Capacitor ESR calculator: dissipation factor, impedance and ripple heating

What ESR is, and what the dissipation factor says about it

A real capacitor turns some of the AC through it into heat. The datasheets describe that loss in two ways that say the same thing. The equivalent series resistance is, in Vishay's words, "the ohmic part of an equivalent series circuit. Its value assumes all losses to be represented by a single resistance in series with the idealized capacitor" (Film Capacitors, General Technical Information, p. 4). The dissipation factor is the ratio of that resistance to the capacitor's reactance: "The dissipation factor (tan δ) is the ratio of the ESR to the capacitive reactance X_C (series capacitance) or the active power to the reactive power at a sinusoidal voltage of a specified frequency" (p. 4). Its reciprocal "is also known as Q-factor".

TDK's General technical information for film capacitors draws the phasor diagram behind the name. Because of the ESR, "the phase shift between voltage and current is slightly less than 90°. The difference between the phase angle θ and 90° is the defect angle δ, which is measured through the dissipation factor tan δ" (p. 14), and it writes the relation the calculator is built on: tan δ = ESR · 2πf · C. KEMET's datasheet for its ESC aluminium electrolytics puts it the same way, "Tan δ = ω x ESC x ESR", with ESC the equivalent series capacitance (p. 12), and turns it round on p. 13: ESR = tan δ/(2πf · ESC).

The one word that matters in all three is frequency. A dissipation factor is measured at one frequency, 120 Hz for KEMET's electrolytics and 1 kHz for film parts, and the ESR it gives is the ESR at that frequency and no other. KEMET: "ESR value depends on frequency and temperature" (p. 13). A figure carried from one frequency to another, with no second datasheet figure to anchor it, gives the wrong ESR, so the calculator keeps every figure tied to the frequency it belongs to.

What the ESR is made of depends on the technology. In a film capacitor Vishay lists "the polarization losses of the dielectric material (R_pol), the losses caused by the resistance of the leads, termination and electrodes (R_s) and the insulation resistance (R_is)" (p. 4). In an electrolytic, KEMET's p. 14 names R_e, the "Resistance of electrolyte and paper mixture", which is why an electrolytic's ESR is so much larger and so much more temperature-dependent than a film part's.

ESR from the dissipation factor: the KEMET ESC series

KEMET's Table 1 (pp. 6–9) is the kind of data a design starts from: for each of 195 parts it prints the rated voltage and capacitance, the case size, the DF at 120 Hz and 20 °C, the impedance Z at 100 kHz and 20 °C, and the ripple current at 100 kHz and 105 °C. The DF is printed once per voltage, from 22 % at 6.3 V down to 8 % at 100 V. The ESR at 120 Hz follows from ESR = tan δ/(2π · 120 Hz · C). Here is every 470 µF part in the series, one per voltage, with Table 1's 100 kHz impedance beside it:

Two things stand out. The ESR at 120 Hz falls with voltage only because the DF does: the capacitance is the same in every row. And at every voltage the 120 Hz ESR is several times the whole impedance at 100 kHz. That is not a contradiction. At 100 kHz the reactance of 470 µF is 3.39 mΩ, so Table 1's Z there is almost all ESR, and the ESR of an electrolytic falls steeply with frequency. KEMET's typical ESR curves on p. 13 fall across the whole range from 0.1 kHz to 100 kHz. A 120 Hz ESR used at a switching frequency, or a 100 kHz impedance used for a 120 Hz rectifier ripple, is wrong by the ratio in the table.

Above 1,000 µF there is a footnote to remember: "When capacitance exceeds 1,000 µF, the DF value (%) is increased by 2% for every additional 1,000 µF." At the top of the range, the 6.3 V 15,000 µF part, the printed 22 % becomes 50 %, and the 120 Hz ESR goes from 19.5 mΩ to 44.2 mΩ. The datasheet does not say how a fraction of 1,000 µF counts: whether 2,200 µF at 16 V carries one step or two. The calculator counts every started 1,000 µF, the higher of the two readings, giving 20 % for that part, and says so beside the result.

A Table 1 DF is a specified value, not a typical one, and the difference is large. KEMET's typical curves on pp. 12–13 include an ESC 22 µF/100 V part, the same rating as ESC226M100AG3 in Table 1, whose 8 % DF gives 4.82 Ω at 120 Hz. Read off the typical curves, that part's DF at 100 Hz is about 1.8 %, which by the same formula is 1.3 Ω, and the ESR curve sits at about the same 1.3 Ω at 100 Hz, 27 % of the table's 120 Hz figure; at 100 kHz it is under 0.4 Ω, against Table 1's 790 mΩ limit for Z. These are readings off log graphs, good to a few per cent, but the two typical curves agree with each other through ESR = tan δ/(2πf·C), and both sit well inside the table. Design to the table; expect the bench to read lower.

Impedance across frequency: the ESR floor and the two slopes

The impedance of the series model is Vishay's and TDK's |Z| = √(ESR² + (2πf·L_S − 1/(2πf·C))²): "the magnitude of the vectorial sum of ESR and the total reactance (inductive reactance minus capacitive reactance)" (TDK p. 20). On log-log axes it is three straight pieces joined by two bends, which is what the figure draws. At low frequency "the capacitive reactance X_C = 1/2πf · C prevails", falling a decade per decade; at very high frequency "the inductive reactance X_L = 2πf · L_S is dominant" and rises a decade per decade; and "When capacitive reactance equals inductive reactance, natural resonance occurs. At this point the reactances cancel each other out and impedance equals ESR" (TDK p. 21). The resonance is f_res = 1/(2π√(L_S·C)).

The flat floor between the two slopes is the ESR, and its width depends on how large the ESR is against the reactances. For a film capacitor the floor is narrow: the ESR is milliohms, and the capacitive slope runs almost into the inductive one. For an electrolytic it is wide, and KEMET's p. 14 explains why its floor is not quite flat. The electrolytic is drawn there as the oxide capacitance C_o in series with R_e, shunted by the electrolyte-soaked paper's own capacitance C_e, then L. Between the two corners "resistance of the electrolyte predominates: Z = R_e", until the reactance of C_e takes over and the impedance falls again towards resonance; "Generally speaking, it can be estimated that C_e ≈ 0.01 C_o." The calculator's model holds the ESR constant, so in KEMET mode it draws the curve through Table 1's 100 kHz point and marks the 120 Hz ESR from the DF as a second, dotted level above it.

The series inductance sets the top end. KEMET gives no ESL for the ESC series, so the calculator leaves it at zero until a measured value is entered. For film parts Vishay states that the inductance of radial types is "typically measured with 2 mm long lead wires. Typical values are less than 1.0 nH per mm of lead length" (p. 4), and TDK that "the maximum value is 1 nH per mm of lead length and capacitor length" (p. 20). The reactance calculator draws the same curves for any R, L and C, and the decoupling calculator puts several such capacitors in parallel against a target impedance.

Worked example: a KEMET ESC 1,000 µF, 16 V electrolytic

The calculator's default is ESC108M016AH4, 1,000 µF at 16 V in a 10 × 20 mm can, on p. 7 of the datasheet, carrying 1 A rms of 120 Hz ripple, the kind a full-wave rectifier on a 60 Hz line puts through a reservoir capacitor. The current is illustrative; everything else is Table 1.

Table 1    ESC108M016AH4: 1,000 µF, 16 V, DF 16 %, Z 39 mΩ, RC 1,250 mA
X_C        1/(2π · 120 Hz · 1,000 µF)                 = 1.326 Ω
ESR        0.16 × 1.3263 Ω                            = 212.2 mΩ
           at C − 20 %: 0.16/(2π · 120 · 800 µF)      = 265.3 mΩ
100 kHz    √((39 mΩ)² − (1.59 mΩ)²)                   = 38.97 mΩ
rating     1,250 mA × 0.90 (120 Hz column)            = 1.125 A
heat       (1 A)² × 0.2122 Ω                          = 212 mW

The ESR at 120 Hz is 212 mΩ, and KEMET's own instruction on p. 13, "Tolerance limits of the rated capacitance must be taken into account when calculating this value", raises it by 25 % to 265 mΩ for a part at the bottom of its ±20 % tolerance (p. 4). The 100 kHz impedance of 39 mΩ leaves 39.0 mΩ of ESR once the 1.59 mΩ of X_C is taken out, so the 120 Hz ESR is 5.4 times the 100 kHz one. The same 1 A at 100 kHz would leave 39.0 mW in the part, where at 120 Hz it leaves 212 mW.

Ripple current: the heat, the rating and the frequency factors

The heat is the simplest part of the page. TDK p. 14: "P = V_ESR²/ESR = ESR · I²", and since the current is the same through every element of a series circuit, that is exact for the model whatever the frequency, provided the ESR is the one at that frequency. For film capacitors TDK and Vishay also write it in terms of the voltage across the whole part, "P = 2πf · C · tan δ · V²" (TDK p. 14), which relies on "tan δ = 2πf · C · ESR << 0.1". In terms of the current that voltage drives, the voltage form is ESR · I² · (1 + tan²δ) (derived), so for a film part the two agree to within tan²δ, 25 parts per million even at polyester's 0.5 %, and for an electrolytic at a high frequency, where KEMET's typical DF climbs past 100 % (p. 12), the voltage form overstates the heat. The calculator uses ESR · I².

What that heat does to an electrolytic is what the ripple-current rating is for. KEMET p. 16 lists what the maximum ripple current depends on: the ambient temperature, the "Surface area of the capacitor (heat dissipation area)", tan δ or ESR, and the frequency; and "The capacitor’s life depends on the thermal stress." Table 1's rating is "specified at 100 KHz 105°C" (p. 5), and the "Compensation Factor of Ripple Current (RC) vs. Frequency" table on p. 4 scales it to lower frequencies, where the ESR is higher. For the default part, in the 331 – 1,000 µF row:

At 120 Hz the part is rated for 1.125 A, and 1 A is 89 % of it. At that rated current, the ESR from the DF puts 269 mW into the part; at its 100 kHz rating of 1.25 A the 100 kHz ESR puts in 60.9 mW. Both heats come from limits, the DF and the Z column, so they are worst cases; KEMET does not say what heat its ratings were set for, and the two figures are not meant to match. Between the printed columns the calculator takes the factor of the column at or below the ripple frequency, the lower of the two, since the factors rise with frequency; below 50 Hz the table has nothing, and the calculator says so.

The calculator does not turn an electrolytic's heat into a temperature rise. The ESC datasheet gives no thermal resistance and no heat transfer coefficient, so there is nothing to divide the watts by, and the ripple-current rating is the thermal limit it does give. What it does give is the life a temperature buys: "L = Lo x 2^(To-T)/10", "applicable between 40°C and To" (p. 16). The default part's can is 10 mm by 20 mm, and cans from 10 × 15 mm up are rated 3,000 hours at 105 °C (p. 5); run at an illustrative 65 °C, the formula gives 48,000 hours. For the ripple a reservoir capacitor sees in the first place, the smoothing capacitor calculator works from the load and the line; for a converter's output, the buck ripple calculator adds the ESR's share to the ripple voltage.

Worked example: TDK's 220 nF film capacitor at 20 kHz

TDK works a film example on pp. 26–27 from a datasheet curve of maximum voltage against frequency: "a 220 nF capacitor has a maximum V_RMS of 100 V at 20 kHz. So under these conditions (100 V_RMS, 20 kHz), if the dielectric is PP, the self-heating (ΔT) will be 10 °C. If the dielectric is PET or PEN, the self-heating will be 15 °C." The current follows from the capacitance alone, and TDK prints it as "100 · 2π · 20000 · 220 · 10⁻⁹ = 2.765A"; the calculator gives 2.765 A. For a 200 nF part between the curves TDK scales by the square root of the capacitance ratio and prints 104.8 V and 2.636 A; the calculator gives 104.88 V and 2.636 A, so the 104.8 is 104.88 truncated rather than rounded. For 75 V instead of 100 V, the self-heating scales with the square of the voltage: TDK's "8.4 °C for PET or PEN dielectric" and "5.6 °C for PP dielectric", which the calculator reproduces as 8.4375 and 5.6250 °C before rounding.

The calculator's film mode runs the same part through Vishay's method instead: the heat from the ESR, then ΔT = P × 1000/(A × α) (p. 3), with A the surface area in cm² and α, the heat transfer coefficient, "0.96 for plastic boxes with a smooth surface". TDK's typical dissipation factor for polypropylene at 1 kHz is 0.0005 (p. 3); the 9 × 18 × 26.5 mm box below is an illustrative size, not a TDK dimension.

TDK        I = 100 V · 2π · 20 kHz · 220 nF           = 2.765 A
ESR        tan δ 0.05 % × X_C 36.17 Ω                 = 18.1 mΩ
heat       (2.765 A)² × 18.09 mΩ                      = 138 mW
box        9 × 18 × 26.5 mm, all six faces            = 17.55 cm²
Vishay     ΔT = 138.2 mW / (17.55 × 0.96)             = 8.2 °C

That is inside TDK's 10 °C for polypropylene. The result moves in proportion to tan δ: with Vishay's own typical PP figure of 0.02 % at 1 kHz (p. 2) it is 3.3 °C, and a polyester part at Vishay's 0.5 % would reach 82 °C in the same box, far beyond the 15 °C TDK allows PET. Both are 1 kHz figures applied at 20 kHz, and TDK says the dissipation factor of a film capacitor rises with frequency: the series component "increases rapidly with frequency until it becomes the dominating component in the tan δ curve for high frequencies" (p. 15). A datasheet's 10 kHz or 100 kHz tan δ, which TDK says is measured for MKT, MFP and MKP parts up to 1 µF (p. 15), is the one to enter. TDK's own warning applies to all of this: α and tan δ "depend on technology, construction, material and geometry of each capacitor", which "complicates the use of these equations", and the datasheet curves were "obtained empirically" (p. 26). The method is for a first estimate; the curve, or a thermocouple on the part, is the answer.

Where the model stops being valid

The ESR moves with frequency. TDK p. 18 describes a film capacitor's ESR across frequency: "At low frequencies, ESR is dominated by the dielectric losses, decreasing roughly as f^-1. At medium to high frequencies, losses in the conductors are dominant and ESR becomes relatively constant. At very high frequencies (>10 MHz) ESR increases by √f due to the skin effect." An electrolytic's falls all the way from 0.1 kHz to 100 kHz in KEMET's typical curves (p. 13). The calculator holds the ESR at the value entered across the impedance plot, so the curve is right at the operating point and an approximation away from it.

The ESR moves with temperature. KEMET p. 15: "R_e is the most temperature-dependent component of an electrolytic capacitor equivalent circuit. Electrolyte resistivity will decrease if temperature rises." The ESC table on p. 4 gives the impedance at 120 Hz at −40 °C as 8 times its 20 °C value in the 6 V column, 6 times at 10 V and 4 times from 16 V up. Table 1's DF and Z are 20 °C figures; a cold start sees several times the ESR, and a hot part less.

Real ripple is not a sine. A rectifier's charging pulses and a converter's triangle both carry harmonics, and each harmonic sees the ESR at its own frequency. TDK p. 27: "the power dissipation must be calculated by Fourier decomposition of the waveform into its harmonics", P_gen = ΣV_RMS,i² · 2πf_i· C · tan δ(f_i). The calculator takes one RMS current at one frequency.

The thermal model is a surface balance. Vishay's ΔT assumes the heat leaves through the surface at α = 0.96 mW/(°C·cm²), and "At pulse or AC load operations the surface temperature may, due to an internal temperature increase, rise above the ambient temperature" (p. 4). TDK defines ΔT at "the hottest part of the capacitor surface in relation to the surrounding atmosphere" (p. 25) and asks for a measurement in any critical case: "a temperature check on the capacitor under working conditions should be carried out" (p. 30). A part next to a hot MOSFET, or boxed in by others, is outside the model.

Above resonance it is an inductor. Vishay p. 4: "Above the resonate frequency the inductive part of the capacitor prevails." Past f_res the ESL, not C, sets |Z|, and adding capacitance does nothing for the impedance there.

Common ESR mistakes

• Using an ESR at the wrong frequency. The default part's 120 Hz ESR is 5.4 times its 100 kHz ESR; carried to 100 kHz it overstates the heat by that factor, and the 100 kHz figure carried to 120 Hz understates it by the same.
• Reading a DF in % as a fraction. 16 % is 0.16; entered as 16 it gives an ESR a hundred times too large.
• Forgetting the footnote above 1,000 µF. At 15,000 µF the printed 22 % DF is really 50 %, and the ESR is 2.27 times what the column suggests.
• Using the nominal capacitance for a worst case. KEMET p. 13 asks for the tolerance; at −20 % the same DF means 25 % more ESR.
• Taking the 100 kHz ripple rating at 120 Hz. The default part is rated 1.25 A at 100 kHz but 1.125 A at 120 Hz, and the smaller parts lose more: the 4.7 µF row's factor at 120 Hz is 0.40.
• Applying P = 2πf·C·tan δ·V² where tan δ is not small. TDK derives it for tan δ << 0.1. ESR · I² has no such condition.
• Treating a typical curve as a specification, or the table as what the bench will show. The table is the limit; the 22 µF/100 V part above reads typically about 27 % of its table ESR at low frequency.

Further reading

• Vishay Roederstein, Film Capacitors, General Technical Information (26033) — dissipation factor, ESR and impedance definitions (p. 4), the power and temperature-rise equations with α = 0.96 (p. 3), and typical film properties (p. 2).
• TDK, Film Capacitors, General technical information — ESR and dissipation factor (pp. 14–18), impedance and resonance (pp. 20–21), and the AC load limits with the 220 nF example (pp. 24–27).
• KEMET (YAGEO) ESC radial aluminium electrolytic capacitors, A4006 — Table 1 of ratings (pp. 6–9), the ripple-current factors (p. 4), and the application notes on DF, ESR, impedance and life (pp. 12–16).
• Reactance calculator — X_C, X_L and the |Z| of a real capacitor or inductor at any frequency.
• Smoothing capacitor calculator — the reservoir capacitance a rectifier needs, and the ripple it leaves.
• Decoupling calculator — capacitors in parallel against a target impedance, where ESR and ESL set the floor.
• Buck ripple calculator — a converter's output ripple, with the output capacitor's ESR in it.