Advanced: mathematics, real capacitors, and applications
The ideal capacitor is simple; the real component is not. Dielectric properties, conductor resistance, package inductance, leakage, temperature, voltage, frequency, aging, and manufacturing tolerances all affect behavior.
Geometry and dielectric constant
For an ideal parallel-plate capacitor:
ε = ε0εr
- A is overlapping plate area.
- d is dielectric thickness.
- ε0 is the permittivity of free space.
- εr is the dielectric's relative permittivity.
Large area, a thin dielectric, or a high-permittivity dielectric increases capacitance. Modern components create large effective area by winding foils, etching rough surfaces, stacking layers, or sintering porous material.
Current and changing voltage
An ideal capacitor's current is proportional to how quickly its voltage changes. A sudden attempt to change capacitor voltage requires a large current. This is why capacitors oppose rapid voltage changes and why inrush current can be important when large capacitors are connected to a low-impedance source.
Rearranging and integrating gives:
Ideal AC impedance
|ZC| = 1 / (ωC)
The ideal current leads voltage by 90 degrees. Real components have losses that reduce this ideal phase relationship.
A practical equivalent circuit
A useful first model contains:
- An ideal capacitance.
- Equivalent series resistance (ESR).
- Equivalent series inductance (ESL).
- A large leakage resistance in parallel with the capacitance.
At low frequency the capacitive term dominates. As frequency rises, impedance falls until ESR limits the minimum. Above the self-resonant region, inductive behavior can dominate.
Self-resonant frequency
A capacitor used for high-frequency decoupling should have useful impedance at the frequencies of interest. Package size, mounting, via placement, and trace length can add as much inductance as the capacitor itself.
Dissipation factor and Q
Dielectric and resistive losses are often expressed by dissipation factor, tan δ. In a simplified series model:
For low-loss capacitors, quality factor is approximately the reciprocal of dissipation factor:
Ripple current and thermal design
Ripple current flowing through ESR produces heat. The temperature rise depends on ESR, current spectrum, thermal resistance, cooling, package geometry, and ambient temperature.
For electrolytic and power capacitors, the datasheet may give ripple limits at specific frequencies and temperatures plus correction factors for other conditions.
RC transient equations
For an initially discharged ideal capacitor charging through a resistor from a DC source:
For discharge from an initial voltage V0:
The time constant τ = RC controls how fast the exponential response occurs. These equations are the foundation of first-order timing and filter circuits.
First-order RC filters
For the simplest RC low-pass or high-pass network, the corner frequency is:
At the corner frequency, the magnitude of an ideal first-order response is about 0.707 of its passband value, corresponding to approximately −3 dB.
Energy and discharge hazards
Because energy increases with the square of voltage, a physically modest high-voltage capacitor can store dangerous energy. A bleeder resistor may be used to discharge a capacitor after power is removed:
Bleeder selection must consider required discharge time, resistor voltage rating, continuous power while energized, and fault conditions.
Dielectric absorption
Some dielectrics exhibit dielectric absorption: after a charged capacitor is discharged, a smaller voltage may slowly reappear. This matters in precision sample-and-hold circuits, integrators, high-voltage service work, and any application that assumes a discharged capacitor stays exactly at zero volts.
Capacitance versus voltage and temperature
Some capacitor technologies are quite stable; others change substantially with bias voltage or temperature. High-capacitance Class II ceramic capacitors can lose a significant portion of their nominal capacitance under DC bias. Electrolytics and tantalums have different temperature and frequency behavior. For precision work, the capacitance at the actual operating conditions matters more than the large number printed on the part.
Decoupling and power-distribution networks
A bypass capacitor is effective only if the complete current loop has low impedance. Good high-speed layout places the capacitor close to the device, minimizes loop area, and uses short, wide connections to power and ground. Several capacitor values may be used, but merely scattering values by decade does not guarantee low impedance; package inductance and board resonances must also be considered.
Snubbers and transient control
RC and RCD snubbers absorb or reshape switching transients. Capacitor selection must account for repetitive pulse current, dv/dt, RMS current, dielectric loss, and failure mode. Film capacitors are common in higher-energy snubber and DC-link service because of their pulse and AC characteristics.
Engineering selection
| Question | Why it matters |
|---|---|
| What capacitance is required at operating voltage? | Some dielectrics lose capacitance under DC bias. |
| What is the maximum steady and transient voltage? | Voltage margin affects reliability and safety. |
| What frequencies and currents are present? | ESR, ESL, ripple, and self-resonance determine heating and impedance. |
| What temperature range is expected? | Capacitance, ESR, leakage, and life may all change. |
| Is polarity fixed? | Polarized capacitors may fail under reverse voltage. |
| What failure mode is acceptable? | Open, short, venting, ignition risk, and stored energy can matter differently. |
| How long must the equipment last? | Electrolytic life and environmental stress can dominate service life. |
Sources and further reading
- TDK — aluminum electrolytic capacitor general technical information
- TDK — power capacitor technical information
- For actual component limits, use the exact manufacturer's current datasheet, reliability guidance, and application notes.