1. INTRODUCTION The pole contour of the gradient magnet lamination is designed to result in a linear distribution of the two dimensional field. The three dimensional fringe fields for the unchamfered magnet will not be linear across the pole. A sharp (900) pole end will saturate at moderate fields, causing the field integral to change with small differences in magnet excitation. The gradient magnet pole ends must be shaped (chamfered) (1) to "soften" the end to avoid saturation effects and (2) to establish the linear distribution of the field integral. The pole end chamfer for the prototype SPEAR3 gradient magnet will need to be developed empirically and its performance evaluated prior to releasing the final geometry for production fabrication. Because of the difficult communications with SLAC with the anticipated development of the prototype and manufacture of the production magnets at lliEP in China, a plan and timeline must be developed for approval process of the final chamfer design. This note is written in order to describe suggested magnet measurement equipment and techniques to evaluate the prototype magnet performance and to describe a suggested procedure and algorithm for calculating the chamfer profile using the results of magnetic measurements.
2. DIFFERENCES BETWEEN THE PROTOTYPE AND PRODUCTION MAGNETS BECAUSE COMMITMENT FOR THE FABRICATION OF PARTS FOR THE PRODUCTION MAGNETS CANNOT be made prior to the evaluation of the Prototype magnet, the iron laminations for the Prototype will be stamped from different material than used for the production magnets. Also, the Prototype design will include removeable pole ends made from solid steel for the top and bottom poles at both ends of the magnet (four pieces). The pole ends for the production magnets will be machined from glued end packs of laminations. Therefore, the Prototype magnet performance which are affected by the B-H curve for the magnets will NOT reflect the subsequentperformance of the production magnets. Despite these differences, the study of the Prototype performance should reflect the transverse distribution of the field integral for the production magnets becausethe mechanical shape of the pole ends will have a much greater affect than the small differences in magnetic iron properties at the field levels in which the studie~will be performed. Cat. Code:S30731 I SSRL # M307 I Page 2Qf9 ': Jack Tanabe Date: Sept. 3, 1999
2.1 Prototype Geometry The difference betweenthe production magnetand prototype configurations is illustrated below.
.' Solid End Plate
, Glued End Stack
ProductionMagnet Configuration PrototypeConfiguration Four machined chamfer inserts will be required for the model. If one wants to preserve the inserts, only two can be used at a time. (The field differences due to changes in two of the chamfers will be half the affect due to changes in all four chamfers.) The general scheme for optimizing the pole contour will use the empirical process described in the following section.
3. MAGNETIC MEASUREMENTS The line integral of the field as a function of the transverse location across the magnet must be measured. The equipment required to perform this measurementsare a line integral coil, stages to move the line integral coil, a datum Hall probe and a data acquisition system. The line integral coil can be configured to measure either the full field or compensated so that the fundamental dipole and quadrupole fields are suppressed. The elements of the measurement system are described in the figures.
Precisionlevel used to assurethat theplane of the magnetgap is hOri~
Line of Sight
--~ [~~~~==~~~=Stage Stage~=~=:~~~J
.I Precisionlevel usedto assurethat the plane of the magnetgap is horizontal
Fixed Offset
Stage TargetsMounted On Coil Supports
The static Hall probe measures the magnetic field due to any variation in the power supply current during the measurement. The data taken from the Hall probe can be used to normalize the measured signal from the measurementcoil to a fixed excitation if there are any significant variations in the power supply during measurements. The stage is a device to move the coil precisely in one dimension across the magnet aperture. Two stages are required, one near each end of the measurementcoil. The stepping motors on each stage may be connected to a single power supply, ensuring that each motor rotates synchronously and the coil moves parallel to itself. Only one stage will require a shaft encoder to measureposition. The data acquisition system will require an integrator (preferably a digital integrator) in addition to the usual components. With some small modifications, the electronic data acquisition system is identical to that required for the rotating coil measurementsfor quadrupoles and sextupoles. Data Acquisition Two Channel Computer Command Integrator DtoA Cat.Code: S30731 SSRL# M307 e 4 of 9 Author s : JackTanabe , 1999
3.1 Coil Configuration(s) The magnetic measurementcoil designed and built at SLAC consists of two coils. When connected in opposition, the fundamental dipole and quadrupole field are suppressed enhancing the sensitivity to higher order multipole errors. If the signal from only one coil is collected, the measurementwill measureall the field components except the dipole field. All measurements required to characterize the unchamfered magnet as well as subsequent measurementsto characterize the field linearity for the various chamfer iterations will use the coil in the uncompensated mode. The evaluation measurementof the final chamfer shape shall use the coil in the compensated mode. The field quality measurements for all the production magnets will be made with the compensatedcoil. The approximate shapeof the coil(s) is described in the figure. ~..:~-:~..I ~~_:~.. .~-=~_:~..
-400 turns -400 turns UncompensatedCoil CompensatedCoil
3.2 Signals The description of the electrical signals from the coil and the algorithms used to compute the various multipole components of the line integral field are described in a separate engineering note. Briefly, one takes the signal from the coil in both the uncompensated (single coil) and compensatedmode (both coils connected in electrical opposition), subtracts the linear drift (due to DC thermocouple voltages) and normalizes for any power supply deviations. A least square fit is performed to determine the coefficients of a polynomial which characterizes the signal. The following is a short description of the expected signal from the coil in each configuration and a "qualitative" explanation of the relationship between the polynomial coefficient and the multipole terms.
3.2.1 Uncompensated Mode The nth multipole coefficients are related to the n-l coefficient of the polynomial fit of the integrated voltage signal from the sweeping uncompensated coil. If the integrator is "zeroed" at the beginning of the measurement,the dipole signal (n=l) is suppressed. In order to measure the dipole field, one must use the uncompensatedcoil in a static positon and ramp the magnet power supply. 3.2.2 Compensated Mode The nth multipole coefficients are related to the n-2 coefficient of the polynomial fit of the integrated voltage signal from the sweeping compensatedcoil. The degree to which the coil is "perfectly compensated" depends on the precision to which the coil is wound (the areas of both coils must be precisely equal). Since perfect precision is impossible to achieve, some linear signal from the integrated quadrupole field (~l % or more) will probably remain in the signal from the compensatedcoil. The sextupole signal (n=3) is characterized by the linear term (n=l) from the compensatedcoil. Since the linear term is "polluted" by both the uncompensatedpart of the quadrupole field and from any linear drift due to DC thermocouple voltages in the coil which cannot be subtracted, the sextupole term will be measured from the uncompensated coil measurement where the Quadratic term (n=2) characterizesthe sextupole error.
3.3 Measurements The distribution of the integrated field for the unchamfered magnets and for each iteration of the chamfer will be needed. These measurements will use the coil in the uncom~ensated mode (the signal from only one of the two coils will be collected). After the final chamfer has been developed, the measurementswill be taken with the coil in both the uncompensated and compensatedmode.
3.3.1 UncompensatedMeasurements The integrator is zeroed and the magnet is ramped with the coil in a fixed position. The integrated voltage during magnet ramping characterizesthe line integral dipole field. The coil is then swept across the magnet aperture. This signal characterizes the integral quadrupole and higher order multipoles. The two signals are combined to determine the fiel integral distribution for the unchamfered and/or iterated chamfer shape(s).
3.3.2 Final Uncompensated Measurement The integrator is zeroed and the magnet is ramped with the coil in a fixed position. The integrated voltage characterizes the line integral dipole field. The coil is then swept across the magnet aperture. The ramped integrated voltage is the final integral dipole field. The linear and quadratic terms from the swept measurements characterize the quadrupole and sextupole terms, respectively.
3.3.3 Final Compensated Measurement With this configuration, the fundamental dipole and quadrupole fields will be suppressed and the higher order error multi poles (n~4) can be more accurately characterized.
4. EMPIRICAL OPTIMIZATION Preliminary measurementof the distribution of the line integral will be made using a full set of unchamfered pole tips. The measuredfield integral will be larger than the desired field since the effective length of the magnet is increased by the fringe field. (The unchamfered iron length is equal to the design length.) It will also most probably be nonlinear and will have the maximum Cat.Code: S30731 SSRL# M307 e 6 of 9 Author s : JackTanabe , 1999
variation from linearity at the center of the pole. This is due to the finite pole width and the curved shapeof the coil at the end. The following illustration is a sample of the measurements one might expect from the preliminary measurement of the unchamfered prototype and is similar to the preliminary measurements(made using a Hall probe map) on the ALS gradient magnet prototype. Desired Field Profile
r ~~ / MeasuredField Profile, M(x) Ramped Magnet .Largest Signal Difference, function of x :t Correction * x
MagnetCenterline The largest difference between measured field and the desired field profile occurrs at the low field region of the map becausethe fringe field largest where the vertical gap is largest and the field is lowest. In the case of ALS, the insert thickness for the prototype was not sufficiently thick to remove the amount of material to compensate for the "largest correction". Therefore, a linear "target profile" whose slope was slightly less than the desired field profile was selected. This is not crucial since any mismatch in the integrated gradient can be tuned out by adjusting the neighboring quadrupole magnets. It is more important to reduce the nonlinearity of the of the field distribution using the chamfers. The algorithm used to determine the depth of the chamfer is developed in the following section.
4.1 Chamfer Depth Algorithm Let O(x)be the measureddifference between the measuredfield integral and the targetlinear function,y=rnx+b. y=rnx+b o(x)= M(x)-y =M(x)-rnx-b The chamferdepth is the o(x) functiondivided by the [linear] magnetcentral field. Az(x) = ~ = M(x)- rnx -~ B( x) B atramp location + GradIent x x FinalChamfer Shape ~'"""'...,;~? l ~~--
4.2 Cutting the Chamfer The chamfer using the simplest mechanical tooling is one with a constant slope since it can be shaped using a conical cutting tool. The three dimensional cut can be made using a two dimensional tool path as illustrated below. A left handed coordinate system is used in the illustration. The two functions h(x) and Az(x) are given by the pole contour and the cut depth formula from the empirical measurements. An expression for the location of the tool point for a given offset of the tool from the magnet centerline, y toolpath'can be easily derived. y toolpath can be any arbitrary function of x. A linear function which has a maximum value at the largest gap location and reduces to some minimum value at the smallest gap location can minimize the required tool size. (It should be noted that the maximum diameter of the tool is determined by the minimum curvature of the desired chamfer shape.) Pole Contour
z Xl X2
tan etool--h(x)-YtooIDath Zpoint ()X -L\z ()x
h,X)-YtooID8d1 Zpomt. ( X ) =AL.lZ( x )+ tan e tool
The inserts can be machined separately or in pairs. If the inserts are machined separately, one should be careful since the two sets of inserts are mirror images of each other. Paired machining of the chamfers is preferable. The paired machining requires tooling so that the inserts are placed together on a plate which will place them in the same relative location as when they are assembled on the prototype magnet. If the inserts are machined in pairs, the same tool path can be used for the lower pole by offsetting the tool by a distance, -y toolpath. For the production magnets, the glued end packs can be laid down on a bed of a tape controlled mill and the same techique used to cut the paired inserts can be used.
4.3 Iterations The magnet should be remeasured with the first set of inserts. The same algorithms should be used to compute the incremental depth required from the first guess of the chamfer shape. This incremental depth should be added to the first depth and the sum should be used to shape a second set of chamfered inserts. If the incremental depth is "deeper" than the first chamfer, the first set of inserts can be reused.
4.4 Under-Relaxation Method Another method for cutting the chamfers for the prototype and preserve the inserts (which are expensive to produce) is to cut the chamfer depth shallower than recommended by the algorithm. That is use (for instance) O.88(x) when computing the chamfer depth for the first iteration. Then, the measurement for the first set of chamfers should result in recommendations for "deeper" chamfers for subsequentiterations.
5. TIMELINE Becauseof the distance betweenSLAC and IHEP, coordination of the optimization of the prototypewill be difficult. The following is recommended. 1. SLAC and IHEP should agreeon the meansfor prototypeoptimization. This includes the following: .Agreement on magnet measurementtechniques and tooling. .Adoption of the algorithm for determining chamfer depth. .Agreement on the method and tooling for machining the chamfer. 2. lliEP will build the prototype including a number of "blank" inserts, make the preliminary measurements. lliEP will evaluate the data, recommend a chamfer shape and send the results of the measurementsalong with the recommended chamfer geometry to SLAC. 3. SLAC will evaluate the data and the chamfer recommendation and either approve of the chamfer shapeor recommend an alternate shape. 4. lliEP will devise the toolpath, cut the inserts, attach the new inserts onto the prototype and remeasurethe magnet. lliEP will send the results of the remeasurementsto SLAC. 5. Steps 3. and 4. are repeateduntil the required field integral uniformity is achieved. 6. The prototype will be sent to SLAC for remeasurementand confirmation that the required field quality has been achieved. The size of the gradient magnet makes it difficult to air ship it to SLAC from China. Therefore, it is expected that up to several months will be required before the magnet is available at SLAC for final evaluation. It is recommended that the results of final prototype measurements from IHEP be used to trigger the approval for production magnet fabrication and assembly.
5.1 109D "First Article" Since it is more likely that the full length first article will reproduce the prototype performance (the prototype is full length), SLAC may recommend that the 3/4 length magnet be built shortly after approval of the final chamfer shape. End packs using the chamfer shape developed from the full length prototype should be used for the 3/4 length "first article". This magnet should be measured at lliEP and the results evaluated at SLAC. It is expected that the field integral uniformity for this 3/4 length "first article" will be satisfactory. However, if there are any significant differences among the performance of the full length prototype and the 3/4 length "first article", one should have the results from the "first article" early.