# Three-Level Hybrid Envelope Tracking Supply Modulator with High-Bandwidth Wide-Output-Swing
## 3 Circuit Design  
### Linear Amplifier  
This section describes a class-AB linear amplifier with rail-to-rail input and output ranges. Figure 4 shows the detailed circuit design of LA. It is worth noting that a source follower (SF) is adopted as a buffer to reduce the output impedance and improve driving capability.  
The input stage operates in parallel with pMOS and nMOS differential input pairs, enabling the input common-mode range to reach both rails of the supply voltage. To achieve this, it is necessary to ensure that the input-stage transconductance (\(g_{m}\)) remains constant over the entire common-mode range. In this structure, the constant transconductance is achieved by using tail current compensation [27]. The class AB output stage can output large current, and has a certain ability to supply power for the load while providing low output impedance.  
The output stage is designed to be a common source amplifier to provide high current driving capability. The buffer stage includes two source followers to reduce the output impedance of the circuit. Figure 5a shows a conventional source follower (CSF) [28], the output impedance is approximately \(1/(g_{m1}+g_{mbl})\). However, the CSF only reduces the output impedance to a certain extent, and it is not suitable for the case if the load is small. Therefore, this paper employs a super source follower (SSF) with feedback loop, as shown in Figure 5b. In this circuit, the feedback loop composed of M\({}_{\text{NF}}\), M\({}_{\text{ND}}\), and M\({}_{\text{PB}}\). This loop controls the output transistors MP1 and MN1. The current \(\text{I}_{\text{b1}}\) is greater than the current \(\text{I}_{\text{b2}}\). Assuming that all transistors are in saturation region, then the current difference between \(\text{I}_{\text{b1}}\) and \(\text{I}_{\text{b2}}\) flows through the input transistor M\({}_{\text{NF}}\) to the drive transistor M\({}_{\text{ND}}\).  
Figure 4: Detailed circuit of the LA.  
Figure 5: The block diagram of source followers: (**a**) conventional source follower; (**b**) super source follower; and (**c**) output impedance calculation circuit of the SSF.  
Using the classical feedback theory to calculate the output impedance of the SSE, it can be expressed as in Equation (7). \(R_{op}\) is the open-loop output impedance, \(A\beta\) is the loop gain. The output impedance of SSF is given by Equation (8).  
\[\frac{V_{x}}{I_{x}}=\frac{R_{op}}{1+A\beta} \tag{7}\]  
\[R_{O}=\frac{1}{g_{mF}\times(1+g_{mD}r_{o2})} \tag{8}\]  
where \(g_{mF}\) is the transconductance of M\({}_{\text{NF}}\), \(g_{mD}\) is the transconductance of M\({}_{\text{ND}}\), and \(r_{o2}\) is the output impedance of current source I\({}_{\text{b2}}\).  
The output impedance of the LA is simulated, and Figure 6a shows the output impedance versus frequency for different common-mode (CM) input voltages. The frequency response of the LA is shown in Figure 6b, which shows the amplitude and phase margin of the amplifier in open-loop condition. It can be seen that the DC gain is 78 dB, with a phase margin of 45 degrees and the unit gain-bandwidth of 171 MHz. Figure 7 shows the transient simulation results, which verifies that the LA in this paper can amplify the 100 MHz envelope signal well.