1. Lecture 7 Jack Tanabe Cornell University Ithaca, NY Magnetic Measurements

2. Introduction Magnetic measurements, like magnet design, is a broad subject. It is the intention of this lecture to cover only a small part of the field, regarding the characterization of the line integral field quality of multipole magnets (dipoles, quadrupoles and sextupoles) using compensated rotating coils. Other areas which are not covered are magnet mapping, AC measurements and sweeping wire measurements.

3. Voltage in a Coil

4. Therefore, substituting; where A, the vector potential is a function of the rotation angle, .

5. Measurement System Schematic

6. Digital Integrator The Digital Integrator consists of two elements. –Voltage to Frequency Converter. –Up-Down (Pulse) Counter.

7. Using an Integrator on a Rotating Coil Using an integrator simplifies the requirements on the mechanical system. The use of an integrator measures the angular distribution of the integrated field independent of the angular rotation rate of the coil.

8. Theory where L is the coil length and A is the vector potential, a function of the rotation angle . The magnetic field can be expressed as a function of a complex variable which can be expressed, in general as ; Rewriting; The Vector Potential is, therefore;

9. Therefore, when we are measuring the integrated Voltage, we are actually measuring the real part of the function of a complex variable. We are measuring the rotational distribution of the integrated Vector Potential, AL. We really want to measure the distribution of the Field Integral.

10. Field Integral Equating the real and imaginary parts of the expression; Let us take just one term of the infinite series.

11. In order to fully characterize the line integral of the magnetic field distribution, we need to obtain only |C n | and  n from the measurement data. The graph illustrates the output from a quadrupole measurement. The integrator is zeroed before the start of measurement and the graph displays the result of a linear drift due to DC voltage generated in the coil.

12. Fourier Analysis In principal, it is possible to mathematically characterize the measured data by performing a Fourier analysis of the data. –The Fourier Analysis is performed after the linear portion of the curve is subtracted from the data. Equating common terms,

13. or, finally, Separately, for the fundamental and error terms;

14. Fundamental and Error Fields In general, the Fourier analysis of measurement data will include as many terms as desired. The number of terms is only limited by the number of measurement points. –Earlier, we introduced the concept of the fundamental and error fields. The Vector potential can be expressed in these terms.

15. Compensated (Bucked) Coil The multipole errors are usually very small compared to the amplitude of the fundamental field. Typically they are < 10 -3 of the fundamental field at the measurement radius. –The accuracy of the measurement of the multipole errors is often limited by the resolution of the voltmeter or the voltage integrator. Therefore, a coil system has been devised to null the fundamental field, that is, to measure the error fields in the absence of the large fundamental signal.

16. Consider the illustrated coil. Two sets of nested coils with M outer and M inner number of turns to increase the output voltage for the outer and inner coils, respectively, are illustrated.

17. Compensated Connection The two coils are connected in series opposition. Define the following parameters: and We define the coil sensitivities; then,

18. Compensation (Bucking) The sensitivities for the fundamental (n=N) and the multipole one under the fundamental (n=N-1) are considered. Why one under the fundamental? Consider the quadrupole, N=2

19. The classical geometry which satisfies the conditions for nulling the N=2 and N=1 field components in the compensated mode have the following geometry. Homework, show that s 1 and s 2 are zero for these values, compute the balance of the sensitivities and compare with the graph.

20. Compensated Measurements Quadrupole measurements using the coil in the compensated configuration are typically as illustrated in the figure.

21. Bucking Ratio In the illustrated example of the compensated measurements, two properties can be readily seen. –The drift is present. Usually, it is a larger portion of the signal than in the uncompensated measurements. This is because the DC voltage, usually due to thermocouple effects, is a larger fraction of the small compensated coil measurements. –The signal is dominated by a quadrupole term. This is because of coil fabrication errors so that the quadrupole sensitivity is only approximately zero. The quality of the compensation is measured as a bucking ratio. Achieving a Bucking Ratio > 100 indicates a well fabricated coil.

22. Uncompensated Measurements The magnet is also measured with the rotating coil wired in the uncompensated condition to measure the fundamental field integral and the multipole one below the fundamental. Where the sensitivities in the uncompensated condition are designated by capital S. For the Quadrupole;

23. Recalling the expression for the magnetic field components, the amplitude of the fundamental field is, Solving,

24. Substituting into the expression for the fundamental amplitude;

25. Normalized Field Errors The separate multipole field errors, normalized to the fundamental field amplitude can be computed from the measurement data. a n and b n are from the compensated measurements and a N and b N are from the uncompensated measurements.

26. Reference Radius The expression for the normalized error multipole is evaluated at the outside radius of the inner coil, r 1. This radius is limited by measurement coil fabrication constraints and, in general, is substantially smaller than the pole radius and generally smaller than the desired radius of the good field region, which might be > 80% of the pole radius. Therefore, the expression for the normalized error multipole is re-evaluated at a reference radius, r 0.

27. The figure illustrates a 35 mm. pole radius quadrupole with a compensated rotating coil installed in the gap. The coil housing is < 35 mm. so that it will fit between the four poles. A half cylinder sleeve is placed around the housing to center the coil. As a result of these mechanical constraints, the maximum coil radius is < 27 mm.

28. The desired good field radius is 32 mm., the maximum 10  beam radius. Therefore, in order to compute the field quality at this radius, the normalized field errors are recomputed at the required r 0. and Therefore, and

29. Dipole Measurements The quadrupole coil configuration can also be used to measure a dipole magnet. Since the coil has no quadrupole sensitivity in the bucked configuration, a quadrupole error must be evaluated using the unbucked configuration. Since a quadrupole multipole is not an allowed multipole for a symmetric dipole magnet, this does not usually present a serious problem. However, if the dipole design constraints requires that the symmetry conditions be violated (ie. a “C” shaped dipole), the evaluation of the small quadrupole error present in this geometry may be marginal.

30. Sextupole Measurements For sextupole measurements, it is desirable to make s 3 and s 2 =0 for the compensated coil. This set of equations is difficult to solve algebraically. Therefore, the equations are solved transcendentally.

31. One of many solutions to these equations are, The compensated sensitivities for these parameters are illustrated.

32. Relative Phase The calculation of the phase angles is based on an arbitrary mechanical angular shaft encoder zero datum, adjusted by aligning the measurement coil. Therefore, a phase of the fundamental field,  N, is always present. This angular offset can introduce large errors since small angular offsets between this datum and the zero phase of the fundamental field can result in large errors in the relative phase of the multipole error with respect to the quadrupole zero datum. Therefore, one normally computes a relative phase with respect to a zero phase for the fundamental field.

33. A one page summary of the multipoles for 15Q-001 measured at approximately 81 Amps is reproduced in the table. These measurements were made at IHEP in the PRC. Sample Quadrupole Measurements

34. Two measurements are made at each current, one with the coil connected in the uncompensated mode and one in the compensated mode. The integrated voltage for each magnet is Fourier analyzed and the amplitudes of each coefficient are listed. The u1 and u2 amplitudes (PHI[n] in 10E-8 V-sec.) are the amplitudes of the coefficients for the cos  and cos 2  terms from the uncompensated measurements. The balance of the amplitudes are the coefficients of the cos n  terms from the compensated coil measurements.

35. The next four columns include measured and computed values. –Angle The absolute phase angle of the nth Fourier term with respect to the shaft encoder zero datum. The same datum is used for both the uncompensated and compensated measurements. –PHI[n]/PHI[2] The ratio of the compensated nth Fourier coefficient to the uncompensated 2 nd Fourier coefficient. –Coil Coef.[n] The coil sensitivities computed from the design radii of the various measurement coil wire bundles. –B[n]/B[2] The computed (using the coil sensitivities) absolute value of the ratio of the multipole amplitude to the quadrupole field amplitude, evaluated at 32 mm.

36. Multipole Spectrum

37. Multipole Errors as Vectors

38. Distribution of n=6 Multipole Errors

39. Distribution of n=10 Multipole Errors

40. Distribution of n=3 First Random Multipole Errors

41. Iso-Errors The normalized multipole errors and their phases provide information regarding the Fourier components of the error fields. Often, however, one wants to obtain a map of the field error distribution within the required beam aperture. This analog picture of the field distribution can be obtained by constructing an iso-error map of the field error distribution. This map can be reconstructed from the normalized error Fourier coefficients and phases.

42. where is the phase angle of the multipole error with respect to the zero phase for the fundamental (quadrupole) field. Therefore,

43. and Where

44. The computations and contour map are programmed using MatLab. 15Q01 at 81 Amps.

45. The iso-error plot is replotted for only the allowed multipoles (n=6, 10, 14 and 18) and the first three unallowed multipoles (n=3,4 and 5). It can be seen that it is virtually identical with the previous plot, indicating that the unallowed multipole errors > 6 are not important. 15Q001 at 81 Amps

46. When the iso-error curve is replotted with the unallowed multipole errors reduced to zero and the allowed multipole phases adjusted to eliminate the skew terms, the  B/B <1x10 -4 region is dramatically increased. This illustrates the importance of the first three unallowed multipole errors which are primarily the result of magnet fabrication and assembly errors. 15Q01 at 81 Amps. Unallowed multipole errors = 0. No skew phases for allowed multipoles.

47. Lecture 8 Lecture 8, describes techniques and principles for core fabrication. These descriptions are extremely important since the performance and quality of the magnetic field are dominated by the iron core of the manufactured magnets. –The requirements are for the full population of magnets required for the synchrotron, not only for the individual magnets. –This important subject is covered in chapter 9 of the text. Lecture 8 also describes magnet assembly and electrical bussing. Finally, fiducialization, installation and alignment are briefly described. These subjects are covered in chapter 12 of the text.