# A High-Efficiency K-band MMIC Linear Amplifier Using Diode Compensation
## 2 Nonlinear Analysis and Diode Compensation  
Table 1 shows the main parameter of the 0.15 \(\upmu\)m GaAs pHEMT process. The equivalent circuit models of pHEMT and a diode are shown in Figure 1.  
Using a diode depletion capacitance model [8], the values of \(C_{gs}\) and \(C_{gd}\) can be derived as  
\[C_{gs}(v_{gs})=\frac{C_{gsb}}{\sqrt{1-\frac{v_{gs}}{v_{bi}}}} \tag{1}\]  
\[C_{gd}(v_{gd})=\frac{C_{gdb}}{\sqrt{1-\frac{v_{gd}}{v_{bi}}}} \tag{2}\]  
The functional model of a single-stage pHEMT amplifier can be represented by Figure 2.  
In this model, \(V_{S}(\omega)\) and \(Z_{S}(\omega)\) constitute the Thevenin equivalent of the excitation source and the input matching network, and \(Z_{L}(\omega)\) is the load impedance of the pHEMT. Assuming that the pHEMT is excited by a voltage, \(v_{gs}(A,t)\):  
\[v_{gs}(A,t)=\sum_{0}^{n}V_{gsk}(A,\omega)\cdot\cos(k\omega_{0}t) \tag{3}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline V\({}_{\text{TH}}\) (V) & \(-1.2\) & \(f_{\text{t}}\) (GHz) & 85 \\ V\({}_{\text{BDG}}\) (V) & 10 & Gm\_ Peak (mS/mm) & 495 \\ I\({}_{\text{dmax}}\) (mA/mm) & 650 & P\({}_{\text{density}}\) (W/mm) & 0.8 \\ I\({}_{\text{dss}}\) (mA/mm) & 500 & C\({}_{\text{MIM}}\) (pF/mm\({}^{2}\)) & 400 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters of the 0.15 \(\upmu\)m GaAs pHEMT process. Figure 1: (a) The equivalent circuit model of pHEMT; (b) the equivalent circuit model of a diode. Figure 2: Single-stage pHEMT amplifier model. where \(V_{ggk}\) represents the Fourier components of \(v_{gs}\) and A is the amplitude. By setting the dynamic charge between the gate and source determined by \(v_{gs}\) as \(Q_{gs}\), \(i_{gs}\) can be derived as  
\[i_{gs}(t)=\frac{\partial Q_{gs}[v_{gs}]}{\partial t}=C_{gs}[v_{gs}]\cdot\frac{ \partial v_{gs}}{\partial t} \tag{4}\]  
Then, we get the frequency domain representation of \(i_{gs}\) as  
\[I_{gs}=[\frac{1}{2}{\sum_{-n}^{n}C_{gsk}(V_{gs},\omega)}]\cdot[\frac{1}{2}{ \sum_{-n}^{n}jk\omega_{0}V_{gsk}}] \tag{5}\]  
where \(C_{gsk}\) is the Fourier components of \(C_{gs}\):  
\[C_{gsk}=\frac{\omega}{2\pi}{\int_{-\pi/\omega}^{\pi/\omega}C_{gs}[v_{gs}]} \cdot e^{-jk\omega_{0}t}dt \tag{6}\]  
Substituting (R4) into (R3), the fundamental component of \(i_{gs}\) would be  
\[I_{gs1}=j\omega_{0}C_{gs0}V_{gs1}+\frac{1}{2}j2\omega_{0}{C_{gs1}}^{*}\cdot V _{gs2}-\frac{1}{2}j\omega_{0}{C_{gs2}}\cdot{V_{gs1}}^{*}+\ldots \tag{7}\]  
As a result, the fundamental \(v_{gs}(t)\) voltage \(V_{gs1}\) (\(A\)) can be derived as  
\[V_{gs1}=V_{S}(\omega)-j\omega Z_{s}(\omega)\{C_{gs0}\cdot V_{gs1}+{C_{gs1}}^{* }\cdot V_{gs2}-\frac{1}{2}{C_{gs2}}\cdot{V_{gs1}}^{*}+\ldots\} \tag{8}\]  
and solving (8) leads to  
\[V_{gs1}=\frac{V_{s}(\omega)}{1+j\omega Z_{s}(\omega)\cdot C_{gs0}} \tag{9}\]  
The term involving \(C_{gs2}\) and \(V_{gs2}\) is neglected as it is significantly smaller than the others. Incorporating the feedback capacitor C\({}_{gd}\) into (9), we get  
\[V_{gs1}=\frac{V_{s}(\omega)}{1+j\omega Z_{s}(\omega)\cdot[C_{gs0}+C_{gd}(1-A_{ v})]} \tag{10}\]  
(10) reveals that \(C_{gs}\) and \(C_{gd}\) influence the phase of fundamental \(v_{gs}(t)\), which leads to AM/PM distortion. Using a similar method, we can derive the fundamental \(v_{ds}\)(t) voltage \(V_{ds1}\) (\(A\)) as  
Figure 1:0  
The term involving \(C_{ds2}\) is neglected as it is significantly smaller than the others. Since the denominator of (11) is approximately equal to unity [9], as a result, there is no significant impact on AM/PM brought about by \(C_{ds}\). From (10) and (11), it can be derived that the main contributors to the AM/PM distortion are the variations of \(C_{gs}\) and \(C_{gd}\). According to the barrier capacitance effect, the value of \(C_{gs}\) is usually much bigger than that of \(C_{gd}\). Therefore, the nonlinear characteristic of pHEMT is mainly contributed by \(C_{gs}\), which approximately equals \(C_{in}\).