Calculating Essential Charge-Pump Parameters

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

Interval Two

Next, we evaluate what happens when the switch flips to position 2. It goes to position 2 at t=DT, where D is the duty cycle of the switching frequency and T is its period. Since we evaluate this interval separately, we must set the appropriate initial conditions for the two capacitors. We can call these initial conditions V3 and V4, but they are actually the final states of the capacitors at the end of interval one, and therefore can be derived from the expressions obtained in Eqs. 1 and 2. These conditions are shown in the schematic of Fig. 3.

Before analyzing the circuit in Fig. 3, we need the initial conditions V3 and V4. They can be obtained by recognizing that V3 is the voltage across C1 at time t = DT, and V4 is the voltage across C2 at t=DT. Therefore, we write the following expressions:

Having obtained the initial conditions for interval two, we can find an expression for voltages across the flying and output capacitors as we did for the first interval. To preserve clarity, we refer to the initial conditions in this interval as V3 and V4, even though they are actually functions of the initial conditions at interval one. Solving for the voltage across output capacitor C2 (V^II_C2), we obtain the following equation (see derivations 1, 2 and 3 online):

where s₁, s₂, A′, B′, C′, and D′are constants that depend on the values of C1, C2, R1, and R2. Because the equations for these constants are rather complex and require a lot of space, they are presented in an appendix at the end of the online version of this article. (See Eqs. A1 through A6 in the appendix.)

The voltage across the flying capacitor during interval two is expressed as a function of the output voltage during that period, and simplified as follows (see derivations 4 and 5):

As for Eq. 5, the terms N₂ and N₄ are constants that depend on the values of C1, C2, R1 and R2. Because of the complexity of these equations, they are presented in the online appendix. (See Eqs. A7, A8, A9 and A10 in the appendix.)

Unifying Initial and Final Conditions

So far, we have equations that describe the voltages across the two capacitors in each of the intervals. Next, we need to equate the initial and final conditions in a way that forces the equations to represent steady-state operation.

To discuss this point further, we first examine the voltage across the output capacitor, which is the output voltage of the charge pump. During interval one, the output capacitor is connected to the load by itself, and its voltage drops from the initial condition V2. During that interval, the flying capacitor charges to a voltage higher than its initial condition V1. During the second interval, the flying capacitor connects to the output capacitor through R1, so the output voltage begins to increase.

For the system to be in equilibrium, note that the average output voltage has to be constant. The only way to achieve that condition is for the output voltage at the end of interval two to equal its initial voltage at the beginning of interval one. The same consideration applies to the conditions for the flying capacitor, because the average charge supplied to the output must be constant. We can write these two conditions in the following way:

Since is a function of V1 and V2, and is also a function of V1 and V2, we can form a system of two equations in two unknowns (V1 and V2), and solve for the equilibrium (steady-state) initial conditions for interval one. When we obtain V1 and V2, we can solve for every other parameter of interest, including the average output voltage, efficiency and output ripple, as a function of capacitor size, duty cycle, switching frequency, series resistance and load.

We now write and solve the system of equations. Eqs. 10 and 14 reduce to:

Plugging Eqs. 3 and 4 into Eqs. 15 and 16, and rearranging the terms slightly, we obtain the final system of two equations in the two unknowns V1 and V2:

To solve these equations, it is convenient to reduce their size by adopting a short notation for some of the constant terms.

We must also bear in mind that a final numerical solution is feasible only with a computer, which further justifies the change in notation. Making substitutions for all constant terms except V1 and V2, and rearranging them (see Eqs. A11 through A17 in the online appendix), we can rewrite the system of equations as follows:

We've now reduced the system of equations to a manageable form that can be solved for V1 and V2:

Eqs. 19 and 20 represent the result for which we have been working. They give the precise initial conditions for the charge-pump model in equilibrium, as a function of all of the parameters we specified at the beginning of this article: capacitor size, load, series resistance, frequency or period, and duty cycle. Knowing the values of V1 and V2, we can easily solve for other parameters of interest. Knowing V1 and V2 also allows us to plot waveforms for the capacitor voltages. Because the equations for V1 and V2 are rather complicated, we can use mathematical software such as Matlab or Excel to give precise numerical values and plot the results.

The latter part of this article presents several such plots, but first we derive some useful and important expressions. One such is the average output voltage.