Δ AB Σ AB R 4sin 2 with x being the calibrated horizontal beam position with reference to the beam pipe center (typically expressed in mm, and α being the coverage angle of the electrodes, intercepting some fraction of the wall (image) current Jw . In general, the position characteristic is non-linear, however, for narrow electrodes sin / 2 / 2 placements x2 + y2 approximatively: Δ2 Σ R with 2 / R = kBPM 1 / kBPM x (12) sometimes called the monitor constant, and expresses the sensitivity of the BPM pickup in the linear region near the beam pipe center and serves as linear calibration factor. From (5) and (7) we can also find a closed form expression for the normalized position characteristic, now in Cartesian coordinates: Δ Σ with: s x yR, , , /2 /2 arctan 2 J R d R x y w ,Φ Φ 2 2 x yR arctan R x y2 x yR being the sensitivity function. An example of the horizontal position characteristic based on (13) and (14) is illustrated in Fig. 4b as parametric plot with contours of constant Δ/Σ, and in Fig. 4c as function of the horizontal beam position. The small covering angle (α = 30°) of the electrodes lead to high non-linear " cushion " effects in the position characteristic, with substantial cross-coupling to the vertical plane, however, most of these non-linear position effects can be corrected in the BPM data post-processing by look-up tables or polynomial-fit functions, assuming also a set of vertical BPM electrodes is available. Historically, the splitplane type BPM pickup, sometimes called " shoe-box " BPM, was used to deliver a perfect linear position characteristic, but has several other drawbacks and is a bit outdated. The analytical expressions given in this section for the position characteristic of a broadband BPM pickup in a beam-pipe of circular cross-section can be modified for a different arrangement of the electrodes, e.g., electrodes rotated by 45°, as well as for a different normalization procedure, e.g., applying the ratio of the two opposite BPM electrodes as difference of the logarithmic BPM electrode signal levels: 26 22 2 22 2 tan / 4 2Ry tan / 4 2Ry (14) i bunch t q t it w (16) which is cancelled by the wall current iw(t), originated by the image charges qw In most other cases the longitudinal distribution of the particles in the RF bucket is approximated by an analytic function ibunch (t) to describe the envelope function of the longitudinal particle current distribution vs. time, e.g., in most cases by a Gaussian, sometimes by a cos2 or similar particle density distribution function (see (17) and Fig. 5, with Fig. 5c indicating the spectral lines, e.g., in this example spaced by 100 MHz, in a logarithmic scaled magnitude display similar to that of a spectrum analyzer when observing a single bunch circulating in a ring accelerator of small circumference). time-domain t i Gauss t zeN e 2t it bb c s 22 cos bb , /2t tt /2 tt Nt 1o 0, 2 cos elsewhere 2t 2 2 frequency-domain 2 I f eNe Gauss t f 2 = −q distributed around the azimuth of the beam pipe wall. In some cases, e.g., very short bunches, this δ-signal approach for the bunch current is an acceptable approximation, with the Dirac (δ) function defined as 1t . AB s x y R s xyR AB s xyR s xy R ,, , ,, , , ,, , ,, (13) << R2 and small beam disthe beam position response follows x higher order terms (11) Δ Σ vv (15) vB 20 log 20 log AB 10 vA 10 20 log10 For cross-section geometries different from the discussed circular BPM pickup, e.g., elliptical, rectangular or arbitrary shape, also to include the details of the shape of the BPM electrodes itself, a numerical analysis is required. Still, in most cases a solution of the electrostatic problem, i.e., solving the Laplace equation for one of the BPM electrodes in two or three dimensions is sufficient, followed by applying Green's reciprocation theorem to compute the scalar potential of a pair or quadruple of symmetric arranged electrodes, thus the position characteristic for the normalized horizontal / vertical BPM signals [14]. Bunched Beam Signals For the analysis of the position characteristic of a broadband BPM pickup, based on the sensitivity function s(x, y), the discussed line charge approach is usually sufficient. However, to estimate the waveform and signal power delivered by the BPM pickup electrode in (3), the bunch current signal ibunch (t) (ω) needs to be known-as discussed in this section- along with the transfer impedance Z(ω) of the pickup electrode - discussed in the next section. A point-like charge q = zeN, with z being the charge state (for ions), e ≈ 1.6 · 10−19 or Ibunch C the elementary charge and N the number of particles, traveling with relativistic velocity β = v / c = 1 in a perfectly conducting vacuum chamber (see Fig. 3d) has a bunch current: I f ft ft 2 eN ft bb sin b 1 (17) IEEE Instrumentation & Measurement Magazine December 2021

Instrumentation & Measurement Magazine 24-9 - Cover2

Instrumentation & Measurement Magazine 24-9 - 1

Instrumentation & Measurement Magazine 24-9 - 2

Instrumentation & Measurement Magazine 24-9 - 3

Instrumentation & Measurement Magazine 24-9 - 4

Instrumentation & Measurement Magazine 24-9 - 5

Instrumentation & Measurement Magazine 24-9 - 6

Instrumentation & Measurement Magazine 24-9 - 7

Instrumentation & Measurement Magazine 24-9 - 8

Instrumentation & Measurement Magazine 24-9 - 9

Instrumentation & Measurement Magazine 24-9 - 10

Instrumentation & Measurement Magazine 24-9 - 11

Instrumentation & Measurement Magazine 24-9 - 12

Instrumentation & Measurement Magazine 24-9 - 13

Instrumentation & Measurement Magazine 24-9 - 14

Instrumentation & Measurement Magazine 24-9 - 15

Instrumentation & Measurement Magazine 24-9 - 16

Instrumentation & Measurement Magazine 24-9 - 17

Instrumentation & Measurement Magazine 24-9 - 18

Instrumentation & Measurement Magazine 24-9 - 19

Instrumentation & Measurement Magazine 24-9 - 20

Instrumentation & Measurement Magazine 24-9 - 21

Instrumentation & Measurement Magazine 24-9 - 22

Instrumentation & Measurement Magazine 24-9 - 23

Instrumentation & Measurement Magazine 24-9 - 24

Instrumentation & Measurement Magazine 24-9 - 25

Instrumentation & Measurement Magazine 24-9 - 26

Instrumentation & Measurement Magazine 24-9 - 27

Instrumentation & Measurement Magazine 24-9 - 28

Instrumentation & Measurement Magazine 24-9 - 29

Instrumentation & Measurement Magazine 24-9 - 30

Instrumentation & Measurement Magazine 24-9 - 31

Instrumentation & Measurement Magazine 24-9 - 32

Instrumentation & Measurement Magazine 24-9 - 33

Instrumentation & Measurement Magazine 24-9 - 34

Instrumentation & Measurement Magazine 24-9 - 35

Instrumentation & Measurement Magazine 24-9 - 36

Instrumentation & Measurement Magazine 24-9 - 37

Instrumentation & Measurement Magazine 24-9 - 38

Instrumentation & Measurement Magazine 24-9 - 39

Instrumentation & Measurement Magazine 24-9 - 40

Instrumentation & Measurement Magazine 24-9 - 41

Instrumentation & Measurement Magazine 24-9 - 42

Instrumentation & Measurement Magazine 24-9 - 43

Instrumentation & Measurement Magazine 24-9 - 44

Instrumentation & Measurement Magazine 24-9 - 45

Instrumentation & Measurement Magazine 24-9 - 46

Instrumentation & Measurement Magazine 24-9 - 47

Instrumentation & Measurement Magazine 24-9 - 48

Instrumentation & Measurement Magazine 24-9 - 49

Instrumentation & Measurement Magazine 24-9 - 50

Instrumentation & Measurement Magazine 24-9 - 51

Instrumentation & Measurement Magazine 24-9 - 52

Instrumentation & Measurement Magazine 24-9 - 53

Instrumentation & Measurement Magazine 24-9 - 54

Instrumentation & Measurement Magazine 24-9 - 55

Instrumentation & Measurement Magazine 24-9 - 56

Instrumentation & Measurement Magazine 24-9 - 57

Instrumentation & Measurement Magazine 24-9 - 58

Instrumentation & Measurement Magazine 24-9 - 59

Instrumentation & Measurement Magazine 24-9 - 60

Instrumentation & Measurement Magazine 24-9 - 61

Instrumentation & Measurement Magazine 24-9 - 62

Instrumentation & Measurement Magazine 24-9 - 63

Instrumentation & Measurement Magazine 24-9 - 64

Instrumentation & Measurement Magazine 24-9 - 65

Instrumentation & Measurement Magazine 24-9 - 66

Instrumentation & Measurement Magazine 24-9 - 67

Instrumentation & Measurement Magazine 24-9 - 68

Instrumentation & Measurement Magazine 24-9 - 69

Instrumentation & Measurement Magazine 24-9 - 70

Instrumentation & Measurement Magazine 24-9 - 71

Instrumentation & Measurement Magazine 24-9 - 72

Instrumentation & Measurement Magazine 24-9 - 73

Instrumentation & Measurement Magazine 24-9 - 74

Instrumentation & Measurement Magazine 24-9 - 75

Instrumentation & Measurement Magazine 24-9 - 76

Instrumentation & Measurement Magazine 24-9 - 77

Instrumentation & Measurement Magazine 24-9 - 78

Instrumentation & Measurement Magazine 24-9 - 79

Instrumentation & Measurement Magazine 24-9 - 80

Instrumentation & Measurement Magazine 24-9 - Cover3

Instrumentation & Measurement Magazine 24-9 - Cover4

https://www.nxtbookmedia.com