NATL INST. OF STAND & TECH MIST

1""""""""" PUBUCATI0N8

National Institute of Standards and Technology

Technology Administration, U.S. Department of Commerce

NIST Technical Note 1532

/oo

Relative Permeability Measurements

for Metal-Detector Research

Michael D. Janezic

James Baker-Jarvis

NIST Technical Note 1532

Relative Permeability Measurements

For Metal-Detector Research

Michael D. Janezic

James Baker-Jarvis

Radio-Frequency Technology Division

Electronics and Electrical Engineering Laboratory

National Institute of Standards and Technology

Boulder, CO 80305

April 2004

U.S. Department of Commerce

Donald L. Evans, Secretary

Technology Administration

Phillip J. Bond, Under Secretaryfor Technology

National Institute of Standards and Technology

Arden L. Bement, Jr., Director
National Institute of Standards U.S. Government Printing Office For sale by the

and Technology Washington: 2004 Superintendent of Documents

Technical Note 1532 U.S. Government Printing Office

Natl. Inst. Stand. Technol. Stop SSOP

Tech. Note 1532 Washington, DC 20402-0001

1 6 Pages (April 2004) Phone: (202) 512-1 800

CODEN: NTNOEF Fax: (202) 512-2250

Internet: bookstore.gpo.gov
Contents

1 Introduction 1

2 Overview and Definitions 2

3 Toroid Meeisurement Technique 4

3.0.1 Relative Permeability Model Excluding Toroid Conductivity 4

3.0.2 Relative Permittivity Model Including Toroid Conductivity 5

4 Measurements of Relative Permeability 7

5 Conclusions 11

6 References 12

Relative Permeability Measurements for Metal-Detector

Research

Michael D. Janezic and James Baker- Jarvis

Electromagnetics Division, National Institute of Standards and Technology,

Boulder, CO 80305

We examine a measurement method for characterizing the low-frequency relative permeability of ferromagnetic metals commonly used in the manufacture of weapons. In addition to the metal's conductivity and geometry, the relative permeability is another

important variable in the detection of metal weapons. We describe a measurement

method for measuring the relative permeability of toroidal metal samples of rectan- gular cross section. Taking into account the finite conductivity of the metal and skin

depth, we present a model for the inductance of a wound toroid from which the permeability is calculated. Relative permeabilities of ferritic and martensitic stainless steels

are presented as a function of frequency and magnetic field strength.

Key words: conductivity; loss factor; magnetic response; metal detector; permeability.

1. Introduction

The purpose of this report is to disseminate information on the characterization of the

relative permeability of ferromagnetic metals for use in metal-detector research. Hand-held

(HH) and walk-through (WT) metal detectors sense a perturbation of the applied magnetic

fields caused by the presence of metallic objects. In order to study the interaction of metal

detectors with metals, reliable data are needed on the conductivity and permeability of

metals used to test the detector. Previously, we have studied the measurement process for

determining the metal conductivity [1]. Although most of the metals considered in Ref. [1]
N V V \

Primary ma^etic Held . ^

Figure 1: Approximate eddy currents on a handgun due to an incident variable magnetic

field.

were noimiagnetic, there were several ferromagnetic metals with magnetic properties. Some

magnetic property data exist for these materials [2]. However, for the low magnetic fields

employed in metal-detection systems, the relative permeabilities of these materials are not

well known as a fimction of frequency or applied magnetic field. The relative permeability

is also a function of temperature. Although we did not attempt to perform temperaturedependent measurements, this aspect of the metal properties is summarized in Ref. [3]. This

note describes a simple method for determining the relative permeability of ferromagnetic

metals and presents measurement results for two ferromagnetic stainless steels commonly

used in the manufacture of weapons.

The first section provides an overview and some definitions necessary for understanding

the basic properties of ferromagnetic metals. The next section describes two theoretical

models for a wound metal toroid from which we can calculate the relative permeability of

the toroid specimen. The first model neglects the effects of the finite conductivity of the

toroid and the skin-depth effect, while the second model takes these effects into account when

determining the magnetic-field distribution in the toroid. In the final section we describe

the measiuement system and present relative permeabilities of ferromagnetic stainless steels

as a function of frequency and applied magnetic field strength.

2. Overview and Definitions

Interactions of magnetic fields with materials in a metal detector are related to the relative

permeability //^, conductivity cr, and the size and shape of the object [4,5]. Time-varying

magnetic fields in a metal detector interact with metallic objects by inducing eddy currents

on the object's surface. The currents in the conductor are generated by the applied magnetic

field as expressed by J = V x H. These eddy currents modify the incident magnetic field

and thereby allow detection by receiving coils (see Figure 1).
Applied magnetic field H

Figure 2: Magnetization curve for a ferromagnetic material.

One of the most important features of ferromagnetic metals, such as nickel, iron and

cobalt, is their large magnetic moment. Since many stainless steel alloys include these

materials, stainless steels can exhibit various levels of magnetic behavior.

A ferromagnetic metal in a demagnetized state is composed of many small magnetized

domains. In this state, these domains are randomly oriented so that the averaged magnetic

moment over the entire sample is zero. However, when a magnetic field is applied to the sample, the magnetic moment of each domain tries to align itself with the magnetic field. Figure

2 shows a B-H magnetization curve for a typical ferromagnetic metal, with the magnetic flux

density B plotted as a function of the applied magnetic field H.

For metal-detector systems, the applied magnetic field is small and near the origin of

the magnetization curve. For this region, the magnetization effects are reversible and, if the

applied field is static, no hysteresis effects exist. As a result, the relationship between B and

H is linear, where B = nayi^H, with ^ = 47r x 10"^ H/m being the permeabihty of free

space. The relative permeability /i^ in this region is also known as the initial permeabihty.

A more in-depth description of ferromagnetic behavior can be found in Ref. [6]

The level of interactions of the applied magnetic field of a metal detector with a ferromagnetic metal depend on the relative permeability /i|., conductivity cr, and the size and

shape of the metal object. Electromagnetic fields are attenuated as they travel through

ferromagnetic metals. The depth of penetration at which the electromagnetic field decreases

to 1/e of its value at the surface is called the skin depth. The skin depth 5 of a conducting

metal is the depth at which a plane wave traveling through the conductor is decreased from
its initial magnitude by a factor of 1/e:

^- / / .

- (1)

where / is the frequency, a is the conductivity of the conductor, and (i^ is the relative permeability of the conductor. We see that the frequency, conductivity, and relative permeability

of the material determine the skin depth. At higher values of frequency, relative permeability, or conductivity, the skin depth decreases, and the attenuation of the wave traveling

through the materials increases [7].

3. Toroid Measurement Technique

The relative permeability of a metal at low frequencies and low magnetic field intensities can be calculated from the measured inductance of a wound toroidal sample of metal of

rectangular cross section. A typical test specimen is shown in Figure 3. A toroid wound with

n turns of insulated copper wire is coimected to an impedance bridge, and the inductance

is measured as a function of frequency and current, which is related to the strength of the

magnetic field generated within the toroid. In order to calculate the relative permeability,

we examined two models that describe the relationship between the measured inductance

and the relative permeability of the toroid. The first model does not include the effects of

skin depth, whereas the second model does include skin depth.

3.0.1 Relative Permeability Model Excluding Toroid Conductivity

In the first model, we assume that a toroid of rectangular cross-section is wound uniformly

with n turns of wire as shown in Figure 3. Another necessary assumption is that the toroid's

relative permeability is much greater than one, allowing us to assume that all the magnetic

flux is confined to the toroid. Finally, for the low magnetic field intensities generated within

the toroid by the current in the windings, we assume that the relative permeability is linear.

For this first model, we neglect the conductivity a of the toroid, thereby neglecting the effect

of skin depth.

From Ampere's law, the magnetic field within the toroid is

^* = t' (^)

where / is the current applied to the terminals of the toroid's windings. The inductance L
-p

t Min turns

(a) (b)

Figure 3: A typical toroid test specimen: (a) cross-section (b) toroid with windings.

of the toroid can be shown to be given by

Pjv j^j^B-ndv = -JJJ |^«M.2^y ^^ liQjir 7^3- I I -^;z~ ) pdpaqxiz = 1 in

27r (3)

where n is the number of windings, a is the inner diameter of the toroid, b is the outer

diameter of the toroid, h is the thickness of the toroid, /i^ is the toroid's relative permeability,

and Ho is the relative permeability of free space. Solving for the relative permeability fir we

obtain

fJ'r (4)

a

3.0.2 Relative Permittivity Model Including Toroid Conductivity

In the previous model, which neglected the toroid's conductivity, we obtained an expression

for the magnetic field H^ that did not include the effects of the metal's skin depth. However,

due to the finite conductivity of the toroid, the effect of skin depth cannot be neglected, and

an improved expression for the magnetic field must be found in order to accurately calculate

the toroid's relative permeabihty. Namjoshi et al. [8] derived an expression for the magnetic

field within a toroid of rectangular cross section, and his model is outlined briefly here.

When the conductivity a is included, the magnetic field H^ within the toroid satisfies

IdH^ m d^H.

where

dp^ p dp (Pk'

+ ^i_q^ _ p//, = en,

dz

JUJfJ^o^irCr,

00, (5)

(6)
and H^o is the incident magnetic field. Using Ampere's law and assuming that the magnetic

field is confined to the toroid, then H^o is given by

H,<Ao nl

27rp'

(7)

Given the value H^o in (7), then (5) can be solved for H^, where

Ji{asp) Yi{asp)

^^y 3=1 n=0 \ h Masb) Yiiasb)

(8)

and where Ji and Fi are Bessel functions of the first and second kind of order one, as is the

sth zero of

Ji{asa)Yi{asb) - Ji{asb)Yi{asa) = 0, (9)

and Agn are constants that are functions of the dimensions, conductivity, permeability of the

toroid, and measurement frequency:

A —

2k'^nl {-lY K,{a — Kb)

-K (2n + l) i2

where

(2n+l)7r

h

K =

+ a2 + A;2(l-'^>^

Ji{asa)Yi{asa),

Ji{asb)

(10)

(11)

Ji{asa)'

The first term for the magnetic field in (8) is identical to that found in (2) , but the second

term contains the effects of skin depth. The inductance of the toroid is then given by

b 2n h

L = - J^B -Hdv = —

J I Jl2oi2rHlpdpd(f>dz, (12)

a

where H^ is given in eq. (8). Solving for p'^. in (12), gives the relative permeability of the

toroid.
Figure 4: Stainless steel toroid samples with and without windings.

4. Measurements of Relative Permeability

We machined toroid samples, as shown in Figure 4, from two stainless steels. The first

stainless material, UNS S43000, is a ferritic stainless steel, while the second material, UNS

S41600, is a martensitic stainless steel. Both ferritic and martensitic steels are ferromagnetic

and have a relative permeability greater than unity.

For all toroid samples, the outer diameter 2a was 6.92 mm, while the inner diameter

26 was 6.54 mm. The height h of the toroid was 1.02 mm. Each toroid was uniformly

wound around the entire circumference with 106 turns of copper wire. The conductivity,

from Ref. [1], of UNS 41600 was 1.53 x 10^ S/m, while the conductivity of UNS 43000 was

1.77 X 10^ S/m. In order to remove any residual magnetization, each toroid was demagnetized

with an ac current source. Before each measurement, the toroid was connected to a current

source, operating at 50 Hz, and the current was slowly decreased from 1 A to 0.01 mA.

After demagnetization, the toroid was connected to the terminals of an impedance bridge

(precision LCR meter), and the inductance of the toroid was measured as a function of

frequency and the current applied to the windings of the toroid. We varied the frequency

from 500 Hz to 10 kHz, while the root-mean-squared (rms) current varied from 0.1 to 10

mA. Figures 5 and 6 show the measured inductance as a function of frequency and applied

current for stainless steels UNS S41600 and UNS S43000.

Using the theoretical model described in Section 3.0.2, we calculated the relative permeability for both the USN 43000 and USN 41600 stainless steels; these data are shown in
10 X

3>

ni

T3C

Frequency (Hz) -•-500 --750

-A- 1000 -¥-2500 —5000 -^-7500 -M- 10000

0.1

2 3 456789 2 3 456789

10 Current I _(mA)

Figure 5: Measured inductance of a toroid of UNS S43000 stainless steel as a function of

frequency and applied current.

13.0

5

« 12.5-

12.0

Frequency (Hz) -•-500 --750

-*- 1000 -¥-2500 —5000 -^7500

-M- 10000

0.1

2 3 456789

1

2 3 456789

10 Current I _..(mA)

Figure 6: Measured inductance of a toroid of UNS S41600 stainless steel as a function of

frequency and applied current.

Figures 7 and 8. In addition to the plots of relative permeability, we plot the magnetic field density B

versus the applied magnetic field H in Figures 9 and 10. As expected, due to the low applied

magnetic field, each of the curves is linear.
80'

03

0)

0) Q.>

TO

0}

70

-t -

Frequency (Hz) -•-500 --750

-*- 1000 -?-2500 -—5000

-•-7500 -M- 10000

60.

>l M

0.1

2 3 456789

1

2 3 456789

10 Current I r„.(mA)

Figure 7: Relative permeability of UNS 43000 stainless steel.

I"

CO9}

Q.I

TO

a:

Frequency (Hz) -•-500

-•-750 -4- 1000 -¥-2500 —5000 --7500

-M- 10000

Current I r„j(mA)

Figure 8: Relative permeability of UNS 41600 stainless steel.
4x10

OQ

COc

Q

Xu

c

o»

10 20

Magnetic Field H (A/m)

Figure 9: Magnetic flux density versus applied magnetic field for UNS 43000 stainless steel.

e

CO

at

tz

<]> O

Li. O

C

4x10 -

Magnetic Field H (A/m)

Figure 10: Magnetic flux density versus applied magnetic field for UNS 41600 stainless steel.

10
5. Conclusions

We have described the procedure for measuring the relative permeabilities of metals at

low frequencies and low applied magnetic fields. Using measurements of the inductance

of a metal toroid of rectangular cross section, we calculated the relative permeability. A

theoretical model for the inductance of a metal toroid of finite conductivity was reviewed.

We presented measurements for the relative permeability of two stainless steels, UNS 41600

and 43000, as a function of frequency and applied magnetic field strength and found that the

toroid method, with the skin-depth correction model, works well for ferromagnetic metals.

Acknowledgments

Funding for this research was provided by the National Institute of Justice through the Oflfice

of Law Enforcement Standards at the National Institute of Standards and Technology.

11
6. References

[1] M.D. Janezic, R.F. Kaiser, J. Baker-Jarvis, and G. Free, "Dc conductivity measurements

of metals," NIST Technical Note 1531, National Institute of Standards and Technology,

2004.

[2] J.R. Davis, Ed., Metals Handbook, ASM International, Materials Park, OH, 1998.

[3] K. Ara, "Magnetic chaxaicteristics of ferromagnetic stainless steels," IEEE Trans. Magn.,

vol. 25, pp. 2617-2622, 1989.

[4] James Balcer-Jarvis, Raian Kaiser, Michael D. Janezic, N. G. Paulter, and K. L. Stricklett,

"Metal detector studies: Research materials," NIST Technical Note 1514, National

Institute of Standards and Techology, 2002.

[5] Sadao Yarnazaki, Hiroshi Nakane, and Akio Tanaka, "Basic analysis of a metal detector,"

in IEEE Instrumentation and Measurement (Technology Conference), 2001, pp. 474-477.

[6] R.M. Bozorth, Ferromagnetism, IEEE Press, Piscataway, NJ, 1978.

[7] M. J. Peters, J. G. Stinstra, and M. Hendriks, "Estimation of the electrical conductivity

of human tissue," Electromagn., vol. 21, pp. 545-557, 2001.

[8] K.V. Namjoshi, J. Douglas Lavers, and P.P. Biringer, "Eddy-cm-rent power loss in

toroidal cores with rectangular cross section," IEEE Trans. Magn., vol. 34, pp. 636-

641, 1998.

12
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