## 2 Design of Non-Foster Circuit  
Unlike conventional circuits, non-Foster circuits generate negative impedance. All-natural impedance follows the Foster's reactance theorem. According to Foster reactance theorem, passive elements with a single port increase reactance as frequency increases [20]. Particularly, in the case of a passive element having a large impedance variation according to a frequency or in the case of a small antenna, the impedance of the antenna is matched in a very narrow bandwidth when using a conventional passive element such as an LC. Unlike the impedance matching method using a passive element, such as the conventional LC matching, the non-Foster circuit can reverse the reactance and reactance slope of a passive element to be matched; therefore, impedance matching can be performed over a wider band than originally. Figure 1 shows Foster impedance matching, which is a narrow band impedance matching, and Figure 2 shows non-Foster impedance matching, which is a wide band impedance matching. The non-Foster circuits are largely classified into two types: The circuit using an op-amps and the circuit using transistors [21]. The two methods differ in the active elements used; however, they are all impedance converter. Each circuit has advantages and disadvantages in design complexity and bandwidth. In this paper, a non-Foster circuit using transistors is designed for matching in a wide band over 50 MHz. The non-Foster circuit is designed by cross-coupled-junction transistors. In the case of a cross-coupled circuit, the phase of the input current is output same, but the phase of the input voltage is output 180\({}^{\circ}\) apart from each other. Because of this characteristic, the input impedance is inverted and becomes negative value. The non-Foster circuit can have any negative impedance value by inverting the impedance value of the passive element by connecting the passive element of the desired value to the circuit [22]. Figure 3 shows an equivalent circuit of a designed non-Foster circuit to increase the bandwidth of a CRLH TL antenna. In this paper, we designed a non-Foster circuit that satisfies the open circuit stable condition to connect the designed CRLH TL antenna in series. The gate and drain of the two transistors were cross-coupled. To optimize the non-Foster circuit element values, we used the Agilent ADS2009 circuit simulator. We used the Avago ATF-53189 transistor with an operating frequency of 50 MHz to 6.5 GHz to consider the transistor's internal parasitic effects and obtain broadband characteristic of the non-Foster circuit [23]. Table 1 shows the parameters of the designed non-Foster circuit. Figure 1: Foster impedance matching. Figure 3: Schematic of non-Foster circuit. Figure 2: Non-Foster impedance matching. It is well known that a non-Foster circuit may exhibit instability due to the positive feedback formed in the transistor-ground loop. Therefore, it is important to verify stability in the non-Foster circuit design [24]. In this paper, we verified stability with two methods: by checking impedance of the non-Foster circuit and the transient response in the time domain. Figure 4 shows the impedance of the non-Foster circuit and load. In order to satisfy the open circuit stable condition, the impedance of the non-Foster circuit should be smaller than the impedance of the load [25] (i.e., \(Z_{\text{Load}}>Z_{\text{NFC}}\)). Both the simulated and measured results show that the proposed non-Foster circuit satisfies the open circuit stable condition throughout the observation frequency range. The differences between the simulation and measurement values originated from the difference between the ideal environment and the real one. The latter includes the effects from lossy and reactive parasitic components. Figure 5 shows the impedance measurement environment.