From: Arthur W. Brooks Sent: Wednesday, July 05, 2006 12:43 PM To: Thomas G. Brown Cc: Wayne T. Reiersen; Michael C. Zarnstorff; Brentley C. Stratton; Bradley E. Nelson Subject: Mechanical Alignment of C1 thru C4 as-built coils to Minimize Resonant Field Errors Attachments: C1-4 realignment test.xls; C1-4 realignment test.pdf Tom, I've gone though the initial process of realigning the C1 thru C4 coils based on the as built current centers. This is to give you an preliminary estimate of the magnitude of coil motions required so you can assess the impact on machine assembly. As expected, the motions required are small. In fact, we may want to question whether it is worth doing at all since small motions in the wrong direction can be very damaging. As you know, the coils are being wound targeting the average current center achieved on prior coils. C1 targeted the design basis, C2 targeted C1, C3 and C4 targeted the average of C1 and C2 (since C3 was not finished prior to starting C4). The final alignment of the coils will presumably be done targeting the average of all coils of each type. For now, I am just targeting the design current center. Since there are 6 degrees of freedom per coil, theoretically we can target 6 modes completely. That is, we could determine an alignment which completely cancels 6 resonance field error modes induced by errors in the current center for each coil. Due to ill-conditioning of the dofs on modes (that is, some dofs have similar effects on particular modes so are not linearly independent) this is not always possible effectively limiting the number of modes which can be targeted (see attached pdf file). Alternately, we can use all 24 dofs for the present 4 coils to target the same or more modes. I've tried both methods here. The results are given in the attached spreadsheet. I have chosen to target the 3/5, 6/10 and 1/2 sine and cosine modes for the reference m50 (aka m50_e04) full current, full beta plasma. This too may change, but it gives us a starting point. (We have in effect built in a symmetric component by trying to make all the coils similar - i.e. by targeting the average current center - its not clear to me yet whether the symmetric component should be targeted or the plasma reoptimized for 'new' coils). In targeting the symmetric modes, it does not matter which position a given coil is in (first, second or third period, either half). In targeting the asymmetric modes it does matter. I have assumed, as we discussed, that C1 and C2 are in the first field period, C3 and C4 the second. Looking at the machine in plan-view, that puts C1 is in the half period from -60 to 0 deg as would be A1 and B1, C2 is in the next half period from 0 to 60 deg, and so on rotating counter-clockwise. Below are the resulting transformations of each coil for the two methods descibed above. Using each coil to independently correct it's own field error leads to larger displacements (.038" at origin) to control just 4 modes than using all four coils to correct 6 modes (.011" at origin): Coil Realignment relative to global coordinate system Each Coil (6 dof) Targeting 4 modes (3/5, 1/2, sin & cos) independently C1 Origin 0.00040934 0.00674867 0.00309482 X axis 1.00000000 -0.00004088 0.00002874 Y axis 0.00004088 1.00000000 0.00007032 Z axis -0.00002875 -0.00007032 1.00000000 C2 Origin -0.00323208 -0.03096036 0.02236041 X axis 0.99999952 0.00033231 -0.00091690 Y axis -0.00033190 0.99999984 0.00045618 Z axis 0.00091705 -0.00045587 0.99999948 C3 Origin -0.01426276 -0.01285752 -0.02457020 X axis 0.99999973 -0.00022982 -0.00069794 Y axis 0.00023020 0.99999983 0.00053686 Z axis 0.00069781 -0.00053702 0.99999961 C4 Origin 0.01163284 0.00620140 0.01452996 X axis 0.99999998 0.00017247 0.00002632 Y axis -0.00017245 0.99999982 -0.00058090 Z axis -0.00002642 0.00058090 0.99999983 All coils (24 dof) targeting 6 modes (3/5, 3/6, 1/2, sin & cos) collectively C1 Origin 0.00496570 -0.00337040 0.00488230 X axis 1.00000000 -0.00002923 0.00001009 Y axis 0.00002923 1.00000000 0.00005889 Z axis -0.00001010 -0.00005889 1.00000000 C2 Origin -0.00302021 -0.00392354 -0.00412903 X axis 1.00000000 -0.00002517 0.00000997 Y axis 0.00002517 0.99999999 0.00013734 Z axis -0.00000997 -0.00013734 0.99999999 C3 Origin 0.00090761 0.00435469 -0.00327171 X axis 1.00000000 -0.00003387 -0.00005282 Y axis 0.00003387 1.00000000 0.00005421 Z axis 0.00005282 -0.00005421 1.00000000 C4 Origin 0.00389920 -0.00001208 -0.01071775 X axis 0.99999998 -0.00003415 -0.00017100 Y axis 0.00003415 1.00000000 -0.00002165 Z axis 0.00017100 0.00002165 0.99999999 The above relate to an unperturbed coil coordinate system with Cx Origin 0.00000000 0.00000000 0.00000000 X axis 1.00000000 0.00000000 0.00000000 Y axis 0.00000000 1.00000000 0.00000000 Z axis 0.00000000 0.00000000 1.00000000 The above data is also included in the attached spreadsheet if it is easier to extract. Regards, Art