A microcomputer-controlled, magnetic field measurement and analysis system

Nuclear Instruments and Methods in Physics Research A294 (1990) 397-410 North-Holland

A MICROCOMPUTER-CONTROLLED, MAGNETIC FIELD MEASUREMENT ANI} ANALYSIS SYSTEM C. HADDOCK *, O. SMITH and J.S.C. McKEE Department of Physics, University of .Manitoba, Winnipeg, Manitoba, Canada R3 T 2N2

Received 28 November 1989 and in revised form 4 May 1990

Current accelerator projects will involve the construction and field measurement of up to 10000 magnets. A statistical analysis has shown that in order to optimize the performance of an accelerator it will be necessary to measure the parameters of field strength, field uniformity and harmonic content for every magnet. If the measurements are performed at the construction site . the magnets which do not meet the required specifications can be repaired immediately . This paper describes a self-contained field measurement and analysis system, based on an IBM microcomputer, which performs all the remote control, data acquisition and data analysis functions automatically. The system is of low cost such that each manufacturer can provide field parameters for each magnet as it is completed thus avoiding a logistical problem at the accelerator site. 1. Introduction Current accelerator projects under consideration for funding will involve the manufacture of up to 10000 magnets. Of these projects the advanced hadron facilities (AHF) involve increases in circulating beam current of 100 times that presently available at a particular energy. The magnetic field produced by these magnets needs to be known more accurately than previously if the loss of beam due to collision with the walls of the vacuum vessel is to be avoided. The information needed to successfully operate these new machines can only be obtained if the magnetic field distribution of each magnet is precisely known. In order to reduce the spilled beam to acceptable levels and to optimize the performance of an accelerator it has been shown that it will be necessary to measure the field parameters of field strength, field uniformity and harmonic content for eve - magnet manufactured [1]. The measurement of these parameters at the construction site will serve to identify magnets which exceed specified tolerances and require reconstruction . Further, such a scheme will avoid the logistical problem lJI 11dV111g to 111CW~1111N x` ialYG 11U111UG9 va airá&.A%~ta me vac. accelerator site in a limited time . e techniques used for magnetic field measurement have change little since the 1970s, however major advances in the fields of instrumentation, data acquisition and data logging using computers can be used synergistically to meet the sophisticated requirements of new * Present address: TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia, Canada V6T 2A3.

machines . This paper describes a field mapping system of low cost, which can be delivered as a turn-key unit to the magnet m: iufacturer. After careful alignment of a measurement field probe all the functions of the system including probe driving, interface control, data acquisition and data analysis are performed automatically . A data set is logged to a hard disk and a hardcopy output summarizing the field parameters is produced for each magnet . The system may be configured to measure dipole, quadrupole or sextupole magnet types, it is described in detail in the following sections, where it is configured to measure the field in quadrupole magnets.

2. Fie-Ad measuring probes Beam optics calculation programs such as TRANSPORT [2] require the effec°ive length and harmonic content of a magnet as input data to model the effect of beam line elements on accelerated beam . These parameters represent integrated values through the length of the element. It is therefore convenient to use probes which measure these parameters directly and quickly. The field measuring probes are of the fluxmeter type, . i.e . they work on the principle of Faraday reduction will be found in operation of such prob ,The theory of appendix B. A description of the cometry of the pe,)bes follows. 2.1 . Differential field probe The purpose of this probe is to measure the uniforma .l . ity of field gradient 8B,lax on the x ams and a

í1l68-94102/90/$03 .50 ~'-1990 - Elsevier Scienc:e Publishers B.V . (North-Holland)

398

C Haddock et cal. / Magnetic field measurement and analysis system

Fig. 1. Differential field probe .

on the y axis of the quadrupole aperture . The probe consists of a pair of long coils, wound on mechanically symmetrical formers . The coils are longer than the axial length of the magnet to be measured so that they also sample the fringing magnetic field . In this way the field gradient measurements are integrated over the effective length of the magnet . The height and width of the coils

are made as small as possible so that an approximation is made to a line integral through the magnet. In practice the formers must accommodate enough windings to give a signal output which is larger than the electronic noise. The coils are placed together side by side and held rigidly together. The windings of the coils are connected

0

Pig. 1. Harmonic measurement field probe.

e. Haddock et al.

/ Magnetic field measurement and analysis system

together in opposition so that their net output is proportional to the field gradient at the position of the coils. In practice the coils are stationary and the flux linking them changes as the current in the magnet coils is ramped . The changing output voltage is recorded by the system. Fig. 1 shows the field gradient probe. 2.2. Harmonic measurement probe This probe consists of an insulating cylinder of plexiglas, which is machined so that its outer surface is accurately symmetrical around the axis of the cylinder. The radius of the probe is chosen to be close to the aperture radius of the magnet. Shallow grooves are milled along the length of the cylinder at angular serarations of 45" . Single wires are placed in the groc.ves in such a way that the winding has the same symm try as the multipole component being measured ; fil,,. 2 shows the harmonic measurement probe . Three separate windings corresponding to dipole, quadrupole and octupole are wound onto the cyWider. These windings are fixed in place in the grooves asing an epoxy resin . The winding terminations are connected to a printed circuit board at the end of the cyhridex Tae probe rotates in a static magnetic field and the alternating voltages produced are recorded by the system . To measure the output voltages from the rotating probe a six conductor ribbon cable is attached to the printed circuit board and wrapped several times around the end of the probe. In operation the cylinder rotates in the field approximately once every 6 s and the required data is

obtained in a few revolutions. The direction of rotation is reversed when it is necessary to rewind the ribbon cable . In this way electrical noise problems associated with rotating contacts are eliminated . 3. Mechanical details The field mapping station is shown in fig. 3. It consists of two linear translation tables mounted one on each side of the magnet to be measured. The tables move on linear bearings parallel to each other along two ground steel rods set perpendicular to the longitudinal axis of the magnet. The ground steel rods ensure that the vertical travel of the tables is reduced to the order of 1/1000th of an inch (0.025 mm) along the entire length of the rods. The base of each translating table consists of a block in. diameter of aluminum tapped to accommodate a of the This rod is fixed to the base frame threaded rod. other end is end via bearings and the station at one a stepping through an antibacklash gear to connected motor which drives the table to the required position. The stepper motor receives 200 pulses for one rotation of its shaft, this results in a table movement of 0.1 in. (2.5 mm) A single pulse from the motor controller therefore results in a linear displacement of 0.0005 in. (0.0125 mm) The positions of the translating tables are determined using commercially available optical graduation type scales . A scale was attached to each of the translating tables so that the position could be read ). at any time with an accuracy of 0.0002 in. (0-005

Fig. 3. Field mapping station.

C. Haddock et al. / Magnetic field measurement and analysis system

For positioning the field mapping probes in the Y (vertical) and Z (longitudinal) directions, micrometer adjustable translation tables of the type found on optical benches were used. Arrays of threaded holes were tapped into the X (horizontal) direction translation tables to provide approximate positioning of the smaller micrometer adjustable tables. Field mapping probes were then supported from holders attached to the smaller translators and the micrometers adjusted to align the probe within the magnet aperture. The base frame of the field mapping station consists of a ground stainless steel one inch plate to which the translating tables and magnet support cradle are fixed. The cradle is designed specifically for a particular set of magnets so that it matches the contours of the magnet and ensures that the median plane is parallel to the surface of the table. Further, it ensures that the longitudinal axis of the magnet is perpendicular to the direction of travel of the translation table. If a single magnet is to be measured, the ground base of the mapping table may be used as a reference surface for a dial indicator . 4. Instrumentation

In recent years advances in the speed, resolution and accuracy of analogue to digital converters have increased dramatically. It was decided that a modern analogue to digital converter (ADC), an integral part of a high-accuracy digital multimeter, could perform data collection quickly and accurately so that recording of pulse information could take place in real time. Thus electronic integrators, which can be inherently unstable and represent the weak link in this type of apparatus are no longer required in the measurement process. This section describes the voltmeter and other instrumentation associated with the measurement system. 4.1 . Voltmeter

The digital voltmeter used in the system is a Hewlett Packard HP3457A, which has a maximum resolution of 6 . digits . In this application the ADC samples at a rate of 50 samples/s which allows 5 4 digit resolution and an accuracy of the voltage recorded of 0.026. The voltmeter is remotely controlled via an IEEE-488 interface. In practice, the instrument makes a number of readings after receiving an external trigger from the magnet current supply, it then transfers these readings to the controlling computer over the bus. In the case of field gradient measurements, static probes are positioned in time-varying magnetic fields. In the case of multipole measurements rotating probes are positioned in static magnetic fields. In either case a time varying voltage is produced by the probe which is sampled by the digital voltmeter.

4.2. Position readout The positions of the motorized X direction translating tables are read out via a Mitutoyo GR15 optical graduation type position encoder. The encoder allows the position of the tables to be read tc an accuracy of 0.005 mm. The position of each table information is sent on demand to the controlling computer via an RS-232C serial interface . 4.3. Motor controller

Stepper motor control is accomplished using an inhouse designed motor controller. The unit can supply the series of pulses required to drive up to three stepper motors . Two of these motors are used to drive the X direction translating tables. A third motor is used during a rotating coil survey to drive the harmonic probe. The motor control unit is computer controller via an 8 bit parallel digital interface. 4.4. Current supply Current is supplied to the magnet from a Bruker Corporation BMN-02 current supply . The unit supplies current via an impedance matching transformer and is stable to 1 part in 106 over a period of 1 h. An IEEE-488 interface is provided for computer control of the current supply . 4.5. Controlling computer A single computer is used for instrumentation control, data acquisition and analysis . It consists of an IBM-PC/XT containing IEEE-488, + RS232C interface cards. A Data Translation DT-2801 card provides an 8 bit digital output. A 25 MB hard disk is used for storage . Graphics display resolution is 640 (horizontal) by 350 (vertical). 5. Interfacing

7'

The standalone instruments, i.e. the linear translation table position readout, the digital voltmeter and the magnet current supply, transfer information to the computer over digital to digtal interface. The computer and the devices assert "handshaking" signals to control the flow of information between them. The interfaces are of two types, a serial interface (RS232) in which information is transferred one bit at a time, and a parallel interface (IEEE-488) which transfers information over 16 parallel lines simultaneously, i.e. one byte at a time. The devices have the advantage that data taken by the standalone instrument may be kept in a small memory buffer within the instrument until requested by the

C Haddock et al. / Magnetic field measurement

computer. As the number of devices and interfaces connected to the computer increases. this feature is particularly useful . Information is transmitted over these interfaces via a set of american-standard-code-for-in formation-interchange (ASCII) characters . The stepping motor controller is computer controlled but contains no information buffer, it is controlled over an 8 bit parallel digital interface . The linear translating table readout sends data to the computer using the RS232 interface . The IEEE-488 interface is a general purpose interface standard for sending byte serial commands and data between instruments. Its main advantage is that is has a bus structure, i .e. several devices can be placed along the same set of parallel lines and may communicate with each other. 5.1. 8 bit parallel interface The stepping motor controller is an in house design for providing power for up to three stepping motors. It is remotely controlled by an eight bit parallel interface using TTL logic levels . The eight lines of input are divided so that each motor may be given single pulses to produce single steps, or a line may be set that drives the motor continuously . Another line on the interface determines the direction of rotation of the shaft of the motor. The interfacing of the instrumentation to the controlling computer is shown schematically in fig . 4. ó. Software The data acquisition and instrument command functions are controlled by software based on a_ commercial software package called ASYST. The syntax of the program is similar to that of FORTH and APL. The main advantages of the software are : (1) it contains a set of interface control subroutines which can be easily called to control the interface cards mounted in the computer ; (2) it has built in analysis functions which can perform, for example, Fourier transforms on the acquired data, and

HP3457A DIGITAL MULTIMETER [_. _ __}_BMN-02 CURRENT SUPPLY

F

_

v F9SI10~ READUCr

á

MOTOR CONTROL 14 a

_ __IDM .PC__XT BU i

_ _ _

Fig. 4. 1werfacing-to-cont roller instrument,

and anahsis

i t has a set of graphics routines which n to quickly display the results of opera acquired data. ASYST is based on the concepts of a stack and routines known as °° m.-ords" . Higher-order words may he defi as a series of lower-level words, p :yVicusly known to the system, these in turn call lower-level words chine language is reached at the lowest level. Words are thus interpreted as they are entered . The system may then be "saved" on disk as a new system to which the added word is "known" . In practice an instruction entered at the terminal, for example READ.X.POSITION will perform a series of functions. In this example the system first raises DTR on the serial port. This has the effect of requesting the multiplexer of the linear table position readout to take a position measurement. The ASCII data are transferred over the serial link and stored in a string array also previously defined within the software system. A word previously defined which translates ASCII data to numerical data is executed, and another word displays the results on the computer terminal and stores them in another previously defined array for readout at the end of a series of measurements . In this way complex sequences of instrument control and data acquisition and manipulation can take place through relatively simple programs .

(3)

7. Field mapping process This section describes in detail the sequence of events which result in a map of the field gradient and harmonic content of the magnet . 7.1. Field gradient mapping process The field gradient measurement probe is aligned in the magnet aperture so that its center line coincides with the mechanical axis of the magnet, and such that it is equidistant from the pole tips of the magnet. 7.1 .1 . Single measurement The sequence of words described below a.Te called by the instruction MAKE-MEAS whose function is to per t_` _t_ . :1 f:lA sseau . 'T'h®norrsman a a- csarmsamsac ..ed reading o_r d:ccs_a... if eaenacaa IUriiD a sin~e reading pulses spaced by 50 + 1 MM sends a series of 80 MOV parallel output port of two lines of the 8 bit ins along the stepping motor which is connected to the computer, repprocessed so that they . These pulses are controller other six lines of the levels 1 and 0. The resent logic the high or low so that are arranged parallel outpui translating of the two result occurs, the driving desired tables simultaneously through + 1 mm. A compensation routine (COMPENSATE) is executed until the positions of the translating tables agree with the required

402

C. Haddock et al. / Magnetic field measurement and analysis system

Fig. 5. Differential field measuring flow chart. position to within 0.005 mm (0.0002 in.). The probe position is then read (READPOS) and stored as the first element in an array . After READPOS and COMPENSATE have completed, further commands and acquisition take place on the GPIB bus . The current supply is instructed to supply current to the magnet on receipt of the command SET1250A. As the current in the coils begins to

ramp, a rapidly increasing voltage is developed across the magnet coils, this voltage is processed to provide an external trigger to the digital voltmeter. This process takes place before the signal from the field probe has risen above the level of the pickup noise. The digital multimeter begins a series of 256 readings which it makes over an interval of 6.0 s. After this time the current supplied to the coils, and the resultant magnetic field are both constant and the output from the differential field probe has reduced to zero. The readings from the digital voltmeter are in ASCII form 40

n

n

20 -i E

X (mm)

Fig . 6. Uniformity of differential field in the aperture of a quadrupole magnet used in a microprobe arrangement .

-40

0

100

200

n

300 400 500 reading number

r

600

Fig. 7. Signal output from quadrupole winding .

700

C. Haddock et al. / Magnetic field measurement and analysis system I

2 .0

4 pole

250

300

50

60

(100%)

ey 1 . 0 U 20 pole 0 .5

44

12 pole 29-

0. 0

10

20

3

30

71

40

multiple of fo

Fig. 8. Fourier transform spectrum of quadrupole winding.

r~mnttt®r mam~rar ,e n the -nd -re saa sent te waav the evutraasinnba~rrw saaaas~ in aaa waaal,rv,.va aaavaaavaJ uu they are taken . It is shown in the appendix that the differential field is proportional to the individual voltages e; read from the probe . This number is evaluated and stored in an array in computer memory where the index of the array is the same as that for the position measurement .

aua

71 .2 . Full aperture measurement . . .. IS i a`.rGasraaa cmcn®cm.®d The probe position cs by v 1s .l!v m taaa° n in .) and the measurement process (MAAKEMEAS) reI

-t~A

tint :!

tha

~,ntirp

maanpt

anprttirp

hac

hPpn

surveyed . 's results in two arrays, one contains the positions of the probe and the second, a voltage proportional to the values of differential field at those positions . After performing the measurements the program plots the differential field vs position data on the computer console . A flow chart of the differential measuring

process is shown in fig. 5 . The individual coils within the probe are symmetrically wound although in practice accurate symmetry is difficult to achieve . To cancel the effect of asymmetries

C. Haddock et al. / Magnetic field measurement and analysis system

a

Fig. 9. Fourier transform spectrum of dipole winding. in the two coils, the measurement is repeated with the coils rotated through 180* . The average of the values corresponding to differential field (at each position X,) for each orientation of the probe is calculated and displayed versus position to show the uniioírnity of differential field in the aperture of the magnet. The differential field uniformity for a quadrupole element used in a proton microprobe arrangement is shown in fig . 6. 7.2. Hannonic content measuring process

The harmonic measuring probe is aligned so that its axis coincides with the mechanical axis of the magnet .

The ASYST word MPOLE calls a series of lower words to perform the multipole measurement . As the probe rotates in a static magnetic field, an alternating signal is produced from each of the windings. Consider the situation where the harmonic probe is positioned in the aperture of the magnet. Let the starting angle of the measurements (0 = 0') be that where the windings of the probe are at their closest position to the pole tips. The number of readings taken by the voltmeter is arranged so that just as the voltmeter has taken 305 readings ; the nrohe -has rotated through 360*. The measurement process is repeated five times over a period of a few minutes. If one assumes that the magnetic field has remained stable over this period (the current supply

C. Haddock et al. / Magnetic field measurement andanalysis system -20

1

405

i

. 20

c.~ c

10

U

.05

.00

Fig. 10. Fourier transform spectrum of octupole winding. is stable to 1 part in 106 over periods of 1 h) then, if the voltmeter trigger pulse occurred at a well defined angular position, the five output waveforms will be identical apart from random electronic noise. Further, if the average of these five signals is taken at each angular position, then the s!ignal-to-noise ratio will be reduced . Finally, if one takes the Fourier transform of the measured waveform, the harmonic content of the magnetic field may be determined . This measurement is performed by the system as described below. The harmonic probe is driven by a stepping motor such that one revolution of the probe occurs in 4 .175 s. The probe is rewound to an angular position of -10 °

(10 ° anticlockwise from the desired starting point). The ASYST word ROTI rotates the probe through an angle of 370 °. For the signal averaging technique to work the t ._te_ t, Le ~: .~ .~ of .+mcisely vt71[id~tctce aácüs~~ vc tir .a.vr~.d a. P..,.."- .~ thesame anQnlar position for each measurement . To accomplish this an angular encoder is mounted on the opposite end of the harmonic probe shaft. The encoder provides a pulse output as the probe passes through 0' . This pulse is fixed with respect to the angular position of the shaft and has a width corresponding to 0.001 ° . The pulse is fed to the external trigger of the voltmeter . As the ASYST word ROTI executes, the pulse triggers the voltmeter which samples the output signal

C. Haddock et al. / Magnetic field measurement and analysis system

416 BEGIN

1

9ETL_T0_ 100A

REWIND2

FOURIER XYGR GRAPH HARMONIC AP-PLIIUDN VS Nßß

SET CURRENT SUPPLY S TO 1UU AMP

GRAPHICS READOUT

j4

READOUT HARMONIC AMPLITUDE AND NUMBER

ROTATE PROBE FROM -10 TO 280 DEG.

LVOLTMVER-ACTIVfYY

ON EXTERMAL TRIGGER TA12 929 READINGS OF HARM . PROBT SIGN

I

. REWINDO

MOVE ANTICLOCKWISE WITH LIMIT SWITCH ON

Fig. 11. Multipole measurement process flow chart.

from the probe over a single rotation . Data from successive revolutions of the probe are added to a numerical array and the average value calculated . After the data have been acquired the ASYST word FOUTIER performs a fast Fourier transform (FFT) on the values stored in the array. For the quadrupole winding for example, the stored signal represents four oscillations of a cosine waveform with some perturbation due to higher order multipoles. The signal output from the quadrupole winding is shown in fig. 7. The Fourier transform of this array is determined and the magnitudes of the Fourier components are plotted versus harmonic number. The heights of the first 20 peaks are read by the program . The Fourier spectrum of the signal from the quadrupole winding is shown in fig . 8. The harmonic probe contains dipole, quadrupole and octupole windings. The process described above is repeated as each winding is connected to the voltmeter . The Fourier spectra of the signals from the dipole and octupole windings are shown in figs . 9 and 10. From these data the harmonic component of the magnetic field may be determined . In practice, the harmonic probe is aligned to the mechanical center of the magnet, this results in the dipole component shown in fig. 9 which is propoYtional to the separation between the magnetic and mechanical centers of the magnet aperture. Furthermore, probes aligned to the mechanical center of the magnet result in " ,-ed down" of harmonics as shown in

fig. 10. The displacement of the octupole coil from the magnetic center produces an apparent quadrupole component of the order of 0.1% of the real quadrupole term. This effect is discussed in detail in the appendix. A flow chart of the multipole measuring process is shown in fig . 11. 8. Summary A field measurement and analysis system has been designed and constructed in the Department of Physics, University of Manitoba. The system can be used to measure the representative parameters of quadrupole focussing magnets, it does so completely automatically once the magnet has been aligned in the system bed. The unit was constructed for a fraction of the price of currently available systems, while performing the same measurements to the same accuracy. Iris reature will make the system attractive when many magnets need to be measured rapidly to high accuracy in a relatively short period of time. Appendix: Introduction to the theory of fhrtxmeter operation

The analysis of the outputs from the field measurement probes is presented in this section .

C. Haddock et al. / Magnetic field measurement andanalvsis system

Appendix A : Differential cost As the current in the magnet coils is ramped, the emf generated in each coil is given by

e(t) - - á ,

(A.1)

0= - fe (t) di .

(A.2)

where io is the magnetic flux linking the coil. It follows that

If the sampling rate of the voltmeter is high enough this expression may be replaced by the expression for the discrete samples n

= 1, ei(t) ät .

(A .3)

i-

If the two coils are connected so that their signals oppose, the difference in the flux linking the two coils may be found : n

$t - ck = 1: (et - eA(t) àt .

(A .4)

Due to asymmetries in the probe construction the measurement of the differential flux is performed twice in each plane (Ozx and Ozy). After the first measurement, the probe is rotated through 180 ° and the measurement repeated. The readings are taken in accurately reproduced positions for zero degrees and 180" orientations, and the signals averaged . In this way the problem of the relative orientation of the mechanical and magnetic median planes is removed . The value calculated is the differential flux linking the two coils. This value can be converted to the field gradient if the distance between the effective centers of the coils, and the effective area of each coil is known. In practice the separation of centers is determined by observing the individual output from each coil as it passes the axis of a "calibration" quadrupole, i.e. one whose axial position is well known [3] . The current in the quadrupole coils is ramped and the measuring coil oriented so that zero output is observed. This position is recorded and the process repeated for the second coil . The distance between the two positions for zero output corresponds to the effective separation of the centers of the two coils. To determine the effective area each coil is placed concentrically within a large coil whose dimensions are accurately known [31 . The two coils are placed in a calibration dipole magnet whose field distribution has been accurately determined using the Nuclear Magnetic Resonance technique. As the current in the dipole windings is ramped the ratio of the outputs of the two coils is equal to the ratio of their area if the number of turns in each coil is equal.

407

A .1. Useful aperture A perfect quadrupole will have a field that varies linearly with radius, i.e. aB,fdr = coast. The useful aperture of a magnet may therefore be consider that area over which the field gradient is essentially constant, or that area over which the effective length of the magnet is constant . Let the useful aperture be an area whose radius represents the point at which the field gradient has fallen to by 1 part in 103 of its central value . Since the field gradient is directly proportional to the differential flux this information may be obtained from the data shown in fig. ó. A.2. Effective length The part of a quadrupole aperture over which the effective length of the magnet is constant is an important parameter in beam optics. The effective length may be determined by analyzing the signal from a single coil of the differential probe. The effective length of a magnet is given by L.

f Bja) d z,

Bn(0)

(A .5)

where B is the component of the field perpendicular to the plane of the measurement . The integral value above may be determined from the sampled data output if the coil dimensions are accurately known. The value of Bn (®) is the component of B perpendicular to the plane of measurement in the center of the magnet and may be determined using a small search coil of accurately known dimensions, cemented to the center of the long coil. Alternately a Hall probe may be used . It is the variation of effective length versus displacement from the axis which is of use, since the flux linking the coil is directly proportional to jB.&. This data also gives the effective length of the magnet. Appendix B: Harmonic probe theory B.1 . Introduction The representation of a magnetic field may he expressed as a function of a complex variable B(r)=B,+IB,,

where _ .. ° .R

,~a - ÎL

T £L

The field rna v ,inularb, he described in cylindrical polar coordinates

(0 .1)

B=B,+iB® .

It is possible to expand the field as a power series of the form [41 B

= F ,8 -,

I

iCnrn

_ , e  n®

C. Haddock et al. / Magnetic field measurement andanalysis system

408

rotating loop, and da = Ldr. Substituting the expression for B from eq. (7) above 0= ƒB -n da = Im J 00

= Im ~ iCn

n

n

1: icn r" -1

n=1

(B.5)

e-i ( ne+4, )

(B.6)

The emf generated is therefore given by

dO,

do _ - C Lr"cos(n9+4n) where 4" is a phase angle for the harmonic n .

e= -

L dr

The amplitude of the voltage signal from rotating loop will be mostly due to that of the mental component N. The major contribution amplitude of the signal from a single wire loop

N

V,= C r N-1Lrw,

(B.7) such a fundato the will be (B.8)

where w is the frequency of rotation of the cylinder . The above equation has been written so that the term CNrN-1 appears and is the value in tesla of the fundamental component of the field. Fig. 12. A single wire loop in the aperture of a magnet [5].

It can be seen that the magnitude of B is constant on the circle of radius r and that the sign of B changes n times as 9 increases from 0 to 21T rad . The term Cn r" -1 is therefore the magnitude of the 2n component of the field and as such is in units of T. A normalized set of harmonic coefficients is defined by Cn r" -1 = Cn r" ®N Cn = (B .3) , CNrN_ 1 CN where N identifies the dominant harmonic. In the case of a quadrupole : N = 2. If all the harmonics are evaluated at the same radius then c" = Cn/CN . B.2. Application to a rotating coil wire loop Consider a single wire mounted parallel to the axis of the multiple magnet, and on the surface of a cylinder of radius r and length l, concentric with the magnet, as shown in fig. 12. A loop of wire is constructed by two radial elements labelled r, and another wire along the magnet axis. The cylinder is placed in the aperture of a magnet as shown in the figure. As the cylinder rotates, the emf produced is given by the Faraday induction law 00

e ( t ) = -di=- il = 1 dt fB-nda

(B.4)

where (A is the flux passing through an area da of the

B.3. Output signal from a single wire loop

Consider the measurement of harmonic coefficients of a quadrupole magnet using the single loop design above. Let the radius of the cylinder be made as close as possible to the radius of the aperture of the magnet . Let the quadrupole and measuring coil have the following parameters : pole tip field = 0.8 T, aperture radius = 3.0 cm, cylinder length = 25.0 cm, and rotation rate = 1 revolution every 4.0 s. The amplitude of the voltage signal corresponding to the fundamental component will be: V, = 9.424 mV. In a well designed magnet, it is desirable that the higher order harmonics have coefficients 10° times less than the fundamental . To verify this by measurement would require measurement of the signal from the coil with an accuracy of 9.424 mV/10000 = 1 pV. The current generation of analogueto-digital converters cannot sample an alternating signal at the required sampling rate with this accuracy, therefore some way must be found to increase the sensitivity to the higher-order harmonics . Various methods exist for increasing the response of u.ao. Jlllb.I -IMP IVE"EIIIE LVl1J. l lEbJV 111\rIUU~.. (i) increasing the number of turns in the loop; (ii) rotating the coil at a higher rotation rate and reading the coil output via a set of slip rings through a narrow band pass filter [5] ; and (iii) a second coil is placed concentrically with the first at a smaller radius, the second coil is less sensitive to the higher-order harmonics and its output is amplified, inverted and added to the to the signal from the main coil so that the fundamental signal is "tucked out" [6].

C. Haddock et al. / Magnetic field measurement andanalysis system

two sets are connected in opposition, the net signal output is zero. The same is true for the sextupole case. In this way, the Morgan coil produces an output for those multipole components which have the same symmetry as the measuring coil and suppresses many other multipole signals . A Morgan coil wound with symmetry 2m is essentially equivalent to 2m single loops wire so that their output signals add. The voltage from such a coil will V, .t= -- (2m)Cr"m'Lrtw (B .q) and the coil responds to odd-multiples-of-2m field harmonics . In practice, separate windings for quadrupole, sextupole and octupole symmetries are wound on the surface of a single cylinder. These additional windings allow measurement of higher-order multipole components which are suppressed, in the manner shown above, by the quadrupole sensitive Morgan coil.

Fig. 13. Morgan harmonic cou.

B.4. Morgan coil design

A more elegant method for measuring higher-order components of a magnetic field consists of winding a coil on the surface of a cylinder in such a way that the coil has the same symmetry as the magnetic multipole component to be measured. This coil design is known as the Morgan coil 171 and is shown schematically in fig . 13. The Morgan coil achieves its suppression of unwanted multipoles by symmetry in angle. Consider fig. 14a; four wires are wound on the surface of a cylinder and are separated azimuthally by 21x/4 rad . The wires form a probe for the measurement of 4pole, 12pole, - - . ,2n(2m + 1)pole (m = 0,1,2, - . . ) harmonic components. In this case for n = 2 (quadrupole winding), the signal from all other multipoles is suppressed as will be shown below. The wires are wound on the surface of the cylinder so that the signals from adjacent wires oppose. Consider again fig 1A. ;n -h;., h the mil is located in the anerture of a quadrupole magnet . As the coil rotates, the signal from winding 1 has the opposite sign from that from winding 2. However, the two windings are connected in opposition so that the net effect is that the signals from windings 1, 2, 3 and 4 all add. Now consider the quadrupole measuring coil in the aperture of sextupole and octupole magnets as shown in figs . 14b and 14c. In fig. 14b the signals from windings 1 and 3 add, as do those from 2 and 4. Because these

Fig. 14. Suppression of unwanted multipolen by a Morgan coil.

C. Haddock et al. / Magnetic field measurement andanalysis system

410

Appendix C: Effect of a misalignment of the rotating probe

additional harmonics will be very small compared to the "real" harmonics due to the Rm term.

B may be expressed in the form of a power series as follows-

Appendix D: Discussion of errors

B=

00

n-1

1Cn r"-1

e

-~(netl~rn)

(C.1)

00

n-1

(C .2)

iC

where Z=x+iy=rc io. Consider a new coordinate system (x, y') whose origin is displaced by R from the original (x, y) origin [8]. In the complex plane Z= Z' + R,

(C .3)

where Z' = r'eie' .

(C .4)

One may write Z" - 1=(Z'+R) " -1=

n-1

~o(nml)(Z') 

-1-mRm. (C.5)

where (

li -1 _ m ) m1(n - 1 - m)!

Substitution into eq. (15) gives 00 n-1 (n _ 1 E iCn E t m )(Z')n_1-mRm . n= 1 rrt=0 The flux o is calculated from

(C .6)

(C .7)

00 n-1 F, iC ~ J rr = 1 m=0 `n m 1 1

X (r')n-1-m eie(n-1-m)R' I L dr '

(C .8)

and, the voltage e is given by - do dt

00

n-1

=- V c v, n=1

It has been shown that, if one assumes only an error in azimuth, the displacement of a conductor winding by 80 results in a voltage relative error given by [7] SE

S8

C 2C.'

(D .2)

where c is the signal from the coil. The angular positioning of the wires therefore limits the accuracy of the probe. In the case of the probe discussed here, the wires are laid in a milled slot which is 0.012 in. wide (0.0030 cm). If one assumes that the maximum displacement of the conductor is half the slot width the value of 80 is given by 80= 0.012 -_ 3 .2 X 10 -3 rad, (D .3) 1 .875 where the radius of the cylinder is 1 .875 in. (4.7625 cm). The geometry of the coil therefore limits the accuracy of the measurements of harmonic components to 1 .6 X 10-3 or 0.16` of the reading. References

o= fB-n da = Im

D.1. Errors due to the construction of the probe During winding of the coil it is possible to displace a wire by Sr radially and 80 azimuthally . The result of which will be equivalent to a perfect coil plus an error loop of area 1/2 . Sa = L [Sr 2 + (r80) 21 (D .1)

~n -1 1r. r,- m P&&(n-1-m) m J

m=0`

0

X R `L d .

The harmonics present in the field will be unaffected by the misalignment . For every harmonic present a second series of harmonics will be be produced. The second series will appear an order lower than the magnetic field harmonic which produced them, e.g. a quadrupole harmonic will generate a dipole term. In general these

[1] B. Berkes, Proc. Kaon Factory Projecq Definition Study Magnet Design Workshop, Vancouver (1988) . [2] K.L. Brown, F. Rothacker, D.C. Carey and C. Iselin, Transport : A computer program for designing charged particle beam systems, SLAG rep . SLAC-91 Rev 2 JC-28. [3] C. Germain, Nucl. Instr. and Meth. 21 (1963) 17. [4] B.C. Brown, Fundamentals of magnetic measurements with illustrations from Fermilab experience, Int . Committee for Future Accelerators Workshop on Superconducting Magnets and Cryogenics, Upton, New York (1986) . [5] W. Lamb and R.J. Lan, Search coil measurements of the hártnonic content of beam transport magnets, Proe. ist int . Symp. on Magnet Technology, Stanford (1965) . [6] B.C. Brown et al., Report on the Production Magnet Measurement System for the Fermilab Energy Saver Superconducting Dipoles and Quadrupoles, Fermi National Accelel ator Laboratory (1983) . [7] G.H. Morgan, Stationary coil for measuring the harmonics in pulsed transport magnets, Proc. 4th Int . Conf. on Magnet Technology, Brookhaven (1972) . [8] R .D. Fyvie and D.E . Lobb, Nucl . Instr . and Meth. 114 (1974) 609.