Task type: Plane-parallel problem of DC magnetics. Geometry Biot-Savart law Two long wires located at a distance of mm from each other in the air carry opposing currents of 10 A. Task Compute magnetic flux density at the point K. Solution The problem can be solved analytically. This is the Biot Savart law statement. Let us examine a long current carrying I wire and also an end P in the space. The current carrying wire is shown in the picture with a particular color.
When the current throughout the tiny length of wire is similar to the current carried by the total wire itself that can be written as.
So this can be written as,. At present, substituting the constant K value in the above expression, we can get the following expression. Therefore, magnetic flux density at P end due to the whole length of the wire can be written as,. As in a conventional receive coil, detuning is achieved when inductor L forms a parallel LC blocking circuit with capacitor C3 when the PIN diode D is turned on.
Capacitor C4 transforms the coil impedance to 50 V. RF, radio frequency. As a benchmark, the DB0 maps are also shimmed using heating. DC shim current is brought to each loop via SH basis sets ranging from the third to sixth order. The twisted pair AWG18 copper wires. Inductive chokes are used to wise and whole-volume basis for each array. Due to space constraints, the two eye loops are joined the brain region on both a global whole-brain 50 slices as a single shim coil, resulting in only 31 independent and slice-optimized basis 1-cm slabs centered on slice shim channels.
The initial and completed coil arrays of interest. Constrained optimization is performed using along with DC feed wires are shown in Figure 2. The spherical harmonic field ampli- flat tray containing the 25 cm of DC twisted pair nearest tudes are unconstrained. For an array with prevent the wires from picking up transmitted RF caus- Nc loop elements, the three-dimensional 3D coil DBo ing heating and reducing the body coil efficiency.
The eigenvalues for R are calculated and sorted. For the twisted- The cumulative sum of eigenvalues is plotted against the pair DC lines on the helmet, care must taken to route the number of coil principal components. A self-shielding toroidal choke geometry Fig.
The channel brain array is built on MHz. Coil loops with a diameter of 9. The voltage across a nal pattern, with critical overlap to decouple neighboring 0. Other details of the RF coil are given in Top and bottom halves of coil before and after conversion to a combined RF-shim array.
Care is taken to route DC lines at least 2 cm from the RF preamps to prevent feedback oscillations. For the final 25 cm before reaching the loops, chokes and LC trap circuits are used on the twisted pair to suppress RF pickup during transmission.
Digitally programmable shim supply boards right-most column provide DC current up to The output op amps are mounted to heat sinks with in-laid piping for optional water cool- ing. DC, direct current; RF, radio frequency. The 7. The shim supply assembly with all 31 boards changes in the current output. The DAC is controlled is placed in the scanner room 2. To assess the signifi- Technology, Westborough, MA that is in turn controlled cance of induced voltage, a test 9.
Schematic for in-house, low-cost, digitally programmable shim supply circuit boards. The boards use paired OPA high- current op amps controlled by current feedback from a 0.
The summing point op amp feedback loop can be operated in two modes: 1 a resistor for zero bandwidth and unconditional stability, and 2 a capacitor that sets an appro- priate bandwidth to permit gradient-induced voltage compensation without allowing oscillations.
Simulated shimming performance is compared for six shim array geometries and spherical harmonics up to sixth order using a reference DB0 brain field map acquired at 3T 50 slices, 2 mm thickness. Three representative slices are shown for both global slice and slice-optimized shimming. For RF-shim helmet arrays, performance improves with array size, with a channel approaching the performance of a fourth-order spherical harmonic basis set. With channels, the RF-shim array rivals fifth- to sixth-order spherical harmonics, providing enough degrees of freedom to mitigate all but the most severe deep sinus B0 inhomogeneity.
Experiments the impact of using different numbers of chokes, the test SNR is assessed in two ways. First, the unloaded-to- is repeated on loops with varying numbers of distributed loaded quality factor ratio Q-ratio is obtained for indi- tuning capacitors.
For comparison, loops tuned for 7T vidual 3T test loops by using a lightly coupled double imaging MHz are also tested. For comparison, an SNR map is also conductivity of 1. The Q- acquired with a geometrically similar channel com- ratio is measured for conventional RF-only loops and for mercial head coil Siemens Healthcare, Erlangen, Ger- RF-shim loops, with chokes added to bridge RF-tuning many.
Dimensions of the two coils are shown in capacitors and with a DC-blocking capacitor added Supporting Figure S2. A proton density weighted gradi- Supp. Noise covariance information is acquired using the same sequence but with the RF transmission disabled. Calibration DB0 field maps in Hz are obtained with mA of current flowing through one coil at a time. Maps are acquired on a cm diameter water-filled bal- loon phantom with a double-echo gradient echo sequence transverse, TE: [5, 7.
Field mapping scan duration is 2 minutes per shim channel—or slightly over 1 hour for the entire array. The background inhomogeneity in the balloon phantom zero shim current is also mapped and subtracted from the coil B0 maps. After the full set of B0 FIG. The size jected to the safety tests described by Keil et al. Eight of the spherical harmonic basis up to the sixth order is indicated healthy volunteers are recruited to test the ability of the for reference.
The channel RF-shim array is comparable to a shim array to compensate B0 inhomogeneity in the brain. Scans are performed in the head- first supine position using the scanner body coil for RF transmit and the combined RF-shim array coil for RF order. The residual field variation is shown in three rep- receive. As expected, the residual off- ping shimming procedure.
Then, the dual-echo gradient resonance diminishes as the number of shim channels is echo sequence is acquired. A 3D brain mask is created by increased due to the greater number of degrees of free- processing magnitude images using the FMRIB Software dom provided by larger arrays.
The FSL array performance is comparable to third-order global Prelude tool 29 is used to perform 3D phase unwrapping shimming and fourth-order slice-optimized shimming. Marked improvements in geneity. Fourier undersampling. Scans are run in both blip-up The PCA analysis of coil B0 field maps, plotted in Fig- posterior—anterior and blip-down anterior—posterior ure 5, illustrates how many independent degrees of free- modes to compare the overlap between the two distorted dom of substantial efficiency are present in each images with and without the MC shims applied.
Baseline simulated coil array. The number of eigenmodes in the field mapping and EPI scans are performed with the SH basis up to sixth order are shown for comparison. The slice-optimized matrix shim is then calculated RF receive arrays can achieve undersampling factors up to optimally compensate off-resonance in a 1-cm slab of to rate 3 over all three dimensions combined ; the PCA interest using at most After MC shim- on B0 shows that an 8-channel shim array has approxi- ming, the residual field variation is measured using a mately 3 independent degrees of freedom for canceling second B0 field map.
Finally, the EPI protocol is rerun to B0 inhomogeneity. Additionally, when the shim current of 2. SNR and interelement coil coupling are compared for the channel RF-shim array before and after conversion, as well as for a size-matched channel commercial receive coil see Supporting Figure S2.
Sagittal SNR maps show a modest loss after the channel RF array is converted into a combined RF-shim array, with the greatest impact at the periphery. However, the array still provides equivalent SNR to the commercial coil. The coil correlation matrix shows that interelement coupling is not degraded by the added shim- ming hardware.
RF, radio frequency; SNR, signal-to-noise ratio. The chokes and DC currents by gradient-induced voltages in the loops. When the entire pling, and preamplifier decoupling. The addition of shim array is tested with 1 A flowing in all channels for chokes causes a modest shift in the resonant frequency of the coil that decreases as the number of distributed tuning capacitors grows Supp. At 7T, the addition of 0. Due to the finite impedance of the chokes, fluctuations in the Q-ratio and coil tuning were initially observed when the twisted pair was moved or handled.
These interactions were largely eliminated by adding 1, pF bypass capacitors connecting the node where the twisted pair meets the chokes to the virtual ground between the split drive-point capacitors, C2 and C3 Supp. DB0 field maps measured in balloon phantom a generated by mA current in eight representative coils encircling the mid- forms at the level of the geometrically similar channel plane of the brain.
The independent spatial profiles of the coils are commercial head coil Fig. The Z2 shim degrades toward the bottom peak. In EPI images acquired using the RF-shim array, of the helmet, where there are fewer coils to support the field. Five brain slices in two volunteers are shimmed by optimizing shim currents for a 1-cm slab around the slice of interest. Close agreement is obtained between predicted and acquired shimmed DB0 maps.
High-resolution EPI scans show markedly less distortion when MC shims are applied, largely bringing features such as the ventricles back into alignment, as indi- cated by the orange lines. Residual inhomogeneity in the anterior-most edge of the brain distorts the edge of the image, highlighting the need to combine MC shimming with parallel imaging approaches e.
The total current used to shim each slice, starting from the top row, is [5. But we confirm that shim coils locations, indicating the need for water cooling if the shim arrayed on the surface of a helmet substrate can be used array is to be operated at full capacity for the duration of to model and thus substitute for low-order SH fields, as typical clinical MRI scan.
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