Capacitors, Capacitance and Charge

Introduction to Capacitors

A capacitor is an electrical device that is used to store electrical energy. A capacitor consists of two metal plates separated by an insulator called a dielectric. It stores electricity in the form of an excess of electrons on one plate and a deficiency on the other. In this state, the capacitor is said to be charged. The charge is achieved by applying a voltage across the plates.

Capacitors are used to smooth rectified AC outputs, they are used in telecommunication equipment such as radio receivers for tuning to the required frequency, they are used in time delay circuits, in electrical filters, in oscillator circuits, in magnetic imaging (MRI) instruments, etc.

Electrostatic Field

The diagram below illustrates two parallel plates X and Y, charged to different potentials:

If an electron that has negative charge is placed between the plates, a force will act on the electron tending to push it away from the negative plate Y towards the positive plate X likewise a positive charge would be acted on by a force tending to move it toward the negative plate. Any space such as that shown between the plates in which an electric charge experiences a force is called an electrostatic field. The direction of the field is defined as that of the force acting on a positive charge placed in the field. For example in the figure above, the direction of the force is from the positive plate to the negative plate.

Such a field can be represented in magnitude and direction by lines of electric force drawn between the charged surfaces. The closeness of the lines is an indication of the field strength. Whenever a voltage is established between two points an electric field will always exist.

Electric lines of force (also called electric flux lines) are continuous and start and finish on point charges. In addition, the lines cannot cross each other. When a charged body is placed close to an uncharged body, an induced charge of opposite sign appears on the surface of the uncharged body. This is because lines of force from the charged body terminate on its surface.

The force of attraction or repulsion between two electrically charged particles is proportional to the magnitude of their charges and inversely proportional to the square of the distance separating them, i.e.

Where constant k ≈ 9 x 10⁹ in air

This is known as coulomb’s law.

Electric Field Strength

Let’s consider two parallel conducting plates separated from each other by air and connected to opposite terminals of a battery of voltage V volts.

There will be an electric field in the space between the plates. If the plates are close together, the electric lines of force will be straight and parallel and equally spaced except near the edge where fringing will occur. Over the area in which there is negligible fringing.

Where, d is the distance between the plates. Electric field strength is also called potential gradient.

Capacitance

Static electric field arises from electric charges, electric field lines beginning and ending on electric charges. Thus, the presence of the field indicates the presence of equal positive and negative electric charges on the two plates as shown in the figure above.

Let the charge + Q coulombs on one plate and – Q coulombs on the other. The property of this pair of plates that determines how much charge corresponds to a given voltage between the plates is called the capacitance.

The unit of capacitance is the farad F (or μF =10^-6 or pF =10^-12 F), which is defined as the capacitance of a capacitor which requires a potential difference of 1 volt to maintain a charge of 1 coulomb on that capacitor.

Therefore, charge = capacitance x volt

Q(C) = C(F) X V(V)

Capacitors

Every system of electrical conductors possesses capacitances; there is capacitance between the conductors of overhead power transmission lines and also between the wires of a telephone cable. In these examples, the capacitance is undesirable but has to be accepted, minimized or compensated for. There are other instances, where capacitance is a desirable property; Devices specifically built to possess capacitance are called capacitors.

The charge Q stored in a capacitor is given by:

Q = I x t coulombs

Where I, is the current in amperes and t is the time in seconds.

Symbols for a fixed capacitor and a variable capacitor used in in electrical circuits are shown below:

Electric Flux Density

Unit flux is defined as emanating from a positive charge of 1 coulomb. Thus electric flux Ψ is measured in coulombs, and for a charge of Q Coulombs, the flux Ψ = Q Coulombs.

Electric flux density D is the amount of flux passing through a defined area A, that is perpendicular to the direction of the flux.

Electric flux density is also termed to as charge density σ.

Permittivity

At any point in an electric field, the electric field strength E maintains the electric flux and produces a particular value of electric flux density D at that point. For a field established in vacuum (assume air for practical purposes); the ratio D/E is a constant ε₀ i.e.

Where ε₀ is termed to as the permittivity of free space or the free space constant; the value of ε₀ is 8.85 x 10^-12 F/M.

When an insulating medium, such as mica, paper, plastic, or ceramic is introduced into region of an electric field the ratio of D/E is modified.

Where ε_r is the relative permittivity of the insulating material indicating its insulating power compared with that of vacuum.

Here ε_r has no unit. The typical values of ε_r include mica, 3.7; air, 1.00; polythene, 2.3; glass, 5-10; water, 80; ceramics 6-1000.

The product ε₀ε_r is called the absolute permittivity, ε.

 ε₀ε_{r =} ε

The insulating medium separating charged surfaces is called a dielectric. Compared with the conductors, dielectric materials have very high resistivity. Hence, they are used to separate conductors at different potentials such as capacitor plates or electric power lines.

Parallel Plate Capacitor

A parallel Plate capacitor is illustrated in the diagram below:

Experiments show that capacitance C is proportional to the Area A of a plate, inversely proportional to the plate spacing d i.e. the dielectric thickness and depends on the nature of dielectric.

Where ε₀ = 8.85 x 10^-12 F/m (constant)

ε_0, = Relative permittivity

A = Area of one of the plates in m²

d = thickness of dielectric in m

Another method employed to increase the capacitance is to interleave several plates as shown in fig Y above. 10 plates are shown, forming 9 capacitors with a capacitance 9 times that of one pair of plates.

If such an arrangement in fig Y is used having n plates, then capacitance;

C α (n-1)

Capacitor Networks

In practical circuits, capacitors are often connected together. They can either be connected in series or parallel.

Capacitors connected in Series

The figure below shows 3 capacitors C₁, C₂ and C₃ connected in series across a supply voltage V.

Let the voltage across the individual capacitors be V₁, V₂ and V₃ respectively.

Let’s assume the charge on plate ‘a’ of capacitor C₁ be +Q coulombs. This induces an equal but opposite charge of –Q coulombs on plate ‘b’. The conductor between plates ‘b’ and ‘c’ is electrically isolated from the rest of the circuit so that an equal but opposite charge of +Q coulombs must appear on plate ‘c’ which in turn induces an equal and opposite charge of –Q coulombs on plate ‘d’ and so on.

Just as the current is common to all parts of series resistive circuit, so charge is common in series capacitive circuit.

That is for a series connected capacitors, the reciprocal of the equivalent capacitance is equal to the sum of the reciprocals of the individual capacitances.

For special case of two capacitors in series:

Capacitors Connected in Parallel

The figure below shows 3 capacitors C₁, C₂ and C₃ connected in parallel with a supply voltage V applied across the arrangement.

When the charging current reaches point A it divides, some flowing into C₁ some flowing into C₂ and some into C₃. Hence, the total charge, Q_T = (I x t) is divided between the 3 capacitors. The capacitors each store a charge and these are shown as Q₁, Q₂, and Q₃.

Q_T = Q₁ + Q₂ + Q₃

But Q_T = CV, Q₁ = C₁V, Q₂ = C₂V, Q₃ = C₃V

Therefore, CV = C₁V + C₂V + C₃V where C is the total equivalent circuit capacitance.

That is, C = C₁+ C₂ + C₃

It then follows that for n parallel-connected capacitors

C = C₁+ C₂ + C₃+…+C_n

That is, the equivalent capacitance of a group of parallel-connected capacitors is the sum of the capacitors of the individual capacitors.

Energy Stored by a Capacitor

The energy W, stored by a capacitor is given by:

W = ½ CV²joules

Dielectric Strength

The maximum amount of field strength that a dielectric can withstand is called the dielectric strength of the material.

Where d is the thickness of the dielectric material. V_m is the voltage applied across the plates.

Capacitance in AC Circuits

In an AC circuit a capacitance has the effect of opposing the voltage, thus causing the circuit current to lead. In a purely capacitive circuit the current leads the voltage by 90°. The waveforms and phasors of this kind of circuit are shown in the following diagrams:

The opposition offered by a capacitor in AC circuit is called the capacitive reactance and is given by:

Where X_C = capacitance reactance (ꭥ)

 f, = frequency of supply (Hz)

C = capacitance (F)

As with the inductive reactance, ohm’s law can be applied:

V = I x X_C

Related: Inductance

Practical Types of Capacitors

The practical types of capacitors are characterized by the main material used for their dielectric. The main types of capacitors are discussed in details as follows:

Variable Air Capacitors

These usually consist of two sets of metal plates (e.g. Aluminium) one plate is fixed, the other variable. The set of moving plates rotate on a spindle as shown below:

As the moving plates are rotated through half a revolution, the meshing and therefore the capacitance, varies from a minimum to a maximum value. The variable air capacitors are employed in Radio and Electronic circuits where very low losses are required or where a variable capacitance is needed. The maximum value of such capacitors is between 500 pF and 100 pF.

Paper Capacitors

An example of a paper capacitor is shown below where the length of the roll corresponds to the capacitance needed.

This whole assembly is usually soaked in oil or wax to exclude moisture, and then placed in a plastic or aluminium container for protection.

Paper capacitors are made in various working voltage up to about 150 kV and are used where loss is not very important.

The maximum value of paper capacitors is between 500 pF and 1000 pF.

Disadvantages of Paper Capacitors

• The variation in capacitance with temperature change.
• Shorter service life than the most types of capacitors.

Mica Capacitors

Usually the whole capacitor is soaked in wax and placed in Bakelite case. Mica is easily obtained in thin sheets and is a good insulator however mica is expensive and is not used in a capacitor above about 0.2 μF.

A modified form of mica capacitor is the silvered mica type. The mica is coated on both sides with a thin layer of Silver, which forms the plates. Capacitance is stable and less likely to change with age. Such capacitors have a constant capacitance with change of temperature, a high working voltage rating, long service life and are used in high frequency circuits with fixed values of capacitance up to about 1000 pF.

Ceramic Capacitors

They are made in various forms, each type of construction depending on the value of capacitance required.

For high values of a capacitance, a tube ceramic material is used. For smaller values the cup construction and disc construction are used. Certain ceramic have a very high permittivity and this enables capacitors of high capacitance to be made which are of small physical size with a high working voltage rating. Ceramic capacitors are available in the range of 1 pF to 0.1 pF and may be used in high frequency electronic circuits subject to a wide range of temperatures.

Electrolytic Capacitors

Their construction is similar to the paper capacitor with aluminium used for the plates and with thick absorbent material such as paper soaked in an electrolyte (ammonium borate) separating the plates. The completed capacitor is usually assembled in an aluminium container and hermetically sealed. Its operation depends on the formation of a thin aluminium oxide layer on the positive plate by the electrolytic action when a direct potential is maintained between the plates. This oxide layer is very thin and forms the dielectric. Note that, the absorbent paper between the plates is conductor and does not act as a dielectric.

These kinds of capacitors must always be used on DC and must be connected with correct polarity; otherwise the capacitor will be destroyed since the oxide layer will be destroyed.

Electrolytic capacitors are manufactured with working voltage from 6 V to 600 V, although accuracy is not very high. These capacitors possess a much larger capacitance than other types of capacitors of similar dimensions due to the oxide film being only a few microns thick.

Limitations of Electrolytic Capacitors

They are only limited to use on DC supplies.

Plastic Capacitors

Some plastic materials such as Polystyrene and Teflon can be used as dielectrics. The construction of plastic capacitors is similar to the paper capacitors but using film instead of paper. Plastic capacitors operate well under conditions of high temperature, provide a precise value of capacitance, serve long life, and have high reliability.

Titanium Oxide Capacitors

They have a very high capacitance with a small physical size when used at a low temperature.

Applications of Capacitors

Capacitors have a wide range of applications in electrical and electronics engineering. For example, they are used in:

• Motor starting.
• Power factor correction.
• Radio interference suppression.
• Minimizing the stroboscopic effects in fluorescent lighting circuits.