## 2 Materials and Methods
### Gapped Microstrips  
The use of gapped microstrips has previously been proposed [40; 41; 42], including demonstrations of the analytical method and use in several applications. In a study by [PERSON] [43], a theoretical expression for the equivalent circuit's parameters was proposed by solving a Green function. The circuit equivalence of a gapped microstrip can be expressed using a pi circuit, where the \(C_{a}\) is the shunt capacitor and the \(C_{b}\) is the series capacitance produced by the coupling effect between nearby strip lines. The equivalent circuit is given by:  
\[C_{e}=C_{a}+2C_{b};\;C_{m}=C_{a} \tag{1}\]  
where the subscript \(e\) and \(m\) indicate the precedence of an electric and magnetic wall in the middle of the gap, respectively. The solution for the capacitance in the strip line was found by solving the Green's equations. A more common approach to solve the capacitance has also been proposed [44; 45; 46], which also describes the problem as two conductor lines with open ends and a series coupled capacitance between them, as shown in Figure 1a. The circuit representation is also a Pi network (Figure 1b), \(C_{s}\) is the gap capacitance, and \(C_{p1}\) and \(C_{p2}\) are the shunt capacitors. The solution to the capacitors is given by:  
\[C_{s}=500\;h\exp\Bigl{(}-1.86\frac{s}{h}\Bigr{)}Q1\Biggl{(}1+4.19\Biggl{(}1- \exp\Biggl{(}-0.785\sqrt{\frac{h}{W1}}\frac{W2}{W1}\Biggr{)}\Biggr{)}\Biggr{)} \tag{2}\]  
\[C_{P1}=C_{1}\frac{Q2+Q3}{Q2+1} \tag{3}\]  
\[C_{p2}=C_{2}\frac{Q2+Q4}{Q2+1} \tag{4}\]  
where \(W1\) and \(W2\) are the width of the conductor lines, \(h\) is the height of the dielectric material, and \(s\) is the size of the gap between the lines. The coefficients \(Q1\) to \(Q5\) are given in Reference [47], which are in relationship the width \(W1\) and \(W2\) of each gap, the height and the effective relative permittivity of the dielectric material, the length of the microstrip and the open capacitance of the microstrip. Figure 1: The model for the analysis (**a**) of gapped microstrips, (**b**) the circuit representation of the gapped microstrip, (**c**) the model of a gapped microstrip used for analysis of the effect of size and distance of the gap; the close-up image shows the variables s and d.  
The previous analysis and theory showed that gapped microstrips are viable, and could be used for our final design. Further, the series capacitance could be used in combination with the loading capacitance to ensure the microstrip lines operated at a specific frequency. A gapped microstrip line can be used in MRI applications if the line resonates at the respective frequency and has a uniform field along the line (y-z plane). The size and position of the gap can determine the performance of the microstrip by fulfilling the previously stated conditions. A large gap prevents the flow of current along the line and can affect the uniformity of the produced magnetic field. With this in mind, we needed to find the optimal position and size of the gap. To this end, we performed numerical analysis using EM simulations to determine the effects of the gap. The simulations showing the effects of gaps in the microstrip were done by modeling a microstrip, as shown in Figure 1c. The length of the strip line was 225 mm and the width was 5 mm. The strip line was placed above a dielectric material of 5 mm height, resulting in a total width and length of 40 \(\times\) 225 mm, respectively, for the ground plane. The dielectric material was assigned as Rogers RT/duroid 6006 with a relative permittivity of 6.45, and a capacitor of 1 pF was set as the loading condition to make the microstrip resonate at 300 MHz. The microstrip was excited with a Gaussian pulse with a central frequency of 300 Mhz and bandwidth of 350 MHz. The grid size was 48 \(\times\) 48 \(\times\) 82 for a total 188.928 k cells; this resolution was enough to ensure the model and the computational model were accurate. The first analysis involved changing the size of the gap from 0.5 to 5 mm, in steps of 0.5 mm each.