2. 1202 Kumar et al.
NF coil ϭ 10 ϫ log͑NPR͓͒dB͔ [4]
The coil NF (NFcoil) in Eq. [4] 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
[5]
FIG. 1. Circuit model of a sample loaded resonant loop (12). L and Q-Factor Prediction
C 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 angular
respectively. Vsig, Vns, and Vnc are, respectively, the detected MR frequency, , Q ϭ L/R. The unloaded coil contains only
signal voltage, the root-mean-square (rms) noise voltage from the coil resistance RC. Under sample-loaded conditions, the
sample, and the rms coil noise voltage produced by the loop resis- series resistance R includes the sum of coil and sample
tance. 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 to
attributable to eddy currents induced in adjacent conduc- tune the loops, and losses in the solder joints. Q-factors of
tors. 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 1
tion of frequency by full-wave electromagnetic numerical Q ind ϭ ,Q ϭ [6]
method-of-moment (MoM) analysis. The SNR of these RL C CRC
coils is then compared to the SNR of lossless coils whose
radii 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 formula
MATERIALS AND METHODS
ͱ
Figure 1 shows the circuit model of a sample-loaded res-
2
onant loop (12). The SNR of the loop detector measured as ␦ϭ . [7]
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 [1] 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
n
Vn 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
[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 the
copper 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 as
Noise Power Ratio and coil NF
meas ϭ R meas ϫ A/L [8]
The noise factor or noise power ratio (NPR) is the ratio of
SNRpwr at the input port of any device to that at the output
port. In the case of an RF coil used for MR signal detection, where A is the cross-sectional area and L is the length of
NPR compares the noise power with the coil noise in- the wire. Applying Eqs. [7] and [8] with measurements for
cluded, to the noise power which would be obtained with- the wire loop (subscript, wire), the copper resistive loss is
out it, and is given by (corrected from Ref. 12): then:
RS ϩ RC Q U/Q L SNR pwr͑unloaded͒ D
NPR ϭ ϭ ϭ [3] R wire ϭ wire ⅐ . [9]
RS Q U/Q L Ϫ 1 SNR pwr͑loaded͒ ␦ wired
3. Noise Figure Limits of Circular Loop MR Coils 1203
Table 1
Relationship 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 the
of 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 [10] Losses in the sample were numerically computed from the
2w␦
real part of the input impedances of loops under loaded
where Dm and w refer to the middle radius and width of the conditions (14,15) using the full-wave electromagnetic
strip, 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 for
modeling software CAPCAD (http://tinyurl.com/3qrg67) single coils, Green’s functions are applied to a semiinfinite
provided by the capacitor manufacturer Dielectric Labora- medium with the electrical properties (conductivity, 0.69
tories (Cazenovia, NY, USA; http://www.dilabs.com/in- S/m Յ Յ 0.82 S/m; dielectric constant, 58 Յ ⑀r Յ 80) of
dex.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 , [11] 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 be
where Qm1 and Qm2 are the Qs measured before and after between 1 mm and 3 mm depending on the size of the
the 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 in
determined 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 [12]
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, including
The inductance of the flat washer loop is determined by the computed losses associated with eddy currents on the
applying the Neumann formula for inductance (13), (Wiki- surface of overlapped loop elements. Loops comprising
pedia, Inductance; http://en.wikipedia.org/wiki/Induc- arrays were configured with the prescribed geometries that
tance) and numerically integrating Eq. [13] 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 [13] 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 outer
radii 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,
[14] Palo Alto, CA, USA) operating in S21 mode, calibrated for
4. 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 , [15]
ͱRC ϩ RS
is calculated as a function of loop radii and frequency, and
FIG. 2. a: Circuit model of a sniffer coil. A gap in the ground shield the loop with the maximum SNR is determined for each
is shown at the top of the loop. The middle gray line is the inner target depth. The coil radii considered in the analysis
conductor; the dark outer lines represent ground shield. b: Two range from 2 mm to 125 mm in the MR frequency range of
sniffer 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 dB
One sniffer loop transmits RF energy to the resonant loop and the
obtained from deploying an optimal coil of radius rR with
other 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 intrinsic
a 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 ͔ [16]
loop and the receive sniffer detects currents excited in the
test loop. The QL measurements for single loops were done
RESULTS
with the coil placed on the thigh muscle of a healthy adult
volunteer 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 for
containing 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 were
single, tuned, resonant coil with the other overlapping distributed at four equally-spaced breaks in the loops. The
loops open-circuited. The open circuit loops simulate per- computed QU values agree with the measured values with
fect decoupling while including the same eddy current a deviation of Յ6% for wire loops and Յ8% for the tape
losses 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. [3] and [4]. quency.
Depth-Optimized Coils Effect of Multiple Parallel Capacitors
We apply the results to determine loop sizes that optimize The loop resonating at 48.5 MHz in Table 2 was tuned with
SNR 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 resulted
in 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 four
lossless 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 yielded
Table 2
Calculated 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
5. 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 the
FIG. 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 the
a Q of 304, which is within 5% of the measured value of
290. Thus, using multiple, parallel capacitors at each coil
break minimizes the total ESR from the tuning capacitors
and increases the QL and QU of the loops.
Figure 3 shows ESR as a function of frequency in linear
scale plotted for 10-, 24-, 68-, 270-, and 470-pF capacitors
computed using the CAPCAD program. The ESR increases
with operating frequency by up to 100% from 64 MHz to
300 MHz.
NF for Single Coils
Figure 4 shows NFcoil for a single wire loop coil and a
single tape loop coil as a function of frequency, coil radius,
and number of capacitor junctions. The MoM calculations
were validated by nine experimental measurements at 64,
130, and 400 MHz for NFcoil ϭ 0.5 dB, and at 64, 130, and
200 MHz for NFcoil ϭ 1.77 dB using loops tuned with two
or four capacitors. The calculated and measured NFcoil
values agree within 10%, the difference again not being
frequency-dependent. The results show loops tuned with
two capacitors perform better with slightly smaller radii at
NFcoil ϭ 1.77 dB as compared to those tuned with four
capacitors (Fig. 4; dashed line). This is validated by mea-
surements at 64, 130, and 200 MHz (Fig. 4b). The NFcoil ϭ
1.77dB coil (with 1/3 of the noise arising from coil losses)
has a radius of 23 mm at 64 MHz, decreasing to about
8 mm at 400 MHz.
NF for Array Coils
NFcoil is plotted for the arrays drawn in Fig. 5a. Figure 5b
shows the results for copper wire loops (d ϭ 46 mm) at
ϳ128 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 a
lapped loops increases. However, NFcoil of the flat strip healthy volunteer’s thigh. The wire diameter was 3.2 mm for loop
loops increases as the number of loops increases, indicat- radius Ͼ15 mm and 2 mm for loop radius Ͻ15 mm. The computed
ing 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) was
arrays are caused by direct E-field interactions or by in- 4 mm.
6. 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 are
b: Effect of array geometry (i–v), on NFcoil for wire loops (D ϭ 46 mm) commonly used for many-coil array applications and field
at 128 MHz, and flat tape loops (outer diameter ϭ 44 mm; inner
diameter ϭ 36 mm) at 128 MHz. The experimental values of NFcoil
for the individual component loops were ϳ0.6 dB and ϳ0.7 dB for
the wire and flat strip coils, respectively. Loops were loaded with
physiologically analogous agar gel phantom (⑀ ϭ 63.5, ϭ 0.72
S/m), and separated by 0.5 mm.
resonant wire loop was analyzed as a function of strip
width. Figure 7 shows that the ESR of the wire loop in-
creases as a function of the width of the overlapping flat
strip loop, as does the resonant frequency of the primary
resonant wire loop.
Depth-Optimized Coils
The coil radii, rR, that yielded the maximum SNR at a
given target depth along the coil axes, as determined
from the full-wave analysis with coil noise included, are
listed in Table 3a. The optimal lossless coil radii derived
from the quasistatic expression, r0 ϭ /͌5, based on the
Bio-Savart Law are also listed. Figure 8 plots the results of FIG. 6. Numerical MoM results illustrating eddy current effects in a
the 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 are
20 mm) reflects the dominance of coil noise. The radius of separated by a 0.5-mm gap, and the current in the wire loop is
coils 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 the
radius at the shallower depth. These two coils also differ middle of the overlapped area is 0.53 mA. The pattern is clearly
from 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.
7. 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. [12]), 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. When
diameter 3.2 mm, as a function of the width of the overlapped flat sample noise is dominant, (NF Ͻ 0.5 dB), the SNR is
tape 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). When
Coil 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 dependence
strengths. The relative noise contribution from each indi- between linear and the 7/4th power. If we include the
vidual coil increases as loop size decreases. Additional change of capacitor ESR values with field strength, the
losses 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 capacitor
terize 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 to
simply 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, as
radius, accounting for the real effects of capacitor ESR, Table 3b shows, the problem of increasing coil noise is not
solder joint loss, and loop overlap, all of which place real at high fields but at low fields. Thus the real limits to coil
limits on the SNR advantages of using ever-increasing performance occur at lower fields and small coil radii, as
numbers 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, large
depth 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 power
the unloaded resonant QU values for individual loops. We capability. The current flow across the capacitor junction
also 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 loop
NF 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 can
for designing the geometry and selecting the materials and cause charge buildup and result in direct E-field losses via
components for fabricating individual surface coils and coupling of the coil to the imaging subject (9), so this
arrays 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-induced
1 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
8. 1208 Kumar et al.
Table 3
Optimal Coil Radii (mm) for Target Depths of Interest, Determined With MoM Full-Wave Simulations (rR), and SNR Loss (dB) of the
Lossy 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 0.02
loop 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 numerically
when 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 width
ple 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 the
a 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-mm
and 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 connecting
FIG. 8. The optimum coil radius, rR (mm), including coil losses, as a
adjacent coils, which may harbor additional eddy current
function 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 degradation
coil (without coil losses) radii, r0, are indicated by horizontal bars in due to coil losses evident in Table 3b, especially at lower
the center of the plot. field strengths, underscores the difficulty of realizing any-
9. Noise Figure Limits of Circular Loop MR Coils 1209
where near either the intrinsic SNR (10) let alone the 2. Pruessmann KP, Weiger M, Scheidegger MB, Boesiger P. SENSE: sen-
ultimate intrinsic SNR (11) as coil radius is reduced below sitivity encoding for fast MRI. Magn Reson Med 1999;42:952–962.
3. Sodickson DK, Manning WJ. Simultaneous acquisition of spatial har-
10 mm, at least for room temperature coils. Not only does monics (SMASH): fast imaging with radiofrequency coil arrays. Magn
SNR decrease because of coil loss, but, in addition, the Reson Med 1997;38:591– 603.
optimal coil size increases, resulting in a further reduction 4. Wiggins GC, Triantafyllou C, Potthast A, Reykowski A, Nittka M, Wald
in SNR as compared to what would have resulted from LL. 32-channel 3 Tesla receive-only phased-array head coil with soc-
cer-ball element geometry. Magn Reson Med 2006;56:216 –223.
using a smaller, optimized, lossless detector.
5. Setsompop K, Wald LL, Alagappan V, Gagoski B, Hebrank F, Fontius U,
Because single coils form the basis of MRI phased arrays, Schmitt F, Adalsteinsson E. Parallel RF transmission with eight chan-
an optimized array must use optimized array elements. nels at 3 Tesla. Magn Reson Med 2006;56:1163–1171.
Figure 5 shows that the additional losses associated with 6. Edelstein WA. Radiofrequency systems and coils for MRI and MRS. In:
assembling the loops into array configurations can be min- Grant D, Harris R, editors. Encyclopedia of nuclear magnetic resonance,
Vol. 6. New York: Wiley; 1996. p 3950 –3955.
imized by using wire coils. Figure 8 indicates that as coil 7. Edelstein WA, Foster TH, Schenck JF. The relative sensitivity of surface
radius become smaller, the size of an optimized element coils to deep lying tissues. In: Proceedings of the 4th Annual Meeting,
differs significantly from the optimum size computed as- Society of Magnetic Resonance, London, UK, 1985. p 964, 965.
suming no coil losses. As a consequence, the increase in 8. Roemer P, Edelstein WA. Ultimate sensitivity limits of surface coils. In:
Proceedings of the SMRM 6th Annual Meeting, New York, NY, USA,
loop coil losses with decreasing coil radius places very
1987. p 410.
real limits on the improvement in SNR performance of 9. Chen CN, Hoult DI. Biomedical magnetic resonance technology.
phased arrays that can be gained by reducing the size of Bristol: Adam Hilger; 1989. p 160 –161, 143 p.
individual array elements. These limits are frequency-de- 10. Edelstein WA, Glover GH, Hardy CJ, Redington RW. The intrinsic
pendent. signal-to-noise ratio in NMR imaging. Magn Reson Med 1986;3:604 –
618.
Thus the linearly dependent SNR seen with head and 11. Ocali O, Atalar E. Ultimate intrinsic signal-to-noise ratio in MRI. Magn
body MRI coils (10)—and large surface coils—is unrealis- Reson Med 1998;39:462– 473.
tic for small coils where coil losses dominate. Our results 12. Edelstein WA. Principles and modeling of radiofrequency coil signal
(e.g., Table 3b) suggest that use of cold or superconducting detection and image production. In: Proceedings of the Weekend Edu-
cational Program on RF Systems Engineering, 14th Annual Meeting of
RF loop coils and tuning elements (20 –22) may be more
ISMRM, Seattle, WA, USA, 2006.
beneficial for shallow target depth and small coils than has 13. Grover FW. Inductance calculations. Mineola, NY: Dover Publications,
heretofore been considered for human subjects, as NFcoil Inc.; 1973.
would approach zero. 14. Kumar A, Bottomley PA. Optimizing the intrinsic signal-to-noise ratio
of MRI strip detectors. Magn Reson Med 2006;56:157–166.
15. Kumar A, Bottomley PA. Optimized quadrature surface coil designs.
MAGMA 2008;21:41–52.
CONCLUSIONS 16. Hoult DI, Lauterbur PC. Sensitivity of the zeugmatographic experiment
involving human samples. J Magn Reson 1979;34:425– 433.
We have shown that it is possible to completely character-
17. Wiggins GC, Alagappan V, Potthast A, Schmitt M, Wiggins CJ, Fischer
ize the noise and SNR performance of real surface loop H, Jahns K, Benner T, Polimeni J, Wald LL. Design optimization and
coils and arrays using the manufacturer’s data on capacitor SNR performance of 3T 96-channel phased array head coils. In: Pro-
losses, measured losses in solder connections, and calcu- ceedings of the 15th Annual Meeting of ISMRM, Berlin, Germany, 2007
lated effects of conductor skin-depth resistive losses as (Abstract 243).
18. Mueller OM; General Electric, assignee. Unconditionally stable ultra
well as tissue losses. Our results enable prediction of coil low noise RF preamplifier. US patent 4835485; 1989.
NFs, NFcoil, for various size loop coils. NFcoil is a key factor 19. Zhu YD, Hardy CJ, Sodickson DK, Giaquinto RO, Dumoulin CL, Ken-
in establishing upper limits to the SNR benefits of very wood G, Niendorf T, Lejay H, McKenzie CA, Ohliger MA, Rofsky NM.
large channel-counts in phased-arrays and massively par- Highly parallel volumetric imaging with a 32-element RF coil array.
Magn Reson Med 2004;52:869 – 877.
allel MRI. This approach should be useful in the design
20. Black R, Early T, Roemer PB, Mueller OM, Mogro-Campero A, Turner
and fabrication of loop MR detectors and transmitters over L, Johnson G. A high-temperature superconducting receiver for nuclear
a wide range of field strengths. magnetic resonance microscopy. Science 1993;259:793–795.
21. Boskamp EB, Lindsay SA, Lorbiecki JE. On the coil noise contribution
to SNR versus coil diameter, temperature, frequency and load distance.
REFERENCES In: Proceedings of the 13th Annual Meeting of ISMRM, Miami Beach,
FL, USA, 2005 (Abstract 916).
1. Roemer PB, Edelstein WA, Hayes CE, Souza SP, Mueller OM. The NMR 22. Darrasse L, Ginefri J-C. Perspectives with cryogenic RF probes in bio-
phased array. Magn Reson Med 1990;16:192–225. medical MRI. Biochemie 2003;85:915–937.
![1202 Kumar et al.
NF coil ϭ 10 ϫ log͑NPR͓͒dB͔ [4]
The coil NF (NFcoil) in Eq. [4] 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
[5]
FIG. 1. Circuit model of a sample loaded resonant loop (12). L and Q-Factor Prediction
C 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 angular
respectively. Vsig, Vns, and Vnc are, respectively, the detected MR frequency, , Q ϭ L/R. The unloaded coil contains only
signal voltage, the root-mean-square (rms) noise voltage from the coil resistance RC. Under sample-loaded conditions, the
sample, and the rms coil noise voltage produced by the loop resis- series resistance R includes the sum of coil and sample
tance. 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 to
attributable to eddy currents induced in adjacent conduc- tune the loops, and losses in the solder joints. Q-factors of
tors. 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 1
tion of frequency by full-wave electromagnetic numerical Q ind ϭ ,Q ϭ [6]
method-of-moment (MoM) analysis. The SNR of these RL C CRC
coils is then compared to the SNR of lossless coils whose
radii 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 formula
MATERIALS AND METHODS
ͱ
Figure 1 shows the circuit model of a sample-loaded res-
2
onant loop (12). The SNR of the loop detector measured as ␦ϭ . [7]
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 [1] 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
n
Vn 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
[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 the
copper 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 as
Noise Power Ratio and coil NF
meas ϭ R meas ϫ A/L [8]
The noise factor or noise power ratio (NPR) is the ratio of
SNRpwr at the input port of any device to that at the output
port. In the case of an RF coil used for MR signal detection, where A is the cross-sectional area and L is the length of
NPR compares the noise power with the coil noise in- the wire. Applying Eqs. [7] and [8] with measurements for
cluded, to the noise power which would be obtained with- the wire loop (subscript, wire), the copper resistive loss is
out it, and is given by (corrected from Ref. 12): then:
RS ϩ RC Q U/Q L SNR pwr͑unloaded͒ D
NPR ϭ ϭ ϭ [3] R wire ϭ wire ⅐ . [9]
RS Q U/Q L Ϫ 1 SNR pwr͑loaded͒ ␦ wired](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)