Synopsis

Keywords: RF Arrays & Systems, RF Arrays & Systems, matching

The noise figure of a matching network between a receive coil and a preamplifier with low input impedance can be estimated using relationship \(F\approx1+R_{11}/R_\mathrm{c}\). It is demonstrated how to use this relationship to choose optimal matching-decoupling network topology in terms of signal-to-noise ratio (SNR). The formula serves as an efficient tool to estimate noise figure of matching networks and is useful in low-field and high-coil-Q applications, where the noise of the matching network becomes comparable to coil noise.

Introduction

Preamplifier decoupling is widely used in receive coil arrays to avoid detuning array elements unwantedly¹. Design equations for ideal three-² and four-element³ networks (or equivalent) have been reported. However, there is no explicit guideline for choosing circuits for better noise performance. Whereas resistive loss of networks can significantly degrade SNR at preamplifiers’ inputs, it largely relies on coil designers’ experience to discern the most SNR-degrading network component and to minimise SNR degradation. A diagram of an array element consisting of a coil, a preamplifier and a matching-decoupling network is illustrated in Figure 1. In the following sections, we show that the noise figure of the matching network can be estimated using $$F\approx1+\dfrac{R_{11}}{R_\mathrm{c}},\qquad(1)$$where $$$R_\mathrm{c}$$$ is the coil resistance and $$$R_{11}=\Re Z_{11}$$$ is the input impedance of the matching network $$$Z_\mathrm{i}$$$ when preamplifier A is disconnected. Equation (1) holds for typical low-input-impedance preamplifiers with $$$X_\mathrm{a}+X_\mathrm{o}=0$$$. It reveals that the SNR degradation is determined by the ratio between component loss defining $$$R_{11}$$$ and coil loss.

Methods

To derive (1), $$$F$$$ is defined as the relation between the SNR before entering preamplifier A and the SNR on coil C: $$$\mathrm{SNR}_\mathrm{a}=\mathrm{SNR}_\mathrm{c}\times F^{-1}$$$. Unless otherwise noted, $$$X=\Im Z$$$, $$$R=\Re Z$$$. At 290 K, $$$F^{-1}=G_\mathrm{av}$$$.⁴ Figure 1 shows the expression of $$$G_\mathrm{av}$$$.⁵ From the relation between $$$Z_\mathrm{i}$$$ and $$$Z_\mathrm{o}$$$, we have $$-Z_{⌀}^{2}=\left(Z_\mathrm{i}-Z_{11}\right)\left(Z_{22}+Z_\mathrm{a}\right)=\left(Z_\mathrm{o}-Z_{22}\right)\left(Z_{11}+Z_\mathrm{c}\right).\qquad(2)$$ From (2) we solve $$Z_{22}=\dfrac{Z_\mathrm{a}\left(Z_{11}-Z_\mathrm{i}\right)+Z_\mathrm{o}\left( Z_\mathrm{c}+Z_{11}\right)}{Z_\mathrm{i}+Z_\mathrm{c}}\,.\qquad(3)$$ Therefore $$Z_\mathrm{o}-Z_{22}=\dfrac{\left(Z_\mathrm{i}-Z_{11}\right)\left( Z_\mathrm{o}+Z_\mathrm{a}\right)}{Z_\mathrm{i}+Z_\mathrm{c}}\,.\qquad(4)$$ For a lossless matching network, $$$\left.Z_{11}\right|_{\mathrm{lossless}}=-\mathrm{j}X_\mathrm{c}$$$ if $$$X_\mathrm{a} + X_\mathrm{o} = 0$$$ and the matching network is correctly tuned. $$$\left.\Im Z_\mathrm{i} \right|_{\mathrm{lossless}}=-X_\mathrm{c}$$$ yields the maximum preamplifier decoupling regardless of $$$X_\mathrm{a}+X_\mathrm{o}$$$.⁶ For low-loss matching networks, it is found in simulation and experiments that, if $$$X_\mathrm{a}+X_\mathrm{o}=0$$$, $$$\Im Z_{11}=-X_\mathrm{c}$$$ still holds well; $$$\Im Z_\mathrm{i}=-X_\mathrm{c}$$$ is also a close condition for maximum preamplifier decoupling. So assume $$$\Im Z_{11}=\Im Z_\mathrm{i}=-X_\mathrm{c}$$$; together with $$$X_\mathrm{a}+X_\mathrm{o}=0$$$, we infer that $$$\Im \left(Z_\mathrm{i}-Z_{11}\right) = 0$$$, $$$\Im \left(Z_\mathrm{o}+Z_\mathrm{a}\right)=0$$$, $$$\Im \left(Z_\mathrm{i}+Z_\mathrm{c}\right) = 0$$$. Then, using (4), $$$\Im \left(Z_\mathrm{o}-Z_{22} \right)=0$$$. We can then write $$$Z_\mathrm{o}-Z_{22}=R_\mathrm{o}-R_{22}$$$ and $$$Z_\mathrm{c}+Z_{11}=R_\mathrm{c}+R_{11}$$$. Substituting these into $$$G_{\mathrm{av}}$$$ in Figure 1, we get $$F=G_{\mathrm{av}}^{-1}=\dfrac{R_\mathrm{c}+R_{11}}{R_\mathrm{c}}\times\dfrac{R_\mathrm{o}}{R_\mathrm{o}-R_{22}}\,.\qquad(5)$$ Recall that $$$Z_{11}$$$ is defined by the input impedance at port 1 with port 2 disconnected, and $$$Z_{22}$$$ is defined by the input impedance at port 2 with port 1 disconnected⁷. Compared with $$$R_\mathrm{o}$$$—often 50 Ω and higher—$$$R_{22}$$$ of a low-loss matching network should be negligible. Neglecting $$$R_\mathrm{o}/\left( R_\mathrm{o}-R_{22}\right)$$$, we arrive at (1), thereby concluding the derivation.
To demonstrate (1) and (5) at 32.13 MHz, the Larmor frequency of ¹³C at 3 T, oval mouse coils shown in Figure 2(a) are matched to WMA32C preamplifiers (WanTCom Inc., Chanhassen, MN, USA). The coil impedance $$$Z_\mathrm{c}=25.7~\mathrm{mΩ}+\mathrm{j}11.7~Ω$$$, output impedance $$$Z_\mathrm{o} =Z_\mathrm{n,opt}=52.04+\mathrm{j}0.07~Ω$$$ and preamplifier impedance $$$Z_\mathrm{a}=2.30-\mathrm{j}0.77~Ω$$$ for initial calculation. It follows that $$$X_\mathrm{a}+X_\mathrm{o}=-0.70~Ω$$$, close to 0. The three-² and four-element³ matching networks are first calculated by corresponding formulae, and then realised by off-the-shelf components. Printed circuit boards (PCBs) are routed and simulated in Advanced Design System (Keysight Technologies, Santa Rosa, CA, USA). $$$F$$$ is extracted from simulations. PCBs are fabricated, and SNR values are measured on a bench top with an SMC100 signal generator (Rohde & Schwarz, Munich, Germany) and an Agilent E4440A spectrum analyser (Keysight Technologies, CA, USA). During simulation and experiments, initially capacitors and inductors are fine-tuned so that the maximum preamplifier decoupling is achieved for the mouse coil, and the output impedance is $$$Z_\mathrm{o}=Z_\mathrm{n,opt}=52~Ω$$$; then the peaks of SNR are tuned to 32.13 MHz. Capacitors are from PPI 1111C (Passive Plus Inc., Huntington, NY, USA). Inductors are from Coilcraft Inc. (Cary, IL, USA). Since our experiment is on a test bench, for simplicity, no active decoupling circuit is implemented.

Results

The measured coil current (preamplifier decoupling) and SNR are shown in Figure 3. The ratio of unloaded-to-loaded Q is shown in Figure 3. The simulated and estimated by (5) at 32.13 MHz are shown in Figure 4.

Discussion

At 32.13 MHz, the measured SNR of the four-element network is higher than the SNR of the three-element network by 6.60 dB, which implies noise figure difference of 6.60 dB as both setups use the same preamplifier. Simulation predicts 7.33 dB SNR difference and 7.17 dB noise figure difference. Both are close to the measured 6.60 dB. As shown in Figure 4, $$$F$$$ estimated by (5) stands very close to simulated $$$F$$$, with errors below 0.29 dB, or 7.0%. It is also clear that the term $$$R_\mathrm{o}/\left(R_\mathrm{o}-R_{22}\right)$$$ in (5) is negligible. Thus in this case (5) can be simplified to (1). Equations (1) and (5) explain the difference between SNR (and $$$F$$$) using the four-element network in Figure 2(d) and the three-element network in Figure 2(c). The inductor in Figure 2(c) contributes significantly to the term $$$R_{11}$$$, whereas the inductors in Figure 2(d) contribute only to $$$R_{22}$$$, and therefore affects $$$F$$$ less. Inductors are typically much lossier than capacitors, so the their places in networks should be carefully chosen to avoid significant contribution to $$$R_{11}$$$.

Conclusion

A formula to predict the noise figure of low-loss matching networks is derived for $$$X_\mathrm{a}+X_\mathrm{o}=0$$$. It helps select amongst network topologies for the highest SNR. The formula predicts noise figure with reasonable accuracy, which is confirmed by simulation and experiments.

Acknowledgements

This work is supported in part by the Danish National Research Foundation under grant DNRF 124.