# A Broadband Analog Predistortion Linearizer Based on GaAs MMIC for Ka-Band TWTAs
## 2 Analog Predistortion MMIC Topology
### Independently Tunable Linearizer Design for Improved Nonlinear Indices  
#### 3.1.1 Gain and Phase Nonlinear Branch Design  
The proposed APD MMIC schematic adopts a reflective APD circuit topology in the gain and phase nonlinear branches, as shown in Figure 1. A Lange coupler is connected with the nonlinear generator unit to achieve good isolation of the input and output signals. The input signals enter the circuit from port 1 of the Lange coupler and arrive at ports 2 and 3 of the coupler with equal gain and a phase difference of 90\({}^{\circ}\). After passing through the Schottky diodes, the signals are reflected back to ports 1 and 4 of the coupler. Finally, the signals returning to port 1 from ports 2 and 3 cancel each other due to their equal gain and phase difference of 180\({}^{\circ}\), while the signals returning to port 4 from ports 2 and 3 are stacked and output due to their equal gain and phase [28].  
A Schottky diode can be considered equivalent to a variable junction resistance and a junction capacitance, and its impedance can be expressed as \(Z_{\mathrm{d}}=R_{d}+jX_{d}\), which is shown in Figure 1. The Schottky diode is grounded in series. The input impedance at point A can be expressed using Equation (3), while the reflection coefficient at the coupler port 2 or port 3 can be expressed as Equation (4). The amplitude and phase of the reflection coefficient are shown in Equations (5) and (6). Since the signals reflected by ports 2 and 3  
\begin{table}
\begin{tabular}{c c c c} \hline
**Diode Size** & \(I_{s}\) **(pA)** & \(R_{s}\) **(\(\Omega\))** & \(n\) \\ \hline One-finger 10 \(\upmu\)m & 7.91 & 0.15 & 1.37 \\ Two-finger 20 \(\upmu\)m & 28.92 & 0.11 & 1.37 \\ Four-finger 40 \(\upmu\)m & 93.91 & 0.09 & 1.37 \\ \hline \end{tabular}
\end{table}
Table 1: DC parameter of diodes with different sizes.  
are superimposed and output at port 4, the transmission characteristics of the entire APD circuit are completely determined by the reflection coefficient \(\Gamma\). The reflection coefficient \(\Gamma\) can be adjusted through an adjustment of the diode's impedance, particularly the nonlinear resistance. This can be realized by selecting the diode size and matching microstrip line. In this paper, the gain (phase) nonlinear branch is designed to have large (small) gain expansion and small (large) phase expansion, and these characteristics vary greatly with the biases \(V_{\mathrm{g}}\) and \(V_{\mathrm{p}}\) of the diodes in the corresponding branches.  
\[Z_{\mathrm{in}}=Z_{\mathrm{d}}=R_{\mathrm{d}}+\mathrm{i}X_{d} \tag{3}\]  
\[\Gamma=\frac{Z_{\mathrm{in}}-Z_{0}}{Z_{\mathrm{in}}+Z_{0}}=\frac{Y_{0}-Y_{ \mathrm{in}}}{Y_{0}+Y_{\mathrm{in}}}=\frac{(G_{0}-G_{\mathrm{in}})-\mathrm{i}B _{\mathrm{in}}}{(G_{0}+G_{\mathrm{in}})+\mathrm{i}B_{\mathrm{in}}} \tag{4}\]  
\[|\Gamma|=\sqrt{\frac{\left(G_{0}-G_{\mathrm{in}}\right)^{2}-B_{\mathrm{in}}^{ 2}}{\left(G_{0}+G_{\mathrm{in}}\right)^{2}+B_{\mathrm{in}}^{2}}} \tag{5}\]  
\[\angle\Gamma=\arctan(\frac{-B_{\mathrm{in}}}{G_{0}-G_{\mathrm{in}}})-\arctan (\frac{B_{\mathrm{in}}}{G_{0}+G_{\mathrm{in}}}) \tag{6}\]