Capacitance and capacitors
A 10 µF X7R ceramic can end up below 3 µF when 10 V is applied to it, and that does not appear in the tolerance printed on it. It is the cause of power supplies that oscillate and filters that do not filter.
01What capacitance is
Two conductors separated by an insulator store charge when a potential difference is applied to them. Capacitance measures how much charge they store per volt, and it depends only on the geometry and on the material in between: not on the voltage or on the charge.
The formula gives the only three ways to increase capacitance: more area, less separation or a better dielectric. The whole history of capacitor technology is a tug-of-war with those three variables, and each family chose one.
02The dielectric, and its limit
When an insulator is placed between the plates, its molecules polarize and create an opposing field that weakens the original one. With the same charge there is less voltage, so the capacitance goes up by a factor of \( \varepsilon_r \), the relative permittivity.
| Dielectric | \( \varepsilon_r \) | Strength in kV/mm | Where it is used |
|---|---|---|---|
| Vacuum or air | 1.0 | 3 | Variable capacitors, transmission lines |
| Polypropylene | 2.2 | 650 | Film capacitors, audio and power filters |
| Impregnated paper | 3.5 | 16 | Motor-start capacitors |
| Mica | 5.4 | 120 | Precision RF capacitors |
| Silicon dioxide | 3.9 | 1000 | The gate oxide of every MOSFET |
| Aluminum oxide | 8.5 | 710 | Electrolytic capacitors |
| Barium titanate | 1000 to 5000 | 10 | High-capacitance ceramics, X7R and Y5V |
Dielectric strength is the maximum field the material withstands before it suddenly conducts. Exceeding it does not degrade the capacitor: it punctures it, and the failure is permanent. The last row shows the central compromise: barium titanate has a very high permittivity and a poor strength, and that is why high-capacitance ceramics are low-voltage.
Build the capacitor by choosing the dielectric, area and separation, and apply a voltage to it. Capacitance, charge, energy and the internal field are calculated, and you are warned if the field exceeds the strength of the material.
03The capacitors you actually buy
The ideal capacitor of the textbooks does not exist. The real one has series resistance, series inductance and leakage, and its capacitance changes with temperature, with voltage and with time.
| Type | Typical range | Its virtue | Its trap |
|---|---|---|---|
| Ceramic C0G/NP0 | 1 pF to 10 nF | Extremely stable, no drift | Little capacitance per volume |
| Ceramic X7R | 1 nF to 10 µF | High capacitance, small and cheap | Loses capacitance with applied voltage |
| Ceramic Y5V | 10 nF to 100 µF | Maximum capacitance per volume | Up to −80% with temperature and more with voltage |
| Aluminum electrolytic | 1 µF to 100 mF | High capacitance and cheap | Polarized, high ESR, dries out over the years |
| Tantalum | 100 nF to 1 mF | Stable and compact | Fails short-circuited, sometimes with flames |
| Polypropylene film | 1 nF to 100 µF | Very low losses, handles current | Bulky |
A 10 µF, 16 V X7R ceramic can end up at less than 3 µF when 10 V is applied to it, and that does not appear in the printed tolerance: it is the voltage coefficient, characteristic of barium titanate, whose permittivity drops as it polarizes. It is the cause of switching power supplies that oscillate and of filters that do not filter, and the error is invisible when measuring the capacitor with an instrument that tests it at one millivolt. The practical solution is to choose a rated voltage two or three times higher than the working voltage.
And on the losses side there is the ESR, the equivalent series resistance, which is what makes a capacitor heat up. In a switching power supply, ripple current flows continuously through the output capacitor, and the power \( I^2 R_{ESR} \) heats it from the inside: that accelerates the drying out of the electrolyte, which in turn raises the ESR. It is the most common failure mechanism of a power supply, and the one the kinetics unit explained with Arrhenius.
04Combinations and the transient
And charging it through a resistor produces the transient that governs half of a digital circuit:
05In the lab
Measure the capacitance of a 10 µF X7R ceramic with an LCR meter and then in a circuit with increasing DC bias voltage, obtaining the capacitance from the cutoff frequency of an RC filter. Plot capacitance against applied voltage.
Charge a known capacitor through different resistors and record the curve with the oscilloscope. Measure \( \tau \) at 63% and compare with \( RC \). Estimate the error introduced by the input impedance of the probe.
Measure the ESR of new electrolytics and of others salvaged from old equipment. Relate it to the condition of the equipment and estimate the heating with a ripple current of 1 A rms.
06Common mistakes
- Believing capacitance depends on the applied voltage. In the model it does not; in a real ceramic, it does a lot.
- Adding capacitances in series. In series the reciprocals add, the opposite of resistors.
- Using \( U = QV \) without the medium. The stored energy is \( \tfrac{1}{2}QV \).
- Reversing an electrolytic. It is polarized, and backwards it is destroyed, often noisily.
- Ignoring the ESR. It is what heats the capacitor and what ends up killing it.
- Choosing the rated voltage with no margin. Two or three times the working voltage is advisable, especially with ceramics.
07Self-assessment
What does the capacitance of a parallel-plate capacitor depend on?
On the area, the separation and the permittivity of the dielectric. Not on the voltage or the charge.
What happens if the dielectric strength is exceeded?
The material suddenly conducts and the capacitor is punctured. The failure is permanent.
Why can a 10 µF X7R end up with 3 µF?
Because of the voltage coefficient of barium titanate: its permittivity drops as it polarizes.
How are capacitances combined in parallel?
They add, because it is equivalent to adding areas.
What fraction of the final voltage does an RC reach in one time constant?
63%. In five time constants it reaches 99.3%.
What is ESR and why does it matter?
The equivalent series resistance. With the ripple current it dissipates \( I^2R \) and heats the capacitor from the inside, which accelerates its degradation.
08Further reading
- Raymond A. Serway and John W. Jewett. Physics for Scientists and Engineers with Modern Physics. 10th ed., Cengage, 2019 (Spanish edition, “Física para ciencias e ingeniería”). Capacitance, dielectrics, stored energy and combinations, with the classic treatment.
- Edward M. Purcell and David J. Morin. Electricity and Magnetism. 3rd ed., Cambridge University Press, 2016 (Spanish edition, “Electricidad y magnetismo”). The energy of the electric field and the polarization of a dielectric, from first principles.
- Paul Horowitz and Winfield Hill. The Art of Electronics. 3rd ed., Cambridge University Press, 2015. The appendix on passive components: which capacitor to use in each application and what the traps of each family are.