Calculating Essential Charge-Pump Parameters

Jul 1, 2006 12:00 PM
By Vladimir Vitchev, Customer Service Engineer, Maxim Integrated Products, Sunnyvale, Calif.

Capacitive charge-pump circuits are used in many applications. And though these circuits appear deceptively simple, engineers working on them need a thorough understanding of how they function. By analyzing the model of a basic charge-pump circuit, it's possible to derive expressions for efficiency and output voltage as functions of the pump's duty cycle, switching frequency, output and flying capacitances, switch and other series resistances, and load.

The model derived in this article enables designers to understand and predict the behavior of charge pumps under a wide variety of conditions. Mathematical derivations are shown in the PDF version of this article to provide a full understanding of the model and to provide a general approach for analyzing other complex power-supply circuits.

Developing the charge-pump model requires some lengthy derivations of key equations. For the complete derivations in their entirety, click here. There is also a spreadsheet that automates the final expressions derived here, allowing a quick calculation of essential charge-pump parameters including output voltage and efficiency.

Charge pumps use a charge-storage element (i.e. a capacitor) to transfer charge from a source to a load. Fig. 1 shows the basic model of a charge pump. It can represent several topologies, such as regulated stepdown charge pumps, inverting regulated charge pumps and inverting unregulated charge pumps. For inverting charge pumps, the output voltage is negative, but the magnitude of that voltage is the same as that predicted by the model. Stepup or boost charge pumps operate in a very similar fashion; however, there are a number of key differences that are not taken into account in the model described in this article. Thus, the model cannot readily be applied to describe their operation.

For unregulated charge pumps, R1 represents the total resistance of the internal switches, which are usually MOSFETs. For regulated charge pumps, the resistor R1 is a variable resistance that can be implemented by varying the bias of the MOSFET switch in the on state.

This resistor regulates the charge pump's output voltage. Unlike inductive dc-dc converters, in which the output voltage is regulated according to the duty cycle of the switch, output voltage for the charge pump is regulated by changing the value of this resistance.

The switch is controlled by an oscillator that alternates between position 1 and position 2 (Fig. 1). We assume for this analysis that the circuit is operating in the steady-state mode. That is the condition used to derive an expression for output voltage as a function of all the variables.

Interval One

Because of the steady-state assumption, each capacitor will be charged to some initial voltage. We can provide those voltages by placing an initial condition on each capacitor (i.e., a voltage source connected in series with the respective capacitor), which we will call V1 and V2. The resulting circuit is shown in Fig. 2.

Having established the initial conditions, let's solve for the two capacitor voltages during the first interval (I), when the switch is connected to position 1. (For interval two [II], the switch is in position 2.) The voltage across C2 is always the output voltage of the charge pump. Solving for the voltage across C1 during interval one, i.e., V^I_C1 (t), we obtain the following expressions:

Eq. 1 gives an expression for voltage across the flying capacitor (C1) during interval one. Note that this voltage is a function not only of the element values, but also of the initial condition V1. By inspection, the output voltage V^I_C2 (t) across output capacitor C2 during interval one is:

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