Introduction

Transmit arrays (TxArrays), using multiple parallel RF transmitting elements which provide the additional degrees of freedom, are extensively examined to enhance field uniformity^1,2, enable RF shimming while intending to mitigate local SAR^3,4, increase power efficiency^5,6, or execute implant-friendly modes⁷. However, TxArray’s performance significantly profits from parallel transmit technology when the coupling between individual array elements is low. Therefore, designing TxArrays, due to the coupling issue, is much more complicated than designing conventional coils.

A variety of methods for designing TxArrays in simulation environments has been proposed, such as the co-simulation strategy⁸, which demonstrated excellent performance. This process replaced all lumped-components with equivalent lumped ports, and calculated TxArrays multi-port scattering matrices, to be used for determining the optimal lumped components. However, for those TxArrays that are not machine-manufactured, measurements with a high probability are different from the simulation results; therefore, special attention needs to be paid to include the exact structure of fabricated TxArray in the design process. Co-simulation may also be used in this case, although it requires numerous measurements, especially for TxArrays with large number of transmitting and lumped elements, to obtain the multi-port $$${\bf{S}}$$$-matrices, which are not feasible.

In this work, we aim to assess the feasibility of using the equivalent circuit model of a manufactured TxArray to reduce the measurement and simulation differences by evaluating the performance of simulated and fabricated 8-channel degenerate birdcage TxArray as an efficient coil for imaging at 123.2MHz. We utilized the equivalent circuit model by considering all magnetic and electrical couplings^9-11 to introduce a practical method for the determination of fabricated TxArray’s inductance and resistance values.

Methods

We simulated an 8-channel TxArray coil (Figure1a) using the co-simulation strategy8. By adjusting the capacitors placed between the nearest neighbors, TxArray’s transmit channels could be decoupled. In the design process, the capacitors were adjusted to minimize the following cost function^11,12:

$$\begin{array}{*{20}{c}}{\begin{array}{*{20}{c}}{\min}&{\frac{1}{8}\sum\limits_{i=1}^8{\sum\limits_{j=1}^8{|{s_{ij}}{|^2}}}+{{\left.{\frac{{P_r^T}}{{P_i^T}}}\right|}_{{\rm{in}}\,{\rm{CP}}\,{\rm{mode}}}}}\end{array}}&{(1)}\end{array}$$

Where $$$P_i^T$$$ and $$$P_r^T$$$indicate the total incident and reflected power from the coil. Furthermore,$$$\sum\limits_{j=1}^8{|{s_{ij}}{|^2}}$$$ can be interpreted as the total reflected to incident power ratio when only one single-channel (SC) is excited. Therefore, Eq.$$$1$$$ intends to minimize the return power for the circularly-polarized (CP) and SC excitation modes as two critical modes of operation. However, the structure of the TxArray supported by two cylindrical plexiglasses (Figure1b) with the capacitors achieved from the simulation was then constructed. As TxArray was not machine-manufactured and therefore not quite the same as the simulated TxArray, we expect that the $$${\bf{S}}$$$-matrices that were simulated or measured would not be the same.

Figure2 illustrates the manufactured TxArray’s equivalent circuit model, which used to provide an efficient and practical strategy in order to determine the suitable capacitors for the fabricated TxArray. The effects of all self/mutual-inductances and self/mutual-resistances¹³ emanating from the coil, the shield, and the load were taken into consideration.

Based on the model, TxArray impedance matrix, which can simply be derived as a function of the free parameters (capacitors) and frequency ($$$\omega$$$), by analyzing the model with Kirchhoff mesh current method, can be described as follows:

$$\begin{array}{*{20}{c}}{{\bf{Z}}(\omega)=\frac{1}{{j\omega }}({{\bf{C}}_{\bf{m}}}+2{{\bf{C}}_{\bf{s}}})+\frac{1}{{{\omega^2}}}{{\bf{C}}_{\bf{m}}}{{\left({j\omega {\bf{L}}+\frac{1}{{j\omega}}{\bf{C}}+{\bf{R}}}\right)}^{-1}}{{\bf{C}}_{\bf{m}}}}&{(2)}\end{array}$$

where $$${\bf{L}}={\left[{{{\rm{L}}_{{\rm{qp}}}}}\right]_{8\times8}}$$$($$${\bf{L}}={\left[{{{\rm{R}}_{{\rm{qp}}}}}\right]_{8\times8}}$$$) represents the inductance (resistance) matrix in which $$${{\rm{L}}_{{\rm{qp}}}}$$$($$${{\rm{R}}_{{\rm{qp}}}}$$$) indicates the mutual-inductance (-resistance) between the $$$q$$$th and $$$p$$$th loops.$$${{\rm{L}}_{{\rm{qq}}}}$$$($$${{\rm{R}}_{{\rm{qq}}}}$$$) refers to the self-inductance (-resistance) of the $$$q$$$th loop.$$${\bf{C}}{\rm{=}}{\left[{{{\rm{C}}_{{\rm{qp}}}}}\right]_{8\times8}}$$$ and $$${{\bf{C}}_{\bf{x}}}{\rm{=}}{\left[{{\rm{C}}_{{\rm{qp}}}^{\rm{x}}}\right]_{8\times8}}$$$ also represent the capacitance matrices where $$${{\rm{C}}_{{\rm{qp}}}}$$$ and $$${\rm{C}}_{{\rm{qp}}}^{\rm{x}}$$$ are defined as:

$$\begin{array}{*{20}{c}}{\begin{array}{*{20}{c}}{{{\rm{C}}_{{\rm{qp}}}}=\left\{{\begin{array}{*{20}{c}}{\frac{1}{{c_d^{q-1}}}+\frac{1}{{c_d^q}}+\frac{1}{{c_t^q}}+\frac{1}{{c_m^q}}}&{q=p}\\{-\frac{1}{{c_d^{\min\{q,\,p\}}}}}&{q-p\equiv\pm1\,\,({\rm{mode8}})}\\0&{{\rm{other}}}\end{array}}\right.}&{{\rm{and}}}&{{\rm{C}}_{{\rm{qp}}}^{\rm{x}}=\left\{{\begin{array}{*{20}{c}}{\frac{1}{{c_x^q}}}&{q=p}\\0&{{\rm{other}}}\end{array}}\right.}\end{array}}&{(3)}\end{array}$$

The capacitors provided by the simulation can be considered as initial capacitors. By measuring the $$${\bf{Z}}$$$-matrix of the fabricated TxArray for these initial capacitors, the inductance and resistance matrices can be calculated as:

$$\begin{array}{*{20}{c}}{{\bf{R}}(\omega)={\rm{real}}\{\frac{1}{{{\omega^2}}}{{\bf{C}}_{\bf{m}}}{{\left({{\bf{Z}}(\omega)-\frac{1}{{j\omega}}({{\bf{C}}_{\bf{m}}}+2{{\bf{C}}_{\bf{s}}})}\right)}^{-1}}{{\bf{C}}_{\bf{m}}}\}}&{(4)}\\{{\bf{L}}(\omega)=\frac{1}{\omega}{\rm{imag}}\{\frac{1}{{{\omega^2}}}{{\bf{C}}_{\bf{m}}}{{\left({{\bf{Z}}(\omega)-\frac{1}{{j\omega}}({{\bf{C}}_{\bf{m}}}+2{{\bf{C}}_{\bf{s}}})}\right)}^{-1}}{{\bf{C}}_{\bf{m}}}\}+\frac{1}{{{\omega^2}}}{\bf{C}}}&{(5)}\end{array}$$

Once the inductance and resistance matrices are determined, the new capacitors can be accomplished to minimize the cost function implemented in Eq.$$$1$$$ for the fabricated TxArray. The new $$${\bf{Z}}$$$-matrix which fulfills the design criteria can be then measured by updating the capacitors.

Results

The performance of the simulated and fabricated TxArray both with nearly the same capacitor values is shown in Figure3. The simulated and measured reflection coefficients, the reflected to incident power ratio in the SC/CP modes, and $$${\bf{S}}$$$-matrices are described in this figure. Initially, the structure of TxArray was assumed to be circular-symmetric, resulting in identical mounted capacitors at similar locations in each channel. As expected, the measurements differ with the simulation results, therefore the fabricated TxArray is not completely circular-symmetric.

By applying the proposed approach to the fabricated TxArray, new quantities of the capacitors were determined. Figure4 compares the performance of the simulated and fabricated TxArrays with the initial and new capacitors, respectively. In this figure, the simulated and measured reflection coefficients, the reflected to incident power ratio in the SC/CP modes, and $$${\bf{S}}$$$-matrices are represented. Figure4 demonstrates that the measurements were approached to the simulation results by mounting the new capacitors.

In order to clarify the proposed method, Figure5 displays a flowchart demonstrating how an optimum capacitor values can be calculated for the manufactured TxArray

Conclusion

Considering the exact manufactured structure of TxArrays in the design process may reduce the error between the measurements and simulations. We developed a practical approach to designing TxArrays precisely by integrating the equivalent circuit model of the manufactured TxArray and the simulation, and then validated it by constructing an 8-channel TxArray and measuring its performance