Buck-boost converter calculator (4-switch, non-inverting)

The power stage of a non-inverting 4-switch buck-boost converter, the kind that holds a rail while its input moves from above it to below it, worked the way TI's SLVA535 works it: the duty cycle in buck mode at the highest input and in boost mode at the lowest, the inductor each mode asks for and the larger of the two, the peak switch current and the load the IC's current limit can deliver in each mode, and the output capacitance for a ripple and an overshoot target. A sweep of the input range shows where each requirement peaks. It does not cover the single-switch inverting buck-boost, whose output is negative.

Fig 1 — V_IN from 2.60 V to 5.00 V into 3.30 V at 2.00 A: buck above V_OUT, D_Buck = 0.710 at 5.00 V; boost below it, D_Boost = 0.330 at 2.60 V. Top, the inductance each input asks for at K_ind = 0.3, against the 1.00 µH fitted; bottom, the peak switch current with it, worst at 3.19 A at 2.60 V against a 4.50 A limit.

Minimum input voltage V_IN,min (V)

Maximum input voltage V_IN,max (V)

Output voltage V_OUT (V)

Maximum output current I_OUT (A)

Switching frequency F_SW (kHz)

Efficiency at V_IN,max, η (buck) (%)

Efficiency at V_IN,min, η (boost) (%)

Ripple coefficient K_ind (× I_L)

Inductance fitted L (0 for the required value) (µH)

Switch current limit I_LIM (0 to skip) (A)

Output ripple target ΔV_OUT (0 to skip) (mV)

Load-release overshoot ΔV_OUT (0 to skip) (mV)

Required inductance L, the larger of Eq 3 and Eq 4: 881 nH, buck mode
Operating modes, V_IN,min to V_IN,max against V_OUT: boost at V_IN,min · buck at V_IN,max
Buck duty D_Buck at V_IN,max, η = 93 % (Eq 1): 0.710 · 0.660 ideal
Boost duty D_Boost at V_IN,min, η = 85 % (Eq 1): 0.330 · 0.212 ideal
Inductance for buck mode, at V_IN,max (Eq 3): 881 nH
Inductance for boost mode, at V_IN,min (Eq 4): 341 nH
Inductor ripple ΔI_max with 1.00 µH (Eq 6, Eq 9): buck 568 mA · boost 405 mA
Peak switch current I_SW,max at each corner (Eq 5, Eq 8): buck 2.28 A · boost 3.19 A
Inductor and switch rating: worst peak across V_IN · inductor RMS: 3.19 A at 2.60 V · 2.99 A
Load the IC can deliver with I_LIM = 4.50 A (Eq 7, Eq 10): buck 4.22 A · boost 2.88 A · covers 2.00 A
Output capacitance for 100 mV of ripple (Eq 14, Eq 17): buck 353 nF · boost 3.11 µF
Output capacitance for 100 mV of overshoot on load release (Eq 16): 545 nF
Minimum output capacitance, after DC-bias derating: 3.11 µF · from Eq 17
Input capacitance: SLVA535 gives no equation: from the IC datasheet

The inductance is set by buck mode, but the peak current by boost mode: rate the inductor and check the IC for 3.19 A at 2.60 V, not for the current at the corner that chose L. SLVA535: "Use the greater of the two switch currents".

How this is calculated

Standard: TI SLVA535B — Basic Calculations of a 4-Switch Buck-Boost Power Stage (Jan 2018, rev July 2018); TI TPS63802 datasheet (SLVSEU9D), §8.5; TI SLVA372 — Basic Calculation of a Boost Converter's Power Stage; TI SNVA559 — Switching Regulator Fundamentals

$D_{Buck} = \frac{V_{OUT}}{V_{IN\,max} \times \eta}, \qquad D_{Boost} = 1 - \frac{V_{IN\,min} \times \eta}{V_{OUT}}$: SLVA535 Equation 1: "the minimum duty cycle for buck mode and maximum duty cycle for boost mode", with η the "estimated efficiency at calculated VIN, VOUT, and IOUT". As printed, the D_Boost line has V_IN max; the text, section 4.2 and the example's 0.330 all put it at V_IN min, as here. The example's D_Buck of 0.614 does not follow from the equation, which gives 0.710; the page's worked example sets out both.
$L > \frac{V_{OUT} \times \left(V_{IN\,max} - V_{OUT}\right)}{K_{ind} \times F_{SW} \times V_{IN\,max} \times I_{OUT}}, \qquad L > \frac{V_{IN\,min}^{2} \times \left(V_{OUT} - V_{IN\,min}\right)}{F_{SW} \times K_{ind} \times I_{OUT} \times V_{OUT}^{2}}$: Equations 3 (buck) and 4 (boost). "Select the largest value of inductance calculated from either Equation 3 and Equation 4." K_ind is the ripple as a fraction of the inductor's average current, "0.2 < Kind < 0.4". The example gives 0.881 µH and 0.341 µH, and fits 1.0 µH.
$I_{SW\,max} = \frac{\Delta I_{max}}{2} + I_{OUT}, \qquad \Delta I_{max} = \frac{\left(V_{IN\,max} - V_{OUT}\right) \times D_{Buck}}{F_{SW} \times L}, \qquad I_{max\,out} = I_{LIM} - \frac{\Delta I_{max}}{2}$: Equations 5, 6 and 7, buck mode: "In buck mode, the maximum switch current is when the input voltage is at its maximum."
$I_{SW\,max} = \frac{\Delta I_{max}}{2} + \frac{I_{OUT}}{1 - D_{Boost}}, \qquad \Delta I_{max} = \frac{V_{IN\,min} \times D_{Boost}}{F_{SW} \times L}, \qquad I_{max\,out} = \left(I_{LIM} - \frac{\Delta I_{max}}{2}\right) \times \left(1 - D_{Boost}\right)$: Equations 8, 9 and 10, boost mode: "In boost mode, the maximum switch current is when the input voltage is at its minimum." These are SLVA372's boost equations, and the calculator runs them through the same code as the boost converter calculator. "Use the greater of the two switch currents."
$C_{OUT\,min1} = \frac{K_{ind} \times I_{OUT}}{8 \times F_{SW} \times V_{OUT\,ripple}}, \qquad C_{OUT\,min2} = \frac{\left(K_{ind} \times I_{OUT}\right)^{2} \times L}{2 \times V_{OUT} \times \Delta V_{OUT}}, \qquad C_{OUT\,min} = \frac{I_{OUT} \times D_{Boost}}{F_{SW} \times \Delta V_{OUT}}$: Equations 14 and 16 (buck mode, ripple and load-release overshoot) and 17 (boost mode). For a converter with external compensation; "Select output capacitance that is larger than both minimum required output capacitance for buck and boost mode operation", after derating for DC bias. The example gives 0.71 µF, 0.55 µF and 3.11 µF.
$V_{IN}^{*} = \tfrac{2}{3}\,V_{OUT}, \qquad I_{L,RMS} = \sqrt{I_L^{2} + \frac{\Delta I^{2}}{12}}$: Derived, not in SLVA535. Equation 4 is V_IN²·(V_OUT − V_IN) over constants, and its derivative is zero at 2/3·V_OUT; when that lies inside the boost range the requirement there exceeds the one at V_IN min. The RMS is that of a triangle of ripple ΔI riding on the average inductor current I_L: I_OUT in buck mode, I_OUT/(1 − D) in boost mode. Equation 1 also marks the edge of buck mode: D_Buck reaches 1 at V_OUT/η, 3.55 V in the example.

Assumptions

Continuous conduction mode, a 4-switch IC with integrated switches, and full load, as SLVA535 states in its abstract.
Buck mode is evaluated at V_IN max and boost mode at V_IN min, as in SLVA535. Inputs close to V_OUT, where Equation 1 gives no valid duty cycle, are the transition region; SLVA535 does not treat it, and neither does the calculator.
The efficiencies are estimates at the two corners, from the IC's datasheet curves. The sweep holds each corner's efficiency across its mode's half of the range; that is this page's assumption.
The inductance is the value at the operating current. SLVA535: "the peak current increases with decreasing inductance", so rate the part above the largest switch current.
Output capacitance is the derated value, at the output voltage. An internally compensated IC wants the L and C its datasheet recommends. The input capacitor comes from the datasheet; SLVA535 gives no equation for it.

What a 4-switch buck-boost converter is, and when it bucks or boosts

A buck-boost converter regulates an output that can sit above or below its input, which is exactly what a battery needs when its voltage crosses the rail it feeds: a Li-ion cell powering 3.3 V starts above the output and ends below it. The kind this calculator covers is the non-inverting, 4-switch type that TI's application note SLVA535 is written for. Its abstract sets the scope: "This application note gives the equations to calculate the power stage of a non-inverting buck-boost converter built with an IC with integrated switches and operating in continuous conduction mode." Its Figure 1 is one inductor between two switch pairs: on the input side SW1 in series and SW3 to ground, arranged like a buck converter's switch and rectifier; on the output side SW4 to ground and SW2 in series, arranged like a boost converter's. "Many of the Advanced Low Power buck-boost converters (TPS63xxx) have all four switches integrated in the IC."

That is a different circuit from the single-switch converter that shares the name. TI's SNVA559 describes that one: "The Buck-Boost or Inverting regulator takes a DC input voltage and produces a DC output voltage that is opposite in polarity to the input. The negative output voltage can be either larger or smaller in magnitude than the input voltage." It uses one switch, one inductor and one diode, and makes a negative rail. SNVA559 describes how it operates but gives no design equations for it, and no other document in this site's library does, so this page does not compute the inverting converter. Everything below is the non-inverting 4-switch converter, whose output has the same polarity as its input.

SLVA535 does not model the converter as one circuit with one duty cycle. It works the power stage twice: once as a buck at the highest input, where the buck duty cycle is smallest, and once as a boost at the lowest input, where the boost duty cycle is largest. "These duty cycles are important because at these duty cycles the converter is operating at the extremes of its operating range." Both come from its Equation 1, with the estimated efficiency η folded in, "estimated efficiency at calculated V_IN, V_OUT, and I_OUT": D_Buck = V_OUT / (V_IN,max·η) and D_Boost = 1 − V_IN,min·η / V_OUT. The calculator treats an input above V_OUT as buck mode and one below it as boost mode, and evaluates each mode where SLVA535 does.

From there every quantity comes in a pair, and the rule for combining them is the same each time. For the inductor, "Select the largest value of inductance calculated from either Equation 3 and Equation 4." For the switch current, "Derive the maximum switch current for both cases. Use the greater of the two switch currents for remainder of this application note." For the output capacitor, "Select output capacitance that is larger than both minimum required output capacitance for buck and boost mode operation." The calculator follows those three rules, and adds a sweep across the whole input range, drawn in the figure, to show where each requirement peaks.

The inductor equations hold the ripple at a fixed fraction K_indof the current the inductor carries. In buck mode that is the output current, and SLVA535's Equation 3 is the same inductor the buck inductor calculator solves. In boost mode the inductor carries the input current, I_OUT·V_OUT/V_IN, and Equation 4 is SLVA372's boost inductor estimate with the ripple taken as K_indof that, which is why the boost half of this calculator runs through the same code as the boost converter calculator. SLVA535's guidance on K_ind is the same in both sections: "A good estimation for the inductor ripple current is 20% to 40% of the output current, or 0.2 < Kind < 0.4."

Buck-boost inductor calculation across a Li-ion cell's range

The chart takes SLVA535's own example, 3.3 V at 2 A, K_ind = 0.3, 93 % efficient in buck mode and 85 % in boost, at the 2.12 MHz its numbers imply, and runs it from a single Li-ion cell instead: 4.2 V, the charge voltage the Li-ion charge calculator works to, down to 2.5 V, the level below which its sources treat a cell as deeply discharged. Each row evaluates the equations of the mode that input falls in, with the efficiency of that mode's corner. The inductor fitted is the requirement from SLVA535's rule for this range, 555 nH.

V_IN	Mode	Duty D	L this input asks for (Eq 3 or 4)	Ripple ΔI with 555 nH	Peak switch current
2.50 V	boost	0.356	360 nH	755 mA	3.48 A
2.80 V	boost	0.279	283 nH	662 mA	3.10 A
3.00 V	boost	0.227	195 nH	579 mA	2.88 A
3.20 V	boost	0.176	73.8 nH	477 mA	2.67 A
3.30 V	—	—	—	—	—
3.50 V	—	—	—	—	—
3.60 V	buck	0.986	216 nH	251 mA	2.13 A
3.90 V	buck	0.910	399 nH	463 mA	2.23 A
4.20 V	buck	0.845	555 nH	645 mA	2.32 A

Three things are worth reading off it. The inductance is set at the top of the range: Equation 3 asks 555 nH at 4.2 V against 360 nH from Equation 4 at 2.5 V, so buck mode chooses the part. The peak current is set at the bottom: 3.48 A at 2.5 V against 2.32 A at 4.2 V, because in boost mode the inductor carries the input current, larger than the output current by 1/(1 − D). And the rows next to 3.3 V are empty or nearly so. At 3.3 V the converter is neither bucking nor boosting; at 3.5 V Equation 1 asks for a buck duty cycle of 1.014, because with 93 % efficiency the input has to exceed 3.3 V / 0.93 = 3.55 V before a buck can hold the output. That band is the transition region, and SLVA535 does not treat it.

Worked example: SLVA535's TPS63802 design

SLVA535's Appendix A designs a converter with the TPS63802 for V_OUT= 3.3 V, I_OUT = 2 A, V_{IN min} = 2.6 V and V_{IN max}= 5.0 V, with efficiencies of 93 % at 5.0 V and 85 % at 2.6 V, K_ind = 0.3 and a 1.0 µH inductor. The appendix prints its results but not the switching frequency, the switch current limit or the ripple targets behind them. Each can be recovered from one printed value and then checked against the others. Solving Equation 3 for F_SW with the printed 0.881 µH gives 2.123 MHz; the calculator's default, 2123 kHz, is that to four figures, and it reproduces every other printed number. The TPS63802 datasheet corroborates it: its electrical characteristics give an "Inductor Switching Frequency, Boost Mode" of 2.1 MHz typical, at V_IN = 2.3 V, V_OUT = 3.3 V, no load, MODE = HIGH and T_J = 25 °C, and the inferred figure is 1.1 % above it. The same table gives 1.6 MHz typical in buck mode, at V_IN = 4.3 V; the example's buck-mode numbers use the same frequency as its boost-mode ones. A 4.5 A limit reproduces both deliverable currents, a 50 mV ripple target reproduces Equation 14's 0.71 µF, and 100 mV reproduces Equations 16 and 17. Those four values are inferred, not printed. The left column is the calculator; the right is what TI prints.

duty      D_Buck  = 3.3 / (5.0 × 0.93)             = 0.710      TI: 0.614
          D_Boost = 1 − 2.6 × 0.85 / 3.3           = 0.330      TI: 0.330
F_SW      Eq 3 solved for F_SW at 0.881 µH         = 2.12 MHz   TI: not printed
inductor  Eq 3 at 5.0 V, K_ind = 0.3               = 0.881 µH   TI: 0.881 µH
          Eq 4 at 2.6 V, K_ind = 0.3               = 0.341 µH   TI: 0.341 µH
          the larger, rounded up to a part: 1.0 µH fitted
buck      Eq 6: 1.7 × 0.710 / (F_SW × 1.0 µH)      = 568 mA     TI: 492 mA
          Eq 6 with D_Buck = 0.614 instead         = 492 mA    
          Eq 5: ΔI/2 + 2 A                         = 2.28 A     TI: 2.24 A
          Eq 7: 4.5 A − ΔI/2                       = 4.22 A     TI: 4.25 A
boost     Eq 9: 2.6 × 0.330 / (F_SW × 1.0 µH)      = 405 mA     TI: 405 mA
          Eq 8: ΔI/2 + 2 A / (1 − 0.330)           = 3.19 A     TI: 3.19 A
          Eq 10: (4.5 A − ΔI/2) × (1 − 0.330)      = 2.88 A     TI: 2.88 A
C_OUT     Eq 14 at a 50 mV ripple                  = 0.71 µF    TI: 0.71 µF
          Eq 16 at a 100 mV overshoot              = 0.55 µF    TI: 0.55 µF
          Eq 17 at a 100 mV ripple                 = 3.11 µF    TI: 3.11 µF

Every boost-mode figure and both inductances match. The buck-mode duty cycle does not. Equation 1 gives 3.3 / (5.0 × 0.93) = 0.710, and the example prints 0.614, which is 3.3 × 0.93 / 5.0 = 0.614: the efficiency on the other side of the fraction. The ideal ratio, 3.3 / 5.0, is 0.660; Equation 1 raises the duty cycle above it to make up the losses, while 0.614 lies below it. The rest of the buck column follows from whichever duty cycle is used. With the printed 0.614, Equation 6 gives the printed 492 mA, and Equation 5 then gives 2.246 A, which the example prints as 2.24 A. With Equation 1's 0.710 the ripple is 568 mA and the peak 2.28 A. The calculator uses Equation 1 as printed. Either way the buck corner is not the one that matters for the switch rating: the boost corner's 3.19 A is larger, and it is the one the note carries forward.

Equation 1 has a second discrepancy with the example, in the other direction. As printed, it puts V_IN,max in D_Boost, which here would give −0.409, a negative duty cycle. The text defines D_Boost as the "maximum duty cycle for boost mode", section 4.2 says "In boost mode, the maximum switch current is when the input voltage is at its minimum", and the example's 0.330 is 1 − 2.6 × 0.85 / 3.3. The calculator uses V_IN,min, as the example does.

The 4.5 A limit is not a datasheet figure. The TPS63802's peak current limit depends on the mode, with V_IN ≥ 2.5 V: 4 A minimum, 5 A typical and 5.75 A maximum in boost mode, 5 A typical in buck-boost mode and 3.8 A typical in buck mode. A real design checks Equations 7 and 10 at the limit for each mode, and at the minimum where one is given. With the boost-mode minimum of 4.00 A, Equation 10 gives 2.54 A; with the buck-mode typical of 3.80 A, Equation 7 gives 3.52 A. Both still cover the 2 A load.

For the output capacitor the note takes the largest of its three results, Equation 17's 3.11 µF, and fits more: "A single 22 µF, 6.3 V, X5R, +/- 20% ceramic capacitor, (MuRata, GRM188R60J226MEA0), was chosen for the output capacitance." It then derates it: "By using the manufacture's provided information, the derated value of the output capacitor is 8.2 µF which is sufficient for the minimum output capacitance calculated in Equation 17." A 22 µF part is worth 8.2 µF once derated, which is why the minimum is compared with the derated value and not the marking. For the input: "A single 10 µF, 6.3 V, X5R ceramic capacitors are chosen for the design." The note's section 5 sizes the feedback divider as well; that is a voltage divider calculation and is left out here.

Buck-boost converter design equations: where they stop being valid

The transition region. SLVA535 evaluates buck mode at V_IN,max and boost mode at V_IN,min, and says nothing about inputs close to V_OUT, where a real 4-switch converter changes over from one mode to the other. Its Equation 1 marks the edge of what it covers: D_Buck reaches 1 at V_IN = V_OUT/η, 3.55 V in the example, and above V_OUT but below that the buck equations have no valid duty cycle. The calculator leaves those inputs out of the sweep and says so if V_IN,max falls among them. How a particular IC behaves there is in its datasheet.

Continuous conduction. The abstract limits the note to a converter "operating in continuous conduction mode". When the ripple exceeds twice the average inductor current, the current reaches zero each cycle and the equations stop applying; the calculator reports it. The results are for the full load entered, not for lighter ones.

The efficiency is an estimate at a corner. SLVA535's η is "estimated efficiency at calculated V_IN, V_OUT, and I_OUT", and its example uses a different figure at each end of the range. It enters only the duty cycles, and through them the ripple and the boost-mode currents. The sweep holds each mode's corner efficiency across that mode's half of the range, which is an assumption of this page, not of the note; the corner values themselves are exactly SLVA535's.

Integrated switches and the datasheet's inductor. The note is written for an IC with the switches inside it, and it puts the datasheet first: "Data sheets often give a range of recommended inductor values. If this is the case, choose an inductor from this range." Equations 3 and 4 are for the other case: "For device datasheets where no inductor range is given, an inductor that satisfies both buck and boost mode conditions must be chosen."

Compensation limits the output capacitor. "With internally compensated converters, the recommended inductor and capacitor values must be used, or the recommendations in the datasheet for adjusting the output capacitors to the application must be followed. This usually involves keeping the same ratio of L × C as the recommended values." Equations 14, 16 and 17 are for a converter with external compensation, and even then "the compensation has to be adjusted for the used output capacitance".

Capacitance is the derated value. "Always account for DC bias capacitance drop and derate the capacitance of the output capacitors for the design calculations." For the input capacitor the note gives no equation at all: "The minimum value for the input capacitor is normally given in the datasheet", and "The dielectric material must be X5R or better. Otherwise, the capacitor loses much of its capacitance due to dc bias or temperature."

ESR ripple. Equations 15 and 18 add the ripple from the output capacitor's ESR in each mode, on top of the capacitive ripple. The calculator does not compute them; the boost converter calculator has the boost-mode ESR term, and the buck ripple calculator the buck-mode one.

Common buck-boost converter mistakes

Sizing the inductor for one mode only. A Li-ion design sized from the boost side alone, 360 nH at 2.5 V, runs its buck corner at 4.2 V with 994 mA of ripple, K_ind = 0.50 instead of 0.3. The requirement has to be computed in both modes and the larger one kept. Equation 3 grows with V_IN, so the buck side asks most at the very top of the range: 555 nH at 4.2 V against 216 nH at 3.6 V.
Rating the inductor at the corner that chose it. In the Li-ion case the inductance comes from buck mode, but the peak current comes from boost mode, 3.48 A against 2.32 A. SLVA535: "the inductor must always have a higher current rating than the largest value of current given from Equation 5 and Equation 8; this is because the peak current increases with decreasing inductance."
Assuming Equation 4 is largest at V_IN,min.It rises with V_IN up to 2/3·V_OUT and only then falls. A Li-ion cell boosted to 5 V needs 491 nH at 2.5 V by the note's rule, but 582 nH at 3.33 V to hold K_indthere, 19 % more. The calculator reports the interior peak whenever it falls inside the range.
Using the ideal duty cycle. Without the efficiency, SLVA535's example gives D_Buck = 0.660 instead of 0.710, and D_Boost = 0.212 instead of 0.330. The boost-mode inductor current, I_OUT/(1 − D), is understated by the same margin.
Checking the IC's current limit in one mode.Equation 10 multiplies Equation 7's form by (1 − D_Boost): the example's 4.5 A limit delivers 4.22 A in buck mode but only 2.88 A in boost mode. "Imax out must be greater than Iout" has to hold in both.
Comparing the capacitance marking with the minimum.The example's 22 µF part is 8.2 µF at its operating point. The minimum from Equations 14, 16 and 17 is a derated value.
Designing an inverting buck-boost with these equations.The single-switch inverting converter has a negative output and different current waveforms. SLVA535 does not cover it, and neither does this calculator.