What is the formula for capacitive reactance?
Xc = 1 / (2πfC), with f in hertz, C in farads and Xc in ohms. Reactance is inversely proportional to both frequency and capacitance: double either one and Xc halves.
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Capacitive reactance calculator
Use this capacitive reactance calculator to find the reactance of a capacitor at any frequency, or the capacitance or frequency for the reactance you need. Add the ESR and ESL from the datasheet to see the real impedance, the self-resonant frequency and the current at an AC voltage.
Solve for
Examples
F
Hz
Ω
Optional: equivalent series resistance from the datasheet.
H
Optional: equivalent series inductance, typically 0.5 to 1 nH for 1 nF to 100 nF ceramics.
V
Optional: gives the current through the capacitor.
Capacitive reactance Xc
1.592 kΩ
- Angular frequency ω
- 6.283 krad/s
Ideal capacitor: Xc halves each time the frequency doubles. Enter the ESR and ESL from the datasheet to see the real impedance and the self-resonant frequency.
100 nF at decade steps around 1 kHz
| Frequency | Xc |
|---|---|
| 1 Hz | 1.592 MΩ |
| 10 Hz | 159.2 kΩ |
| 100 Hz | 15.92 kΩ |
| 1 kHz | 1.592 kΩ |
| 10 kHz | 159.2 Ω |
| 100 kHz | 15.92 Ω |
| 1 MHz | 1.592 Ω |
How to calculate capacitive reactance
Use Xc = 1 ÷ (2πfC). A 100 nF capacitor has a reactance of 1.592 kΩ at 1 kHz and 1.592 Ω at 1 MHz: reactance falls in proportion to frequency, and the current through the capacitor leads the voltage by 90°. A real capacitor stops following this line at its self-resonant frequency, where its ESL takes over.
Capacitive reactance formula
Reactance is the ratio of the AC voltage across a capacitor to the AC current through it. The faster the voltage changes, the more charge has to flow to follow it, so the reactance falls as frequency rises.
Xc = 1 ÷ (2π × f × C) = 1 ÷ (ωC)
- Xc
- capacitive reactance in Ω
- f
- frequency in Hz (ω = 2πf in rad/s)
- C
- capacitance in F
Rearranged for the capacitance or the frequency:
C = 1 ÷ (2π × f × Xc) f = 1 ÷ (2π × C × Xc)
A real capacitor behaves like C in series with its ESR and ESL. Its impedance, self-resonant frequency and losses are:
|Z| = √(ESR² + (2πf × ESL − Xc)²) fr = 1 ÷ (2π√(ESL × C)) DF = ESR ÷ Xc
- ESR
- equivalent series resistance in Ω
- ESL
- equivalent series inductance in H
- fr
- self-resonant frequency in Hz
- DF
- dissipation factor (tan δ); Q = 1 ÷ DF
Capacitive reactance examples
Example 1: a 100 nF capacitor at 100 kHz and 1 MHz
Xc = 1 ÷ (2π × 100 kHz × 100 nF) = 15.92 Ω. Ten times the frequency gives a tenth of the reactance: 1.592 Ω at 1 MHz. The current leads the voltage by 90° at both frequencies.
Example 2: current through a 470 nF capacitor on the mains
At 50 Hz, Xc = 1 ÷ (2π × 50 × 470 nF) = 6.773 kΩ, so 230 V RMS drives 34 mA RMS through it. At 60 Hz the reactance drops to 5.644 kΩ, and 120 V drives 21.3 mA. This is the current a capacitive dropper or an across-the-line (X) capacitor carries; it shifts with mains voltage, frequency and the capacitor’s tolerance.
Example 3: a 100 nF decoupling capacitor with 0.5 nH ESL and 20 mΩ ESR
The self-resonant frequency is 1 ÷ (2π√(0.5 nH × 100 nF)) = 22.5 MHz. At 10 MHz the ideal Xc is 159.2 mΩ, but the ESL cancels part of it and |Z| = 129.3 mΩ. At 100 MHz the ESL reactance (314.2 mΩ) dominates and |Z| = 298.9 mΩ at a phase of +86.2°: the part is an inductor there, and a smaller package with lower ESL, not more capacitance, is what lowers the impedance.
Capacitive reactance vs frequency table
Xc = 1 / (2πfC) for common capacitors at common frequencies. These are ideal values: above a part’s self-resonant frequency (1.59 MHz for 10 µF with 1 nH of ESL) its real impedance rises instead of falling.
| f \ C | 1 nF | 10 nF | 100 nF | 1 µF | 10 µF | 100 µF |
|---|---|---|---|---|---|---|
| 50 Hz | 3.18 MΩ | 318 kΩ | 31.8 kΩ | 3.18 kΩ | 318 Ω | 31.8 Ω |
| 60 Hz | 2.65 MΩ | 265 kΩ | 26.5 kΩ | 2.65 kΩ | 265 Ω | 26.5 Ω |
| 1 kHz | 159 kΩ | 15.9 kΩ | 1.59 kΩ | 159 Ω | 15.9 Ω | 1.59 Ω |
| 10 kHz | 15.9 kΩ | 1.59 kΩ | 159 Ω | 15.9 Ω | 1.59 Ω | 159 mΩ |
| 100 kHz | 1.59 kΩ | 159 Ω | 15.9 Ω | 1.59 Ω | 159 mΩ | 15.9 mΩ |
| 1 MHz | 159 Ω | 15.9 Ω | 1.59 Ω | 159 mΩ | 15.9 mΩ | 1.59 mΩ |
| 10 MHz | 15.9 Ω | 1.59 Ω | 159 mΩ | 15.9 mΩ | 1.59 mΩ | 159 µΩ |
| 100 MHz | 1.59 Ω | 159 mΩ | 15.9 mΩ | 1.59 mΩ | 159 µΩ | 15.9 µΩ |
Self-resonant frequency by capacitance and ESL
fr = 1 / (2π√(ESL·C)). Texas Instruments gives 0.5 to 1 nH as the typical ESL of 1 nF to 100 nF decoupling capacitors; mounting pads and vias add to it.
| C \ ESL | 500 pH | 1 nH | 2 nH |
|---|---|---|---|
| 1 nF | 225 MHz | 159 MHz | 113 MHz |
| 10 nF | 71.2 MHz | 50.3 MHz | 35.6 MHz |
| 100 nF | 22.5 MHz | 15.9 MHz | 11.3 MHz |
| 1 µF | 7.12 MHz | 5.03 MHz | 3.56 MHz |
| 10 µF | 2.25 MHz | 1.59 MHz | 1.13 MHz |
What the formula leaves out
The nominal capacitance is not what the circuit sees. Class 2 ceramics (X5R, X7R) lose a large share of their capacitance with DC bias and temperature, so Xc at the working voltage can be well above the value computed from the label. ESR is not constant either: in ceramics the dielectric loss dominates at low frequency and the metal loss at RF, and aluminum electrolytics can show ten times their room-temperature impedance at −40 °C. On a board, the pads, vias and planes add inductance in series with the part’s own ESL, which lowers the self-resonant frequency further. When several capacitors of very different values sit in parallel, their resonances can create an impedance peak (anti-resonance) between them, so check the combined curve, not each part alone.
Glossary
- Capacitive reactance (Xc)
- Also called capacitor reactance: the opposition of a capacitor to alternating current, 1 / (2πfC), in ohms. It stores energy rather than dissipating it.
- Impedance (Z)
- The complex ratio of voltage to current: resistance plus reactance. Its magnitude |Z| is what limits the current.
- ESR
- Equivalent series resistance: the dielectric and metal losses of a capacitor lumped into one resistor in series with C.
- ESL
- Equivalent series inductance: the inductance of the electrodes, terminations and current loop of the part.
- Self-resonant frequency (SRF)
- Where Xc and the ESL reactance cancel. |Z| is at its minimum (equal to the ESR) and the part turns inductive above it.
- Dissipation factor (DF, tan δ)
- ESR divided by Xc at a stated frequency. Q is its reciprocal.
Related guides
- Inductive reactance calculator: XL = 2πfL, the inductor counterpart.
- Resonant frequency calculator: where an L and a C have equal reactance.
- RC low pass filter calculator: Xc and R as a frequency-dependent divider.
Frequently asked questions
What is the impedance of a capacitor?
For an ideal capacitor the impedance magnitude equals its reactance, Xc = 1 / (2πfC), and the current leads the voltage by 90°. A real capacitor adds ESR and ESL, and the capacitor impedance formula becomes |Z| = √(ESR² + (2πf·ESL − Xc)²).
What is the reactance of a 100 nF capacitor?
1.592 kΩ at 1 kHz, 15.92 Ω at 100 kHz and 1.592 Ω at 1 MHz. Above a few MHz the real part departs from these numbers, because its ESL starts to cancel the reactance.
What is the difference between capacitive reactance and impedance?
Reactance is the imaginary part of the impedance, the part that stores energy instead of dissipating it. A real capacitor also has ESR and ESL, so its impedance magnitude is √(ESR² + (2πf·ESL − Xc)²), which is never below the ESR.
What is the self-resonant frequency of a capacitor?
The frequency where the reactance of C and of its ESL cancel: fr = 1 / (2π√(ESL·C)). There |Z| equals the ESR; above it the part behaves as an inductor. A 100 nF ceramic with 0.5 nH of ESL resonates at 22.5 MHz.
How much current flows through a capacitor on AC mains?
I = V / Xc. A 470 nF capacitor across 230 V at 50 Hz has Xc = 6.773 kΩ and passes 34 mA RMS; on 120 V at 60 Hz it passes 21.3 mA. The current leads the voltage by 90°, so it dissipates almost nothing apart from I² × ESR.
How do I size a coupling capacitor with capacitive reactance?
A series coupling capacitor and the load resistance form a high-pass filter whose −3 dB point is where Xc equals the load. For a 10 kΩ input and a 20 Hz low end, C = 1 / (2π × 20 Hz × 10 kΩ) = 796 nF; use the next standard value up (1 µF) or larger.
What is the dissipation factor of a capacitor?
DF = tan δ = ESR / Xc, the ratio of the power lost in the ESR to the reactive power in C, and Q = 1 / DF. Datasheets quote it at one frequency: Knowles gives about 1 to 2 % at 1 kHz for X7R ceramics, while C0G parts reach a Q of 600 to 1000 at 1 MHz.
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Sources
- OpenStax, University Physics Volume 2, 15.2 Simple AC Circuits (capacitive reactance, Example 15.1)
- ROHM application note 63AN091E, Impedance Characteristics of Bypass Capacitor
- Knowles (Syfer) AN0036, ESR Measurements of High Q MLCCs (Q, DF, ESR and power)
- ATC / Kyocera AVX Circuit Designer’s Notebook, ESR Loss Factors
- Texas Instruments SPRAC76, Sitara Processor Power Distribution Networks (typical ESR and ESL of decoupling capacitors)
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