In this case, the magnetic induction and the magnetic field vectors do not necessarily have the same direction. Some materials have anisotropic magnetic properties that make these two vectors point in different directions. The magnetization vector can consist of permanent and induced magnetization components. The permanent magnetization vector does not depend on the presence of an external field. The induced magnetization vector does depend on an external magnetic field and only exists while the inducing field is present.
Magnetic materials can be loosely classified as magnetically ―soft‖ or magnetically ―hard.‖ In a mag-netically hard material, the permanent magnetization component dominates (a magnet is an example). Magnetization in a soft magnetic material is largely induced and is described by the following equation:
r
r
M???H (48.5)
where ??is called the material’s magnetic susceptibility. In an isotropic material (magnetic properties are not direction dependent), ??is a scalar quantity, and the magnetization and field vectors are proportional and aligned. In anisotropic material (magnetic properties depend on direction), ??is a tensor represented by a 3 ??3 matrix; therefore, the magnetization vector magnitude and direction depend on the direction and strength of the inducing field. As a result, the magnetization vector will not always align with the magnetization inducing field vectors. Equation 48.5 can be modified for magnetically ―soft‖ material to the following:
r
B???0??1???H??0?H
r r
(48.6)
where ??is called the relative permeability of the material. ??????
A magnetized object with a magnetic moment m will experience torque T in the presence of a uniform???magnetic field H. Equation 48.7 expresses this relationship.
T??m ??H
r
r r
(48.7)
Torque is the cross-product of the magnetic moment and field vectors. The magnitude equation is:
T ??mH sin??
???
???
(48.8)
where ??is the angle between the direction of m and H .
There is an intimate relationship between electric and magnetic fields. Oersted discovered that passing a current through a wire near a compass causes the compass needle to rotate. The compass was the first magnetic field strength sensor. Faraday found that he could produce an electric voltage at the terminals of a loop of wire if he moved a magnet near it. This led to the induction or search coil sensor.
Magnetic fields are produced by the flow of electric charge (i.e., electric currents). In effect, a magnetic field is a velocity-transformed electric field (through a Lorentz transformation). Current flowing through a straight wire, a loop of wire, or a solenoid will also produce a magnetic field as illustrated in Figure 48.3. Units are always a problem when dealing with magnetic fields. The Gaussian cgs (centimeter, gram, and second) system of units was favored for many years. Since ?0 = 1 in the cgs system, magnetic field and flux density have the same numeric value in air, and their units (oerstedt for field and gauss for flux density) are often indiscriminately interchanged. This has led to great confusion. The cgs system has now been replaced by the International System of Units (SI). The SI system uses, among others, the meter (m), kilogram (kg), second (s) and ampere (A) as the fundamental units. Payne [4] gives a very good explanation of the differences between these systems of units as they relate to magnetic fields. Table 48.2 summarizes the relationships between the two systems of units. The SI system of units is used throughout this chapter.
? 1999 by CRC Press LLC
FIGURE 48.3 Magnetic fields are also produced by electric currents.
TABLE 48.2 Factors for Converting from cgs to SI Magnetic Field Units Description
Magnetic induction Magnetic field strength Magnetization
Magnetic dipole moment Magnetic flux
Magnetic pole strength Permeability of free space
Symbol
B H M m ??p
SI unit Tesla A m–1 A m–1 A m2
Weber (Wb) A m
??0 H m –1?
Gaussian cgs unit gauss
oerstedt (oe) emu m3 emu maxwell emu
–7
4 ?????10
Multiply by 104
4?????10–3 10–3 103 108
1
48.2 Low-Field Vector Magnetometers
The Induction Coil Magnetometer
The induction or search coil, which is one of the simplest magnetic field sensing devices, is based on Faraday’s law. This law states that if a loop of wire is subjected to a changing magnetic flux, ?, through the area enclosed by the loop, then a voltage will be induced in the loop that is proportional to the rate of change of the flux:
e?t???????d t???
d (48.9)
? 1999 by CRC Press LLC
FIGURE 48.4 Induction or search coil sensors consist of a loop of wire (or a solenoid), which may or may not surround a ferromagnetic core. (a) Air core loop antenna; (b) solenoid induction coil antenna with ferromagnetic core.
???
???
Since magnetic induction B is flux density, then a loop with cross-sectional area A will have a terminal
voltage: r r
d ??
????? B A e t ????????? dt?(48.10)
for spatially uniform magnetic induction fields.????????
Equation 48.10 states that a temporal change in B or the mechanical orientation of A relative to B???will produce a terminal voltage. If the coil remains fixed with respect to B, then static fields cannot be???detected; but if the loop is rotated or the magnitude of A is changed, then it is possible to measure a static field. The relationship described by Equation 48.10 is exploited in many magnetic field measuring instruments (see [1]).
Figure 48.4 shows the two most common induction coil configurations for measuring field strength: the air core loop antenna and the rod antenna. The operating principle is the same for both configurations. Substituting ?0?eH(t) for B in Equation 48.10 and, assuming the loop is stationary with respect to the field vector, the terminal voltage becomes:
dH ??t?? e t ??????????n?A?0 e dt
(48.11)
where n is the number of turns in the coil, and ?e is the effective relative permeability of the core. The core
of a rod antenna is normally constructed of magnetically ―soft‖ material so one can assume the flux density in the core is induced by an external magnetic field and, therefore, the substitution above is valid. With an air (no) core, the effective relative permeability is one. The effective permeability of an induction
? 1999 by CRC Press LLC
FIGURE 48.5 The induction coil equivalent circuit is a frequency-dependent voltage source in series with an inductor, resistor, and lumped capacitor.
coil that contains a core is usually much greater than one and is strongly dependent on the shape of the core and, to some extent, on the configuration of the winding.
Taking the Laplace transform of Equation 48.11 and dividing both sides by H, one obtains the following transfer function T(s) for an induction coil antenna:
T?s ???????nAs ?????Ks ??VmA ???1 ?
0 e
(48.12)
where E(s) = T(s) H(s), E(s) and H(s) are the Laplace transforms of e(t) and H(t), and s is the Laplace
transform operator. Inspection of Equation 48.12 reveals that the magnitude of the coil voltage is pro-portional to both the magnitude and frequency of the magnetic field being measured. The coil constant or sensitivity of the loop antenna is:
K??????nA ??
e
0
VsmA??1
?
(48.13)
Figure 48.5 is the equivalent circuit for an induction coil antenna. The actual voltage measured at the
terminals of the loop is modified by the inductance L, resistance R, and the distributed stray and shield capacitances represented by the lumped capacitor C. These circuit parameters depend on the geometry of the core, coil, and winding.
The electrostatic shield made of nonmagnetic material shown in Figure 48.4 is an important element in the design of an induction coil. It prevents coupling of electric fields into the coil, thereby assuring that the signal seen at the coil terminals is only that due to a magnetic field. The shield should not be placed too close to the winding since it contributes to coil capacitance and noise. The Air Core Loop Antenna
The air core loop antenna consists of a circular or rectangular loop containing one or more turns of wire and no magnetic core. The diameter of the loop is usually much greater than the dimensions of the winding cross-section. The sensitivity of a circular loop antenna with a winding inside diameter d and rectangular cross-section is approximately:
? 2 ?
??t ???? 3 ???t ??? ?? 4 1 2??
4?? ???????d ???????
????d????
??
where t is the thickness of the winding and n is the number of turns. The resistance of the coil is:
d?K??n???
2
0
(48.14)
d
R?4n
? 1999 by CRC Press LLC
???t ???
?????????1 ??d d2 w
?????
(48.15)
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