# A Wideband Hybrid Envelope Tracking Supply Modulator with Slew-Rate-Enhanced Linear Amplifier
## 2 Circuit Design
### LA Design with an SRE Structure
#### 2.2.2 AC Analysis of the LA with the Proposed SRE Circuit  
The SRE circuit in Figure 5b can be extracted to a corresponding small-signal model, as shown in Figure 8. In the small-signal models below, unless otherwise specified, \(g_{mi}\), \(R_{Oi}\), and \(C_{i}\) are used to represent the transconductance \(g_{m}\), output resistance \(r_{ds}\), and output capacitance \(C_{d}\) of transistor \(M_{i}\), respectively, where \(i\) denotes the subscript of the transistor. Due to the symmetrical design of the circuit, we have \(g_{m3P}=g_{m4P}\) and \(g_{m3N}=g_{m4N}\). The small-signal model in Figure 8 can be simplified to a one-stage amplifier model within the red dashed box in Figure 9. We can derive the relationship given in Equation (13):  
\[\left\{ \begin{array}{l}g_{mSR}=2k\frac{g_{m3P}g_{m4N}}{g_{m3P}+g_{m3N}}\\ R_{OSRP}=R_{Ox2}\big{|}\big{|}R_{Oy2},R_{OSRN}=R_{Ox3}\big{|}\big{|}R_{Oy3}\\ C_{SRP}=C_{x2}+C_{y2},C_{SRN}=C_{x3}+C_{y3}\end{array} \right. \tag{13}\]  
The detailed derivation process of the small-signal LA model is provided in Supplementary Material. As derived in Supplementary Material, we can combine the input stage of LA with the SRE circuit to simplify it into the first-stage amplification circuit model shown on the left side of Figure 10. We can derive the relationship shown in Equation (14):  
\[\left\{ \begin{array}{l}g_{m1}{=}g_{mSR}+g_{m1 Conv}=2k\frac{g_{m3P}g_{m4N}}{g_ {m4P}+g_{m3N}}+\frac{g_{m4P}g_{m1b}}{g_{m4P}+g_{m4N}}\\ R_{O1P}=R_{OSRP}||R_{O2b},R_{O1N}=R_{OSRN}||R_{O2a}\\ C_{1P}=C_{SRP}+C_{2b}+C_{gs3a},C_{1N}=C_{SRN}+C_{2a}+C_{gs3b}\end{array} \right. \tag{14}\]Figure 8: A small-signal model of the SRE circuit. Figure 10: The small-signal model of the entire LA. Figure 9: A small-signal model of the proposed LA’s input stage. The small-signal model of the entire LA after the aforementioned simplification process is shown in Figure 10. If we incorporate the following steps into the LA design, we can achieve symmetry in the circuit to a large extent:  
(1) Let \(k_{NP}=\frac{R_{01N}}{R_{01P}}\). (2) Select the sizes of M3a and M3b, such that \(C_{1P}=k_{NP}C_{1N}\). (3) Choose the R1/C1 and R2/C2 that satisfy the relation. Specifically, when \(R_{01N}=R_{01P}\), we have \(k_{NP}=1\). In this case, we derive that \(V_{GP}-V_{GN}=0\). When \(V_{GP}=V_{GN}\), the effects of \(M_{3c}\) and \(M_{3d}\) cancel each other out because the current \(g_{m3d}V_{GN}\) is equal to the current \(g_{m3c}V_{GP}\), as shown in Figure 10, and the current through the impedance \(Z_{TL}\) equals zero. Then, due to symmetry, the LA circuit can be simplified to that shown in Figure 11. We find that the small-signal model of the proposed amplifier can be simplified to the well-known two-stage operational amplifier model, simply by substituting the parameters with those in parentheses shown in Figure 11. We can derive the relationship shown in Equation (15):  
\[\left\{ \begin{array}{l}g_{ml}=2g_{m1}=4k\frac{g_{m3d}g_{m3d}}{g_{m2d}+g_{m3d }}+2\frac{g_{m4}g_{m1b}}{g_{m4}+g_{m4b}}\\ R_{I}=R_{01N}||R_{01P}=(R_{0 SRN}||R_{02a})||(R_{OSRP}||R_{02b})\\ C_{I}=C_{1P}+C_{1N}=(C_{SRP}+C_{2b}+C_{gs3a})+(C_{SRN}+C_{2a}+C_{gs3b})\\ R_{c}=R_{1}||R_{2},C_{c}=C_{1}||C_{2}\\ g_{mll}=g_{m3a}+g_{m3b}\\ R_{II}=R_{03d}||R_{03b},C_{II}=C_{3a}+C_{3b}+C_{L}\end{array} \right. \tag{15}\]  
If the poles are widely spaced, then we can derive the expressions for the zeroes and poles as follows:  
\[\left\{ \begin{array}{l}p_{1}\cong\frac{-1}{g_{mll}R_{I}C_{c}}\\ p_{2}\cong\frac{-g_{mll}}{C_{II}}\\ p_{3}=\frac{-1}{R_{c}C_{I}}\\ z_{1}=\frac{1}{C_{c}(1/g_{mll}-R_{z})}\end{array} \right. \tag{16}\]  
In the AC analysis, the recommended SRE circuit is equivalent to a stage that is parallel to the input stage of the amplifier and can be integrated into the input stage circuit. Therefore, when selecting the compensation capacitor and resistor to adjust the stability of the amplifier, the impact of the SRE circuit must be considered.