## 2 A Rapid Output Matching Optimization Method with Dual-Objective Orientation
### Determination of the Optimal Impedance Domain  
Subsequently, the optimal impedance domain (OID) is defined as the intersection region of the P1 dB contour of 31 dBm and the PAE contour of 40%, as highlighted by the gray shading in Figure 1a. With an increasing frequency, the P1 dB and PAE contours gradually shrink, and both rotate counterclockwise toward the real axis of the Smith chart, \begin{table}
\begin{tabular}{c c c c} \hline \(f_{\rm i}\) **(GHz)** & \(U_{1}(f_{\rm i})\) **(\Omega)** & \(U_{2}(f_{\rm i})\) **(\Omega)** & \(Z_{\rm opt,s}(f_{\rm i})\) **(\Omega)** \\ \hline
22 & 15.6 + j12.3 & 15.4 + j16.8 & 15.5 + j15.3 \\
24 & 14.1 + j13.1 & 13.9 + j16.4 & 14.0 + j15.3 \\
26 & 14.3 + j11.2 & 12.9 + j15.5 & 13.4 + j14.1 \\
28 & 12.8 + j10.4 & 11.5 + j14.2 & 11.9 + j12.9 \\
30 & 12.7 + j9.8 & 10.9 + j12.5 & 11.5 + j11.6 \\
32 & 12.2 + j8.6 & 9.3 + j11.2 & 10.3 + j10.3 \\ \hline \end{tabular}
\end{table} 
Table 1: The optimal load impedances obtained via load-pull simulations and calculations. Figure 1: (**a**) Load-pull simulation results at the 1 dB compression point and the determination of OID; (**b**) OPG contours of 6.85 dB within the \(BW_{\rm d}\) and the determination of P-OID. reflecting the trajectory of \(Z_{\text{opt}}(f_{\text{i}})\). This trend indicates that the transistor's performance deteriorates at higher frequencies. Approximately two-thirds of the OID boundary is delineated by the contours at 32 GHz. The operating power gain (OPG) is also a key metric that should not be overlooked, as achieving the target gain with fewer amplification stages can greatly improve the overall PAE. As depicted in Figure 1b, a value of 6.85 dB is chosen for the OPG because the corresponding narrowest contour at 32 GHz and the OID together create a new overlap area that still contains every \(Z_{\text{opt,s}}(f_{\text{i}})\), without significantly reducing the original OID, which is termed the preferred OID (P-OID). As long as all \(Z_{\text{omn}}(f_{\text{i}})\) across \(BW_{\text{d}}\) fall inside the P-OID, the transistor's large-signal output capability is assured, eliminating lengthy HB simulations. The OMN parameter optimization can then be focused on minimizing the offset between each \(Z_{\text{omn}}(f_{\text{i}})\) and \(Z_{\text{opt,s}}(f_{\text{i}})\). When the transistor enters the gain compression region, its transconductance \(g_{\text{m}}\) and \(C_{\text{ds}}\) exhibit strong nonlinearity. Therefore, based on the derived P-OID, two-tone load-pull simulations can be further performed to obtain the third-order intermodulation distortion (IMD3) contours, leading to a stricter definition of the OID and thus providing more comprehensive guidance for optimizing the parameters of the OMN. IMD3 is a major concern in linearity requirements because it lies near the operating frequencies and causes spectral regrowth. Assume that the two-tone excitation \(v_{\text{in}}\) (\(t\)) is expressed as  
\[v_{\text{in}}(t)=A\cos(\omega_{1}t)+B\cos(\omega_{2}t) \tag{4}\]  
where \(A\) and \(B\) represent the amplitudes of two fundamental components. Neglecting memory effects and higher-order nonlinearities, the drain current equations for IMD3 frequencies are  
\[I_{\text{d}}=I_{\text{d0}}+\sum_{k=1}^{n}g_{\text{mk}}V_{\text{gs}}^{k} \tag{5}\]  
\[i_{(2\omega_{1}-\omega_{2})}=\frac{3}{4}g_{\text{m3}}A^{2}B\cos[(2\omega_{1}- \omega_{2})t] \tag{6}\]  
\[i_{(2\omega_{2}-\omega_{1})}=\frac{3}{4}g_{\text{m3}}AB^{2}\cos[(2\omega_{2}- \omega_{1})t] \tag{7}\]  
where \(I_{\text{d}}\) and \(I_{\text{d0}}\) represent the total drain current and the DC drain current, respectively, while \(V_{\text{gs}}\) denotes the RF voltage applied across the gate-source terminals. The parameter \(k\) indicates the nonlinear order, and \(g_{\text{mk}}\) is the \(k\)-th order transconductance. For every frequency component present at the device ports, \(g_{\text{mk}}\) values are associated with the load impedance, as they are defined by the combinations of the drain and gate voltages with the current during an RF swing [26].