Charge pump calculator: inverter and voltage doubler

How a charge pump makes −VIN or 2 VIN from two capacitors

A charge pump moves charge with capacitors and switches instead of an inductor. TI's LM2662/LM2663 datasheet describes its own four switches in the plainest terms (p. 10): "When S1 and S3 are closed, C1 charges to the supply voltage V+. During this time interval switches S2 and S4 are open. In the second time interval, S1 and S3 are open and S2 and S4 are closed, C1 is charging C2. After a number of cycles, the voltage across C2 will be pumped to V+. Since the anode of C2 is connected to ground, the output at the cathode of C2 equals −(V+)". The capacitor that moves, C1, is the pumping or flying capacitor; C2 holds the output up between transfers. The figure above draws both halves of the cycle.

The same four switches make a positive doubler when the pins are reconnected: the input goes on the GND pin, the output comes off V+, and OUT and LV are grounded (Figure 27). C1 charges to V_INin one half and is then stacked on top of V_IN in the other, so the output rises towards twice the input. The datasheet says the doubler's "unloaded output voltage is twice of the input voltage and is not reduced by the diode D1's forward drop" (p. 16): the Schottky diode in that circuit "is only needed for start-up".

Neither output is regulated. The datasheet's model for the inverter is "an ideal voltage source in series with a resistor. The voltage source equals −(V+)" (p. 12), and this page treats the doubler the same way, with 2 V_IN as the source. Everything this calculator reports follows from that resistance, R_out: "The output voltage drop is the load current times the output resistance" (p. 17), the loss in it is I_L²R_out, and the load line in the figure falls from the no-load voltage with R_out as its slope.

The output resistance equation and what sets it

The datasheet's Equation 1 (p. 12) breaks R_out into four terms, "A good approximation": twice R_SW, "the sum of the ON resistance of the internal MOS switches"; 2/(f_osc × C1), the part that falls with frequency and capacitance; four times the ESR of C1; and once the ESR of C2. The factor of four is explained on the same page: "Since the switching current charging and discharging C1 is approximately twice as the output current, the effect of the ESR of the pumping capacitor C1 is multiplied by four in the output resistance. The output capacitor C2 is charging and discharging at a current approximately equal to the output current, therefore, its ESR only counts once in the output resistance."

Two of the four terms are fixed by the parts, and the third by the choice of capacitor type. Only 2/(f_osc × C1) responds to more capacitance or a faster clock, and it responds with diminishing returns. The datasheet: "Instead of increasing the capacitance, the oscillator frequency can be increased to reduce the 2/(fosc × C1) term. Once this term is trivial compared with RSW and ESRs, further increasing in oscillator frequency and capacitance will become ineffective." The calculator reports each term separately so that you can see which one you are paying for, and says so when the capacitor term has become trivial.

The datasheet does not give R_SW as a number. It does give every other term of Equation 1 in Figure 26 (p. 15), four curves of output resistance against oscillator frequency, each for a stated pair of capacitors and their ESR. Taking the capacitor terms of Equation 1 off each curve leaves 2R_SW between about 2.1 and 2.5 Ω at every frequency on every curve; the mean is the calculator's default, R_SW = 1.15 Ω. The figure also confirms the ESR terms on their own: its two 10 µF curves differ only in ESR, 0.025 Ω against 0.4 Ω, and Equation 1 puts them 1.875 Ω apart at every frequency. Read off the figure, the gap is between 1.76 and 1.95 Ω.

Output resistance against frequency and capacitor: a table

Equation 1 with R_SW = 1.15 Ω, for Figure 26's four capacitor pairs (C1 = C2, each with the ESR shown), across the range the LM2662 can be set to. The last column is the floor 2R_SW + 4 ESR_C1 + ESR_C2 that no capacitance or frequency goes below.

The 47 µF row shows the datasheet's own remark on Figure 26 at work: "Once the frequency is increased to some point (such as 100 kHz for the 47-μF capacitors), the output resistance is dominated by the ON resistance of the internal switches and the ESRs of the external capacitors." At 100 kHz the capacitor term is 0.426 Ω of3.23 Ω. The 10 µF ceramic row is the other remark: "in higher frequency range, the output resistance using the 10-μF ceramic capacitors is close to these using higher value tantalum capacitors", because its ESR is a quarter of the 47 µF tantalum's.

Against Figure 26 itself, read at four frequencies, with Equation 1's value after each reading:

The largest difference in the table is 0.19 Ω. The readings were taken from the figure's bitmap in the PDF, locating the gridlines by pixel and the curve by the centre of its stroke, which is about 0.16 Ω thick; the page's tests hold Equation 1 to all 36 readings within 0.3 Ω.

Output ripple

The datasheet's Equation 2 (p. 13) gives the peak-to-peak ripple as "determined by the oscillator frequency, and the capacitance and ESR of the output capacitor C2": I_L/(f_osc × C2) + 2 × I_L × ESR_C2. The first term is the charge the load takes from C2 between top-ups; the second is the step the output takes across C2's ESR as its current reverses. "Again, using a low ESR capacitor will result in lower ripple." At 200 mA:

The 10 µF tantalum makes the point: at 150 kHz its 0.4 Ω of ESR contributes 160 mV of the 293 mV, and no amount of extra capacitance would touch that part. The datasheet notes that "A low value, smaller size capacitor usually has a higher ESR compared with a bigger size capacitor of the same type. Ceramic capacitors can be chosen for their lower ESR." The ripple here is the datasheet's switching ripple; any spikes from the switching edges depend on layout, which the datasheet's layout section addresses with placement rather than a number.

Efficiency: the loss in Rout and the IC's own supply current

The datasheet's Equation 6 (p. 17) is the efficiency as output power over output power plus two losses: I_L²R_out, "the conversion loss associated with the switch on-resistance, the two external capacitors and their ESRs", and I_Q(V+), "the quiescent power loss of the IC device". The first grows with load, the second does not, so a charge pump is least efficient at light load and at heavy load and best in between. The table uses the defaults (5 V, 47 µF, 0.1 Ω), at each FC setting with its own typical I_Qfrom the table, 1.3 mA at 150 kHz and 0.3 mA at 20 kHz.

At 200 mA the faster clock wins, because it lowers R_out; at a few milliamps the slower one wins, because the oscillator's own current is then most of the loss. The datasheet says as much when it introduces the FC pin (p. 10): "A higher oscillator frequency allows smaller capacitors to be used for equivalent output resistance and ripple, but increases the typical supply current from 0.3 mA to 1.3 mA." Its Figure 25 plots efficiency against oscillator frequency at three loads, and each curve has a peak: the lighter the load, the lower the frequency the peak is at.

Worked example: the datasheet's test conditions

The electrical characteristics table (p. 5) is measured with "V+ = 5 V, FC = Open, C1 = C2 = 47 μF", and its footnote adds that "capacitors C1 and C2 are 47-μF, 0.2-Ω maximum ESR capacitors". It prints an output resistance of 3.5 Ω typical and 7 Ω maximum at I_L = 200 mA, and a power efficiency of 86 % typical at 200 mA. The calculator's defaults are those conditions with FC tied to V+ (150 kHz) and the 0.1 Ω ESR Figure 26 gives for 47 µF tantalums:

switches  2R_SW = 2 × 1.15 Ω                       = 2.30 Ω
pump      2/(150 kHz × 47 µF)                      = 0.284 Ω
ESR       4 × 0.1 Ω + 0.1 Ω                        = 0.50 Ω
Rout                                               = 3.084 Ω
VOUT      −(5 V − 0.2 A × 3.084 Ω)                 = −4.383 V
ripple    0.2/(150 kHz × 47 µF) + 2 × 0.2 × 0.1    = 68.4 mV
losses    0.2² × 3.084 Ω + 1.3 mA × 5 V            = 123 mW + 6.50 mW
η         877 mW / 1.01 W                          = 87.1 %

Equation 1 gives 3.08 Ω, below the table's 3.5 Ω typical and well inside its 7 Ω maximum, and close to Figure 26's own reading for this curve at 150 kHz, 3.06 Ω. Put the table's 3.5 Ω into Equation 6 with the 0.3 mA supply current of its FC-open column and the efficiency is 85.9 %, the table's 86 %.

The table's header says FC = open, which is 20 kHz, and there the arithmetic does not close. At 20 kHz the capacitor term is 2.128 Ω, Equation 1 gives 4.93 Ω with 0.1 Ω capacitors and 5.43 Ω with the footnote's 0.2 Ω maximum, and Figure 26 reads about 4.7 Ω at 20 kHz for 47 µF. The table's 3.5 Ω and 86 % sit much closer to the 150 kHz result. The datasheet does not say which frequency those two rows were measured at beyond the header, so the calculator shows both: switch the oscillator to 20 kHz and it reports 4.93 Ω, −4.01 V and 80.2 %.

The table also gives the oscillator's minimum, 7 kHz with FC open and 55 kHz with FC tied to V+, and R_out depends on it directly. For the same 47 µF parts, Equation 1 gives 3.57 Ω at 55 kHz and 8.88 Ω at 7 kHz, which is why the calculator reports R_out at the table's minimum beside the typical figure.

Worked example: a 5 V rail from 3.3 V with the doubler

The datasheet's Figure 23 (p. 15) is "Generating +5 V From +3.3 V Input Voltage": the LM2662 as a doubler with C1 = C2 = 47 µF, followed by an LP2986 low-dropout regulator set to 5 V at 200 mA, which the text describes as "+5 V output from an input as low as +3.3 V". The regulator needs its input above 5 V by its dropout. From 3.3 V at 200 mA:

• At 150 kHz: R_out = 3.08 Ω, the doubler's output is 5.98 V, 0.98 V above 5 V.
• At 20 kHz: R_out = 4.93 Ω, the output is 5.61 V, 0.61 V above 5 V.

Whether that is enough depends on the regulator's dropout at 200 mA, which is in the LP2986's datasheet rather than this one; theLDO dropout calculator takes it from there. Two things move the margin the wrong way. The doubler draws its input current at about twice the load current, 402 mA here, from the 3.3 V rail. And R_SW is higher at 3.3 V than at 5 V: the datasheet's Figure 3 draws the output resistance rising from about 3 Ω at 5 V to about 4.5 Ω at 1.5 V. Enter a larger R_SW to see what that costs.

Sizing the capacitors for a target

Solved for C1, Equation 1 gives the pumping capacitor for a target output resistance, and solved for C2, Equation 2 gives the output capacitor for a target ripple. Both have a floor. For 47 µF-class tantalums of 0.1 Ω, R_out cannot go below 2.80 Ω however large C1 is, and the calculator says so rather than returning an absurd capacitance.

For R_out = 4 Ω at 20 kHz with 0.1 Ω parts, C1 must be 83.3 µF; the next E12 value, 100 µF, gives 3.80 Ω. For 50 mV of ripple at 200 mA and 150 kHz with a 0.025 Ω ceramic, C2 must be 33.3 µF; 39 µF gives 44.2 mV. The 10 µF ceramic of Figure 26 at 150 kHz gives R_out = 3.76 Ω against the 47 µF tantalum's 3.08 Ω, which is often the better trade on a small board.

Past 200 mA, or for a lower R_out than one device can reach, the datasheet parallels devices (Equation 3, p. 13): "Each device must have its own pumping capacitor C1, while only one output capacitor Cout is needed", and the output resistance is one device's divided by their number. Two of the default devices give 1.54 Ω. For a larger output voltage it cascades them, with the warning that "the number of n is practically limited since the increasing of n significantly reduces the efficiency and increases the output resistance and output voltage ripple."

Where the model stops being valid

R_SW is not a constant. Equation 1 treats the switches as a fixed resistance, but the datasheet's own curves show it moving. Figure 3 draws the output resistance against supply voltage, higher at low V+; Figures 4 and 5 draw it against temperature, and Figure 5, with 47 µF electrolytic capacitors, climbs from about 3.5 Ω at room temperature past 9 Ω at −40 °C, where Figure 4, the same measurement with OS-CON capacitors, stays below 3 Ω at 5 V. The default R_SW is fitted at the conditions of Figure 26; for a cold or low-voltage design, enter a larger one.

Equation 1 and Equation 2 are written for the inverter. The doubler uses the same switches and capacitors, and the calculator applies the same two equations to it. The datasheet does not state that they hold for the doubler; its doubler section gives the no-load output and the start-up diode, and nothing more on R_out.

Load current. The table specifies the output at 200 mA, and 250 mA is the absolute-maximum continuous output current. The output may be shorted, within limits: "OUT may be shorted to GND for one second without damage. However, shorting OUT to V+ may damage the device and should be avoided. Also, for temperatures above 85°C, OUT must not be shorted to GND or V+, or device may be damaged." (p. 4)

Frequency. An external clock on OSC works in the inverter only and is "limited to 150 kHz"; in the doubler "OSC cannot be driven by an external clock" (p. 10), and an external capacitor on OSC only lowers the frequency.

Supply range. The inverter runs from 1.5 to 5.5 V, but "For a supply voltage less than 3.5 V, the LV pin must be connected to ground to bypass the internal regulator circuitry" (p. 12). The doubler takes 2.5 to 5.5 V, and needs its start-up diode: "Voltage across V+ and LV must be larger than 1.5 V to insure the operation of the oscillator. During start-up, D1 is used to charge up the voltage at V+ pin to start the oscillator; also, it protects the device from turning-on its own parasitic diode and potentially latching-up." (p. 16)

Heat. The SOIC package is 170 °C/W junction to ambient, and the absolute-maximum dissipation of 735 mW at 25 °C is (T_Jmax − T_A)/R_θJA = 735 mW. The calculator's junction estimate puts every watt of loss in the package, including the part dissipated in the capacitors' ESR, so it is an upper bound; at the defaults it is 47 °C.

Common charge pump mistakes

• Using the switching frequency in Equation 1 or Equation 2. Both are written in the oscillator frequency, and "The output switches operate at one half of the oscillator frequency, ƒOSC = 2ƒSW" (p. 5). At 150 kHz with 47 µF, putting 75 kHz into Equation 1 doubles the capacitor term to 0.567 Ω, and Equation 2 reports 96.7 mV of ripple instead of 68.4 mV.
• Counting C1's ESR once. It counts four times; a 0.4 Ω tantalum as C1 adds 1.6 Ω to R_out before the switches are counted.
• Treating the output as regulated. It is a source behind R_out: −5.0 V with no load and −4.38 V at 200 mA in the default. Where the rail must hold its value, the datasheet adds an LDO after it (Figures 22 and 23).
• Buying capacitance to fix ESR. Once the capacitor term is small, more capacitance or a faster clock "will become ineffective"; a lower ESR is what moves R_out and ripple then.
• Running the fast oscillator at light load. At 2 mA the default circuit is 60.5 % efficient at 150 kHz and 86.8 % at 20 kHz.
• Leaving out the doubler's diode, or leaving LV open below 3.5 V in the inverter. Both are in the datasheet's circuit for a reason quoted above.

Further reading

• TI LM2662/LM2663 Switched Capacitor Voltage Converter datasheet (SNVS002E) — the switch sequence (p. 10), Equation 1 and the inverter (p. 12), ripple, paralleling and cascading (p. 13), Figure 26 (p. 15), the doubler (p. 16) and efficiency (p. 17).
• LDO dropout calculator — the headroom a regulator after the charge pump needs.
• LDO thermal calculator — junction temperature from dissipation and θJA, the same arithmetic as the heat estimate here.
• Boost converter calculator and buck-boost converter calculator — the inductor-based alternatives, for currents or ratios a charge pump cannot reach.
• Smoothing capacitor calculator — ripple on a capacitor that holds a rail up between top-ups, for a rectifier.
• E-series calculator — the preferred values the sized capacitors are rounded to.
• Op-amp gain calculator — the datasheet lists op-amp power supplies among the part's applications; a negative rail from −V_IN is the usual reason to fit one.