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Active Vibration Control of Gradient Coils toReduce Acoustic Noise of MRI Systems

N.B. Roozen, A.H. Koevoets and A.J. den Hamer.

Abstract—Lorentz-force induced vibrations in Magnetic res-onance imaging (MRI) systems cause significant acoustic noiselevels during scanning, the main acoustic noise source being thevibrating gradient coil. In this paper a novel active vibrationcontrol technique is presented to reduce vibrations of the gradientcoil and hence achieve a reduction of acoustic noise duringscanning. The active vibration control technique uses seismicmasses that are actuated by means of piezo actuators to createforces on the gradient coil counteracting its vibrations. Using4 seismic mass actuators, a vibration reduction of 3 to 8 dBat resonance frequencies is achieved, giving an overall vibrationreduction of 3 dB for a typical FE-EPI gradient sequence, assubstantiated by measurements. Using 8 actuators, an overallvibration reduction of 5 dB is predicted for this sequence.

Index Terms—Acoustic noise, Active vibration control, Mag-netic Resonance Imaging.

I. INTRODUCTION

MAGNETIC resonance imaging (MRI) scanners arewidely used in hospitals for both medical diagnosis

and clinical research. Nowadays magnetic field strengths ashigh as 3 Tesla are common, allowing a high spatial imagingresolution using fast scanning sequences. However, with in-creasing magnetic field strength, the acoustic noise generatedduring scanning increases also.

The acoustic noise produced by MRI systems is a causefor growing concern. Price et al. [1] give an overview ofthe acoustic noise produced by commercially available MRIsystems ranging in field strength from 0.2T to 3T. The acousticnoise levels reported varied from 85dB(A) for 0.2 - 0.5Tsystems to 115dB(A) for 3T systems in the case of fast pulsedsequences. They also point out that the increasing gradientfield levels and slew rates are pushing these values even higher,and that levels up to 130dB(A) have been reported for 3Tsystems. Needless to say, a reduction of the acoustic noiselevels is desirable to avoid discomfort or even hearing loss ofboth medical personnel and patients.

Various noise reduction techniques have been proposedin literature. Mannsfield et al. [2] proposed a revolutionarygradient coil design, using additional coils to counteract thevibrations of the gradient coil carrier. Special care has tobe taken to stiffen the gradient coil carrier, to avoid adverseeffects. Current state of the art of this technology is that a

N.B. Roozen is with Royal Philips Electronics, Philips Applied Technolo-gies, High Tech Campus 5, 5656AE Eindhoven, The Netherlands and with theEindhoven University of Technology, Department of Mechanical Engineering,P.O. Box 513, 5600 MB Eindhoven, The Netherlands.

A.H. Koevoets is with Royal Philips Electronics, Philips Applied Technolo-gies, High Tech Campus 5, 5656AE Eindhoven, The Netherlands.

A.J. den Hamer is with the Eindhoven University of Technology, Depart-ment of Mechanical Engineering, P.O. Box 513, 5600 MB Eindhoven, TheNetherlands.

significant acoustic noise reduction can be obtained of anunprecedented 50 dB within a relatively narrow frequencyrange for a given coil carrier material [3].

Edelstein et al. [4] as well as Katsunuma et al. [5] pro-pose the design of a vibration-isolated gradient coil assemblycontained in an airtight enclosure. Katsunuma et al. [5] claiman acoustic noise reduction of more than 10 dB, with noisereductions up to 23 dB if the coil is independently supported.Edelstein et al. [4] also used a low-eddy-current radio fre-quency (RF) coil and a non-conducting inner bore cryostat toreduce eddy current induced vibrations.

Moelker et al. [6] used passive sound insulation to reducethe acoustic noise radiating from the patient bore covers,leading to noise reductions up to 18.8 dB.

Active noise cancelation was first proposed by McJury etal. [7], followed shortly by Goldman et al. [8] and later onby Chen et al. [9]. The proposed technique uses anti-phaseacoustic waves to create a zone of destructive interference.In order to reduce electro-magnetic interference (EMI) andradio-frequency interference (RFI), the use of metal parts suchas copper leads, even small ones, need to be minimized. Arecent development is the use of an opto-acoustical transducerthat operates on the principle of light modulation [10]. Theseare immune from, and do not create, EMI and RFI, which isespecially important for functional MRI.

Usually feedback controllers are used for active noise can-celation. The time required for the control signal to propa-gate from the loudspeaker to the subject, and the necessityfor maintaining stability, means that the upper frequencyof control for feedback systems is commonly limited to afew hundred Hertz [10]. For this reason noise cancelationsystems are combined with passive hearing protection deviceswhich give a good attenuation at high frequencies, which iscomplementary to the noise reductions that can be obtainedby active noise control systems at low frequencies.

The effectiveness of present active noise cancelation sys-tems in conjunction with hearing protection devices that reducesound transmission through the ear canal is limited by noiseconducted through the head and body that bypasses thesetreatments [11].

An alternative approach is to use active structural acousticcontrol, in which the sound radiating structure is controlledin an active manner, leading to acoustic noise reduction in aglobal sense, and thus not suffering from the limitations dueto head and body conduction. Though there is a vast literatureon this subject in general, only a few research groups havepublished on this in relation to an MRI scanner. Trajkov etal. [12], [13], [14] report on a study to control the vibrationsof funnel-shaped covers, such as those of an MRI system,

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Fig. 1. Gradient coil assembly (coordinate system shown also)

by means of piezo-electric patches. Koevoets et al. [15], [16]have reported on measurement based techniques for improvedmodelling of piezo-electric patches in similar applications.

Roozen et al. [17] implemented an active isolation systemto support a gradient coil assembly. The goal of this workwas to reduce the acoustic noise radiated by the main magnetstructure that is excited via a structure-borne path by thevibrating gradient coil, reporting reductions of 10 dB of saidstructure-borne noise transmission path.

This paper describes a novel active vibration control systemto suppress the vibrations of a gradient coil assembly, leadingto a reduction of the radiated acoustic noise levels in a globalsense.

The paper is organized in the following manner: Sec. IIdescribes the main principles of structural excitation in MRIsystems. Analytical and experimental analysis of the structuraldynamical behavior of the gradient coil assembly is used toidentify the most relevant resonances of the gradient carrier.In Sec. III, a novel MRI compatible actuator is presented thatis used to reduce the vibrations of the gradient coil. A feed-forward control design strategy based on measured frequencyresponse function (FRF) data is presented that is able toachieve vibration reduction in a global manner. Experimentalresults presented in Sec.III-C show that a reduction of thestructural vibration levels of the gradient coil up to 10 dB atthe structural resonances of the gradient coil carrier can beachieved with the proposed approach.

II. GENERATION OF ACOUSTIC NOISE

To understand how acoustic noise is generated in a MRIsystem, some basic knowledge about a MRI system is re-quired. Basically, a MRI system consists of a main magnetsystem, a gradient coil system and a RF-coil system. Thesemain components will be discussed shortly.

The purpose of the main magnet is to create a staticmagnetic field. Extreme care is taken to ensure the uniformityof this static field in the scanning volume.

The gradient coil system (shown in Fig. 1) creates a highlylinear magnetic gradient field. Three gradient fields can becreated, i.e. the X- Y- and Z-gradient fields, which varylinearly across the X-, Y- and Z-direction, respectively. TheX-gradient coil is basically a three-dimensional saddle coil.Fig. 2 shows the windings of an X-gradient coil without carrier

Fig. 2. X Gradient coil, conductor shown only

structure. The Y-gradient coil is similar to the X-gradient coil,rotated 90 degrees. The X- and Y-gradient coils cause magneticgradients in the X- and Y-directions, respectively. The Z-coilis a rotationally symmetric coil, causing gradients in the Z-direction. By means of these gradient fields, specific slices ofthe human body can be selected and the magnetic resonancesignal can be localized [18].

Finally, the RF-coil system creates a radio frequency signalin the order of tens of MHz (e.g. 64 MHz for a 1.5 Tesla MRIsystem, its value being dependent upon the static magneticfield of the system), which excites the atomic nuclei of theselected slice at their resonance frequency.

During scanning the magnetic gradient field is switched onand off by means of alternating gradient coil currents, calledgradient sequences. As the main magnetic field is in the orderof a number of Tesla’s (currently 3T for most machines), andthe gradient coil currents in the order of 600 Amps, the Lorentzforces acting upon the gradient coil windings are in the orderof 2000 N per meter coil winding. Knowing that the total

Fig. 3. Lorentz forces on X-coil

Fig. 4. Lorentz forces on Z-coil

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Air borne soundradiation

Structure bornesound radiation

Magnetic bornesound radiation

Magneticmain field

Magneticgradient field

Gradient coilvibrations

Fig. 5. Noise transmission paths

winding length of an X- or Y-coil amounts approximatelyhundred meters, the Lorentz forces are huge, resulting invibrations with acceleration levels in the order of 100 ms−2.As a result acoustic noise is caused during scanning.

A. Noise transmission paths

The gradient coil vibrations cause acoustic noise to beradiated by the gradient coil system directly, via the air bornenoise transmission. The direct transmission path is shielded bythe covers of the system, which is in most cases insufficient.

Besides the direct radiation of the gradient coil also indirecttransmission paths exists. Via structure borne paths the cryostatstructure that supports the gradient coil is brought into vibra-tion, which then radiates acoustic noise. Attenuation of thistransmission path can be obtained by a compliant mountingof the gradient coil, which is not always possible because ofthe significant mass of the gradient coil and requirements withrespect to the position of the gradient coil relative to the patientand cryostat. An alternative approach is to use active vibrationisolation control [17].

A third transmission path exists in MRI systems, which isa magnetic borne path. The main function of the gradient coilis to create time varying gradient fields. These gradient fieldswill cause Eddy currents in electrically conducting parts ofthe MRI system, which can cause significant Lorenz forcesdue to the presence of a very strong static magnetic fieldemanating from the main magnet. Especially the vibratingcryostat structure, encompassing the main magnet, will vibrate

1

XY

Z

Fig. 6. Finite element model of half a gradient coil structure

significantly because of this transmission path. Tackling themagnetic borne path was done by Biloen et al. [19] andEdelstein et al. [20] using a copper screen wrapped aroundthe gradient coil to shield the cryostat structure from magneticfields radiated by the gradient coil. The effectiveness of themagnetic shielding was about 20dB for the cryostat borestructure, and about 10dB for the front and back sections ofthe cryostat structure. In Fig. 5 the three transmission pathsare summarized.

The relative importance of the transmission paths changeswith system design. Whilst for systems with an aluminiummagnet housing the magnetic transmission path can be signi-ficant, for systems having a stainless steel magnet housing thistransmission path is less strong. The direct acoustic radiationof the gradient coil strongly depends upon the central borecovers. For most MRI systems the direct air borne transmissionpath is the most dominant path as the relatively thin coversused in the patient bore can not easily reduce the acousticnoise. Therefore the focus of this paper is on the air bornetransmission path. The aim is to reduce the vibrations of thegradient coil itself by means of an active vibration suppressionsystem, so as to reduce the usually dominant air borne noisetransmission path.

B. Gradient coil dynamics and acoustic noise

In order to design the active vibration suppression system,the excitation of the structural dynamics of the gradient carrieris analyzed. This knowledge will be used in Sec. III-A foractuator placement.

For analysis of the structural dynamics of the gradientcarrier, the concepts of structural mode-shapes is used [21]. Inorder to apply this technique, it is assumed that the structuraldynamic behavior is lightly damped or satisfies Raleigh-damping. As a result, the dynamic response can be writtenas a super-position of structural mode-shapes:

Yi(jω) =∑

k

φiφj

s2 + 2ξkωnk+ ω2

nk

Uj(jω) (1)

where φj and φi represent respectively the mode-shape atthe location with index j where force Uj(jω) is appliedand the location with index i where the displacement Yi(jω)is measured. ωnk

and ξk represent respectively the modal

Fig. 7. Gradient coil bending mode shape

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Fig. 8. Gradient radial mode shape

eigenfrequency and relative damping of mode k. It can beobserved from Eq.(1) that the frequency response of a certainpoint depends on the inner-product between a force distribu-tion and the mode-shape, and the inner-product between theobservation point and the mode-shape. Since we are interestedin vibration suppression over the entire structure, we willmainly focus on the first part, i.e. the inner-product betweenforce distribution and structural mode-shape. Though a largenumber of structural modes exist, only a few modes of thegradient coil structure are excited by the Lorentz forces dueto the specific spatial distribution of these forces. The mode-shapes that are excited most dominantly will be discussedshortly. This analysis is based on numerical simulations usingthe finite element method (FEM).

The Galerkin finite element method is used to predict thedynamic behavior of the gradient coil structure. Half thestructure is modeled with 4-noded isoparametric tetrahedralelements (ANSYS element type ”Solid92”), exploiting sym-metry in Y-direction (see Fig. 6). Using tetrahedral solidelements, the high stiffness of the gradient coil structure istaken into account correctly. As the gradient coil structureconsists of many different materials, each material layer ismodeled separately, having appropriate material propertiesassigned to it. Some of the structural eigenmodes that wereextracted with this model are shown in Fig. 7, Fig. 8 andFig. 9.

The Lorentz force distribution being developed in the X- andZ-coil windings are schematically depicted in Fig. 3 and Fig. 4,respectively. This explains the relative effective excitation ofthe mode as shown in Fig. 7 by the Lorentz force distributionof the X- and Y- coils since the inner product of Lorentz forcesfor the X- and Y-coil and this bending mode is quite high. Twobending modes exists for a gradient coil structure, one whichis vibrating predominantly in the X-direction and the otherwhich is vibrating predominantly in the Y-direction. Besidesthe bending modes also other modes are excited by the X- andY-coils. One such mode is shown in Fig. 9, which has a modeshape which shows both bending and radial movements.

The Lorentz force distribution of the Z-coil typically excitesthe radial mode as shown in Fig. 8. This mode is an axi-symmetric mode, which is one of the reasons why this modeis excited by the axi-symmetric Lorentz forces created by theZ-coil. The Lorentz forces acting upon the Z-windings are

Fig. 9. Gradient coil square mode shape

shown in Fig. 4. It is again the inner product of the Lorentzforces of the Z-coil and this radial mode which is quite high,resulting in a strong excitation of this specific mode by theZ-coil.

Experiments were carried out to allow a comparison of thenumerical results to experimental results. Frequency responsefunction estimates (FRF’s) were measured from the X-, Y- andZ- gradient coil currents to the structural velocity at 120 posi-tions equally distributed on the inner wall of the gradient coilcarrier. The measurements were performed using a 4 channelSiglab digital signal analyzer, a Bruel and Kjaer conditioningamplifier type Nexus and three Bruel and Kjaer accelerometerstype 4393V. The FRF’s are presented in arbitrary units (a.u.)for both the frequency axis and the response axis.

The dominance of the bending mode and combinedbending-radial mode for the X- and Y-coil, as shown in Fig. 7and in Fig. 9 respectively, as well as the dominance of theradial mode for the Z-coil excitation as shown in Fig. 8is clearly visible in the measured structural response of thegradient carrier. Fig. 10 shows the measured FRF’s from theY-gradient coil current to the structural velocity at the 120positions on the gradient coil. The spectra for the X-coilexcitation are quite similar. Fig. 11 shows the FRF from theZ-coil input to the gradient coil acceleration. In Fig. 10 andFig. 11, a number of dominant peaks in the spectrum aremarked with the letters A through E, which correspond tothe operational deflection shapes (ODS) given in Fig. 12 andFig. 13. These figures show the response of the gradient coil atthe specified resonance frequencies A through E as measuredat the 120 postions on the inner side of the gradient coil carrier.

The resonance peaks A and B correspond to bending modes,as shown in Fig. 7. The asymmetry caused by the gradientsupport structure, which was not incorporated in the analyticalmodel, produces two bending modes having different eigen-frequencies. Their operating deflection shapes for Y-gradientcoil actuation are shown in Fig. 12.

Resonance peak C corresponds to the combined bending-radial mode shape, its operational deflection shape shown inFig. 13 and the corresponding numerical mode shown in Fig. 9.The symmetry of the combined bending-radial mode shownin Fig. 9 suggests that this mode will hardly be excited bythe X- or Y-coils due to reasons of symmetry, causing theinner product of Lorentz forces for the X- and Y-coil and the

5

Frequency [a.u.]

Str

uctu

ral v

eloc

ity [a

.u.] 10 dB A

BC

D

Fig. 10. FRF’s from Y-gradient current to structural velocity at 120 differentpositions on the gradient coil carrier in radial direction, in arbitrary units (a.u.)

Frequency [a.u.]

Str

uctu

ral v

eloc

ity [a

.u.] 10 dB

E

Fig. 11. Frequency response functions from Z-gradient current to structuralvelocity at 120 different positions on the gradient coil carrier in radial direction

square mode to vanish. In practice it is excited, again due tothe asymmetry caused by the gradient support structure. Notethat peak C is less significant than peak B or D.

Resonance D corresponds to a second order bending mode,having a higher spatial wavenumber in the axial direction, asshown in Fig. 13.

Resonance E (Fig. 11) corresponds to the radial mode ofwhich its shape does not vary as function of the circumferentialcoordinate. Its operation deflection shape for Z-coil excitationis given in Fig. 13. The corresponding mode shape obtainedvia FEM analysis is shown in Fig. 8.

III. ACTIVE VIBRATION CONTROL GRADIENT COIL

The design of an active vibration control system needs tofulfill a number of requirements for application in an MRIsystem. First of all the actuator needs to be MR compati-ble, meaning that it may not disturb the magnetic field asthis would degrade image quality. This requires the use ofnon-magnetic materials, such as aluminum or specific, non-magnetic, stainless steel materials and plastic materials ifpossible.

In addition, the currents that are possibly used for actuationmay not disturb the magnetic field as well. Bearing in mindthat the forces required to counteract the vibrations of thegradient coil are in the order of a few hundred Newtons, this

A BFig. 12. Operational Deflection Shape at frequencies A and B, showingbending mode responses

C D

EFig. 13. Operational Deflection Shape at frequencies C, D and E, showinghigher order bending and radial mode responses

will not be an easy task. Lorentz actuators are for this reasonnot permitted. Piezo actuators, however, are MRI compatible,as the currents that are used to actuate the piezo are very small.Piezo’s have a very high electrical impedance, and it is onlythe electrical charge that is actuating the piezo. The PZT piezomaterial as such also appears to be MRI compatible.

Furthermore the actuator needs to be compact. The firstreason for this requirement is a practical reason: there islittle space available. Covers are often close fitted around thegradient coil and cryostat, limiting the space to centimeters.Another reason for this requirement is that a compact designwill not suffer from parasitic resonances in the frequencyrange of operation as these resonances will degrade actuatorefficiency.

The actuator should be able to deliver a force of sufficientmagnitude over a broad frequency range. As argued earlierthe actuation force is in the order of a few hundred Newtons.This force should be delivered at frequencies between 500 and1500 Hz. In practice the strongest acoustic noise levels occurwithin the mentioned frequency range.

Finally, the design should be such that it is ”force balanced”,meaning that the forces that are created by the actuator areactuating upon the gradient coil only, and not upon other partsof the MRI system. The reason for this requirement is thatsignificant vibrations in the ”other parts” of the MRI systemcould occur otherwise, resulting in a significant contributionto the acoustic noise levels, which is obviously not desired.

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Fig. 14. CAD drawing of seismic mass piezo actuator

Fig. 15. Individual parts seismic mass piezo actuator

Fig. 16. Seismic mass piezo actuator mounted on gradient coil carrier

All the above requirements led to the design of a seismicmass piezo actuator, which is described in the next section.

A. Seismic mass piezo actuator design

The basic principle used in the seismic mass piezo actuatordesign is the principle of inertia. A piezo actuator is mountedbetween the gradient carrier and a seismic mass, its acceler-ation causing an inertial force. This design is compact, forcebalanced and MRI compatible.

Fig. 14 shows a CAD drawing of the seismic mass piezoactuator. The base plate is mounted on the gradient coil ends.Actuating the piezo will cause a force being transferred to thegradient coil through the base plate. Guidance plates are used

to guide the seismic mass during its movements. Fig. 15 showsthe individual parts of the actuator.

The locations of the actuators on the gradient carrier struc-ture are chosen such that all dominant modes, as discussedin Sec. II-B are controllable with the actuators, i.e. the inner-product between the mode-shape and the force generated bythe actuators is relatively large (see Eq.(1)). However, practicalconsiderations constrain the placement of the actuators. Due toavailability of construction space in axial direction, placementof the actuators in the Z-direction is preferred.

Force actuation in axial direction seems a little bit strangeat first sight. It should however be mentioned that all relevantmodes of the gradient coil structure, such as the bendingmode shown in Fig. 7, clearly vibrate in both radial and axialdirections and are therefore controllable via forces in axialdirection.

Four seismic mass piezo actuators were mounted on thegradient coil, two mounted at the front and two mounted atthe back of the gradient coil, one at the left hand and one atthe right hand side. Fig. 16 shows the front right actuator. Theforces exerted on the gradient coil are directed in axial (Z)direction.

B. Controller designThe amount of acoustic noise radiated from the gradient

coil is directly related to the velocities at the surface of thegradient coil [22]. The squared velocity integrated over the coilsurface and time is therefore chosen as a relevant measure forthe amount of radiated noise. Hence, the following objectiveis used for control design:

minupiezo-act(t)

∫ ∞

−∞

∫ π

−π

∫ zmax

zmin

v(φ, z, t)2 dφ dz dt (2)

where v(φ, z, t) represents the velocity at the surface of thecoil at radial location φ, axial position z at time t andupiezo−act is the vector of input signals to the piezo-actuators.

In order to achieve this goal, both feedback and feedforwardcontrol strategies for vibration suppression can be considered[23], [24], [25]. Due to several restrictions imposed by feed-back (summarized below), it is motivated that feedforwardtechniques better fit the characteristics of this particular ap-plication. A summary of properties of feedback systems forthis application is given.• Analytical properties of closed-loop feedback systems,

e.g. Bode’s sensitivity integral, make that disturbances aretypically amplified in the high frequent region, i.e. abovethe bandwidth. This requires closed-loop bandwidthswhich are significantly higher than the frequency contentof the sequences applied on the gradient coils, typicallywithin the frequency range from a few hundred Hertz upto 1500 Hz. The need for a high bandwidth poses tightrequirements on the dynamics of the equipment used inthe feedback loop and is expected to result in high cost.

• Creating a sensor that represents a relevant performancemeasure for noise radiation is not straightforward.

• Knowledge about the main disturbance source permitsnon-causal control strategies which is not exploited withfeedback.

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• The high modal density observed in the feedback loopmakes that feedback control design for performance (oreven stability) is not straightforward.

Due to the complexity of the system, a data-based feedforwardapproach is pursuit such that modeling of the system can beomitted. To enable practical evaluation, Eq.(2) is discretizedin space where the coil is subdivided in a grid of 12 segmentsin φ direction and 10 in z direction:

minupiezo-act(t)

12∑ı=1

10∑

j=1

∫ ∞

−∞v(φi, zj , t)2 dt (3)

It is expected that the resulting distance between grid-pointsis small compared to the modal wavelengths in the relevantfrequency region, such that all relevant modal shapes are wellobserved.

For control purpose, linearity of the system and especiallyof the piezo actuators is assumed. It will be seen later that thisrestriction can be relaxed for the piezo actuators. As a result,Eq.(3) can be written in the frequency domain via Parseval’stheorem:

minupiezo-act(t)

12∑ı=1

10∑

j=1

∫ ∞

−∞|V (φi, zj , ω)|2 dω (4)

where V (jω) represents the Fourier transformed signal v(t).It will be derived that for repeating sequences, the feedfor-

ward signal to the piezo-actuators, upiezo-act(t) can be computedbased on the measured FRF-data and knowledge of the gra-dient sequences. The resulting inputs are optimal in the senseof Eq.(4).

According to the frequency separation principle (no feed-back is applied such that Bode’s sensitivity integral does nothold), the objective given in Eq.(4) can be minimized perfrequency:

minupiezo-act(t)

V̄ †(jω)V̄ (jω) ∀ ω (5)

where V̄ (jω) is a vector of velocities over the grid of 120points, V̄i·j = V (φi, zj) and .† represents the conjugatetranspose of (.).

The output of the system V̄ (jω) can be written as functionof the inputs to the gradient coils and piezo-actuator via:

V̄ (jω) = Hcoil(jω)

Ux(jω)Uy(jω)Uz(jω)

+Hpiezo(jω)

U1(jω)...

Un(jω)

(6)

where Hcoil(jω) and Hpiezo(jω) represent the frequency re-sponse function matrix between actuator inputs and velocitiesat the coil surface which can be obtained from measurements.[Ux, Uy, Uz]T and [U1, . . . , Un]T represent respectively theinput to the gradient coils and the piezo actuators. Substitutionof Eq.(6) in Eq.(5) gives:

minUpiezo-act(t)

∥∥∥∥∥Hcoil(jω)

Ux(jω)Uy(jω)Uz(jω)

︸ ︷︷ ︸−b

+Hpiezo(jω)︸ ︷︷ ︸A

U1(jω)...

Un(jω)

︸ ︷︷ ︸x

∥∥∥∥∥2

(7)

Since the input-signals to the coils [Ux, Uy, Uz]T are knowna priori, Eq.(7) can be solved via a least-squares approximationproblem of the form minx ‖Ax − b‖2. The solution is givenby (for compactness, dependence on jω is omitted):

U1

...Un

= −(H†

piezoHpiezo)−1H†piezoHcoil

Ux

Uy

Uz

(8)

If the signals are repeating over time, which is mostcommon for MRI sequences, the vector [Ux, Uy, Uz]T canbe described by a finite number of frequency domain co-efficients and thus has a finite length. As a result, Eq.(8)can be solved off-line. The feedforward signal that minimizesEq.(3) is obtained by applying a fast fourier transform on[ux(t), uy(t), uz(t)]T over the repeating time-interval. Thisdata is substituted in Eq.(8) and the resulting [U1, . . . Un]T

are transformed back into the time domain via an inverse fastfourier transform to obtain upiezo-act(t).

1) Remarks: In this work we chose to minimize thequadratic sum of the structural velocities as this is a reasonablemeasure for the acoustic noise radiated by the gradient coil.A better acoustic performance could result by weighing thevelocities by the acoustic radiation modes, a concept that wasintroduced by Cunefare et al. [26],[27],[28], and first appliedto the acoustic radiation from MRI gradient coils by Kuijperset al. [29].

Active suppression of gradient coil vibrations during typicaloperation using gradient coil currents of 100 Amps requiressignificant actuator forces. Depending on the frequency ofactuation, these forces demand piezo displacements in theorder of 10 µm. Such high levels of displacements can berealized by means of piezo stacks, but generally give riseto non-linear behavior of the piezo. In [30] a lifted Itera-tive Learning Control approach was introduced to cope withthis problem. This technique suppresses the super-harmonicresponse of the seismic mass piezo actuator by exploiting therepetitive behavior of the gradient coil sequences. Using thisiterative learning control approach, it can be assumed that theresponse of the piezo stack behaves linearly after a few trials.Hence, even under high excitation conditions of the piezo’s,Eq.(8) can be applied.

C. Experimental results

Four seismic mass piezo actuators were used in the exper-iments to reduce the vibration levels of the gradient coil. Thefront right seismic mass piezo actuator is shown in Fig. 16.Similar actuators were implemented at the front left, rear leftand rear right of the gradient coil.

In order to apply Eq.(8), the FRF’s from [Ux, Uy, Uz] to V̄and [U1, . . . , Un] to V̄ are measured [31]. This is performedusing white noise excitation on the input of the piezo actu-ator while measuring the velocities with an accelerometer.A dSpace control system combined with high-voltage piezoamplifiers is used to generate the feedforward signals anddrive the piezo actuators. In Fig. 17 the FRF’s from onepiezo actuator to the accelerations in radial direction of all

8

Frequency [a.u.]

Str

uctu

ral v

eloc

ity [a

.u.] 10 dB

Fig. 17. FRF’s from the front left piezo actuator to the structural velocityat 120 different positions on the gradient coil carrier, in radial direction

Frequency [a.u.]

Str

uctu

ral v

eloc

ity [a

.u.]

10 dB

piezo actuatorZ−gradient

Fig. 18. FRF’s of the seismic mass piezo actuator (in red solid) and Z-gradient coil (in blue dashed) to the structural velocity of the gradient coilcarrier

120 grid points are plotted in arbitrary units (a.u.) for both thefrequency axis and the response axis. It can be observed thatHpiezo(jω) increases with frequency. This is caused by thefact that the piezo stack can be seen as position actuator due toits high stiffness. As a result, the inertia forces created by theseismic mass increase with frequency for a fixed maximumdisplacement of the piezo actuators. This property suits theapplication well since Hcoil(jω) also shows an increase asfunction of frequency (see Fig. 10 and Fig. 11).

Minimizing the objective function as described by Eq.5, using four seismic mass piezo actuators simultaneously,reduces the spatially averaged response of the gradient coilvibrations for Y-gradient fields by 3 to 8 dB at the dominantfrequencies (see Fig. 19). For Z-coil gradients the reductionsare in the order of 3 to 4 dB at the dominant frequencies, asshown in Fig. 21.

The reduction of the structural vibrations that can beachieved by means of active vibration control is limited fortwo reasons: the limited stroke of the actuators and the factthat the actuating forces are discrete in nature. By means ofthe four seismic mass actuators discrete forces are introduced,which will excite other structural modes of the gradientcoil carrier than the gradient coil current induced Lorentzforces do. In general, more modes will be excited by thepoint forces introduced by the seismic mass actuators. These

Frequency [a.u.]

V̄† (jω)V̄

(jω)

[a.u

.]

5 dB

Y−gradient

Y−gradient+4 piezo actuators

Fig. 19. FRF’s of Y-gradient coil current to the squared sum of gradientcarrier velocities, uncontrolled (red solid) and controlled using 4 actuators(blue dashed)

Frequency [a.u.]

V̄† (jω)V̄

(jω)

[a.u

.]

5 dB

Y−gradient

Y−gradient+8 piezo actuators

Fig. 20. FRF’s of Y-gradient coil current to the squared sum of gradientcarrier velocities, uncontroled (red solid) and controlled using 8 actuators(simulated, blue dashed)

additional modes have a higher spatial wavenumber, comparedto the gradient coil current induced vibrational modes. Thisis illustrated in Fig. 18, which shows the structural responseof the gradient coil carrier for Z-gradient current excitationand for piezo actuator excitation. From this figure it can beseen that the gradient current induced Lorentz forces excitea limited number of eigenmodes, whereas the piezo actuatorsexcite many more modes, having high spatial wavenumbers.Whereas in theory the modal contribution of a specific modecan be canceled completely, in practice this is of little value asother modes will then more strongly contribute to the structuralresponse. In effect, the reduction of the spatially averagedstructural response is limited, as shown in Fig. 19 and Fig. 21.

Increasing the number of seismic mass piezo actuators willresult in higher reductions of the spatially averages structuralgradient coil response. To quantify this effect a number ofsimulations, based on measurement data, have been performedusing virtual piezo actuators. To this end the FRF’s betweenthe force exerted by a virtual piezo actuator and the struc-tural response was experimentally determined by means ofan impact hammer. Using these FRF’s, combined with theactual data of the four physically present seismic mass piezoactuators, the vibration reduction that can be achieved usingeight actuators was computed. The results are shown in Fig. 20

9

Frequency [a.u.]

V̄† (jω)V̄

(jω)

[a.u

.]

5 dB

Z−gradient

Z−gradient+4 piezo actuators

Fig. 21. FRF’s of Z-gradient coil current to the squared sum of gradient coilcarrier velocities, uncontrolled (red solid) and controlled using 4 actuators(blue dashed)

Frequency [a.u.]

V̄† (jω)V̄

(jω)

[a.u

.]

5 dB

Z−gradient

Z−gradient+8 piezo actuators

Fig. 22. FRF’s of Z-gradient coil current to the squared sum of gradient coilcarrier velocities, uncontrolled (red solid) and controlled using 8 actuators(simulated, blue dashed)

and Fig. 22 for Y- and Z-coil excitation, respectively. Fromthese figures it can be seen that the achievable reductions rangefrom 7 to 10 dB for Y-coil excitation and from 3 to 7 dB forZ-coil excitation.

In the preceding text the reduction of the gradient coilstructural vibrations was discussed in terms of frequencyresponse functions. To estimate the reduction of the gradientcoil structural vibrations in practice, the response of thegradient coil was calculated for a typical gradient coil currentas function of time. For this purpose an EPI sequence wasemployed (see Fig. 23) which is known for its high acousticnoise levels [5]. The resulting response of the gradient coilstructure in the frequency domain is shown in Fig. 24, withoutand with control, using 4 actuators. The cumulative powerspectrum shows a reduction for the entire frequency rangestudied of 3 dB. Using 8 actuators 5 dB reduction is obtained.

IV. CONCLUSION

Noise generation in MRI systems is a growing cause forconcern. As magnetic field strengths and gradient slew ratesincrease, so does the noise produced by the system. The rootcause of the noise generation lies in the interaction betweenthe alternating currents in the gradient coil system and thestatic main magnet field. The Lorenz forces exerted on the

Time [a.u.]

Gra

dien

t coi

l cur

rent

[a.u

.]

Fig. 23. Typical gradient coil sequence (FE-EPI, Y-coil)

V̄† (jω)V̄

(jω)

[a.u

.]

5 dB

Frequency [a.u.]

cum

sum

[a.u

.]

Y−gradient FE−EPI

Y−gradient FE−EPI + 4 piezo actuators

Fig. 24. Gradient coil carrier response due to FE-EPI input sequence, velocityautospectrum (top) and cumulative power spectrum (bottom)

gradient coils lead to vibrations and, via various transmissionpaths, to noise radiation. In most MRI systems the acousticnoise radiated by the gradient coil carrier itself, the so-calleddirect transmission path, is most dominant.

In this paper, the design of an active vibration suppressionsystem is proposed which is applied directly to the gradientcarrier and hence reduces acoustic noise emission in a directmanner. As a first step, the structural dynamics of the gradientcoil carrier are analyzed to identify the dominant structuraldynamics and hence determine the optimal actuator placement.In order to generate the forces that counteract the gradientvibrations, a MRI compatible actuator is proposed whichconsist of a piezo actuator mounted between an inertial massan the gradient coil. This setup is compact and well suited todeliver forces in the relevant frequency region between 500Hz and 1500 Hz.

A feedforward control design strategy is described thatexploits the repetitive behavior of MRI sequences to generatean optimal input signal for the seismic mass piezo actuatorsbased on measured frequency response data solely such thatmodeling can be omitted.

Experimental results using 4 actuators show that significantreduction of the vibration levels can be achieved ranging from3 to 8 dB at the resonance peaks, resulting in a reduction of 3dB for the entire frequency range studied, using a typical FE-EPI input sequence. The proposed approach however has thepotential to achieve even better performance if more actuatorsor more distributed force patterns are applied. More seismicactuators will represent the Lorenz force distribution better,and therefore counteract the modes excited by the Lorenzforces more effectively while exciting less other modes. A

10

simulation study with 8 actuators showed reductions rangingfrom 7 to 10 dB at the resonance peaks, resulting in a reductionof 5 dB for the entire frequency range.

ACKNOWLEDGMENT

The authors would like to thank Philips Medical Systems,and in particular Kees Ham, Hans Tuithof and Peter van derMeulen, for facilitating and financing the research describedin this paper.

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N.B. (Bert) Roozen received the Masters degree inmechanical engineering in 1986 from the EindhovenUniversity of Technology, The Netherlands. From1986 to 1995 he worked at Fokker Aircraft inAmsterdam, were he also performed his PhD studyon the sound transmission through aircraft structures.He obtained his PhD in technical sciences fromthe Eindhoven University of Technology in 1992.Since 1995 he works for Royal Philips Electronicsin Eindhoven, the Netherlands, starting at the PhilipsResearch Laboratories and now working at Philips

Applied Technologies as chief technologist. Since 2003 he serves as a part-time professor Acoustics and Noise Control at the Eindhoven University ofTechnology. His research interests are in the field of noise source identi-fication by means of nearfield acoustical holography, numerical simulationof structural-acoustic interactions, tyre-road noise, active structural-acousticcontrol and noise control engineering in general. He is President of theAcoustical Society of the Netherlands (NAG) and Director of the InternationalInstitute of Acoustics and Vibration (IIAV).

A.H. (Marco) Koevoets received the M.Sc. degreein mechanical engineering from the Eindhoven Uni-versity of Technology in 2001. He is employed asTechnologist at Philips Applied Technologies withinthe System Analysis Group since 2001. His firstresearch area was active vibration control for thereduction of sound in MRI scanners. Later he par-ticipated in the development of a next generationLithography Scanners focusing on the system dy-namics and control aspects. His latest research in-terest includes compensation of thermo-mechanical

induced deformation in high precision devices. He has (co-) authored severaltechnical papers and patent filings in these fields.

A.J. (Arjen) den Hamer received his M.Sc. De-gree (Cum Laude) in Mechanical Engineering fromthe Eindhoven University of Technology in 2005.He performed his M.Sc. project in the area ofcontrol design for active vibration suppression sys-tems. Acoustic noise reduction of MRI systems wasamong the application fields of this project. SinceSeptember 2005, he is working as a PhD studentin the Control Systems Technology group at theEindhoven University of Technology in the field ofnon-parametric frequency domain optimal controller

synthesis, granted by Philips Applied Technologies.