeds the given value, the program will start to save pictures (see the block diagram in the appendix C).
Figure 37 shows examples of spark pictures taken. The web camera gives coloured pictures, but its resolution is not good. The Basler camera has a better resolution, but can not take coloured pictures.
Figure 37: left: picture of a 440 kW spark in dry air taken with the web camera; right: picture of a 830 kW spark in dry air taken with the Basler camera
6.3 Observing time of the spark The power when sparking occurs was read out by a standard power meter (Agilent). The re ected power was also observed via a USB measuring adapter. The time of the spark within the 1 ms pulse was noted for every spark. Figure 38 shows the power meter display, while the RF is on.
48 6.3 Observing time of the spark 6 MEASUREMENT ASSEMBLY
Figure 38: Spark timing observed with the power meter (pulse length: 1 ms)
When a spark occurs, the sound sensor system switches o the RF and the power meter freezes the last pulse. Figure 39 shows a breakdown pulse with a length of 250 µs.
Figure 39: Freezed pulse due to a spark with a pulse length of 250 µs
49 7 MEASUREMENT RESULTS
7 Measurement results
Two measurement periods are presented in this chapter. In the rst measurement period (16.06 - 15.07.2008), one measurement consists of eight sparks. At least two (some days four measurements) of these `8 sparks` measurements have been done. One measurement always gives the daily sparking level of dry air (standard conditions) and is compared with the other measurement, where the investigated parameter has been changed. This rst period gives a tendency about the behaviour of the investigated parameter.
During the second measurement period (17.07 - 29.07.2008), only one measurement (consisting of 34 sparks) per day has been done. These measurements allow a statistical investigation. The investigated parameters for these `34 sparks` measurements are the same as for the rst period.
7.1 Data analysis The behaviour of sparks has a stochastical component like all collision processes in na- ture. In [Kahle] is mentioned, that the most common description of electrical breakdown is to use the (Gaussian) normal distribution with a mean value P¯ (sparking level) and a standard deviation σ. The spark density distribution f(P ) for the power P can be described with the mean value P¯ and its standard deviation σ by following formula:
¯ 1 − 1 ( P −P )2 f(P ) = √ e 2 σ (26) 2πσ This assumption will be checked (for the `34 sparks` measurements) with the so called χ2-test.
¯ is given by the arithmetic mean value ( ¯ 1 Pi=1 ) of each measurement (n = P P = n Pi n q 34 sparks) and the standard deviation is given by 1 Pi=1 ¯ 2. σ = n−1 n (Pi − P ) The spark density distribution describes the number of sparks in dependence of the power in a histogram. To get the breakdown probability in dependence of the power, this distribution has to be integrated from 0 to the maximum measured sparking power
(Pmax).
Z Pmax ¯ 1 − 1 ( P −P )2 F (P ) = √ e 2 σ dP (27) 2πσ 0 This integral can also be expressed with the so called error function Erf(x), which is nearly similar to the probability distribution of the normal distribution:
1 P − P¯ P¯ F (P ) = (Erf( max√ ) + Erf(√ )) (28) 2 2σ 2σ
50 7.2 Results for the `8 sparks` measurements 7 MEASUREMENT RESULTS
The comparison of the measured probability with the error function tted for these values gives a hint about the Gaussian (or not Gaussian) behaviour.
With the help of the so called χ2-test, it is tested, if the Gaussian distribution will describe the sparking behaviour for this measurement setup.
The χ2-test shows whether a random distribution of n (measured) values, which can be put into m classes (intervals) ts to an assumed mathematical distribution (for more 2 information see [Lang/Pucker]). The χ -test compares the number ni of measured values in the given intervals with the assumed (Gaussian) distribution ni0 = f(ni) (in this case the normal distribution).
i=1 2 X (ni − ni0) χ2 = (29) n m i0 The value χ2 has to be compared with the value χ2(1 − α, m − 1) given by the χ2- distribution, where α gives the con dence probability for this test. The test fails for χ2 > χ2(1 − α, m − 1) and the random distribution does not t to the assumed distri- bution.
A probability of 95 % has been used and the 34 sparks have been seperated into √ m = 6 (≈ 34) intervals. In this case the χ2-distribution gives a value of χ2(0.05, 5) = 1.15 (see [Wegmann/Lehn]).
7.2 Results for the `8 sparks` measurements The mean values and standard deviations for this measurement period are listed in tables for all measurement days. The measurement value for one measurement series (measurement day) is the mean value for this day (mean value for 8 sparks). In addition to the mean value its standard deviation is listed.
At the bottom of each table is the mean value of all measurement days given with its uncertainty u. This main mean value is the measurement value. The measurement uncertainty u for the measurement value has been calculated using: 1 u = √ σc¯ (30) n where c is the con dence coe cient, given by the Student`s t-distribution (see [Witting] or [Rabinovich]).
σ¯ is the mean value of the daily standard deviation listed in the tables.
51 7.2 Results for the `8 sparks` measurements 7 MEASUREMENT RESULTS
All measurement uncertainties are calculated for a con dence probability of 95 %. Thus c is set to 1.86 for a value given by n = 8 sparks (for 34 sparks it is about 1.69).
As a reference, the measured values for dry air are always given in addition. At the end of each table the ratio of the measurement value to the value for standard condition of dry air is listed.
7.2.1 Humidity measurement The in uence of the relative humidity has been investigated for 3 dierent relative hu- midity magnitudes (13 %, 40 % and 60 %).
13 % is the humidity of the used compressor (without dry air machine), 40 % is an average value for the hall humidity and 60 % should represent a typical value for outside conditions (`Hamburger wheather`). At all humidity measurement days at least one (sometimes two) dry air measurement has been done.
Table 9 presents the results for the 8 measurement days.
52 measurement dry air stand. dev. compressor air stand. dev. `normal` air stand. dev. `wet` air stand. dev. dry air stand. dev. series 1 % [MW] 1 % [MW] 13 % [MW] 13 % [MW] 40 % [MW] 40 % [MW] 60 % [MW] 60 % [MW] 1 % [MW] 1 % [MW] day 1 1.24 0.58 1.45 0.54 1.36 0.56 1.30 0.56 day 2 0.88 0.21 1.29 0.45 1.23 0.57 0.94 0.37 day 3 1.11 0.53 1.34 0.60 0.91 0.25 1.20 0.47 day 4 1.36 0.63 1.18 0.37 day 5 0.94 0.26 1.28 0.50 day 6 1.06 0.23 1.08 0.43 day 7 1.01 0.41 1 0.25 day 8 0.86 0.21 1.01 0.44 mean value ± uncertainty 1.06 ± 0.09 1.12 ± 0.15 1.27 ± 0.17 1.17 ± 0.14 1.15 ± 0.18 ratio to dry air 1 1.06 1.2 1.1 1.08
Table 9: Results for dierent humidity values. The uncertainty quoted have been calculated using (30). (detailed explanation in chapter 7.2) 7.2 Results for the `8 sparks` measurements 7 MEASUREMENT RESULTS
The results show, that no investigated humidity value decreases the power capability. The main result for the analysed humidity range is that it is at least comparable to dry air and for some humidity values even better (for example at 40 %). This can be explained with the electronegativity of the water molecules (see [Köhrmann]). This measurement also shows the large deviation of the measured power values. The ratio between the dry air measurements is 1.08.
7.2.2 Temperature measurement The temperature measurement has been done with dry air from the compressor with a ow of 1500 l/h. Two temperatures have been compared, 28-30 ◦C (hall temperature) and 48 ◦C. This is the interesting temperature area, because for a bigger duty cycle the average power is larger and the waveguide can be heated up to this temperature. A heating coil has been used to reach higher temperatures. It is possible with this setup to get 60 ◦C at the outer waveguide wall beside the coil. This corresponds to a temperature of about 48 ◦C at the post and the inner wall. Figure 40 shows an infrared picture of the waveguide and the post when it is heated by the coil at the outer side up to 60 ◦C.
Figure 40: Infrared picture of the waveguide heated up to 60 ◦C, at the right bottom there is a temperature sensor to prove the IR-camera measurement
The results for both temperatures are given in table 10. There seems to be no dierence for the investigated temperature area. The ratio (dry air 30 ◦C to 60 ◦C) of 0.95 is unfortunately not signi cant because the daily deviation of the power capability is too large. At two days the power capability for 60 ◦C was higher than for 30 ◦C. The ratio to dry air should be theoretically in the order of 0.9 (with gure 12).
54 7.2 Results for the `8 sparks` measurements 7 MEASUREMENT RESULTS
measurement 30 ◦C stand. dev. 60 ◦C stand. dev. series [MW] [MW] [MW] [MW] day 1 0.83 0.36 0.90 0.37 day 2 0.91 0.41 0.96 0.29 day 3 1.24 0.55 1.11 0.40 day 4 1.29 0.60 1.06 0.36 mean value ± uncertainty 1.07 ± 0.16 1.01 ± 0.12 ratio to 30 ◦C 1 0.95
Table 10: Measurement results for dierent air temperatures. The uncertainty quoted have been calculated using (30). (detailed explanation in chapter 7.2)
7.2.3 Pressure measurement The pressure measurement has been done with dry air at 1.3 bar. It is not necessary to make the waveguides gas tight with gaskets for this pressure range.
Figure 41: Side view of the measurement setup. The two used pressure sensors are visible. The lower part of the klystron is in the background
55 7.2 Results for the `8 sparks` measurements 7 MEASUREMENT RESULTS
At Figure 41 the output for the dry air with a valve which has been closed for the pressure measurement is visible. The two used pressure sensors are also visible, one analogue sensor near the valve and one sensor with a readout via USB at the waveguide.
The power capability should increase for an ideal gas with a square (see equation (21)). This would give a ratio for the sparking level of 1.3 bar 2 = 1.69 for the mentioned ( 1 bar ) pressure. In practice the exponent is not two, but assumes values like 3/2 or 4/3 (see chapter 3.6).
Table 11 shows the results for the pressure measurements.
measurement dry air 1 bar stand. dev. dry air 1.3 bar stand. dev. series [MW] [MW] [MW] [MW] day 1 1.45 0.44 1.81 0.78 day 2 1.19 0.63 1.43 0.44 day 3 1.84 0.60 2.26 0.40 day 4 1.29 0.48 1.59 0.62 mean value ± uncertainty 1.44 ± 0.18 1.77 ± 0.18 ratio to dry air 1 bar 1 1.23
Table 11: Measurement results for pressurised air as compared to air at normal pressure. The uncertainty quoted have been calculated using (30). (detailed explanation in chapter 7.2)
The ratio of the sparking levels is 1.23. So the power capability is about 20 % better for even a small overpressure. This result gives a good hint, how to increase the power capability of an established waveguide system with little technical eort.
7.2.4 Nitrogen measurement The nitrogen measurements consist of two experiments. One is the investigation of ni- trogen with a small ow (300 l/h) at normal pressure (1 bar) and the other is nitrogen with pressure (1.3 bar). The humidity of the nitrogen given by the nitrogen bottle is about 1 %. The measurement procedure is similar to air. After owing through the waveguide the exhaust nitrogen ows through a pipe out of the hall.
The nitrogen measurement has been compared every day with the daily dry air measurement. The results are shown in table 12. In principle nitrogen should behave like air. In [Gilden/Gould] it is mentioned, that with `Paschen curve behaviour` nitrogen should have about 90 % of the power capability of air.
56 measurement dry air 1 bar stand. dev. nitrogen 1 bar stand. dev. nitrogen 1.3 bar stand. dev. dry air 1bar stand. dev. series [MW] [MW] [MW] [MW] [MW] [MW] [MW] [MW] day 1 1.35 0.40 0.54 0.11 0.58 0.16 1.13 0.48 day 2 1.43 0.44 0.73 0.23 0.42 0.04 1.41 0.34 day 3 1.36 0.44 0.53 0.15 0.45 0.08 0.99 0.38 day 4 1.56 0.73 0.38 0.15 0.48 0.17 1.65 0.69 day 5 1.31 0.61 0.39 0.14 0.41 0.15 mean value ± uncertainty 1.40 ± 0.15 0.51 ± 0.05 0.47 ± 0.03 1.30 ± 0.16 ratio to dry air 1 0.36 0.31 0.96
Table 12: Measurement results for nitrogen as compared to dry air for dierent pressures. (detailed explanation in chapter 7.2) 7.2 Results for the `8 sparks` measurements 7 MEASUREMENT RESULTS
The result shows a very strange behaviour. The dierence between air and nitrogen is enormous. It is unusual that the power capability of nitrogen with a pressure of 1.3 bar is worse than for nitrogen with 1 bar. This might be a problem with the measure- ment procedure. That is why the start value for the power sweep of the signal generator has been changed to 90 kW (for the last two measurements). The measurement with 34 sparks shows that this assumption is correct (table 15).
In principal nitrogen behaves like from the MWS simulations expected. It is not clear, why air has such a `good` power capability!
Figure 42 shows two sparks in nitrogen taken with the Basler camera.
Figure 42: left: picture of a 450 kW spark in nitrogen with pressure 1.3 bar; right: picture of a 350 kW spark in nitrogen (1 bar)
7.2.5 Synthetic air measurement To check the setup for the nitrogen measurement some measurements with a bottle of synthetic air (instead of nitrogen) have been done. Synthetic air consists of 20.5 % oxygen and 79.5 % nitrogen (no noble gases). This measurement also shows if there are in uences by dirt from the compressor for the dry air measurements. The synthetic air has been lled in the waveguide with a ow of 800 l/h. As usual it has been compared with dry air (standard conditions).
The results in table 13 shows, that there is no dierence between dry air and syn- thetic air. So there are no in uences to the measurement by the used setup (compressor, distribution lines, etc.). This measurement also shows, that even the smaller ow of 800 l/h (compared to 1500 l/h for dry air) has no in uence.
Figure 43 shows a 2.82 MW (relative strong) spark in synthetic air. The post structure can only be guessed behind the light spot.
58 7.3 Results for the `34 sparks` measurements 7 MEASUREMENT RESULTS
measurement dry air stand. dev. synthetic air stand. dev. series [MW] [MW] [MW] [MW] day 1 0.93 0.35 0.88 0.33 day 2 1.39 0.60 1.11 0.66 day 3 1.67 0.64 1.89 0.84 mean value ± uncertainty 1.33 ± 0.20 1.29 ± 0.23 ratio to dry air 1 0.97
Table 13: Measurement results for synthetic air as compared to dry air. The uncertainty quoted have been calculated using (30). (detailed explanation in chapter 7.2)
Figure 43: Picture of a 2.82 MW spark in synthetic air.
7.3 Results for the `34 sparks` measurements √ The measured values have been subdivided in 6 (≈ 34) intervals for the `34 sparks` measurements. The number of sparks within these intervals have been counted, thus a histogram can be developed. The normal distribution has been calculated with the mean value and the standard deviation of the 34 sparks and normalised to the maximum value for the six interval centers. With these numbers the χ2-test has been done. The rst measurement with 34 sparks has been done for dry air. Figure 44 shows a histogram for the 34 sparks with dry air. The intervals for the histogram have a width of 600 kW.
59 7.3 Results for the `34 sparks` measurements 7 MEASUREMENT RESULTS
The red curve connects the six points calculated by a assumed normal distribution with a mean value of 1.59 MW and a standard deviation of 0.63 MW.
Figure 44: Histogram for dry air, tted with a normalised Gaussian distribution (red)
At rst sight it may look like a Gaussian distribution, but the χ2-test result fails. Figure 45 shows the spark probability in dependence of the power. It is tted with the error function for the assumed Gaussian distribution with the mentioned mean value and standard deviation.
Figure 45: Spark probability for dry air in the measurement setup (blue) and calculated error function (red) of the assumed Gaussian distribution
60 7.3 Results for the `34 sparks` measurements 7 MEASUREMENT RESULTS
Table 14 shows the results of the χ2-test for all `34 measurements`. The test will be passed, if the χ2 is smaller than 1.15 (see explanation for the χ2-test). No measurement passes the test. So the probability for sparking does not follow a Gaussian distribution for these measurements. This result is signi cant (con dence probability: 95 %).
measurement χ2 fail/pass test
dry air (standard conditions) 3.05 fail dry air with pressure 1.3 bar 9.63 fail compressor air 1 bar (humidity: 13 %) 5.50 fail compressor air 1.3 bar (humidity: 13 %) 12.99 fail normal air without ow (humidity: 30 %) 4.96 fail nitrogen 1 bar ( ow: 300 l/h) 5.83 fail nitrogen 1.3 bar 37.23 fail synthetic air 1 bar ( ow: 800 l/h) 4.47 fail synthetic air 1.3 bar 8.06 fail
Table 14: Results of the χ2-test
The test also shows, that the ow measurements are closer to the border value of 1.15. By contrast the pressurized measurements have no chance to pass the test. This can also be seen in the histograms of the pressurized measurements. Figure 46 shows the histogram for the pressurized synthetic air as example. So the spark kinetic behaves `more` Gaussian for a gas with ow, than for stagnant (pressurized) gas.
Figure 46: Spark histogram for pressurized synthetic air (1.3 bar)
61 7.3 Results for the `34 sparks` measurements 7 MEASUREMENT RESULTS
Table 15 shows the mean values for the 34 sparks, uncertainties, the ratios to dry air (standard conditions) and the ratios to the ow measurement.
ratio to ow measurement mean value uncertainty ratio to measurement [MW] [MW] dry air of the same gas dry air (standard conditions) 1.59 0.18 1 - dry air with pressure 1.3 bar 2.12 0.24 1.34 1.34 compressor air 1 bar 1.77 0.18 1.1 - compressor air 1.3 bar 2.17 0.25 1.37 1.25 normal air without ow 1.82 0.2 1.1 - nitrogen 1 bar ( ow: 300 l/h) 0.41 0.05 0.25 - nitrogen 1.3 bar 0.45 0.04 0.29 1.13 synthetic air 1bar ( ow: 800 l/h) 1.66 0.22 1.05 - synthetic air 1.3 bar 2.02 0.22 1.28 1.22
Table 15: Mean values and uncertainties for the `34 sparks` measurements
The results are a little bit dierent to the `8 sparks` measurements, but the tendency is similar. The ratios to the ow measurement are about 1.2 - 1.3. The compressor air with humidity is a little bit better and synthetic air behaves like dry air. The nitrogen measurements have again the lowest power capability with ratios in the order of 0.3. So the power capability of nitrogen seems to be three times smaller than the capability of air.
62 7.4 Spark timing 7 MEASUREMENT RESULTS
7.4 Spark timing The observation of the re ected power shows that the spark probability for every time within the pulse is equiprobable. Figure 47 shows a histogram for 536 sparks from both measurement periods.
Figure 47: Distribution of the spark timing within the observed 1 ms pulses. Interval: 100 µs
7.5 Pulse length and repetition rate A small investigation within the `34 sparks` measurements is analysing the in uence to the power capability concerning the pulse length and the repetition rate. Three dierent pulse lengths have been investigated (0.5 ms, 1 ms, 1.4 ms) and two dierent repetition rates have been available (5 Hz & 10 Hz) to analyse. All six measurements have been realized with dry air (standard conditions).
Figure 48 displays the six measured points. The power capability decreases for in- creasing pulse length. This decrease is theoretically expected and should go continuously to the point, where CW - conditions are reached. The same behaviour of decreasing power capability can also be seen for the repetition rate. The power capability is nearly 1.5 times higher for 5 Hz, than for 10 Hz.
63 7.6 Systematic errors 7 MEASUREMENT RESULTS
Figure 48: Dependency of pulse length in between 0.5 ms and 1.4 ms for 5 Hz and 10 Hz
7.6 Systematic errors The systematic errors given by the measurement devices and the setup are very small compared to the standard deviation of the measurement itself.
The used Agilent power meter (E4417A) has a relative error for the measured power levels ( 90 dBm) of about 1 %. The accuracy of the used cables and attenuators is ± 0.02 dB. The systematical error for the frequency (1.3 GHz) and the pulse modulation given by the signal generator (Rohde & Schwarz) is also not worth mentioning (< 1 % relative error).
The measurement of the gas ow has been done with a classical ow meter (KROHNE) with an uncertainty of 1 %, corresponding to ± 20 l/h. The used WIKA pressure sensor has an relative error of 0.2 %. A value of 1 % can also be taken for the humidity sensors (KAESER) of the dry air machine. Nitrogen and synthetic air have a quality of 5.0 (99.999 % of the bottle volume is nitrogen/synthetic air) and humdity of about 1 %. The systematic errors of the dierent gas distribution lines for air and nitrogen is also not relevant. This can be seen by comparing the results for dry air and synthetic air. A systematic error due to surface problems after many sparks with the post or the waveguide have not been recognized. A post exchange (14.07.2008) validates, that there are no systematic problems concerning the surface.
All the mentioned systematic errors are very small compared with the statistical uctuation and should not aect the results.
64 8 SUMMARY AND CONCLUSION
8 Summary and Conclusion
8.1 A model of power limitation In chapter 4.5 some theoretical hints are given on how to calculate the practical power capability of a waveguide. Formula (24) is a rst model to describe the power limitation. Each eect gives its own factor to the power limitation. The structures of these factors give an idea about the importance of various eects. Equation (24) can be modi ed to determine the maximum power capability attain- able in practice:
1 VSWR + 1 600 P = P ( )2 p4/3 h ( )2 (31) theo f 2β2 2 · VSWR T + 300
Ptheo is the theoretical power capability of the waveguide using (13). The following factors are dimensionless.
f is the factor for the eld enhancement determined by the waveguide geometry and β is the eld enhancement factor given by the surface of the waveguide. Typical values for f are given by numerical simulations: 1 for straight waveguide, 1.5 for E-bend, 2 for bellows (see chapter 5.1). The factor β is in the order of 2 (see chapter 4.3).
The Voltage Standing Wave Ratio (VSWR) describes the standing wave parts in the waveguide, which limits the power capability (for details see chapter 4.5.1).
p is the air pressure in units of bar. This factor shows that the power capability increases for overpressure (for details see chapter 3.6).
The empirical factor h describes the measured in uence of the humidity. It is h = 1.2, if the relative humidity is between 16 % and 40 % otherwise h = 1.
The limitation by the temperature T (in K) can be considered by the theoretical factor 600 2, where the relative power capability is normalised to 300 K (see chapter ( T +300 ) 3.6). In practical the in uence by the temperature is only interesting for waveguide applications heated by large average RF power. This factor could not be veri ed experi- mentally within the present investigation since the temperature range was rather small.
Equation (31) can explain the big dierence between practical and theoretical power capability. The values for β are 2 for a typical waveguide distribution. In a waveguide line with normal air conditions (p ≈ 1 bar, T ≈ 300 K, h > 40 %) the 3 `gas condition factors` can be set to 1. So there is no in uence by the air for the mentioned conditions. The reduction of power is by choosing f = 1.5 and VSWR = 1.2 :
1 1.2+1 2 = 0.093 . 22·1.52 · ( 2·1.2 ) Ptheo ·Ptheo
65 8.2 Summary of the measurement results 8 SUMMARY AND CONCLUSION
With (31) the theoretical power capability Ptheo of 58 MW (for a WR650 waveguide) is reduced to 5.4 MW for the discussed waveguide distribution.
Formula (31) is only an approximation to demonstrate the in uence of several eects, but it is not accounting for all eects of power limitation. This problem would exceed the scope of this thesis.
8.2 Summary of the measurement results The results of the measurements show some expected behaviour, but also give some strange outcomes.
The humidity measurements show that humidity increases the power capability (com- pared with dry air) a little bit. This can be explained with the electro negativity of the water molecules. To use this eect, the air should not be over saturated. The measure- ments show that the power capability increases most with humidity values between 16 % and 40 %.
The measured dependence on pressure is in accordance with expectations. The re- sults of this measurement combined with the humidity results give a good hint on how to increase the power capability of an existing waveguide system. For FLASH this could be an option for the future to use compressed air with the humidity given by the com- pressor. This can be done with little technical eort.
Within the range of temperatures applied, no in uence on the power capability was observed. But there could be a dierence between an arti cially heated waveguide (by a heating coil) and a waveguide heated by high average RF power. Maybe the used temperature dierence was too small to see the theoretical behaviour described in equation (31).
The measurement with nitrogen gives a very strange result. The power capability of nitrogen is two times smaller (with the `8 sparks` measurements) than for air. This is totally unexpected and so far there is no explanation for this dierence. Nitrogen better ts with the calculated value for the power capability of the measurement device, so the question should be: Why has air such a good power capability compared to nitrogen for this setup ?
66 8.2 Summary of the measurement results 8 SUMMARY AND CONCLUSION
The synthetic air measurement shows the same behaviour like the reference measurement with dry air, which was expected. This measurement has just been done to prove that there are no systematical errors in the measurement setup (especially with the nitrogen distribution line). The synthetic air measurement also shows that the mag- nitude of the gas ow has no in uence.
It could be proved with the help of the χ2-test that sparking is not a Gaussian dis- tributed event for this measurement setup.
The observation of the timing of the spark shows that the spark could occur at each point within the pulse with the same probability for the measured range. The investi- gation of the pulse length and the repetition rate results in the expected decreasing of the power capability.
The measurements for the dierent gas conditions give very interesting results and good hints on where it is useful to make further investigations. Especially for the nitro- gen measurements it would be interesting to prove these results by another experimental setup.
Many of the mentioned aspects concerning power limitation like for example HOMs and the in uence of the surface were only slightly touched and should also be investigated in more detail.
67 References
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[Anderson/Lisak] D. Anderson, M. Lisak: Breakdown in air- lled microwave waveguides during pulsed operation, Chalmers University of Technology, Göteborg, 1984
[Collin] R. E. Collin: Foundations for microwave engineering, McGraw-Hill Book Company, 1966
[FLASH] FLASH homepage: http:// ash.desy.de/
[Forbes] R. G. Forbes: Use of a spreadsheet for Fowler-Nordheim equation calculations, School of Electrical Engineering, University of Surrey, 1998
[Gilden/Gould] M. Gilden, L.Gould: Handbook on high power capabilities of waveguide systems, Department of the Navy, Electronics Division, 1963
[Habermann] T. Habermann: Rastermikroskopische Untersuchung der Feldemission von Metall- und Diamantkathoden, Dissertation, Bergische Universität Wuppertal, 1999
[Harvey] A. F. Harvey: Microwave Engineering, Academic Press London and New York, 1963
[Hess] H. Hess: Der elektrische Durchschlag in Gasen, Friedr. Vieweg & Sohn, Braunschweig, 1976
[Jüttner] B. Jüttner: Über die Natur des Vordurchschlages im Hochvakuum, Physikalisch Technisches Institut der Deutschen Akademie der Wis- senschaften, Berlin, 1965
[Kahle] M. Kahle: Elektrische Isoliertechnik, Springer Verlag, Berlin, 1989
[Kind/Kärner] D. Kind, H. Kärner: Hochspannungs - Isoliertechnik, Friedr. Vieweg & Sohn, Braunschweig, 1982
[Köhrmann] W. Köhrmann: Der Ein uss der Luftfeuchtigkeit auf den elektrischen Durchschlag, Annalen der Physik 6. Folge, Band 18, 1956
[Laska] L. Laska: Dielectric proberties of SF6 containing oxgen and other gases, Czech. J. Phys. B 34 page 1038, 1984 [Lang/Pucker] C. B. Lang, N. Pucker: Mathematische Methoden in der Physik, Spektrum Akademischer Verlag, München, 2005
[Lorrain/Corson] P. Lorrain, D.R. Corson: Electromagnetic elds and waves, Freeman and Company, San Francisco, 1970
[Meinke] H. H. Meinke: Felder und Wellen in Hohlleitern, Oldebourg Verlag, München, 1949
[Mortimer] R. G. Mortimer: Physical chemistry, Academic Press San Diego, 2000
[Nolting] W. Nolting: Grundkurs Theoretische Physik 3, Springer Verlag, Berlin, 2004
[Rabinovich] S. G. Rabinovich: Measurement errors and uncertainties, Springer Ver- lag, New York, 2000
[Rajtsin] D. G. Rajtsin: Breakdown strength of RF structure, Sovetskoe radio, Moscow, 1977
[Sucher/Fox] M. Sucher, J. Fox: Handbook of microwave measurements (Volume II), Polytechnic press of the Polytechnic Institute of Brooklyn, New York, 1963
[Schneidmiller/Saldin/Yurkov] E. Schneidmiller, E. Saldin, M. Yurkov: The Physics of Free Electron Lasers, Springer Verlag, Berlin, 2000
[Wegmann/Lehn] H. Wegmann, J. Lehn: Einführung in die Stochastik, Vandenhoeck & Ruprecht, 1984
[Witting] H. Witting: Mathematische Statistik, Teubner Studienbücher, Stuttgart, 1966 A SURFACE INVESTIGATIONS Appendix
A Surface investigations
Table 16 shows the measured roughness for a 200 mm straight waveguide, an E-bend and a H-bend (mean value for 4 measurements). The pro lometer scans a distance of 4 mm and has an accuracy of ± 0.01 µm. element Ra Ry Rz Rq straight [200 mm] 2.02 13.95 9.37 2.61 H-bend 3.42 19.83 15.28 4.59 E-bend 9.73 45.58 36.83 12.13
Table 16: Roughness measured with the (Mitutoyo SJ-301) pro lometer
where Ra is the arithmetic mean value of the norm of the y-coordinate (height) [ 1 PN ], Yi Ra = n i=1 |Yi|
Rq is the square root mean value of the y-coordinate [ 1 PN 2 1/2], Yi Rq = ( n i=1 Yi )
Ry is the sum of the largest (Yl) and the deepest y-coordinate (Yd)[Ry = Yl + Yd],
and Rz is the sum of the mean values of the 5 largest (Yli) and the 5 deepest y- coordinates ( )[ 1 P5 1 P5 ] Ydi Rz = 5 i=1 Yli + 5 i=1 Ydi The roughness Ry of the E-bend is nearly four times higher than that of the straight. Figures 49 and 50 show the roughness of a H-bend and an E-bend.
This dierence concerning the surface can also be seen by calculating the surface eld enhancement factor β. It is for the H-bend and the straight in the order of 2, while the E-bend has a β of 3 (compare gures 49 and 50).
Figure 49: Surface pro le of an H-bend; vertical: 10 µm/scale division, horizontal: 100 µm/scale division (total 4mm)
i B TABLES FOR THE HOM SIMULATION OF THE E-BEND
Figure 50: Surface pro le of an E-bend; vertical: 10 µm/scale division, horizontal: 100 µm/scale division (total 4 mm)
B Tables for the HOM simulation of the E-bend
The two tables for the higher modes of 3.9 GHz and 5.2 GHz are given below. The modes show the same behaviour like the 2.6 GHz mode. (for more information see 5.4 or 4.5.2)
wave type cut o frequency VSWR at 3.9 GHz power ratio H [GHz]
TE10 0.908 1.11 1 TE01 1.817 1.04 1.85 TE20 1.817 1.06 0.92 TE11 2.032 1.05 1.13 TM11 2.032 1.14 0.85 TE21 2.057 1.05 1.64 TM21 2.570 1.06 0.91 TE30 2.726 1.11 0.93 TE31 3.726 1.02 0.81 TM31 3.276 1.04 0.85 TE40 3.634 1.09 0.65 TE02 3.634 1.20 0.93 TE12 3.746 1.50 0.32 TM12 3.746 1.48 0.34
Table 17: Results for the 14 propagable wave types of the 3.9 GHz modes
ii C LABVIEW PROGRAM FOR TAKING SPARK PICTURES
wave type cut o frequency VSWR at 5.2 GHz power ratio H [GHz]
TE10 0.908 1.20 1.00 TE01 1.817 1.10 2.15 TE20 1.817 1.04 1.02 TE11 2.032 1.03 1.31 TM11 2.032 1.10 0.62 TE21 2.570 1.10 1.39 TM21 2.570 1.15 0.75 TE30 2.726 1.09 1.05 TE31 3.276 1.10 1.06 TM31 3.276 1.07 0.82 TE40 3.634 1.07 0.97 TE02 3.634 1.05 2.19 TE12 3.746 1.05 1.22 TM12 3.746 1.06 0.39 TE41 4.063 1.07 1.14 TM41 4.063 1.25 0.36 TE22 4.063 1.01 1.21 TM22 4.063 1.06 0.92 TE50 4.543 1.04 0.80 TE32 4.543 1.04 0.96 TM32 4.543 1.11 0.41 TE51 4.893 1.03 0.55 TM51 4.893 1.05 0.72 TE42 5.139 1.90 0.35 TM42 5.139 1.90 0.24
Table 18: Results for the 25 propagable wave types of the 5.2 GHz modes
C Labview program for taking spark pictures
Figure 51 shows the block diagram of the Labview program for the web camera, which has been used to take pictures of sparks. The block diagram for the Basler camera program has the same structure. The whole time the camera acquires pictures by the USB acquisition in the outer For-loop and compares the mean light intensity with an adjusted value. If the comparison gives `false`, the acquisition will go on. The inner case structure is started for `true` and saves a number (de ned by the user) of pictures in the adjusted folder.
iii D SPARK PICTURES
Figure 51: Block diagram of the used Labview program
D Spark pictures
All spark pictures show that the main part of the spark is going from the end of the post to the bottom of the waveguide. Some pictures also show very interesting rami cation. Figure 52 shows a 390 kW spark in nitrogen with pressure, where a part of the avalanche goes back to the post. The right picture in gure 52 shows a spark in dry air, taken with the web camera during the measurement with a repetition rate of 10 Hz and a pulse length of 1 ms.
iv D SPARK PICTURES
Figure 52: left: 390 kW spark in nitrogen (1.3 bar) taken with the Basler camera; right: 1.5 MW spark in dry air taken with the web camera
The strength of the light intensity of the sparks is not very constant. The right picture of gure 53 shows a relative weak lighted spark in synthetic air, although the sparking power is 600 kW. The left picture shows a 440 kW spark in dry air. This spark has a strange rami cation with a part going up.
Figure 53: left: 440 kW spark in dry air (standard conditions) taken with the web camera; right: 600 kW spark in synthetic air taken with the Basler camera
v Acknowledgments
At this point I would like to thank all people supporting me during the time creating this thesis.
A special thank goes to Valery Katalev, who gave many suggestions and hints. In particular for the measurements and the setup he has avoided no trouble and eort to support me with many helpful advices.
I am also very grateful to Stefan Choroba, who made this thesis possible and had also many advices. He made many suggestions concerning contents and was very helpful with his correction proposals especially during the nal stage of this thesis.
I would like to thank Frank Eints for constructing a HOM lter and helping to make the SF6 section at klystron 6 gas tight. The measurements could not have been done in time without his help. I also thank Vladimir Zhemanov for his help during the measurement period. He helped to establish the setup and helped with many measure- ments.
My thanks also go to the people of the group MHF-p (Richard Wagner, Torsten Grevsmühl, Igor Sokolov, Thomas Froelich) for helping to operate the modulator/klystron. They also handled any computer troubles at the test stand.
I would like to thank Jörg Rossbach for his support during the nal stage of this thesis and especially for the possibility to attend the Cern Accelerator School.
I wish to thank Hauke Langkowski, due to his correction proposals.
Finally I am also indebted to my mother Susanne Göller for proofreading this thesis and making correction proposals.