## 3 Circuit Design  
### Principle of Distributed Amplifier  
According to the analysis of the transistor small-signal model in the previous section, the parasitic capacitance of the transistor has an adverse impact on the gain and bandwidth of the amplifier. The basic principle of the DA is to connect the gate and drain parasitic capacitors of a certain number of transistors side-by-side with inductive devices, so that the gate and drain ATLAS are constructed to overcome the gain roll-down caused by transistor parasitic capacitance. Therefore, on one hand, the parasitic capacitance of the transistor is absorbed by the inductive element, which effectively inhibits the high frequency gain drop caused by the parasitic capacitance of the amplifier. On the other hand, the parasitic capacitance and the inductive element form a T-type matching network greatly expand the working frequency band of the amplifier [12]. As shown in Figure 4, the DA is composed of a gate ATL and a drain ATL, and the characteristic impedance of the transmission line is \(Z_{0}\). The gate input and drain output ATLAS are coupled through the transconductance of the transistors. Figure 4: Topology of a typical DPA. Figure 3: GaN HEMTs equivalent small-signal model for transistors. As described above, the DA is based on the transmission line theory. Take a double transmission line with length \(\Delta Z\); as long as \(\Delta Z\) tends to be infinitesimal, the equivalent lumped parameter circuit of this transmission line can be considered to have infinitesimal resistance, conductivity, inductance, and capacitance. The equivalent model circuit is shown in Figure 5.  
For the limited transmission lines, it can be considered as a distributed parameter circuit cascaded by several \(\Delta Z\)-length transmission lines. Using [PERSON]'s law of voltage and current for the circuit shown in Figure 5b, we can obtain Formulas (1) and (2). The specific meaning of the variables involved in the formula will not be repeated. Formulas (3) and (4) are finally obtained by solving simultaneous equations, where \(\gamma\) is the complex propagation constant, which is a function of frequency, and its expression is Formula (5). \(\alpha\) is the attenuation constant and \(\beta\) is the phase shift constant, as is explained in detail in [12]. \[v(z)-R\Delta z\cdot i(z) -jwL\Delta z\cdot i(z)-v(z+\Delta z)=0 \tag{1}\] \[i(z)-G\Delta z\cdot v(z+\Delta z) -jwC\Delta z\cdot v(z+\Delta z)-i(z+\Delta z)=0\] (2) \[v(z) =V_{0}^{+}e^{-\gamma z}+V_{0}^{-}e^{\gamma z}\] (3) \[i(z) =I_{0}^{+}e^{-\gamma z}+I_{0}^{-}e^{\gamma z}\] (4) \[\gamma=\alpha+j\beta=\sqrt{(R+jwL)(G+jwC)} \tag{5}\]  
Formula (6) denotes the characteristic impedance \(Z_{0}\) of the transmission lines. At this point, the voltage and current on the transmission line are correlated by characteristic impedance, that is, \(Z_{0}=\frac{V_{0}^{+}}{I_{0}^{-}}=-\frac{V_{0}^{-}}{I_{0}}\). \[Z_{0,n}=\frac{R+jwL}{\gamma}=\sqrt{\frac{R+jwL}{G+jwC}} \tag{6}\]  
According to the transmission line theory, the propagation constants of the gate and drain ATLAS of the DA can be written as \(\gamma_{g}=\alpha_{g}+j\beta_{g}\) and \(\gamma_{d}=\alpha_{d}+j\beta_{d}\). In the case of the lossless transmission line \(\alpha_{g}=\alpha_{d}=0\), then \(\gamma_{g}=j\beta_{g}\)\(\gamma_{d}=j\beta_{d}\). Figures 6 and 7 show the equivalent circuit model of the gate and drain ATLAS. As shown in Figure 6, after the RF signal is transconducted to the drain transmission line, the signal can be transmitted from the left and right directions, so the gain of the DA is divided into forward gain and reverse gain. The forward gain is the gain from the signal input port to the signal output port, by adjusting the propagation constant and impedance of each transmission line to make the phase velocity of the signal propagation on the input and output transmission lines equal, as the forward traveling wave amplified by each stage of transistor can be added in phase at the drain transmission line port.