# Background and Definitions

The definitions are presented for ${B}_{1}^{+}$ , ${B}_{1}^{-}$ and the intrinsic signal-to-noise-ratio.

${B}_{1}^{+}$ is a component of the magnetic radio frequency (RF) field. This component rotates in a plane perpendicular to the static magnetic field in an MRI machine. If the Z axis is defined as the direction of the static magnetic field, as is customary, then ${B}_{1}^{+}$ is defined as

(1) ${B}_{1}^{+}=\frac{\left({B}_{x}+j{B}_{y}\right)}{2}$ , where $j=\sqrt{-1}$ .

The rotating RF field is essential in producing an MRI image. MRI coil designers need to be able to plot this quantity. In addition, they need to be able to compare it with the anti-rotating component ${B}_{1}^{-}$ , defined as

(2) ${B}_{1}^{-}=\frac{\left({B}_{x}^{*}+j{B}_{y}^{*}\right)}{2}$ .

In an efficient coil, ${B}_{1}^{+}$ significantly exceeds ${B}_{1}^{-}$ . This can be visualised in the third quantity of interest, which is defined as

(3) $\mathrm{ratio}=\frac{|{B}_{1}^{+}|}{|{B}_{1}^{-}|}$ .

The application macro also calculates the absolute intrinsic signal-to-noise ratio, the SNR excluding MRI system losses and conductive losses in the detector, using the formula1

(4) $\mathrm{ISNR}=\frac{\left(\omega \Delta V{M}_{0}|{B}_{t}^{-}|\right)}{\sqrt{4kT\Delta f{R}_{L}}}$ .

$\omega =2\pi f$ is the Larmor frequency, $\Delta V$ is the voxel volume, ${M}_{0}$ is the magnetisation, ${B}_{t}^{-}$ is the magnitude of the left-hand circularly polarised component of the transverse RF magnetic field produced by the coil with a one ampere current in the loop, $k$ is Boltzmann's constant (1.38E-23 J/K), $T$ is the absolute temperature, $\Delta f$ is the bandwidth of the receiving system and ${R}_{L}$ is the noise resistance of the detector.

1 A. Kumar and P.A. Bottomley, Optimized Quadrature Surface Coil Designs, Magn. Reson. Mater. Phy., vol. 21, no.1-2, pp. 41-52, March 2008.