Capacitance and Charge




 
 Capacitors store electrical energy on their plates in the form of an
electrical charge


Capacitance is the measured value of the ability of a capacitor to store an
electric charge. This capacitance value also depends on the dielectric
constant of the dielectric material used to separate the two parallel
plates. Capacitance is measured in units of the Farad (F), so named after
Michael Faraday.

Capacitors consist of two parallel conductive plates (usually a metal)
which
are prevented from touching each other (separated) by an insulating
material
called the “dielectric”. When a voltage is applied to these plates an
electrical current flows charging up one plate with a positive charge with
respect to the supply voltage and the other plate with an equal and
opposite
negative charge.

Then, a capacitor has the ability of being able to store an electrical
charge Q (units in Coulombs) of electrons. When a capacitor is fully
charged
there is a potential difference, (p.d.) between its plates, and the larger
the area of the plates and/or the smaller the distance between them (known
as separation) the greater will be the charge that the capacitor can hold
and the greater will be its Capacitance.

The capacitors ability to store this electrical charge ( Q ) between its
plates is proportional to the applied voltage, V for a capacitor of known
capacitance in Farads. Note that capacitance C is ALWAYS positive and never
negative.

The greater the applied voltage the greater will be the charge stored on
the
plates of the capacitor. Likewise, the smaller the applied voltage the
smaller the charge. Therefore, the actual charge Q on the plates of the
capacitor and can be calculated as:



	Charge on a Capacitor
	
	capacitance and charge on a capacitor

Where: Q (Charge, in Coulombs) = C (Capacitance, in Farads) x V (Voltage,
in
Volts)

It is sometimes easier to remember this relationship by using pictures.
Here
the three quantities of Q, C and V have been superimposed into a triangle
giving charge at the top with capacitance and voltage at the bottom. This
arrangement represents the actual position of each quantity in the
Capacitor
Charge formulas.

	
	capacitor charge
and transposing the above equation gives us the following combinations of
the same equation:

	
	capacitor charge equation

Units of: Q measured in Coulombs, V in volts and C in Farads.

Then from above we can define the unit of Capacitance as being a constant
of
proportionality being equal to the coulomb/volt which is also called a
Farad, unit F.

As capacitance represents the capacitors ability (capacity) to store an
electrical charge on its plates we can define one Farad as the
“capacitance of a capacitor which requires a charge of one coulomb to
establish a potential difference of one volt between its plates” as
firstly described by Michael Faraday. So the larger the capacitance, the
higher is the amount of charge stored on a capacitor for the same amount of
voltage.

The ability of a capacitor to store a charge on its conductive plates gives
it its Capacitance value. Capacitance can also be determined from the
dimensions or area, A of the plates and the properties of the dielectric
material between the plates. A measure of the dielectric material is given
by the permittivity, ( ε ), or the dielectric constant. So another way of
expressing the capacitance of a capacitor is:



Capacitor with Air as its dielectric
	
	air dielectric capacitance
Capacitor with a Solid as its dielectric
	
	solid dielectric capacitance

Where A is the area of the plates in square metres, m2 with the larger the
area, the more charge the capacitor can store. d is the distance or
separation between the two plates.

The smaller is this distance, the higher is the ability of the plates to
store charge, since the -ve charge on the -Q charged plate has a greater
effect on the +Q charged plate, resulting in more electrons being repelled
off of the +Q charged plate, and thus increasing the overall charge.

ε0 (epsilon) is the value of the permittivity for air which is 8.854 x 10
-12 F/m, and εr is the permittivity of the dielectric medium used between
the two plates.



Parallel Plate Capacitor

	
	parallel plate capacitor

We have said previously that the capacitance of a parallel plate capacitor
is proportional to the surface area A and inversely proportional to the
distance, d between the two plates and this is true for dielectric medium
of
air. However, the capacitance value of a capacitor can be increased by
inserting a solid medium in between the conductive plates which has a
dielectric constant greater than that of air.

Typical values of epsilon ε for various commonly used dielectric materials
are: Air = 1.0, Paper = 2.5 – 3.5, Glass = 3 – 10, Mica = 5 – 7 etc.

The factor by which the dielectric material, or insulator, increases the
capacitance of the capacitor compared to air is known as the Dielectric
Constant, (k). “k” is the ratio of the permittivity of the dielectric
medium being used to the permittivity of free space otherwise known as a
vacuum.

Therefore, all the capacitance values are related to the permittivity of
vacuum. A dielectric material with a high dielectric constant is a better
insulator than a dielectric material with a lower dielectric constant.
Dielectric constant is a dimensionless quantity since it is relative to
free
space.



Capacitance Example No1

A parallel plate capacitor consists of two plates with a total surface area
of 100 cm2. What will be the capacitance in pico-Farads, (pF) of the
capacitor if the plate separation is 0.2 cm, and the dielectric medium used
is air.
	
	capacitance value
then the value of the capacitor is 44pF.



Charging & Discharging of a Capacitor

Consider the following circuit.
	
	capacitor circuit
Assume that the capacitor is fully discharged and the switch connected to
the capacitor has just been moved to position A. The voltage across the
100uf capacitor is zero at this point and a charging current ( i ) begins to
flow charging up the capacitor exponentially until the voltage across the
plates is very nearly equal to the 12v supply voltage. After 5 time
constants the current becomes a trickle charge and the capacitor is said to
be “fully-charged”. Then, VC = VS = 12 volts.

Once the capacitor is “fully-charged” in theory it will maintain its
state of voltage charge even when the supply voltage has been disconnected
as they act as a sort of temporary storage device. However, while this may
be true of an “ideal” capacitor, a real capacitor will slowly discharge
itself over a long period of time due to the internal leakage currents
flowing through the dielectric.

This is an important point to remember as large value capacitors connected
across high voltage supplies can still maintain a significant amount of
charge even when the supply voltage is switched “OFF”.

If the switch was disconnected at this point, the capacitor would maintain
its charge indefinitely, but due to internal leakage currents flowing across
its dielectric the capacitor would very slowly begin to discharge itself as
the electrons passed through the dielectric. The time taken for the
capacitor to discharge down to 37% of its supply voltage is known as its
Time Constant.

If the switch is now moved from position A to position B, the fully charged
capacitor would start to discharge through the lamp now connected across it,
illuminating the lamp until the capacitor was fully discharged as the
element of the lamp has a resistive value.

The brightness of the lamp and the duration of illumination would ultimately
depend upon the capacitance value of the capacitor and the resistance of the
lamp (t = R*C). The larger the value of the capacitor the brighter and
longer will be the illumination of the lamp as it could store more charge.



Capacitance Example No2

Calculate the charge in the above capacitor circuit.

	
	capacitance and charge equation
then the charge on the capacitor is 1.2 millicoulombs.



Current through a Capacitor
Electrical current can not actually flow through a capacitor as it does a
resistor or inductor due to the insulating properties of the dielectric
material between the two plates. However, the charging and discharging of
the two plates gives the effect that current is flowing.

The current that flows through a capacitor is directly related to the charge
on the plates as current is the rate of flow of charge with respect to time.
As the capacitors ability to store charge (Q) between its plates is
proportional to the applied voltage (V), the relationship between the
current and the voltage that is applied to the plates of a capacitor
becomes:


Current-Voltage (I-V) Relationship
	 
	current and voltage of a capacitor	current through a capacitor


As the voltage across the plates increases (or decreases) over time, the
current flowing through the capacitance deposits (or removes) charge from
its plates with the amount of charge being proportional to the applied
voltage. Then both the current and voltage applied to a capacitance are
functions of time and are denoted by the symbols, i(t) and v(t).

However, from the above equation we can also see that if the voltage remains
constant, the charge will become constant and therefore the current will be
zero!. In other words, no change in voltage, no movement of charge and no
flow of current. This is why a capacitor appears to “block” current flow
when connected to a steady state DC voltage.



Capacitance Value – The Farad
We now know that the ability of a capacitor to store a charge gives it its
capacitance value C, which has the unit of the Farad, F. But the farad is an
extremely large unit on its own making it impractical to use, so sub-multiple’s
or fractions of the standard Farad unit are used instead.

To get an idea of how big a Farad really is, the surface area of the plates
required to produce a capacitor with a value of just one Farad with a reasonable
plate separation of just say 1mm operating in a vacuum. If we rearranging the 
equation for capacitance above this would give us a plate area of:
	A = Cd ÷ 8.85pF/m = (1 x 0.001) ÷ 8.85×10-12 = 112,994,350 m2
or 113 million m2 which would be equivalent to a plate of more than 10
kilometres x 10 kilometres (over 6 miles) square. That’s huge.

Capacitors which have a value of one Farad or more tend to have a solid
dielectric and as “One Farad” is such a large unit to use, prefixes are
used instead in electronic formulas with capacitor values given in micro
-Farads (μF), nano-Farads (nF) and the pico-Farads (pF). For example:


Capacitance Sub-units of the Farad
	
	capacitance prefixes

Convert the following capacitance values:
 
a) 22nF to μF, b) 0.2μF to nF, c) 550pF to μF

	
a)  22nF = 0.022μF
	
b)  0.2μF = 200nF
	
c)  550pF = 0.00055μF 
 

While one Farad is a large value on its own, capacitors are now commonly
available with capacitance values of many hundreds of Farads and have names
to reflect this of “Super-capacitors” or “Ultra-capacitors”.

These capacitors are electrochemical energy storage devices which utilise a
high surface area of their carbon dielectric to deliver much higher energy
densities than conventional capacitors and as capacitance is proportional to
the surface area of the carbon, the thicker the carbon the more capacitance
it has.

Low voltage (from about 3.5V to 5.5V) super-capacitors are capable of
storing large amounts of charge due to their high capacitance values as the
energy stored in a capacitor is equal to 1/2(C x V2).

Low voltage super-capacitors are commonly used in portable hand held devices
to replace large, expensive and heavy lithium type batteries as they give
battery-like storage and discharge characteristics making them ideal for use
as an alternative power source or for memory backup. Super-capacitors used
in hand held devices are usually charged using solar cells fitted to the
device.

Ultra-capacitor are being developed for use in hybrid electric cars and
alternative energy applications to replace large conventional batteries as
well as DC smoothing applications in vehicle audio and video systems. Ultra
-capacitors can be recharged quickly and have very high energy storage
densities making them ideal for use in electric vehicle applications.



Energy in a Capacitor

When a capacitor charges up from the power supply connected to it, an
electrostatic field is established which stores energy in the capacitor. The
amount of energy in Joules that is stored in this electrostatic field is
equal to the energy the voltage supply exerts to maintain the charge on the
plates of the capacitor and is given by the formula:
energy stored in a capacitor
	
so the energy stored in the 100uF capacitor circuit above is calculated as:
	
	energy in a capacitor

The next tutorial in our section about Capacitors, we look at Capacitor
Colour Codes and see the different ways that the capacitance and voltage
values of the capacitor are marked onto its body.