Magnetic - Field - Measurement

where dw is the diameter of the bare wire and ??is its resistivity in ??m (1.7 ??10–8???m for copper).

The inductance of the coil is more difficult to compute since it depends heavily on the geometry of the coil. Those who are interested in computing very accurate inductance values for a wide variety of coil shapes should consult [5]. Equation 48.16 is a general expression that gives a good approximation for the inductance of a circular air core coil.

2

??d k??L n ??

???????w

2

H (48.16)

where w is the width of the winding, d is the average diameter, and k is Nagaoka’s constant:

1 k

t d t 1 ??0.45 ??0.64 ??0.84

(48.17)

w d w

The distributed capacitance of the coil contributes the most to the overall antenna capacitance. The

parasitic capacitances can usually be ignored. Equation 48.18 can be used to estimate the distributed capacitance of a coil.

??????????????0.018544dw n????????w 1 ???1 Cd 2

n ???t ???t w 1 1 1 w ???

(48.18)

where ?w is the dielectric constant of the wire insulation, ?l is the dielectric constant of the interlayer

insulation if any, tw is the thickness of the wire insulation, tl is the thickness of the interlayer insulation, and nl is the number of layers. Adding a second layer to a single-layer coil significantly increases the capacitance but, as the number of layers increases, the capacitance decreases.

The air core loop antenna is particularly useful for measuring magnetic fields with frequencies from 100 Hz to several megahertz. Because it has a linear response to magnetic field strength, it has virtually no intermodulation distortion. On the negative side, the size of the sensor can get quite large for applications that require high sensitivities at low frequencies. The Rod Antenna

The rod antenna is a good alternative to an air core loop antenna. It is smaller in size than a loop antenna with the same sensitivity, and it can be designed to operate at lower frequencies. Unfortunately, its response to magnetic field strength can be nonlinear and the core adds noise.

Figure 48.4(b) is a typical configuration for a rod antenna. It is basically a solenoid with a magnetic core. The core can have a circular or rectangular cross-section and can be made from a ferrite, a nickel-iron alloy, an amorphous metal glass alloy, or some other material with high relative permeability. The winding can be wound directly on the core or on a form through which the core is inserted. Insulation is sometimes placed between layers of the winding to reduce distributed capacitance. An electrostatic shield is placed around the winding to attenuate any electric field coupling into the signal. The shield has a gap that runs the length of the winding. This prevents circulating currents in the shield from attenuating the magnetic field within the coil.

The most common rod antenna configuration is a core with a circular cross-section and a tightly coupled winding that runs most of the length of the core. The sensitivity of the rod antenna is computed by substituting ?e in Equation 48.13 with the following:

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??????

??1??dc e ?????????d?t ????

??

2

(48.19)

TABLE 48.3 Demagnetizing Factors, N for Rods and Ellipsoids Magnetized Parallel to Long Axis

Dimensional ratio (length/diameter)

0 1 2 5 10 20 50 100 200 500 1000 2000

Rod 1.0 0.27 0.14 0.040 0.0172 0.00617 0.00129 0.00036 0.000090 0.000014 0.0000036 0.0000009

Prolate ellipsoid 1.0 0.3333 0.1735 0.0558 0.0203 0.00675 0.00144 0.000430 0.000125 0.0000236 0.0000066 0.0000019

Oblate ellipsoid 1.0 0.3333 0.2364 0.1248 0.0696 0.0369 0.01472 0.00776 0.00390 0.001576 0.000784 0.000392

where dc is the core diameter and ?– is the core average effective permeability. The core effective or apparent permeability depends on its geometry and initial permeability, as well as the winding length relative to the core length. A rod becomes magnetized when a magnetic field is applied to it. In response, a magnetic field is created within the rod that opposes the externally applied field and reduces the flux density. The demagnetizing field is proportional to the magnetization and the net field H in the core is:

H??H????NM (48.20)

where H? is the applied external field, N is the demagnetizing factor, and M is the magnetization. The

apparent relative permeability of a core is the ratio of the flux density B in the middle of the core to the flux density in air:

B

a

?????????H ??????0

??i

1????????N 1 ?

i

(48.21)

where ?i is the initial relative permeability of the core material. Initial relative permeability is the slope

of the B–H magnetization curve near zero applied field for a closed magnetic path.

The value of N is shape dependent. As the length-to-diameter ratio m of a rod increases, N decreases and the apparent relative permeability approaches the initial permeability. Table 48.3, which is reproduced from [6], lists demagnetizing factors for a rod, prolate ellipsoid (cigar shape), and oblate ellipsoid (disk shape).

Equation 48.22 can be used to approximate the value of N for cylindrical rods with m > 10 and ?i > 1000:

2 01 ??.

N?

10 ??0 46 log m . m 2

(48.22)

The apparent permeability of a rod with a small m and large ?i is almost exclusively determined by m alone. Table 48.4 lists the magnetic properties of several ferromagnetic materials that can be used to construct a core.

Bozorth [7] found that the apparent permeability of a rod is not constant throughout the length of the rod. It reaches a maximum at the center of the rod and continuously drops in value until the ends of the rod are reached. The variation in permeability can be approximated by:

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TABLE 48.4 Magnetic Properties of Typical Core Material Name Mild steel Silicon iron CN20 MN60 ―49‖ Alloy 2605S-2

4-79 Permalloy Mumetal HyMu ―80‖ 2826MB

Composition

0.2 C, 99 Fe 4.0 Si, 96 Fe Ni-Zn Ferrite Mn-Zn Ferrite 48 Ni, 52 Fe

Fe-based amorphous alloy 4 Mn, 79 Ni, 17 Fe

5 Cu, 2 Cr, 77 Ni, 16 Fe 4.2 Mo, 80 Ni, 15 Fe

NiFe-based amorphous alloy

Manufacturer

?i?120 500 800 5000 6500 10,000 20,000 20,000 50,000 100,000

?max?2000 7000 4500 10,500 75,000 600,000 100,000 100,000 200,000 800,000

Ceramic Magnetics Ceramic Magnetics Carpenter Allied-Signal Magnetics Magnetics Carpenter Allied-Signal

Note: ?i is the slope of the magnetization curve at the origin. ?max is the maximum incremental slope of the magnetization curve.

??2????????????????????(48.23)

l 1 a??F l ??

????????l?????

??????

where l is the distance from the center of the rod to the measurement point, l0 is the half length of the rod, and F is a constant that varies from 0.72 to 0.96. The average permeability seen by the coil is the integral of Equation 48.23 over the length of the coil:

2 ???????????????????w ??

??1 F l ?(48.24)

??a ??????? l c ???

???

where lw is the length of the winding and lc is the length of the core. Equation 48.24 is substituted into Equation 48.19 to compute the rod’s effective relative permeability which is used in Equation 48.13 to compute sensitivity.

The inductance of the rod antenna can be computed using the following equations:

e

L ??

?0?????????????n d t l 2 w 4lc

2

2

(48.25)

??????

??1??dc

????????????e ??d?t ????f l l

1

(48.26)

3

f lw lc ??.

??1 90880 8672 ?1 1217 0 8263??w c. ??l l . ??w c???l l ??w c???. l l? ??2

(48.27)

The function f(lw/lc) accounts for the variation in flux density from the middle of the winding to its ends

and assumes the winding is centered about the middle of the core.

Equations 48.15 and 48.16 can be used to compute the resistance and capacitance of a rod antenna. Signal Conditioning

To be useful, the induction coil signal must be conditioned using either a voltage or a current amplifier. Figure 48.6 illustrates the circuit configurations for both of these signal conditioning methods. The voltage

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FIGURE 48.6 (a) The amplitude of a voltage-amplified induction coil signal is proportional to the frequency and strength of the field. (b) The amplitude of a current-amplified induction coil signal is only proportional to field strength beyond its L/R corner frequency.

amplifier can have either a single-ended or differential input and it can be tuned or untuned. The signal output of the voltage amplifier is proportional to the magnitude and frequency of the field for frequencies well below resonance. Its output will peak at the resonant frequency of the coil or at the tuning frequency. Because its output signal depends on both the frequency and strength of the magnetic field, the voltage amplifier is more suited to narrow band or tuned frequency applications.

In the current amplifier configuration, the induction coil terminals are connected to a virtual ground. As long as the product of the amplifier forward gain and the coil ohmic resistance is much greater than the feedback resistor, the output signal magnitude is independent of the frequency of the magnetic field beyond the R/L (rad s–1) corner of the coil. This remains true up to the coil’s resonant frequency. For this reason, the current amplifier configuration is particularly suited to broadband magnetic field strength measurements. The current amplifier configuration minimizes intermodulation distortion in induction coils with magnetic cores. The current flowing through the coil produces a magnetic field that opposes the ambient field. This keeps the net field in the core near zero and in a linear region of the B–H curve.

Current-amplifier-based induction coil magnetometers have been built that have a flat frequency response from 10 Hz to over 200 kHz. Some magnetometers designed for geophysical exploration applications have low frequency corners that extend down to 0.1 Hz. For further information on this subject, see [8, 9].

The Fluxgate Magnetometer

The fluxgate magnetometer has been and is the workhorse of magnetic field strength instruments both on Earth and in space. It is rugged, reliable, physically small, and requires very little power to operate. These characteristics, along with its ability to measure the vector components of magnetic fields over a

? 1999 by CRC Press LLC