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Nuclear Instruments and Methods in Physics Research A 489 (2002) 570–576

The linear variable differential transformer (LVDT)position sensor for gravitational wave interferometer

low-frequency controls

Hareem Tariqa,b,*, Akiteru Takamoria,c, Flavio Vetranod,e, Chenyang Wanga,Alessandro Bertolinia,f, Giovanni Calamaid,g, Riccardo DeSalvoa,

Alberto Gennaih, Lee Hollowayi,h, Giovanni Losurdod, Szabolcs M!arkaa,Massimo Mazzonid,j, Federico Paolettih, Diego Passuelloh, Virginio Sannibalea,

Ruggero Stangad,j

aLIGO project, California Institute of Technology, 1200 E. California Blvd, Pasadena, CA 91125, USAbFlorida Institute of Technology, 150 W. University Bl., Melbourne, FL 32901, USA

cDepartment of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japand INFN Sezione di Firenze, via G. Sansone 1, 50019 Sesto Fiorentino, Firenze, Italy

e Istituto di Fisica, Universita di Urbino, via S. Chiara 27, 61019 Urbino, ItalyfDipartimento di Fisica, Universit !a di Pisa, via F. Buonarroti, 4, 56100 Pisa, Italy

gOsservatorio di Arcetri, Largo E. Fermi 4, 51125 Firenze, Italyh INFN Sezione di Pisa, via Livornese 1291, S. Piero a Grado, 56100 Pisa, Italy

iDepartment of Physics, University of Illinois at Urbana-Champaign, 901 West Illinois Street, Urbana, IL 61801, USAjDipartimento di Astronomia e Scienze dello Spazio, Universit !a di Firenze, Largo E. Fermi 2, 51125 Firenze, Italy

Received 5 December 2001; received in revised form 10 February 2002; accepted 17 February 2002

Abstract

Low-power, ultra-high-vacuum compatible, non-contacting position sensors with nanometer resolution and

centimeter dynamic range have been developed, built and tested. They have been designed at Virgo as the sensors

for low-frequency modal damping of Seismic Attenuation System chains in Gravitational Wave interferometers and

sub-micron absolute mirror positioning. One type of these linear variable differential transformers (LVDTs) has been

designed to be also insensitive to transversal displacement thus allowing 3D movement of the sensor head while still

precisely reading its position along the sensitivity axis. A second LVDT geometry has been designed to measure the

displacement of the vertical seismic attenuation filters from their nominal position. Unlike the commercial LVDTs,

mostly based on magnetic cores, the LVDTs described here exert no force on the measured structure. r 2002 Elsevier

Science B.V. All rights reserved.

PACS: 04.80.Nn; 95.30.Sf; 95.55.Ym

Keywords: Gravitational wave interferometers; Seismic isolation; Viscous modal damping; Position sensors; LVDT

*Corresponding author. LIGO project, California Institute of Technology, 1200 E. California Blvd, Pasadena, CA 91125, USA.

E-mail address: [email protected] (H. Tariq).

0168-9002/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 1 6 8 - 9 0 0 2 ( 0 2 ) 0 0 8 0 2 - 1

1. Introduction

One of the most simple and effective Seismicnoise Attenuation Systems for Gravitational WaveDetection interferometers [1–3] is a chain ofpendula [4–7] suspended from a very low-fre-quency stage called Inverted Pendulum (IP). TheIP [8,9] has the triple function of:

* pre-filtering low-frequency seismic noise todiminish the excitation of the rigid body motionresonance of the attenuation chain;

* providing a suitable platform to damp theabove-mentioned resonance.

* providing the mechanical compliance to allowprecision positioning of the mirror suspendedfrom the chain using small forces.

The attenuation chains provide seismic attenua-tion in the frequency region of interest forgravitational wave detection and are capable ofdelivering the required seismic attenuation factors(B10�10) above 5–10Hz, depending on thependula lengths.

The IP is composed of three flex joints, eachsupporting a leg. At the top, the three legs areconnected to a rigid table by means of smallflexures. The IP has two translational modes andone torsional mode.

A triplet of accelerometers [10,11] is mounted at1201, in pinwheel geometry, at the periphery of theIP top table. It is designed to detect the accelera-tion in the IP’s three degrees of freedom. Theaccelerometer signal is fed to actuators [12] todamp the rigid body resonant modes of the chain[13–15], and hence reduce the mirror r.m.s.residual motion of the suspended mirror to a fewtens of nanometers, in the frequency band between10mHz and 5Hz.

The accelerometers are inertial sensors, totallyinsensitive to velocity, and, therefore, cannotprovide positioning of the IP table (and of themirror suspended from it) with respect to a localframe. Additionally, at low frequency (typicallybelow 100mHz, depending on the mechanicsprecision and ground stability) inertial sensorscan be more sensitive to tilts than to realaccelerations. In order to take full advantage ofthe accelerometer performance it is necessary to

complement them with a set of high-precisionposition sensors. The specifications of these posi-tion sensors are as follows:

* linearity within 1% over 710mm, to match thetypical IPs dynamic range;

* position resolution B10 nm r.m.s. to allow thereduction of mirror residual motion;

* insensitivity to movements in the directionsorthogonal to the sensing axis, to allow for freemovements of the IP on the horizontal planeand not to re-inject seismic signals from theorthogonal directions in the feed back loop;

* force-free and contact-free, to avoid shortcircuiting of the seismic excitation to the IP table;

* only passive components inside the vacuumsystem, for reliability reasons; and

* fully Ultra-High-Vacuum (UHV) compatible,to prevent the contamination of the vacuumconditions.

Special linear variable differential transformer(LVDT) position transducers have been designedfor the GW interferometer Virgo, to satisfy all theabove requirements. Also, all the electroniccircuitry is placed outside the vacuum enclosureto satisfy the passive instrument requirement. TheLVDT signals are driven and read-out at the endof 20 or more meters of twisted shielded cablewithout loss of resolution. These transducers turnedout to be so versatile that a modified, smallerversion (referred in this paper as the ‘‘second type’’)has been designed, which is being used in severalother locations of the GW interferometer seismicattenuation systems. For example, LVDTs arebeing used as a sensor for all the Virgo super-attenuator filter vertical working points, as a sensorin the TAMA MGASF top filter, and as genericprecision position sensor in mechanics R&D [16].

2. The LVDT working principle

The LVDT is composed of three coils [17] whosecross-section is shown in Fig. 1. The centralemitter coil, driven with a sinusoidal signal at afrequency between 10 and 20 kHz, is mountedbetween two larger receiver coils; the two receivercoils are identical, counter-wound and connected

H. Tariq et al. / Nuclear Instruments and Methods in Physics Research A 489 (2002) 570–576 571

either in series or in parallel. The emitter ismounted on the IP table while the two receiversare attached on a reference structure such that theemitter is exactly in the mid-point between thetwin receiver coils; therefore, no net signal isinduced. When the table displacements move theemitter coil in a direction, a sinusoidal signalappears on the receiver coils. This signal hasamplitude roughly proportional to the displace-ment from the LVDT geometrical center. If thecoil is moved in the opposite direction the sign ofthe induced sinusoid is changed.

By a careful choice of the coil geometry, it waspossible to get better than 1% linearity over 25mmof movement range with less than a percentsensitivity to transversal movements. Also thesophisticated design of the lock in amplifier circuitallowed for better than 10 nm r.m.s. positionresolution over the aforementioned range.

The coils are made of Kapton-coated copperwire, wound around glass or Peek supports, and

are, therefore, completely UHV compatible.Although the coils were accurately spooled, thelumped coil of this design makes it quite insensitiveto the spooling precision.

Unlike conventional magnetic core LVDTs [18],this arrangement has no ferromagnetic compo-nents and does not generate unwanted forces whenexposed to external magnetic fields.

The 1% linearity and transversal motion in-sensitivity requirements are met by using twosimple circular coils wired in series and placed in aMaxwell Pair (MP) configuration.

The MP configuration is related to the better-known Helmoltz Coils (HC) configuration. TheHC coil is optimized to deliver a magnetic fieldintensity as uniform as possible over a volume aslarge as possible at the center of the pair. The MPis designed to deliver a maximally uniform fieldgradient over the same volume. The MP config-uration is slightly less compact than the HC one; inthe MP, the coil separation s is

s ¼ rffiffiffi3

p

where r is the coil radiusThe MP pairs also differ from the HC because

the current flows in opposite directions in the tworeceiver coils to generate the linear position signalof the LVDT. If the two MP coils had parallel andsame direction field, the emitter coil would inducea signal insensitive to position. Opposite currentflow in the windings induces null signal at thecentral position while a remarkably linear signal isobtained along the displacement when the emittercoil moves along the MP symmetry axis. Theinstrument remains insensitive to movementsalong other two axes. The MP has a reasonablylarge sweet spot (B20% of the volume). As long asthe emitter coil remains in the sweet spot, goodlinearity and insensitivity to transversal motion isachieved. An example of the calculated linearity isshown in Fig. 2.

3. LVDT measured performances

The LVDT is expected to be linear well below apercent over a range of 30mm and it is quiteinsensitive to movements in the other two

Fig. 1. Schematic view, sectioned along the symmetry axis, of a

LVDT position transducer. The three coils appear as the six

darkly shaded rectangles. The coil spools are made of either

glass or Peek, the coil supports are made of slotted and

mutually isolated aluminum parts to avoid eddy current loops.

No effort is made to make thermally compensated supports

because they are intended for use in a thermally stabilized

vacuum envelope.

H. Tariq et al. / Nuclear Instruments and Methods in Physics Research A 489 (2002) 570–576572

orthogonal directions. The expected linearity isconfirmed by the calibration of the actual instru-ments. The emitter coil is moved along the LVDTaxis with the help of a micrometric positioningstage. A calibration constant of about 500mV/mmwas obtained as shown in Fig. 3 (top), witho71% deviation from the linearity illustrated inFig. 3 (bottom). The very symmetric and almostsinusoidal residuals point either to geometricaleffects present due to the non-negligible thicknessof the three coils or to the imperfect spacing of thetwo receiver coils.

Alignment errors of the emitting coil with thereceiving coils’ symmetry axis affect the LVDTsensitivity only to the second order and mayintroduce some unwanted sensitivity to transversalmotions. The insensitivity of the LVDT to thetransversal motion is shown in Fig. 4. Only thetrend of the measured distribution is important.The equations of magnetic field in vacuum requiresmoothness of the LVDT response, which isincompatible with the measured scatter. Thedispersion of the measurement points is due to asmall wobble of the micrometric sled in themagnetic field gradient direction. The flat trendof the measurement proves that the actual

sensitivity of the LVDT to transversal motion isvery small, up to a radial distance of 12mm.

In principle, the LVDT position resolution isonly limited by the electronics noise. The electro-nics noise is kept low by using a simple phaselocked signal detection technique. The signal sentto the emitter coil is also used to synchronize a lownoise, double input linear gate. The amplifiedsignal of the receiver coils is sent with oppositepolarity to two alternate gate inputs. For a moredetailed discussion of the electronics see Ref. [19].The actual circuit can be found in Ref. [20].

In the IP configuration of the LVDTs, describedabove, a resolution of 20 nm r.m.s. was achieved,as shown in Fig. 5. Further electronics noiseimprovements have led to LVDT resolution twiceas good as what is shown in Fig. 5.

-1

-0.8

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per

cen

tag

e d

evia

tio

n f

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co

nst

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gra

die

nt

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Fig. 2. Percentage deviation of the magnetic field gradient

along the sensitive axis of the LVDT.

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Fig. 3. Calibration measurement of an IP LVDT (top) and its

residuals from a linear fit (bottom). The error bars (voltmeter

resolution) are within the marker size.


4. Other applications of LVDTs

A second, smaller version of the LVDT (theactual mechanical design of this LVDT can befound in Ref. [22]), with a different geometry,

shown in Fig. 6 (the actual mechanical design canbe found for example in Ref. [21]), has been usedto generate diagnostics of passive filters of theattenuation chains in the Virgo superattenuators[4] and position feedback signals in the top filterboth in the superattenuators and in the TAMA-SAS [22]. They have been used also as positionsensor in R&D creep measurements [16,23].

These smaller LVDTs have a simpler, mono-lithic geometry and rely on a different geometry toachieve good linearity. They are made with long,thin, uniformly wound cylindrical coils rather thanwith short, lumped, loop-like coils. Good linearityis achieved by choosing the right coil aspect ratios(Fig. 6). While mechanically much simpler andself-supporting, a feature quite useful for smallgeometries, these coils are more sensitive totransversal motions and to the smoothness of thecoiling of the wire around the spool. Even smallimperfections of coiling are easily detectable aslocal perturbations on the linearity of the re-sponse.

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read

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t va

riat

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(m

m)

transversal displacement (mm)

Fig. 4. Sensitivity of a LVTD to transversal motion. The error

bar shown is the resolution of the voltmeter used in this

measurement.

Fig. 5. Spectral r.m.s. position resolution of the LVDT. The

apparent drop off of the residual noise above 50Hz is generated

by the low-pass filters after the demodulation and it is not an

indicator of improved sensitivity.Fig. 6. Geometry of a small LVDT. The emitter coil is 9mm in

diameter and 18mm long. The receiver coils are continuously

wound on a spool 24mm in diameter, 28mm long; the first

14mm are wound in one direction and the remaining 14mm are

counter-wound; in the figure a gap is shown in the center to

show where the winding direction changes. The receiver coils

are wound in a single layer without any space between each wire

turn. In the emitter more than one layer is often wound to

improve the sensitivity.


The LVDTs are obviously very sensitive to thecoil spool stability. In a creep measurement, wherenylon spools were used, the creep signal wasactually dominated by the spool shrinking whilethe spool itself progressively lost humidity in thedry thermally stabilized measurement chamber[23]. Ceramics or Peek coil spools have beensuccessfully used as a replacement to avoid thisproblem and to achieve the desired signal stability[16].

While the linear dynamic range of these smallerLVDTs is similar to that of the bigger LVDTversions (Fig. 7), they do not allow much lateralmovement and are used only where the mechanicalconfiguration allows only linear motion. Thesesmall LVDTs are routinely used with higher gain(in part obtained with more coil layers in theemitter and in part electronically) to drive their

position resolution below nanometer level. This isobtained at the price of a dynamic range reducedto a few mm only. Position resolution at the10�12m level has been obtained with LVDTs atthe price of proportionally, narrower dynamicranges [24].

5. Problems due to electronics

The LVDT drivers are sensitive to variations ofamplitude of the excitation signal. Special care istaken in designing the LVDT driver card in orderto generate a suitably stable excitation signal. Theinternal frequency generators for the LVDTdrivers are perfectly adequate when absolutepositioning resolution is not an issue, like forexample in mode damping feedback circuits, wheredrifts of 5mHz are irrelevant.

Thermal stabilization of the frequency generatorand of the LVDT driver card is needed whenabsolute precision is necessary. For example,thermal drifts have been found to be the dominat-ing noise source in long-term precision positionmeasurements [16].

For very high stability use, the LVDT drivercard is provided with an input for an external,amplitude-stabilized oscillator.

6. Conclusion

High-resolution and high-linearity non-contact-ing position sensors have been developed to satisfythe requirement of GW detector passive SeismicAttenuation Systems. They are fully UHV compa-tible. Only passive LVDT coils are mounted insidethe vacuum enclosure, thus guaranteeing a veryhigh level of reliability. Position resolutions belowthe nanometer, within useful ranges of a fewmillimeter and percent linearity have beenachieved. The low sensitivity of these LVDTs tomovements in the orthogonal degrees of freedomfrom the symmetry axis is particularly well suitedfor use in feedback loops. This precision positiondetector, designed for position feedback in GWSeismic Attenuation Systems, has proved useful

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m)

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als

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)

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(a)

(b)

Fig. 7. Linearity performance of the small LVDT of Fig. 6. The

error bars (voltmeter resolution) are within the marker size. The

mechanical movement used has an error of 10 m.


for many other non-contacting position measure-ments.

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