**Capacitance and Capacitors**

**I. Elementary Characteristics**

In its most elementary state a capacitor

consists of two metal plates separated by

a certain distance d, in between the plates

lies a __dielectric __material
with dielectric

constant
έ = ε_{o}ε , where ε_{o }
is the dielectric of air.

The __dielectric
material __allows for charge to accumulate between the capacitor plates.
Air (actually vacuum)

has the lowest
dielectric value of __
ε _{o }
= __

All other materials have higher dielectric values, since they are higher in density and can therefore accumulate

more charge.

**Capacitance** is defined to be the amount of
charge Q stored in between the two plates for a potential difference

or voltage V existing across the plates. In other words:

**The capacitance C is
given by C = Q/V
(electrical definition)**

The Physical meaning of capacitance can be seen by relating it to the physical characteristics of the two plates,

so that, the capacitance is related to the dielectric of the material in between the plates, the square area of a

plate and the distance between the plates by the formula:

__C =
ε _{o}ε A/d
__(physical definition)

Clearly, the larger the area of the plate the more charge can be accumulated and hence the larger the

capacitance. Also, note that as the distance d increases the Capacitance decreases since the charge cannot be

contained as 'densely' as before.

Both definitions of Capacitance are compatible, although for our purposes we will be referring mostly to the

__
electrical definition.__

**Capacitance and
Energy**

__
__

Perhaps the two most important differences between capacitors and inductors and resistors are that both capacitors and inductors values depend on frequency whereas resistors don't, and they can both store energy whereas resistors dissipate the energy. This is what the picture on the right means to illustrate since touching such a

capacitor will certainly transfer a great deal of charge into our bodies!!

As the capacitor is fully charged the energy is given by:

__
W = 1/2 * C*V^{2 }__ , where W stands as
the symbol for energy and V is the voltage across the plates.

**
Types of Capacitors**

The simple two-plate capacitor model falls short in representing all capacitors since we have different types

such as: ceramic disc capacitors, electrolytic capacitors, polyester capacitors, tantalum capacitors and surface mount capacitors. Each type is selected according to several criteria, essentially: the maximum voltage the capacitor can hold, the value of the dielectric, dimensions and tolerance ratings.

Polyester type Electrolytic type surface mount ceramic disc caps.

The surface mount capacitors are about the size of a dime or smaller and therefore certainly used

with great advantage on the layout of any PC Board requiring many components.

Note that although the Physical definition for different types of capacitors will change depending on their

structure and
dimensions, however the Electrical definition ** C = Q/V **holds
for any capacitor type.

**II.
Capacitor Circuits**

__1. Basic Capacitor
Circuit__

Here, the basic electrical parameters are given.

The voltage V which can be seen as the battery or voltage across

the capacitor, the capacitance C, and the current I.

Before the battery is connected across the capacitor the voltage across
the capacitor is 0 volts. Once the battery is connected the capacitor will
start to charge up and since __I = Q/T (the current is the charge going
through a wire over a period of time, by definition) , __a small change
in charge over a small period in time can be expressed as dQ/dT or dq/dt and
hence the current must be expressed more properly as I = dq/dt at a __specific__
__point of time.__ But, since C = Q/V or Q = C*V a very small change in
charge will become dq = C*dV, that is

the capacitance times a small change in voltage. Therefore the current I can be expressed as:

__I = C*dV/dt which is a basic equation__ we will need to come back to later
on. Once the capacitor is fully charged we can state that V = Q/C where Q
is the total final charge and obviously this will have to equal the

voltage across the battery.

Of course we have described here only the ideal situation since any capacitor has losses and more realistically

a resistor should be included in series with the capacitor. Also, we have omitted to include a switch in the circuit to now when the capacitor starts charging up.

__2.
Capacitors in Parallel__

For two capacitors in parallel the voltage across either one is

the same, namely **V**.
The charge in C1 is Q1 and the charge in

capacitor C2 is Q2. By the electrical definition of capacitance

we can also state that: C1 = Q1/V and C2 = Q2/V.

The total capacitance of the circuit Ctotal will be given by the total charge Qtotal over the voltage or Qtotal/V.

But the total charge for this circuit is the sum of Q1 and Q2 since both capacitors are in parallel and the charge

becomes additive.
Hence, the total capacitance will be given by: __ Ctotal = (Q1+Q2)/V
__ or equivalently

**Ctotal = Q1/V
+ Q2/V = C1 + C2.**

In the case of more than two capacitors in parallel the total charge Ctotal = (Q1 + Q2 +Q3 + ... Qn)/V

where n denotes the total number of capacitors in the circuit. This equation becomes:

__Ctotal = C1 + C2 +
C3 + ... + Cn .__

Note that capacitors in parallel add in the same way that resistor add in series!!

**3. Capacitors in
Series**

For the case of capacitors in series the total voltage V splits

into the voltage V1(across C1) and the voltage V2(across C2).

The total charge Q will be the charge on the total capacitance.

As in any series circuit the current I is the same throughout.

By definition again the total Capacitance Ctotal = Q/V or Q/(V1 + V2). By taking the reciprocal, i.e:

inverting both sides, we get that: 1/Ctotal = (V1 + V2)/Q = V1/Q + V2/Q or: 1/(Q/V1) + 1/(Q/V2);

(May want to grab a piece of paper and pencil to work through the Math!)

Hence__
1/Ctotal = 1/C1 + 1/C2 __and for the
case of more than two capacitors, we have that:

**1/Ctotal
= 1/C1 + 1/C2 + 1/C3 + ... + 1/Cn **, where n is
the total number of capacitors.

Thus, we NOTE that capacitors in series add as resistors in parallel !!!

Again, these are only some of the basics for capacitor circuits and certainly we will have to take a look at

the capacitor and resistor in series so as to complete our exploration of Capacitor Circuits in D.C.

electronics.