3.2 Capacitance

Capacitance measures a system's ability to store electric charge. It's central to understanding how energy gets stored in electric fields and how capacitors behave in circuits.

Capacitors show up everywhere in electronics. They store energy, filter signals, and create time delays. This topic covers capacitor types, construction, energy storage, and how capacitors behave in different circuit configurations.

Fundamentals of capacitance

Capacitance is about storing electrical energy in an electric field. Along with resistors and inductors, capacitors form the core building blocks of circuit design. Getting comfortable with capacitance gives you a solid foundation for understanding charge storage, electric fields, and potential differences.

Definition and basic concepts

Capacitance is the ratio of stored charge to the voltage applied across the system:

$C = Q/V$

It depends on the geometry of the capacitor and the materials used. The SI unit is the farad (F), but real-world capacitors are usually much smaller, with values in the picofarad (pF) to microfarad (μF) range.

Capacitors vs resistors

These two components do fundamentally different things:

• Capacitors store energy in electric fields; resistors dissipate energy as heat.
• Capacitors block DC current but allow AC current to pass. Resistors impede current regardless of type.
• A capacitor's opposition to current (reactance) changes with frequency. A resistor's resistance stays constant.
• Capacitors introduce a phase shift between voltage and current. Resistors keep voltage and current in phase.

Units of capacitance

One farad equals one coulomb per volt. That's actually a huge amount of capacitance, so you'll almost always see smaller prefixes:

• $1 \text{ F} = 10^6 \text{ μF} = 10^9 \text{ nF} = 10^{12} \text{ pF}$

You can also express the farad in terms of other electrical units: $1 \text{ F} = 1 \text{ C/V} = 1 \text{ s/Ω}$.

Types of capacitors

Different capacitor geometries suit different applications. The three classic types you need to know are parallel plate, cylindrical, and spherical.

Parallel plate capacitors

Two flat, parallel conductive plates separated by a dielectric material. Capacitance is directly proportional to plate area and inversely proportional to the distance between plates:

$C = \varepsilon A / d$

where $\varepsilon$ is the permittivity of the dielectric, $A$ is the plate area, and $d$ is the separation. This is the most common model for understanding capacitance and appears frequently in introductory problems.

Cylindrical capacitors

Two concentric cylindrical conductors separated by a dielectric. You'll find this geometry in coaxial cables and some high-voltage applications like power transmission lines. The capacitance is:

$C = 2\pi\varepsilon L / \ln(b/a)$

where $L$ is the cylinder length, $a$ is the inner radius, and $b$ is the outer radius.

Spherical capacitors

Two concentric spherical shells with a dielectric between them. These show up in specialized applications requiring symmetrical charge distribution. The capacitance is:

$C = 4\pi\varepsilon / (1/a - 1/b)$

where $a$ is the inner radius and $b$ is the outer radius. The electric field strength decreases with distance from the center, following the spherical symmetry.

Capacitor construction

How a capacitor is built determines its capacitance, voltage rating, size, and cost. Three factors matter most: the dielectric material, the plate geometry, and the spacing between plates.

Dielectric materials

The insulating material between the plates increases capacitance beyond what you'd get with just air or vacuum. Common dielectrics include paper, plastic films, ceramics, and metal oxides.

• Dielectric strength sets the maximum electric field the material can handle before breaking down.
• Dielectric constant (relative permittivity, $\varepsilon_r$) determines how much the material boosts capacitance.

Plate geometry and spacing

Larger plate area increases capacitance; greater plate separation decreases it. Many real capacitors use interleaved plate designs to pack more effective area into a compact volume. At very small plate separations, edge effects and fringing fields become significant and the simple formulas lose accuracy.

Capacitor size vs capacitance

Modern miniaturization techniques can pack high capacitance into tiny packages. Surface-mount capacitors, for example, provide impressive capacitance density for circuit boards. There are always trade-offs, though: shrinking the size while increasing capacitance means pushing closer to the dielectric's breakdown voltage, and cost goes up.

Calculating capacitance

Each geometry has its own formula. The key to all of them is understanding what physical quantities drive the capacitance up or down.

Parallel plate capacitor formula

$C = \varepsilon A / d$

The permittivity $\varepsilon$ breaks down into two parts: $\varepsilon = \varepsilon_0 \varepsilon_r$, where $\varepsilon_0 = 8.85 \times 10^{-12} \text{ F/m}$ is the permittivity of free space and $\varepsilon_r$ is the relative permittivity of the dielectric.

This formula assumes a uniform electric field and neglects fringing at the edges. It's accurate when the plate separation is much smaller than the plate dimensions.

Capacitance in cylindrical geometry

$C = 2\pi\varepsilon L / \ln(b/a)$

Here $L$ is the length of the cylinders, $a$ is the inner conductor radius, and $b$ is the outer conductor radius. This is the formula you'd use for a coaxial cable or any concentric-cylinder arrangement.

Capacitance in spherical geometry

$C = 4\pi\varepsilon / (1/a - 1/b)$

The variables $a$ and $b$ are the inner and outer sphere radii. This assumes perfect spherical symmetry. If you let $b \to \infty$, you get the capacitance of an isolated sphere: $C = 4\pi\varepsilon_0 a$.

Energy storage in capacitors

Capacitors store energy in the electric field between their plates. This is what makes them useful for applications ranging from camera flashes to power supply smoothing.

Electric field energy

The energy stored comes from the work done to build up the electric field. For a parallel plate capacitor, the field strength between the plates is:

$E = V/d$

The energy density (energy per unit volume) in that field is proportional to the square of the field strength.

Potential energy in capacitors

The total energy stored in a charged capacitor can be written three equivalent ways:

$U = \frac{1}{2}CV^2 = \frac{1}{2}QV = \frac{Q^2}{2C}$

Pick whichever form is most convenient based on what quantities you know. Notice that energy increases with the square of voltage or charge, so doubling the voltage quadruples the stored energy.

Energy density of capacitors

Energy density tells you how much energy is stored per unit volume:

$u = \frac{1}{2}\varepsilon E^2$

To get higher energy density, you need stronger electric fields or materials with higher permittivity. Supercapacitors achieve high energy density by using porous electrodes with enormous surface area.

Capacitors in circuits

Capacitors behave very differently from resistors in circuits. They introduce time-dependent effects during charging and discharging, and their response depends on frequency in AC circuits.

Charging and discharging processes

When you connect a capacitor to a voltage source through a resistor:

1. Current flows onto the plates, building up charge.
2. As charge accumulates, the voltage across the capacitor rises.
3. The rising capacitor voltage opposes the source, so current decreases exponentially.
4. Eventually the capacitor voltage equals the source voltage, and current drops to zero.

Discharging is the reverse: connect the charged capacitor across a load, and current flows out, decaying exponentially as the stored charge depletes.

RC time constant

The time constant $\tau = RC$ controls how fast a capacitor charges or discharges in an RC circuit.

• After one time constant ($t = \tau$), the capacitor has charged to about 63.2% of the source voltage.
• After $5\tau$, the capacitor is effectively fully charged (over 99%).

This parameter is essential for designing timing circuits and analyzing transient behavior.

Steady-state behavior

Once transients die out:

• In DC circuits, a fully charged capacitor acts as an open circuit (no current flows).
• In AC circuits, the capacitor has a frequency-dependent impedance: $Z_C = 1/(j\omega C)$, where $\omega$ is the angular frequency.

At higher frequencies, the impedance drops, so more current passes through. Ideal capacitors produce a 90° phase shift where current leads voltage.

Capacitors in series and parallel

Combining capacitors lets you achieve capacitance values you can't get from a single component. The rules are the opposite of what you might expect from resistors.

Equivalent capacitance in series

For capacitors in series, the reciprocals add:

$1/C_{eq} = 1/C_1 + 1/C_2 + \cdots + 1/C_n$

The total capacitance is always less than the smallest individual capacitor. Each capacitor in a series chain stores the same charge $Q$, but the voltage divides among them.

Equivalent capacitance in parallel

For capacitors in parallel, the capacitances add directly:

$C_{eq} = C_1 + C_2 + \cdots + C_n$

The total is always greater than the largest individual capacitor. All parallel capacitors share the same voltage, but charge distributes among them in proportion to their capacitance.

Voltage and charge distribution

This is the opposite pattern from resistors in series and parallel, which is a common source of confusion.

Dielectrics and capacitance

Inserting a dielectric material between the plates of a capacitor changes its behavior significantly. This is one of the main ways engineers control capacitance in practice.

Dielectric constant

The dielectric constant (relative permittivity, $\varepsilon_r$) measures how well a material stores electric field energy compared to vacuum. Vacuum has $\varepsilon_r = 1$. Common materials range from about 2 (Teflon) to over 1000 (certain ceramics like barium titanate).

Effect on capacitance

Inserting a dielectric multiplies the capacitance:

$C = \varepsilon_r C_0$

where $C_0$ is the capacitance with vacuum between the plates. This means you can get the same capacitance in a physically smaller package by choosing a high-$\varepsilon_r$ material. The trade-off is that every dielectric has a maximum voltage it can handle before it breaks down.

Polarization of dielectrics

Here's why dielectrics increase capacitance. When an external electric field is applied, the molecular dipoles in the dielectric partially align with the field. This alignment creates an internal electric field that opposes the applied field, reducing the net field between the plates. With a weaker net field for the same charge, the voltage drops, and since $C = Q/V$, the capacitance goes up.

Different materials exhibit different types of polarization: electronic (shifting of electron clouds), ionic (displacement of ions), and orientational (rotation of permanent dipoles).

Applications of capacitors

Capacitors are among the most versatile components in electronics. The specific type you choose depends on the application's requirements for capacitance, voltage rating, frequency response, and size.

Energy storage devices

• Power supplies use capacitors to smooth out voltage fluctuations after rectification.
• Backup power systems in electronics use large capacitors to maintain operation during brief outages.
• High-power delivery applications like camera flashes and defibrillators charge a capacitor slowly, then discharge it rapidly to deliver a burst of energy.

Filtering in electronics

• Coupling capacitors block DC while passing AC signals between circuit stages.
• Smoothing capacitors reduce ripple in rectified power supplies.
• Bypass capacitors shunt high-frequency noise to ground, keeping it out of sensitive circuits.
• Combined with inductors, capacitors form resonant circuits that select specific frequencies.

Timing circuits

In RC circuits, the predictable exponential charging curve makes capacitors ideal for creating time delays. LC circuits (inductor + capacitor) generate oscillations at a frequency set by the component values. These principles underlie electronic timers, clock circuits, and microcontroller timing.

Advanced capacitor concepts

Beyond standard capacitors, several specialized technologies extend what's possible with capacitive energy storage and circuit design.

Variable capacitors

These allow you to adjust capacitance within a circuit. Mechanical types (like air variable capacitors) change the plate overlap area. Electronic types called varactors are semiconductor diodes whose capacitance changes with applied voltage. Variable capacitors are used in radio tuning circuits and other frequency-selective applications.

Supercapacitors

Also called ultracapacitors or electric double-layer capacitors (EDLCs), these achieve capacitance values of farads or even thousands of farads. They use porous electrodes with extremely large surface areas. Supercapacitors bridge the gap between conventional capacitors (fast charge/discharge, low energy) and batteries (slow charge/discharge, high energy).

Quantum capacitance

In nanoscale materials like graphene, the finite density of electronic states adds an extra capacitance term in series with the geometric capacitance. This quantum capacitance can become the dominant factor at the nanoscale, and it must be accounted for when designing nanoscale electronic devices and sensors. It's calculated using quantum mechanical principles rather than classical electrostatics.