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Proceedings of the 2005 IEEE
Engineering in Medicine and Biology 27th Annual Conference
Shanghai, China, September 1-4, 2005
A New Definition of Mutual Impedance between
Two Coils for Simultaneous MRI Signal
Reception
Bing Keong Li1, Hon Tat Hui1 and Stuart Crozier1
1
School of Information Technology and Electrical Engineering, University of Queensland, Qld, Australia
Abstract - A new definition of mutual impedance for two coils
used for simultaneous signal reception in MRI is introduced.
The new mutual impedances are used to accurately quantify
the mutual coupling effect between the two coils so that it can
be removed from the measured terminal voltages. Using the
new mutual impedances, a new decoupling method can be
formulated to obtain the uncoupled voltages. Numerical results
on two square coils are obtained and the uncoupled coil
voltages calculated based on them indicate extremely small
errors when compared with the ideal uncoupled voltages. The
new mutual impedances are useful for MRI phased array
designs.
I. INTRODUCTION
Phased array coils have been suggested for use to
increase the signal-to-noise ratio (SNR) in magnetic
resonance imaging (MRI) [1]. They are an alternative RF
resonator for high-field imaging and superconductor
receiving coils in increasing the low SNR in MRI.
However, it is found that simultaneous reception of signal
from the phased array coils is hindered by a critical
problem. This is the mutual coupling effect between
receiving coils. To mitigate this problem, two popular
methods for mutual decoupling are suggested by Roemer
and collabarators [1]. The first method is to overlap adjacent
coils and the second method is to use extremely low input
impedance preamplifiers as buffers for the array coils. Both
these methods have drawbacks in that overlapping will
reduce the imaging area while low input impedance
preamplifiers will reduce the available signal power. Hence,
their decoupling powers may not be sufficient in some
cases.
In this paper, we consider this problem by
introducing a new mutual impedance, which is the key
parameter to be used in a new decoupling method to
compensate for the mutual coupling effect [2, 3]. We show
that in MRI, due to the available knowledge of the signal
source (the active slice of spinning protons, a new mutual
impedance can be defined very conveniently and accurately
to quantify the mutual coupling effect between two closely
placed surface coils. This greatly increases the accuracy in
the removal of the coupling voltages from the measured
voltages on the coil terminals. Numerical results for the
new mutual impedances for different separation distance
between coils have been obtained and the use of the new
0-7803-8740-6/05/$20.00 ©2005 IEEE.
mutual impedances for the formulation of the terminal
voltages of the coils will be introduced. In the new
expressions of the terminal voltages, the coupling voltages
can be easily identified and removed. The initial findings
presented herein, indicate that by using the new mutual
impedance, the coupling voltages can almost completely be
removed.
II. METHODOLOGY
The conventional definition of the mutual
impedance between two coils is defined with one coils being
in the transmitting mode connected to a current source while
the other is in the receiving mode and open-circuited [4, 5].
The mutual impedance, Z 12 is then calculated as the ratio of
the voltage induced across the open-circuit terminal of the
first coil (excited one) to the excitation current flowing
through the short-circuit terminal of the 2nd coil (exciting
one). (This ratio is negative if the current direction is
defined differently [4]) To calculate the open-circuit
voltage across the first coil (in the receiving mode),
reciprocity theorem is usually used with an assumption of a
(single-phase) sinusoidal current distribution flowing with
its terminals shorted [5]. Z 21 is calculated in a similar way
with the positions of coil 1 and 2 interchanged. Obviously,
the conventional definition of mutual impedance does not
properly model the interactions between the two coils for
MRI application because one coil has to be in the
transmitting mode whereas in MRI, the two coils are used
simultaneously either in the transmitting or receiving mode.
Another drawback of the conventional definition is the lack
of source reference. This implies that the conventional
definition of mutual impedance is self-contained. It ignores
the external signal source, which actually strongly affects
the interaction between the two coils. A third deficiency of
the conventional definition is that it does not take into
account the terminal loads, which actually affect the current
distributions and consecutively affect the mutual impedance
so calculated.
To remedy this situation, we propose a new definition
of the mutual impedance between two coils for signal
reception in MRI systems, which is useful for future study
of phased array coil in MRI application. Consider two
square coils used as RF resonator in MRI as shown in Fig. 1.
The two coils are designed square in shape and constructed
using metallic strip with a side length L and strip width W.
The distance between the centers of the two coils is denoted
by d, which can be changed to vary the amount of mutual
coupling. The coils are aligned along the Y axis on the Y-Z
plane and equidistant from the Z axis. The two coils are not
exactly on the same plane but offset by a small separation
dx along the X axis so that they can be overlapped with each
other. Four distributed capacitors and one tuning capacitor
are placed along the coils to reduce frequency shifts
associated with dielectric loading and to tune the coil to the
desired resonance frequency fo, which in this work is tuned
to 85MHz for 2T MRI systems. The distributed capacitors
are labeled as C1, C2, C3, and C4 and the tuning capacitor is
denoted as Ct. When the coils are placed alone, they
resonate at the frequency fo with a very small self-resistance
r. An external matching circuit is required to match the
small resistance r to the system impedance of 50Ω. To
define the new mutual impedance, an external signal source
is required. In MRI, the external signal source is the active
slice in the sample which is placed in the region with
negative X coordinates. A rectangular slice is considered in
this work (or otherwise it can be arbitrary) and its
dimensions along the X, Y, and Z axes are denoted
respectively by a, b, and c. The relative positions of the
slice from the two coils are indicated by the distances of the
slice from the origin which are denoted by px, py, and pz as
shown in Fig. 1.
ρ
The induced magnetization M from the active slice is
then defined as
ρ
M = [M x ( x, y )xˆ + jM y ( x, y ) yˆ ]
z
b
c
active slice
pz
py
px
coil 1
y
coil 2
L
d
x
V1
Z t12 =
V1
I2
(2)
Note that, unlike the conventional definition of mutual
impedance, the definition in (2) depends on the terminal
load Z L being connected to the two coils. That is, the two
coils are loaded with Z L rather than open-circuited or
shorted as in the conventional definition. Both V1 and I2 are
terminal quantities, which can easily be measured and thus
making it simpler in designing the measurement procedure
for the new impedance. Note that whether coil 1 is also
excited or not by the same active slice is not considered in
this definition. Only the induced voltage V1 which is
excited by the current distribution on coil 2 is taken into
account in (2). In general Z12
and Z21
are different
t I2
t I1
unless the two coils are exactly the same and the current
distributions i1=i2. This is because in the definition of Z12
t
and Z 21
, there is an external source. This is different from
t
the two conventional mutual impedances Z12 and Z 21 ,
which are equal for any reciprocal devices since they do not
require an external source in their definition.
(1)
where Mx(x,y) and My(x,y) are respectively the magnitudes
of the magnetization along the X and Y directions with unit
vectors denoted by x̂ and ŷ , respectively. Mx(x,y) and
My(x,y) are in general spatially and frequency dependent,
however for mutual coupling analysis, they are assumed to
take a constant unit magnitude of 1 A/m.
a
We now assume that coil 2 is excited by the active slice
so that a current distribution is induced on it. The value of
this current distribution at the terminal load is indicated by
I2. This induced current distribution then re-radiates and
induces (couples) a current distribution on coil 1 which
produces a voltage V1 across the terminal load Z L = r . The
new mutual impedance Z t12 is then defined as
V2
Fig. 1. The two square coils for simultaneous signal reception in MRI.
III. RESULTS AND DISCUSSION
Using the new definition in (2), the mutual impedance
of the two square coils for various separation distant d is
calculated and tabulated in Table I. It can be seen that the
new mutual impedances are very small for d > 1.0L and
decreases with increasing separation distance d. For d <
1.0L, the new mutual impedances decreases again with
decreasing separation distant. The conventional mutual
impedance shows a similar trend with the magnitude of the
mutual impedance reaching a peak almost at d = 1.0L. The
decreasing of mutual impedance with decreasing separation
for d < 1.0L justifies the use of overlapping of adjacent coils
for reducing the mutual coupling effect but judging from the
mutual impedance, the amount of coupling is still
substantial at d = 0.9L and both the new and conventional
mutual impedances increase again at d = 0.8L. It can also
be noticed from Table I that the new mutual impedances are
very different from the conventional mutual impedance.
TABLE I.
ª
ª U1 º « 1
« » = « 21
«¬ U 2 »¼ «− Z t
« Z
¬
L
The new mutual impedances of the two coils calculated at various
separations distant, d. The coil dimensions are L=120mm, W=10mm, and
dx=1.5mm. The capacitors values are C1=C2=C3=C4=60pF and Ct=56.5pF
and fo is set to 85MHz. The self-impedance of the stand-alone coil is
−
º
Z12
t
»
Z L » ª V1 º
« »
»
1 » «¬V2 »¼
¼
(5)
0.04+j0.04Ω. The active slice used in the calculation has dimensions of
a=200mm, b=400mm, and c=2.5mm and is positioned at px=-10mm, py=0,
and pz=0.
New Mutual Impedances
d
Conventional
Mutual
Impedances
12
t
Z
0.8 L
(Ω
Ω)
-5.73-j6.40
Z
21
t
(Ω
Ω)
4.43-j4.95
Z12 = Z 21
(Ω
Ω)
49.03-j129.73
0.9 L
0.77+j0.93
0.97+j1.09
-0.98+j11.85
1.0 L
-20.068-j39.82
-75.09-j17.92
3.63-j1103.37
1.1 L
5.38+j6.06
4.96+j5.64
-1.31+j1.77
1.2 L
3.24+j3.67
3.12+j3.56
-1.39-j1.53
1.3 L
2.18+j2.49
2.13+j2.45
-1.10-j2.16
1.4 L
1.57+j1.80
1.54+j1.78
-0.74-j2.19
1.5 L
1.17+j1.36
1.16+j1.35
-0.45-j1.98
1.6 L
0.90+j1.06
0.90+j1.05
-0.26-j1.70
1.7 L
0.71+j0.84
0.71+j0.84
-0.15-j1.44
1.8 L
0.57+j0.68
0.57+j0.68
-0.08-j1.21
1.9 L
0.47+j0.56
0.46+j0.56
-0.04-j1.02
2.0 L
0.38+j0.47
0.38+j0.47
-0.01-j0.86
Using (5), we find that the uncoupled voltages can be
obtained very accurately with percentage errors generally
smaller than 10−6% when compare with the ideal uncoupled
voltages, which are obtained when coils are receiving in
isolation. These extremely small percentage errors indicate
the accurate quantification of the mutual coupling effect by
the new mutual impedances and justify their usefulness. It
is interesting to note that the expressions in (5) do not even
require the knowledge of the self-impedance of the coils
which to some extent simpler than the conventional
formulation on open-circuit voltage method [6].
III. CONCLUSION
A new definition of mutual impedance for two coils used for
simultaneous signal reception in MRI is introduced. The
differences between the new mutual impedance and the
conventional mutual impedance are discussed in this work
and detailed calculation of the new mutual impedance is
given. In additional, numerical values for two square coils
are obtained and the method for the use of the new mutual
impedance to compensate for the coupling voltages is
provided.
Now consider expressing the terminal voltages on
the coils using the new mutual impedances. Using the
superposition principle, the terminal voltages V1 and V2
across the terminal loads of the two coils can be expressed
as the sum of two parts. The voltages induced by the active
slice U1, U2 and the voltage induced by the current on the
other coil, which are given as
REFERENCES
[1]
B. Roemer, W. A. Edelstein, C. E. Hayes, S. P. Souza, and O. M.
Mueller, “The NMR phased array,” Magn. Reson. Med., vol. 16, pp.
192-225, 1990.
[2]
H. T. Hui, “Improved compensation for the mutual coupling effect in
a dipole array for direction finding,” IEEE Trans. Antennas
Propagat., vol. 51, pp. 2498-2503, 2003.
V1 = ZLI1 = U1 + Z I
(3)
V2 = ZLI2 = U2 + Zt21I1
(4)
12
t 2
[3]
H. T. Hui, “A new definition of mutual impedance for application in
dipole receiving antenna arrays,” IEEE Antennas and Wireless
Propagation Letters, vol. 3, pp. 364-367, 2004.
[4]
where I1 and I2 are the terminal currents on the two coils,
and Z 21
are the coupled voltages. Hence by
Z12
t I2
t I1
, Z 21
and the
knowing the new mutual impedances Z12
t
t
terminal voltages V1 and V2, we can easily calculate the
uncoupled voltages U1 and U2 as
E. C. Jordan, Electromagnetic waves and radiating systems, New
York: Prentice-Hall Inc; 1968, Chapter 11.
[5]
R. E. Collin, Antennas and radiowave propagation, New York:
[6]
I. J. Gupta and A. A. Ksienski, “Effect of mutual coupling on the
McGraw-Hill, 1985.
performance of adaptive arrays,” IEEE Trans. Antennas Propagat.,
vol. 31, pp. 785-791, 1983.