3.1 Capacitors

Capacitor fundamentals

A capacitor stores energy in the electric field between two conductors. This makes it one of the most important components in circuit analysis, since capacitors show up everywhere from power supplies to signal processing.

Definition and purpose

A capacitor is an electrical component with two conductive surfaces separated by an insulating gap. When you connect it to a voltage source, charge builds up on the plates: positive on one side, negative on the other. That separated charge creates an electric field, and energy is stored in that field.

A few key behaviors to know:

• A capacitor temporarily holds electric charge and releases it when needed
• It blocks DC current while allowing AC current to pass through
• It can smooth out voltage fluctuations in power supplies

Basic structure

Every capacitor has the same core design: two conductive plates separated by an insulating material called a dielectric. The plates can be metal foils, thin films, or other conductors. The dielectric can be air, paper, ceramic, or various polymers.

The physical dimensions of the capacitor directly determine how much charge it can store. Larger plate area means more room for charge to accumulate. Smaller separation between plates means a stronger electric field for the same voltage.

Capacitance units

Capacitance is measured in farads (F), named after Michael Faraday. One farad equals one coulomb of charge per volt of potential difference:

$C = \frac{Q}{V}$

where $C$ is capacitance, $Q$ is the stored charge, and $V$ is the voltage across the plates.

One farad is actually a huge amount of capacitance. Most real capacitors are measured in much smaller units:

• Microfarads ($\mu F$) = $10^{-6}$ F
• Nanofarads (nF) = $10^{-9}$ F
• Picofarads (pF) = $10^{-12}$ F

Types of capacitors

Different capacitor geometries produce different electric field patterns and different capacitance formulas. You'll encounter three main types in this course.

Parallel plate capacitors

This is the most common and most straightforward configuration: two flat conductive plates separated by a uniform dielectric. The capacitance is directly proportional to plate area and inversely proportional to plate separation. Most textbook problems start here because the electric field between the plates is approximately uniform, which simplifies the math considerably.

Cylindrical capacitors

These consist of two concentric cylindrical conductors with a dielectric filling the gap between them. Think of a cable with an inner wire and an outer sheath. The capacitance depends on the radii of the inner and outer cylinders and the length of the capacitor. You'll often see these in high-voltage applications like power transmission lines.

Spherical capacitors

Two concentric spherical shells with dielectric between them form a spherical capacitor. The capacitance depends on the radii of the inner and outer spheres. These show up less often in practical electronics but are useful in physics problems because of their high symmetry.

Capacitance calculation

Parallel plate formula

For a parallel plate capacitor, the capacitance is:

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

where $\varepsilon$ is the permittivity of the material between the plates, $A$ is the area of each plate, and $d$ is the separation between them.

This formula assumes a uniform electric field between the plates and ignores fringing effects at the edges. It's accurate when the plate dimensions are much larger than the separation distance.

Notice what this tells you: to increase capacitance, you can increase the plate area, decrease the separation, or use a material with higher permittivity.

Dielectric effects

Inserting a dielectric material between the plates increases the capacitance. The dielectric's molecules partially align with the applied field, which reduces the net electric field inside the capacitor. This means you can store the same charge at a lower voltage, or more charge at the same voltage.

The formula becomes:

$C = \frac{\kappa \varepsilon_0 A}{d}$

where $\kappa$ (kappa) is the dielectric constant (also called relative permittivity) and $\varepsilon_0$ is the permittivity of free space ($8.85 \times 10^{-12}$ F/m). Since $\kappa > 1$ for all real dielectrics, inserting one always increases capacitance. For example, paper has $\kappa \approx 3.7$, so a paper-filled capacitor stores about 3.7 times more charge than an air-filled one of the same dimensions.

Dielectrics also increase the maximum voltage the capacitor can handle before breakdown occurs.

Capacitors in series vs parallel

Capacitor combination rules work opposite to resistor rules, which trips up a lot of students.

Series (capacitors connected end-to-end):

$\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \ldots$

The total capacitance decreases. Each capacitor shares the same charge, but the voltage divides across them.

Parallel (capacitors connected across the same two nodes):

$C_{eq} = C_1 + C_2 + C_3 + \ldots$

The total capacitance increases. Each capacitor sees the same voltage, and the charges add up.

A quick way to remember: capacitors in parallel plus (add directly), capacitors in series use the reciprocal formula.

Energy storage in capacitors

Electric field energy

A charged capacitor stores energy in the electric field between its plates. The energy isn't on the plates themselves; it's distributed throughout the field in the gap. The energy density (energy per unit volume) at any point is:

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

where $E$ is the electric field strength. Higher permittivity materials store more energy per unit volume for the same field strength.

Potential energy formula

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

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

Use whichever form is most convenient based on what quantities you know. Notice that energy increases with the square of voltage. Doubling the voltage across a capacitor quadruples the stored energy.

Example: A $10 \, \mu F$ capacitor charged to $100$ V stores:

$U = \frac{1}{2}(10 \times 10^{-6})(100)^2 = 0.05 \text{ J}$

Charge vs voltage relationship

For an ideal capacitor, charge and voltage have a perfectly linear relationship: $Q = CV$. If you plot $Q$ vs $V$, you get a straight line whose slope equals the capacitance. The energy stored equals the area under this $Q$-$V$ curve, which is a triangle with area $\frac{1}{2}QV$.

Real capacitors can deviate slightly from this linear behavior due to effects like dielectric absorption and leakage current, but for this course, the ideal model is what you'll use.

Charging and discharging

RC circuits

When you connect a capacitor to a voltage source through a resistor, you get an RC circuit. The resistor limits how fast charge can flow onto the plates, so the capacitor charges gradually rather than instantly.

• Charging: Connect the capacitor to a voltage source through a resistor. Current starts high and decreases as the capacitor charges.
• Discharging: Disconnect the source and connect the charged capacitor across a resistor. Current starts high (opposite direction) and decreases as the capacitor loses its stored energy.

Time constant

The time constant $\tau$ controls how fast the charging or discharging happens:

$\tau = RC$

where $R$ is resistance in ohms and $C$ is capacitance in farads. The units work out to seconds.

After one time constant ($t = \tau$):

• During charging, the capacitor reaches about 63.2% of its final voltage
• During discharging, the capacitor drops to about 36.8% of its initial voltage

After $5\tau$, the capacitor is considered fully charged or fully discharged (it's reached about 99.3% of the way).

Exponential behavior

Both charging and discharging follow exponential curves:

Charging:

$V(t) = V_0\left(1 - e^{-t/RC}\right)$

Discharging:

$V(t) = V_0 \, e^{-t/RC}$

The current in both cases is also exponential, starting at a maximum value of $I_0 = V_0 / R$ at $t = 0$ and decaying toward zero. The key idea is that the rate of change is always proportional to how far the capacitor is from its final state.

Dielectrics in capacitors

Dielectric materials

A dielectric is an insulating material placed between the capacitor plates. Common examples include ceramics ($\kappa \approx 10\text{–}12{,}000$), plastics like polyethylene ($\kappa \approx 2.3$), paper ($\kappa \approx 3.7$), and even air ($\kappa \approx 1.0006$).

The dielectric constant $\kappa$ tells you how much the material multiplies the capacitance compared to a vacuum. Higher $\kappa$ means higher capacitance for the same physical size, which is why high-$\kappa$ ceramics are used when you need large capacitance in a small package.

Polarization mechanism

When you apply an electric field across a dielectric, the molecules inside respond by partially aligning with the field. This is called polarization, and it comes in several forms:

• Electronic polarization — the electron cloud around each atom shifts slightly relative to the nucleus. This happens in all dielectrics.
• Orientation polarization — polar molecules (like water) physically rotate to align with the field.
• Ionic polarization — positive and negative ions in a crystal shift in opposite directions.

All of these create tiny internal dipoles that produce a field opposing the applied field. The net effect is a reduced electric field inside the dielectric, which is why the capacitance increases.

Dielectric strength

Dielectric strength is the maximum electric field a material can withstand before it breaks down and starts conducting. It's measured in volts per meter (V/m). For example, air breaks down at about $3 \times 10^6$ V/m, while mica can handle around $200 \times 10^6$ V/m.

This property determines the maximum voltage rating of a capacitor. Engineers always apply a safety margin below the breakdown threshold. Temperature, humidity, and impurities can all reduce dielectric strength.

Applications of capacitors

Energy storage devices

Capacitors provide quick bursts of energy when needed. Camera flashes use a capacitor that charges slowly, then dumps all its energy into the flash tube in milliseconds. Defibrillators work the same way, delivering a controlled energy pulse to the heart. In power supplies, capacitors maintain voltage during brief interruptions. They also serve as backup power for memory retention in electronic devices.

Filtering in circuits

Capacitors are widely used as filters because they block DC and pass AC. In a DC power supply, a capacitor smooths out the AC ripple after rectification. In audio circuits, capacitors can separate high-frequency signals from low-frequency ones, which is how crossover networks in speaker systems direct bass to woofers and treble to tweeters.

Timing applications

Since RC circuits charge and discharge at predictable rates, they're natural choices for timing. The time constant $\tau = RC$ sets the delay. This principle is used in blinking turn signals, windshield wiper timing, clock signal generation in digital circuits, and oscillator circuits in synthesizers.

Capacitor combinations

Series connections

In a series connection, capacitors are linked end-to-end so the same charge $Q$ appears on each capacitor. The voltages add up across the chain:

$V_{total} = V_1 + V_2 + V_3 + \ldots$

The equivalent capacitance is always less than the smallest individual capacitor:

$\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \ldots$

Why use series? It increases the overall voltage rating. If each capacitor can handle 100 V, two in series can handle 200 V (though total capacitance drops).

Parallel connections

In a parallel connection, all positive plates are connected together and all negative plates are connected together. Every capacitor sees the same voltage, and the total stored charge is the sum of individual charges:

$C_{eq} = C_1 + C_2 + C_3 + \ldots$

Use parallel connections when you need more total capacitance or more charge storage capacity.

Equivalent capacitance

For complex networks with both series and parallel combinations, simplify step by step:

1. Identify groups of capacitors that are purely in series or purely in parallel.
2. Replace each group with its equivalent capacitance.
3. Redraw the simplified circuit.
4. Repeat until you have a single equivalent capacitance.

This is the same strategy you use for resistor networks, just with the series/parallel formulas swapped.

Capacitor limitations

Breakdown voltage

Every capacitor has a maximum voltage rating. If you exceed it, the dielectric breaks down and becomes conducting, which usually destroys the capacitor permanently. The breakdown voltage depends on the dielectric material, its thickness, and environmental conditions. Always choose a capacitor with a voltage rating comfortably above your circuit's operating voltage.

Leakage current

No dielectric is a perfect insulator. A small leakage current flows through the dielectric even under constant voltage, causing the capacitor to slowly lose its charge over time. This current increases with temperature and applied voltage. Leakage matters most in low-power circuits and applications where a capacitor needs to hold its charge for extended periods.

Temperature effects

Capacitance values can drift with temperature. Each capacitor has a temperature coefficient that describes how much its capacitance changes per degree. Dielectric properties also shift with temperature, affecting both capacitance and breakdown voltage. For reliable operation, check that your capacitor is rated for the temperature range of your application.

Advanced capacitor concepts

Variable capacitors

Some capacitors allow you to adjust their capacitance. Mechanical variable capacitors use overlapping plates that can be rotated to change the effective area. Electronic variable capacitors called varactors use a reverse-biased semiconductor junction whose capacitance changes with applied voltage. Classic applications include tuning a radio to different stations and voltage-controlled oscillators.

Supercapacitors

Supercapacitors (also called ultracapacitors or EDLCs) achieve enormously high capacitance values, sometimes hundreds or thousands of farads. They store energy through ion adsorption at an electrode-electrolyte interface rather than through a traditional dielectric. They fill the gap between conventional capacitors (which charge/discharge very fast but store little energy) and batteries (which store lots of energy but charge/discharge slowly).

Quantum capacitance

At the nanoscale, quantum effects introduce an additional capacitance term called quantum capacitance. It arises because low-dimensional materials like graphene have a finite density of electronic states, which limits how much additional charge can be added at a given energy. Quantum capacitance adds in series with the geometric capacitance, reducing the total. This concept is relevant to nanoelectronics and emerging quantum computing devices, though it's well beyond typical circuit analysis.