## 3 Design of a Resonator and Experimental Results  
The proposed resonator is composed of a dual split-ring structure and an open-loaded T-type stub, as shown in Figure 2. From the figure, \(Z_{ex}\) and \(Z_{in}\) are the characteristic impedances of the external and internal split-ring structures, while \(Z_{T}\) and \(Z_{s}\) are those of the series transmission line and open-loaded stub in the T-type structure. \(Z_{p1}\) and \(Z_{p2}\) are the characteristic impedances of the gaps represented by gap sizes \(g_{ex}\) and \(g_{in}\), while \(Z_{qp}\) is that of an open-loaded T-type stub. In addition, \(l_{TH}\), \(l_{TV}\), \(l_{ex1}\), and \(l_{ex2}\) are the physical lengths of the external split-ring structure, and \(l_{in,m,n}\) (\(m\) = 1 to 3, \(n\) = 4 to 5) refers to the set of physical lengths of the internal split-ring structure. \(l_{T}\) and \(l_{s}\) are the physical lengths of the series transmission line and open-loaded stub in the T-type structure, respectively. The symbols \(w_{ex}\), \(w_{in}\), \(w_{T}\), and \(w_{s}\) denote the widths of the external and internal split-ring structures, the series transmission line, and the open-loaded stub in the T-type structure, respectively, while \(g_{ex}\) and \(g_{in}\) indicate the coupling gaps between the split-ring structures and the open-loaded T-type stub, respectively. \(\theta_{TH}\), \(\theta_{TV}\), \(\theta_{ex1}\), and \(\theta_{ex2}\) are the electrical lengths of the external split-ring resonator and \(\theta_{in,m,n}\) (\(m\) = 1-3, \(n\) = 4-5) is that of the internal split-ring structure, while \(\theta_{T}\) and \(\theta_{s}\) are those of the series transmission line and open-loaded stub in the T-type structure. In the equivalent circuit depicted in Figure 2, \(L_{ex}\) and \(L_{in,m,n}\) (\(m\): 1-3, \(n\): 4-5) are the inductances of the external and internal split-ring structures. Then, \(1/L_{in}\) can be expressed as \(1/L_{in}+1/L_{in}=L_{in}\) (for in for in, \(m\): 1 to 3, \(n\): 4 + 5). \(L_{T}\) and \(L_{s}\) are the inductances of the series transmission line and open-loaded stub, respectively, while \(C_{ex}\), \(C_{in}\), \(C_{T}\), and \(C_{S}\) are the capacitances of the external and internal split-ring structures, the series transmission line, and the open-loaded stub, respectively. \(C_{op}\) is the mutual capacitance between the transmission line and the ground plane, with \(C_{p1}\) and \(C_{p2}\) being those of the coupling gap between the split-ring structure and the open-loaded T-type stub, respectively. Lastly, port 1 and port 2 are the input and output ports, and the characteristic impedance of the ports is 50 \(\Omega\), which is called the feeding-line in the resonator structure of Figure 2. Then, the gm is the gap size between the resonator and feeding line. The physical structure of \(C_{p2}\) (see equivalent circuit) is carried out the same as with \(g_{in}\). Figure 1: Oscillator and sensor for tumor tracking. To design the resonator, the capacitance is given by Equations (1)-(6), where \(t\) is the thickness (0.018 \(\upmu\)m) of the transmission line at the substrate and \(\varepsilon_{0}\) and \(\varepsilon_{r}\) are, respectively, the free-space and relative permittivity (\(\varepsilon_{0}=8.854\times 10^{-12}\) F/m) [22]. \(N\) is the number of turns (\(N=3\)) in the split-ring structure, while \(C_{m}\), the parasitic capacitance of the dielectric in the substrate, equaling 0.423 \(\upmu\)F [22]. Finally, \(A\) is the metal dimension of the substrate, while \(d\) is the distance between the metal and the ground in the substrate [22].