The coupling between any two loops in an MRI RF phased-array coil may be reduced by using a capacitive decoupling network (CDN.) CDN is compatible with.
Design Guidelines for the Capacitive Decoupling Networks J. Jevtic1, V. Pikelja1, A. Menon1, D. Seeber1, N. Tatum1, W. Johnson1 1
IGC-Medical Advances, Inc., Milwaukee, Wisconsin, United States Synopsis The coupling between any two loops in an MRI RF phased-array coil may be reduced by using a capacitive decoupling network (CDN.) CDN is compatible with parallel imaging, transmit receive operation, and other decoupling techniques. A generalized CDN may be significantly simplified in many cases of practical interest by selecting only a portion of the full generalized circuit. We will present an efficient and simple design procedure for the CDN, including an overview of practical layouts, closed form expressions for the capacitor values, iterative calculation strategy, as well as the results of numerical modeling and coil measurements. Introduction The capacitive decoupling network1 (CDN) is capable of reducing the coupling between the channels of an MRI phased-array RF coil. The technique is compatible with non-overlapping loops2 and high power requirements, which are desirable in parallel imaging and transmit/receive coils, respectively. In addition, it may be used in conjunction with preamplifier decoupling and coil overlap for added decoupling performance. Fig.1 shows 3 stages of an unrolled generalized cylindrical CDN with 4 capacitors per stage, capable of inductively decoupling up to 9 identical loops laid on a cylinder, such as in a head coil, for example. The initial experience of several practitioners, as well as our own, has been that the conventional semi-empirical adjustment of the capacitors is slow and tedious, due to the large number of components and their mutual interdependence. We have since used the CDN in many coil designs and were able to develop several efficient and simple ways of selecting the network layout and calculating the component values. Design Procedure Fig. 1 Generalized CDN Our design approach starts with a layout selection for the simplified CDN, followed by the determination of the coupling coefficients, calculation of the capacitor values, and measurement of the residual couplings. A simplified CDN may be obtained from the generalized network in Fig. 1 by setting certain capacitor values to zero or infinity, and replacing the components with open and short circuits, respectively. Several examples of simplified CDNs are illustrated in Fig.2. Once the layout of the loops and the CDN is selected we build either a physical or a computer model of the coil. An example of the computer model in NEC3 is shown in Fig. 4. Next, for each pair of resonant loops, i and j, we determine the coupling coefficient kij by measuring the eigen-frequencies f+ and f- of the two resonant modes: k=(f+2-f-2)/(f+2+f-2).The knowledge of the coupling coefficients allows us to calculate the required capacitor values in the CDN by using either closed form expressions, iterative solution of a set of nonlinear equations, or by using a numerical field/circuit solver. Closed Form Expressions First circuit in Fig. 2 is a special case of CDN in Fig. 1 for C3=inf and C4=0. We use Cr to denote the value of a single capacitor which resonates a loop. The second circuit is an alternative CDN for four identical loops (C1=C4=0). Similar layout can be used even for an 8 channel coil, as in the third circuit, provided the coupling between all non-adjacent loops is nearly equal in value to kavg. Finally, the fourth circuit allows for decoupling between 4 loops which are not necessarily identical.
C1 =
Cr k21 − k31 k 21 k 21 + k31
C2 =
Cr k31 2k21 k21 + k31
Cr 4k31 Cr C3 = k21 − k31
C2 =
Cr 8k avg Cr C3 = k12 − kavg
C2 =
C1 =
C1r C3r 2k13
C2 =
C2 r C4 r 2k24
4 4
C2 r C4r k13
C1r C3r k 24 + 4 C2r C4 r k13 4
4
C1r C3r k24
C1r C3r k24 + 4 C2 r C4 r k13
kij Iterative Calculation Approach k13k24 1 , i≠ j = − Fig. 2 Simplified CDNs with corresponding design formulas. The derivation and use of closed form Cij Cir C jr 4 C1r C2 r C3r C4r expressions becomes impractical when the CDN has more than 2 capacitors per stage. Instead, the components of the generalized circuit in Fig.1 may be determined using an iterative 0 C3 + 2C4 −C [ a] = 4 [b] nonlinear equation solver, such as a multi-variate relaxed Newton-Raphson − − C3 0 C 2 method. The decoupling condition for a given set of capacitor values a b [ ][ ] C1,C2,C3, and C4 can be expressed as x2 i+1 – x2 i-1 = ki1 /Cr for i=2,...[N/2]+1 − C4 0 −C [ A] = ... [b ] = 3 where N is the number of identical loops and xi can be calculated from 2 0 − C2 C + C + C [ a][b] 1 2 3 [x]=[A]-1[c] where [A] is a block diagonal matrix specified in Fig. 3. T [a ] [c] = [ −1 0 1 0 0 ... 0] Numerical Field/Circuit Simulation For more than 4 non-identical loops, the computational electromagnetics Fig. 4 Visualization of Fig. 3 Circuit matrix for CDN in Fig. 1. codes, such as NEC3 or FastHenry4, allow for an efficient and fairly reliable coupling in NEC. trial-and-error determination of the CDN components in addition to determination of coupling coefficients between the loops before the physical model is even built. The absolute difference between the calculated and measured coupling coefficients is usually better than 0.005. Fig. 4 shows the output of NEC simulation for a complex 8 channel coil made out of triangular loops before it was decoupled using a CDN. Two non-adjacent loops are coupled as evidenced by the color scale which reflects the intensity of the RF currents. Summary Capacitive decoupling networks can be efficiently designed with the aid of closed form expressions, iterative non-linear solvers, and numerical field/circuit solvers. References: [1] Proc.ISMRM 9:17 (2001); [2] MRM 45:495 (2001); [3] NEC, Lawrence Livermore Lab (1981); [4] IEEE/MTT 42:1750 (1994)
Proc. Intl. Soc. Mag. Reson. Med. 11 (2003)
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