Research Article Optimal Value of Series Capacitors for ...downloads.hindawi.com/journals/js/2016/3480965.pdf · Transmission lines are o en used as coils in high eld magnetic resonance

Research ArticleOptimal Value of Series Capacitors for Uniform FieldDistribution in Transmission Line MRI Coils

Vitaliy Zhurbenko

Technical University of Denmark 2800 Kongens Lyngby Denmark

Correspondence should be addressed to Vitaliy Zhurbenko vzelektrodtudk

Received 13 August 2015 Revised 20 October 2015 Accepted 4 November 2015

Academic Editor Guiyun Tian

Copyright copy 2016 Vitaliy Zhurbenko This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Transmission lines are often used as coils in high field magnetic resonance imaging (MRI) Due to the distributed nature oftransmission lines coils based on them produce inhomogeneous field This work investigates application of series capacitorsto improve field homogeneity along the coil The equations for optimal values of evenly distributed capacitors are derived andexpressed in terms of the implemented transmission line parameters The achieved magnetic field homogeneity is estimated underquasistatic approximation and compared to the regular transmission line resonator Finally a more practical case of a microstripline coil with two series capacitors is considered

1 Introduction

Magnetic resonance (MR) imaging at high field strength canpotentially offer higher spatial andor temporal resolutionthen conventional (low field) MRI systems These benefitswill facilitate a significant increase in diagnostic accuracyfor certain medical applications There are however severaltechnological challenges associated with high field operationOne of them is high magnetic resonance frequency whichleads to inhomogeneity of the alternating magnetic field dueto the short wavelength of the field This calls for technolog-ical improvements to meet the demands for magnetic fielddistribution

The purpose of MR coils is to generate and sense alter-nating magnetic fields in MR systems While loop coils are aconventional choice for low field (low frequency)MR systems[1ndash3] transmission line sections operating in standing wavemode are often used in high field systems [4ndash6]Themaximaand minima of the standing wave result in inhomogeneousfield generated by the coil Such inhomogeneity leads todegradation of MR image quality

Several approaches to improve homogeneity have beensuggested in the literature These include implementation ofalternating impedance transmission lines [7 8] metamaterialinspired structures [9 10] and substituting coil with severalshorter coils with individual feeding [11] In this work the

approach of inserting series capacitors into transmission linecoil is investigated Series capacitors compensate for the phaseshift in self-inductance of the transmission line section Thiscompensation results in a more uniform current distributionon the lineTheuniform current distribution in its turn leadsto homogeneous magnetic field The equations for optimalvalues of evenly distributed series capacitors are derivedTheachieved homogeneity of themagnetic field is estimated usingBiot-Savartrsquos law and compared to a regular transmissionline resonator An example of a transmission line coil usingsuspended microstrip technology is considered

2 Coil Design for UniformCurrent Distribution

In order to boost the sensitivity of transmission line coilsthey are usually operated as resonators Hence the loadingof transmission line coils is either open-circuit short-circuitor purely reactive Such a loading results in infinite stand-ing wave ratio (assuming lossless case) where the distancebetween the consecutive minima or maxima is one-half awavelength [12] To avoid dark spots on MR image andachieve reasonable homogeneity current minima should beavoided For that reason coils based on transmission linesare almost always shorter than half a wavelength and proper

Hindawi Publishing CorporationJournal of SensorsVolume 2016 Article ID 3480965 7 pageshttpdxdoiorg10115520163480965

2 Journal of Sensors

capacitive loading is used to achieve symmetric currentdistribution along the line [7 8 13ndash15] An example ofa normalized current distribution |119868(119911)| on a 25 cm longtransmission line at 298MHz is shown in Figure 1 (dashedred curve)

The current reduces at the ends of the line due todestructive interference of the incident and reflected waves

It is however preferable to have a uniform currentdistribution That would result in a homogeneous magneticfield along the line

A more uniform current distribution can be achievedby inserting series capacitors 119862

119904into the transmission line

(Figure 1(b)) These capacitors can compensate for self-inductance of the transmission line by introducing oppositephase shift The loading capacitor 119862

119871 in its turn should

be chosen to provide symmetry of the current distributionThis structure (Figure 1(b)) results in a smaller standingwave ratio than in the conventional case (Figure 1(a)) Anexample of current distribution on a line using two seriescapacitors is shown in Figure 1(c) (solid blue curve) As canbe seen the resulting uniformity of the current distributionis considerably improved

Obviously the higher the number of capacitors themore uniform the current distribution can be achieved andconsequently the higher the magnetic field homogeneity willbe

The value of the series capacitor119862119904should be chosen such

that it compensates the phase shift introduced by one sectionof the transmission line (Appendix B)

119862119904=

sin (120573119897119873)

412058711989101198850(1 minus cos (120573119897119873))

(1)

where 119897 is the total length of the transmission line coil 119873 isthe number of transmission line sections after inserting seriescapacitors (eg using two series capacitors will result in 119873 =

3) 120573 is the phase constant1198850is the characteristic impedance

of the implemented transmission line and 1198910is the operating

frequency A frequency of 298MHz is used in all exampleshere which is close to operating frequency ofMR systems forhydrogen imaging with 7 T magnetic field

The expression for the loading capacitor which will resultin the optimal current distribution is derived imposing cur-rent symmetry condition for 119897119873 length line (Appendix A)

119862119871= 2119862119904=

sin (120573119897119873)

212058711989101198850(1 minus cos (120573119897119873))

(2)

For example for 50Ω line the calculated values of thecapacitors using (1) and (2) at 119891

0= 298MHz 120573 = 728 and

119873 = 3 are 119862119871asymp 342 pF and 119862

119904asymp 171 pF

Having the values of the capacitors and knowing theparameters of the transmission line current distribution canbe foundusingwell-developed transmission line analysis [12]The current on 119899th transmission line section

119868119899(119911) =

119881+

0119899

1198850

(119890minus119895120573(119911+(119873minus119899)119897119873)

minus Γ119871119899

119890119895120573(119911+(119873minus119899)119897119873)

) (3)

where Γ119871119899

and 119881+

0119899are the load reflection coefficient and

amplitude of the incident wave at the end of 119899th transmission

0

IN Z0 CL

minuslz

(a)

0

IN Z0 Z0Z0

minusl

n = 1 n = Nn = N minus 1

ZinN

minus2lN

z

minuslN

CL

CsCs

middot middot middot



(b)

2

07

08

09

06

10

|I||I|m

ax

minus26 minus22 minus18 minus14 minus10 minus6 minus2

With 2 series capacitors Cs (Figure 1(b) (N = 3))Without series capacitors (Figure 1(a))

z (cm)

(c)

Figure 1 (a) Regular transmission line resonator (b) Transmissionline coil with series capacitors 119862

119904 (c) Corresponding normalized

magnitude of the current distribution Here 119911 is the longitudinalcoordinate 119897 is the total length of the coil and 119873 is the number ofsections

line section respectively 119899 = 1 sdot sdot sdot 119873 The total current 119868(119911)is then a combination of currents on all transmission linesections 119868

119899(119911)

The amplitude of the incident wave at the end of 119899thtransmission line section

119881+

0119899=

119881119892

2

(1 minus Γ119892) 119890minus119895120573119897119873

1 minus Γ119892Γ119871119899

119890minus1198952120573119897119873 when 119899 = 1

119881in119899

119890119895120573119897119873 + Γ119871119899

119890minus119895120573119897119873 when 2 le 119899 le 119873

(4)

Here 119881119892

is the voltage of the generator which excitesthe transmission line coil 119881in119899 is the total voltage at theinput terminals of 119899th transmission line section Generatorreflection coefficient Γ

119892= (119885119892minus1198850)(119885119892+1198850) where119885

119892is the

impedance of the generator Reflection coefficient at the end

Journal of Sensors 3

of 119899th transmission line section Γ119871119899

= (119885119871119899

minus1198850)(119885119871119899

+1198850)

where

119885119871119899

=

1

11989521205871198910119862119871

when 119899 = 119873

1

11989521205871198910119862119904

+ 1198850

119885119871119899+1

+ 1198951198850tan (120573119897119873)

1198850+ 119895119885119871119899+1

tan (120573119897119873) when 1 le 119899 le 119873 minus 1

(5)

The field strength due to 119868(119911) can be estimated using Biot-Savart law

|H (119911 119903)| =119903

4120587int0

minus119897

10038161003816100381610038161003816119868 (1199111015840

)100381610038161003816100381610038161198891199111015840

(1199032 + [1199111015840 minus 119911]2

)32

(6)

where 119903 is the perpendicular distance to the conductorcarrying the current from the point of observation Since themagnetic flux density is proportional to the magnetic field(B = 120583H where 120583 is the permeability) it is possible toestimate the produced B field Integration in (6) is limited tominus119897 sdot sdot sdot 0 since there is no current outside this region

Equation (6) is used to estimate the field distributionsdue to currents in Figure 1(c) This would allow assessing theachieved field homogeneity for the conventional case and thecase with series capacitors The results for 119903 of 3 cm and 6 cmare shown in Figure 2

The presented data indicates that the insertion of seriescapacitors considerably improves homogeneity of the fieldFor example at 119903 = 3 cm homogeneity which is defined as aratio

min |B (119911)|

max |B (119911)| (7)

increases from 31 to 49Since the field strength reduces at the ends of the line

(Figure 2) a higher degree of homogeneity can be achievedby reducing the field of view (the imaging domain abovethe coil) For example limiting the field of view to 119911 =

minus5 cm sdot sdot sdot minus 20 cm will result in increase in homogeneity from81 (Figure 4(a)) to 95 (Figure 4(b))

It is important to keep in mind that the presented fielddistribution in Figure 2 is not for the entire transmission linebut only due to the current distribution |119868(119911)| in Figure 1 Anaccurate field distribution for more practical coil realizationsshould be obtained by full-wave simulations An example ofa microstrip line coil with two distributed series capacitors isconsidered in the following section

3 Example of a Microstrip Line Coil

Thedesign approach described above is in general applicableto any coil based on TEM or quasi-TEM transmissionline A more realistic example of a transmission line coilusing suspended microstrip technology is considered inthis section The impedance of the implemented suspended

microstrip transmission line is 50Ω The height (distancebetween the ground plane and the suspended Printed CircuitBoard (PCB)) ℎ is chosen such that the PCB is convenientlysupported by the feeding SMA connector and is equal to415mm The implemented PCB is based on FR4 laminatesheet with thickness ℎ

119904= 155mm The top and side views

of the regular 50Ω line using this technology are shown inFigure 3(a)

The width 1198821of the 50Ω line is found using commer-

cially available models for microstrip lines on multilayersubstrate and is equal to 225mm The effective dielectricconstant of such a microstrip line is approximately 136The total length of the coil is 25 cm which corresponds toelectrical length of nearly 03120582

A transmission linewith two series capacitors119862119904is shown

in Figure 3(b)The length of the first and the last transmissionline sections 119897

1is one-third of the line in Figure 3(a) The

conductor of the second section (the one in the middle) isprinted on the opposite side of the PCB In thisway capacitorscan be formed by overlapping adjacent transmission linesections Obviously the propagation constant is different forthe second transmission line section To compensate for thatthe physical length 119897

2had to be slightly increased such that

the electrical lengths (and current distribution) of all threesections are identical The corresponding parameters of thetransmission line are 119882

2= 181mm and 119897

2= 895mm

The width of the capacitor plates is chosen the sameas the width of the first line 119882

1in order to minimize

the discontinuity effect associated with the line-capacitorinterconnect

The length of the capacitor is estimated using expressionfor the capacitance of flat parallel metallic plates

119897119888=

119862119904ℎ119904

11988211205760120576119903

(8)

Substituting 119862119904= 171 pF ℎ

119904= 155mm 119882

1= 225mm

and 120576119903

= 455 in this equation results in 119897119888

asymp 293mmApparently the length of the capacitor is a significant fractionof the transmission line length therefore the capacitor canbarely be considered as a lumped capacitor This will ofcourse have impact on the overall field distribution and itsresemblance to the ideal case considered in the previoussection

Equation (1) indicates that the higher number of capac-itors (smaller 119897119873 ratio) leads to higher value of 119862

119904 Should

we have chosen 119873 gt 3 it would be difficult to realize larger


0

02

04

06

08

1

r = 6 cm homogeneity 33


0z (cm)

minus25 minus20 minus15 minus10 minus5

|B|

max|B(r=3

cm)|

(a)

002

04

06

08

1

z (cm)




|B|

max|B(r=3

cm)|

(b)

Figure 2 Normalized magnitude of B field distribution (a) due tocurrent on a regular transmission line (dashed red curve in Figure 1)and (b) for the case with two series capacitors (solid blue curve inFigure 1)

capacitors in the present configuration and would requireimplementation of lumped capacitors instead

Full-wave simulations ofmicrostrip structures in Figure 3can take into account the distributed behavior of the capaci-tors as well as the influence of the ground plane current andthe current through the loading capacitor Initial simulationsshowed slight asymmetry in current distribution due tofringing effects at the ends of the microstrip lines nonidealmicrostrip mode excitation and current through the load-ing capacitor (oriented normally to the ground plane) Tocompensate for that the value of the loading capacitor 119862

119871of

the regular transmission line resonator (Figure 3(a)) had tobe reduced to 72 pF (the initial value calculated from (2) for119873 = 1 is 83 pF) The corresponding value for the microstripline with two series capacitors (Figure 3(b)) 119862

119871= 22 pF The

value of the series capacitors has not been changedThe magnetic field profiles for the coils in Figure 3 are

illustrated by full-wave simulations in free space using com-mercial finite integration technique solver [16] (Figure 4)Thesinusoidal current distribution in the regular transmissionline resonator equates to theBfield distribution in Figure 4(a)with maximum at the center As it was predicted a moreuniformcurrent profile on the linewith series capacitors leadsto more uniform field behavior as seen in Figure 4(b)

To evaluate the efficiency of the designed coil in thepresence of lossy tissue phantom studies were conductedThe simulated phantom has a shape of rectangular cuboid

Top view

Side view

Ground plane Suspended PCB

W1

hs h

CL

(a)


Top view

Side view

W1 W1W2

CL

CsCs

l1l1 l2

lc

(b)

Figure 3 (a) Sketch of a regular suspendedmicrostrip transmissionline resonator (b) Transmission line coil with two series parallelplate capacitors119862

119904 Air gap ℎ = 415mm PCB height ℎ

119904= 155mm

1198971

= 833mm 1198972

= 895mm 119897119888

= 293mm 1198821

= 225mm and1198822= 181mm

(a)

(b)

Figure 4 Comparison of the normalized magnitudes of the mag-netic field distribution between (a) the regular line and (b) the linewith two series parallel plate capacitors in free space

and placed 5mm above the coil Dielectric properties of thephantom material correspond to average properties of brainat 289MHz (1205761015840

119903asymp 52 120576

10158401015840

119903asymp 33 built-in tissue model in

[16]) As can be seen from the simulation results presentedin Figure 5 both coils exhibit comparable maximum fieldgenerated by a unit of the RF input signal

The coil with two series capacitors still offers betterhomogeneity despite exhibiting slight field asymmetry due todistributed nature of losses in the phantom This asymmetrycould beminimized by slightly adjusting the series capacitorswhen designing for a particular imaging application


Regular line resonator

Coil with 2 series capacitors

0

(T radic

W)

1e minus 06909e minus 07818e minus 07727e minus 07636e minus 07545e minus 07455e minus 07364e minus 07273e

eminus 07

182 minus 07909e minus 08

0

(T radic

W)


eminus 07


(a)



0

(T radic

W)

0

7e minus 007

6e minus 007

5e minus 007

4e minus 007

3e minus 007

2e minus 007

1e minus 007


z (cm)

(b)

Figure 5 Simulated B1

+ field in [119879radic119882] generated by a unit of theRF input signal in presence of a phantom (a) Slice along the line and(b) comparison of two coils at 119903 = 3 cm

Loop for magneticfield mapping

Transmission line coil

Figure 6 Experimental setup for magnetic field mapping

The coil with series capacitors was fabricated and its fieldwas mapped in free space (Figure 6)

A lumped capacitor was used for119862119871in the fabricated coil

The coil was tuned to 298MHz andmatched to 50Ω A smallloop (15mm in diameter) was implemented for mapping themagnetic field distribution by moving it along the microstripas illustrated in Figure 6 A vector network analyzer was usedto measure coupling between the line and the loop probeAssuming that the loop is sensitive only to the magnetic fieldthe measured |119878

21| will be proportional to the magnetic field

generated by the coil In this way the profile of the magneticfield distribution is obtained with 1 cm step as illustrated inFigure 7

The data in Figure 7 presents comparison between thesimulated and measured magnetic field distributions 3 cmand 6 cm above the transmission line surface The field

00

02

04

06

08

10

12

000 500z (cm)

r = 6 cm measuredr = 6 cm simulated


|b|

minus3000 minus2500 minus2000 minus1500 minus1000 minus500

Figure 7 Normalized magnitude of the B field distribution for themicrostrip transmission line coil with two series capacitors

profile exhibits two peaks due to distributed behavior of theimplemented series capacitors The achieved homogeneity ofthe coil for 119903 = 3 cm is 55

The coil with evenly distributed capacitors has beenconsidered in thiswork It is expected to achieve better homo-geneity by adjusting the length of particular transmission linesections This approach however has been left out of thescope of this work

4 Conclusion

It was demonstrated that the magnetic field homogeneity forthe transmission line MR coils can be improved by insertingseries capacitors The higher the number of capacitors is themore uniform the field can be generated Theoretically anydegree of homogeneity can be achievedThe only limitation isthe finite length of the coil (which can be overcome by reduc-ing the field of view) and potentially losses in the lumpedcapacitors Another benefit of using series capacitors is thecapability to provide in a simplemanner the required uniformfield over a very long length and construct transmission linecoils which can be even longer than one-half of a wavelength

It is shown that the profile of the current distributioncan be changed inserting series capacitors Values for evenlydistributed capacitors have been derived in this work (equalvalue capacitors inserted between equal length line sections)It is expected that even higher homogeneity over a widerfield of view could be achieved using unevenly distributedcapacitors with variable values

The approach developed here to control magnetic fielddistribution would be useful in the design of coils for highfield systems for imaging in large in terms of wavelengthobjects

The derived equations have been used in the designexample of a suspended microstrip coil with two seriescapacitors This particular coil realization required slightreduction of the loading capacitor value 119862

119871in order to

compensate for nonideal behavior and achieve symmetricfield distribution


Appendices

A Deriving Equation for Loading Capacitor

The current distribution has highest homogeneity when themaximum appears at the center of the transmission linesection Considering the current on the last transmission linesection ((3) when 119899 = 119873) the maximum of the currentmagnitude will appear at 119911 = minus1198972119873 if 119890119895(120579119871+2120573119911) = minus1 where120579119871is the phase of the load reflection coefficient Consequently

120579119871minus 119897120573119873 = minus120587Load reflection coefficient in terms of normalized reac-

tance of the load 119909119871is

Γ119871=

119895119909119871minus 1

119895119909119871+ 1

=1199092

119871minus 1

1199092119871+ 1

+ 1198952119909119871

1199092119871+ 1

(A1)

where 119909119871= minus12120587119891

01198621198711198850 Consider

tan 120579119871=Im Γ119871

Re Γ119871

=2119909119871

1199092119871minus 1

= minus tan(minus120573119897

119873+ 120587)

= tan(120573119897

119873)

(A2)

Solving this quadratic equation for 119909119871

119909119871=

1 plusmn radic1 + tan2 (120573119897119873)

tan (120573119897119873)=

1 plusmn sec (120573119897119873)

tan (120573119897119873)

=cos (120573119897119873) plusmn 1

sin (120573119897119873)

(A3)

We choose solution with ldquominusrdquo since 119909119871is negative which

leads to the following equation for the loading capacitor

119862119871=

minus sin (120573119897119873)

212058711989101198850(cos (120573119897119873) minus 1)

(A4)

B Equation for Series Capacitor

To derive equation for series capacitor119862119904the last (119873th) trans-

mission line section in Figure 1 is considered The foregoingsection should be loaded with 119885

119871in order to achieve current

distribution identical to distribution on the last section Thisis described by the following equality119885in119873minus1198952120587119891

0119862119904= 119885119871

where

119885in119873 = 1198850

119885119871+ 1198951198850tan (120573119897119873)

1198850+ 119895119885119871tan (120573119897119873)

(B1)

is the input impedance of the last (119873th) transmission linesection Using this equation

minus119895

21205871198910119862119904

=119885119871(1198850+ 119895119885119871tan (120573119897119873)) minus 119885

0(119885119871+ 1198951198850tan (120573119897119873))

1198850+ 119895119885119871tan (120573119897119873)

=119895 tan (120573119897119873) (119885

2

119871minus 1198852

0)

1198850+ 119895119885119871tan (120573119897119873)

(B2)

Substituting (A4)

minus119895

21205871198910119862119904

=119895 tan (120573119897119873) (minus119885

2

0(cos (120573119897119873) minus 1)

2

sin2 (120573119897119873) minus 1198852

0)

1198850+ 1198951198850((cos (120573119897119873) minus 1) sin (120573119897119873)) tan (120573119897119873)

=1198951198850(minus (cos (120573119897119873) minus 1)

2

minus sin2 (120573119897119873))

sin (120573119897119873)

= 1198951198850

2 cos (120573119897119873) minus 2

sin (120573119897119873)=

minus1198952

21205871198910119862119871

(B3)

from where

119862119904=

119862119871

2 (B4)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author would like to thank Danish National ResearchFoundation (Grant DNRF124) for partial support of theactivities

References

[1] J J H Ackerman T H Grove G G Wong D G Gadian andG K Radda ldquoMapping of metabolites in whole animals by 31PNMRusing surface coilsrdquoNature vol 283 no 5743 pp 167ndash1701980

[2] J T Vaughan and J R Griffiths Eds RF Coils for MRI JohnWiley amp Sons New York NY USA 2012

[3] D T P Nilsson J J Mohr and V Zhurbenko ldquoPractical aspectsof 13C surface receive coils with active decoupling and tuningcircuitrdquo in Proceedings of the 42nd European Microwave Con-ference (EuMC rsquo12) pp 65ndash68 Amsterdam The NetherlandsNovember 2012

[4] J T Vaughan H P Hetherington J O Otu JW Pan andGMPohost ldquoHigh frequency volume coils for clinical NMR imagingand spectroscopyrdquoMagnetic Resonance in Medicine vol 32 no2 pp 206ndash218 1994

[5] B A Baertlein O Ozbay T Ibrahim et al ldquoTheoretical modelfor an MRI radio frequency resonatorrdquo IEEE Transactions onBiomedical Engineering vol 47 no 4 pp 535ndash546 2000

[6] R F Lee C R Westgate R G Weiss D C Newman andP A Bottomley ldquoPlanar strip array (PSA) for MRIrdquo MagneticResonance in Medicine vol 45 no 4 pp 673ndash683 2001

[7] I A Elabyad and A Omar ldquoAn investigation of alternatingimpedance microstrip transceiver coil arrays for MRI at 7Trdquoin Proceedings of the IEEE MTT-S International MicrowaveSymposium (IMS rsquo11) pp 1ndash4 IEEE Baltimore Md USA June2011

[8] C E Akgun L Delabarre H Yoo et al ldquoStepped impedanceresonators for high-field magnetic resonance imagingrdquo IEEETransactions on Biomedical Engineering vol 61 no 2 pp 327ndash333 2014


[9] A Rennings J Mosig A Bahr C Caloz M E Ladd andD Erni ldquoA CRLH metamaterial based RF coil element formagnetic resonance imaging at 7 teslardquo in Proceedings of the3rd European Conference on Antennas and Propagation (EuCAPrsquo09) pp 3231ndash3234 Berlin Germany March 2009

[10] A Senn A Peter and J G Korvink ldquoAn 8-channel metama-terial T-R coil at 94Trdquo in Proceedings of the ISMRM AnnualMeeting p 1 Montreal Canada 2011

[11] X Yan J O Pedersen L Wei X Zhang and R Xue ldquoMul-tichannel double-row transmission line array for human MRimaging at ultrahigh fieldsrdquo IEEE Transactions on BiomedicalEngineering vol 62 no 6 pp 1652ndash1659 2015

[12] D Pozar and D M Pozar ldquoTransmission line theoryrdquo inMicrowave Engineering chapter 2 pp 48ndash94Wiley 4th edition2011

[13] Y Pang Z Xie D Xu et al ldquoA dual-tuned quadrature volumecoil with mixed 1205822 and 1205824 microstrip resonators for multinu-clear MRSI at 7 Trdquo Magnetic Resonance Imaging vol 30 no 2pp 290ndash298 2012

[14] R Ludwig G Bodgdanov J King A Allard and C F FerrisldquoA dual RF resonator system for high-field functional magneticresonance imaging of small animalsrdquo Journal of NeuroscienceMethods vol 132 no 2 pp 125ndash135 2004

[15] X Zhang K Ugurbil R Sainati and W Chen ldquoAn inverted-microstrip resonator for human head proton MR imaging at 7teslardquo IEEE Transactions on Biomedical Engineering vol 52 no3 pp 495ndash504 2005

[16] CST Microwave Studio 2015 httpswwwcstcom

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capacitive loading is used to achieve symmetric currentdistribution along the line [7 8 13ndash15] An example ofa normalized current distribution |119868(119911)| on a 25 cm longtransmission line at 298MHz is shown in Figure 1 (dashedred curve)

The current reduces at the ends of the line due todestructive interference of the incident and reflected waves

It is however preferable to have a uniform currentdistribution That would result in a homogeneous magneticfield along the line

A more uniform current distribution can be achievedby inserting series capacitors 119862

119904into the transmission line

(Figure 1(b)) These capacitors can compensate for self-inductance of the transmission line by introducing oppositephase shift The loading capacitor 119862

119871 in its turn should

be chosen to provide symmetry of the current distributionThis structure (Figure 1(b)) results in a smaller standingwave ratio than in the conventional case (Figure 1(a)) Anexample of current distribution on a line using two seriescapacitors is shown in Figure 1(c) (solid blue curve) As canbe seen the resulting uniformity of the current distributionis considerably improved

Obviously the higher the number of capacitors themore uniform the current distribution can be achieved andconsequently the higher the magnetic field homogeneity willbe

The value of the series capacitor119862119904should be chosen such

that it compensates the phase shift introduced by one sectionof the transmission line (Appendix B)

119862119904=

sin (120573119897119873)

412058711989101198850(1 minus cos (120573119897119873))

(1)

where 119897 is the total length of the transmission line coil 119873 isthe number of transmission line sections after inserting seriescapacitors (eg using two series capacitors will result in 119873 =

3) 120573 is the phase constant1198850is the characteristic impedance

of the implemented transmission line and 1198910is the operating

frequency A frequency of 298MHz is used in all exampleshere which is close to operating frequency ofMR systems forhydrogen imaging with 7 T magnetic field

The expression for the loading capacitor which will resultin the optimal current distribution is derived imposing cur-rent symmetry condition for 119897119873 length line (Appendix A)

119862119871= 2119862119904=

sin (120573119897119873)

212058711989101198850(1 minus cos (120573119897119873))

(2)

For example for 50Ω line the calculated values of thecapacitors using (1) and (2) at 119891

0= 298MHz 120573 = 728 and

119873 = 3 are 119862119871asymp 342 pF and 119862

119904asymp 171 pF

Having the values of the capacitors and knowing theparameters of the transmission line current distribution canbe foundusingwell-developed transmission line analysis [12]The current on 119899th transmission line section

119868119899(119911) =

119881+

0119899

1198850

(119890minus119895120573(119911+(119873minus119899)119897119873)

minus Γ119871119899

119890119895120573(119911+(119873minus119899)119897119873)

) (3)

where Γ119871119899

and 119881+

0119899are the load reflection coefficient and

amplitude of the incident wave at the end of 119899th transmission

0

IN Z0 CL

minuslz

(a)

0

IN Z0 Z0Z0

minusl

n = 1 n = Nn = N minus 1

ZinN

minus2lN

z

minuslN

CL

CsCs




(b)

2

07

08

09

06

10

|I||I|m

ax

minus26 minus22 minus18 minus14 minus10 minus6 minus2

With 2 series capacitors Cs (Figure 1(b) (N = 3))Without series capacitors (Figure 1(a))

z (cm)

(c)

Figure 1 (a) Regular transmission line resonator (b) Transmissionline coil with series capacitors 119862

119904 (c) Corresponding normalized

magnitude of the current distribution Here 119911 is the longitudinalcoordinate 119897 is the total length of the coil and 119873 is the number ofsections

line section respectively 119899 = 1 sdot sdot sdot 119873 The total current 119868(119911)is then a combination of currents on all transmission linesections 119868

119899(119911)

The amplitude of the incident wave at the end of 119899thtransmission line section

119881+

0119899=

119881119892

2

(1 minus Γ119892) 119890minus119895120573119897119873

1 minus Γ119892Γ119871119899

119890minus1198952120573119897119873 when 119899 = 1

119881in119899

119890119895120573119897119873 + Γ119871119899

119890minus119895120573119897119873 when 2 le 119899 le 119873

(4)

Here 119881119892

is the voltage of the generator which excitesthe transmission line coil 119881in119899 is the total voltage at theinput terminals of 119899th transmission line section Generatorreflection coefficient Γ

119892= (119885119892minus1198850)(119885119892+1198850) where119885

119892is the

impedance of the generator Reflection coefficient at the end



= (119885119871119899

minus1198850)(119885119871119899

+1198850)

where

119885119871119899

=

1

11989521205871198910119862119871

when 119899 = 119873

1

11989521205871198910119862119904

+ 1198850

119885119871119899+1

+ 1198951198850tan (120573119897119873)

1198850+ 119895119885119871119899+1


(5)


|H (119911 119903)| =119903

4120587int0

minus119897

10038161003816100381610038161003816119868 (1199111015840

)100381610038161003816100381610038161198891199111015840

(1199032 + [1199111015840 minus 119911]2

)32

(6)




min |B (119911)|

max |B (119911)| (7)


















2= 181mm and 119897

2= 895mm





119897119888=

119862119904ℎ119904

11988211205760120576119903

(8)


119904= 155mm 119882

1= 225mm

and 120576119903




119904 Should



0

02

04

06

08

1



0z (cm)


|B|

max|B(r=3

cm)|

(a)

002

04

06

08

1

z (cm)




|B|

max|B(r=3

cm)|

(b)




119871of


119871= 22 pF The




Top view

Side view


W1

hs h

CL

(a)


Top view

Side view

W1 W1W2

CL

CsCs

l1l1 l2

lc

(b)



119904= 155mm

1198971

= 833mm 1198972

= 895mm 119897119888

= 293mm 1198821

= 225mm and1198822= 181mm

(a)

(b)



119903asymp 52 120576

10158401015840







0

(T radic

W)


eminus 07


0

(T radic

W)


eminus 07


(a)



0

(T radic

W)

0

7e minus 007

6e minus 007

5e minus 007

4e minus 007

3e minus 007

2e minus 007

1e minus 007


z (cm)

(b)












00

02

04

06

08

10

12

000 500z (cm)



|b|





4 Conclusion





119871in order to



Appendices





Γ119871=

119895119909119871minus 1

119895119909119871+ 1

=1199092

119871minus 1

1199092119871+ 1

+ 1198952119909119871

1199092119871+ 1

(A1)

where 119909119871= minus12120587119891

01198621198711198850 Consider

tan 120579119871=Im Γ119871

Re Γ119871

=2119909119871

1199092119871minus 1


119873+ 120587)

= tan(120573119897

119873)

(A2)


119909119871=

1 plusmn radic1 + tan2 (120573119897119873)

tan (120573119897119873)=

1 plusmn sec (120573119897119873)

tan (120573119897119873)

=cos (120573119897119873) plusmn 1

sin (120573119897119873)

(A3)



119862119871=

minus sin (120573119897119873)

212058711989101198850(cos (120573119897119873) minus 1)

(A4)






0119862119904= 119885119871

where

119885in119873 = 1198850

119885119871+ 1198951198850tan (120573119897119873)

1198850+ 119895119885119871tan (120573119897119873)

(B1)


minus119895

21205871198910119862119904

=119885119871(1198850+ 119895119885119871tan (120573119897119873)) minus 119885

0(119885119871+ 1198951198850tan (120573119897119873))

1198850+ 119895119885119871tan (120573119897119873)

=119895 tan (120573119897119873) (119885

2

119871minus 1198852

0)

1198850+ 119895119885119871tan (120573119897119873)

(B2)

Substituting (A4)

minus119895

21205871198910119862119904

=119895 tan (120573119897119873) (minus119885

2

0(cos (120573119897119873) minus 1)

2

sin2 (120573119897119873) minus 1198852

0)

1198850+ 1198951198850((cos (120573119897119873) minus 1) sin (120573119897119873)) tan (120573119897119873)

=1198951198850(minus (cos (120573119897119873) minus 1)

2

minus sin2 (120573119897119873))

sin (120573119897119873)

= 1198951198850

2 cos (120573119897119873) minus 2

sin (120573119897119873)=

minus1198952

21205871198910119862119871

(B3)

from where

119862119904=

119862119871

2 (B4)



Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











= (119885119871119899

minus1198850)(119885119871119899

+1198850)

where

119885119871119899

=

1

11989521205871198910119862119871

when 119899 = 119873

1

11989521205871198910119862119904

+ 1198850

119885119871119899+1

+ 1198951198850tan (120573119897119873)

1198850+ 119895119885119871119899+1


(5)


|H (119911 119903)| =119903

4120587int0

minus119897

10038161003816100381610038161003816119868 (1199111015840

)100381610038161003816100381610038161198891199111015840

(1199032 + [1199111015840 minus 119911]2

)32

(6)




min |B (119911)|

max |B (119911)| (7)


















2= 181mm and 119897

2= 895mm





119897119888=

119862119904ℎ119904

11988211205760120576119903

(8)


119904= 155mm 119882

1= 225mm

and 120576119903




119904 Should



0

02

04

06

08

1



0z (cm)


|B|

max|B(r=3

cm)|

(a)

002

04

06

08

1

z (cm)




|B|

max|B(r=3

cm)|

(b)




119871of


119871= 22 pF The




Top view

Side view


W1

hs h

CL

(a)


Top view

Side view

W1 W1W2

CL

CsCs

l1l1 l2

lc

(b)



119904= 155mm

1198971

= 833mm 1198972

= 895mm 119897119888

= 293mm 1198821

= 225mm and1198822= 181mm

(a)

(b)



119903asymp 52 120576

10158401015840







0

(T radic

W)


eminus 07


0

(T radic

W)


eminus 07


(a)



0

(T radic

W)

0

7e minus 007

6e minus 007

5e minus 007

4e minus 007

3e minus 007

2e minus 007

1e minus 007


z (cm)

(b)












00

02

04

06

08

10

12

000 500z (cm)



|b|





4 Conclusion





119871in order to



Appendices





Γ119871=

119895119909119871minus 1

119895119909119871+ 1

=1199092

119871minus 1

1199092119871+ 1

+ 1198952119909119871

1199092119871+ 1

(A1)

where 119909119871= minus12120587119891

01198621198711198850 Consider

tan 120579119871=Im Γ119871

Re Γ119871

=2119909119871

1199092119871minus 1


119873+ 120587)

= tan(120573119897

119873)

(A2)


119909119871=

1 plusmn radic1 + tan2 (120573119897119873)

tan (120573119897119873)=

1 plusmn sec (120573119897119873)

tan (120573119897119873)

=cos (120573119897119873) plusmn 1

sin (120573119897119873)

(A3)



119862119871=

minus sin (120573119897119873)

212058711989101198850(cos (120573119897119873) minus 1)

(A4)






0119862119904= 119885119871

where

119885in119873 = 1198850

119885119871+ 1198951198850tan (120573119897119873)

1198850+ 119895119885119871tan (120573119897119873)

(B1)


minus119895

21205871198910119862119904

=119885119871(1198850+ 119895119885119871tan (120573119897119873)) minus 119885

0(119885119871+ 1198951198850tan (120573119897119873))

1198850+ 119895119885119871tan (120573119897119873)

=119895 tan (120573119897119873) (119885

2

119871minus 1198852

0)

1198850+ 119895119885119871tan (120573119897119873)

(B2)

Substituting (A4)

minus119895

21205871198910119862119904

=119895 tan (120573119897119873) (minus119885

2

0(cos (120573119897119873) minus 1)

2

sin2 (120573119897119873) minus 1198852

0)

1198850+ 1198951198850((cos (120573119897119873) minus 1) sin (120573119897119873)) tan (120573119897119873)

=1198951198850(minus (cos (120573119897119873) minus 1)

2

minus sin2 (120573119897119873))

sin (120573119897119873)

= 1198951198850

2 cos (120573119897119873) minus 2

sin (120573119897119873)=

minus1198952

21205871198910119862119871

(B3)

from where

119862119904=

119862119871

2 (B4)



Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

02

04

06

08

1



0z (cm)


|B|

max|B(r=3

cm)|

(a)

002

04

06

08

1

z (cm)




|B|

max|B(r=3

cm)|

(b)




119871of


119871= 22 pF The




Top view

Side view


W1

hs h

CL

(a)


Top view

Side view

W1 W1W2

CL

CsCs

l1l1 l2

lc

(b)



119904= 155mm

1198971

= 833mm 1198972

= 895mm 119897119888

= 293mm 1198821

= 225mm and1198822= 181mm

(a)

(b)



119903asymp 52 120576

10158401015840







0

(T radic

W)


eminus 07


0

(T radic

W)


eminus 07


(a)



0

(T radic

W)

0

7e minus 007

6e minus 007

5e minus 007

4e minus 007

3e minus 007

2e minus 007

1e minus 007


z (cm)

(b)












00

02

04

06

08

10

12

000 500z (cm)



|b|





4 Conclusion





119871in order to



Appendices





Γ119871=

119895119909119871minus 1

119895119909119871+ 1

=1199092

119871minus 1

1199092119871+ 1

+ 1198952119909119871

1199092119871+ 1

(A1)

where 119909119871= minus12120587119891

01198621198711198850 Consider

tan 120579119871=Im Γ119871

Re Γ119871

=2119909119871

1199092119871minus 1


119873+ 120587)

= tan(120573119897

119873)

(A2)


119909119871=

1 plusmn radic1 + tan2 (120573119897119873)

tan (120573119897119873)=

1 plusmn sec (120573119897119873)

tan (120573119897119873)

=cos (120573119897119873) plusmn 1

sin (120573119897119873)

(A3)



119862119871=

minus sin (120573119897119873)

212058711989101198850(cos (120573119897119873) minus 1)

(A4)






0119862119904= 119885119871

where

119885in119873 = 1198850

119885119871+ 1198951198850tan (120573119897119873)

1198850+ 119895119885119871tan (120573119897119873)

(B1)


minus119895

21205871198910119862119904

=119885119871(1198850+ 119895119885119871tan (120573119897119873)) minus 119885

0(119885119871+ 1198951198850tan (120573119897119873))

1198850+ 119895119885119871tan (120573119897119873)

=119895 tan (120573119897119873) (119885

2

119871minus 1198852

0)

1198850+ 119895119885119871tan (120573119897119873)

(B2)

Substituting (A4)

minus119895

21205871198910119862119904

=119895 tan (120573119897119873) (minus119885

2

0(cos (120573119897119873) minus 1)

2

sin2 (120573119897119873) minus 1198852

0)

1198850+ 1198951198850((cos (120573119897119873) minus 1) sin (120573119897119873)) tan (120573119897119873)

=1198951198850(minus (cos (120573119897119873) minus 1)

2

minus sin2 (120573119897119873))

sin (120573119897119873)

= 1198951198850

2 cos (120573119897119873) minus 2

sin (120573119897119873)=

minus1198952

21205871198910119862119871

(B3)

from where

119862119904=

119862119871

2 (B4)



Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












0

(T radic

W)


eminus 07


0

(T radic

W)


eminus 07


(a)



0

(T radic

W)

0

7e minus 007

6e minus 007

5e minus 007

4e minus 007

3e minus 007

2e minus 007

1e minus 007


z (cm)

(b)












00

02

04

06

08

10

12

000 500z (cm)



|b|





4 Conclusion





119871in order to



Appendices





Γ119871=

119895119909119871minus 1

119895119909119871+ 1

=1199092

119871minus 1

1199092119871+ 1

+ 1198952119909119871

1199092119871+ 1

(A1)

where 119909119871= minus12120587119891

01198621198711198850 Consider

tan 120579119871=Im Γ119871

Re Γ119871

=2119909119871

1199092119871minus 1


119873+ 120587)

= tan(120573119897

119873)

(A2)


119909119871=

1 plusmn radic1 + tan2 (120573119897119873)

tan (120573119897119873)=

1 plusmn sec (120573119897119873)

tan (120573119897119873)

=cos (120573119897119873) plusmn 1

sin (120573119897119873)

(A3)



119862119871=

minus sin (120573119897119873)

212058711989101198850(cos (120573119897119873) minus 1)

(A4)






0119862119904= 119885119871

where

119885in119873 = 1198850

119885119871+ 1198951198850tan (120573119897119873)

1198850+ 119895119885119871tan (120573119897119873)

(B1)


minus119895

21205871198910119862119904

=119885119871(1198850+ 119895119885119871tan (120573119897119873)) minus 119885

0(119885119871+ 1198951198850tan (120573119897119873))

1198850+ 119895119885119871tan (120573119897119873)

=119895 tan (120573119897119873) (119885

2

119871minus 1198852

0)

1198850+ 119895119885119871tan (120573119897119873)

(B2)

Substituting (A4)

minus119895

21205871198910119862119904

=119895 tan (120573119897119873) (minus119885

2

0(cos (120573119897119873) minus 1)

2

sin2 (120573119897119873) minus 1198852

0)

1198850+ 1198951198850((cos (120573119897119873) minus 1) sin (120573119897119873)) tan (120573119897119873)

=1198951198850(minus (cos (120573119897119873) minus 1)

2

minus sin2 (120573119897119873))

sin (120573119897119873)

= 1198951198850

2 cos (120573119897119873) minus 2

sin (120573119897119873)=

minus1198952

21205871198910119862119871

(B3)

from where

119862119904=

119862119871

2 (B4)



Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Appendices





Γ119871=

119895119909119871minus 1

119895119909119871+ 1

=1199092

119871minus 1

1199092119871+ 1

+ 1198952119909119871

1199092119871+ 1

(A1)

where 119909119871= minus12120587119891

01198621198711198850 Consider

tan 120579119871=Im Γ119871

Re Γ119871

=2119909119871

1199092119871minus 1


119873+ 120587)

= tan(120573119897

119873)

(A2)


119909119871=

1 plusmn radic1 + tan2 (120573119897119873)

tan (120573119897119873)=

1 plusmn sec (120573119897119873)

tan (120573119897119873)

=cos (120573119897119873) plusmn 1

sin (120573119897119873)

(A3)



119862119871=

minus sin (120573119897119873)

212058711989101198850(cos (120573119897119873) minus 1)

(A4)






0119862119904= 119885119871

where

119885in119873 = 1198850

119885119871+ 1198951198850tan (120573119897119873)

1198850+ 119895119885119871tan (120573119897119873)

(B1)


minus119895

21205871198910119862119904

=119885119871(1198850+ 119895119885119871tan (120573119897119873)) minus 119885

0(119885119871+ 1198951198850tan (120573119897119873))

1198850+ 119895119885119871tan (120573119897119873)

=119895 tan (120573119897119873) (119885

2

119871minus 1198852

0)

1198850+ 119895119885119871tan (120573119897119873)

(B2)

Substituting (A4)

minus119895

21205871198910119862119904

=119895 tan (120573119897119873) (minus119885

2

0(cos (120573119897119873) minus 1)

2

sin2 (120573119897119873) minus 1198852

0)

1198850+ 1198951198850((cos (120573119897119873) minus 1) sin (120573119897119873)) tan (120573119897119873)

=1198951198850(minus (cos (120573119897119873) minus 1)

2

minus sin2 (120573119897119873))

sin (120573119897119873)

= 1198951198850

2 cos (120573119897119873) minus 2

sin (120573119897119873)=

minus1198952

21205871198910119862119871

(B3)

from where

119862119904=

119862119871

2 (B4)



Acknowledgment


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Optimal Value of Series Capacitors for ...downloads.hindawi.com/journals/js/2016/3480965.pdf · Transmission lines are o en used as coils in high eld magnetic resonance