-|(-
electronic component chosen for relatively high capacitance with accompanying low resistance and inductance.

Capacitors have a capacitance rating and a voltage rating. If the voltage rating is grossly exceeded, they will burn out, melt down or explode. Electrolytic capacitors also have a polarity, and tend to explode if hooked up backwards.

Blocks DC, passes AC. Connect one to ground to bleed noise off of your circuits.

A capacitor is defined as two conductors seperated by an insulator. This could be as simple as two sheets of tin foil held apart and connected to a power source.

The capacitance of a specific capacitor is defined as:
keoA
---------
d

Where k is the dielectric constant of the insulator, eo is the permisstivity of free space, A is the overlapping area of the plates, and d is the distance between the plates. Capacitors store electric charge.

The capacitor can be used by itself as a filter. If connected in shunt with the circuit it is a low-pass filter. If connected in series with a circuit it is a high-pass filter.

To read six-dot capacitors, use the following table:

```
Start at Arrow Side:
Type  1st Digit 2nd Digit
1       2        3
______________________
/                      \
|   o>      o>      o>   |
|                        |
------|                        |--------
|                        |
|   o       o        o   |
\______________________/

6       5        4
Class          Multiplier
Tolerance

Type   Colour   1st    2nd    Mult    Tolerance
1       2      3      4       5         6

Mica   Black     0      0      100        1%
Brown     1      1      101        2%
Red       2      2      102        3%
Orange    3      3      103        4%
Yellow    4      4      104        5%
Green     5      5      105        6%
Blue      6      6      106        7%
Violet    7      7      107        8%
Gray      8      8      108        9%
Mica   White     9      9      109
Gold                    0.1       10%
Paper  Silver                  0.01      20%

```
The character or class dot concerns temperature coefficient and testing methods.
Some equations relating to capacitors:

When a capacitor is discharged to produce an electric current, the decrease in the charge stored in the capacitor is exponential - it has a constant half-life. The equation for this discharge is:

x = x0 e-(t / CR)

t is the time during which the capacitor discharges, C is the capacitance of the capacitor, and R is the resistance of the circuit connected to it. x can stand for the charge in the capacitor, Q, and it can also stand for the current and voltage of the resulting electric flow, I and V. Therefore as a capacitor discharges, charge in the capacitor and current and voltage in the circuit all decrease exponentially.

C × R, capacitance times resistance, is known as the time constant of a capacitor and is represented by τ, the letter tau. It is equal to the time, in seconds, taken by the values of Q, I and V (represented by x in the above equation) to decrease by a factor of e, the exponential function.

If capacitors are placed in parallel, the total capacitance is the sum of the individual capacitances: Ctotal = C1 + C2 + C3 etc. If they are placed in series, the total capacitance decreases according to the equation: 1/Ctotal = 1/C1 + 1/C2 + 1/C3 etc. Note that is the reverse of the case for resistors, which become less effective in parallel and more effective in series.

Given its large variety of applications, the capacitor is probably one of the the most used electronic component. Like inductors, they can store energy for later use, for example your camera's flash charges a capacitor that unleashes current when a snapshot is taken and memory chips use them to keep track of information (see MOS capacitor). They are also widely used in filters, for example decoupling capacitors or AM demodulation.

Capacitor quick facts

The electronic symbol for a capacitor is one of the following, depending on it having polarity or not :

`--| |--` or `--| (--`

The main equations used to describe a capacitor's behaviour is :

I = C.dU/dt
Z = 1/jCω (complex impedance)

` i   C`
`-<--| |----`
`   ---->`
`     U`

where i is the current through the capacitor (in Ampere), U is the voltage (in Volt) and C is a constant characteristic of the capacitor called capacity and expressed in Farad. The main recommendation is to keep in mind that the voltage of a capacitor is continuous.

Capacitors have voltage recommendations and some (for example electrolytic capacitors for obvious reasons) have polarity. If those recommendations are exceeded by too much the capacitor will probably melt down, leak (electrolytic capacitors contain a liquid) or even explode.

The electric charge and energy stored in the capacitor is given by :

q = C.V
E = 1/2.C.V2 = 1/2.q2/C

where q is the charge of half of the capacitor (in Coulomb) and E is the energy stored in it (in Joule).

Capacitors can be associated in different ways. Serial and parallel associations of two capacitors can be replaced by one equivalent capacitor :

`---| |---| |--- ~ ---| |---` 1/C = 1/C1 + 1/C2
```    C1    C2          C```

`     C1`
`  +-| |-+            C`
`--|     |--   ~   ---| |---` C = C1 + C2
`  +-| |-+`
`     C2`

Usage

Let's see how a capacitor charges and discharges in a resistor. Consider the following circuit :

```       i
+->--[ R ]----+    ^
|             |    |
E+ -----          --- C |
E-  ---           ---   | U
|             |    |
+-------------+    |```

There are two elements in the circuit, let's write two equations :

i = C.dU/dt (the capacitor)
E = U + R.i (the resistor)
Hence di/dt + 1/(R.C).i = 0

i is solution of a linear differential equation with constant coefficients. The general form of solutions is i = I0.exp(-t/τ), with τ = 1/(R.C). τ is called the capacitor time constant, it is the characteristic duration of charge. In about 4 to 5 times τ the capacitor is either charged or discharged at 99% (e-4=1.8%, e-5=0.67%).

U = E - R.i = E - U0.exp(-t/τ). Below is the rough graph of a capacitor charge and discharge. During the first period, E>0 so the capacitor is charged. During the second period E=0, the capacitor is discharged in the resistor.

```
U
^
E_|               _____
|         . ¨¨       .
|     .               .
|   .                   .
| .                        .
|.                            - .  ____
0 +---------------------------------------> time
```

As you can see, the voltage slowly decreases or increases to limit values. Hence one application of a capacitor is temporization. This was used in many christmas tree garlands.

A capacitor has a certain inertia in changing its state. Because of this, low frequencies will be transmitted when the capacitor shunts the circuit since the capacitor has time to charge/discharge in one period. But it will block high frequencies since it will not have sufficient time to change its state during one period.

The following are two famous filters. The first one shunts low frequencies (high-pass filter) and the second one shunts high frequencies (low-pass filter). With τ = R.C (capacitor time constant), the turnover frequency for both filters is f = 1/τ = 1/(R.C). For example with the first filter, frequencies way below f will be attenuated, those way above f have an attenuation that is asymptotic to 0 dB.

```
^  ---| |---+---   ^  ---|R|-+---  ^
|      C    |      |         |     |
|          +-+     |        ---    |
U|          |R|    V|        --- C  |W
|          +-+     |         |     |
|           |      |         |     |
|  ---------+---   |  -------+---  |```

Say Z is the complex impedance of the capacitor, we have : V = R.U/(R+Z) hence H1 = V/U = R/(R+Z) for the first diagram, and W = Z.V/(R+Z) hence H2 = Z/(Z+R) for the second diagram. Since Z = 1/jCω, Z is small for high frequencies (high values of ω) and large for low frequencies. Thus the transfer function H1 approaches 0 for low frequencies (this is a high-pass filter) and the transfer function H2 approaches 0 for high frequencies (this is a low pass filter).

A famous application of filters is AM demodulation. you can build a little AM receiver with as less as a diode, a resistor and a capacitor. Here is the circuit :

```  antenna ------->|---+-----+  headphone
diode  |     |
+-+   ---
|R|       C
+-+   ---
|     |
------------+-----+```

This circuit only allows low frequencies and blocks high ones thus it eliminates the carrier and keeps only the signal. Of course the signal should be amplified for the receiver to be really useful but you can really have fun with this circuit and a pair of headphones.

Capacitor geometry

There are many different kinds of capacitors. The most important ones are ceramic capacitors and electrolytic capacitors. What is a capacitor anyway ? It's just two conductors placed in front of eachother, separated by an insulator. The most basic model is called the plane plate capacitor. It consists of two metal plates of surface S facing eachother at a distance d in free space.

```
|          |
|          |
A|          |B
---+          +---
|          |
|          |
|          |

d
<---------->
```

To find the capacity of such a capacitor, just use the formula C = q/V. Say plate A has charge +q and plate B has charge -q. Gauss' theorem states that the electric field between the plates is constant (side effects neglected) and E = q/(2.S.ε0). Hence V = VB - VA = E.d = q.d/(2.S.ε0). Finally :

C = q/V = 2.S.ε0/d.

You can play around with this formula for a while : for example how to increase a capacitor's capacity ? Either use larger plates or put them closer. You can't do this infinitely though because there is radiation pressure on the surface of conductors.

Another way to build a capacitor with a variable capacity is to use two 3/4 disks, place them in front of each other. When you make one turn, a variable surface faces the other disk which makes the capacity change. Put this sort of capacitor in the little AM demodulator I've given above and you'll be able to change your radio's tuning.

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