1202 Kumar et al. NF coil ϭ 10 ϫ log͑NPR͓͒dB͔  The coil NF (NFcoil) in Eq.  measures the SNR reduc- tion attributable to losses in the coil. The reduction in the voltage SNR (SNRV) is: SNR V͑unloaded͒ ͫ SNR pwr͑unloaded͒ ͬ 1/2 SNR V͑loaded͒ ϭ SNR pwr͑loaded͒ ϭ ͱNPR FIG. 1. Circuit model of a sample loaded resonant loop (12). L and Q-Factor PredictionC are the inductance and capacitance of the resonant loop, respec-tively. Rs and Rc are the noise resistances due to sample and coil, For a loop with inductance L resonating at an angularrespectively. Vsig, Vns, and Vnc are, respectively, the detected MR frequency, , Q ϭ L/R. The unloaded coil contains onlysignal voltage, the root-mean-square (rms) noise voltage from the coil resistance RC. Under sample-loaded conditions, thesample, and the rms coil noise voltage produced by the loop resis- series resistance R includes the sum of coil and sampletance. resistances RC and RS. To calculate Q for a loop, it is necessary to determine its resistive losses. These include copper conductive losses, the ESR of capacitors used toattributable to eddy currents induced in adjacent conduc- tune the loops, and losses in the solder joints. Q-factors oftors. The depth-optimized radii, rR, of individual loops inductors and capacitors are deﬁned respectively as:that produce the optimum SNR for a given target depth ,including all of the coil losses, are determined as a func- L 1tion of frequency by full-wave electromagnetic numerical Q ind ϭ ,Q ϭ method-of-moment (MoM) analysis. The SNR of these RL C CRCcoils is then compared to the SNR of lossless coils whoseradii are determined by the quasistatic optimization rule, where RL and RC refer to the ESR of inductor L, and capac-r0 ϭ /͌5 (7–9). itor C. Copper conductive losses are calculated for the loops using the skin depth formulaMATERIALS AND METHODS ͱFigure 1 shows the circuit model of a sample-loaded res- 2onant loop (12). The SNR of the loop detector measured as ␦ϭ . a power ratio is given as 0 2 V sig where ϭ 2.2 ϫ 10– 8 ⍀-m is the copper resistivity from our SNR power ϭ 2  own measurements, and 0 ϭ 4 ϫ 10Ϫ7 H/m is the Vn magnetic permeability of free space.where Vsig is the signal power and V2 is the noise power. 2 We have studied loops made from both round 3.2-mm nVn includes the noise contributions from the sample and (1/8-inch) hollow copper wire tubing and 4-mm-wide,from the loop: 0.015-mm-thick ﬂat copper strip. Each loop was tuned using high Q ceramic chip capacitors (Series 11; Dielectric Laboratories, NJ, USA) with Ϯ2% tolerance. Copper resis- V n ϭ V nc ϩ V ns ϭ 4͑R c ϩ R s͒kT⌬f 2 2 2  tivity was determined experimentally, using a four-termi- nal measurement, by applying a known direct current (DC) Rc and Rs are, respectively, resistive losses from the of ϳ1 A; to the conductors (wire, strip) and measuring thecopper loop and the sample volume; k, is Boltzmann’s voltage drop at the current injection points.constant; T is the sample and coil temperature (assumed to The DC resistivity for the round wire loops was calcu-be the same) in K; and ⌬f is the receiver bandwidth in Hz. lated from the measured resistance asNoise Power Ratio and coil NF meas ϭ R meas ϫ A/L The noise factor or noise power ratio (NPR) is the ratio ofSNRpwr at the input port of any device to that at the outputport. In the case of an RF coil used for MR signal detection, where A is the cross-sectional area and L is the length ofNPR compares the noise power with the coil noise in- the wire. Applying Eqs.  and  with measurements forcluded, to the noise power which would be obtained with- the wire loop (subscript, wire), the copper resistive loss isout it, and is given by (corrected from Ref. 12): then: RS ϩ RC Q U/Q L SNR pwr͑unloaded͒ D NPR ϭ ϭ ϭ  R wire ϭ wire ⅐ .  RS Q U/Q L Ϫ 1 SNR pwr͑loaded͒ ␦ wired
Noise Figure Limits of Circular Loop MR Coils 1203Table 1Relationship Between Coil NF, NFcoil, and Coil Properties Used in the Analysis Noise ﬁgure Noise power Coil noise Sample noise Qu/QL ratio NF (dB) ratio (NPR) power (%) power (%) 0.2 1.05 22.22 4.50 95.50 0.5 1.12 9.20 10.87 89.13 1.0 1.26 4.86 20.57 79.43 1.8 1.50 2.99 33.47 67.53 3.0 2.00 2.00 49.88 50.12 5.0 3.16 1.46 68.38 31.62 Here, D is the diameter of the loop and d is the diameter The results from Eqs. [9 –14] were used to compute theof the wire. The resistance of the ﬂat strip loop is similarly unloaded coil Qs.calculated as: Full-Wave Numerical MoM Analysis D m R flat ϭ flat ⅐  Losses in the sample were numerically computed from the 2w␦ real part of the input impedances of loops under loadedwhere Dm and w refer to the middle radius and width of the conditions (14,15) using the full-wave electromagneticstrip, respectively. MoM (FEKO; EM Software and Systems, South Africa; The ESRs for the capacitors are obtained from capacitor www.feko.info). In the numerical MoM computations formodeling software CAPCAD (http://tinyurl.com/3qrg67) single coils, Green’s functions are applied to a semiinﬁniteprovided by the capacitor manufacturer Dielectric Labora- medium with the electrical properties (conductivity, 0.69tories (Cazenovia, NY, USA; http://www.dilabs.com/in- S/m Յ Յ 0.82 S/m; dielectric constant, 58 Յ ⑀r Յ 80) ofdex.aspx). muscle tissue over the MR frequency range 42.6 to The RF resistances of the solder joints are determined by 400 MHz (Gabriel C, Tissue Dielectric Properties; http://cutting the resonant loop, soldering it back together and niremf.ifac.cnr.it/tissprop).measuring the change in Q. The resistance is An acrylic insulation layer (⑀r ϭ 2.2) was placed be- tween the coil elements and the medium to model the ͩ ͪ surface coil’s housing, which is used to support and sep- 1 1 R solder ϭ L Ϫ ,  arate it from the sample. The acrylic layer also reduces the Q m2 Q m1 direct electric (E) ﬁeld interactions between the coil and the sample. The insulation thickness was adjusted to bewhere Qm1 and Qm2 are the Qs measured before and after between 1 mm and 3 mm depending on the size of thethe added solder joint(s). loop. Each loop was tuned with two to seven tuning ca- The inductances of the wire loops and ﬂat strip loops are pacitors, depending on the size of the loop, distributed indetermined from analytical formulas whose accuracy we order to minimize E-ﬁeld coupling with the sample vol-validated by experiment. The inductance of the round ume.wire loop is (13) The array computations assumed a semiinﬁnite medium with constant ϭ 0.72 S/m and ⑀ ϭ 63.5 to coincide with L ϭ 0 ͩ ͪͫ ͩ ͪ ͬ D 2 ln 8D d Ϫ2  that of a saline phantom used for experimental validation. Loaded Q values for single loops, and for extended 1D and 2D coplanar loop arrays were then determined, includingThe inductance of the ﬂat washer loop is determined by the computed losses associated with eddy currents on theapplying the Neumann formula for inductance (13), (Wiki- surface of overlapped loop elements. Loops comprisingpedia, Inductance; http://en.wikipedia.org/wiki/Induc- arrays were conﬁgured with the prescribed geometries thattance) and numerically integrating Eq.  using Mathcad minimize coupling (1).(PTC, Needham, MA, USA; http://tinyurl.com/3at3ws): The NFs of loop coils made of copper wire and strips were also computed, alone and in arrays. The relationship ͵͵͵ 2 a2 a2 between NFcoil, NPR, QU/QL ratio, and the percentage of the 2 Lϭ f͑r 1,r 2,͒dr 1dr 2d  total noise attributable to the coils and the sample are ͑a 2 Ϫ a 1͒ 2 illustrated in Table 1. At a 3-dB NFcoil level, coil and 0 a1 a1 sample noise contributions are equal.where: a1, and a2 are, respectively, the inner and outerradii of the ﬂat washer loop; is the polar coordinate; and Measurement and Validation ͩ ͪ The predicted coil Qs were validated by measurements 0 cos͑͒ f͑r 1,r 2,͒ ϭ ͑2r 1r 2͒ . performed with loosely coupled “sniffer” coils, as shown 4 ͓͑r1 cos͑͒ Ϫ r2 ͒2 ϩ ͑r1 sin͑͒͒2 ͔0.5 in Fig. 2, using an HP 4395A network analyzer (Agilent,  Palo Alto, CA, USA) operating in S21 mode, calibrated for
1204 Kumar et al. to the transverse ﬁeld, B1 (16). For each target depth , we calculated the coil losses RC analytically. We then com- puted both the sample losses RS and the B1 ﬁeld for a series of loop radii r using the full-wave electromagnetic numer- ical MoM. SNR, given by B1 SNRϱ ,  ͱRC ϩ RS is calculated as a function of loop radii and frequency, andFIG. 2. a: Circuit model of a sniffer coil. A gap in the ground shield the loop with the maximum SNR is determined for eachis shown at the top of the loop. The middle gray line is the inner target depth. The coil radii considered in the analysisconductor; the dark outer lines represent ground shield. b: Two range from 2 mm to 125 mm in the MR frequency range ofsniffer coils made of semirigid (UT-300) coaxial cable mounted on a 43 MHz to 400 MHz.ring stand. The coils are inductively decoupled by overlapping them. For a given target depth, the difference in SNR in dBOne sniffer loop transmits RF energy to the resonant loop and the obtained from deploying an optimal coil of radius rR withother functions as a pickup loop. coil loss included, as compared to the SNR of the optimal coil of radius r0 with sample losses only—the intrinsica two-port, one-path measurement. The sniffer coils are SNR (7,11)—is computed as:also overlapped to the extent required for inductive de-coupling (1). The transmit sniffer excites the resonant test SNR loss(dB) ϭ 10 ϫ log͓͑SNR͑rR ͒/SNR͑r0 ͒͒2 ͔ loop and the receive sniffer detects currents excited in thetest loop. The QL measurements for single loops were done RESULTSwith the coil placed on the thigh muscle of a healthy adultvolunteer with 3-mm-thick acrylic mounting sheet sepa- Table 2a and Table 2b list the calculated effective resis-rating the muscle tissue and coil. tances along with the computed and measured unloaded Q For the arrays, QL is measured on an agar gel phantom values for wire loops fabricated with d ϭ 3.2 mm and Dm(width ϭ 220 mm, depth ϭ 300 mm, height ϭ 120 mm) ϭ 50 mm and tuned to 48.5, 64, 124, and 207 MHz, and forcontaining a 35 mM NaCl solution with 1% agar by weight ﬂat strip loops with w ϭ 4 mm and Dm ϭ 40 mm tuned to( and ⑀ as above). The test arrays are comprised of a 49, 65, 128, and 200 MHz. The tuning capacitors weresingle, tuned, resonant coil with the other overlapping distributed at four equally-spaced breaks in the loops. Theloops open-circuited. The open circuit loops simulate per- computed QU values agree with the measured values withfect decoupling while including the same eddy current a deviation of Յ6% for wire loops and Յ8% for the tapelosses that would occur if all coils were resonant. NFcoil loops. The deviations do not vary systematically with fre-was calculated from the measured Qs via Eqs.  and . quency.Depth-Optimized Coils Effect of Multiple Parallel CapacitorsWe apply the results to determine loop sizes that optimize The loop resonating at 48.5 MHz in Table 2 was tuned withSNR as a function of depth . For sample-noise dominated four sets of two parallel capacitors (270 pF and 220 pF)loops, the optimal radius r0 ϭ /͌5 (7) at low frequencies positioned at each of four breaks in the loop. This resultedin which wavelength effects are negligible. in a net capacitor ESR at each coil break of approximately When coil noise is added, the optimal diameter changes, half the ESR of a single 490-pF capacitor in the same place.and the SNR realized is less than would be achieved with The unloaded Q calculated when using the ESRs for fourlossless coils. We calculate the optimal coil radii including individual 490-pF capacitors in series was 235. The calcu-coil losses as follows. The signal strength is proportional lation with the parallel capacitor pair arrangement yieldedTable 2Calculated and Measured Qs Frequency Skin depth Qloop Qloop Cap (pF) Rwire (⍀) Rcap (⍀) Rsol (⍀) (MHz) (mm) (calculated) (measured) a. Calculated and measured Qs of circular wire loop (d ϭ 3.2 mm, D ϭ 50 mm) 49 490 9.423e-3 0.028 0.039 0.017 304 290 64 270 8.203e-3 0.033 0.091 0.023 242 250 124 68 6.50e-3 0.052 0.132 0.047 345 358 207 24 5.09e-3 0.066 0.174 0.075 369 390 b. Calculated and measured Qs of ﬂat circular strip loop (w ϭ 4 mm, Dm ϭ 40 mm) 49 530 10.86e-3 0.033 0.040 0.037 224 204 65 300 9.42e-3 0.038 0.046 0.042 257 230 128 78 6.73e-3 0.053 0.073 0.059 345 310 203 32 5.38e-3 0.067 0.108 0.074 411 375
Noise Figure Limits of Circular Loop MR Coils 1205 duced eddy current losses, we applied the numerical MoM analysis to calculate currents induced on the conductors of a series of ﬂat strip loops overlapped with a driven 3.2- mm-diameter wire loop carrying 1 A; at 64 MHz. The wire and strip are separated by an 0.5-mm air gap. Flat strip widths of 2 mm, 5 mm, 7 mm, and 10 mm were evaluated with a constant 1-A current applied to the wire loop. Figure 6 shows the detailed current ﬂow on a 10-mm- wide strip from the driven, overlapping wire loop. The net current ﬂow through the center of the strip is 0.53 mA, which would result in a dissipation of ϳ5 nW compared to the total dissipation of 2 mW calculated for all currents. The currents in the individual strip mesh elements varied up to 250 mA. The current distribution is clearly circula- tory rather than ﬂowing in one direction along the strip. The circulatory current pattern is consistent with theFIG. 3. Effective series resistances in ohms (ESR) of 10-pF, 24-pF, losses being dominated by magnetically-induced circulat-68-pF, 270-pF, and 470-pF capacitors as a function of frequency ing eddy currents rather than from capacitance-coupled(MHz) as determined by the CAPCAD program. Points are at 25- (driving loop to ﬂat strip) E-ﬁeld induced currents.MHz intervals from 25 MHz to 500 MHz. The effect of the overlapping open-circuited ﬂat strip on both the noise resistance and the resonant frequency of thea Q of 304, which is within 5% of the measured value of290. Thus, using multiple, parallel capacitors at each coilbreak minimizes the total ESR from the tuning capacitorsand increases the QL and QU of the loops. Figure 3 shows ESR as a function of frequency in linearscale plotted for 10-, 24-, 68-, 270-, and 470-pF capacitorscomputed using the CAPCAD program. The ESR increaseswith operating frequency by up to 100% from 64 MHz to300 MHz.NF for Single CoilsFigure 4 shows NFcoil for a single wire loop coil and asingle tape loop coil as a function of frequency, coil radius,and number of capacitor junctions. The MoM calculationswere validated by nine experimental measurements at 64,130, and 400 MHz for NFcoil ϭ 0.5 dB, and at 64, 130, and200 MHz for NFcoil ϭ 1.77 dB using loops tuned with twoor four capacitors. The calculated and measured NFcoilvalues agree within 10%, the difference again not beingfrequency-dependent. The results show loops tuned withtwo capacitors perform better with slightly smaller radii atNFcoil ϭ 1.77 dB as compared to those tuned with fourcapacitors (Fig. 4; dashed line). This is validated by mea-surements at 64, 130, and 200 MHz (Fig. 4b). The NFcoil ϭ1.77dB coil (with 1/3 of the noise arising from coil losses)has a radius of 23 mm at 64 MHz, decreasing to about8 mm at 400 MHz.NF for Array CoilsNFcoil is plotted for the arrays drawn in Fig. 5a. Figure 5bshows the results for copper wire loops (d ϭ 46 mm) atϳ128 MHz and ﬂat washer loops (od ϭ 44 mm, id ϭ FIG. 4. Loop detector radius vs. frequency for various NFcoil con-36 mm) at ϳ128 MHz. The NFcoil for copper wire loops tours, numerically calculated by the full-wave MoM analysis for (a) wire loops, and (b) strip loops and experimental validation. Mea-does not increase signiﬁcantly as the number of over- sured values were determined from QU and QL measurements on alapped loops increases. However, NFcoil of the ﬂat strip healthy volunteer’s thigh. The wire diameter was 3.2 mm for looploops increases as the number of loops increases, indicat- radius Ͼ15 mm and 2 mm for loop radius Ͻ15 mm. The computeding that coupling losses accumulate as coils are added (17). dashed line and measured crosses (experimental) are for two ca- To investigate whether the additional losses in ﬂat loop pacitor breaks on the loop, instead of four. Strip width in (b) wasarrays are caused by direct E-ﬁeld interactions or by in- 4 mm.
1206 Kumar et al. For target depths of up to 100 mm the plots show little ﬁeld dependence, indicating that the losses are ﬁxed. For the range 20 Ͻ Յ100 mm, sample losses dominate, wave- length effects are negligible, and the deviation of rR from r0 is minimal. Thus, the quasistatic design rule r0 ϭ /͌5 is suitable for designing real coils with target depths in this range. Only at Ϸ 150 mm does rR vary signiﬁcantly with increasing ﬁeld, introducing some frequency dependence to the optimum design rule. Here, where the dimensions become comparable to the wavelength in the media, rR Ͻ r0 for ﬁelds above 7T, while at lower ﬁelds rR Ͼ r0, where the maximum sensitivity is displaced asymmetrically from the coil axis (14,15). Table 3b lists the SNR penalty incurred from use of the lossy depth-optimized loop coils, as compared with depth- optimized lossless coils with radii r0 ϭ /͌5. Signiﬁcant losses of Ͼ1.5 dB are apparent at target depths of Ͻ10 mm at every ﬁeld strength studied. DISCUSSION Coil preampliﬁers generally have an NF of about 0.5 dB (18). Manufacturers typically aim for a system NF of about 1 dB. For example, using the “hot-cold” resistor method (Ref. 9; p. 196), we have measured a 1.3-dB system NF on a Philips 3T Achieva at Johns Hopkins. It would therefore be desirable to limit the additional NF contributions of coils to about 1 dB.FIG. 5. a: Array geometries (i–v) for wire loops and ﬂat tape loops. Circular loop surface coils with small diameters areb: Effect of array geometry (i–v), on NFcoil for wire loops (D ϭ 46 mm) commonly used for many-coil array applications and ﬁeldat 128 MHz, and ﬂat tape loops (outer diameter ϭ 44 mm; innerdiameter ϭ 36 mm) at 128 MHz. The experimental values of NFcoilfor the individual component loops were ϳ0.6 dB and ϳ0.7 dB forthe wire and ﬂat strip coils, respectively. Loops were loaded withphysiologically analogous agar gel phantom (⑀ ϭ 63.5, ϭ 0.72S/m), and separated by 0.5 mm.resonant wire loop was analyzed as a function of stripwidth. Figure 7 shows that the ESR of the wire loop in-creases as a function of the width of the overlapping ﬂatstrip loop, as does the resonant frequency of the primaryresonant wire loop.Depth-Optimized CoilsThe coil radii, rR, that yielded the maximum SNR at agiven target depth along the coil axes, as determinedfrom the full-wave analysis with coil noise included, arelisted in Table 3a. The optimal lossless coil radii derivedfrom the quasistatic expression, r0 ϭ /͌5, based on theBio-Savart Law are also listed. Figure 8 plots the results of FIG. 6. Numerical MoM results illustrating eddy current effects in athe full-wave analysis. The difference between rR and the 10-mm-wide open-circuited ﬂat strip loop, overlapping a 23-mm-quasistatic optimum, r0, at shallow target depths ( Յ radius wire loop with a wire diameter of 3.2 mm. The coils are20 mm) reﬂects the dominance of coil noise. The radius of separated by a 0.5-mm gap, and the current in the wire loop iscoils optimized for ϭ 10 mm approaches that for ϭ constant at 1 A. The local currents on the strip coil are indicated by vectors whose lengths are linearly proportional to the current am-5 mm, as the relative increase in coil losses for the smaller plitude on the mesh elements. The vector value denoting the max-coil erodes the SNR advantage of further reductions in coil imum current (0.25 A) is annotated. The net current ﬂow in theradius at the shallower depth. These two coils also differ middle of the overlapped area is 0.53 mA. The pattern is clearlyfrom the others in that they are each tuned with only two circulatory, consistent with the losses being dominated by magnet-distributed capacitors instead of four, because of their ically-induced circulating eddy currents rather than from capaci-small size. tance-coupled (driving loop to ﬂat strip) E-ﬁeld induced currents.
Noise Figure Limits of Circular Loop MR Coils 1207 with four capacitor junctions. For a given NFcoil, the coil radius decreases as the resonant frequency increases. Thus, the radius of a wire 1-dB noise coil decreases to about 18 mm at 128 MHz (3T) and to about 11 mm at 400 MHz. The coil radius for a given NFcoil level decreases on average 40% from 64 MHz to 400 MHz for both wire loops and ﬂat strip loops. The wire loop radius at each frequency is slightly higher for a given NFcoil than the radius of the ﬂat strip loop, which means that a wire loop is slightly less efﬁcient as a single loop element than is a strip loop. On the other hand, surface coils made with ﬂat copper conductors were ad- versely affected by eddy current losses when arranged in arrays, whereas the wire coils were not so affected (Figs. 5 and 6). Increasing the dimensions of the individual 3.2- mm-diameter wire and 3– 4-mm-wide strips utilized in the present analysis would not signiﬁcantly affect our results. This is because conductor losses decrease as conductor area increases, but inductance only changes as the loga- rithm of the conductor diameter (Eq. ), while capaci- tance and solder joint losses would remain essentially the same. Knowing the NFcoil as a function of loop radius is valu- able for estimating SNR during reception, and also for transmission efﬁciency and consequent speciﬁc absorp- tion rate (SAR) when the loops are used for excitation.FIG. 7. a: Noise resistance (⍀) of the 23 mm wire loop of wire NFcoil is related to the ﬁeld dependence of SNR. Whendiameter 3.2 mm, as a function of the width of the overlapped ﬂat sample noise is dominant, (NF Ͻ 0.5 dB), the SNR istape loop (mm). b: Resonant frequency (MHz) of the wire loop as a approximately equal to the intrinsic SNR and varies ap-function of the width of the overlapped ﬂat tape loop width (mm). proximately linearly with ﬁeld strength B0 (10). WhenCoil spacing is 0.5 mm. NFcoil is large and coil noise dominates, SNR ϳ B0 (16). 7/4 Intermediate NFcoil values give rise to an SNR dependencestrengths. The relative noise contribution from each indi- between linear and the 7/4th power. If we include thevidual coil increases as loop size decreases. Additional change of capacitor ESR values with ﬁeld strength, thelosses arise from overlapping coils in arrays. In this work, capacitor ESR ϰ f1/2, which is the same as for the skin-we introduced the concept of a coil NF, NFcoil, to charac- depth losses of the wire. Therefore, including the capacitorterize the noise contribution from each coil relative to the losses with the conductor losses under coil noise-domi-total noise. NFcoil can be treated as a conventional NF and nant conditions also causes the SNR dependence on B0 tosimply added to the MRI system NF to totally characterize approach its 7/4th power.the noise properties. We then determined the practically Note that as coil size is increased above the sizes re-achievable loop surface coil NFcoil as a function of loop ported here, the NFcoil becomes negligible (Fig. 4). Also, asradius, accounting for the real effects of capacitor ESR, Table 3b shows, the problem of increasing coil noise is notsolder joint loss, and loop overlap, all of which place real at high ﬁelds but at low ﬁelds. Thus the real limits to coillimits on the SNR advantages of using ever-increasing performance occur at lower ﬁelds and small coil radii, asnumbers of ever-smaller detector elements in phased ar- reported herein.rays (4,19). We have shown also that using multiple parallel capac- Moreover, we have shown that, by determining the skin- itors per capacitance location rather than a single, largedepth conduction losses in wires and ﬂat strips, capacitor capacitor reduces total coil losses. Multiple parallel capac-ESRs and solder joint losses, one can accurately predict itors per junction also increases loop coil transmit powerthe unloaded resonant QU values for individual loops. We capability. The current ﬂow across the capacitor junctionalso showed that numerical full-wave MoM computations is distributed, thus avoiding capacitor heating and capac-of B1 and the losses in nearby body tissue, combined with itor burnout when applying powerful RF pulses. Minimiz-estimation of all of the coil losses, yield QL and hence coil ing the number of capacitor breaks in each loop—for ex-NFcoil values that are consistent with measured values over ample, using two capacitance positions vs. four capaci-the range 50 MHz to 200 MHz (Fig. 4). Thus the SNR and tance positions— can also help decrease NFcoil for loopNF performance of loop and array detectors can now be coils as coil radius is reduced (Fig. 4).characterized in advance, and this information can be used However, too few distributed capacitor junctions canfor designing the geometry and selecting the materials and cause charge buildup and result in direct E-ﬁeld losses viacomponents for fabricating individual surface coils and coupling of the coil to the imaging subject (9), so thisarrays that maximize SNR performance. tradeoff must be taken into account. For example, the The results show that at 64 MHz (1.5T) NFcoil contributes computed ratio of the direct E-ﬁeld losses to B1-induced1 dB to the total NF for a 24-mm-radius wire coil (Fig. 4) eddy-current losses in the sample for a 1-cm-diameter wire
1208 Kumar et al.Table 3Optimal Coil Radii (mm) for Target Depths of Interest, Determined With MoM Full-Wave Simulations (rR), and SNR Loss (dB) of theLossy Coils of Radii rR a. Optimal coil radii (mm) for target depths of interest, determined with MoM full-wave simulations (rR), as compared with quasistatic optimum radii (r0) without coil losses Target depth Optimal radius, rR (mm) with coil loss (full-wave method) r0 (mm) (mm) 1T 1.5T 3T 4.7T 7T 9.4T 5 2.2 5.4 5.3 5.0 4.6 4.6 4.5 10 4.4 9.0 8.0 7.8 7.5 5.6 5.3 20 8.9 15.2 14.7 13.7 11.9 11.3 11.1 50 22.0 25.8 25.7 23.3 21.3 22.4 22.5 75 33.5 34 34.5 33.8 34.4 35 34.5 100 45.0 46.5 47.5 48.0 48.3 46.6 44.8 150 67.0 79.4 78.4 78.0 76.0 67.5 62.0 b. SNR loss (dB) of the lossy coils of radii rR listed in (a), as compared to the SNR of lossless coils with radii r0 chosen to satisfy the quasi-static expression, r0 ϭ /͌5 Target depth, SNR loss (dB) (mm) 1T 1.5T 3T 4.7T 7T 9.4T 5 6.0 5.8 4.7 3.9 2.8 2.1 10 4.8 4.7 3.4 2.2 2.2 1.6 20 3.1 2.0 1.0 0.6 0.3 0.2 50 0.8 0.6 0.3 0.15 0.1 0.05 75 0.62 0.35 0.12 0.07 0.04 0.03 100 0.32 0.19 0.07 0.03 0.02 0.02 150 0.14 0.1 0.05 0.03 0.02 0.02loop with four capacitors is just 2% at 3T, while this ratio We observed no discernible change in NFcoil for 3.2-mm-increases to 80% for the two-capacitor 1-cm loop. The diameter wire loops arranged in overlapping arrays (Fig.series capacitor ESR effect plays an increasingly signiﬁ- 5b). However, ﬂat washer loops in a two-coil array exhib-cant role at higher frequencies (e.g., above 100 MHz; Fig. 3) ited a 0.3-dB increase in NFcoil, close to the numericallywhen loop sizes are small and additional series capaci- simulated 0.2-dB value. We attributed the increase in NF-tances increase NFcoil. coil in arrays of ﬂat coils to signiﬁcant eddy current losses When coil sizes are large, the interaction with the sam- (Fig. 6), which increase linearly with conductor widthple is strong, and distributing additional series capaci- (Fig. 7a). In addition, if the overlapping coils are ﬂat loops,tances does not signiﬁcantly affect the net losses. This there is an increase in the resonant frequency of the pri-works in favor of the coil designer, since large coils require mary loop (Fig.7b) due to an inductance decrease from thea small net capacitance for tuning, and distributing the shadowing effect of the overlapped loop. These computa-capacitances is necessary anyway to avoid E-ﬁeld losses tions were done for coils separated by a thin 0.5-mmand parasitic capacitance effects between coil conductors, insulating gap.and between the coil and the sample. While the magnitude of the intercoil losses from both E-ﬁeld and eddy-current losses decreases as the insulating gap is increased, the E-ﬁeld losses decrease faster than the eddy-current losses. Thus, the conclusion that eddy-cur- rents represent the dominant intercoil loss mechanism for ﬂat coils in phased-arrays is unaltered by increasing the gap. As is evident from Fig. 5b, the eddy current losses for the ﬂat washer loops also compound as the number of overlapping elements increases. Although we have been able to account for essentially all of the losses in single coils and the eddy current losses in arrays, it is certainly possible to produce coils that underperform relative to those analyzed here. For exam- ple, the coils and arrays we investigated were somewhat idealized in that they were conﬁgured with perfect decou- pling. We have excluded losses in the cables connectingFIG. 8. The optimum coil radius, rR (mm), including coil losses, as a adjacent coils, which may harbor additional eddy currentfunction of ﬁeld strength (T) for target depths of 10 mm, 20 mm,50 mm, 75 mm, 100 mm, and 150 mm, as determined by full-wave losses depending on how they are conﬁgured.numerical MoM. The data are ﬁt linearly. The quasistatic optimum The consequences of the substantial SNR degradationcoil (without coil losses) radii, r0, are indicated by horizontal bars in due to coil losses evident in Table 3b, especially at lowerthe center of the plot. ﬁeld strengths, underscores the difﬁculty of realizing any-
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