1.2 Poles and Zeros

Understanding the poles and zeros of the ACOT control loop is essential for developing an effective control-loop design strategy. While the S-domain representation provides the mathematical framework for modeling the system, poles and zeros offer intuitive insight into how the loop gain behaves across frequency. Poles introduce phase lag and reduce gain slope, whereas zeros provide phase boost and help counteract the effects of dominant poles, particularly the double pole introduced by the output LC filter.

By appropriately placing these poles and zeros, the designer can shape the loop response to achieve a desired crossover frequency, sufficient phase margin, and stable operation. In ACOT control, where ripple injection replaces conventional compensation networks, the locations of poles and zeros are directly governed by the ripple injection and feedback components. Therefore, identifying and positioning these poles and zeros correctly is a critical step in ensuring both stability and optimal transient performance of the converter.

To quantitatively identify and locate these poles and zeros, the ACOT control loop must be expressed in terms of its S-domain impedance model. By representing each circuit element using its equivalent impedance, the ripple injection network, feedback network, and power stage can be combined into a unified loop-gain expression. This formulation allows the complex switching behavior of ACOT control to be translated into a continuous frequency-domain model, from which the dominant poles, zeros, and gain characteristics can be extracted.

Based on the S-domain equivalent circuit shown in Figure 1-1, the following impedance relationships and loop-gain expressions are used to derive the standard pole-zero form of the ACOT control loop.

The basic S-domain impedance relationships used for this analysis are:

$$\begin{gathered} \mathrm{ZR}\left(\mathrm{s}\right)=\mathrm{R} \\ \mathrm{ZL}\left(\mathrm{s}\right)=\mathrm{sL} \\ \mathrm{ZC}\left(\mathrm{s}\right)=\frac{1}{\mathrm{sC}} \end{gathered}$$

For the ACOT control-loop model, the ripple-injection impedance is formed by R_INJ and C_INJ:

$\mathrm{ZB}\left(\mathrm{s}\right)={\mathrm{R}}_{\mathrm{INJ}}+\frac{1}{{\mathrm{sC}}_{\mathrm{INJ}}}$

This can be rewritten as:

$\mathrm{ZB}\left(\mathrm{s}\right)=\frac{1+{\mathrm{sR}}_{\mathrm{INJ}}{\mathrm{C}}_{\mathrm{INJ}}}{{\mathrm{sC}}_{\mathrm{INJ}}}$

The feed-forward feedback impedance is formed by R1 and C_FF:

$$\begin{gathered} \mathrm{ZF}\left(\mathrm{s}\right)=\mathrm{R}1\parallel \frac{1}{{\mathrm{sC}}_{\mathrm{FF}}} \\ \mathrm{ZF}\left(\mathrm{s}\right)=\frac{\mathrm{R}1}{1+\mathrm{sR}1{\mathrm{C}}_{\mathrm{FF}}} \end{gathered}$$

The output capacitor, inductor and load impedances are:

$$\begin{gathered} \mathrm{ZC}\left(\mathrm{s}\right)=\frac{1}{{\mathrm{sC}}_{\mathrm{OUT}}} \\ \mathrm{ZL}\left(\mathrm{s}\right)=\mathrm{sL} \\ \mathrm{ZLOAD}\left(\mathrm{s}\right)={\mathrm{R}}_{\mathrm{LOAD}} \end{gathered}$$

Recalling Equation 1-5, ACOT loop gain can be expressed as:

$\mathrm{G}\left(\mathrm{s}\right)\cong -\frac{\mathrm{ZB}}{\mathrm{ZF}}·\frac{\mathrm{ZC}\parallel \mathrm{ZLOAD}}{\mathrm{ZC}\parallel \mathrm{ZLOAD}+\mathrm{ZL}}$

The transformation of G(s) from this impedance-based expression into the final pole-zero form is provided in Appendix 2: Step-by-Step Derivation of ACOT Loop Gain in S-Domain. This final form allows the dominant poles and zeros of the ACOT control loop to be identified and used for loop design.

$\mathrm{G}\left(\mathrm{s}\right)\cong \frac{{\displaystyle \frac{1+{\mathrm{sR}}_{\mathrm{INJ}}{\mathrm{C}}_{\mathrm{INJ}}}{{\mathrm{sC}}_{\mathrm{INJ}}}}}{{\displaystyle \frac{\mathrm{R}1}{1+\mathrm{sR}1{\mathrm{C}}_{\mathrm{FF}}}}}·\frac{{\displaystyle \frac{{\mathrm{R}}_{\mathrm{LOAD}}}{1+{\mathrm{sR}}_{\mathrm{LOAD}}{\mathrm{C}}_{\mathrm{OUT}}}}}{{\displaystyle \frac{{\mathrm{R}}_{\mathrm{LOAD}}}{1+{\mathrm{sR}}_{\mathrm{LOAD}}{\mathrm{C}}_{\mathrm{OUT}}}+\mathrm{sL}}}$

This expression can be arranged into standard pole-zero form:

$\mathrm{G}\left(\mathrm{s}\right)=\frac{\mathrm{K}\left(1+{\displaystyle \frac{\mathrm{s}}{{\mathrm{\omega }}_{\mathrm{Z}1}}}\right)\left(1+{\displaystyle \frac{\mathrm{s}}{{\mathrm{\omega }}_{\mathrm{Z}2}}}\right)}{\mathrm{s}\left[1+{\displaystyle \frac{\mathrm{s}}{{\mathrm{Q\omega }}_{\mathrm{P}}}}+{\left({\displaystyle \frac{\mathrm{s}}{{\mathrm{\omega }}_{\mathrm{P}}}}\right)}^2\right]}$

From this form, the gain constant, poles and zeros can be identified as follows:

$$\begin{gathered} \mathrm{K}=\frac{1}{\mathrm{R}1{\mathrm{C}}_{\mathrm{INJ}}} \\ {\mathrm{\omega }}_{\mathrm{P}}=\frac{1}{2\mathrm{\pi }\sqrt{{\mathrm{LC}}_{\mathrm{OUT}}}} \\ {\mathrm{\omega }}_{\mathrm{Z}1}=\frac{1}{{\mathrm{R}}_{\mathrm{INJ}}{\mathrm{C}}_{\mathrm{INJ}}} \\ {\mathrm{\omega }}_{\mathrm{Z}2}=\frac{1}{\mathrm{R}1{\mathrm{C}}_{\mathrm{FF}}} \end{gathered}$$