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
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
Sensors for Mechatronics. DOI: 10.1016/B978-0-12-391497-2.00006-6
2012 Elsevier Inc. All rights reserved.
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
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:
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:
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
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).
The magnetic quantities H and B are related by the equation
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)
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:
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.
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 e