Charge pumps & capacitive DC-DC converters

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charge pumps are capacitive DC-DC converters. The two most common switched capacitor voltage converters are the voltage inverter and the voltage doubler circuit

image-20241014211627207

voltage doubler

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output buffer capacitor

To achieve a stable DC output voltage

Step-Wise Ramp-Up

without load

VinCp + Vout, n − 1Co = (Vout, n−Vin)Cp + Vout, nCo

We derive a recursive equation that describes the output voltage Vout, n after the nth clock cycle $$ V_{out,n} = \frac{2V_{in}C_p + V_{out,n-1}C_o}{C_p + C_o} $$

Voltage Ripple & Droop

ripple_droop.drawio

$$\begin{align} (V_t - V_h)(C_p + C_o) &= \frac{I_{load}}{2f_{sw}} \\ (V_h - V_b)C_o &= \frac{I_{load}}{2f_{sw}} \end{align}$$

we obtain $$ V_t - V_b = \frac{I_{load}}{f_{sw}C_o}\left(1 - \frac{C_p}{2(C_p + C_o)}\right) $$ That is, peak-to-peak ripple $$ \Delta V_{out,p2p} \approx \frac{I_{load}}{f_{sw}C_o} \space\space\space\space \text{if}\space\space C_o \gg C_p $$

Then, with aforementioned Step-Wise Ramp-Up equation, $V_t = \frac{2V_{in}C_p + V_bC_o}{C_p + C_o}$ $$\begin{align} V_b &= 2V_{in} - \frac{I_{load}}{f_{sw}C_p}\left(1 + \frac{C_p}{2C_o}\right) \\ V_t &= 2V_{in} - \frac{I_{load}}{f_{sw}C_p}\left(1 - \frac{C_p}{2(C_p+C_o)}\right) \end{align}$$

Therefore, average output voltage $\overline{V}_{out}$ in steady-state is $$ \overline{V}_{out} = \frac{V_t+V_b}{2}=2V_{in} - \frac{I_{load}}{f_{sw}C_p}\left(1 + \frac{C_p^2}{4C_o(C_p+C_o)}\right) \approx 2V_{in} - \frac{I_{load}}{f_{sw}C_p} $$ which results in a simple expression for the output voltage droop

$$ \Delta V_{out} = \frac{I_{load}}{f_{sw}C_p} $$

The charge pump can be modeled as a voltage source with a source resistance Rout. Therefore, ΔVout can be seen as the voltage drop across Rout due to the load current:

$$ R_{out} = \frac{\Delta V_{out}}{I_{load}} = \frac{1}{f_{sw}C_p} $$ image-20241015072846141

multiphase CP

multiphaeCP.drawio

(Vt−Vb)(Cp+Co) = IloadΔt

Therefore peak-to-peak ripple $$ \Delta V_{out,p2p} = \frac{I_{load}\Delta t}{C_p+C_o} = \frac{I_{load}\Delta t}{C_{tot}} $$

where Ctot = Cp + Co

with $$ \left\{ \begin{array}{cl} V_b &= 2V_{in} - \frac{I_{load}\Delta t}{C_p} \\ V_t &= 2V_{in} - \frac{I_{load}\Delta t}{C_p} + \frac{I_{load}\Delta t}{C_p+C_o} \end{array} \right. $$

Then $$ \overline{V}_{out} = \frac{V_t+V_b}{2}=2V_{in} - \frac{I_{load}\Delta t}{C_p}\cdot \frac{C_p+2C_o}{2C_p+2C_o} \approx 2V_{in} - \frac{I_{load}\Delta t}{C_p} $$ That is output voltage droop $$ \Delta V_{out} = \frac{I_{load}\Delta t}{C_p} $$

reference

Bernhard Wicht, “Design of Power Management Integrated Circuits”. 2024 Wiley-IEEE Press

Breussegem, T. v., & Steyaert, M. (2013). CMOS integrated capacitive DC-DC converters. Springer

Zhang, Milin, Zhihua Wang, Jan van der Spiegel and Franco Maloberti. “Advanced Tutorial on Analog Circuit Design.” (2023).

Anton Bakker, Tim Piessens., ISSCC2014 T9: Charge Pump and Capacitive DC-DC Converter Design

Wicht, B., ISSCC2020 T2: Analog Building Blocks of DC-DC Converters [https://www.nishanchettri.com/isscc-slides/2020%20ISSCC/TUTORIALS/T2Visuals.pdf]

Hoi Lee, ISSCC2018 T8: Fundamentals of Switched-Mode Power Converter Design [slides,transcript]