actuator and sensor motion proximity sensors

Actuators & Sensors in Mechatronics:
Analog Sensors for Motion
K. Craig
1
• Introduction
– Importance of Accurate Measurement in Control Systems
– Sensitivity Analysis
• Motion Transducers
– General Discussion
• Types of Motion Transducers
– Potentiometers (resistively-coupled transducer)
– Variable-Inductance Transducers
– Eddy-Current Transducers
– Variable-Capacitance Transducers
– Piezoelectric Transducers

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2
Introduction
• Measurement of plant outputs is essential for
feedback and feedforward control, as well as for
performance evaluation of a process.
• The measurement subsystem in a control system
contains sensors and transducers that detect
measurands and convert them into acceptable
signals – typically voltages. These voltages are
then appropriately modified using signal-
conditioning hardware such as filters, amplifiers,
demodulators, and A/D converters.

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• Impedance matching might be necessary to
connect sensors and transducers to signal-
conditioning hardware.
• Accuracy of sensors, transducers, and associated
signal-conditioning devices is important in control
system applications for two main reasons:
– The measurement system in a feedback control system
is situated in the feedback path of the control system.
Even though measurements are used to compensate for
the poor performance in the open-loop system, any
errors in measurements themselves will enter directly
into the system and cannot be corrected if they are
unknown.

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– Furthermore, it can be shown that sensitivity of a
control system to parameter changes in the
measurement system is direct. This sensitivity cannot
be reduced by increasing the loop gain, unlike the case
of sensitivity to the open-loop components.
• Accordingly, the design strategy for closed-loop
control is to make the measurements very accurate
and to employ a suitable controller to reduce other
types of errors.
• Most sensor-transducer devices used in feedback
control applications are analog components that
generate analog output signals, that then require
A/D conversion to obtain a digital representation
of the measured signal.

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Sensitivity Analysis
• Consider the function y = f(x). If the parameter x changes
by an amount ∆x, then y changes by the amount ∆y. If ∆x
is small, ∆y can be estimated from the slope dy/dx as
follows:
• The relative or percent change in y is ∆y/y. It is related to
the relative change in x as follows:
dy
y x
dx
∆ = ∆
y dy x x dy x
y dx y y dx x
 
∆ ∆ ∆
= =  
 

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• The sensitivity of y with respect to changes in x is given
by:
• Thus
• Usually the sensitivity is not constant. For example, the
function y = sin(x) has the sensitivity function:
y
x
x dy dy/ y d(ln y)
S
y dx dx / x d(ln x)
= = =
y
x
y x
S
y x
∆ ∆
=
( )
( )
( ) ( )
y
x
xcos x
x dy x x
S cos x
y dx y sin x tan x
= = = =

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• Sensitivity of Control Systems to Parameter
Variation and Parameter Uncertainty
– A process, represented by the transfer function G(s), is subject to a
changing environment, aging, ignorance of the exact values of the
process parameters, and other natural factors that affect a control
process.
– In the open-loop system, all these errors and changes result in a
changing and inaccurate output.
– However, a closed-loop system senses the change in the output due
to the process changes and attempts to correct the output.
– The sensitivity of a control system to parameter variations is of
prime importance.
– Accuracy of a measurement system is affected by parameter
changes in the control system components and by the influence of
external disturbances.

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– A primary advantage of a closed-loop feedback control system is
its ability to reduce the system’s sensitivity.
– Consider the closed-loop system shown. Let the disturbance D(s)
= 0.
– An open-loop system’s block diagram is given by:
+
-
Σ
C(s)
R(s)
Gc
(s) G(s)
H(s)
E(s)
B(s)
D(s)
Σ
+
+
C(s)
R(s)
Gc
(s) G(s)

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– The system sensitivity is defined as the ratio of the percentage
change in the system transfer function T(s) to the percentage
change in the process transfer function G(s) (or parameter) for a
small incremental change:
– For the open-loop system
T
G
C(s)
T(s)
R(s)
T/T T G
S
G /G G T
=
∂ ∂
= =
∂ ∂
c
T
G c
c
C(s)
T(s) G (s)G(s)
R(s)
T/T T G G(s)
S G (s) 1
G /G G T G (s)G(s)
= =
∂ ∂
= = = =
∂ ∂

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– For the closed-loop system
– The sensitivity of the system may be reduced below that of the
open-loop system by increasing GcGH(s) over the frequency range
of interest.
– Similarly
( )
c
c
T
G
c
2
c
c c
c
C(s) G (s)G(s)
T(s)
R(s) 1 G (s)G(s)H(s)
T/T T G
S
G /G G T
G G 1
G G
(1 G GH) 1 G GH
1 G GH
= =
+
∂ ∂
= =
∂ ∂
= =
+ +
+
( )
c
T c
G
c c c c
T/T T G 1
S
G /G G T 1 G GH
∂ ∂
= = =
∂ ∂ +

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( )
c
c
T
H
2
c c
2
c
c c
c
C(s) G (s)G(s)
T(s)
R(s) 1 G (s)G(s)H(s)
T/T T H
S
H/H H T
(G G) H G GH
G G
(1 G GH) 1 G GH
1 G GH
= =
+
∂ ∂
= =
∂ ∂
− −
= =
+ +
+
– The sensitivity of the closed-loop system to changes in the feedback
element H(s) is:
– When GcGH is large, the sensitivity approaches unity and the
changes in H(s) directly affect the output response. Use feedback
components that will not vary with environmental changes or can be
maintained constant.

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( )
( )
( ) ( ) ( )
( )
c
G s
C s D s
1 G s G s H s
=
+
– As the gain of the loop (GcGH) is increased, the sensitivity of the
control system to changes in the plant and controller decreases, but
the sensitivity to changes in the feedback system (measurement
system) becomes -1.
– Also the effect of the disturbance input can be reduced by
increasing the gain GcH since:
• Therefore:
– Make the measurement system very accurate and stable.
– Increase the loop gain to reduce sensitivity of the control system to
changes in plant and controller.
– Increase gain GcH to reduce the influence of external disturbances.

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• In practice:
– G is usually fixed and cannot be altered.
– H is essentially fixed once an accurate measurement system is
chosen.
– Most of the design freedom is available with respect to Gc only.
• It is virtually impossible to achieve all the design
requirements simply by increasing the gain of Gc. The
dynamics of Gc also have to be properly designed in order
to obtain the desired performance of the control system.

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Motion Transducers
• By motion we mean the four kinematic variables:
– Displacement (including position, distance, proximity,
and size or gage)
– Velocity
– Acceleration
– Jerk
• Each variable is the time derivative of the
preceding one.
• Motion measurements are extremely useful in
controlling mechanical responses and interactions
in dynamic systems.

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• A one-to-one relationship may not always exist
between a measuring device and a measured
variable.
– For example, although strain gages are devices that
measure strains (and, hence, stresses and forces), they
can be adapted to measure displacements by using a
suitable front-end auxiliary sensor element, such as a
cantilever or spring.
• Furthermore, the same measuring device may be
used to measure different variables through
appropriate data-interpretation techniques. For
example:

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16
– Resolver signals that provide angular displacements are
differentiated to get angular velocities.
– Optical encoders can serve as both displacement and
velocity transducers depending on whether the number
of pulses generated is counted or the pulse rate is
measured (either by counting the number of pulses
during a unit interval of time or by gating a high-
frequency clock signal through the pulse width).
– In principle, any force sensor can be used as an
acceleration sensor, velocity sensor, or displacement
sensor, depending on whether an inertia element
(converting acceleration into force), a damping element
(converting velocity into force), or a spring element
(converting displacement into force), respectively, is
used as the front-end auxiliary sensor.

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• Why do we need separate transducers to measure
the four kinematic variables because any one is
related to any other through integration or
differentiation? It should be possible, in theory, to
measure only one of these four variables and use
either analog or digital processing to obtain any of
the remaining motion variables.
• The feasibility of this approach is highly limited
and depends crucially on several factors:
– The nature of the measured signal (e.g., steady, highly
transient, periodic, narrow-band, broad-band)
– The required frequency content of the processed signal
(or the frequency range of interest)

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– The signal-to-noise ratio (SNR) of the measurement
– Available processing capabilities (e.g., analog or digital
processing, limitations of the digital processor, and
interface, such as the speed of processing, sampling
rate, and buffer size)
– Controller requirements and the nature of the plant
(e.g., time constants, delays, hardware limitations)
– Required accuracy in the end objective (on which
processing requirements and hardware costs will
depend)
• For example, differentiation of a signal (in the
time domain) is often unacceptable for noisy and
high-frequency, narrow-band signals.

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19
• Rules of Thumb:
– In low-frequency applications (on the order of 1 Hz),
displacement measurements generally provide good
accuracies.
– In intermediate-frequency applications (less than 1
kHz), velocity measurement is usually favored.
– In measuring high-frequency motions with high noise
levels, acceleration measurement is preferred.
– Jerk is particularly useful in ground transit (ride
quality), manufacturing (forging, rolling, and similar
impact-type operations), and shock isolation ( delicate
and sensitive equipment) applications.

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Typical Specifications for Analog Motion Transducers

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Potentiometers
• The potentiometer is a displacement transducer.
– It is an active transducer that consists of a uniform coil
of wire or a film of high-resistive material (e.g., carbon,
platinum, conductive plastic) whose resistance is
proportional to its length.
– A fixed voltage vref is applied across the coil or film
using an external, constant DC voltage supply.
– The transducer output signal vo is the DC voltage
between the moving contact (wiper arm) sliding on the
coil and one terminal of the coil.
– Slider displacement is proportional to the output
voltage, which assumes that the output terminals are
open-circuit: o
v kx
=

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– When we assume that the output terminals are open-
circuit, we are assuming an infinite-impedance load (or
resistance in the present DC case) present at the output
terminals, so that the output current is zero.
– In actual practice, the load (the circuitry into which the
potentiometer is fed) has a finite impedance and so the
output current (through the load) is nonzero.
– The output voltage thus drops, even if the reference
voltage vref is assumed to remain constant under load
variations (i.e., the voltage source has zero output
impedance).
– This consequence is known as the loading effect of the
transducer and the linear relationship is no longer valid.
An error in the displacement reading results.

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– Loading can effect the transducer reading in two ways:
• By changing the reference voltage, i.e., loading the voltage
source
• By loading the transducer
– To reduce these effects, one needs:
• A voltage source that is not seriously affected by load
variations (i.e., a power supply with a low output impedance)
• Data-acquisition and signal-conditioning circuitry that has a
high input impedance
– Remember: A perfect measuring device should have
the following dynamic characteristics:
• Output instantly reaches the measured value (fast response)
• Transducer output is sufficiently large (high gain or low output
impedance)

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• Output remains at the measured value (without drifting or
being affected by environmental effects and other undesirable
disturbances and noise) unless the measurand itself changes
(stability)
• The output signal level of the transducer varies in proportion to
the signal level of the measurand (static linearity)
• Connection of the measuring device does not distort the
measurand itself (loading effects are absent and impedances
are matched)
• Power consumption is small (high input impedance)

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– Choose resistance of a potentiometer with care.
• High resistance is preferred as this results in reduced power
dissipation for a given voltage which also results in reduced
thermal effects.
• However, increased resistance increases the output impedance
of the potentiometer and results in loading nonlinearity error
unless the load resistance is also increased proportionately.
• Low-resistance potentiometers have resistances less than 10 Ω.
• High-resistance potentiometers can have resistances on the
order of 100 kΩ.
• Conductive plastics can provide high resistances (e.g., 100 Ω
per mm) and have reduced friction, reduced wear, reduced
weight, and increased resolution.
– Potentiometers that measure angular displacements are
more common and convenient than rectilinear
potentiometers.

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Schematic Diagram of a
Potentiometer
Potentiometer Loading

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Practical Potentiometer
Configurations
Rectilinear Motions Angular Motions

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C
max
R R
θ
θ
=
θ
– Consider a rotary potentiometer and let’s discuss the
significance of the loading nonlinearity error caused by
a purely resistive load connected to the potentiometer.
• For a general position θ of the potentiometer slider arm,
suppose that the output segment of the coil is Rθ. Assuming a
uniform coil:
• RC is the total resistance of the potentiometer coil.
• Current balance at the sliding contact point gives:
• RL is the load resistance.
ref o o o
C L
v v v v
R R R R
θ θ
−
= +
−

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• Combining equations results in:
• Loading error appears to be high for low values of the RL/RC
ratio. Good accuracy is possible for RL/RC > 10, particularly
for small values of θ/θmax.
• Hence to reduce loading error in potentiometers: (1) Increase
RL/RC (increase load impedance, reduce coil impedance); and
(2) Use potentiometers to measure small values of θ/θmax (or
calibrate only a small segment of the element for linear
reading).
( )
( )( )
( ) ( ) ( )
ref o o o
max max L C
max L C
o
2
ref L C max max
v v v v
1 / / R / R
/ R / R
v
v R / R / /
−
= +
− θ θ θ θ
θ θ
=
+ θ θ − θ θ

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• The loading nonlinearity error is defined by:
– Using only a segment of the resistance element as the
range of the potentiometer is similar to adding two end
resistors to the elements. It is known that this tends to
linearize the potentiometer.
– If the load resistance is known to be small, a voltage
follower may be used at the potentiometer output to
virtually eliminate loading error. This arrangement
provides a high load impedance to the potentiometer
and a low impedance at the output of the amplifier.
( ) ( )
o ref max
max
v / v /
e 100%
/
− θ θ
=
θ θ

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– Three obvious disadvantages of this resistively-coupled
transducer:
• The force required to move the slider arm comes from the
motion source, and the resulting energy is dissipated through
friction. This energy conversion involves relatively high
forces and the energy is wasted rather than being converted
into the output signal of the transducer.
• The electrical energy from the reference source is dissipated
through the resistor coil (or film) resulting in an undesirable
temperature rise.
• Finite resolution in coil-type potentiometers, where resolution
is determined by the number of turns in the coil. Infinitesimal
resolutions are now possible with high-quality resistive-film
potentiometers that use conductive plastics. In this case,
resolution is limited by other factors (e.g., mechanical
limitations and signal-to-noise ratio).

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32
A Rotary Potentiometer with a Resistive Load

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Loading Nonlinearity in a Potentiometer

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Loading Nonlinearity in a Potentiometer
Loading Nonlinearity Error in Potentiometer
For θ / θmax = 0.5
-2.4%
10.0
-20%
1.0
-71.4%
0.1
Loading Nonlinearity
Error e
Load Resistance Ratio
RL / RC

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– Limitations of Potentiometers as Displacement-
Measuring Devices:
• The force needed to move the slider (against friction and arm
inertia) is provided by the displacement source. The
mechanical loading distorts the measured signal itself.
• High-frequency (or highly transient) measurements are not
feasible because of such factors as slider bounce, friction and
inertia resistance, and induced voltages in the wiper arm and
primary coil.
• Variations in the supply voltage cause error.
• Electrical loading error can be significant when the load
resistance is low.
• Resolution is limited by the number of turns in the coil and by
the coil uniformity. This limits small displacement
measurements.

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• Wearout and heating up (with associated oxidation) in the coil
(film) and slider contact cause accelerated degradation.
– Advantages Associated with Potentiometer Devices:
• They are relatively less costly.
• Potentiometers provide high-voltage (low impedance) output
signals, requiring no amplification in most applications.
Transducer impedance can be varied simply by changing the coil
resistance and supply voltage.
– Although potentiometers are primarily displacement
transducers, they can be adapted to measure other types of
signals, such as pressure and force, using appropriate
auxiliary sensor (front-end) elements.

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• Problem
– A potentiometer circuit with element resistance RC and
equal end resistors Re is shown. Derive the necessary
input/output relations. Show that the end resistors can
produce a linearizing effect in the potentiometer. At
half the maximum reading of the potentiometer,
calculate the percentage loading error for the three
values of the resistance ratio RC/Re = 0.1, 1.0, 10.0,
assuming that the load resistance RL is equal to the
element resistance. Compare the results with the
corresponding value for Re = 0. Finally, choose a
suitable value for RC/Re and plot the curve of
percentage loading error versus fractional displacement
x/xmax. From the graph, estimate the maximum loading
error.

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38
Potentiometer Circuit with End Resistors

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Variable-Inductance Transducers
• Motion transducers that employ the principle of
electromagnetic induction are termed variable-
inductance transducers.
– When the flux linkage through an electrical conductor
changes, a voltage is induced in the conductor. This, in
turn, generates a magnetic field that opposes the primary
field. Hence, a mechanical force is necessary to sustain
the change of flux linkage. If the change in flux linkage
is brought about by a relative motion, the mechanical
energy is directly converted into electrical energy. This
is the basis of electromagnetic induction and the
principle of operation of variable-inductance transducers.

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– In these devices, the change of flux linkage is caused
by a mechanical motion and the mechanical-to-
electrical energy transfer takes place under near-ideal
conditions.
– The induced voltage or change in inductance may be
used as a measure of the motion. Variable-inductance
transducers are generally electromechanical devices
coupled by a magnetic field.
– There are many different types of variable-inductance
transducers. Three primary types can be identified:
• Mutual-Induction transducers
• Self-induction transducers
• Permanent-magnet transducers

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– Variable-inductance transducers that use a non-
magnetized ferromagnetic medium to alter the
reluctance (magnetic resistance) of the flux path are
known as variable-reluctance transducers. Some of the
mutual-inductance transducers and most of the self-
inductance transducers are of this type. Permanent-
magnet transducers are not considered variable-
reluctance transducers.

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42
• Mutual-Induction Transducers and Differential
Transformers
– The basic arrangement of a mutual-induction transducer
constitutes two coils, the primary winding and the
secondary winding.
– The primary winding carries an AC excitation that
induces a steady AC voltage in the secondary winding.
– The level of the induced voltage depends on the flux
linkage between the coils.
– In mutual-induction transducers, a change in the flux
linkage is effected by one of two common techniques.
• One technique is to move an object made of ferromagnetic
material within the flux path. This changes the reluctance of
the flux path, with an associated change of the flux linkage in
the secondary coil.

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43
• This is the operating principle of the linear variable differential
transformer (LVDT), the rotary variable differential
transformer (RVDT), and the mutual-induction proximity
probe. All of these are in fact variable-reluctance transducers.
• The other common way to change the flux linkage is to move
one coil with respect to the other. This is the operating
principle of the resolver. This is not a variable-reluctance
transducer.
• The motion can be measured by using the secondary signal in
several ways: (1) The AC signal in the secondary winding may
be demodulated by rejecting the carrier frequency (primary-
winding excitation frequency) and directly measuring the
resulting signal, which represents the motion. This method is
particularly suitable for measuring transient motions; (2) the
amplitude of the secondary (induced) voltage may be
measured; (3) measure the change of inductance in the
secondary circuit directly, by using a device such as an
inductance bridge circuit.

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• Linear Variable Differential Transformer (LVDT)
– The LVDT is a displacement-measuring device that
overcomes most of the shortcomings of the
potentiometer.
– It is considered a passive transducer because the
measured displacement provides energy for “changing”
the induced voltage, even though an external power
supply is used to energize the primary coil which in
turn induces a steady carrier voltage in the secondary
coil.
– The LVDT is a variable-reluctance transducer of the
mutual induction type.

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Schematic Diagram of a LVDT

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Typical LVDT
Operating Curve

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– In its simplest form, the LVDT consists of a cylindrical,
insulating, nonmagnetic form that has a primary coil in
the mid-segment and a secondary coil symmetrically
wound in the two end segments.
– The primary coil is energized by an AC supply of
voltage vref. This will generate by mutual induction an
AC of the same frequency in the secondary winding.
– A core of ferromagnetic material is inserted coaxially
into the cylindrical form without actually touching it.
– As the core moves the reluctance of the flux path
changes. Hence, the degree of flux linkage depends on
the axial position of the core.

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48
Series Opposition Connection of Secondary Windings

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49
– Since the two secondary coils are connected in series
opposition, so that the potentials induced in the two
secondary coil segments oppose each other, the net
induced voltage is zero when the core is centered
between the two secondary winding segments. This is
known as the null position.
– When the core is displaced from this position, a
nonzero induced voltage will be generated. At steady-
state, the amplitude vo of this induced voltage is
proportional, in the linear (operating) region, to the core
displacement x. Consequently, vo may be used as a
measure of the displacement.
– Because of opposed secondary windings, the LVDT
provides direction as well as the magnitude of the
displacement.

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– For an LVDT to measure transient motions accurately,
the frequency of the reference voltage (the carrier
frequency) has to be at least 10 times larger than the
largest significant frequency component in the
measured motion. For quasi-dynamic displacements
and slow transients on the order of a few hertz, a
standard AC supply (at 60 Hz line frequency) is
adequate. The performance (particularly sensitivity and
accuracy) is known to improve with the excitation
frequency, however.

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51
– Advantages of the LVDT include:
• It is essentially a non-contacting device with no frictional
resistance. Near-ideal electromechanical energy conversion
and light-weight core result in very small resistive forces.
Hysteresis (both magnetic hysteresis and mechanical backlash)
is negligible.
• It has low output impedance, typically on the order of 100 Ω.
Signal amplification is usually not needed.
• Directional measurements (positive/negative) are obtained.
• It is available is small size, e.g., 1 cm long with maximum
travel of 2 mm.
• It has a simple and robust construction (inexpensive and
durable).
• Fine resolutions are possible (theoretically, infinitesimal
resolution; practically, much better than a coil potentiometer).

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• Rotary Variable Differential Transformer (RVDT)
– The RVDT operates using the same principle as the
LVDT, except that in an RVDT, a rotating
ferromagnetic core is used.
– The RVDT is used for measuring angular
displacements.
– The rotating core is shaped so that a reasonably wide
linear operating region is obtained.
– Advantages of the RVDT are essentially the same as
those cited for the LVDT.
– The RVDT measures angular motions directly, without
requiring nonlinear transformations (as is the case for
resolvers). The linear range is typically ± 40°, with a
nonlinearity error less than 1 percent.

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Schematic Diagram of the RVDT
Operating Curve

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54
• In variable-inductance devices, the induced
voltage is generated through the rate of change of
the magnetic flux linkage. Therefore,
displacement readings are distorted by velocity;
similarly, velocity readings are affected by
acceleration.
• For the same displacement value, the transducer
reading will depend on the velocity at that
displacement. This error is known to increase
with the ratio (cyclic velocity of the core / carrier
frequency). Hence, these rate errors can be
reduced by increasing carrier frequency.

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• Mutual-Induction Proximity Sensor
– This displacement operates also on the mutual-
induction principle.
– The insulating core carries the primary winding in its
middle limb. The two end limbs carry the secondary
windings that are connected in series.
– Unlike the LVDT and the RVDT, the two voltages
induced in the secondary winding segments are additive
in this case.
– The region of the moving surface (target object) that
faces the coils has to be made of ferromagnetic material
so that as it moves, the magnetic reluctance and the flux
linkage will change.

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56
– This, in turn, changes the induced voltage in the
secondary windings, and this change is a measure of the
displacement.
– Unlike the LVDT, which has an axial displacement
configuration, the proximity probe has a transverse
displacement configuration. It measures transverse
displacements or proximities of moving objects.
– The displacement-voltage relation of a proximity probe
is nonlinear. Hence, these proximity sensors should
only be used for measuring very small displacements,
unless accurate nonlinear calibration curves are
available.
– The proximity sensor is a non-contact device and so
mechanical loading is negligible.

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– Because a ferromagnetic object is used to alter the
reluctance of the flux path, the mutual-inductance
proximity sensor is a variable-reluctance device.
– Proximity sensors are used in a wide variety of
applications pertaining to non-contacting displacement
sensing and dimensional gaging, e.g., level detection,
angular speed measurement at steady state, detecting
surface irregularities in machined parts, measurement
and control of the gap between a robotic welding torch
head and the work surface, gaging the thickness of
metal plates in manufacturing operations.

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Schematic Diagram of the Mutual-Induction Proximity Sensor
Operating Curve

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• Resolver
– This mutual-induction displacement transducer depends
on relative motion between the primary coil and the
secondary coil to produce a change in flux linkage.
– It is not a variable-reluctance transducer because it does
not employ a ferromagnetic moving element.
– It is widely used for measuring angular displacements.
– The rotor contains the primary coil. It consists of a
single two-pole winding element energized by an AC
supply voltage vref. The rotor is directly attached to the
object whose rotation is being measured.
– The stator consists of two sets of windings placed 90°
apart.

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– If the angular position of the rotor with respect to one
pair of stator windings is denoted by θ, the induced
voltage in this pair of windings is given by:
– The induced voltage in the other pair of windings is
given by:
– Note that these are amplitude-modulated signals; the
carrier signal vref is modulated by the motion θ. The
constant parameter a depends primarily on geometric
and material characteristics of the device.
– Either of the two output signals may be used to
determine the angular position in the first quadrant (0 ≤
θ ≤ 90°).
o1 ref
v av cos
= θ
o2 ref
v av sin
= θ

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– Both signals are needed, however, to determine the
displacement (direction as well as magnitude) in all
four quadrants (0 ≤ θ ≤ 360°) without causing any
ambiguity.
– As for differential transformers, transient displacement
signals can be extracted by demodulating the
modulated outputs. This is accomplished by filtering
out the carrier signal, thereby extracting the modulating
signal.
– The output signals of a resolver are nonlinear
(trigonometric) functions of the angle of rotation.

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62
– The primary advantages of the resolver include:
• Fine resolution and high accuracy
• Low output impedance (high signal levels)
• Small size
• Simple and robust operation
– Its main limitations are:
• Nonlinear output signals (an advantage in some applications
where trigonometric functions of rotations are needed)
• Bandwidth limited by supply frequency
• Slip rings and brushes needed (which adds mechanical loading
and also creates wearout, oxidation, and thermal and noise
problems).

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Schematic of a Resolver

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64
• Self-Induction Transducers
– These transducers are based on the principle of self-
induction. Unlike mutual-induction transducers, only a
single coil is employed. This coil is activated by an AC
supply voltage vref.
– The current produces a magnetic flux, which is linked
with the coil. The level of flux linkage (or self-
inductance) can be varied by moving a ferromagnetic
object within the magnetic field. This changes the
reluctance of the flux path and the inductance of the
coil. This change is a measure of the displacement of
the ferromagnetic object. The change in inductance is
measured using an inductance measuring circuit.

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65
– Note that self-inductance transducers are usually
variable-reluctance devices.
– A self-induction proximity sensor can be used as a
displacement sensor for transverse displacements, e.g.,
the distance between the sensor tip and the
ferromagnetic surface of a moving object can be
measured.
– High-speed displacement measurements can result in
velocity error (rate error) when the variable-inductance
displacement sensors (including self-induction
transducers) are used. This effect may be reduced, as in
other AC-powered variable-inductance sensors, by
increasing the carrier frequency.

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66
Diagram of a Self-Inductance Proximity Sensor

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67
• Permanent-Magnet Transducers
– A distinctive feature of permanent-magnet transducers
is that they have a permanent magnet to generate a
uniform and steady magnetic field.
– A relative motion between the magnetic field and an
electric conductor induces a voltage that is proportional
to the speed at which the conductor crosses the
magnetic field.
– In some designs, a unidirectional magnetic field
generated by a DC supply, i.e., an electromagnet, is
used in place of a permanent magnet. Nevertheless,
this is generally termed a permanent-magnet
transducer.

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68
• Permanent-Magnet Speed Transducers
– The principle of electromagnetic induction between a
permanent magnet and a conducting coil is used in
speed measurement by permanent-magnet transducers.
– Depending on the configuration, either rectilinear
speeds or angular speeds can be measured.
– Note that these are passive transducers, because the
energy for the output signal vo is derived from the
motion (measured signal) itself.
– The entire device is usually enclosed in a steel casing to
isolate it from ambient magnetic fields.

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69
– In the rectilinear velocity transducer, the conductor coil
is wrapped on a core and placed centrally between two
magnetic poles, which produce a cross-magnetic field.
– The core is attached to the moving object whose
velocity must be measured. The velocity v is
proportional to the induced voltage vo.
– A moving-magnet and fixed-coil arrangement can also
be used, thus eliminating the need for any sliding
contacts (slip rings and brushes) for the object leads,
thereby reducing mechanical loading error, wearout,
and related problems.
– The tachometer is a very common permanent-magnet
device. Here the rotor is directly connected to the
rotating object.

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– The output signal that is induced in the rotating coil is
picked up as a DC voltage vo using a suitable
commutator device – typically consisting of a pair of
low-resistance carbon brushes – that is stationary but
makes contact with the rotating coil through slip rings
so as to maintain the positive direction of induced
voltage throughout each revolution.
– The induced voltage is given by:
– h is the coil height, 2r is the coil width, n is the number
of turns in the coil, β is the flux density of the uniform
magnetic field, and ωc is the angular speed.
( )
o c
v 2nhr
= β ω

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71
– When tachometers are used to measure transient
velocities, some error will result from the rate
(acceleration) effect. This error generally increases
with the maximum significant frequency that must be
retained in the transient velocity signal.
– Output distortion can also result because of reactive
(inductive and capacitive) loading of the tachometer.
Both types of error can be reduced by increasing the
load impedance.

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Permanent Magnet Transducers:
(a) Rectilinear velocity transducer
(b) DC tachometer-generator

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73
• Eddy-Current Transducers
– If a conducting (i.e., low-resistivity) medium is
subjected to a fluctuating magnetic field, eddy currents
are generated in the medium. The strength of eddy
currents increases with the strength of the magnetic
field and the frequency of the magnetic flux.
– This principle is used in eddy-current proximity
sensors. Eddy-current sensors may be used as either
dimensional gaging devices or displacement sensors.
– Unlike variable-inductance proximity sensors, the
target object of the eddy-current sensor does not have
to be made of a ferromagnetic material. A conducting
target object is needed, but a thin film conducting
material is adequate.

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– The probe head has two identical coils, which form two
arms of an impedance bridge. The coil closer to the
probe face is the active coil. The other coil is the
compensating coil. It compensates for ambient
changes, particularly thermal effects.
– The other two arms of the bridge consist of purely
resistive elements.
– The bridge is excited by a radio-frequency voltage
supply, the frequency ranging from 1 MHz to 100
MHz. This signal is generated from a radio-frequency
converter (an oscillator) that is typically powered by a
20-volt DC supply.
– In the absence of the target object, the output of the
impedance bridge is zero, which corresponds to the
balanced condition.

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75
– When the target object is moved close to the sensor,
eddy currents are generated in the conducting medium
because of the radio-frequency magnetic flux from the
active coil. The magnetic field of the eddy currents
opposes the primary field that generates these currents.
Hence, the inductance of the active coil increases,
creating an imbalance in the bridge. The resulting
output from the bridge is an amplitude-modulated
signal containing the radio-frequency carrier. This
signal is demodulated by removing the carrier. The
resulting signal (modulating signal) measures the
transient displacement of the target object.
– Low-pass filtering is used to remove high-frequency
leftover noise in the output signal once the carrier is
removed.

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– For large displacements, the output is not linearly
related to the displacement. Furthermore, the
sensitivity of the eddy-current probe depends
nonlinearly on the nature of the conducting medium,
particularly the resistivity (for low resistivities,
sensitivity increases with resistivity; for high
resistivities, sensitivity decreases with resistivity).
– The facial area of the conducting medium on the target
has to be slightly larger than the frontal area of the
eddy-current probe head. If the target area has a curved
surface, its radius of curvature has to be at least four
times the diameter of the probe.
– Eddy-current sensors are medium-impedance devices;
1000 Ω output impedance is typical.

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– Sensitivity is on the order of 5 V/mm.
– Since the carrier frequency is very high, eddy-current
devices are suitable for highly transient displacement
measurements (e.g., bandwidths up to 100 kHz).
– Another advantage of the eddy-current sensor is that it
is a non-contacting device; there is no mechanical
loading on the moving (target) object.

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78
Eddy-Current Proximity Sensor

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79
Impedance Bridge

actuator and sensor motion proximity sensors

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