## 2 Circuit Design
### Class-J Output Matching Network  
The continuous Class-J mode specifies the optimum load impedances at the fundamental frequency and the second harmonic frequency, as follows [26; 27].  
\[Z_{L}(f_{0})=(1+j\gamma)R_{opt} \tag{1}\]  
\[Z_{L}(2f_{0})=-\mathrm{j}(3\pi/8)\gamma R_{opt} \tag{2}\]  
For the third- and higher-order harmonics, the optimal load impedances are considered to be short circuits in the continuous Class-J mode. The term \(R_{opt}\) represents the optimal load resistance of the transistor. The parameter \(\gamma\) is derived from the voltage and current waveform and is utilized to determine the load impedance (\(Z_{L}\)). If the value of \(\gamma\) falls within the range of \(-1\) to \(1\), it indicates that the continuous Class-J mode can be sustained. Consequently, it becomes possible to attain output power, efficiency, and bandwidth.  
Figure 7a illustrates the OMN design, in which \(R_{L}\) represents the load impedance of \(50\)\(\Omega\). The capacitor \(C_{out}\) represents the transistor's drain-source parasitic capacitance. The resonant circuit \(L_{2}//C_{2}\) is designed to function as an open circuit at the second harmonic frequency. Additionally, the capacitor \(C_{4}\) serves as a DC block in the circuit.  
Figure 7b displays the OMN equivalent circuit at the fundamental frequency. In order to compensate for the presence of wire bond (WB) between GaN and GaAs IPD chips, the capacitor \(C_{1}\) is designed so that \(C_{out}\), WB, and \(C_{2}\) form a combination to achieve an equivalent capacitive reactance, denoted as \(C_{eq}\). Subsequently, the inductor \(L_{1}\) is designed to resonate with the \(C_{eq}\) at the fundamental frequency. Therefore, the OMN functions as an L-type impedance matching network, utilizing the combination of \(L_{eq}\) and \(C_{3}\) to convert the load impedance \(R_{L}\) to \(R_{opt}\).  
Figure 6: (**a**) The bias and stability network; (**b**) [PERSON]’s stability factor (K), \(\Delta\) and stability measure (B1); (**c**) large-signal K.  
Figure 7c depicts the equivalent circuit of the OMN at the second harmonic frequency. The resonant circuit \(L_{2}//C_{2}\) functions as an open circuit at the second harmonic frequency. The reactance of \(L_{1}\) increases as the frequency rises. Thus, the L\({}_{1}\)//C\({}_{\text{eq}}\) is designed to be equivalent to a capacitor, denoted as \(C_{\omega 2}\). As a result, the load impedance \(Z_{L}(2f_{0})\) could achieve purely imaginary impedance as described in the Equation (2). The capacitor \(C_{3}\) plays a crucial role in reducing the load \(R_{L}\) and eliminating the resistances. This enables the resonant circuit \(L_{2}//C_{2}\) to behave more like an open circuit. Consequently, it becomes easier to achieve a small and purely imaginary impedance, which is advantageous compared to the approach described in [28].  
The optimum load of 40 \(\Omega\) is obtained through load-pull simulation along with the equivalent output capacitance (\(C_{out}\)) of 974 fF. Figure 8a illustrates the frequency response of the designed OMN in a rectangular plot, while Figure 8b represents the same response in a Smith chart. At the fundamental frequency, the real part of output impedance is approximately 42 \(\Omega\) and the parameter \(\gamma\) ranges from \(-\)0.31 to 0.29. At the second harmonic frequency, the real part of output impedance approaches zero, resulting in a purely imaginary impedance. The simulated result validates that the power amplifier operates in the continuous Class-J mode.  
Figure 8: Impedance at the intrinsic plane of the output stage transistor: (**a**) rectangle plot; (**b**) Smith chart.  
Figure 7: (**a**) The designed OMN of the proposed quasi-MMIC Pas; (**b**) equivalent circuit at the fundamental frequency; (**c**) equivalent circuit at the second harmonic frequency.