# 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
#### 3.1.2 Independently Tunable Linearizer Mechanism Analysis  
The independently tunable linearizer (ITL) decouples the gain and phase expansion curves to a certain extent, and thus allows for better separate compensation of the gain and phase compression of the TWTA. The APD circuit is realized via the combination of the two nonlinear branches, and the schematic diagram is shown in Figure 1. Assuming that the signal entering the APD circuit is Equation (7), the signal arriving at the gain branch after passing through the Wilkinson power divider 1 can be expressed through Equation (8), where \(G_{\mathrm{g}}\) and \(\varphi_{\mathrm{g}}\) are the signal's gain and phase changes. The signal arriving at the phase branch can be expressed as shown in Equation (9), and its gain and phase changes can be expressed as \(G_{\mathrm{p}}\) and \(\varphi_{\mathrm{p}}\). The two nonlinear branch signals are combined into the output signal in vector mode at the Wilkinson power divider 2, and the output signal can be expressed as \(U_{\mathrm{out}}=U_{\mathrm{g}}+U_{\mathrm{p}}\). The gain and phase change in the signals through the APD circuit can be expressed by \(G\) and \(\Phi\), expressed as Equations (10) and (11), respectively.  
\[U_{\mathrm{in}}=G_{0}\exp[j(wt+\varphi_{0})] \tag{7}\]  
\[U_{\mathrm{g}}=\frac{G_{0}G_{\mathrm{g}}}{2}\exp[j(wt+\varphi_{0}+\frac{\pi} {2}+\varphi_{\mathrm{g}})] \tag{8}\]  
\[U_{\mathrm{p}}=\frac{G_{0}G_{\mathrm{p}}}{2}\exp[j(wt+\varphi_{0}+\frac{\pi}{2 }+\varphi_{\mathrm{p}})] \tag{9}\]  
\[G=\left|\frac{U_{\mathrm{out}}^{2}}{U_{\mathrm{in}}^{2}}\right|=\frac{1}{2} \sqrt{G_{\mathrm{g}}^{2}+G_{\mathrm{p}}^{2}+2G_{\mathrm{g}}G_{\mathrm{p}}\cos (\varphi_{\mathrm{g}}-\varphi_{\mathrm{p}})} \tag{10}\]  
\[\Phi=\angle U_{\mathrm{out}}-\angle U_{\mathrm{in}}=\arctan(\frac{G_{\mathrm{ g}}\sin\varphi_{\mathrm{g}}+G_{\mathrm{p}}\sin\varphi_{\mathrm{p}}}{G_{\mathrm{g}} \cos\varphi_{\mathrm{g}}+G_{\mathrm{p}}\cos\varphi_{\mathrm{p}}}) \tag{11}\]  
This circuit structure allows for the independent tuning of the gain and phase expansion through reasonable design decisions. The bias voltage \(V_{\mathrm{g}}\) of the Schottky diodes in the gain (phase) nonlinear branch is adjusted to achieve a wide gain (phase) expansion variation range and a constant phase (phase) expansion. The gain (phase) expansion variation with \(V_{\mathrm{g}}\) (or \(V_{\mathrm{p}}\)) is realized via a small-signal gain (phase) change, while the saturation gain (phase) remains unchanged.  
The phases of the two nonlinear branches under large-signal excitation are aligned through a delay line, so \(\varphi_{\mathrm{sat\_g}}=\varphi_{\mathrm{sat\_p}}\) holds due to the phase align design. In this case, using \(\mathrm{G_{sat\_g}}\), \(\mathrm{G_{sat\_g}}\), \(\mathrm{\varphi_{sat\_g}}\), \(\mathrm{\varphi_{sat\_p}}\) for the corresponding values of (10) and (11), Equations (12) and (13) are obtained.  
\[G_{\mathrm{sat}}=\frac{1}{2}(G_{\mathrm{sat\_g}}+G_{\mathrm{sat\_p}}) \tag{12}\]  
\[\mathrm{\Phi_{sat}}=\mathrm{\varphi_{sat\_g}}=\mathrm{\varphi_{sat\_p}} \tag{13}\]  
Using small signals to excite two nonlinear branches, \(\mathrm{G_{ss\_g}}<\mathrm{G_{sat\_g}}\) and \(\mathrm{\varphi_{ss\_p}}<\mathrm{\varphi_{sat\_p}}\) hold due to the nonlinear branch characteristics. In this case, using \(\mathrm{G_{ss\_g}}\), \(\mathrm{G_{ss\_p}}\), \(\mathrm{\varphi_{ss\_g}}\), \(\mathrm{\varphi_{ss\_p}}\) for (10) and (11), it is obvious that \(\mathrm{G_{ss}}<\mathrm{G_{sat}}\) and \(\mathrm{\Phi_{ss}}<\mathrm{\Phi_{sat}}\). This means that the synthesized signal has both expanded gain and phase. A similar approach can be used to analyze the gain and phase expansion differences when \(V_{\mathrm{g}}\) and \(V_{\mathrm{p}}\) changes. The principle of the independent tuning of gain and phase conversion is shown in Figure 4.