## 2 Analysis of the Proposed Design Strategy  
The voltage waveform of Class-J mode [PERSON] (2009) is defined by Equation (1):  
\[V_{J}(\theta)=(1-cos\theta)\cdot(1-\alpha\ sin\theta) \tag{1}\]  
where \(\alpha\) (\(-1\leq\alpha\leq 1\)) represents the continuous mode parameter and \(\theta\) represents the phase angle of the output voltage related to frequency. As long as the value of \(\alpha\) is within the range of [\(-1\),\(1\)], the efficiency remains at the ideal Class-J efficiency of 78.7%, while the \(P_{\text{sat}}\) remains unchanged. The voltage waveform of the RRCJ mode [12] adds a resistive factor \((1+\beta\ cos\theta)\) to Equation (1), forming Equation (2). \[V_{RRCJ}(\theta)=(1-cos\theta)\cdot(1-\alpha\ sin\theta)\cdot(1-\beta\ cos\theta) \tag{2}\]Here, \(\beta\) (\(0\leq\beta\leq 0.38\)) represents the resistive modulation coefficient. The addition of the resistive factor extends the fundamental impedance of Class-J to the left from the center, while the second harmonic impedance is extended to the right from the edge of the Smith chart, thereby merging the fundamental and second harmonic impedance spaces. The voltage waveform of the CCBJ mode is given by Equation (3):  
\[V_{CCBJ}(\theta)=(1-cos\theta)\cdot(\gamma-\alpha\ sin\theta)\cdot(1-\beta\ cos\theta) \tag{3}\]  
which adds the \(\gamma\) factor (\(1\leq\gamma\leq 1/(1-\beta)\)) to the RRCJ mode, extending the fundamental impedance space from the center to the right. The value of \(\gamma\) controls the extent of the impedance space expansion. The drain-source current (\(i_{\mathrm{ds}}\)) for the three modes mentioned above is defined as Equation (4). \[i_{\mathrm{ds}}(\theta)=1+\sum_{k=1}^{\infty}i_{k}cos\ k\theta=1+\frac{\pi}{2} cos\theta+\frac{2}{3}cos2\theta\ldots \tag{4}\]  
According to formulae (3) and (4) in Ref. [12], the fundamental and second harmonic impedances of the CCBJ mode are derived as shown in Equation (5). \[Z_{1}=R_{opt}[\gamma(1-\beta)+j\alpha(1-0.25\ \beta)],\ \ \ Z_{2}=\frac{3\pi R_{opt}}{8}[ \gamma\beta+j\alpha(1-\beta)] \tag{5}\]  
The corresponding impedance space is illustrated in Figure 1, with parameter ranges defined as \(-1\leq\alpha\leq 1,0\leq\beta\leq 0.38,1\leq\gamma\leq 1.6\). The yellow area represents the fundamental impedance, defined as \(Z_{1}\), and the blue area represents the second harmonic impedance, defined as \(Z_{2}\). Figure 1: The impedance space of CCBJ and the load-pull simulation results. Although efficiency within the impedance space has been extensively studied, output power analysis remains limited. As shown in Figure 1, the red circles represent power contours from load-pull simulations at 12 GHz using the GaN transistor employed in this work (normalized to \(R_{\rm opt}\)), with power steps set to 0.3 dB and the harmonic impedance configured as open. Within the CCBJ-defined impedance space, saturated power fluctuations exceed \(\pm\) 1.2 dB. This variation causes significant power fluctuations across the amplifier's bandwidth. To address this, an analysis based on the classic load-line theory is conducted. The concept of optimal power impedance (\(R_{\rm opt}\)) was first introduced in the load-line theory proposed in Ref. [13] and has since been widely applied in various power amplifier theories. During PA operation, the output voltage is limited by the drain supply voltage (\(V_{\rm dc}\)), while the maximum drain-source current (\(I_{\rm max}\)) is constrained by the product of transistor size and its power density. When either the voltage or current exceeds these physical limits, power compression occurs. Assuming that the output voltage (\(V_{\rm DS}\)) and current (\(I_{\rm D}\)) expressions are given as in Equation (6):  
\[V_{\rm DS}=V_{\rm dc}(1-\sin\theta),I_{\rm D}=\frac{I_{\rm max}}{2}(\sin\theta+ 1),\;\;I_{\rm dc}=\frac{I_{\rm max}}{2} \tag{6}\]  
a load impedance, \(R_{\rm opt}\), exists that allows both the voltage and current amplitudes to reach their maximum values, thereby delivering the maximum power \(P_{\rm opt}\), as shown in Equation (7).