Realistic Model of the Solenoid Magnetic Field Paul S Miyagawa, Steve Snow University of Manchester Objectives Closed-loop model Field calculation corrections Helical model Realistic model Results Conclusions + further work

7 March 2006Mini-workshop on Magnetic Field Modelling2/17 Objectives A useful test of the Standard Model would be measurement of W mass with uncertainty of 25 MeV per lepton type per experiment. Momentum scale will be dominant uncertainty in W mass measurement. Momentum accuracy depends primarily on alignment and B-field. Solenoid field mapping team targetting an accuracy of 0.05% on sagitta to ensure that B-field measurement is not the limiting factor on momentum accuracy. Mapping machine will take measurements of the solenoid B-field. Simulations of the machine performance have been based on a simple model of the B-field. Want to test the performance with a more realistic model. Realistic model seeks to improve on several aspects: –Calculating the field from a curved current path –Modelling the actual current path –Modelling the shape of the solenoid

7 March 2006Mini-workshop on Magnetic Field Modelling3/17 Closed-Loop Model Previously, the solenoid was modelled as a series of closed circular loops of infinitely thin wires evenly spaced in z. For calculating the field, each loop approximated as 800 straight-line segments, and Biot-Savart law applied to each segment. Added in field due to magnetised iron (4% of total field). Model is symmetric in  and even in z. current7600 ANominal current at 2 T number of loops 1154Number of electrical turns half z- length mmIncludes shrinkage due to cold and excitation radius1246 mmAverage of internal and external radii

7 March 2006Mini-workshop on Magnetic Field Modelling4/17 Field calculation corrections (1) We approximate the true current path, which is helical, with straight-line segments whose end points lie on the true path. This leads to two types of error: Inscribed polygon error If a circle is replaced by an inscribed polygon, the average radius of the polygon is less than the radius of the circle. This effect can be corrected by increasing the radius by 2/3 of S where S is the sagitta between the arc and the chord. Line segment integration error The natural approximation is to use the midpoint of the segment to calculate r between the segment and the measurement point. This leads to an error because r is different for each point on the line. This error is more significant than the inscribed polygon error. A correction factor can be added to the magnitude of the field:

7 March 2006Mini-workshop on Magnetic Field Modelling5/17 Field calculation corrections (2) z = 0, r = 0 z = 1, r = 0 z = 0, r = 0.8 z = 1, r = steps most likely sufficient, but we use 200 to be safe.

7 March 2006Mini-workshop on Magnetic Field Modelling6/17 Helical Model First step towards a realistic model is to replace the series of closed loops with a helical coil. The current starts at (0,R,h z ), winds in anti-clockwise direction, terminates at (0,R,- h z ). Note that this does not form a closed path. The overall dimensions of the coil are the same as for the closed-loop model. Model is neither symmetric in  nor even in z. Main differences with closed- loop model are at the ends of the coil due to difficulty in lining up the ends.

7 March 2006Mini-workshop on Magnetic Field Modelling7/17 Realistic Model The ATLAS solenoid consists primarily of four main coil sections. The last loop from each section is welded to the first loop of the next section to form a single coil. At end A, an extra cable is welded to the last loop, and the two cables are routed up through the services chimney. At end C, an extra cable is welded to the last loop. The two cables form the return coil, which is routed along the cryostat surface to end A and up into the chimney. Cables in the chimney are magnetically shielded. The realistic model models the current through these main components.

7 March 2006Mini-workshop on Magnetic Field Modelling8/17 Main Coil Sections From end A to end C, the four coil sections are labelled H-A, F-A, F-C and H-C. Each section is modelled as a shorter version of the helical coil with its own length and number of loops. The loops at either end of each section are treated in the welds, so are not included in the main sections. SectionLength (mm)Number of loops H-A F-A F-C H-C

7 March 2006Mini-workshop on Magnetic Field Modelling9/17 Internal Welds Physically, the last loop from one section lies parallel to the first loop of the next section. These two loops are welded together along their entire length so that electrically they are one loop. Two models of these welds can be used: –The physical representation uses two loops running in parallel. The current gradually transfers from the first loop to the second. –The electrical representation uses a single loop of double pitch. This loop carries the full current over its entire length.

7 March 2006Mini-workshop on Magnetic Field Modelling10/17 End Welds At each end of the solenoid, the extra cable is welded to the main cable over a 30-cm portion. The current starts entirely in the main cable, and 20% is transferred to the extra cable by the end of this portion. The two cables follow the curvature of the solenoid over 4.5°, but are electrically isolated from each other. They carry the current in the ratio 80:20. The two cables then curve away from the solenoid until they are tangent to the chimney angle (11.25° from vertical). The radius of curvature is 165 mm. As the cables are still electrically isolated, they carry the current in the same 80:20 ratio.

7 March 2006Mini-workshop on Magnetic Field Modelling11/17 Return Coil The return coil consists of the two cables from the end C weld running flat along the cryostat surface. The extra cable rests on top of the main cable. The cables are electrically isolated, so they carry the current in ratio 80:20. In the realistic model, the cables connect to the cables from the end A weld so as to form a closed current path for the entire solenoid.

7 March 2006Mini-workshop on Magnetic Field Modelling12/17 Shape Deformations The dimensions given are for a warm coil in quiescent conditions. The real coil will be deformed: –Shrinkage due to cool down. This is modelled as an overall scale reduction of 0.41%. –Bending due to field excitation. This is modelled as the radius r being a parabolic function of z. At the centre, Δr = 0.90 mm; at the coil ends, Δr = 0.38 mm, Δz = 1.09 mm.

7 March 2006Mini-workshop on Magnetic Field Modelling13/17 Results (1)

7 March 2006Mini-workshop on Magnetic Field Modelling14/17 Conclusions

7 March 2006Mini-workshop on Magnetic Field Modelling15/17 Further Work