## 2 Principle of Linearity Improvement
### Phase Condition and Amplitude Analysis of \"Reverse Uplifi\"  
From Equation (1) and its expansion, we can approximate that \(\alpha_{1}\) characterizes the fundamental. However, \(\alpha_{2}\) and \(\alpha_{3}\) cannot be regarded as coefficients of the second and third harmonics simply, because they also characterize the DC and fundamental components at the same time. Therefore, for Equation (11), it is difficult for us to judge the positive and negative of each parameter independently, and it is not necessary. This condition is derived from \(T_{1}T_{3}<0\), implying the presence of gain compression. Therefore, it is only necessary to ensure the stability of the amplifier during design, which can be carried out through simulation, so Equation (11) can be regarded as a known condition. In addition, condition (12) can be transformed into a magnitude of \"reverse uplift\" greater than 0, which can be expressed as  
\[ \begin{split} Gain_{max}-\alpha_{1}=T_{1}-&\frac{{ T_{2}}^{2}}{4T_{3}}-\alpha_{1}\\ &=-\frac{\left(\frac{3}{4}\alpha_{3}\right)^{2}+\frac{{k_{1}}^{2} \alpha_{2}^{2}}{4}(1+k_{1}\alpha_{1})^{2}+\frac{3}{4}k_{2}\alpha_{2}\alpha_{3} (1+k_{1}\alpha_{1})}{\frac{3}{2}k_{1}k_{2}\alpha_{2}\alpha_{3}}>0\end{split}  \tag{13}\]  
As mentioned above, in order to ensure that the amplifier generates the strongest second harmonic to ensure the emergence of the second operating mode, the base/gate voltage should be set in the sub-threshold region. At this time, the third-order transconduc  
Figure 5: Simulated gm and gm\({}_{2}\) versus V\({}_{\text{b}}\) (V\({}_{\text{g}}\)). (**a**) Negative-positive-negative BJT (NPN-BJT), (**b**) N-channel metal-oxide-semiconductor (NMOS) transistor. Simulated gm\({}_{3}\) versus V\({}_{\text{b}}\) (V\({}_{\text{g}}\)). (**c**) NPN-BJT, (**d**) NMOS transistor. tance value is small and close to 0, so the effect of third-order nonlinear coefficient \(\alpha_{3}\) is ignored reasonably to simplify the analysis. Then, Equation (13) can be simplified to  
\[-\frac{\frac{k_{2}^{2}\alpha_{2}}{4}^{2}\left(1+k_{1}\alpha_{1}\right)^{2}}{ \frac{3}{2}k_{1}k_{2}\alpha_{2}\alpha_{3}}>0 \tag{14}\]  
If \(\alpha_{1}<0\), then from Equation (11) we know that \(k_{1}k_{2}\alpha_{2}\alpha_{3}>0\), but this requires the numerator to be less than 0 to satisfy the condition, which cannot be established obviously. If \(\alpha_{1}>0\), and then \(k_{1}k_{2}\alpha_{2}\alpha_{3}<0\), the magnitude of the \"reverse uplift\" can be controlled by the value of the numerator, which is exactly what is needed. Note that the magnitude of \"uplift\" should be limited, otherwise the improving of linearity will be meaningless because the compression curve is too steep. Take the common-source/common-base amplifier as an example; the output signal is opposite to the input signal generally, which means \(\alpha_{1}<0\). However, under this condition, the \"reverse uplift\" cannot be achieved, which requires the introduction of additional phase control. As shown in Figure 6, in the half Gilbert-cell, we introduce a phase-adjusting inductor at the emitter of the transconductance stage. At 27 GHz, as the value of the inductor increases, the phase difference of fundamental between collector and base decreases from 127\({}^{\circ}\) to below 90\({}^{\circ}\), which means that the output signal transitions from reverse to forward gradually. It should be added that the output phase has been shifted by 180\({}^{\circ}\) due to the effect of parasitic capacitors at high frequencies. Due to the selection of the bias voltage, \(|\alpha_{2}|\) inevitably increases, so in order to control the amplitude, the fundamental gain \(|\alpha_{1}|\) should be reduced as much as possible and seek a trade-off between gain and linearity. The phase-adjusting inductor can achieve this requirement through the control of phase and gain. In addition, by controlling the input impedance of the switching stage of the Gilbert-cell, the fundamental gain can also be reduced properly.