## 2 HSM for Base Station PA
### Linear Amplifier Design
#### 2.3.3 Frequency Response of LA  
Modern communication signals have wide bandwidths, requiring an LA with a wide bandwidth for accurate envelope tracking. Typically, the LA bandwidth must be 1.5 to 3 times larger than the envelope signal bandwidth to maintain sufficient tracking accuracy in HSM [18]. In the proposed design, two factors mainly limit the LA bandwidth. First, the floating current source introduces output impedance at nodes X and Y, connected to the gate of the PMOS and NMOS output transistors, respectively. Figure 5 illustrates the small-signal test setup used to analyze the impedance seen from node X of the floating current source. By applying a test voltage \(\mathrm{v}_{\mathrm{test}}\), a current \(\mathrm{i}_{\mathrm{test}}\) flows through the network. The small-signal drain currents \(\mathrm{i}_{\mathrm{d,n}}\) and \(\mathrm{i}_{\mathrm{d,p}}\) flow through MP12 and MN16, respectively, and the remaining current \(\mathrm{i}_{\mathrm{test}}-\mathrm{i}_{\mathrm{d,p}}-\mathrm{i}_{\mathrm{d,n}}\) flows through the parallel output resistances \(\mathrm{r}_{\mathrm{o,P12}}\parallel\mathrm{r}_{\mathrm{o,N16}}\). From this configuration, the node voltage \(\mathrm{V}_{\mathrm{test}}\) is given in (6), and the drain currents \(\mathrm{i}_{\mathrm{d,p}}\) and \(\mathrm{i}_{\mathrm{d,n}}\) are defined in (7) and (8), respectively. Inserting (7) and (8) into (6) gives (9), which simplifies (9a). The resistance of the floating current source seen from node X is given in (10) and further simplified to (10a) when \(\mathrm{g}_{\mathrm{mP}}\) = \(\mathrm{g}_{\mathrm{mN}}\)  
\[\mathrm{v}_{\mathrm{test}}=(\mathrm{r}_{\mathrm{o,P12}}\parallel\mathrm{r}_{ \mathrm{o,N16}})\cdot(\mathrm{i}_{\mathrm{test}}-\mathrm{i}_{\mathrm{d,p}}- \mathrm{i}_{\mathrm{d,n}})+\mathrm{r}_{\mathrm{o,N15}}\mathrm{i}_{\mathrm{ test}} \tag{6}\]  
\[\mathrm{i}_{\mathrm{d,p}}=\mathrm{g}_{\mathrm{mP}}\cdot\mathrm{v}_{\mathrm{test}} \tag{7}\]  
\[\mathrm{i}_{\mathrm{d,n}}=\mathrm{g}_{\mathrm{mN}}\cdot(-\mathrm{i}_{\mathrm{ test}}\mathrm{r}_{\mathrm{o,N15}}) \tag{8}\]  
\[\mathrm{v}_{\mathrm{test}}=(\mathrm{r}_{\mathrm{o,P12}}\parallel\mathrm{r}_{ \mathrm{o,N16}})\cdot(\mathrm{i}_{\mathrm{test}}-\mathrm{g}_{\mathrm{mP}} \mathrm{v}_{\mathrm{test}}+\mathrm{g}_{\mathrm{mN}}\mathrm{i}_{\mathrm{test}} \mathrm{r}_{\mathrm{o,N15}})+\mathrm{r}_{\mathrm{o,N15}}\mathrm{i}_{\mathrm{ test}} \tag{9a}\] \[(1+\mathrm{g}_{\mathrm{mP}}\mathrm{r}_{\mathrm{o,P12}}\parallel\mathrm{r}_{ \mathrm{o,N16}})\mathrm{v}_{\mathrm{test}}=[(\mathrm{r}_{\mathrm{o,P12}} \parallel\mathrm{r}_{\mathrm{o,N16}})(1+\mathrm{g}_{\mathrm{mN}}\mathrm{r}_{ \mathrm{o,N15}})+\mathrm{r}_{\mathrm{o,N15}}]\mathrm{i}_{\mathrm{test}}\] (10b) \[\frac{\mathrm{v}_{\mathrm{test}}}{\mathrm{i}_{\mathrm{test}}} \approx\frac{\mathrm{g}_{\mathrm{mN}}}{\mathrm{g}_{\mathrm{mP}}} \mathrm{r}_{\mathrm{o,N15}}\] (10c) \[\approx\mathrm{r}_{\mathrm{o,N15}} \tag{10d}\]  
Figure 5: Small-signal impedance of the floating current source at node X [19]. The resistance at node X is approximately \(\mathrm{r_{o,P11}}\parallel\mathrm{r_{o,N15}}\) and a large gate capacitance \(\mathrm{C_{g,P17}}\) (\(\sim\)100 pF) is also present. Therefore, a dominant pole appears at node X, as shown in (11). Similarly, the pole at node Y is given in (12). The dominant pole of the system is determined by the lower frequency between the poles at nodes X and Y.  
\[\omega_{\mathrm{p,X}}\approx\frac{\mathrm{(r_{o,P11}}\parallel\mathrm{r_{o,N15 }})\cdot\mathrm{C_{g,P17}}}{\mathrm{(r_{o,P11}}\parallel\mathrm{r_{o,N15}}) \cdot\mathrm{C_{g,N21}}} \tag{11}\]  
\[\omega_{\mathrm{p,Y}}\approx\frac{\mathrm{1}}{\mathrm{(r_{o,P11}}\parallel \mathrm{r_{o,N15}})\cdot\mathrm{C_{g,N21}}} \tag{12}\]  
A second pole appears at the output.