AD-A286 081

ARMY RESEARCH LABORATORY

The Calculation andMeasurement of Helmholtz Coil

Fieldsby David J. DeTroye and

Ronald J. Chase

ARL-TN-35 November 1994

DTICSELECTE•NONV 1 4 1994

F

• 94-34824

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Da%~ id iI )* Iroi e and Ronald I ( haw

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13 ABSTRACT a'24 .a

Helmholtz (coils are used to develop unitorm magnetic tields tor a variety of research applica-tions. Helmholtz coils can be easily constructed, and the fields easily calculated. This makes themespecially useful in calibrating sensors and numerous other low frequency magnetic field testingapplications. This particular report will discuss the calculation and measurement of Helmholtz coilfields to be used for testing of composite materials to low frequency magnetic fields.

14 SUBJECT TERMS 15. NUMBER OF PAGES

21Testing, Helmholtz coils, composite, magnetic field 16. PRICE CODE

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Contents

Page

1. Backgrou ....................................................................................................................... 5

2. Introduct . ................................................................................................................. 5

3. Test Configuration ......................................................................................................... 5

4. M easurem ent Loop Calibration ................................................................................... 7

5. Helm holtz Coil Predicted Results ................................................................................ 8

6. M easured Results ........................................................................................................... 12

7. Conclusions ......................................................................................................................... 17

D istrib u tio n .............................................................................................................................. 2 1

4ppendix

A. M atlab Program Listing for Fie~c .alculations .......................................................... 19

-iigures

1. Helm holtz coil construction ..................... .........................................................................2. Test configuration .............................................................................................................. 63. M easurem ent loop construction ....................................................................................... 74. M easured and calculated responses ......................................... ..................................... 85. Coordinates and variables describing a circular loop used to e'ýpress EM fields ....... 106. M easured resonance of Helmholtz coil ......................................................................... 137. Three-dimensional plots of magnitude of magnetic field B(z) versus B and p ...... 148. M agnetic field B(z) on centerline as a function of z .................... ................................ 159. Contour plots of B(z) versus z and p .............................................................................. 16

Table

1. Loop calibration results ......................................................................................................... 8A,•.t •o f-or

NTIS CRA&IDTIC 1ALi -Ur-•a::" • ,. ,.k _

B y . .. .. ..............

Dit ' S ic I

V-I

1. BackgroundComposite materials are being increasingly used throughout theArmy as a means of decreasing the weight and increasing thestrength of Army systems. With the increased use of compositematerials, the question of electromagnetic shielding must beanswered. The Army Research Laboratory (ARL) is currentlyinvolved in a composite shielding program whose objective is toexamine the composites used in Army systems and characterizethem electromagnetically. The approach used has been to develop acomposite characterization laboratory that will measure the electri-cal properties of composites. The Helmholtz coil is one tool that willbe used in the characterization laboratory.

2. Introduction

Helmholtz coils are used to develop uniform magnetic fields for avariety of research applications. Helmholtz coils can be easily con-structed and the fields easily calculated. This makes them especiallyuseful in calibrating sensors and numerous other low-frequencymagnetic-field testing applications. This report discusses the calcula-tion and measurement of Helmholtz coil fields to be used for testingof composite materials to low-frequency magnetic fields. Futurecomposite testing of this nature will be based on American NationalStandard ANSI/ASTM A698-74, "Standard Test Method for Mag-netic Shield Efficiency in Attenuating Alternating Magnetic Fields."

3. Test Configuration

The Helmholtz coils were constructed in accordance with the guide-lines in the ASTM 698-74 standard. They were fabricated from ¾/4-in.plywood, 1/2-in. Extren® threaded rod, and 14 AWG copper (coated)wire. Each coil had a diameter of approximately 47.5 in. (1.219 in),with 48 turns each of the copper wire. The coils are shown in figure1. Sixteen 1/2-in. Extren threaded rods were used to form the circularshape of the coil. The magnetic fields associated with the Helmholtzcoils were measured with two single-turn shielded loops. Both wereconstructed from 0.141-in. copper jacketed cable, one with a diam-eter of 2 in. and the other of 3 in. Additional information on the cali-bration of the loops is provided in section 4.

A Hewlett Packard (HP) 3577A network analyzer was used to meas-ure the magnetic fields of the Helmholtz coil. The output of the ana-lyzer was connected to an audio amplifier, whicn was connected to

5

the Helmholtz coil. The reference channel monitored the current inthe loop. Channel A recorded the voltage associated with the 2- or 3-in. loops placed inside the coil. The complete test configuration isshown in figure 2.

Figure 1. Helmholtzcoil construction.

Helmholtz coilsFigure 2. Testconfiguration.

Measurement loop

Network a Current probeanalyzer

Amplifier

6

4. Measurement Loop Calibration

Two electrostatically shielded single-turn loops (one 2-in., one 3-in.)were used to measure the fields associated with the Helmholtz coils.Both were constructed as shown in figure 3. The open-circuit voltageat the loop output terminals can easily be calculated by

Vc = 27rfNAB , (1)

where

Voc = open-circuit voltage at loop's output terminals,f = frequency (in hertz),

N = number of turns in loop (one turn),A = area of loop (in meters), andB = magnetic flux density (in teslas).

The loops were calibrated in a transverse electromagnetic (TEM) cellknown as a Crawford cell. Using an HP 3577A network analyzer, wemeasured the loop voltage as a function of output voltage applied tothe TEM cell versus frequency. Knowing the physical dimensions ofthe TEM cell and the measured area of the loops, we could deter-mine the ratio of the measured versus reference voltages; to deter-mine the calculated response, we applied this ratio to

Rc= 20 log(V,./V.f) , (2)

Figure 3. Measurementloop construction. Inner

conductor

conductor Shield

Coaxial line I i

La

where

RC = calibrated response,V,, = open-circuit voltage at loop's output terminals, andVref = reference voltage applied to the TEM cell.

The measured and calculated results are shown in table I for 100kHz. Figure 4 compares the measured and calculated responses for afrequency range of 100 kHz to 10 MHz for both the 2- and 3-in.loops. One can see that the calculated and measured responses fol-low each other to well within 0.5 dB. Based on equation (1), it isassumed that both of the loops would continue to fall at 20 dB perdecade to dc. Because of limitations in the measurement system, theloops could not be calibrated below 100 kHz.

5. Helmholtz Coil Predicted Results

The magnetic fields generated by a circular wire loop can be derivedaccording to the following procedure. The electromagnetic (EM)fields generated by a circular wire loop carrying current I will satisfythe Maxwell equations. For a wire loop (or a coil) excited with low-frequency current, almost all the energy is stored in the magneticfield. This energy is determined from the concepts and rules of cir-cuit theory. Circuit theory can be regarded as describing a specialclass of solutions of the Maxwell equations that result from using the

Table 1. Loop Loop Measured Calculated TEM cell measuredcalibration results. dimension area response response Difference

(in.) (m) (dB) (dB) (dB)

2 2.545 -95.1511 -94.9368 0.21433 5.572 -88.347 -87.8906 0.4541

(a) (b) __ _ _ _ _ _ _ _ _ _ _ _ _

-40 -40

-50 -50

O_ -60 -60

"o-70 ( -70

E -80 c-80

-90 -90-100 •-100

105 106 W 105 106 107

Frequency (Hz) Frequency (Hz)

Figure 4. Measured (solid) and calculated (dotted) responses of (a) 2- and (b) 3-in. loops.

8

first term of a power series solution for the fields. The fieldscomputed with this approximation are called quasi-static. EMphenomena such as radiation are neglected in the quasi-staticapproximation.

1

A general rule of thumb for circuit elements excited by sinusoidalvoltage or current is that tile quasi-static approximation will beextremely good when the phy3ical size of the element is sufficientlysmall with respect to the wavelength of the source excitation. Thisrestriction will be satisfied if the phase of the voltage (current) isapproximately the same over the spatial extent of the element. For acircular loop, this condition requires that the current be nearly con-stant over the circumference of the loop. The current can change intime, but at each instant of time, the current in every part of the loopshould have the same value. Therefore, the quasi-static approxima-tion will be very good as long as 21ra << A (or in terms of frequencyinstead of wavelength, v << c/lOa), where a is the loop radius and cis the velocity of light in free space.

One can find the EM fields generated by a loop in this order ofapproximation by determining the current in the loop employingcircuit theory and using that uniform current as the source of thevector potential from which the fields can easily be derived. This isthe approach followed below.2

Figure 5 shows the coordinates and variables for a loop of radius alocated in the xy plane. The loop is assumed to have a total current I.The vector potential for a general current distribution can then bewritten as

4T fdwhere A is the vector potential and I is the current in the loop. Fromthe direction of the current, A has only a 0 component, and by sym-metry A is independent of the variable 0. In setting up the integra-tion, we arbitrarily set 0 = 0 to simplify the results. With the symbolsgiven in figure 5, the following can be derived:

dl = (-a sin 0', a cos 0', 0)dO'S=(rsin0, 0, rcos0) ,i'= (a cos 0', a sin 0', 0)

I = j .r'= ./r2+a2 - 2ra sin 0 cos 0'

1R. M. Fano, L. J. Chu, and R. B. Adler, Electromagnetic Fields, Energy, aid Forces, John Wiley & Sons, Inc. (1960).2William R. Smythe, Static and Dynamic ElectricitY, Hemisphere Publishing Corp. (1989).

9

Figure 5. Coordinatesand variablesdescribing a circularloop used to expressEM fields.

Substituting these variables in the integral, we note that the integralinvolving sin 0' vanishes (it is odd), as it should since only an AOcomponent must survive, and the integral involving cos 0' (even)allows the range of integration to be halved. This results in

A,0Ia cos O' do'A2(r,6) f r2r + r-+a 2 -2rasin 0cos 0'

It is more convenient to express this integral in terms of cylindrical

coordinates using r2 = p 2 + z2 and sin 0 = pl/p 2 + z2 , which gives

AO ,z)= 41A - cos 0' do',) f ,p2 + a - 2ap cos 0'

This integral does not have a closed form; however, there is a trans-formation that results in tabulated functions. Change the variable tomake the upper limit of the integral ir/2, by using 0' = ir + 20. Theintegral becomes

O •,Z = ola 2 (2 sin 2 0 -_ ) doA( 7 - - (a + p) 2 + z2 - 4ap sin2 0

If we define k2 = 4ap the integral is transformed into(a + z2 '

A,(p,4) = -- a -1- I k2)K(k) - E(k)]

10

with

K~k) do and E(k)=f I- k-siný0 dOK 1 -k- sin2 0

K is the complete elliptic integral of the first kind, and E is the com-plete elliptic integral of the second kind. Both are tabulatedfunctions.

The magnetic fields are computed from the vector potential by

B = V x A, giving two components:

aAeB•(p,z) =a

=- O and B.(p.z)= -(pA 0 )

To calculate these derivatives, we will need the derivatives of K, E,and k, since they are functions of p and z. The math is straightfowardand the results are summarized below:

aK E K and E =E Kak k(l-k 2 ) k ak k kDk k k3 k3 an k = -Zk3

p2p 4"p 4and a-=4ap

The final expression for the magnetic field components becomes-Z a2_+p+-_2_ _Bp(p,z) = Ara - '" E(k)- K(k)

2rp7(=p_*+a)+,2 [(a-p)2+z-

+( z--[a - p 2 2i (p 1+ a + z [(a-._ - -E (k) - K (k)2:pz) = (_P + a)f+z- 7.(a - p)" + ,

For a single loop in the xy plane, we can find the magnetic field com-ponents at any location p and z by first determining k from the abovedefinition, and then substituting into the above formulas.

We can check the accuracy of these expressions for the simple case ofthe magnetic field components along the axis of a loop. On the axis,p = 0, which implies k = 0, and hence K(0) = E(0) = ir/2. The expres-sion for Bp becomes indeterminate, but application of l'Hopital's

rule gives Bp = 0. The expression for Bz becomes 0.5pua 21/ /(a-2 + Z,)3,

as expected.

Calculating the magnetic fields from a Helmholtz coil requires fieldscomputed for two loops having the same axis, located in the planes

11

at z = -d and z = +d. The separation distance 2d is related to the loopradius by d = a/2. The expression for the total magnetic field writtenin terms of the above derivation is just the sum of the individualfields from each loop. Using the notation and results derived earlierfor an individual loop, one can write

B."al(p~z) = B,(,,Y,z + d) + B,,(p~z - dl),Bf"`Iakz) = Bz(pz + d) + Bo(p.z - d)

Once the solutions were obtained, the variables were substituted forthe actual Helmholtz coil. The diameter of the coil was measured at1.219 m, the average current measured (see sect. 6) was 0.019052 A,and the number of turns was 48.

6. Measured Results

Numerous magnetic field measurements were taken on the test con-figuration described in section 3. The objective of the magnetic fieldmeasurements was to verify the predicted response and determinethe size of the test volume. This was done by essentially mappingthe inside of the Helrnholtz coil. A cylindrical coordinate system wasused for consistency with the predicted results. Both the p and zcoordinates were varied in 2-in. increments. Because of symmetry,the 0 component was not recorded, and only one quadrant wasmeasured.

Figure 6 is a measurement of the Helmholtz coil's fields as a functionof frequency. Figure 6 clearly shows that the coil has a resonance atapproximately 58 kHz, which would mean that the usable frequencyrange of the Helmholtz coil is between dc and 40 kHz. Frequenciesabove 40 kHz should not be used.

The measured results compared very well to the calculated :esults,as is shown in figures 7 to 9. Figure 7 shows a three-dimensionalcomparison between the measured and calculated z-component ofthe magnetic field as a function of p and z. One can see that the twocompare very well in wave shape and magnitude. Figure 8 shows atwo-dimensional comparison of the measured versus calculatedmagnetic field as a function of z on the centerline (p = 0).

The peak magnitude of the measured field is 1.345 x 10-6 T (teslas),while the calculated is 1.348 x 10-6 T. Both are essentially the samemagnitude. The measured data have a noticeable peak at z = 0,where the calculated data are flat out to z = ±0.1. This difference(0.37 percent) cannot be fully explained, but one possibility is thatthe measurement loop was not perpendicular to the incident mag-

12

Figure 6. Measured 60resonance ofHelmholtz coil.

40,

* 20 .

0 0

-20 -

-40

-60 .

-80102 103 104 105 106

Frequency (Hz)

netic field. Equation (1) assumes that the magnetic field is normal tothe plane of the measurement loop. When the measurement loop isnot normal to the incident field, equation (1) has to be multiplied bythe sine of the angle between the plane of the loop and the incidentfield. A difference of 0.37 percent is equivalent to the 2-in. loop beingoff less than 50 or 3/S2 in. above and below the normal plane. Whilegreat care was taken to map the fields inside the coils, the measure-ment loops were only sighted in for being perpendicular, and errorsof this magnitude were possible. Figure 9 is another representationof the same data, shown as contour plots. The contour plots showthat the field is uniform at 1.3 x 10-6 T for most of the inner region.Again, one can see that the measured data compare very well withthe calculated.

13

Figure 7. Three- (adimensional plots ofmagnitude ofmagnetic field B(z)3versus B and p-.(a) measured and 2.5(b) calculated.P

o 2

S1.5E

0.501

(IIs(

R2.5

600

14

Figure s. Magnetic (a) 1.35field B(z) oncenterline as a 1.34function of z:(a) measured and 1.33(b) calculated.

1.32

p1.31

E 1.32

E 1.29

1.28

1.27

1.26

1.25 -0.2 -0.-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

z-axis (m)

(b) 1.35

1.34

1.33

S1.32

M8 1.31

E

S1.3 /1.29 -

1.28 /1.27

--0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4z-axis (m)

15

Figure 9. Contour (a) 03 3 1 0plots of B:) versus: 1 4 10-andp p Ia) measured . -and ib) calculated. 1 6 10-_

0.2 / 1

0.1\

E 1 x 10-6

0

/ 1.2x 10-

-0.1

1.4 x 10-6-0.2

1.x 10-6

-0.3 .8X10- , _ _ I-0.4 -0.2 0 0.2 0.4

p-axis (m)

(b) 0.3 )T'0••1."4x10-6

0,2 1 .6<e10--

0.1

1 x 10-

-0.)1.2 x 10-.6

-0.2

1.6x 1.0-

1.8 6x 10 -6

-0.3 1 1-0.4 -0.2 0 0.2 0.4

p-axis (m)

16

7. ConclusionsThe objective of this report is to provide a technical guideline for theuse of the Helmholtz coil that was designed and built for the ARLelectromagnetic composite testing laboratory. The experimental ver-sus theoretical results were compared for the primary field compo-nent and proved to show excellent correlation. By measuring thecurrent being applied to the coils and using the equations in section5, the user can predict the magnetic field at any point inside thecoils. To assist the user in determining the magnetic fields, we haveincluded appendix A, which is a MatlabC program listing that canbe used to generate the graphs as seen in secton 6.

The size of the test volume is also an important con, -ation whenworking with a Helmholtz coil. The ANSI/ASTM t 4 standardindicates that the test volume is related to the coil meter. Thestandard states, "The Helmholtz coil diameter shall be at least threetimes the length of the test specimen or four times its diameter, orboth."3 Figure 9 confirms this test volume, in that the field is uni-form to just over ±0.21 m on the centerline of the p-axis (0.34 percentof the diameter) for all values of z. However, figure 9 also shK vs thatthe test volume depends on the geometry of the test object. Forobjects that are less than 0.28 m high (z-axis) and on the centerline,the test area extends to ±0.4 m (p-axis). Objects that are more than0.28 m high have a test area of ±0.21 m (p-axis). As a rule of thumb,the coil diameter should be at least three times the length of the testspecimen. The test area can be expanded out to about ±0.35 m(p-axis) on centerline for all values of z, if a 2-dB change in field uni-formity can be tolerated at the comers of the test volume. The readeris referred elsewhere 4 for a complete discussion of how one canextend test volume beyond the above limits for the testing of largeobjects.

The loading of the coils by the test object has not been addressed inthis paper. The user must be aware that a large object of magneticmaterial placed inside the coil will load the coil and concentrate thefields on the test object. The resultant field will be distorted whencompared to the field when the coil was empty. It is recommendedthat the user measure the current in the empty coil and verify that itdoes not change when the test object is inserted. If the current doeschange, the amplifier must be adjusted to the original level.

3ANSI/ASTM A698-74, American National Standard, Standard Test Method for Magnetic Shield Efficiency inAttenuating Alternating Magnetic Fields (1974)."4E. L. Bronaugh, Helmholtz Coils for EMI Immunity Testing: Stretching the Uniform Field Area, Seventh Interna-tional Conference on Electromagnetic Compatibility (Conf. Publ. No. 326).

17

Appendix A. Matlab Program Listing for FieldCalculations

The following is a Matlab program used to calculate the fields at anypoint inside a Helmholtz coil of Radius a and current I.

S Like loops.u but symetric plane through z axis% Computes the a field from 2 thin loops of radium a,% located at z - +d and -d, (d-a/2, felaboltz Coils), and% current 1.

LN - 48 % Number of turns in the loops"mo - 4*pi*le-7;1-.019052;a-.6096;d-a/2;

% Define the plane over which fields are computed% N must be odd to include the point (0,0)

14-26; % No. of points along the rho axisN-51; % No. of points along the z axisPl-linspace(O,aX); % rho always positiveP - ( PI(M:-I:2) PlI]

Z-linspace(-d,d,N);(p,z] - msuhgrid(P1,2); I rho-O, p(:,1); a- -d, z(1,:)

t Determine modulus and Elliptic integralskl=(4*a*p)./( (a+p).^2 + (a-d).2 );

[K1,31J - ellipke(kl);k2-(4*a'p)./( (a+p).^2 + (z+d).^2);[K2,92) ellipke(k2);

I Compute 3-rho fieldsbpl - ((z-d)./(p.'sqrt( (a+p).^2 + (z-d).^2))) *

( ( (a^2 + p.-2 + (a-d).^2)./((a-p).^2 + (z-d).^2 ).1Zl - Ki );bp2 - ((z+d)./(p.*sqrt( (a+p).^2 + (z+d).-2 ))) .- ...( ( (a-2 + p.-2 + (X+d).^2)./((a-p).^2 + (z+d).^2 ).592 - K2 );bp - ( (LN'muo'I)/(2*pi) )-(bpl + bp2);bp(l:N) - zeros(N,1); % Remove NaN's from rho-0 axisbp = [ bp(:,N:-1:2) bp ]; % Make it mymetric

I Compute 3-a fieldsbzl - ( I ./sqrt( (a+p).^2 + (z-d).*2 ) ) ..

( ( (a^2 - p.^2 - (z-d).A2)./((a-p).A2 + (z-d).'2 ) ).*Zl + KI );bz2 - ( 1 ./sqrt( (a+p).^2 + (z+d).^2 ) ) .* ...( ( (a^2 - p.^2 - (z+d).^2)./((a-p).^2 + (z+d).42 ) ).*E2 + K2 );bz - ( (LN*uo*I)/(2*pi) )*(bzl + bz2);bz = [ bz(:,M:-1:2) bz 1; % Make it symmetric

I Compute the total B fieldbt - sqrt(bp.A2 + bz.^2);

% test along the z axis where we know the solutionbzz- 1 ./(sqrt(a-2 + (Z-d)."2)).-3 + 1 ./(sqrt(a"2 + (Z+d)."2))."3;bzz - 0.5*LN*muo*l*a^2*bzz;

19

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US Army Rsrch Lab Attn AMSRL-WT-W ChfAttn AMSRL-OP-SD-TA Mail & Records

MgmtAttn AMSRL-OP-SD-TL Tech Library

(3 copies)

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