The CLICopti RF structure parameter estimator

CERN – EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH

CLIC – Note – 1031

THE CLICOPTI RF STRUCTURE PARAMETER ESTIMATOR Kyrre Ness Sjobak (Department of Physics, University of Oslo, Norway and CERN) and Alexej Grudiev (CERN, Geneva, Switzerland)

01/06/2014

CERN-OPEN-2014-036

Abstract This document describes the CLICopti RF structure parameter estimator. This is a C++ library which makes it possible to quickly estimate the parameters of an RF structure from its length, apertures, tapering, and basic cell type. Typical estimated parameters are the input power required to reach a certain voltage with a given beam current, the maximum safe pulse length for a given input power and the minimum bunch spacing in RF cycles allowed by a given long-range wake limit. The document describes the implemented physics, usage of the library through its Application Programming Interface (API) and the relation between the different parts of the library. Also discussed is how the library is checked for correctness, and the example programs included with the sources are described.

Geneva, Switzerland June 2014

The CLICopti RF structure parameter estimator Kyrre Ness Sjobak ([email protected]), Alexej Grudiev May 2014

Abstract This document describes the CLICopti RF structure parameter estimator. This is a C++ library which makes it possible to quickly estimate the parameters of an RF structure from its length, apertures, tapering, and basic cell type. Typical estimated parameters are the input power required to reach a certain voltage with a given beam current, the maximum safe pulse length for a given input power and the minimum bunch spacing in RF cycles allowed by a given long-range wake limit. The document describes the implemented physics, usage of the library through its Application Programming Interface (API) and the relation between the different parts of the library. Also discussed is how the library is checked for correctness, and the example programs included with the sources are described.

1

Contents Abstract

1

Contents

2

1

2

3

Overview and installation 1.1 Basic structure of the code 1.2 Overview of the method . 1.3 Installation & file structure 1.4 Example programs . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

6 6 8 8 9

Handling of cell parameters 2.1 The data type struct Cell . . . . . . . . . . . . . . . . . . . . . . 2.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 double getByOffset (const Cell& cell, const size_t off) . . . . . 2.2.2 double getByOffset (const Cell* const cell, const size_t off) . . 2.2.3 Cell Cell_TD_30GHz_v1_fileParse (std::string& line) . . . . . 2.2.4 Cell Cell_TD_12GHz_v1_fileParse (std::string& line) . . . . . 2.2.5 std::ostream& operator 0.85 P0 [6], as shown in Figure 4.3. This is done though a scaling law [4, 3, 12] connecting the gradient G, pulse length τ and breakdown rate BDR G30 τ 5 =C, (4.30) BDR where C is a scaling parameter strongly influenced by structure geometry. By rescaling data for breakdown rate, gradient, and pulse length from structure tests to a standard breakdown rate BDR0 = 10−6 breakdowns/pulse/meter and pulse length τ0 = 200 ns, it is seen that in most cases the structures reach similar peak fields. Using this data, limits are set for the rescaled peak fields, such that the limits are is slightly above the average reached values. ˆ0 = 220 MV/m, Sˆc0 = 4.0 MW/mm2 , and (P/C)0 = 2.3 MW/mm. These limits can be These limits are E used together with slightly modified version of Equation (4.30) to compute the maximally allowed pulse lengths at BDR = BDR0 as ˆ 6 · τ0 E 0 , ˆ E 3 Sˆc0 · τ0 ≤ and ˆ Sc (P/C)0 · τ0 ≤ . max(P/C)

τE ≤

(4.31)

τSc

(4.32)

τP/C

(4.33)

The maximally allowed pulse length τ , defined as shown in Figure 4.3, is thus found by taking the smallest of the three scaling law pulse length estimates
τ ≤ min τE , τSc , τP/C . (4.34) The maximum beam time tb may then be found by subtracting the time where P (t) ≥ 0.85P0 and t < t2 or t > t3 . This “wasted time” is defined as    tf + tr 1 − 0.85P0 Pstart  0.85P0 < Pstart  tw ≡ τ − tb = 0.85P 0 −Pstart tf 1 − 0.85P0 ≥ Pstart P0 −Pstart (4.35) ( 0.85P0 −P0 +Pstart tr + tf 0.85P0 < P0 − Pstart Pstart −P0 + . 0 tr P0 −0.85P 0.85P0 ≥ P0 − Pstart Pstart ˆ Sˆc and max(P/C) is thus tb = τ − tw , which must then be The actual maximum beam time estimate from E, compared with the estimate from pulsed surface heating. The maximum pulse length τ can be interpreted as a local limit on the local breakdown rate, as Equation (4.30) can be rewritten as G30 τ 5 . (4.36) BDR = C Here C is assumed to be known and a function of z and BDR has units breakdowns per pulse per unit length. However, the data from which the limits are derived are all using the global breakdown rate of the structure and the total structure length, not the local breakdown rates. Thus the same approximation is made here. 37

130

120

G(z) [MV/m]

110

100

90

80

700

50

100 z [mm]

150

200

240

500

220

450 Hˆ (z) [kA/m]

Eˆ (z) [MV/m]

(a) Gradient profile.

200 180 160 1400

400 350 300

50

100 z [mm]

150

2500

200

5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.50

100 z [mm]

150

200

(c) Peak surface magnetic field profile.

3.5 3.0 P/C [MW/mm]

Sˆ c (z) [W/um2 ]

(b) Peak surface electric field profile.

50

2.5 2.0 1.5 1.0

50

100 z [mm]

150

0.50

200

(d) Peak modified Poynting vector profile.

50

100 z [mm]

150

200

(e) P/C profile.

Figure 4.7: Peak field profiles in CLIC_G R05 [7] with 26 cells, with input power such that the average gradient is hGi = 100 MV/m at beam loading I = 1.19 A. The unloaded steady-state profiles are showed in blue/solid, and the loaded profiles in red/dashed.

38

4.1.6.2

Pulse length limits from pulsed surface heating

ˆ the criterion used is that the For calculating the maximum pulse length from the peak surface magnetic field H, peak surface pulse heating ∆T ≤ 50 K [3]. Further, ∆T can be calculated as [12] Z t 2 √ µ0 ω G (z, t0 ) dt0 ¯ 2 (z) √ , (4.37) H ∆T (z, t) = √ 2 ρcε πk2σ t − t0 0 where ρ = 8.95 · 103 kg/m3 is the material density, c = 385 J/(kg · K) is the specific heat capacity, and k = 391 W/(m · K) is the thermal conductivity. The gradient G must then be found as a function of time and position, which is done through the assumption of Equation (4.29), meaning that G2 (z, t) ∝ G2 (z) · P (t). The integral for ∆T (z, t) may thus be written as Z √ µ0 ω G2 (z) t P (t0 ) dt0 2 ¯ √ . (4.38) ∆T (z, t) = √ H (z) P0 2 ρcε πk2σ t − t0 0 This integral can be solved analytically for t2 < t < t3 , using the following partial solutions found using Mathematica (assuming3 0 ≤ tA ≤ tB ≤ t): Z tB
√ √ dt0 √ = 2 t − tA − t − tB (4.39) 0 t−t tA Z tB 0 0

√ √ √ √ t dt 2 √ = t t − t + 2t t − t − t − t − t t − t (4.40) A A A B B B 3 t − t0 tA Thus the integral in Equation (4.38) can be written on closed form as 0 0 Z t Z t1 Z t2 Z t 1 Pstart ttr 0 Pstart + (P0 − Pstart ) t −t P (t0 ) dt0 P tr 0 √ √ √ √ 0 dt0 = dt + dt + 0 0 0 t − t t − t t − t t − t0 0 0 t1 t2     √ √ √ Pstart 2 2t 2 − 2t t − tr − tr t − tr = tr 3 p
√ t − tr − t − tr − tf +2Pstart p
√ P0 − Pstart + 2tr t − tr − t − tr − tf tf  p
√ √ P0 − Pstart 2 tr t − tr + 2t t − tr − t − tr − tf − + tf 3 p
 (tr + tf ) t − tr − tf p +2P0 t − tr − tf .

(4.41)

As an example, Figure 4.8 shows ∆T as a function of tb for the peak surface magnetic field of a structure. In order to estimate a maximum tb allowed for the structure from this expression, a bracket and bisection method is used on Equation (4.38) to estimate the longest tb which keeps ∆T ≤ 50 K. The final estimate has a ˆ Sˆc and max(P/C), precision of one RF cycle. This estimate for tb is then combined with the estimates from E, and the smallest of all four estimates used.

4.1.7

Transverse wakefields

The long range transverse wakefields in the structure are estimated as W⊥ (s) =

i