Koray Ertan^{1,2}, Soheil Taraghinia^{1,2}, and Ergin Atalar^{1,2}

Gradient array systems recently have gained attention due to their various flexibilities and capabilities in different applications. Reducing the mutual-coupling between the coil elements is one of the constraints during the process of the coil design. However, by determining any existing coupling value between the array elements, required decoupling can be achieved. For a typical trapezoidal gradient current waveform, desired voltage values during rise/fall times, are recalculated considering all mutual-couplings between the array elements. This method is evaluated experimentally for different trapezoidal current combinations and can be used in any gradient array system with mutually coupled elements.

For a typical trapezoidal gradient current waveform with determined flat top/bottom values (), required voltage to get the desired rise/fall time (m’th channel), can be achieved using Eq. 1. In this equation Rm and Mm are resistance and self-/mutual-coupling values respectively for N number of channels.

$$V_{p_m} = R_m\cdot I_m +\frac{1}{Δt_m}\sum_{n=1}^{N}M_{mn}\cdot I_n m=1,...,N; Eq.1$$

For effective compensation, we consider the same rise/fall times
() for all channels regardless of their current values, which can
be found by using Eq. 2, in which, is the available voltage
from GPA (V_{PS} = 40V, I_{PS} = 15A).

$$\underset{1 \le m \le N}{\operatorname{max}} \{ \frac{\sum_{n=1}^N M_{mn}\cdot I_n }{V_{PS}-\mid R_m \cdot I_n\mid} \} Eq.2$$

After specifying Δt_{array}, required voltages for each channel
considering mutual-couplings can be achieved for all channels in rising and
falling edges of the trapezoidal current waveform where there is coupling
between the coils. Measured resistance is about 1.4 Ω for all channels and mutual-inductance
matrix is given in Fig. 1.

Table 1 contains normalized weightings of the currents in each channel generating two linear gradient fields in on-center and off-center (5 cm shifted) z locations. Both fields has an elliptical homogeneity volume of 20 × 15 cm in radius and length respectively. Magnetic field profiles of the on-center and off-center volumes in simulation are demonstrated in Fig.2 by applying current values in Table 1.

1.Smith, E., Freschi, F., Repetto, M., & Crozier, S. (2017). The coil array method for creating a dynamic imaging volume. Magnetic resonance in medicine, 78(2), 784-793.

2.Littin, S., Jia, F., Layton, K. J., Kroboth, S., Yu, H., Hennig, J., & Zaitsev, M. (2017). Development and implementation of an 84‐channel matrix gradient coil. Magnetic Resonance in Medicine.

3. K. Ertan, S.Taraghinia, A. Sadeghi, E. Atalar “A z-gradient array for spatially oscillating magnetic fields in multi-slice excitation” Magn Reson Mater Phy (2016) 29: 1. doi:10.1007/s10334-016-0568-x, Abstract No:81

4.Taraghinia, S., Ertan, K., Yardim, A.B. and Atalar, E., Efficient Ripple Current Reduction in Gradient Array System Using Optimized Phase Control Signals with One Stage LC Filter, ESMRMB, Barcelona, Spain, 2017

Figure 1:
(a) Home-built
H-bridge GPAs with 40V and 15A VI capabilities which are controlled using an
FPGA (Xilinx/Virtex-5) with 50 us PWM period. (b) Measured
mutual-coupling matrix of 9 channel array prototype.

Table 1.
Current ratios between 9 channels generating linear gradient fields in large
(a=20 cm, b=15 cm) and elliptical VOI and large volume shifted 5 cm in z
direction.

Figure 2: Optimized Field profiles for (a) on-center and (b) off-center
cases. (Solid line shows the design region for the homogenous gradient)

Figure 3.
Current waveforms of 4 channels measured experimentally before and after
compensation for (a) Large, (b) Small and (c) off-center VOIs with maximum
current of 15 A. In some waveforms scaling is adjusted for better
visualization. Ripple currents on the flat top/bottoms can be reduced digitally
using optimum-phase control signals^{4.}

Figure 4: MR images in (a)
coronal and (b) transverse planes
demonstrating the slice selection using on-center
linear gradient field with mutually coupled gradient amplifiers