# S-Band Class-C-F Power Amplifier with 2 nd Harmonic Control at the Input
## 2 Theory of the Gate Harmonic Control  
The key to realizing high efficiency PA is to increase the fundamental power delivered to the load, while decreasing both the dissipating power and those delivering to the load at the harmonic frequencies. Harmonic manipulated PAs, based on III-V technology, such as GaAs or GaN, are usually based on the assumption that the active device acts as a current source controlled by the input signal. In particular, referring to FET (Field-effect transistor) devices, the current at the intrinsic drain node is mainly controlled by the voltage signal at the gate node, while the output voltage can be eventually shaped with proper harmonic load conditions. It implies that the performances of a harmonic tuned PA depend not only on the proper impedance provided by the output matching network (OMN) at the relevant frequencies, but also on the amplitude and phase relationship between the harmonics and the fundamental frequency of the drain current, which in turns depend on the voltage signal presented at the gate node. Therefore, the gate bias condition, the input signal, and the loading conditions at the input determine the output current waveform, and in turn, the idea is implementable to perform a suitable output voltage waveform, shaping it in order to improve device performances. For example, let us consider a 3 rd harmonic tuned PA, also known as class-F PA. The drain voltage can be written as [2]:  
\[v_{DS}=V_{DD}-V_{1}\cdot\big{[}\cos(\theta)+k_{3}\cdot\cos(3\theta)\big{]}, \tag{1}\]  
where \(V_{DD}\) is the biasing voltage, \(V_{1}\) is the fundamental component, and \(k_{3}=V_{3}/V_{1}\) is the ratio of the 3 rd harmonic component \(V_{3}\) with respect to the fundamental one \(V_{1}\). According to the theory of harmonic tuned PA, the optimum condition for achieving high efficiency is proceeding with \(k_{3}=-1/6\), resulting in the theoretical improvement of 15%. It means that the 3 rd harmonic voltage has to be out of phase with respect to the fundamental one. When the device is driven by sinusoidal input signal, such a condition implies that the device has to be biased in class-B (theoretically) or in class-AB (practically). In fact, referring to the current harmonic components, reported in Figure 1, only for conduction angle from \(\pi\) (e.g., class-B) to \(2\pi\) (e.g., class-A), the 3 rd harmonic current is out of phase with the fundamental one, which is suitable for the realization of a class-F PA. Thus, since the output voltage harmonics is related to the relevant current ones through the passive OMN, the phase condition on the former can be attained only if the same phase condition holds for the current harmonics. If the active device is biased in a class-C biasing range, the 3 rd harmonic component of the intrinsic current is always in phase with the fundamental one; in other words, \(k_{3}\) could hardly be negative in this condition. Such an improper phase relationship results in a detrimental effect to the performance of PA, which cannot be recovered by passive OMN, if a class-F output matching strategy alone is adopted. However, things may be different if an alternative design strategy is adopted. One possible method is to form a rectangular current wave at the intrinsic current plane, in which case, the harmonics generated can be expressed as follows:  
\[I_{r,n}= \begin{cases} I_{Max}\cdot\dfrac{\Phi}{2\pi}&n=0\\ \dfrac{2I_{Max}}{\pi}\dfrac{\sin(\dfrac{\Phi}{n})}{n}&n\geq 1,\end{cases}  \tag{2}\]  
where \(\Phi\) is the conductive angle. As shown in Figure 2, with the different excitation, the biasing range where the 3 rd harmonic is out of phase with respect to the fundamental one is from \(\frac{2}{3}\pi\) to \(\frac{4}{3}\pi\). Part of the conductive angle belongs to the class-C biasing range, which means that class-C-F PA is available under these conditions.