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6 Inductive and Magnetic Sensors

Inductive sensors employ variables and parameters like magnetic induction B, mag-

netic flux , self-inductance L, mutual inductance M or magnetic resistance Rm. Bya particular construction of the device, these quantities are made dependent on an

applied displacement or force. First we review various magnetic quantities and

their relations. Next the operation and the specifications of the major types of mag-

netic and inductive sensors are reviewed. Special attention is given to transformer-

type sensors. The chapter concludes with a section on applications.

6.1 Magnetic and Electromagnetic Quantities

6.1.1 Magnetic Field Strength, Magnetic Induction and Flux

The magnetic field strength H, generated by a flow of charged particles, is defined

according to

I5

IC

HUdl 6:1

where I is the current passing through a closed contour C (Figure 6.1A). The quan-

tities H (A/m) and dl (m) are vectors. For each configuration of conductors carrying

an electric current the field strength in any point of the surrounding space can be

calculated by solving the integral equation (6.1). A current I through a long,

straight wire produces a magnetic field with strength H5 I/2r at a distance r fromthe wire. So the field strength is inversely proportional to the distance from the

wire. The field lines form concentric circles around the wire so the vector direction

is tangent to these circles (Figure 6.1B).

Only for structures with a strong symmetry, simple analytical solutions can be

obtained. Magnetic fields of practical devices and shapes are studied using FEM

(finite element method). The space is subdivided in small (triangular) areas and for

each area the equations are solved numerically. FEM programmes calculate the

field strength and direction over the entire region of interest; the results are visual-

ized in colour or grey-tone pictures or with field lines.

As an illustration Figure 6.2 shows the magnetic field of a permanent magnet

and of a wire loop with a DC current. The FEM programme also shows the numeri-

cal values of the field quantity in each point of the space enclosed by a specified

boundary.

Sensors for Mechatronics. DOI: 10.1016/B978-0-12-391497-2.00006-6

2012 Elsevier Inc. All rights reserved.

http://dx.doi.org/10.1016/B978-0-12-391497-2.00006-6

Evidently a stronger field strength can be obtained by increasing the current.

However a more efficient method is to make multiple turns of the wire. Each turn

carries the current I, thus contributing to the field strength. For example the field

inside a coil with n turns is proportional to the product of n and I. The product n I(expressed as ampere-turns) is a measure for the strength of such a magnetic

source.

Other quantities that describe magnetic and induction phenomena are the mag-

netic induction B (unit Tesla, T, kg/As2) and the magnetic flux (unit Weber,Wb5 kg m2/As25Tm2). By definition the flux is:

5ZZ

S

BUdA 6:2

In words the flux is the inner vector product of the magnetic induction vector

and a surface patch dA, integrated over the total area S for which the flux is

Figure 6.2 Examples of magnetic fields obtained by FEM analysis: (A) a permanent magnet

and (B) a single turn with DC current 1 A.

Figure 6.1 (A) Magnetic field generated by current I, (B) calculation of field strength due to

a straight wire carrying a current I and (C) calculation of magnetic flux.

126 Sensors for Mechatronics

calculated (Figure 6.1C). In particular for a homogeneous field that makes an angle

with the normal on a flat surface A, the flux is:

5BUAUcos 6:3

When the surface is in parallel to the field (5/2), there is no flux throughthat surface. The flux is maximal through a surface perpendicular to the field

(5 0).The magnetic field strength of (permanent) magnets is expressed in terms of

magnetic induction B (so in Tesla) rather than of H. For instance the Earths mag-

netic field strength is about 60 T, and the strengths of permanent magnets rangefrom 0.01 to 1 T.

Free charges moving in a magnetic field experience a Lorentz force, driving

them into a direction according to the well-known right-hand rule:

Fl 5 qv3B 6:4

This happens also with the free electrons in a conductor that moves in a mag-

netic field. The movement results in a potential difference, the induction voltage,

across the conductor and satisfies the equation

Vind 52ddt

6:5

which is the induction law of Faraday. In an open wire loop, moving in a magnetic

field, the induction voltage appears between both ends of the wire; the current

through the wire is zero. In a closed loop, the induced voltage causes a current

equal to Vind/R, where R is the resistance of the wire loop. From Eq. (6.5) it follows

that the induced voltage differs from zero only when the flux changes with time.

At constant flux, the induced voltage is zero. The definition of flux in Eq. (6.5)

links the units volt (V) and Weber (Wb).

6.1.2 Permeability

The magnetic quantities H and B are related by the equation

B5H50rH 6:6

The quantity is the (magnetic) permeability; 0, the permeability of freespace, equals 4 1027 Vs/Am by definition. The relative permeability r is a mate-rial property (compare r for a dielectric material). For vacuum r5 1, for gasesand many nonferrous materials it is very close to 1. The permeability of ferromag-

netic materials is much higher, but strongly non-linear; at higher values of H the

material shows saturation and hysteresis. Table 6.1 shows some values of the static

permeability for several materials, as well as values for the saturation induction.

127Inductive and Magnetic Sensors

Mumetal is often used for shielding system parts that are sensitive to magnetic

fields. The non-linearity of r is employed in, for instance, fluxgate sensors(Section 6.2.3).

6.1.3 Eddy Currents

Any conductor in a non-stationary induction field experiences induction voltages.

This holds not only for wires (where it is used for the generation of electric cur-

rents) but also for bulk material (as the iron cores of transformers and electric

machines). The induced currents through such material follow more or less circular

paths; therefore, they are called eddy currents. They produce unwanted heat, so

normally they are minimized by, for instance, increasing the resistance of the bulk

material. In constructions with a lot of iron (e.g. transformers and electric

machines), this is accomplished by laminating the material: instead of massive

material the construction is built up of a pile of thin iron plates (lamellae) packed

firmly together. Eddy current can only flow in the plane of the plates, but cannot

cross the boundary between two adjacent plates. A useful application of eddy cur-

rents for sensors is described in Section 6.3.3, the eddy current proximity sensor.

6.1.4 Magnetic Resistance (Reluctance) and Self-Inductance

The analogy between the description of magnetic circuits and electrical circuits is

demonstrated by the equations in Table 6.2.

Equation (6.7) links the intrinsic and extrinsic field variables in the electrical

and magnetic domain, respectively (Chapter 2). The electrical conductivity 5 1/opposes the magnetic permeability . With Eq. (6.8) the field quantities E and Hare converted to the circuit quantities V and I. Equation (6.9) defines density prop-

erties, and Eq. (6.10) defines electrical and magnetic resistances, respectively. The

latter is also called reluctance. Finally, Eq. (6.11) expresses the electric and

magnetic resistances in terms of material properties and shape parameters: l is the

length of a device with constant cross section and A its cross-section area.

In an electric circuit consisting of a series of elements, the current through each

of these elements is the same. Analogously the flux through a series of magnetic

Table 6.1 Permeability of Various Construction Materials [1,2]

Material r (max) Bsat (T)

Vacuum 1

Pure iron 5,000 2.2

Transformer steel 15,000 2.0

Mumetal (Fe17Ni56Cu5Cr2) 100,000 0.9

Supermalloy (Fe16Ni79Mo5) 1,000000 0.8

128 Sensors for Mechatronics

elements is the same. So the resistances (reluctances) of these elements can simply

be summed to find the total reluctance of the series circuit.

The self-inductance of a magnetic circuit with coupled flux is found as follows.

The induced voltage equals Vind 5 nUd=dt (when there are n turns). Substitutionof using Eq. (6.9) yields Vind 5 n2=RmUdI=dt and since V5 L(dI/dt) the self-inductance is:

L5n2

Rm5 n2U

Al

6:12

So the coefficient of self-inductance (unit Henry, H5Wb/A) is proportional tothe square of the number of turns and inversely proportional to the reluctance.

Several sensors, based on a change in self-inductance and reluctance, will be fur-

ther discussed in this chapter.

6.1.5 Magnetostriction

All ferromagnetic materials exhibit the magnetostrictive effect. Basically it is the

change in outer dimensions of the material when subjected to an external magnetic

field. In the absence of an external field the magnetic domains (elementary mag-

netic dipoles) are randomly oriented. When a magnetic field is applied, these

domains tend to line up with the field, up to the point of saturation. The effect is

not strong: materials with a large magnetostriction (for instance Terfenol-D) show

Table 6.2 Comparison Between the Electrical and the Magnetic Domain

Electrical Domain Magnetic Domain

E51

UJ H5

1

UB 6:7

V 5

EUdl

nUI5HUdl 6:8

I5

ZZJUdA

5ZZ

BUdA 6:9

V 5ReUI nUI5RmU 6:10

Re 51

Ul

A Rm 51

Ul

A6:11

129Inductive and Magnetic Sensors

a sensitivity of about 5 strain per kA/m, with a maximum strain between 1200 and1600 strain at saturation [3].

The inverse magnetostrictive effect is called the Villari effect: a change in mag-

netization when the material is stressed. This effect is used in magnetostrictive

force sensors, discussed in Section 6.3.6.

6.2 Magnetic Field Sensors

This section presents various sensors for the measurement of magnetic field

strength or magnetic induction. In most mechatronic applications the magnetic field

is not the primary measurement: combined with a magnetic source (e.g. permanent

magnet and coil) they are used to measure displacement quantities and (with an

elastic element) force quantities. Such sensors are discussed in Section 6.3. The

sensors described in this section are: coil, Hall sensors, fluxgate sensors and mag-

netostrictive sensors. One of the most sensitive magnetic field sensors is the

Superconducting QUantum Interference Device (SQUID). This sensor operates at

cryogenic temperature (liquid helium or liquid nitrogen) and is used mainly in med-

ical applications and for material research. They are rarely used in mechatronics.

Most research on magnetic field sensors is focussed on Hall sensors and fluxgate

sensors, in particular to reduce dimensions and fabrication costs, by applying

MEMS technology and integration with interface electronics (see further

Section 6.4). Sometimes innovative concepts are introduced, but the application to

mechatronic systems requires further development. An example of such a new prin-

ciple is given in Ref. [4]. The sensor proposed herein consists of two thin flexible

plates (cantilevers); when magnetized by an external field, the repulsive force

causes a displacement of one of the plates relative to the other, similar to the clas-

sic gold-leaf electrometer. In force equilibrium the displacement is a measure for

the external magnetic field.

Combining Eqs (6.3) and (6.6) gives for the flux through a magnetic circuit with

n windings and area A

t5 nUAtUtUHt 6:13

where all parameters may vary with time due to a time-varying quantity. A mag-

netic or inductive sensor can be based on a change in each of these parameters,

resulting in an induction voltage equal to

Vt5 dtdt

5 nU H@At@t

1AH@t@t

1A@Ht@t

6:14

The remainder of this section deals with three types of magnetic field sensors:

the coil, Hall sensors, fluxgate sensors and magnetostrictive sensors. Their principle

of operation and performance are essentially based on this equation.

130 Sensors for Mechatronics

6.2.1 Coil

The most straightforward method for the transduction from magnetic field to an

electric voltage is a coil: Eq. (6.5) relates the induced voltage in a coil to the mag-

netic flux. At first sight, only AC fields can be measured in this way since the

induced voltage is proportional to the rate of change in flux. Static fields can never-

theless be measured, just by rotating the coil. Let the surface area of the coil be A

and the frequency of rotation , then for a homogeneous induction field B, theinduced voltage equals:

Vt52BUAU d sin tdt

52BUAUUcos t 6:15

With a rotating coil very small induction fields can be measured. Disadvantages

of the method are movable parts, the need for brushes to make electrical connection

to the rotating coil and for an actuator to procure rotation.

6.2.2 Hall Plate Sensors

The Hall plate is based on the magnetoresistive effect. In 1856 W. Thomson (Lord

Kelvin) discovered that a magnetic field influences the resistivity of a current-

conducting wire (see also Section 4.5 on magnetoresistive sensors). Later this effect

was named the Gauss effect. Only after the discovery of the Hall effect in 1879, by

the American physicist E.F. Hall, could the Gauss effect be explained. Both the

Gauss and the Hall effects are remarkably stronger in semiconductors, so they

became important for measurement science only after the development of semicon-

ductor technology.

The Hall effect is caused by the Lorentz forces on moving charge carriers in a

solid conductor or semiconductor, when placed in a magnetic field (Figure 6.3).

The force Fl on a particle with charge q and velocity v equals:

Fl 5 qv3B 6:16

The direction of this force is perpendicular to both B and v (right-hand rule). As

a result the flow of charges in the material is deflected and an electric field E is

built up, perpendicular to both I and B. The charge carriers experience an electric

Figure 6.3 Principle of the Hall sensor.

131Inductive and Magnetic Sensors

force Fe5 qE that, in the steady state, counterbalances the Lorentz force: Fe5Fl.Hence:

E5 v3B 6:17

Assuming all charge carriers have the same velocity, the current density J equals

n q v with n the particle density. When B is homogeneous and perpendicular to v(as in Figure 6.3), the electric field equals simply E5 JB/nq. Finally withI5 b d J and V5E b, the voltage across the Hall sensor becomes:

V 51

nqUIB

d5RHU

IB

d6:18

The factor 1/nq is called the Hall coefficient, symbolized by RH. In p-type semi-

conductors holes are majority carriers, so q is positive. Obviously the Hall voltage

is inversely proportional to the thickness d. A Hall sensor, therefore, has often the

shape of a plate (as in Figure 6.3), explaining the name Hall plate for this type of

sensor. The Hall voltage has a polarity as indicated in Figure 6.3. For n-type semi-

conductors, the charge carriers (electrons) are negative thus the polarity is just the

inverse. The Hall coefficient...