Calibration method for electrode gains in an axially symmetric stripline BPM

Calibration method for electrode gains in an axially symmetric stripline BPM 1 first-author 1 1 1 corresp Corresponding author, bgsun@ustc.edu.cn Corresponding author, bgsun@ustc.edu.cn bgsun@ustc.edu.cn 1 1 1 1 1 National Synchrotron Radiation Laboratory & School of Nuclear Science and Technology, University of Science and Technology of China, Hefei230029, China National Synchrotron Radiation Laboratory & School of Nuclear Science and Technology, University of Science and Technology of China, Hefei230029, China The four electrodes in the stripline beam position monitor (BPM) for Hefei Light Source (HLS II) storage ring are of axially symmetric type. We have derived a new calibration method of electrode gains for this type stripline BPM. The gain fit error of different data grids was analyzed, and the ± 5 mm by ± 5 mm grid is the best. The electrode gains of two stripline BPMs (HLS II SR-BD-STLB1 and HLS II SR-BD-STLB2) were obtained based on offline calibrated data. The results show that data after fitting gains are improved, with the electrode gains being between 0.94 and 1.15. Stripline beam position monitor Electrode gain Axially symmetric Calibration I. INTRODUCTION 1 The Hefei Light Source (HLS) is being upgraded to HLS II. Three stripline BPMs are designed to measure the beam position, emittance and momentum dispersion in the storage ring. To decrease beam emittance, the circular vacuum chamber of the storage ring will be changed to an octagonal type. A cross-section of the stripline BPM is shown in Fig. 1 Fig. 1 (in mm). Due to mechanical errors of the stripline BPM, the four axially symmetric electrodes, labeled with # R, # L, # T and # B to denote the horizontal (left and right) and vertical (top and bottom) electrodes, do not have the same relative gain. The differences in electrode gain will couple the measured horizontal position with the vertical position, resulting in measurement errors [ 1 1 ]. To measure beam parameters correctly, the electrode gains of a stripline BPM shall be measured. Electrode gain for button BPM was measured by Rubin et al . [ 1 1 ]. This method requires mirror symmetry geometry of the four BPM electrodes. Geometry of the four BPM electrodes can be mirroring symmetric or axially symmetric. The measurement method of electrode gains for stripline BPM of the mirroring symmetric type is the same as the method for button BPMs. The geometry of BPM electrodes on the HLS II storage ring is axially symmetric, and an offline measurement method is developed to measure the electrode gains, which uses three electrode gains and a correction factor. II. METHODS 1 The quadrupole component with difference/sum method for the stripline BPM can be expressed as Eq. (1): <math display="block" id="M1"><mrow><msub><mi>Q</mi><mrow><mi>Δ</mi><mo>/</mo><mstyle displaystyle="true"><mo>∑</mo></mstyle></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>V</mi><mtext>R</mtext></msub><mo>+</mo><msub><mi>V</mi><mtext>L</mtext></msub><mo>−</mo><msub><mi>V</mi><mtext>T</mtext></msub><mo>−</mo><msub><mi>V</mi><mtext>B</mtext></msub></mrow><mrow><msub><mi>V</mi><mtext>R</mtext></msub><mo>+</mo><msub><mi>V</mi><mtext>L</mtext></msub><mo>+</mo><msub><mi>V</mi><mtext>T</mtext></msub><mo>+</mo><msub><mi>V</mi><mtext>B</mtext></msub></mrow></mfrac><mn>,</mn></mrow></math> where V R , V L , V T and V B are electrode signals for the four electrodes. Assuming that the # R, # L, # T and # B electrodes are in X and Y axis, as shown in Fig. 1 Fig. 1 , and ignoring the higher order components, the quadrupole component for stripline BPM of axially symmetric type for circular vacuum chamber [ 2 2 , 3 3 ] can be calculated by Eq. (2): <math display="block" id="M2"><mrow><msub><mi>Q</mi><mrow><mi>Δ</mi><mo>/</mo><mstyle displaystyle="true"><mo>∑</mo></mstyle></mrow></msub><mo>=</mo><msub><mi>S</mi><mtext>Q</mtext></msub><mrow><mo>(</mo><mrow><msubsup><mi>x</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><msubsup><mi>y</mi><mn>0</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mi>x</mi><mn>2</mn></msubsup><mo>−</mo><msubsup><mi>σ</mi><mi>y</mi><mn>2</mn></msubsup></mrow><mo>)</mo></mrow><mn>,</mn></mrow></math> where ( x 0 , y 0 ) is the beam position in the BPM, ( σ x , σ y ) is the beam transverse size, and S Q is the quadrupole component sensitivity, which is relevant to azimuthal opening angle of the electrodes and distance from electrode to the BPM center. For a stripline BPM of the axially symmetric type on an octagonal vacuum chamber, the boundary element method [ 4 4 ] and gaussian weighted method of 2D-grid structure [ 5 5 ] are used to simulate beam moving in the BPM. The model for stripline BPMs in the HLS II storage ring was established based on the boundary element method using Matlab. The quadrupole component Q Δ/∑ for the stripline BPM on the HLS II storage ring can be calculated as: <math display="block" id="M3"><mrow><msub><mi>Q</mi><mrow><mi>Δ</mi><mo>/</mo><mstyle displaystyle="true"><mo>∑</mo></mstyle></mrow></msub><mo>=</mo><msub><mi>Q</mi><mn>0</mn></msub><mo>+</mo><msub><mi>S</mi><mtext>Q</mtext></msub><mrow><mo>(</mo><mrow><msubsup><mi>x</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><msubsup><mi>y</mi><mn>0</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mi>x</mi><mn>2</mn></msubsup><mo>−</mo><msubsup><mi>σ</mi><mi>y</mi><mn>2</mn></msubsup></mrow><mo>)</mo></mrow><mn>.</mn></mrow></math> As the distance between horizontal electrodes differs from that between vertical electrodes, there exists a non-zero component Q 0 , compared to the stripline BPM on a circular vacuum chamber. Ignoring the higher order components, the electrical position ( P x , P y ) can be obtained using the difference/sum method: <math display="block" id="M4"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>P</mi><mi>x</mi></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>V</mi><mtext>R</mtext></msub><mo>−</mo><msub><mi>V</mi><mtext>L</mtext></msub><mo stretchy="false">)</mo><mo>/</mo><mo stretchy="false">(</mo><msub><mi>V</mi><mtext>R</mtext></msub><mo>+</mo><msub><mi>V</mi><mtext>L</mtext></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>S</mi><mi>x</mi></msub><msub><mi>x</mi><mn>0</mn></msub><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>P</mi><mi>y</mi></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>V</mi><mtext>T</mtext></msub><mo>−</mo><msub><mi>V</mi><mtext>B</mtext></msub><mo stretchy="false">)</mo><mo>/</mo><mo stretchy="false">(</mo><msub><mi>V</mi><mtext>T</mtext></msub><mo>+</mo><msub><mi>V</mi><mtext>B</mtext></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>S</mi><mi>y</mi></msub><msub><mi>y</mi><mn>0</mn></msub><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> Combining Eqs. (3) and (4) to eliminate x 0 and y 0 , we can obtain an expression that relates the electrode signals and beam transverse size: <math display="block" id="M5"><mrow><msub><mi>Q</mi><mrow><mi>Δ</mi><mo>/</mo><mstyle displaystyle="true"><mo>∑</mo></mstyle></mrow></msub><mo>−</mo><msub><mi>Q</mi><mn>0</mn></msub><mo>=</mo><msub><mi>S</mi><mtext>Q</mtext></msub><mrow><mo>[</mo><mrow><msup><mrow><mo stretchy="false">(</mo><msub><mi>P</mi><mi>x</mi></msub><mo>/</mo><msub><mi>S</mi><mi>x</mi></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><msub><mi>P</mi><mi>y</mi></msub><mo>/</mo><msub><mi>S</mi><mi>y</mi></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msubsup><mi>σ</mi><mi>x</mi><mn>2</mn></msubsup><mo>−</mo><msubsup><mi>σ</mi><mi>y</mi><mn>2</mn></msubsup></mrow><mo>]</mo></mrow><mn>.</mn></mrow></math> By using point charge to substitute gaussian beam, ( σ x , σ y ) becomes (0, 0), and Eq. (5) changes to: <math display="block" id="M6"><mrow><msub><mi>Q</mi><mrow><mi>Δ</mi><mo>/</mo><mstyle displaystyle="true"><mo>∑</mo></mstyle></mrow></msub><mo>−</mo><msub><mi>Q</mi><mn>0</mn></msub><mo>=</mo><msub><mi>S</mi><mtext>Q</mtext></msub><mrow><mo>[</mo><mrow><msup><mrow><mo stretchy="false">(</mo><msub><mi>P</mi><mi>x</mi></msub><mo>/</mo><msub><mi>S</mi><mi>x</mi></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><msub><mi>P</mi><mi>y</mi></msub><mo>/</mo><msub><mi>S</mi><mi>y</mi></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mo>]</mo></mrow><mn>.</mn></mrow></math> HLS is being upgraded, so the electrode gains can be measured offline, rather than online. Two stripline BPMs (HLS II SR-BD-STLB1 and HLS II SR-BD-STLB2) were calibrated [ 6 6 ] by using antenna method, with an antenna of 0.2-mm tungsten filament. This situation can be regarded as that the beam passes the stripline BPM with ( σ x , σ y ) being (0.2 mm, 0.2 mm) and <math id="M7"><mrow><msubsup><mi>σ</mi><mi>x</mi><mn>2</mn></msubsup><mo>−</mo><msubsup><mi>σ</mi><mi>y</mi><mn>2</mn></msubsup><mo>=</mo><mn>0</mn></mrow></math> , so the electrode gains can be calculated using Eq. (6). Before measuring the electrode gains, the parameters ( S x , S y , Q 0 , S Q ) shall be calculated. The boundary element method [ 4 4 ] was used to simulate a point charge moving in a grid of ± 5 mm by ± 5 mm with a 0.5 mm step. The four electrode signals of V R , V L , V T and V B were obtained to calculate the beam positions ( Fig. 2 Fig. 2 ) using Eq. (4). The position sensitivities ( S x and S y ) can be obtained by fitting the simulation data. As shown in Fig. 2 Fig. 2 , S x = 0.0773 mm and S y = 0.0764 mm. The quadrupole component ( Q Δ/∑ ) calculated using Eq. (1) at the point charge position ( x 0 , y 0 ) from (0, 0) to (5, 5) only, due to symmetry, is shown in Fig. 3 Fig. 3 . By plotting and fitting the data of ( Q Δ/∑ - Q 0 ) vs [( P x / S x ) 2 -( P y / S y ) 2 ] for Q Δ/∑ at ( x , y ) from (-5, -5) to (5, 5), the parameters of Q 0 and S Q can be obtained using Eq. (6). Since electrode gains would be offline measured based on Eq. (6), in order to directly obtain constraints on the four electrode signals and decrease calculation error, <math id="M8"><mrow><mrow><mo>(</mo><mrow><msubsup><mi>x</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><msubsup><mi>y</mi><mn>0</mn><mn>2</mn></msubsup></mrow><mo>)</mo></mrow></mrow></math> should be replaced by [( P x / S x ) 2 -( P y / S y ) 2 ]. As shown in Fig. 4 Fig. 4 , ( Q Δ/∑ - Q 0 ) varies linearly with [( P x / S x ) 2 -( P y / S y ) 2 ], and Q 0 = -0.7832 and S Q = 0.0012 mm -2 by the fitting data. III. GAIN ERROR OF THE BPM ELECTRODES 1 In practice, due to mechanical errors of stripline BPM, the four electrodes do not have the same gain, hence the failure of the connection between electrodes defined by Eq. (6). The effect of gain errors was simulated by an 8% reduction of the signal on electrode # R. Fig. 5(a) Fig. 5(a) shows ( Q Δ/∑ - Q 0 ) vs [( P x / S x ) 2 -( P y / S y ) 2 ] under this condition for ± 5 mm by ± 5 mm grid simulated data. The relationship between ( Q Δ/∑ - Q 0 ) and [( P x / S x ) 2 -( P y / S y ) 2 ] is no longer linear, deviating from zero (marked by the + sign). The BPM(HLS II SR-BD-STLB1) was calibrated offline [ 6 6 ]. The results are shown in Fig. 5(b) Fig. 5(b) . The situation is similar to Fig. 5(a) Fig. 5(a) . Therefore, the four electrodes of this BPM have different gains. Due to the gain error of the four electrodes of the BPM, Eq. (6) cannot be applicable. So, a nonlinear least square fitting method was used to get the electrode gains ( g R , g L , g T and g B ). The merit function is <math display="block" id="M9"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mi>χ</mi><mn>2</mn></msup><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover></mstyle><mrow><mo>{</mo><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>g</mi><mtext>R</mtext></msub><msubsup><mi>V</mi><mtext>R</mtext><mi>i</mi></msubsup><mo>+</mo><msub><mi>g</mi><mtext>L</mtext></msub><msubsup><mi>V</mi><mtext>L</mtext><mi>i</mi></msubsup><mo>−</mo><msub><mi>g</mi><mtext>T</mtext></msub><msubsup><mi>V</mi><mtext>T</mtext><mi>i</mi></msubsup><mo>−</mo><msub><mi>g</mi><mtext>B</mtext></msub><msubsup><mi>V</mi><mtext>B</mtext><mi>i</mi></msubsup></mrow><mrow><msub><mi>g</mi><mtext>R</mtext></msub><msubsup><mi>V</mi><mtext>R</mtext><mi>i</mi></msubsup><mo>+</mo><msub><mi>g</mi><mtext>L</mtext></msub><msubsup><mi>V</mi><mtext>L</mtext><mi>i</mi></msubsup><mo>+</mo><msub><mi>g</mi><mtext>T</mtext></msub><msubsup><mi>V</mi><mtext>T</mtext><mi>i</mi></msubsup><mo>+</mo><msub><mi>g</mi><mtext>B</mtext></msub><msubsup><mi>V</mi><mtext>B</mtext><mi>i</mi></msubsup></mrow></mfrac><mo>−</mo><msub><mi>Q</mi><mn>0</mn></msub></mrow><mo>)</mo></mrow><mo>−</mo><mi>c</mi><mo>⋅</mo><msub><mi>S</mi><mtext>Q</mtext></msub><mo>⋅</mo><mrow><mo>[</mo><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><msub><mi>S</mi><mi>x</mi></msub></mrow></mfrac><mfrac><mrow><msub><mi>g</mi><mtext>R</mtext></msub><msubsup><mi>V</mi><mtext>R</mtext><mi>i</mi></msubsup><mo>−</mo><msub><mi>g</mi><mtext>L</mtext></msub><msubsup><mi>V</mi><mtext>L</mtext><mi>i</mi></msubsup></mrow><mrow><msub><mi>g</mi><mtext>R</mtext></msub><msubsup><mi>V</mi><mtext>R</mtext><mi>i</mi></msubsup><mo>+</mo><msub><mi>g</mi><mtext>L</mtext></msub><msubsup><mi>V</mi><mtext>L</mtext><mi>i</mi></msubsup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><msub><mi>S</mi><mi>y</mi></msub></mrow></mfrac><mfrac><mrow><msub><mi>g</mi><mtext>T</mtext></msub><msubsup><mi>V</mi><mtext>T</mtext><mi>i</mi></msubsup><mo>−</mo><msub><mi>g</mi><mtext>B</mtext></msub><msubsup><mi>V</mi><mtext>B</mtext><mi>i</mi></msubsup></mrow><mrow><msub><mi>g</mi><mtext>T</mtext></msub><msubsup><mi>V</mi><mtext>T</mtext><mi>i</mi></msubsup><mo>+</mo><msub><mi>g</mi><mtext>B</mtext></msub><msubsup><mi>V</mi><mtext>B</mtext><mi>i</mi></msubsup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mo>]</mo></mrow></mrow><mo>}</mo></mrow><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> where c is a coefficient to correct S Q . The best fit gains ( g R , g L , g T and g B ) and c are obtained at a minimized χ 2 . The method was used by Rubin et al . [ 1 1 ], and they got their desirable results by controlling the gain equals to 1 every time for each of the # R, # L, # T and # B electrodes, and averaging the data from the fitting measurement. They found also that different grid data affected the gain fit algorithm. So, gain fit error was analyzed with different grids. Fig. 6 Fig. 6 shows that gain fit error for three different grids, with 1%–12% reduction of the electrode signal. For each fit, the relative difference of the fitted gains from the real gains was calculated by Δ g = (| g fitted - g real |/ g real )× 100%. From Fig. 6 Fig. 6 , the difference in data grid size affected the gain fit algorithm. The ± 3 mm by ± 3 mm grid is too small in size, hence the biggest gain fit errors, while the ± 8 mm by ± 8 mm grid is too big to ignore the higher order components and to justify Eqs. (3) and (4), which can be satisfied only in certain range of the stripline BPM. We found that of all the data grids, the ± 5 mm by ± 5 mm grid has the least gain fit error, so it was chosen to measure electrode gains. IV. RESULTS OF CALIBRATION 1 Two BPMs, HLS II SR-BD-STLB1 and HLS II SR-BD-STLB2, were calibrated offline [ 6 6 ]. The beam was simulated by using 0.2 mm tungsten filament. The filament relative to the BPM was moved by stepper motor in a ± 5 mm by ± 5 mm grid, the data was acquired by Libera Brilliance [ 7 7 ]. The fitting results for HLS II SR-BD-STLB1 are given in Fig. 7 Fig. 7 , which shows that the data points after gain fitting are linear and pass through zero (0, 0). Fitted gains for the two BPMs are shown in Fig. 8 Fig. 8 , where the electrode gains are between 0.94 and 1.15. V. CONCLUSION 1 We have derived offline measurement method of electrode gains for axially symmetric type stripline BPM. The simulation results show that fitted electrode gain error is the least in a ± 5 mm by ± 5 mm grid, ( Q Δ/∑ - Q 0 ) vs [( P x / S x ) 2 -( P y / S y ) 2 ] curve after gain fitting is linear and passes through zero. In the future, online measurement of the electrode gains will be performed and the results be compared with the offline results. References Rubin D L , Billing M , Meller R , et al . Phys Rev Spec Top-AC , 2010 , 13 : 092802 . Phys Rev Spec Top-AC Miller R H , Clendenin J E , James M B , et al . Nonintercepting Emittance Monitor, SLAC-PUB-3186 , 1983 . Nonintercepting Emittance Monitor, SLAC-PUB-3186 Fang J , Sun B G , Lu P , et al . Atom Energ Sci Technol , 2011 , 44 : 511 - 516 . (in Chinese) Atom Energ Sci Technol Olmos A , Pérez F , Rehm G . Matlab Code for BPM Button Geometry Computation . Proceedings of DIPAC 2007 , 186 - 188 . Venice, Itally , 2007 . Matlab Code for BPM Button Geometry Computation Russell J S , Gilpatrick D J , Power J F , et al . Characterization of beam position monitor for measurement of second moment . Proceedings of PAC 1995 , 2580 - 2582 . Dallas, USA , 1995 Characterization of beam position monitor for measurement of second moment Wu F F , Zhou Z R , Sun B G , et al . High Power Laser Part Beam , 2011 , 23 : 2971 - 2975 . (in Chinese) High Power Laser Part Beam Ma T J , Yang Y L , Sun B G , et al . Nucl Sci Tech , 2012 , 23 : 261 - 266 . Nucl Sci Tech The authors wish to thank the HLS beam measurement group for discussion and suggestion. 10.13538/j.1001-8042/nst.25.050102 Supported by the Natural Science Foundation of China (Nos. 11175173, 11375178 and 11005105) 2014-03-04 2014-05-01 2014-10-06 2014-10-20 2021-04-11 2014 Nuclear Science and Techniques 5 25