MLCC DC Bias Derating: Estimating the Capacitance Left at Your Voltage

The result

• The loss follows the field in the dielectric layers, and the label gives you that field. Take the voltage times the square root of capacitance per unit volume. It predicts the loss about four times better than the applied voltage as a fraction of rated. On X7R parts the fit wasn't trained on, it lands within 7.4 points of the vendor's curve for four readings in five.
• Density splits the parts into two groups. Above about 1 µF/mm³, a part reads about 24% below its marked value at a converter's ripple level, before any bias is applied. At −40 °C it loses a further 18%. Below that density, both effects are small. The temperature code doesn't predict either.
• Three rules of thumb fail on the data. Running at half the rated voltage doesn't protect the capacitance. A higher voltage rating in the same case usually buys nothing. X7R doesn't reliably lose less than X5R. A bigger case does help, but only when everything else is held the same.

In the worked example, seven 22 µF 1210 parts are needed where the label suggests three and the vendor's 25 °C curve suggests four.

Why the label predicts the loss

A multilayer capacitor is a stack of $N$ dielectric layers, each of thickness $t$ and overlap area $A$. Its capacitance is $C = \varepsilon_0 \varepsilon_r N A / t$, and its active volume is about $N A t$. Eliminate $N A$ and the layer thickness is

where $\varepsilon_0$ is the permittivity of free space, $\varepsilon_r$ is the dielectric's relative permittivity, $\mathrm{vol}$ is the active volume and $C$ is the capacitance. The field in each layer is the voltage over the thickness. Let the package volume stand in for the active volume, and leave out the constants. That gives a field proxy you can read off the datasheet:

where:

• $E^*$ is the field proxy, in V·√(µF/mm³).
• $V$ is the DC voltage across the capacitor, in V.
• $C$ is the marked capacitance, in µF.
• $\mathrm{vol}$ is the package volume, length × width × thickness, in mm³.

The ferroelectric material responds to the field, not to the voltage rating. So two parts at the same $E^*$ should lose about the same share, whatever their case, rating or vendor. Fig. 1 tests that on the 1,764 X7R parts of 100 nF and above whose height the catalogue holds.

Against the fraction of rated voltage, the rank correlation with the retained capacitance is −0.52. Against $E^*$ it's −0.97. Fitted separately on all their X7R parts, Murata's measured curves and AVX's simulated ones put $E_0$ within 1% of each other. The catalogue holds no heights for KEMET's parts, so they can't go on this plot. Use footprint area in place of volume, and KEMET's medians fall within 7 points of the other two vendors' in every bin.

Earlier work says density matters. KEMET's high-reliability catalogue says stability "is related inversely to the packaging efficiency" [1]. Novak found a thinner part of the same footprint lost more [2]. Würth fits one part's curve from that part's own data [3]. Jiang et al. infer the dielectric thickness from a measured curve [4], which runs the opposite way. I haven't found a source that predicts one part's loss from its label.

What you need

From the datasheet's first page:

• The marked capacitance $C$.
• The length, width and thickness, nominal values in mm. Use the thickness for your exact part number, because one case size comes in several heights.
• The dielectric code: X7R, X5R, X6S, X7S or X7T.

From the design: the DC voltage $V$ across the capacitor, and the temperature range it works over.

Worked example: the output capacitor of a 48 V to 12 V buck

The converter. 48 V to 12 V at 500 kHz and 200 W, with a 3.3 µH inductor and 5.45 A of ripple current. It's the converter from Dead Time in a Synchronous Buck: Choosing Both Edges from the Datasheet. It works from −40 °C to 100 °C.

The requirement. Hold the capacitive part of the output ripple to 30 mV peak to peak:

where:

• $C_{out}$ is the effective output capacitance, which has to hold at every temperature.
• $\Delta I_L$ is the inductor's peak-to-peak ripple current.
• $f_{sw}$ is the switching frequency.
• $\Delta V_{out}$ is the allowed peak-to-peak ripple voltage.

The part. A Murata GRM32ER71E226KE15: 22 µF, 25 V, X7R, 1210, 3.2 × 2.5 × 2.7 mm.

1. Field and density. vol = 21.6 mm³. $E^*$ = 12 × √(22/21.6) = 12.11. $\rho_C$ = 22/21.6 = 1.02 µF/mm³, so the part is dense.
2. Bias. $R$ = 1/(1 + (12.11/13.34)^1.77) = 0.543.
3. Ripple level. $C_{25}$ = 0.76 × 0.543 × 22 µF = 9.0 µF.
4. Temperature. At −40 °C, $k_T$ = 0.82, so $C_{eff}$ = 7.4 µF. At 100 °C, $k_T$ = 1.03, so $C_{eff}$ = 9.3 µF. Cold is the worst case, at 7.4 µF.
5. Count. 45.4/7.4 = 6.1, so fit seven parts, 154 µF marked.

Against Murata's measured curves. The part's own family gives 7.3 µF at −40 °C, 9.1 µF at 25 °C and 9.6 µF at 100 °C. The estimate is within 0.3 µF at all three. Seven parts give 50.9 µF at −40 °C.

Against the shortcuts.

• The "derate to half rated voltage" rule says the part is safe at 12 V, so the label's 22 µF stands. That's three parts. At −40 °C they give 22 µF, and the ripple reaches 62 mV.
• Reading the vendor's bias curve at 25 °C, normalised to 0 V, gives 11.9 µF. That's four parts. At −40 °C they give 29 µF, and the ripple reaches 47 mV.

A bigger case instead. A 2220 part at 3.9 mm, the KCM55QR71E226KH01, has 126 mm³ including its metal frame. That's 0.17 µF/mm³, a low-density part. The law gives $R$ = 0.85, and the worst case is 15.9 µF at 100 °C, so it takes three. They lose far less, but they take 97 mm² of board against 56 mm² for seven 1210s. The 1210s win on area here, even though each one keeps less.

Worked example, run backwards: the input capacitor

The same converter needs ceramic capacitance on its 48 V input. Take a 4.7 µF 100 V X7R, and aim to keep half of it at 48 V. Step 5 gives $E^*_{max}$ = 13.34, so the part needs 4.7 × (48/13.34)² = 60.8 mm³.

• 1210 at 2.8 mm (22.4 mm³) falls short. The law gives 0.29, and the KYOCERA AVX KAF32LR72A475KU's curve gives 0.34, so 1.6 µF.
• 1812 at 2.8 mm (40.3 mm³) also falls short. The law gives 0.41.
• 2220 at 2.7 mm (77.0 mm³) clears it. The law gives 0.55, and the Murata GRM55ER72A475KA01's curve gives 0.59, so 2.8 µF.

Both are well below 1 µF/mm³, so the ripple-level factor is close to 1. In the filter of Input Filter Damping: Sizing Rd and Cd to an Impedance Target, the same specification gives 2.4 µF at 48 V. That's the large-case end of this range.

What the data says about the rules of thumb

Four rules of thumb come up again and again in articles on DC bias. The catalogue can test each one on matched parts.

"Run it at half its rated voltage." That rule protects the part's life, not its capacitance. At half rated voltage, 40% of the characterised parts have lost more than 20%, and 18% have lost more than half. The usual pick is a part of 1 µF or more, rated at one to two times the rail. Its median retention is 0.87 at 5 V, 0.62 at 12 V, 0.43 at 24 V and 0.29 at 48 V.

"Choose a higher voltage rating to keep capacitance." Take 10,097 pairs with the same vendor, case, dielectric and capacitance, and compare them at half the lower rating. 80% are within 5 points. 17% gain more than 5 points and 3% lose. Half the pairs have the same curve in volts to within 1%. That's the same dielectric stack, sold at two ratings. For 10 µF at 5 V, the median rises from 0.66 at a 6.3 V rating to 0.94 at 25 V. But that's only across case sizes. In 0805 it stays between 0.57 and 0.68. In 1206 it holds at 0.92 up to a 16 V rating, then falls to 0.83 at 25 V. A higher-rated part helps only when the vendor built it with thicker layers. If it's the same height in the same case, it usually doesn't. This matches Fortunato's finding that within a package and dielectric the rating "seems often to have no effect" [9].

"Choose a bigger case." Matched, this one holds. Take 4,890 pairs with the same vendor, dielectric, capacitance and rating. The larger case keeps more than 5 points more in 59% of pairs, and less in 1%. The median gain is 8 points. The catalogue reproduces the shape of Fortunato's 4.7 µF X7R table at 12 V [9], on 197 parts. The medians are 0.47 in 0805, 0.80 in 1206, 0.90 in 1210, 0.91 in 1812 and 0.96 in 2220. Across the whole catalogue the case predicts very little, because it's only one of the three things in $E^*$.

"X7R loses less than X5R." Take 429 pairs with the same vendor, case, capacitance and rating. The median difference is zero. 72% of pairs are within 5 points, 17% favour X7R and 11% favour X5R. Novak's measurements found no consistent advantage either [2]. The temperature code describes the material's drift with temperature at 0 V. Under bias, the field and the density decide.

Where the numbers come from

Every curve comes from a vendor's own tool, as each vendor publishes it [5], [10], [11]. None was read off a PDF. For scale, the largest earlier multi-vendor measurement covered "more than two dozen" models from six vendors [2].

Left out are X8G, which is Class I in EIA terms, and 6,063 parts whose curves keep more than 99% at rated voltage. Nearly all of those (5,796) are KEMET's. They sit at a low field, where the law predicts more than 0.95 anyway. The law's constants come from the 8,453 Murata and AVX parts whose heights the catalogue holds. The ripple-level and temperature factors come from Murata alone, because only its curves state the AC level and give the bias curve at each temperature.

Murata's measurement conditions give 60 s under bias [12]. The other two tools don't say. The curves are typical, not limits, and none says how long after the last de-ageing it was taken. Under constant bias, capacitance keeps falling past 1,000 hours [13].

What this means for switchmode.io

This note gets you a case and a part count before you've chosen a part, to within about 10 points on X7R. It can't tell you what each capacitor in a finished design holds at every operating corner. That takes each part's own curve, read at every corner, and switchmode.io is being built to do it. In the current alpha, its part picker already lists each ceramic capacitor at its capacitance at your design's bias. It reads that from the vendor's curve, and filters on it. So asking for "at least 20 µF at 12 V" searches Murata, KEMET and KYOCERA AVX parts together. The vendors' own tools each search one catalogue.

For what the capacitance tolerance, bias and temperature do to a control loop's phase margin, see Worst-Case Phase Margin: What Component Tolerance Does to Your Buck Converter. For a filter capacitor taken at the bus voltage, see Input Filter Damping: Sizing Rd and Cd to an Impedance Target.