A Double Cosine Theta Coil Prototype

Transcription

1 A Double Cosine Theta Coil Prototype Elise Martin and Chris Crawford October 28, 21 1 Introduction A cosine theta coil is a coil that produces a dipole field by arranging current density on a shell in a cosine theta distribution. Inside the shell the field will be constant. The field outside the coil can be cancelled to zero by adding a second coil with the right surface current. A physical prototype of the shell coil is pictured in figure 2. The inner coil acts as a flux return and has current running along the front face, down the coil and along the back face, then back, making a circuit. This augments the field produced at the center by the shell coil and matches the boundary conditions between regions. Theoretically the addition of the fields created by these two coils should give a zero field external to the coils and a uniform, constant field inside. (See C. B. Crawford, Yunchang Shin, A method for designing coils with arbitrary fields, Technical Note. This was tested by programming the boundary conditions in COMSOL, which output the locations of the equipotentials. The ROOT programming environment was then used to simulate discretized current elements placed at the locations of the equipotentials, and then the Biot-Savart formula (1.1) was used to find the B-fields at any location.(see figure 1) In addition, a prototype outer coil was constructed from milled copper PCB. (See figure 2.) The copper was milled using a computerized router along the equipotentials. The boards were then soldered together at the edges with wire, current applied, and the resulting B-field mapped. B = µ I 4π l r r 3 (1.1) 2 Simulated Coil We can test the predictions of the electromagnetic theory by simulating a double square cosine theta coil numerically. The COMSOL differential equation solver outputs the locations of the equipotential lines that should give the required constant inner and zero outer B- fields. Both the outside shell and inside septum coils were modeled by plotting discretized vectors along the equipotentials. The fields were then calculated in two ways. The first was by creating polygon-shaped differential area elements between equipotential lines on the 1

2 x xo z y Figure 1: Ideal coil. Top half of coil simulated for ROOT analysis. Equipotentials are seen. Figure 2: Prototype coil. A prototype of the outer coil, built from milled copper PCB and planes soldered together. 2

3 shell, integrating the Biot-Savart formula and doing a Taylor series expansion to obtain the approximate B-field contribution from each differential area (see (2.1) and figure 3 (a)). The second way was by creating discretized current elements along the equipotentials and again expanding and integrating Biot-Savart to get the B-field(see (2.6) and figure 3 (b)). Three variables were then varied to get the get the field to converge and to see how each variable affected the field, the number of length divisions per wire segment, the number of wires/length divisions for the inner coil, and the number of equipotentials on the shell. The shell is 4cm long and centered on zero. The shell is 3 cm wide and 5 cm thick, leaving room for the 2 cm wide by 4 cm long inner coil. Figures 4-6 show the fields as these three variables are varied for the plane method. Figure 7 shows variation for wire method. Figure 8 shows how the field changes depending on how many terms in the expansion of the Biot-Savart integral are used. Zeroeth order is just the first term of eqn 2.1. Second order is the first two lines ( ). Fourth order is all of the terms in eqn 2.1 ( ). The figure shows that using many terms in the equation is more accurate. We can conclude from this simulation that using 2 length divisions on the shell and 6 inner wires makes a abetter field. We also see that using current planes or current wires makes no difference, but the more equipotentials used, the closer to ideal the field is, since the shell approaches a continuous current density. B = µ ( I l 4πr 3 r 1 12 l w 1 4 ( +l r w2 l2 +l r r 2 ( 7 8r 2 + ( l r l w ) r w r (r w ) 2 8 (r) (r l ) 2 8 (r) 2 2 ( r ) 2 w ( r 2 r w r 2 32 ( ( 3 l w 5 r w 16 r 2 ( ) ) 2 r 2 r 2 l w + 1 r w l r 4 r 2 (2.1) ) (2.2) ( 28 r ) 4 w (2.3) r 2 ) ( ( 7 r ) 2 ( ) ) w w (2.4) r 2 ) 2 ( ) )) w 2 (2.5) r 2 r 2 ( B = µ I (r dl) 2 1 4πr 3 8 r 4 8 ) dl 2 dl r (2.6) r 2 3

4 l 1 r l t w l Δl l 1 l 2 (a) r dl (b) Figure 3: (a) shows the differential area element used to solve for the B-field of discrete planes. l 1 and l 2 are the equipotentials, l t is an arbitrary transverse vector in the area, r is the vector to the where the field is found. (b) shows the differential line element used to find the B-field for differential wires. r is the vector to where the field is found 3 Data with Simulated Inner Coil A prototype outer shell coil was constructed with the same dimensions as the simulation and 1 Amp of current was passed through it. (See figure 2). Three dimensional B-field measurements were then taken using a fluxgate magnetometer. All B-fields shown are in Gauss, and all dimensions are in cm. We look at a transverse plane near the center of the box. The plane is both inside and extends outside the box. Figures 1-14 compare the measured fields to the calculated fields and show the residuals for the comparisons. We then look at a plane near the front of the prototype, where the wires to the power supply are attached. Figures show the comparisons for the front plane. 4

5 length divisions per segment 1 length divisions per segment 9 length divisions per segment 8 length divisions per segment 5 length divisions per segment Figure 4: The discretization along the current direction for each wire segment was varied. There are four segments per wire and 1 wires per half shell. The inner coil was kept at 1 wires. It is shown that the field is limited by this variable by 2 divisions. 4π was subtracted from the field inside the coil. Thus this plot shows the residual field. Field shown is in the y-direction, or across the face of the coil, while the z-axis is along the longitudinal length of the coil. 3.1 Possible Sources of Error The simulation of current planes using 1 equipotential lines shows that using only 1 equipotential divisions gives an inherent extra field of about.2 Gauss as compared to 1 equipotential divisions. Another possible source of error is that the soldered wires along the edges do not lie flat. Wires were soldered between faces of the box that connect traces. If these are not perfectly flat, each non-flat wire would create a small dipole. This field would go as B = µ ((m ˆr)ˆr m) (3.1) 4πr3 where m = IA and r is the vector between where the field is measured and where the dipole is located at (r meas r dip ). Figure 17 shows the field along a longitudinal line at the center of the box due to soldered wires that have raised about 1mm. Only the joints along the horizontal edges of the coil will contribute to a field in the horizontal direction. This field will be in the opposite direction of the By field produced by the coil. This effect was approximated by simulating 3 of the 6 dipoles at the center of the edge, 15 at 6 cm left of center, and 15 at 6 cm right of center, for an outside edge. For an inside edge, 3 dipoles were placed at the center, 15 at 5 cm left of center, and 15 at 5 cm right of center. This was done for the four edges at the front 5

6 By (gauss) Entries Mean x Mean 1 equipotentials y RMS x 25 equipotentials RMS y 5 equipotentials 1 equipotentials Figure 5: The number of discretizations perpendicular to the current direction (number of equipotentials) were varied. The number of wires/ length discretizations on the inner coil kept at 1 and the number of length discretizations on the shell kept at 2 per wire segment. It is shown that the greater the number of equipotentials, the closer the field is to the expected result. 4π was subtracted from the field inside the coil. Thus this plot shows the residual field. Field shown is in the y-direction, or across the face of the coil, while the z-axis is along the longitudinal length of the coil. face of the coil. The effects of this approximation, as seen in figure 17, show that there are high field contributions at the longitudinal edges of the coil. These could account for the abnormal field spikes at the same locations seen in the data, i.e. figure Conclusions To find the tolerances of the field, begin with the equation of the B-field. B = µ (m ˆr) (4.1) 4πr3 B x x = µ 4πz (18m xx + 6m 5 y y + 6m z z) (4.2) we know that the constraints in x,y, and z are x = y = 15cm (4.3) z = 2m (4.4) 6

7 By (gauss).2 Entries 2 inner wires 1 inner wires Mean x 6 wires Mean y inner.1 RMS x 1 inner wires RMS y Figure 6: The number of discretizations both along the direction of the current and perpendicular to it were varied. The number of equipotentials on the shell was kept at 1 per half shell and the number of length discretizations kept at 2 per wire segment. The field reaches its best value by about 6 wires, but 2 wires is also very close. 4π was subtracted from the field inside the coil. Thus this plot shows the residual field. Field shown is in the y-direction, or across the face of the coil, while the z-axis is along the longitudinal length of the coil. and we know the field here is This gives a field tolerance of B x = 4mG (4.5) and dipole tolerances of 1 B x B x < 1 6 /cm (4.6) m x < 47 A cm 2 (4.7) m y < 14 A cm 2 (4.8) m z < 16 A cm 2 (4.9) It is concluded that the greater the number of equipotential discretizations used, i.e. the more wires wound, the more closely the coil gives the desired field, since a continuous current density is approximated. There are several errors present in the measurement of the prototype coil. Some of this error can likely be explained by dipoles created by soldering wires between the traces. 7

8 Entries Mean x Mean y RMS x RMS y 25 wires 5 wires 1 wires 1 wires Figure 7: The discretization perpendicular to the current direction (the number of wires) was varied for the continuous wires method. Very little difference is seen between this and the same changes in the equivilant plots for the planes method (see figure 5). 4π was subtracted from the field inside the coil. Thus this plot shows the residual field. Field shown is in the y-direction, or across the face of the coil, while the z-axis is along the longitudinal length of the coil. By (Gauss) Entries Mean x Mean y RMS x RMS y terms to 4th order terms to th order terms to 2nd order Figure 8: Shows By in z for zeroeth, second, and fourth order expansions of Biot-Savart for finite area of current on the coil. 4π was subtracted from the field inside the coil. Thus this plot shows the residual field. Field shown is in the y-direction, or across the face of the coil, while the z-axis is along the longitudinal length of the coil. 8

9 inner coil outer coil total field Figure 9: Field from the prototype outer coil, field from the simulated inner coil, and the sum of the two. This shows how the addition of the two feels gives the desired total field. By (Gauss) x (cm) Figure 1:.Field along vertical line. Red is simulated field and blue is measured field. 9

10 y (cm) Figure 11:.Field along horizontal line. Red is simulated field and blue is measured field. 1

11 y (cm) By (Gauss) y (cm) Figure 12: Field long several parallel horizontal lines. Red is simulated field and blue is measured field. Black is residual. 11

12 x (cm) x (cm) Figure 13: Field long several parallel vertical lines. Red is simulated field and blue is measured field. Black is residual. 12

13 By residual (Gauss) Figure 14: Field long a line down the center longitudinal axis. Red is simulated field and blue is measured field. Black is residual. 13

14 x (cm) by+fieldx(-y+2.73,-x-2.6,-z+65.5) by+fieldx(-y+2.73,-x-2.6,-z+65.5):(x+2.6) (x+2.6) Figure 15: Fields along parallel vertical lines near the front of the prototype. Red is simulated field and blue is measured field. Black is residual. 14

15 y (cm) By residual (Gauss) y (cm) Figure 16: Fields along parallel horizontal lines near the front of the prototype. simulated field and blue is measured field. Black is residual. Red is 15

16 solder_field(x,y,z) solder_field(x,y,z) x (cm) Figure 17: Horizontal fields along the vertical (left) and longitudinal (right) center lines. The coil extends from -2 to +2 along z. In x, the inside of the coil is between -1 and +1, and the actual coil takes up the space 5 cm beyond these. 16