7 Appendix 1: Derivation of ACOT Loop Gain

There are two components contributing to the overall feedback: V_FB1 and V_FB2. V_FB1 occurs when V_SW is high (almost equal to V_IN), while V_FB2 contributes when V_SW is essentially grounded.

So,

Equation 7-1.

V_{FB} = V_{FB 1} + V_{FB 2}

Referring to Figure 1-1, V_FB1 and V_FB2 can be expressed as:

Equation 7-2.

V_{FB 1} = V_{SW} \cdot \frac{R 2 ∥ ZF}{ZB + R 2 ∥ ZF} = V_{SW} \cdot \frac{R 2 \cdot ZF}{(ZB + ZF) \cdot R 2 + (ZB \cdot ZF)}

Equation 7-3.

V_{FB 2} = V_{O} \cdot \frac{R 2 ∥ ZB}{ZF + R 2 ∥ ZB} = V_{O} \cdot \frac{R 2 \cdot ZB}{(ZB + ZF) \cdot R 2 + (ZB \cdot ZF)}

And from Figure 1-1, V_SW can be expressed as:

Equation 7-4.

V_{SW} = - AV \cdot V_{FB}

Using Equation 7-2, Equation 7-3 and Equation 7-4, V_FB can be expressed as:

Equation 7-5.

V_{FB} = V_{FB 1} + V_{FB 2} = [(- AV \cdot V_{FB}) \cdot \frac{R 2 \cdot ZF}{(ZB + ZF) \cdot R 2 + (ZB \cdot ZF)}] + (V_{O} \cdot \frac{R 2 \cdot ZB}{(ZB + ZF) \cdot R 2 + (ZB \cdot ZF)})

Rearranging Equation 7-5, we can obtain V_FB/V_O as in below:

Equation 7-6.

\frac{V_{FB}}{V_{O}} = \frac{1}{1 + \frac{ZF}{R 2} + (1 + AV) \cdot \frac{ZF}{ZB}}

Now placing Equation 7-6 in loop gain expression (Equation 1-4), we obtain:

Equation 7-7.

G = \frac{- 1}{(1 + \frac{ZF}{R 2} + \frac{ZF}{ZB}) \cdot \frac{1}{AV} + \frac{ZF}{ZB}} \cdot \frac{ZC ∥ ZLOAD}{ZC ∥ ZLOAD + ZL}

By design and from test results, it is known that Av acts as a high gain amplifier at low frequencies. With this practical assumption (

\frac{1}{A V}

is almost equal to zero), we could rewrite the expression for Loop Gain (G) as:

Equation 7-8.

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