# Ultra-Generalized Continuous Class F Power Amplifier with Finite Third-Harmonic Load Impedance
## 2 Analysis of the Proposed UCCF Mode
### Brain Current Waveform of the Proposed UCCF Mode  
The drain current waveform of the proposed UCCF mode (\(i_{d,UJF}\)) in one signal period (Figure 4) can be expressed as  
\[ \begin{split}& i_{d,UJF}=\\ &\left\{\begin{array}{l}I_{\text{max}}\times pcr\times\dfrac{ \left(\cos(\theta)-\cos\Bigl{(}\dfrac{\alpha}{2}\Bigr{)}\Bigr{)}}{1-\cos \Bigl{(}\dfrac{\alpha}{2}\Bigr{)}},-\dfrac{\alpha}{2}<\theta<-\dfrac{\beta}{2} \\ \\ I_{\text{max}},\hskip 56.905512 pt-\dfrac{\beta}{2}<\theta<\dfrac{\beta}{2} \\ \\ I_{\text{max}}\times pcr\times\dfrac{\left(\cos(\theta)-\cos\Bigl{(}\dfrac{ \alpha}{2}\Bigr{)}\Bigr{)}}{1-\cos\Bigl{(}\dfrac{\alpha}{2}\Bigr{)}},\ \ \ \dfrac{\beta}{2}<\theta<\dfrac{\alpha}{2}\\ \\ 0,\hskip 56.905512 pt-\pi<\theta<-\dfrac{\alpha}{2},\dfrac{\alpha}{2}<\theta< \pi\end{array}\right.\\ \\ \end{split}  \tag{4}\]  
where  
\[\dfrac{\beta}{2}\ =\ \arccos\left(\dfrac{1-\cos(\dfrac{\alpha}{2})+pcr\times \cos\Bigl{(}\dfrac{\alpha}{2}\Bigr{)}}{pcr}\right) \tag{5}\]  
and \(I_{\text{max}}\) is the maximum drain current amplitude, and \(\beta\) is the angle at which the current waveform reaches \(I_{\text{max}}\). Figure 4 depicts the drain current waveform state for various modes. When \(pcr\) ranges from 1 to 2 and \(\alpha\) ranges from \(\pi\) to 2\(\pi\) in (4), the GCCF mode is present, and the drain current waveform is peak\(-\)clipped with a drain current conduction angle \(\alpha\) of greater than \(\pi\). When \(pcr\) equals 1 and \(\alpha\) ranges from \(\pi\) to 2\(\pi\) in (4), the GCCF  
Figure 3: Gate input voltage of the UCCF mode (\(v_{GS,norm}\)). (**a**) \(pcr\) equals 1, and \(\alpha\) changes from \(\pi\) to 2\(\pi\). (**b**) \(pcr\) equals 1.5, and \(\alpha\) changes from \(\pi\) to 2\(\pi\). mode is present, and the drain current waveform forms a half-sinusoidal wave with \(\alpha\) greater than \(\pi\). When \(pcr\) ranges from 1 to 2 and \(\alpha\) equals \(\pi\) in (4), the PCC-CF\({}_{3}\) mode is present, and the drain current waveform exhibits a peak-clipped drain current waveform. When \(pcr\) equals 1 and \(\alpha\) equals \(\pi\) in (4), the continuous class F mode is present, and the drain current waveform is an ideal half-sinusoidal wave. The Fourier-series expansion of \(i_{d,ULF}\)[11] can be calculated as functions of \(pcr\) and \(\alpha\) by (6)-(8),  
\[i_{d,ULF}=I_{dc}+\sum_{n=1}^{\infty}I_{n}\cdot\cos(n\theta) \tag{6}\]  
where  
\[I_{dc}=\frac{1}{2\pi\cdot I_{\max}}\cdot\int_{0}^{2\pi}i_{d,ULF} \cdot\cos(n\theta)d\theta,(n=0) \tag{7}\] \[I_{n}=\frac{1}{\pi\cdot I_{\max}}\cdot\int_{0}^{2\pi}i_{d,ULF} \cdot\cos(n\theta)d\theta,(n=1,2,3\ldots) \tag{8}\]  
\[\left\{ \begin{array}{ll}I_{dc}&=\ \frac{\csc^{2}\left[\frac{\alpha}{4} \right]\left(\beta+\left(\beta(-1+pcr)-\alpha\cdot pcr\right)\cos\left[\frac{ \alpha}{2}\right]-2 pcr\cdot\sin\left[\frac{\beta}{2}\right]+2 pcr\cdot\sin\left[ \frac{\alpha}{2}\right]\right)}{4\pi}\\ I_{1}&=\ \frac{8\sin\left[\frac{\beta}{2}\right]-pcr\cdot\csc^{2}\left[ \frac{\alpha}{4}\right]\left(\beta-\alpha-4\cos\left[\frac{\alpha}{2}\right] \sin\left[\frac{\beta}{2}\right]+\sin\left[\beta\right]+\sin\left[a\right] \right)}{4\pi}\\ I_{2}&=\ \frac{3\sin[\beta]+pcr\cdot\csc^{2}\left[\frac{\alpha}{4} \right]\cdot\sin^{2}\left[\frac{\beta-\alpha}{4}\right]\left(-3\sin\left[ \frac{\beta}{2}\right]+2\cdot\sin\left[\beta+\frac{\alpha}{2}\right]+\sin \left[\frac{\beta}{2}+\alpha\right]\right)}{3\pi}\\ I_{3}&=\ \frac{pcr\cdot\csc^{2}\left[\frac{\alpha}{4}\right]\left(-3(1+\cos[\beta]) \sin[\beta]+4\cos\left[\frac{\alpha}{2}\right]\sin\left[\frac{3\beta}{2} \right]\right)+\left(8\sin\left[\frac{3\beta}{2}\right]+pcr\cdot\csc^{2} \left[\frac{\alpha}{4}\right]\sin\left[a\right]\right)}{12\pi}\end{array} \right. \tag{9}\]  
Figure 4: Time domain drain current waveforms with different modes. Through (9), the DC (\(I_{dc}\)), fundamental (\(I_{1}\)), second-harmonic (\(I_{2}\)), and third-harmonic (\(I_{3}\)) components can be computed. Figure 5 illustrates the \(I_{dc}\), \(I_{1}\), \(I_{2}\), and \(I_{3}\) trends in the UCCF mode with \(pcr\) and \(\alpha\).