## 3 Broadband PA Implementation
### The Synthesized Transformer Output Matching Network  
A large-signal output model of the transistor can be approximated using a parallel \(R_{\text{opt}}C_{\text{out}}\) network, where \(C_{\text{out}}\) is the main contributor to the frequency-dependent nature of \(Z_{\text{opt}}(f_{\text{i}})\) and is the key limitation on the bandwidth. To mitigate this, a synthesized transformer matching technique is applied to absorbing \(C_{\text{out}}\) into the OMN [28; 29]. This method not only eliminates the need for extra compensation components but also leverages \(C_{\text{out}}\) as a beneficial element, resulting in a simpler MN, a lower insertion loss, and improved broadband matching. The synthesized transformer network (STN) originates from a magnetic coupling resonance model, illustrated in Figure 4a, where both LC resonators resonate at the same angular frequency \(\omega_{0}\). The transformer, with an ITR of \(T\) and mutual inductance \(M\), can be equivalently represented as the T-type inductor network described in Figure 4b, where \(M\) is determined by the coupling coefficient \(k\). By setting the terminal resistances of the STN to \(R_{\text{opt}}\) and \(R_{\text{L}}\) and assigning \(C_{2}\) to \(C_{\text{out}}\), the following relationships are obtained:  
\[T=\frac{R_{\text{L}}}{R_{\text{opt}}}=\frac{L_{1}}{L_{2}}=\frac{C_{\text{out} }}{C_{1}} \tag{14}\]  
\[M=k\sqrt{L_{1}L_{2}} \tag{15}\]  
\[\omega_{0}=1/\sqrt{L_{1}C_{1}}=1/\sqrt{L_{2}C_{\text{out}}} \tag{16}\]  
\[\left\{ \begin{array}{l}L_{1}-M>0\\ L_{2}-M>0\end{array} \right. \tag{17}\]  
Taking the red dashed plane outlined in Figure 4b as the reference boundary, the admittances seen from both sides of the plane are made conjugate, as expressed in Equation (18), where \(Y_{\text{L}}=1/R_{\text{L}}\). This condition enables the maximum power transfer. By substituting Equations (14)-(16) into Equation (18) and performing algebraic manipulation, the lower and upper resonance frequencies (\(f_{\text{L}}\) and \(f_{\text{H}}\)) of the STN can be derived, as given by Equations (19) and (20), where \(Q_{\text{d}}=1/(\omega_{0}\text{ }R_{\text{opt}}\text{ }C_{\text{out}})\). \[Y_{\text{A}}=\left(\frac{1}{R_{\text{opt}}}\right)=Y_{\text{A}}^{*}=Y_{\text{ B}}=j\omega C_{\text{out}}+\frac{1-\omega^{2}L_{1}C_{1}+j\omega L_{1}Y_{\text{L}}}{ j\omega L_{2}-\omega^{2}(j\omega C_{1}+Y_{\text{L}})(L_{1}L_{2}-M^{2})} \tag{18}\]  
\[f_{\text{L}}=\frac{\omega_{0}}{2\pi}\sqrt{\frac{-Q_{\text{d}}^{2}+Q_{\text{d} }^{2}k^{2}+2-\sqrt{\left[Q_{\text{d}}^{2}(1-k^{2})-2\right]^{2}-4(1-k^{2})}}{ 2(1-k^{2})}} \tag{19}\]  
\[f_{\text{H}}=\frac{\omega_{0}}{2\pi}\sqrt{\frac{-Q_{\text{d}}^{2}+Q_{\text{d }}^{2}k^{2}+2+\sqrt{\left[Q_{\text{d}}^{2}(1-k^{2})-2\right]^{2}-4(1-k^{2})}}{ 2(1-k^{2})}} \tag{20}\]Assigning/\(I_{\text{L}}\) and \(f_{\text{H}}\) to the lower and upper edge frequencies of the \(BW_{\text{d}}\) (\(f_{\text{min}}\) = 22 GHz, \(f_{\text{max}}\) = 32 GHz) and using the known design parameters (\(R_{\text{L}}\), \(R_{\text{opt}}\), and \(C_{\text{out}}\)), together with the relevant derived equations, the remaining component values in the STN can be solved. The results comply with the physical implementation condition described in Equation (17). Subsequently,\(f_{\text{L}}\) and \(f_{\text{H}}\) are simultaneously moved toward the center frequency \(f_{\text{c}}\) (27 GHz) in small steps of 0.05 GHz, with the STN parameters updated in tandem. The iteration proceeds until the S\({}_{11}\) of the STN at \(f_{\text{min}}\),\(f_{\text{c}}\), and \(f_{\text{max}}\) becomes equal. The STN parameters before and after the adjustment of the resonance frequencies are summarized in Table 3. The comparison curves in Figure 5 reveal that the STN provides the optimal S\({}_{11}\) response at \(f_{\text{L}}\) and \(f_{\text{H}}\), and a larger separation between the two resonance frequencies results in a more pronounced upward curvature in the central part of the curve, with its peak appearing at \(f_{\text{c}}\).