8 Appendix 2: Step-by-Step Derivation of ACOT Loop Gain in S-Domain

The small-signal loop gain of the ACOT control loop can be written as:

Equation 8-1.

G ≅ - \frac{ZB}{ZF} \cdot \frac{ZC ∥ ZLOAD}{ZC ∥ ZLOAD + ZL}

Where:

ZB = ripple injection path impedance
ZF = feedback path impedance
ZC = output capacitor impedance
ZLOAD = load impedance
ZL = output inductor impedance

The ripple injection path consists of R_INJ in series with C_INJ. Therefore:

Equation 8-2.

ZB = R_{INJ} + \frac{1}{{sC}_{INJ}}

Rearranging:

Equation 8-3.

ZB = \frac{1 + {sR}_{INJ} C_{INJ}}{{sC}_{INJ}}

The feedback path impedance is formed by R1 in parallel with C_FF. Therefore:

Equation 8-4.

ZF = R 1 ∥ \frac{1}{{sC}_{FF}} ZF = \frac{R 1 \cdot \frac{1}{{sC}_{FF}}}{R 1 + \frac{1}{{sC}_{FF}}}

Simplifying:

Equation 8-5.

ZF = \frac{R 1}{1 + sR 1 C_{FF}}

Substituting ZB and ZF:

Equation 8-6.

\frac{ZB}{ZF} = \frac{\frac{1 + {sR}_{INJ} C_{INJ}}{{sC}_{INJ}}}{\frac{R 1}{1 + sR 1 C_{FF}}}

Therefore:

Equation 8-7.

\frac{ZB}{ZF} = \frac{(1 + {sR}_{INJ} C_{INJ}) (1 + sR 1 C_{FF})}{{sC}_{INJ} R 1}

The output capacitor impedance is:

Equation 8-8.

ZC = \frac{1}{{sC}_{OUT}}

The load impedance is:

Equation 8-9.

ZLOAD = R_{LOAD}

Therefore:

Equation 8-10.

ZC ∥ ZLOAD = \frac{\frac{1}{{sC}_{OUT}} \cdot R_{LOAD}}{\frac{1}{{sC}_{OUT}} + R_{LOAD}}

Simplifying:

Equation 8-11.

ZC ∥ ZLOAD = \frac{R_{LOAD}}{1 + {sR}_{LOAD} C_{OUT}}

The output inductor impedance is:

Equation 8-12.

ZL = sL

The output stage gain is:

Equation 8-13.

\frac{ZC ∥ ZLOAD}{ZC ∥ ZLOAD + ZL}

Substituting the impedance expressions:

Equation 8-14.

\frac{ZC ∥ ZLOAD}{ZC ∥ ZLOAD + ZL} = \frac{\frac{R_{LOAD}}{1 + {sR}_{LOAD} C_{OUT}}}{\frac{R_{LOAD}}{1 + {sR}_{LOAD} C_{OUT}} + sL}

Multiplying numerator and denominator by (1 + sR_LOADC_OUT):

Equation 8-15.

= \frac{R_{LOAD}}{R_{LOAD} + sL (1 + {sR}_{LOAD} C_{OUT})}

Expanding:

Equation 8-16.

= \frac{R_{LOAD}}{R_{LOAD} + sL + s^{2} {LR}_{LOAD} C_{OUT}}

Dividing numerator and denominator by R_LOAD:

Equation 8-17.

= \frac{1}{1 + \frac{sL}{R_{LOAD}} + s^{2} {LC}_{OUT}}

Starting from:

Equation 8-18.

G ≅ - \frac{ZB}{ZF} \cdot \frac{ZC ∥ ZLOAD}{ZC ∥ ZLOAD + ZL}

Substituting the derived expressions:

Equation 8-19.

G (s) ≅ - \frac{(1 + {sR}_{INJ} C_{INJ}) (1 + sR 1 C_{FF})}{{sC}_{INJ} R 1} \cdot \frac{1}{1 + \frac{sL}{R_{LOAD}} + s^{2} {LC}_{OUT}}

Therefore, the ACOT loop gain is:

Equation 8-20.

G (s) = - \frac{(1 + {sR}_{INJ} C_{INJ}) (1 + sR 1 C_{FF})}{{sC}_{INJ} R 1 (1 + \frac{sL}{R_{LOAD}} + s^{2} {LC}_{OUT})}

The negative sign indicates the feedback polarity of the control loop. For gain magnitude analysis, the loop gain magnitude is commonly considered as:

Equation 8-21.

| G (s) | = \frac{(1 + {sR}_{INJ} C_{INJ}) (1 + sR 1 C_{FF})}{{sC}_{INJ} R 1 (1 + \frac{sL}{R_{LOAD}} + s^{2} {LC}_{OUT})}