Miheer Mayekar^{1}, Tejkiran Patil^{1}, Tapas Bhuiya^{1}, and Rajesh Harsh^{1}

Previous studies have shown that the coil matching network (CMN) and preamplifier input matching network (IMN) can be combined to match the coil loop to the transistor in the preamplifier directly. Through simulations we have shown that the artificial line ‘π’ LC BALUN can be used to match the coil loop to the transistor in the preamplifier directly. Though ‘L’ section LC BALUN is widely used in surface array coils to create high blocking impedance across coils, it has several limitations. Hence, artificial line LC BALUN can be used as impedance transformation network and as a common-mode choke simultaneously.

In a surface array coil, coupling
between adjacent coil elements can be reduced by creating high blocking
impedance at each coil elements. Each coil element in an array has three noise
signal components viz. thermal noise (V_{n_th}), coupled signal (V_{cp_signal}) and coupled thermal noise (V_{cp_n_th}) from the adjacent coil element. Total noise
voltage of a coil element in a two-element array is given by equation 1.

$$V_{noise_total_coil 1 }= V_{cp_signal_coil 2} + sqrt((V_{n_th}^2) + (V_{cp_n_th_coil 2}^2)) = ((ω*L_{coil}*k*V_{coil 2})/(R_{2} + Z_{block})) + sqrt((4*K*T*Δf*R_{coil 1}) + (4*K*T*Δf*((L_{coil}^2)*(k^2)/(R_{coil 2} + Z_{block}))))$$ 1

Where,k is coefficient of coupling between coil 1 and
coil 2 placed close to each other. According to equation 1, as Z_{block} increases, total noise voltage of the coil
approaches its minimum value which increases the SNR of an array^{1}.

Additionally, by reducing
inter-element coupling, high blocking impedance across coil elements also helps
to optimize the coil array for parallel imaging. Critically overlapped coil
elements in an array is another way to reduce inter-element coupling. However,
critical overlapping is not desirable from the perspective of the algorithms
used to construct MR images (SMASH/GRAPPA). In parallel imaging, the sensitivity
difference between two adjacent coils is inversely proportional to the noise in
the reconstructed image. Hence, to increase the SNR of parallel imaging, under-lapping
(maintaining a gap between two coils) or in some cases extra-lapping (more than
critical overlapping) is the optimum solution^{2,3}.

Conventionally, high blocking impedance at the
coil elements is created by using coil matching network (CMN). Past studies
have used ‘L’ section of inductor and capacitor as a CMN to produce a blocking impedance
that is on the order of 100 to 500 Ω;
which is directly proportional to the coil size and depends on coil loading^{4}.
Several attempts have been made to create blocking
impedance using ‘L’ sections LC BALUN^{5,6}, however they created
blocking impedance of conventional values i.e. 50, 75 and 300 Ω. Previous attempts also imposed restrictions of
low input impedances on the preamplifier to create high blocking impedance and
included a step-down transformer to bring down the input impedance of
preamplifier even further to increase the blocking impedance^{6}.
Several studies also used other methods for improving decoupling in a surface
coil array e.g. capacitive and inductive decoupling^{7,8}, decoupling
through induced current compensation or elimination (ICE)^{9} etc. But
these methods impose restrictions on coil’s fabrication and coil’s placement,
decreasing the flexibility required for parallel imaging coil optimisation.

*Cao* *et al.*^{10}
combine the coil matching network (CMN) and input matching network (IMN) of the
preamplifier (LNA) to create a high blocking impedance (Z_{block}) across the coil
and simultaneously terminate the LNA with its optimum input impedance (Z_{opt}) for noise
matching. Such a combination will eliminate the need of two separate networks
and reduces the noise figure of the entire receiver chain. We can form an artificial
line LC BALUN^{11} with such networks which can act as impedance
transformation networks as well as a common-mode choke.

METHODS:

$$L = sqrt(Z_{coil}*Z_{opt}^*)/ω_{O}$$ 2

$$C = 1/(ω_{O}*sqrt(Z_{coil}*Z_{opt}^*))$$ 3

According to equation
2 and 3, the product of Z_{coil} and Z_{opt}* has to be real and positive to get real values of BALUN components.
According to *Li et al.*^{12}, Addition of extra inductor or
capacitor on source side or load side of BALUN to get real values of BALUN
components, degrades the desired S-parameters of BALUN. An artificial line ‘π’
LC BALUN achieves matching without the degradation.

$$Z_{a} = Z_{b} = (-sin(β)*2*Z_{gate}*((j*2*Z_{gate}*sin(β)) - (Z_{coil}/2*cos(β))))/(j*(cos(β)^2*(Z_{coil}/2) - (Z_{coil}/2)) + (sin(β)*cos(β)*2*Z_{gate}))$$ 4

Equation 4 shows the relationship between Z_{coil}, Z_{gate} and Z_{opt} (Z_{a}||Z_{b}) for BALUN shown in figure 2.

DISCUSSION:

According to figure 3, S-parameters of artificial line π LC BALUN were matched with the desired S-parameters of general LC BALUN network i.e. return loss at differential ports is -6 dB, insertion loss at both differential ports is -3 dB, return loss at single ended port is as low as possible and phase difference between differential ports is 180⁰. According to figure 4 impedances have been transformed as per equation 4 and table 1.CONCLUSION:

Artificial line LC BALUNs, can present high input impedance of the LNA’s first stage FET (Z- Tobgay S. Novel concepts for RF surface coils with integrated receivers, M.S. Thesis. Worcester Polytechnic Institute at Worcester; April 2004.
- PD Dr. med. Stefan O. Schoenberg, Dr. rer. nat. Olaf Dietrich, et al., Parallel Imaging in Clinical MR Applications, Springer-Verlag Berlin Heidelberg, 2007, eBook ISBN: 978-3-540-68879-2.
- G. R. Duensing, S. Vijayakumar, S. B. King, The Effect of Over/Underlap of Surface Coil Elements on AP Direction Acceleration, Proc. ISMRM 14 (2006).
- US Patent: US 8.207,736 B2, Jun. 26, 2012.
- de Zwart, J. A., Ledden, P. J., Kellman, P. , van Gelderen, P. and Duyn, J. H. (2002), Design of a SENSE‐optimized high‐sensitivity MRI receive coil for brain imaging. Magn. Reson. Med., 47: 1218-1227. doi:10.1002/mrm.10169
- G. C. Nascimento, F. F. Paiva, and A. C. Silva, An Inductively Decoupled Coil Array for Parallel Imaging of Small Animals at 7T, Proc. Intl. Soc. Mag. Reson. Med. 16 (2008).
- B. Wu, P. Qu, Y. Pang, G. X. Shen, LC Decoupling Circuit for Arbitrarily placed Coils, Proc. Intl. Soc. Mag. Reson. Med. 14 (2006).
- G. C. Nascimento, F. F. Paiva, A. C. Silva, Inductive Decoupling of RF Coil Arrays: A Study at 7T, Proc. Intl. Soc. Mag. Reson. Med. 14 (2006).
- Li, Y. , Xie, Z. , Pang, Y. , Vigneron, D. and Zhang, X. (2011), ICE decoupling technique for RF coil array designs. Med. Phys., 38: 4086-4093. doi:10.1118/1.3598112
- Xueming Cao, Elmar Fischer, Jürgen Hennig, Maxim Zaitsev, Direct matching methods for coils and preamplifiers in MRI, Journal of Magnetic Resonance, Volume 290, Pages 85-91, 2018.
- Fredrick E. Terman, Electronic and Radio Engineering, McGraw-Hill 4th Ed. 1955.
- Richard Chi-Hsi Li, RF Circuit Design, John Wiley & Sons, Inc., 2009.

Figure 1. ‘L’ section LC BALUN matched to coil and FET of preamplifier directly. According
to the impedance transformation equation for ‘L’ section LC BALUN^{12 }Z_{gate}^{*} = L/C*Z_{block}, Z_{gate} has to be very low, which is achieved by
preamplifier input matching network, to create high Z_{block}.

Figure 2. Artificial line LC ‘π’ BALUN matched to coil and FET of preamplifier directly.
Impedance looking in to the coil is parallel combination of Z_{a} and Z_{b} which should be equal to Z_{opt}^{*} of the LNA for noise matching. Looking in to
the preamplifier, Z_{block} is four times the Z_{gate} of LNA FET. Table 1 shows the variation of Z_{a} or Z_{b} with phase β⁰ by keeping 2Z_{gate} and Z_{coil}/2 constant.

Table 1. 2Z_{gate} = 1000 Ω, Z_{coil}/2 = 5 Ω. As seen from table 1, for β = 80⁰ and -100⁰ individual artificial line ’π’ sections of BALUN
create (165.68 - i5666.58) Ω impedance from 5 Ω as opposed to ‘L’ section BALUN
which creates 200 KΩ impedance from 5 Ω with C = 2.49 pF,
L = 2.49 µH,
f = 63.87 MHz

Figure 3
S-parameters of π BALUN in figure 1 with 2Z_{gate} = 1000 Ω, Zcoil/2 = 5 Ω, β = 80⁰ for C_{a}, C_{b} = 2.09 pF, L_{C} = 2.45 µH and β = -100⁰ for L_{a}, L_{b} = 2.09 µF, C_{C} = 2.53 pF. S45, S46 = -3 dB, S55, S66
= -6 dB, S44 = -65.4 dB and S46 – S45 = 180⁰.

Figure 4. Z_{gate} ≈ 500 Ω, when 2Z_{gate} = 1000 Ω. b) Z_{opt}^{*} ≈ 82.84+i2833.29 Ω when Z_{coil}/2 = 5 Ω.