Capacitors

A capacitor is a device used to store electrical energy.

The plates of a capacitor is charged and there is an electric field between them. The capacitor will be discharged if the plates are connected together through a resistor.

Charge of a Capacitor

The charge of a capacitor can be expressed as

Q = I t                   (1)

where

Q = charge of capacitor (coulomb, C, mC)

I = current (amp, A)

t = time (s)

The quantity of charge (number of electrons) is measured in the unit Coulomb - C - where

1 coulomb = 6.24 10¹⁸ electrons

The smallest charge that exists is the charge carried by an electron, equal to -1.602 10^-19 coulomb .

Example - Quantity of Electricity Transferred

If a current of 5 amp flows for 2 minutes, the quantity of electricity - coulombs - can be calculated as

Q = (5 A) (2 min) (60 s/min)

= 600 C

or, in electrons:

(600 C) ( 6.24 10¹⁸ electrons / C)

= 3.744 10²¹ electrons

Electric Field Strength (Dielectric Strength)

If two charged plates are separated with an insulating medium - a dielectric -  the electric field strength (potential gradient) between the two plates can be expressed as

E = U / d                        (2)

where

E = electric field strength (volts/m)

U = eletrical potential (volt)

d = thickness of dielectric, distance between plates (m)

Example - Electric Field Strength

The voltage between two plates is 230 V and the distance between them is 5 mm . The electric field strength can be calculated as

E = (230 V) / ((5 mm) (10^-3 m/mm))

= 46000 volts/m

= 46 kV/m

Electric Flux Density

Electric flux density is the ratio between the charge of the capacitor and the surface area of the capacitor plates:

D = Q / A                       (3)

where

D = electric flux density (coulomb/m²)

A = surface area of the capacitor (m²)

Charge and Applied Voltage

Charge in a capacitor is proportional to the applied voltage and can be expressed as

Q = C U                        (4)

where

C = constant of proportionality or capacitance (farad, F, µF )

A farad is an enormous capacitance so it is common to deal with microfarads (μF), nanofarads (nF) or picofarads (pF).

Capacitance

From (4) the capacitance can be expressed as

C = Q / U                         (5)

One farad is defined as the capacitance of a capacitor when there is a potential difference across the plates of one volt when holding a charge of one coulomb.

It is common to use µF (10^-6 F) .

Example - Voltage over a Capacitor

A 5 µF capacitor is charged with 10 mC . The voltage across the capacitor can be calculated by modifying (4) to

U = Q / C

= (10 mC) (10^-3 C/mC) / ((5 µF) (10^-6 F/µF)

= 2000 V

= 2 kV

Absolute Permittivity

The ratio of electric flux density to electric field is called absolute permittivity - ε - of a dielectric and can be expressed as

ε = D / E                     (6)

where

ε = absolute permittivity (F/m, farad/m)

The absolute permittivity of free space or vacuum - also called the electric constant - ε₀ - is 8.85 10^-12F/m .

Relative Permittivity

Relative permittivity - also called the dielectric constant ε _r - is the ratio between the flux density of the field in an actual dielectric - ε - and the flux density of the field in absolute vacuum - ε ₀ .

ε _r = ε / ε ₀                    (7a)

The actual permittivity can be calculated by transforming (7a) to

ε = ε _r ε ₀                      (7b)

Parallel Plate Capacitor

The capacitance of a plate capacitor - as shown in the figure above - is proportional with the area A of the plate. The capacitance can be expressed as

C = ε _r ε₀ A / d                   (8)

where

A = area of plate (m²)

d = thickness of dielectric, distance between plates (m)

For a plate capacitor with multiple plates the capacitance can be expressed as

C = ε _r ε₀ A (n - 1) / d                      (8b)

where

n = number of plates

Example - Capacitance of a Plate Capacitor

The capacitance of a plate capacitor with area 5 cm², 10 plates and distance 0.1 mm between the plates - with ceramic dielectric with relative permittivity 30 between the plates - can be calculated as

C = ( 8.85 10^-12F/m ) (30) (5 cm²) (10^-4 m²/cm²) (10 - 1) / ((0.1 mm) (10^-3 m/mm))

= 11 10^-9F

= 11 pF

Typical commonly used Capacitors

Typical capacitors are

• variable air capacitors
• mica capacitors
• paper capacitors
• ceramic capacitors
• plastic capacitors
• titanium oxide capacitors
• electrolytic capacitors

Capacitor as Frequency-dependent Resistor

Since a capacitor looks like a short circuit at higher AC frequencies - capacitors can be considered as simply frequency-dependent resistors that allow you to make frequency-dependent voltage di-
viders.