6.1 Capacitors and capacitance

Capacitor Fundamentals

Capacitors store energy in electric fields and release it when needed. They show up everywhere in circuits, from smoothing out power supply ripple to setting the timing in oscillator circuits. This section covers how they work, how they're built, and the key equations you need to know.

Definition and Concepts

A capacitor is two conductors (called plates) separated by an insulating material called a dielectric. When you apply a voltage across the plates, an electric field forms in the dielectric, and that field stores energy.

Capacitance ($C$) measures how much charge a capacitor can store for a given voltage. It depends on three physical factors: the area of the plates, the distance between them, and the properties of the dielectric material.

• The unit of capacitance is the farad (F): $1 \text{ F} = 1 \text{ C/V}$
• One farad is a huge amount of capacitance. In practice, you'll mostly see values in picofarads (pF, $10^{-12}$ F), nanofarads (nF, $10^{-9}$ F), and microfarads (μF, $10^{-6}$ F).

Charge Storage and Energy

When voltage is applied across a capacitor, positive charges accumulate on one plate and negative charges on the other. The charge stored is directly proportional to the applied voltage:

$Q = CV$

where $Q$ is charge in coulombs, $C$ is capacitance in farads, and $V$ is voltage in volts. So a 10 μF capacitor charged to 5 V stores $Q = 10 \times 10^{-6} \times 5 = 50 \text{ μC}$ of charge.

The energy stored in a capacitor is:

$E = \frac{1}{2}CV^2$

Notice the $V^2$ term. Doubling the voltage quadruples the stored energy, which is why exceeding a capacitor's voltage rating can be so destructive.

Capacitor Construction

Parallel Plate Capacitor

The simplest capacitor model is two parallel conductive plates separated by a dielectric. Its capacitance is given by:

$C = \frac{\varepsilon A}{d}$

where $A$ is the plate area, $d$ is the separation distance, and $\varepsilon$ is the permittivity of the dielectric material.

This formula tells you three ways to increase capacitance:

• Increase plate area ($A$) — more surface means more room for charge to accumulate
• Decrease plate separation ($d$) — bringing the plates closer strengthens the electric field for a given voltage
• Use a dielectric with higher permittivity ($\varepsilon$) — the material between the plates matters a lot

Dielectric Materials

The dielectric is the insulating layer between the plates. Common dielectric materials include air, paper, plastic (like polyester or polypropylene), ceramic, and metal oxides.

Permittivity ($\varepsilon$) describes how easily a dielectric material polarizes in an electric field. It's usually expressed as:

$\varepsilon = \varepsilon_r \varepsilon_0$

where $\varepsilon_0$ is the permittivity of free space ($8.854 \times 10^{-12}$ F/m) and $\varepsilon_r$ is the relative permittivity (also called the dielectric constant). For example, air has $\varepsilon_r \approx 1$, while ceramic dielectrics can have $\varepsilon_r$ values in the thousands, which is why ceramic capacitors can pack a lot of capacitance into a small package.

Dielectric strength is the maximum electric field a dielectric can handle before it breaks down and starts conducting. Breakdown destroys the capacitor, so dielectric strength directly determines the voltage rating.

Capacitor Specifications

Voltage Rating and Breakdown

Every capacitor has a voltage rating, which is the maximum voltage you can safely apply across it. Exceeding this rating causes the electric field inside the dielectric to surpass the dielectric strength, leading to breakdown and permanent failure.

The voltage rating depends on two things:

• Dielectric thickness — a thicker dielectric can withstand more voltage
• Dielectric strength of the material — materials with higher dielectric strength tolerate stronger electric fields

There's a design tradeoff here: increasing $d$ raises the voltage rating but decreases capacitance (since $C = \varepsilon A / d$). Engineers balance these competing demands based on the application.

Electric Field Considerations

The electric field between the plates of a parallel plate capacitor is:

$E = \frac{V}{d}$

where $E$ is the field strength in volts per meter (V/m), $V$ is the applied voltage, and $d$ is the plate separation.

This field must stay below the dielectric strength at all points inside the capacitor. In a perfect parallel plate geometry, the field is uniform. In real capacitors, edges and irregular geometry can create localized regions where the field is stronger than average. These "hot spots" are where breakdown tends to happen first, so capacitor design aims to minimize field non-uniformities through careful geometry and construction.