Before the presentation: a short introduction My home institution is the University of Bath in the UK, where I am studying for a Masters in Mathematics and Physics I am a technical student here for one year, beginning in July

What have I been working on? Since arriving in July, besides from learning, I have been working on: Modelling beamline equipment (specifically the QuattroTank) with transmission line networks. Developing software to assist in this modelling Attempting to understand resonances in the context of transmission lines – e.g. RLC circuits and shunt impedance The negative effect of SUCOBoxes on wire measurements; seen by taking wire measurements of the QuattroTank The Panofsky-Wenzel theorem More recently: Comparing results from HFSS and CST (work in progress) Looking at some formulae from transmission line theory which are used in the wire method (e.g. improved log formula and Wang-Zhang formula) – the scope of each formula and where they might be incorrect. Possible development of new fomulae

This presentation will be about two points in particular: The negative effect of SUCOBoxes on wire measurements Transmission line formulae used with wire measurements

Negative effect of a SUCOBox on wire measurements SUCOBox connection The ideal setup The actual setup In practice, stretching a wire through a piece of equipment and securing it in place is difficult. We used a so-called SUCOBox to assist in the actual placement of the wire: Direct connection

Negative effect of a SUCOBox on wire measurements Despite the small size of the SUCOBox compared to the wavelengths used in measurement, it can introduce considerable distortion in reflected and transmitted signal. E.g. at 0.8 GHz, the 𝜆= 𝑐 𝑓 ≅35 cm, 10 times larger than a SUCOBox. Wire measurements (orange) of a QuattroTank with one SUCOBox were taken – and it is clear that there is considerable distortion in the reflection on the side of the SUCOBox ( 𝑆 11 )[8] . Blue is a transmission line model of the device. Reflection from a SUCOBox Reflection directly from the chamber face Transmission signal – to some extent the oscillations are caused by the SUCOBox (to be investigated more thoroughly)

Arnold with thanks to O. Berrig Comparison of impedance formulae in transmission line theory and beam impedance theory Arnold with thanks to O. Berrig

Why should we use transmission line theory? It is standard practice to measure beam impedance with the stretched wire method, and it has been used for many years in the context of accelerator physics. Many papers are available on the topic[1][2][3]. The stretched wire setup forms a two-conductor transmission line. We must keep in mind that even if the radius of the wire vanishes, it is still a two-conductor system, which means the system with a real beam and the system with a wire are not exactly equivalent.

Using transmission line theory to measure beam impedance Here, we will cover some ways to calculate 𝑚=0 longitudinal impedance, using a the wire method. In this presentation we will focus on the use of scattering parameters, first for transmission ( 𝑆 21 , 𝑆 12 ) and then for reflection ( 𝑆 11 , 𝑆 22 )

Transmission formulae

The use of the Lumped, Log and Improved Log formulae To apply these formulae, our Device Under Test (DUT) and Reference (REF) lines must be matched; there can be no internal reflections caused by the transition from the REF or DUT to the 50 Ω cable. Matching resistor Lumped formula Log formula Improved log formula 𝑍=−2 𝑍 0 ln 𝑆 21 𝐷𝑈𝑇 𝑆 21 𝑅𝐸𝐹 1+ 𝑖 ln 𝑆 21 𝐷𝑈𝑇 𝑆 21 𝑅𝐸𝐹 2 𝜔𝑙 𝑐 𝑍=2 𝑍 0 𝑆 21 𝑅𝐸𝐹 𝑆 21 𝐷𝑈𝑇 −1 𝑍=2 𝑍 0 ln 𝑆 21 𝑅𝐸𝐹 𝑆 21 𝐷𝑈𝑇 Where 𝑆 21 𝑅𝐸𝐹 is the transmission coefficient of some reference pipe; usually something like a smooth, perfectly conducting beampipe. The ratio 𝑆 21 𝐷𝑈𝑇 𝑆 21 𝑅𝐸𝐹 is independent of the matching network

Results of common transmission formulae and a lumped impedance We will apply these formulae to a lumped impedance 𝑍 𝐿 inside a lossless reference line (with propagation velocity 𝑐): 𝑍 0 𝑍 𝐶 𝑍 𝑚 𝑍 𝐿 DUT 𝑍 𝑚 = 𝑍 0 - 𝑍 𝐶 is the matching impedance 𝑍 𝐶 is the characteristic impedance of the cables to the network analyser The DUT in this case is modelled as a lumped impedance in the centre of a beampipe of characteristic impedance 𝑍 0 .

Comparing the three formulae for a lumped impedance 𝑆 21 𝑅𝐸𝐹 = 𝑒 − 𝑖𝜔𝑙 𝑐 Since the reference line is lossless, 𝑆 21 𝐷𝑈𝑇 = 2 𝑍 0 2 𝑍 0 + 𝑍 𝐿 𝑒 − 𝑖𝜔𝑙 𝑐 We can also calculate the transmission of the DUT: Lumped formula Log formula Improved log formula 𝑍= 𝑍 0 ln 1+ 𝑍 𝐿 2 𝑍 0 1+ 𝑖 ln 1+ 𝑍 𝐿 2 𝑍 0 2 𝜔𝑙 𝑐 𝑍=2 𝑍 0 ln 1+ 𝑍 𝐿 2 𝑍 0 𝑍= 𝑍 𝐿 Exact, as expected. For small 𝑍 𝐿 compared to 𝑍 0 this is quite accurate This is extremely problematic. The dependence on 𝑙 is totally unphysical here, and even worse, 𝑙→0 makes a singularity.

These impedance formulae are only valid for a constant REF and DUT cross-section The paper by Hahn and Pedersen[6] shows that a change in cross-section cannot be treated the same as a lumped or distributed impedance in the chamber wall. The following few slides show an attempt to use reflection coefficients to derive the impedance of a small cavity (at low frequency), which by definition has a changing cross-section.

Reflection formulae (Applied for a small cavity)

Longitudinal beam impedance of a small cavity Given a situation where 𝑙≤𝑑≤𝑏, for a cavity in a cylindrical chamber, the low-frequency longitudinal beam impedance is given by[4]: N.B. In A. Chao’s original formula, the impedance is the negative of the quantity on the left. This is only due to a difference in convention. (I 𝛼 𝑒 𝑖𝜔𝑡 versus I 𝛼 𝑒 −𝑖𝜔𝑡 ) This is derived directly from considerations of the interaction of a particle beam and the cavity at low frequency. 𝑍 𝐹 here denotes the impedance of free space.

Transmission line theory: impedance of a cavity There is a simple transmission line model for a cavity: It is also possible to model the cavity as a lumped impedance 𝑙 𝑑 𝑏 2𝑟 With the characteristic impedances given by the standard formula: Where 𝑍 0 = 𝑍 𝐹 2𝜋 ln 𝑏 𝑟 , and 𝑍 𝐿 is some quantity to be determined. 𝑍 0 = 𝑍 𝐹 2𝜋 ln⁡ 𝑏 𝑟 𝑍 1 = 𝑍 𝐹 2𝜋 ln⁡ 𝑏+𝑑 𝑟 with some inner conductor radius r.

Comparing the reflection coefficients from a cavity and a lumped impedance Here, we will set the reflection coefficient 𝑆 11 =Γ= 𝑏 1 𝑎 1 | 𝑎 2=0 equal for both cases, then let 𝑟→0. 𝑎 1 and 𝑏 1 are scattering parameters of the two-port network, which is the cavity or lumped impedance. Two-port network This is actually the reflection coefficient from the start of the cavity. A more correct formula would refer to reflection from the centre of the cavity. However, at low frequency they both give the same result. Γ 𝐶𝑎𝑣𝑖𝑡𝑦 = (𝑍 1 2 − 𝑍 0 2 )tan⁡ 𝜔𝑙 𝑐 (𝑍 1 2 + 𝑍 0 2 ) tan 𝜔𝑙 𝑐 −2𝑖 𝑍 0 𝑍 1 Γ 𝐿𝑢𝑚𝑝𝑒𝑑 = 𝑍 𝐿 2 𝑍 0 + 𝑍 𝐿 Using the low frequency approximation, this yields the formula 𝐿= 𝑍 1 2 − 𝑍 0 2 𝑍 1 𝑙 𝑐 𝑍 𝐿 =𝑖𝜔𝐿 where

Comparing the reflection coefficients from a cavity and a lumped impedance 𝑍 𝐿 =𝑖𝜔 𝑍 1 2 − 𝑍 0 2 𝑍 1 𝑙 𝑐 𝑍 0 = 𝑍 𝐹 2𝜋 ln 𝑏 𝑟 𝑍 1 = 𝑍 𝐹 2𝜋 ln 𝑏+𝑑 𝑟 After expressing the characteristic impedances 𝑍 0 and 𝑍 1 in terms of the geometry, and letting 𝑟 go to zero: 𝑍 𝐿 =𝑖𝜔 𝑍 𝐹 𝑙 𝜋𝑐 ln 𝑏+𝑑 𝑏 Then, the logarithm is expanded linearly, because we assume 𝑑 is small compared to 𝑏. So, 𝑍 𝐿 =𝑖𝜔 𝑍 𝐹 𝑙𝑑 𝜋𝑏𝑐 = 2𝑍 𝐶ℎ𝑎𝑜 ∥

Deriving the impedance of a pipe diameter step This technique was also applied to a step change of the vacuum chamber diameter: Again, comparing the reflection coefficients of the step and some lumped impedance 𝑍 𝐿 : = 𝑏 1 𝑏 2 𝑍 𝐿 2 𝑍 0 + 𝑍 𝐿 = 𝑍 1 − 𝑍 0 𝑍 1 + 𝑍 0 ⇒ 𝑍 𝐿 = 𝑍 𝐹 2𝜋 ln⁡ 𝑏 2 𝑏 1 From Chao’s book: So this time, both approaches are in agreement: 𝑍 𝐶ℎ𝑎𝑜 ∥ = 𝑍 𝐹 2𝜋 ln⁡ 𝑏 2 𝑏 1 𝑍 𝐿 = 𝑍 𝐶ℎ𝑎𝑜 ∥

Deriving the beam impedance directly from characteristic impedances. One very simple approach is to simply consider the cavity as the beam pipe with an additional distributed impedance 𝑍. 𝑍= 𝑅+𝑖𝜔𝐿 𝐺+𝑖𝜔𝐶 for a general line[5] (The conductance 𝐺 is zero for a vacuum dielectric) Cavity Beampipe 𝑍 1 = 𝑍 𝐶𝑎𝑣𝑖𝑡𝑦 = 𝑍+𝑖𝜔𝐿 𝑖𝜔𝐶 𝑍 0 =𝑍 𝑃𝑖𝑝𝑒 = 𝐿 𝐶 𝑍= 𝑍 1 2 − 𝑍 0 2 𝑍 0 𝑖𝜔 𝑐 Solving for the extra impedance: Multiplying 𝑍 by the length of the cavity gives us an effective lumped impedance 𝑍 𝐿 𝑍 𝐿 = 𝑍 1 2 − 𝑍 0 2 𝑍 0 𝑖𝜔𝑙 𝑐 =2 𝑍 𝐶ℎ𝑎𝑜 ∥ This result is obtained by taking the same steps as the previous slide.

General reasons for the differences between transmission line theory and beam theory In transmission lines, it is possible for there to be some current to be reflected on the wire at a discontinuity. In beam theory, it is assumed the whole beam travels straight through the equipment. The addition of an inner conductor changes the EM boundary conditions - altering the fields most strongly near the wire. Shrinking the radius of the wire to zero does not solve this problem; the chamber still remains a two-conductor system at the limit.

Future investigations Attempt to understand exactly why a change in cross-section is forbidden in transmission formulae – and find out to what extent, if any, we can change the cross-section with acceptable errors. Find or develop formulae that can deal with a changing cross-section. Quantify how much SUCOBoxes can distort transmission signals, and how to get rid of SUCOBoxes altogether.

Conclusions Using a SUCOBox distorts the measurements of the wire method. A connector directly attached to the equipment would in principle give much better results. It should be investigated how to overcome the practical difficulties of a direct connection. When using formulae for calculating impedances in a DUT, we must be selective and use the appropriate formula, e.g. use the improved log formula for distributed impedances like in a collimator or kicker. Use the lumped formula for lumped impedances. We must take care when dealing with changing cross-sections – it is not the same situation as an impedance in a chamber wall when it comes to choosing a formula to use.

Thank you for your attention!

References [1] AN ANALOG METHOD FOR MEASURING THE LONGITUDINAL COUPLING IMPEDANCE OF A RELATIVISTIC PARTICLE BEAM WITH ITS ENVIRONMENT A. Faltens, E. C. Hartwig, D.Mohl, and A. M. Sessler August 2, 1971 https://publications.lbl.gov/islandora/object/ir%3A103718/datastream/PDF/download/citation.pdf [2] A.Mostacci: http://pcaen1.ing2.uniroma1.it/mostacci/wire_method/care_impedance.ppt https://indico.cern.ch/event/503166/ [3] Fritz Caspers http://cas.web.cern.ch/cas/UK- 2007/Afternoon%20Courses/RF/wire_impedance_measurementsRFcourse.pdf [4] Chao, A. (1993) Physics of Collective Beam Instabilities in High Energy Accelerators, page 87. Wiley. http://www.slac.stanford.edu/~achao/WileyBook/WileyChapter2.pdf [5] https://en.wikipedia.org/wiki/Coaxial_cable [6] Hahn, H. and Pedersen, F. (1978) ‘On coaxial wire measurements of the longitudinal coupling impedance’, Particle Accelerators and High-Voltage Machines. https://lib-extopc.kek.jp/preprints/PDF/1978/7810/7810003.pdf

References [7] Wang, J.G. and Zhang, S.Y. (2001) ‘Measurement of coupling impedance of accelerator devices with the wire-method’, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 459(3), pp. 381–389. \\cern.ch\dfs\Websites\o\OEBerrig\Impedance\Transverse_wire_measurement.pdf [8] A. Arnold (2016) Transmission line model for bench measurements of the QuattroTank, to be published

Let 𝑟→0 and ln 1+ 𝑑 𝑏 ≅ 𝑑 𝑏 , so ln 𝑏 𝑟 ln 𝑏+𝑑 𝑟 →1 and finally Page 17&18 Derivation 𝑍 𝐿 2 𝑍 0 + 𝑍 𝐿 = (𝑍 1 2 − 𝑍 0 2 )tan⁡ 𝜔𝑙 𝑐 (𝑍 1 2 + 𝑍 0 2 ) tan 𝜔𝑙 𝑐 −2𝑖 𝑍 0 𝑍 1 = 𝑖 (𝑍 1 2 − 𝑍 0 2 )tan⁡ 𝜔𝑙 𝑐 𝑍 1 𝑖 (𝑍 1 2 − 𝑍 0 2 )tan⁡ 𝜔𝑙 𝑐 𝑍 1 +2 𝑍 0 ⇒ 𝑍 𝐿 = 𝑖 (𝑍 1 2 − 𝑍 0 2 )tan⁡ 𝜔𝑙 𝑐 𝑍 1 Taking the first term in the Taylor series of tan ( tan 𝑥 =𝑥+ 1 3 𝑥 3 + 2 15 𝑥 5 +…) 𝑍 𝐿 = (𝑍 1 2 − 𝑍 0 2 ) 𝑍 1 𝑖𝜔𝑙 𝑐 = 𝑍 𝐹 2𝜋 𝑖𝜔𝑙 𝑐 ln 𝑏+𝑑 𝑟 2 − ln 𝑏 𝑟 2 ln 𝑏+𝑑 𝑟 = 𝑍 𝐹 2𝜋 𝑖𝜔𝑙 𝑐 ln 𝑏+𝑑 𝑏 1+ ln 𝑏 𝑟 ln 𝑏+𝑑 𝑟 Let 𝑟→0 and ln 1+ 𝑑 𝑏 ≅ 𝑑 𝑏 , so ln 𝑏 𝑟 ln 𝑏+𝑑 𝑟 →1 and finally 𝑍 𝐿 = 𝑖𝜔𝑙 𝑍 𝐹 𝜋𝑐 𝑑 𝑏

Page 20 Derivation 𝑍 1 = 𝑍+𝑖𝜔𝐿 𝑖𝜔𝐶 = 𝐿 𝐶 1+ 𝑍 𝑖𝜔𝐿 = 𝑍 0 1+ 𝑍 𝑖𝜔𝐿 𝑍 1 = 𝑍+𝑖𝜔𝐿 𝑖𝜔𝐶 = 𝐿 𝐶 1+ 𝑍 𝑖𝜔𝐿 = 𝑍 0 1+ 𝑍 𝑖𝜔𝐿 So, 𝑍=𝑖𝜔𝐿 𝑍 1 𝑍 0 2 −1 . 𝑐= 1 𝐿𝐶 (phase velocity of signal in cavity) and 𝑍 0 = 𝐿 𝐶 ⇒𝐿= 𝑍 0 𝑐 ∴𝑍⋅𝑙= 𝑍 𝐿 = 𝑍 1 2 − 𝑍 0 2 𝑍 0 𝑖𝜔𝑙 𝑐 The steps of the previous slide can now be applied almost identically to get the result 𝑍 𝐿 = 𝑖𝜔𝑙 𝑍 𝐹 𝜋𝑐 𝑑 𝑏

Reflection coefficients 𝑉 0 , 𝐼 0 𝑉 𝑅 , 𝐼 𝑅 𝑉 𝑇 , 𝐼 𝑇 𝑍 0 Lumped Cavity The voltage drop across the impedance is given by The input impedance of a cavity (called a stepped line in many books) is given by: 𝑍 𝑖𝑛 = 𝑍 1 𝑍 0 +𝑖 𝑍 1 tan 𝑖𝜔𝑙 𝑐 𝑍 1 +𝑖 𝑍 0 tan 𝑖𝜔𝑙 𝑐 𝑉= 𝐼 𝑇 𝑍 𝐿 = 𝑉 0 + 𝑉 𝑅 − 𝑉 𝑇 And to conserve current, 𝐼 0 = 𝐼 𝑇 + 𝐼 𝑅 And the reflection off the cavity is given by By definition of impedance, 𝑉 𝑘 = 𝐼 𝑘 𝑍 0 for all 𝑘 Γ= 𝑍 𝑖𝑛 − 𝑍 0 𝑍 𝑖𝑛 + 𝑍 0 = (𝑍 1 2 − 𝑍 0 2 )tan⁡ 𝜔𝑙 𝑐 (𝑍 1 2 + 𝑍 0 2 ) tan 𝜔𝑙 𝑐 −2𝑖 𝑍 0 𝑍 1 𝑉 0 𝑍 0 − 𝑉 𝑅 𝑍 0 𝑍 𝐿 + 𝑍 0 = 𝑉 0 + 𝑉 𝑅 So, Solve for 𝑉 𝑅 𝑉 0 =Γ ∴Γ= 𝑍 𝐿 2 𝑍 0 + 𝑍 𝐿