Designing and Building an Eddy Current Position Sensor

September 1, 1998 By: Steven D. Roach

The experience essential to successfully designing eddy current sunsors can be readily acquired because printed circuit sensors are easily and cheaply prototyped and only basic electronic instruments are needed for testing.


Steven D. Roach, Hewlett-Packard, Electronic Measurements Division


Photo 1. In the radial and thrust magnetic bearings shown here, three
differential eddy current probes from Kaman Instrumentation Corp. (Colorado
Springs, Colorado) monitor X,Y, and Z motion to a precision of <0.0001
in. The visible components are (from left) motor jack shaft; flexible coupling
(black); left thrust housing and stator; thrust disc with shims for setup;
right thrust housing, stator, and thrust probes; radial eddy current probe
assembly; radial magnetic bearing housing and stator; auxiliary bushing
assembly; and pump drive shaft. (Photo by George Gillies, courtesy of ROMAC).

Eddy current sensors are widely used for noncontact position, displacement,
and proximity measurement. Operating on the principle of magnetic induction,
these detectors can precisely measure the position of a metallic target,
even through intervening nonmetallic materials such as plastics, opaque
fluids, and dirt [1]. The sensors are inherently rugged and can operate
over wide temperature ranges in contaminated environments that would defeat
optical, acoustic, and even capacitive devices. One of the more exotic applications
of eddy current sensors is that of supplying position feedback in magnetic
bearings (see Photo 1 and sidebar). In a magnetic bearing assembly, a rotating
shaft is suspended in a magnetic field to achieve wear-free and nearly frictionless
rotation. For bearing stability, eddy current sensors are frequently used
to provide position feedback to the magnet controller.

Practical eddy current sensors vary in diameter from a few millimeters
to a meter and have maximum sensing ranges roughly equal to the radius of
the coil. Linearity is typically 1% of the sensing range and noise levels
of 1 ppmrms/√Hz are common. Temperature drift ranges from ~100 ppm/°C
to 1000 ppm/°C. A small sensor can resolve nanometer-size displacements
and measure a 1 mm span to 10 microns total accuracy. Bandwidths of 50 kHz
are readily achieved.


Figure 1. An AC current in the sensor coil generates an oscillating magnetic
field, which induces eddy currents in the surface of the target. The coil
impedance changes with standoff and is converted to a linear output by the
sensor and signal processing electronics.

Physics and Behavior

An eddy current displacement sensor consists of four components (see
Figure 1): the sensor coil, the target, the sensor drive electronics, and
a signal processing block, which can be a circuit or a microprocessor algorithm.

When the sensor coil is driven by an AC current, it generates an oscillating
magnetic field that induces eddy currents in any nearby metallic object,
designated the target. The eddy currents circulate in a direction opposite


Figure 2. An eddy current sensor can be modeled as a transformer (A)
with a coupling coefficient that depends on standoff. The model can be simplified
to an inductor and resistor (B) that both depend on standoff, x.

that of the coil, reducing the magnetic flux in the coil and thereby its
inductance. The eddy currents also dissipate energy, increasing the coil's
resistance. As shown in Figure 2(A), the coil and the target constitute
the primary and (shorted) secondary of a weakly coupled air-core transformer.
Movement of the target changes the coupling, and this movement is reflected
as an impedance change at the terminals of the coil.

The air-core transformer model is physically accurate, but for purposes
of circuit design, a lossy inductor is simpler and more useful. In Figure
2(B), the complex impedance of the sensor coil is represented a series LR
circuit. Both inductance, L, and resistance, R, change with target position,
or standoff. As the target approaches the coil, the inductance goes down
and the resistance usually goes up. The inductance changes by as much as
a factor of five, providing the physical basis for sensing the target position.

The inductance and resistance are important characteristics of the sensor
because they correspond directly to physical mechanisms and can be used
directly in circuit design and simulation. However, the quality factor,
Q, is more directly connected to the ultimate performance of the sensor.
Q is defined as:

   equation (1)


ω = operating frequency of the sensor in radians per second

Q depends on the standoff, x, because both L and R are functions of displacement.
The higher the value of Q, the more purely reactive the sensor. High Q leads
to high accuracy and stability. The specific value of L is of secondary


Figure 3. Measurements on a 38 mm dia. PC sensor at 1 MHz show how inductance,
resistance, and Q are affected by target displacement. The inductance changes
rapidly when the target is close but the response decays significantly at
one coil radius.

importance since it is constrained only by the need for a manufacturable
coil and a practical circuit design that burns a reasonable amount of energy
at a reasonable frequency. The specific value of R is of even less interest
because R is an undesirable parasitic effect in the first place.

Figure 3 shows how L, R, and Q depend on target standoff. The
data come from real measurements on a 38 mm dia., air-core PC sensor with
an aluminum target. The measurements were made on Hewlett-Packard's HP4285A
LCR meter at 1 MHz. Note that the displacement has been normalized to the
sensor radius (19 mm) so that the graph qualitatively depicts the behavior
of sensors of all sizes. As standoff increases, the inductance increases
by a factor of four, the resistance decreases slightly, and as a consequence
the Q increases. The change in all three parameters is highly nonlinear
and each curve tends to decay roughly exponentially as standoff increases.
The rapid loss of sensitivity with distance strictly limits the range of
an eddy current sensor to ~1/2 the coil diameter and constitutes the most
important limitation of this type of sensing.


Figure 4. Target conductivity affects the response of the 38 mm PC sensor
(A). A stainless steel target produces much higher eddy current losses and
higher resistance when the standoff, x, is small. The inductance is much
less sensitive to conductivity In (B), the Q values of aluminum and stainless
steel are plotted against standoff, x.

The coil's impedance is also affected by:

  • Target size, flatness, and thickness
  • Target material properties, especially conductivity and magnetic permeability
  • Temperature of the target and the coil
  • Coil geometry and DC resistance
  • Operating frequency

Figure 4 shows the effect of target material for an aluminum
and a stainless steel target. The steel target has 1/28 the conductivity
of aluminum, resulting in higher eddy current losses and higher resistance,
especially at close spacing. The target conductivity scarcely affects the
inductance. Neither resistance nor inductance depends strongly on the target
at large sensing distances where the coil interacts only weakly with the

Figure 5 shows the sensor's response to frequency. First note
the resonant behavior at 7 MHz, which is caused by the cable and interwinding
capacitance. The frequency where the inductance peaks is called the self-resonant
frequency (SRF), and the sensor must be operated below it to look like an


Figure 5. The frequency dependence of the 38 mm PC sensor shows that
the Q increases with frequency until the coil becomes self-resonant. With
this graph we can find an operating frequency that maximizes Q while staying
well below self-resonance.

inductor at all. We want to operate the sensor at high frequency to maximize
Q, but the frequency must remain at least a factor of three below the SRF.
Since these are opposing requirements, the choice of operating frequency
is clearly an opportunity for optimization. Practical frequency values for
air core coils typically lie between 100 kHz and 10 MHz.

Temperature drift proves to be the critical source of error in eddy current
sensors and arises from a complex set of factors. Both the inductance and
resistance have positive temperature coefficients that are dependent on
frequency. For the 38 mm PC coil operating at 1 MHz, for example, inductance
increases at 88 ppm/°C. and resistance increases at 2400 ppm/°C.
(Temperature drift will be examined in detail in a later article.)

Target Selection

The response of an eddy current sensor depends on both the conductivity
and the magnetic permeability of the target. In some applications the designer
is free to choose the target material; in others, the target is part of
a machine or assembly over which the designer has no control. High-conductivity,
nonmagnetic metals such as aluminum or copper make the best targets. Magnetic
metals are also good choices, even though the sensor response is a mixture
of eddy current and magnetic reluctance. Magnetic reluctance describes the
way in which magnetic material modifies the effective permeability in a
magnetic circuit. As a magnetic target approaches the coil, eddy currents
reduce the inductance, while reluctance increases the inductance. Since
these effects are in opposite directions, they may cancel each other. The
net result is an easily avoidable null point in the sensor response at small
standoff values.

The skin effect, or tendency for AC current to flow in the surface of
a conductor, applies to both the coil and the target. Due to the skin effect,
the current density in the target drops exponentially with distance from
the surface. Skin effect is characterized by the skin depth, the distance
at which the current density drops to 1/e of its surface value. The expression
for skin depth in a conductor is given by:

   equation (2)


δ = skin depth (meters)

ω = radian frequency (radians/second)

μ = magnetic permeability (henries/meter)

σ = conductivity (siemans/meter)

For nonmagnetic materials, µ = µo = 1.26 x 10-6 H/m. Table 1
shows skin depths in several materials for frequencies of interest
in sensor design. Notice that the lower the conductivity and frequency,
the more deeply the eddy currents penetrate the target. If the target is
at least two skin depths thick, thickness is essentially eliminated as a
factor in the measurement [2]. This rule-of-thumb, however, is extremely


Skin Depth (delta) in Microns for Various Metals and Frequencies


(x 106 S/m)


relative to Cu

Skin depth (µm)
10 kHz
100 kHz
1 MHz
10 MHz
304 SS
Titanium Alloy

Several target characteristics other than thickness, conductivity, and
permeability affect sensor behavior in ways that are difficult to predict
analytically. Differences are seen if the lateral dimensions of the target
are less than twice the sensor diameter, the target is curved, or its surface
roughness is comparable to the skin depth. Though these situations are difficult
to model, all can be accommodated with measurements by building a prototype
sensor and measuring its response with an LCR meter. A versatile multifrequency
instrument such as the HP4285A is extremely valuable for sensor design because
it displays L, R, and Q directly with enough sensitivity to measure even
minute effects such as temperature drift.

Sensor Design

The design of a nearly optimal eddy current sensor is surprisingly easy.
We first need to establish some intuitive guidelines and a set of simplified
design equations. Measurements of the prototype sensor with an LCR meter
then provide highly accurate and detailed data for the design of the electronics.

If small size is the primary goal, sensors based on wirewound coils are
the best choice. The fine magnet wire used in wirewound coils increases
the sensor Q by maximizing the number of turns while minimizing the resistance.
The magnet wire must be packed in perfect layers (usually by hand) under
a microscope. The coils are then inserted in a machined protective holder
and potted with epoxy to mechanically stabilize the windings. A wirewound
sensor is a labor intensive product, and typically costs more than $100.

When cost is the primary consideration, coils printed on a PCB deliver
excellent performance for a fraction of the cost of a wire wound sensor.
The advantages of printing the sensor coil include:

  • Prototyping. Tooling costs are only a few hundred dollars. A
    single prototype run can produce a large number of variations on the sensor
  • Manufacturing. PCB processes are highly automated, standardized,
    and widely available. PC board processing scales well from tens to millions
    of sensors.
  • Cost. Coils cost less than $1.00 in volume, but the cabling
    and connectors can easily dominate the overall price.
  • Performance. The performance competes with wirewound sensors
    for larger diameters (>10 mm).
  • Special configurations. Arrays of sensors and sensors on flex
    circuit material create unique application opportunities.

The primary tradeoff with printed sensors is size. It is difficult to
print a coil <10 mm dia. on a PCB and still match the performance of
a wirewound sensor. Table 2 outlines the most important factors
and tradeoffs in the design of an eddy current sensor.


Factors in Sensor Design [2]


High-quality factor, or Q

Fundamental determinant of temperature stability, power consumption, noise; do anything possible to increase Q

Increase frequency and inductance, decrease resistance; use a wire-wound coil for small sensors

Difficult to achieve high Q with small sensors

High inductance

Increases Q and reduces power

Use more turns in the coil, a larger diameter sensor, or a ferrite core

High inductance is often accompanied by low self-resonant frequency

Low resistance

Increases Q and reduces power

Use more conductive coil windings, larger wire or thicker traces, keep temperatures low

Frequency trades off with the need for high inductance

Optimal coil size

Maximizes sensitivity to displacement

For highest accuracy and stability choose radius=3 x range; a small sensor has small range, too large a sensor has reduced sensitivity

The larger the sensor the larger the "spot size" or area of the target measured by the sensor; the larger the sensor, the larger the required target dimensions

Flat, disc-shaped coil configuration

Maximizes sensitivity by placing all of the turns close to the target

Use a printed coil

Flat coil limits the number of turns, reducing inductance

Operate at high frequency

Increases Q and redeuces power in the sensor; increases sensing bandwidth

Reduce interwinding and cable capacitance

Must stay below the self-resonant frequency; high frequency usually increases power in the electronics

Minimize cable length

Reduces cable noise, temperature drift, and cost of cabling; increases the self-resonant frequency; allows use of lower quality, cheaper cabling

Place the critical circuits in a pod close to the sensor or load electronics on back of printed coil

Placing electronics on or near the sensor limits the temperature range

A complete sensor typically has a cable joining the coil and the electronics.
The cable can be coaxial, twisted pair, ribbon, or traces on a PCB. The
cable affects system design and performance in several ways because all
cables have inductance, capacitance, and DC resistance. The inductance of
the cable adds to that of the sensor. Because the cable inductance is static
(not sensitive to displacement), it reduces the sensor's net sensitivity.
The cable capacitance forms part of the resonant circuit network, so any
instability in the cable capacitance degrades measurement accuracy. Changes
in cable capacitance with temperature and cable movement produce measurement
errors. A common problem with eddy current sensors is noise traceable to
cable vibration. Eddy current sensors can be so sensitive that a hand moving
near a twisted-pair shows up as an observable displacement error, so the
highest performance sensors require fully shielded coaxial cable. The resistance
of the cable is in series with the sensor coil and contributes to a reduction
in Q and to temperature drift. For the best performance, we must use high-quality
microwave coaxial cable such as RG-178 or RG-316. These cables have a highly
stable Teflon dielectric and low capacitance.

When price is an important consideration, and some stability can be sacrificed,
twisted-pair or ribbon cabling is adequate and can be used with very inexpensive
connectors. Twisted-pair can be soldered to a printed coil sensor directly,
eliminating at least one set of connectors. Measurement errors due to cabling
can be greatly reduced by placing the sensor interface circuitry on the
back of a printed circuit coil. Even though the circuit operates in the
coil's magnetic field and the coil "sees" the circuitry, performance
is not seriously affected. It is important, though, to avoid a ground plane
or any large loop areas in the circuit because the sensor would see these
as a second target. With the sensor circuitry integrated with the coil,
we still must deliver power and retrieve the output signal through cabling,
but this cable in no way affects the accuracy of the measurement.

To establish guidelines for the design of the sensor we need a preview
of the sensor drive circuitry. The circuits developed in the "Circuit
Design" section of this article place the sensor in an oscillator loop
where it is resonated with a capacitor. The frequency of oscillation (resonant
frequency) is dependent on target displacement and is the output of the
circuit. In an LC oscillator the frequency is proportional to 1/√LC. The
design of an air core sensor with a printed circuit coil follows these steps:

  1. Set the outer coil radius, ro. We need:

ro > 2 x sensing range for a typical design

ro = sensing range for an aggressive design

  2. Choose a manufacturing technology for the coil. The manufacturing
technology determines the width, w, thickness, t, and pitch, p, of the traces
that make up the coil windings.

  3. Calculate the number of turns that can be packed on a single layer.

The outer coil radius, ro, was determined in an earlier step.

The inner coil radius, ri, should be at least 1/3 the outer radius.

The number of turns per layer is:

   equation (3)

  4. Calculate the unloaded inductance using Burkett's inductance equation
[3]. The unloaded inductance is the inductance of the free coil with no
target. Assume one layer for now.

   equation (4)

  5. Calculate the cable capacitance, CCABLE, from the cable length and the manufacturer's specification of capacitance per unit length. Assume that the interwinding capacitance, CIWC, is zero for a single layer coil. For a two-layer coil, estimate CIWC to be ¼ the parallel plate capacitance,
calculated as if the windings were solid sheets of metal. Thus, for a two-layer coil:

   equation (5)


εr = relative dielectric constant (a dimensionless number; εr = 4.2 for FR-4 PC board)

εo = permittivity of free space = 8.86 × 10-12 F/m

h = thickness of coil (meters)

  6. Estimate the self-resonant frequency:

   equation (6)

  7. Choose the operating frequency. For the LC oscillator circuit, the minimum frequency coincides with the maximum inductance and occurs at maximum standoff. The maximum inductance is given approximately by Equation 4. We want the maximum frequency, which is typically about twice the minimum, to be no more than 1/3 the SRF so that the cable and interwinding-capacitance have only a small effect. With these considerations we have:

   equation (7)

   equation (8)

   equation (9)


C = total parallel resonant capacitance and L comes from Equation 4. Equation 9 can be used to calculate the capacitance required to resonate the sensor at the desired frequency.

  8. Calculate the DC resistance of the coil:

   equation (10)


l = length of the coil windings

ρ = 1/σ = resistivity of the coil windings

w = width of traces of coil windings

t = thickness of traces in coil windings

  9. Estimate the worst-case AC resistance, which is higher than the DC resistance because of the skin effect and proximity effect.

   RAC = 2RDC (11)

  10. Calculate the unloaded Q. (The Q of the free coil with no target.)

   equation (12)

We need:

Unloaded Q > 15 for a typical design

Unloaded Q > 5 for a design with a Q >15

If Q is not high enough, consider adding additional layers to the coil.
Doubling the number of layers doubles the number of turns, which quadruples
the inductance while only doubling the resistance. The Q therefore roughly
doubles. Also, follow the guidelines in Table 2 to increase Q.

  11. Is the unloaded inductance acceptable? If the inductance is too high (a rare situation except for large coils), the resonant capacitance may be too low, allowing cable and interwinding capacitance to dominate the sensor behavior. If the inductance is too low (a common problem, especially
for small coils), the circuits may burn excessive power or require an impractically high operating frequency for acceptable Q.

We need:

1 µH < L < 100 µH for a typical design (but aim for L > 10 µH)

L < 1 µH for an aggressive design

If the inductance is too low, increase the number of turns by using a
finer pitch PC technology, adding layers, or using hybrid integrated circuit

  12. Build a prototype of the coil and measure it on an LCR meter to confirm
the design and to collect data on L(x) and R(x) (x = standoff). The data
will be used to design the electronics. To make these measurements, attach
the target and sensor to a micrometer with an appropriate fixture.

Circuit Design


Figure 6. Adding a capacitor to the sensor creates a resonant sensor
network that magnifies the sensitivity of the impedance to target displacement.
(jω) = V(jω) / I(jω) = the complex impedance of the circuit.

The challenge in circuit design for eddy current sensors is to develop
an output that is linearly proportional to standoff and independent of temperature.
The sensor converts displacement to a change in impedance, and the circuit
converts the change in impedance to another electrical parameter such as
the amplitude, phase, or frequency of an RF carrier. This signal must be
demodulated, temperature-compensated, linearized, offset, and scaled. These
steps may be carried out by a circuit, an algorithm, or a combination of

The basis for nearly all eddy current sensor circuits is shown in Figure
6, where the sensor coil is resonated with a capacitor. Resonance
causes abrupt changes in the circuit impedance Z(jω), as shown in Figure
7, in which the middle point is x = 4 mm. In Figure 7(A), the
magnitude peaks at the resonant frequency and the height of the peak depends
on the Q, which in turn depends on the target displacement. In Figure 7(B),
the phase shifts from +90 at low frequency to ­90 at high frequency.


Figure 7. The impedance, Z, of the resonant sensor network in Figure
6 depends on frequency and target displacement. Target standoff, x, changes
the frequency at which the magnitude peaks (A) and the phase crosses zero
(B), providing bases for position sensing.

The frequency of the phase transition depends on the inductance, which depends
on the target position.

A common way to convert the displacement to voltage is simply to drive
the resonant circuit with a current source at a fixed frequency and demodulate
either the amplitude or the phase of the voltage that appears across the
sensor. Both amplitude and phase detection are complex, requiring an independent
oscillator, phase detector, low-pass filter, and analog postconditioning

With the goal of designing a position sensor of the lowest possible cost,
consider the self-oscillating circuit of Figure 8. It is a straightforward
gate oscillator using low-voltage CMOS logic gates. The two inverters produce


Figure 8. An LC gate-oscillator circuit with an eddy current sensor generates
a frequency output that depends on the target standoff. A microcontroller
can directly digitize the frequency by counting the output pulses.

a large, positive voltage gain, so the circuit oscillates at a frequency
where the phase shift of the resonant sensor network is zero. The output
is a square wave whose frequency is a function of displacement and which
changes ~2:1 over the full practical range of standoff. Assuming that a
microcontroller is already part of the host system, the advantages of the
circuit in Figure 8 include:

  • The output can be connected directly to the counter-timer port
    of the microcontroller and digitized simply by counting pulses. The microcontroller
    digitally linearizes, offsets, and scales the output using constants stored
    during calibration.
  • Because every sensor component (the coil and all electronics)
    is under calibration, no costly, high precision devices are needed.
  • The circuit operates on a single supply voltage at a few hundred
    microamps of current.
  • The circuit requires 7 electronic components at a material cost
    of about $0.35. A PCB coil is on the order of $0.50 in volume. With the
    addition of the load and test costs, the cable and assembly labor, the entire
    sensor subsystem adds as little as $2.00 to the cost of a high-volume, microcontroller-based

The circuit in Figure 8 oscillates at the resonant frequency given by:

   equation (13)

To find the component values for Figure 8 we first determine the desired
operating frequency from the procedure described earlier. Then we solve
Equation 13 for Cp and choose the nearest standard value. Rs determines
the amplitude of the signal at the input of the first inverter as well as
the power consumption. To minimize noise it is desirable to have the largest
signal possible at the inverter input. The smaller the value of Rs, the
larger the signal, but the greater the power consumption. At resonance,
the impedance of the of the LC tank is purely resistive and is equal to:

   equation (14)

Rs and Rres form a voltage divider, so the amplitude is given by:

   equation (15)


4/π • VDD is the amplitude of the fundamental of the square-wave
output signal. We need only consider the fundamental component because Rs
and the LC tank form a bandpass filter centered on the frequency of oscillation.
This filter attenuates all but the fundamental component to a negligible
level. A good rule of thumb is that the peak-to-peak amplitude in Equation
15 should be at least half the supply voltage, VDD, when the coil is unloaded
(i.e., when the target is not present). With this guideline we would set
Rs ≈ Rres, with Rres found for the unloaded case.

The higher the resistance at resonance (Equation 14), the less power
it takes to drive the sensor. Notice that the resistance goes up and the
power goes down for higher frequencies, higher inductances, and higher values
of Q.

CMOS gate oscillators such as that in Figure 8 exhibit curious power
dissipation characteristics. The first inverter operates part of the time
with its input at the logic threshold. This places it in the so-called linear
region, where it acts like a class B linear amplifier and conducts current
directly from the power supply to ground. The higher the supply voltage,
the higher the class B current, so the power in the circuit of Figure 8
increases rapidly with supply voltage. For example, going from VDD = 2.0
V to 2.5 V increases the current by a factor of four. By placing a resistor
between VDD and the inverter power pin the circuit can be operated on higher
supply voltages while maintaining low power consumption. The resistor drops
the voltage on the inverters and provides negative feedback to limit the
class B current. Of course, the decoupling capacitor should still be attached
to the power supply pin of the chip.

A Model Sensor System Design


Figure 9. This 38 mm diameter PC sensor was designed and prototyped in
~2 hr. with the aid of a PC milling machine. Built on 0.76 mm (0.030 in.)
circuit board, the traces are 0.5 mm wide with a 0.76 mm pitch, yielding
a total of 34 turns on two layers.

The sensor shown in Figure 9 was fabricated to illustrate the
complete design of a low-cost sensor system. It is an octagonal spiral milled
on a numerically controlled milling machine designed specially for rapid
prototyping of printed circuits. With this machine, a sensor coil can be
designed and fabricated in <1 hr. The LCR measurements for this sensor
with an aluminum target are shown in Figure 3. The circuit used is shown
in Figure 8 with Rs = 10 kΩ , Cp = 910 pF, and VDD = 2.0 V. Rs is a 1%, 100
ppm/°C metal film resistor, though a 5%, 200 ppm/°C resistor would
have negligible effect on overall performance. Cp is a 5% NPO capacitor
with ±30 ppm/°C temperature coefficient. The initial accuracy
of Cp affects only the initial frequency and is normally calibrated out.
The drift of Cp is very important, however, so an NPO capacitor or similarly
stable capacitor must be used.


Figure 10. The frequency output of the gate oscillator changes by almost
2:1, or 1 MHz, over the 19 mm sensing range. This example uses the 38 mm
PC sensor and an aluminum target.

Figure 10 shows the frequency output for the model sensor system.
The data were taken with the aid of an HP53132A frequency counter. Since
the total inductance shift in Figure 3 is 4:1, the frequency shift is the
square root of that or 2:1. Operating between 1 MHz and 2 MHz, there is
an impressive total frequency shift of almost 1 MHz.

The frequency is a highly nonlinear function of displacement, so linearization
is essential. The empirically derived function:

   equation (16)


Figure 11. For a range of 19 mm, the gate oscillator produces excellent
linearity of 1% after correction.

with fo = 981 kHz produces the linearized output of Figure 11.
The linearity error is <1% of the 20 mm F.S. range, or 200 µm. Even better
linearity could be attained by calibrating at a large number of points and
using piecewise correction. The advantage of linearizing with Equation 16
over piecewise linear correction is that the constants, m, b, and fo, can
be obtained with only three calibration points: the end points and a point
near the middle of the range. It is not even necessary to know the midrange
point exactly since the calibration procedure need only place this point
on a straight line between the end points.


Figure 12. At 280 mA and 2 V the gate oscillator burns less than 600
µW, qualifying as a micropower sensor. Eddy currents degrade the sensor
Q, increasing the power and decreasing the amplitude when the target is
close to the sensor.

Figure 12 shows the sensor amplitude and power supply current.
Notice that the amplitude decreases for small values of standoff because
the Q is dropping rapidly. The reduced Q also requires more drive current
from the power supply. With a supply voltage of 2.0 V the worst case power
dissipation is <600 µW. The power is so low that the capacitance of a
15 pF scope probe on the output increases the power by 20% due to charge

The frequency output of the sensor is highly nonlinear, changing some
125:1 in slope over the full. If the circuit noise and quantization error
are roughly constant in terms of frequency, then they will not be constant
in terms of displacement. In terms of displacement, both the noise and resolution
vary in inverse proportion to the slope of the frequency curve, so they
will vary by 125:1 also. If we measure frequency by counting N pulses over
a gate time of Tg, the estimated frequency is fest = N/Tg. The quantization
error is ±1 count, so the frequency resolution is Δf = 1/Tg. Clearly,
the longer the gate time (and the slower the sample rate), the higher the
resolution. Arbitrarily high resolutions can be reached by counting for
longer periods. Noise is reduced at the same time because the counting process
acts as a digital low-pass filter.

The resolution in terms of displacement depends on the slope of the frequency

   equation (17)


Figure 13. Displacement noise and quantization error increase as the
target moves away from the sensor because the inductance change decays with
distance. The noise and quantization are relative to a full-scale standoff
of one coil radius and the frequency counter gate time is 100 ms.

Noise follows the same form as Equation 17, increasing for larger values
of standoff. Figure 13 shows the resolution and noise of the model
sensor for a gate time of 100 ms, or a digitizing rate of 10 sps.. The noise
is ~20 Hz p-p as measured with the HP53132A counter with a 100 ms gate time.
Most of the noise is at very low frequency (<1 Hz) and can be caused
by vibration or temperature changes due to air currents. Notice that higher
resolution and lower noise can be attained by simply limiting the maximum

A very good way to observe the noise is to display the output square
wave on a digital oscilloscope and delay the scope by 1 ms. With the sensor
operating at 1 MHz (a 1 µs period), the scope displays the phase noise accumulated
over 1000 cycles. With the digital scope set to infinite persistence, the


Figure 14. The temperature stability of the gate oscillator compares
favorably with some of the best commercial sensors. The temperature coefficient
is taken relative to one coil radius.

trace will "paint" a band several nanoseconds wide. The width
of this band divided by the delay time of 1 ms gives the relative noise.
For example, 20 nsp-p/1 ms is 20 ppm or 20 Hz at 1 MHz.

Figure 14 shows the temperature stability of the sensor and
the electronics. The stability is poorer at large displacements because
at large values of standoff the temperature stability suffers the same disadvantage
as the resolution and noise. Limiting the sensing range can greatly improve
temperature stability. Table 3 summarizes the measured performance
of the model sensor for two values of full-scale range.


Summary of Performance of the Model Sensor System

Performance Parameter
10 mm Range
20 mm Range

Frequency range















of range

Quantization error

  1 sps

  10 sps

  100 sps











Noise, peak-to-peak

  1 sps

  10 sps

  100 sps











Temperature drift










ppm of


Power supply




  Power supply rejection

















Eddy current sensors are advantageous for precision, noncontact displacement
sensing where the range is relatively small. They are especially useful
where the environment is dirty or sensing through intervening materials
is required. In applications where a microcontroller is available, a high-performance
eddy current sensor can be added to the system for a few dollars, using
printed circuit sensors and manufacturing methods common to all electronic

Experience is essential to the successful design of eddy current sensors,
but it is quickly gained because printed circuit sensors are easily and
cheaply prototyped and only basic electronic instruments are needed for
testing. The essential design calculations can be done with a desktop computer
and a spreadsheet program. Future articles will investigate advanced computer-aided
design methods for optimizing the sensor and advanced circuit designs for
improved temperature stability.


  1. Scott D. Welsby and Tim Hitz. Nov. 1997. "True Position Measurement
with Eddy CUrrent Technology," Sensors, Vol. 14, No. 11:30-40.

  2. J.H. Smith and C.V. Dodd. 1-5 Oct 1973. "Optimization of eddy-current
measurements of coil-to-conductor spacing," Proc Annual Fall Conference
of the American Society for Nondestructive Testing, Chicago, IL:1-15.

  3. Frank S. Burkett, Jr. 12-12 May 1971. "Improved designs for thin
film inductors," Proc 21st Electronic Components Conference, Washington,


The ROMAC Research Facility

Lloyd Barrett, the University of Virginia


In a fault tolerance test platform, a 2-m-long mockup pump shaft is supported
in three radial magnetic bearings and one magnetic thrust bearing. This
apparatus is used to study fault tolerance strategies and has successfully
demonstrated its ability to tolerate multiple simultaneous power amplifier
failures as well as controller CPU failures. Seven single-sided eddy current
probes made by Bently Nevada (Minden, Nevada) are used to measure shaft
motion at the bearings for feedback to the controller. The controller bandwidth
is ~1 kHz. (Photo by George Gillies, courtesy of ROMAC.)

The Rotating Machinery and Controls (ROMAC) Industrial Program supports
cooperative research efforts conducted by faculty, staff, and students in
the Mechanical, Aerospace, and Nuclear Engineering and the Electrical Engineering
departments at the University of Virginia. The ROMAC program emphasizes
theoretical and experimental research in general areas of rotor dynamics,
turbomachinery, structural dynamics, magnetic bearings, the application
of automatic controls to the dynamics of rotating machinery, internal incompressible



The left and right end radial magnetic bearings on a fault test facility
are shown in a close-up view. Two single-sided eddy current probes monitor
X and Y motion to a precision of <0.0001 in. at a bandwidth of ~2 kHz.
(Photos by George Gillies, courtesy of ROMAC.)

professionals through the medium of ROMAC provides the university researchers
with an understanding of practical industrial problems with rotating machinery
while the industrial participants obtain very timely research results.

The magnetic bearing research program, housed in the ROMAC laboratories,
is one of the largest in the world. Experimental platforms include 12 magnetic
bearing rigs ranging from a tiny artificial heart pump up to a 200 HP centrifugal
compressor. The program's research covers the full spectrum of the technology,
from hardware issues (system dynamics, magnetics, amplifiers, sensing, and
controllers) to the the development of control methods.

Lloyd E. Barrett is Professor and Laboratory Director, ROMAC Laboratories,
Dept. of Mechanical, Aerospace, and Nuclear Engineering, the University
of Virginia, Thornton Hall, McCormic Rd., Charlottsville, VA 22903-2442;
804-924-3292, fax 804-982-2037.

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