1.4 The Crossover Frequency

The crossover frequency (F_C) is one of the most critical parameters in control loop design, as it defines the effective bandwidth of the converter and directly determines the trade-off between transient response and stability. A higher crossover frequency generally enables faster transient response, while a lower crossover frequency improves stability margin. Therefore, accurate estimation and proper placement of F_C is essential in achieving an optimal balance between dynamic performance and robust operation in ACOT-controlled converters.

The determination of F_C can be understood from the loop gain profile shown in Figure 1-3. Starting from the low-frequency gain A₀, the loop exhibits a –20 dB/decade slope, which is introduced by the dominant low-frequency pole associated with the ripple injection network. At the first zero frequency (F_Z1), this slope is canceled, resulting in a flat gain region (0 dB/decade) between F_Z1 and the LC double pole frequency (F_P1 = F_P2).

Beyond this point, the output LC filter introduces a –40 dB/decade slope, causing a rapid reduction in loop gain. The second zero (F_Z2), introduced by the feed-forward capacitor, provides phase boost and changes the slope to –20 dB/decade. The crossover frequency F_C is defined as the point where this overall gain curve intersects the 0 dB line. Thus, F_C is determined by the cumulative gain changes across these frequency regions.

To derive F_C, the logarithmic gain relationship is used:

${\mathrm{Gain}}_2={\mathrm{Gain}}_1+\left(\mathrm{Slope}\right)\cdot \log \left(\frac{{\mathrm{F}}_2}{{\mathrm{F}}_1}\right)$

Substituting backward and simplifying these relationships leads to an expression for crossover frequency in terms of system parameters after rearranging and substituting the pole and zero locations:

${\mathrm{F}}_{\mathrm{Z}1}=\frac{1}{2{\mathrm{\pi R}}_{\mathrm{INJ}}{\mathrm{C}}_{\mathrm{INJ}}}, {\mathrm{F}}_{\mathrm{Z}2}=\frac{1}{2\mathrm{\pi R}1{\mathrm{C}}_{\mathrm{FF}}}, {\mathrm{F}}_{\mathrm{P}}=\frac{1}{2\mathrm{\pi }\sqrt{{\mathrm{LC}}_{\mathrm{OUT}}}}$

A detailed derivation, including application of logarithmic gain relationships and substitution of pole-zero locations, is provided in Appendix 3: Derivation of Crossover Frequency. After performing these steps and simplifying the expressions, the crossover frequency can be expressed as:

${\mathrm{F}}_{\mathrm{C}}=\frac{{\mathrm{R}}_{\mathrm{INJ}}\cdot {\mathrm{C}}_{\mathrm{FF}}}{2{\mathrm{\pi LC}}_{\mathrm{OUT}}}$

Further, using the ripple injection relationship:

$∆{\mathrm{V}}_{\mathrm{FB}}=\frac{{\mathrm{V}}_{\mathrm{IN}}·\mathrm{D}·\left(1-\mathrm{D}\right)}{{\mathrm{R}}_{\mathrm{INJ}}·{\mathrm{C}}_{\mathrm{FF}}·{\mathrm{F}}_{\mathrm{SW}}}$

and rearranging and substituting into Equation 1-19, the crossover frequency can be rewritten as:

${\mathrm{F}}_{\mathrm{C}}=\frac{{\mathrm{V}}_{\mathrm{O}}·(1-\mathrm{D})}{2{\mathrm{\pi LC}}_{\mathrm{OUT}}·∆{\mathrm{V}}_{\mathrm{FB}}·{\mathrm{F}}_{\mathrm{SW}}}$