1.1 Loop Gain Modeling
Deeper analysis often necessitates circuit modeling, to provide greater insight. However, conventional modeling methods are typically designed for converters operating at fixed frequencies. This poses a challenge for ripple-based variable-frequency converters like those employing COT or ACOT control, which dynamically adjust their frequency in response to load and input voltage variations. However, for the modeling and analysis purpose, we make some practical assumptions to develop an ACOT and ripple-injection circuit model as illustrated in Figure 1-1 and the following derivations.
| ZL = Impedance of Inductor | AV = Combined Gain of Modulator and MOSFET Power Stage |
| ZB = Impedance of RINJ and CINJ | VSW = Voltage at SW (switching node), FSW = Switching Frequency |
| ZF = Impedance of CFF and R1 | VFB = Voltage at Feedback |
| ZC = Impedance of Output Capacitor | VO = Output Voltage |
| ZLoad = Impedance of load | VREF = Reference to the Error Comparator |
We define AV as the combination of the modulator gain and MOSFET power stage gain for analysis purposes. The feedback circuit gain can be expressed as:
From Figure 1-1, the LC filter stage gain can be expressed as:
From Equation 1-1, Equation 1-2 and Figure 1-1, the Overall Loop Gain (G) can be expressed as:
Next, we aim to derive an expression for VFB/VO in relation to the impedances of the ripple injection circuit, allowing us to establish a correlation between the ripple injection components and loop gain. The detailed mathematical derivation for VFB/VO is provided in Appendix 1: Derivation of ACOT Loop Gain. In summary, the Overall Loop Gain (G) can be expressed as:
As stated in Figure 1-1, ZB is the impedance of RINJ and CINJ, which can be expressed as:
And ZF is the impedance of R1 and CFF, which can be expressed as:
From Equation 1-5, it is evident that the factor of ZB/ZF, that is, the ratio of the impedances of RC ripple injection and feed-forward circuits, plays a crucial role in enhancing the loop gain. From the basics of control theory, it is understood that the higher loop gains improve overall regulation (both DC set point accuracy and transient performance).