## 2 Technology and Methods
### The Switching Converter and the Hysteresis Comparator  
The hysteresis comparator plays an essential role in achieving high efficiency as shown in Figure 8. Hysteresis control has the advantages of fast response speed, no need for an additional stable compensation circuit, simple implementation, and a small area. By detecting the output current between the linear amplifier and the switching converter, the linear amplifier is indirectly controlled by the hysteresis comparator to generate additional output current to offset the amplification error current that is generated by the switching converter.  
When the detection current is too small, the switching frequency will become higher, which results in a larger switching loss. When the detection current is too large, the switching frequency will become smaller, which results in more current being provided by the linear stage and worse linearity. Therefore, the circuit parameters (\(R_{sense}\), \(L\), \(h\)) need to be optimized to achieve a circuit design of high efficiency and high linearity.  
Figure 8: The control logic of the hysteresis comparator.  
Figure 7: Circuit diagram of the linear amplifier.  
current sensing resistor, \(L\) is a filter inductance, \(h\) is the hysteresis width, and, as shown in Equations (6)-(11), the switching converter \(f_{sw}\) is related to \(R_{sense}\), \(L\), \(h\), \(V_{dd}\), and \(V_{out}\).  
\[\Delta i_{SW}=-\Delta i_{linear} \tag{6}\]  
\[\Delta i_{SW}=k_{1}T_{1}=\frac{V_{dd}-V_{out}}{L}T_{1} \tag{7}\]  
\[\Delta i_{linear}=\frac{V_{sense}}{R_{sense}} \tag{8}\]  
\[T_{1}=\frac{2V}{R_{sense}(V_{dd}-V_{out})}L \tag{9}\]  
\[T_{2}=\frac{2h}{R_{sense}V_{out}} \tag{10}\]  
\[f_{SW}=\frac{R_{sense}(V_{dd}-V_{out})}{2V_{dd}HL} \tag{11}\]  
The hysteresis comparator includes an input stage amplifier, pre-judgment, and an output stage. The prediction gain is improved by the cross-connection of the positive feedback transistors, M40, and M41. Assuming that \(i_{p}\) is much greater than \(i_{n}\), M39 and M41 are turned on and M40 and M42 are turned off, as shown in Figure 9. The width-to-length ratio of M39 and M42 is \(\alpha\); while the width-to-length ratio of M40 and M41 is \(\beta\). The current of M41 is the mirror of the current of M39. As shown in Equation (13), wherein gm is the trans-conductance of the transistor M34, when \(\alpha\) is not equal to \(\beta\), the comparator functions as a hysteresis comparator. The upper corner voltage of the comparator is \(V_{h}\), the lower corner voltage is \(V_{l}\), and \(I_{b}\) is the current of M36 [20].  
\[i_{p}=\frac{\alpha}{\beta}i_{n} \tag{12}\]  
\[V_{h}=\frac{I_{b}}{g_{m}}\frac{\frac{\beta}{\alpha}-1}{\frac{\beta}{\alpha}+ 1},\text{ when }\beta\geq\alpha \tag{13}\]  
\[V_{h}=-V_{1} \tag{14}\]  
The efficiency of the power supply modulator also depends on the application. For broadband applications, the hysteresis width should be minimized--although this is the opposite for narrowband applications in some cases because a small hysteresis bandwidth will cause a higher switching frequency and more losses [39]. How to improve the efficiency of power supply modulators with different input signals has become a focus of research. In  
Figure 9: Circuit diagram of the hysteresis comparator.  
order to solve this problem, some researchers have changed the controllers or the number of switching converters. In this paper, optimizing the supply voltage of the switching converters improves efficiency. Optimizing the supply voltage in steps of \(\Delta\)V (V\({}_{1}-V_{\text{n}}\)) indirectly increases the switching frequency [15], as shown in Figure 10. In Figure 10, the reference voltage VDD2 min is defined as the average value of 2*vip, ti (i = 1, 2,...) is the time period of i, and Wt (t = 1, 2,...) is a series of subsections of the output of the linear amplifier.