# Electronics/Capacitors

## Capacitors

Model of a capacitor

A capacitor (historically known as a "condenser") is a device that stores energy in an electric field, by accumulating an internal imbalance of electric charge. It is made of two conductors separated by a dielectric (insulator). Using the same analogy of water flowing through a pipe, a capacitor can be thought of as a tank, in which the charge can be thought of as a volume of water in the tank. The tank can "charge" and "discharge" in the same manner as a capacitor does to an electric charge. A mechanical analogy is that of a spring. The spring holds a charge when it is pulled back.

When voltage exists one end of the capacitor is getting drained and the other end is getting filled with charge.This is known as charging. Charging creates a charge imbalance between the two plates and creates a reverse voltage that stops the capacitor from charging. As a result, when capacitors are first connected to voltage, charge flows only to stop as the capacitor becomes charged. When a capacitor is charged, current stops flowing and it becomes an open circuit. It is as if the capacitor gained infinite resistance.

You can also think of a capacitor as a fictional battery in series with a fictional resistance. Starting the charging procedure with the capacitor completely discharged, the applied voltage is not counteracted by the fictional battery, because the fictional battery still has zero voltage, and therefore the charging current is at its maximum. As the charging continues, the voltage of the fictional battery increases, and counteracts the applied voltage, so that the charging current decreases as the fictional battery's voltage increases. Finally the fictional battery's voltage equals the applied voltage, so that no current can flow into, nor out of, the capacitor.

Just as the capacitor charges it can be discharged. Think of the capacitor being a fictional battery that supplies at first a maximum current to the "load", but as the discharging continues the voltage of the fictional battery keeps decreasing, and therefore the discharge current also decreases. Finally the voltage of the fictional battery is zero, and therefore the discharge current also is then zero.

This is not the same as dielectric breakdown where the insulator between the capacitor plates breaks down and discharges the capacitor. That only happens at large voltages and the capacitor is usually destroyed in the process. A spectacular example of dielectric breakdown occurs when the two plates of the capacitor are brought into contact. This causes all the charge that has accumulated on both plates to be discharged at once. Such a system is popular for powering tasers which need lots of energy in a very brief period of time.

## Capacitance

The capacitance of a capacitor is a ratio of the amount of charge that will be present in the capacitor when a given potential (voltage) exists between its leads. The unit of capacitance is the farad which is equal to one coulomb per volt. This is a very large capacitance for most practical purposes; typical capacitors have values on the order of microfarads or smaller.

$C = \frac{Q}{V}$

Where C is the capacitance in farads, V is the potential in volts, and Q is the charge measured in coulombs. Solving this equation for the potential gives:

$V=\frac{Q}{C}$

## Capacitor & Direct Current Voltage (DC)

Charge Building

When a Capacitor is connected with electricity source V . Charge will build up on each plates of capacitor of the same amount of charge but different in polarity . This process is called Capacitor Charging

Storing Charge

When both plates are charged up to voltage V then there is no difference in voltage between capacitor's plates and electricity source therefore no current flow in the circuit. This is called Storing Charge

Charge discharge

When the capacitor is connected to ground, current will flow from capacitor to ground until the voltage on capacitor's plates are equal to zero.

Therefore, a Capacitor is a device that can Build up Charge , Store Charge and Release Charge

## Capacitor & Alternating Current Voltage (AC)

### Voltage

$V = \frac{1}{C}\int I dt$

### Current

$I = C \frac{dV}{dt}$

### Reactance

Reactance is defined as the ratio of Voltage over Current

$X_C = \frac{1}{\omega C} \angle -90 = \frac{1}{j \omega C} = \frac{1}{sC}$

### Impedance

Impedance is defined as the sum of Capacitor's Resistance and Reactance

$R_C + X_C = R_C \angle 0 + \frac{1}{\omega C} \angle -90 = R_C + \frac{1}{j \omega C} = R_C + \frac{1}{sC}$

### Angle of Difference between Voltage and Current

For Lossless Capacitor

Current will lead Voltage an angle 90 degree

For Lossy Capacitor

Current will lead Voltage an angle θ degree where
Tan θ = $\frac{1}{\omega CR_C} = \frac{1}{2\pi f CR_C} = \frac{t}{2\pi CR_C}$

Changing the value of R and C will change the value of Phase Angle, Angular Frequency, Frequency and Time

$\omega = \frac{1}{Tan \theta CR_C}$
$f = \frac{1}{2\pi Tan \theta CR_C}$
$t = 2\pi Tan \theta CR_C$

### Time Constant

$T = RC$

## Capacitor Labeling

Capacitors are labelled in several different ways.

## Capacitor Types

Capacitors come in many different styles and must be carefully selected for the application as the variation in performance and available characteristics (capacitance range, size, voltage rating, temperature and impedance effects) may vary widely between types.

In addition to performance characteristics there may be other important criteria and special functions such as environmental conditions, safety ratings, life expectancy, self discharging, temperature compensating, adjustable (mechanically or voltage adjustable).

### Ceramic Disc

Sometimes labeled implicitly, usually labeled with number scheme (223 = 22 000 pF, where 3 represents the number of "0" for instance) The letters "mfd" are often used in place of "µF".

### Ceramic Dipped

Ceramic-dipped capacitors.

These usually use the number code. In the above example, the smallest one says "104". This means 10 0000 pF = 100,000 pF. M is a tolerance. The middle one is labeled 393. This means 39 000 pF. The last is 223, meaning 22 000 pF. K is the tolerance. It also has a 100 V working voltage labeled. Colours on capacitor are an indication for capacitance. Red dipped caps are ___

Green dipped caps are ___

Mylar or polyester film?

Resin-potted mylar/polyester

"Box-style" capacitors.

These are usually labeled explicitly, as there is lots of surface area to write on. This one is 4700 pF, 250 V, 5 kV test. The frequency f0 = 21 MHz is the frequency at which it stops behaving like a capacitor, and more like an inductor.

### Electrolytic

Axial 1000 µF capacitor with a maximum voltage rating of 35 V (black), and radial 10 µF capacitor with a maximum voltage rating of 160 V (blue).
This image shows a pair of radial electrolytic capacitors with leads. The industry standard, as shown, is that the positive lead is longer than the negative lead.

Usually electrolytic caps are labeled explicitly, making identification easy.

Electrolytics are available in axial and radial-leaded packages. In axial-leaded parts, the negative terminal is indicated by a minus sign printed on the package, or by a shorter lead.

Radial-lead parts often uses color code like resistors. The polarity is usually indicated by arrows on a stripe pointing to the negative terminal.

Aluminum electrolytic capacitors of the "snap-in" type. The leads are angled so that they may be "snapped" into a printed circuit board with holes of the correct spacing.
Warning: You should never connect an electrolytic capacitor in such a way that a negative voltage is applied across the terminals from positive to negative. It will explode.

### Tantalum

Tantalum capacitors have high capacitance and low ESR, but low operating voltages. When tantalum capacitors fail, it tends to be "spectacular," they essentially blow up.

## Construction

The capacitance of a parallel-plate capacitor constructed of two identical plane electrodes of area A at constant spacing D is approximately equal to the following:

$C = \epsilon_0 \epsilon_r \frac{A}{D}$

where

C is the capacitance in farads
ε0 is the Permittivity of Space,
εr is the Dielectric Constant
A is the area of the capacitor plates, and D is the distance between them.

A dielectric is the material between the two charged objects. Dielectrics are insulators. They impede the flow of charge in normal operation. Sometimes, when a too large voltage has been reached, charge starts flowing. This is called dielectric breakdown and destroys the capacitor. Beginners sometimes misunderstand this. Timing circuits do measure the rate at which a capacitor charges, but they measure a threshold voltage instead of allowing the voltage to build up until dielectric breakdown. (A device which does function this way is a spark gap.)

No charge should ever flow from one plate to the other. Although a current does flow through the capacitor, charges are not actually moving from one plate to the other. As charges are added to one plate, their electric field displaces like charges off of the other plate. This is called a displacement current.

## Materials

Capacitors can be made either polarized or non-polarized. A polarized capacitor requires that the capacitor be hooked up such that the voltage is always biased in one direction. Hooking a polarized capacitor backwards will result in the capacitor exploding, sometimes releasing harmful fumes. Non-polarized capacitors can be biased in either direction without harm to the capacitor. Polarized and non-polarized capacitors have an upper limit of voltage, where the material will break down and the capacitor will no longer function. This can also cause fumes to be released depending on the type of material. Different materials and their properties.

Ceramic

These are normally low capacitance (between ~1pF to ~1μF). Ceramic capacitors have a very low inductance due to the shape. This means that the capacitance value continues into extremely high frequencies, making them perfect for RF applications. However, ceramic capacitors tend to vary their capacitance with temperature.
• C0G or NP0 - Typical 4.7 pF to 0.047 µF, 5%. High tolerance and temperature performance. Larger and more expensive.
• X7R - Typical 3300 pF to 0.33 µF, 10%. Good for non-critical coupling, timing applications.
• Z5U - Typical 0.01 µF to 2.2 µF, 20%. Good for bypass, coupling applications. Low price and small size.

Polystyrene

Slightly larger than ceramic, but still has small values (usually in the picofarad range).

Polyester

from about 1 nF to 1 μF

Polypropylene

low-loss, high voltage, resistant to breakdown

Tantalum

These are polarized capacitors that are still small enough to be surface mount. Normally the dielectric breakdown voltage is rather low -- typically less than 20 volts -- so the capacitors are not suitable for high voltage applications. Tantalum capacitors have a stable capacitance across varying temperatures, but higher (worse) ESR than any other capacitor material except for electrolytic. Tantalum capacitors have the highest (best) energy density of any material.

Electrolytic

These are also polarized, are much larger than tantalums. The dielectric strength is much higher in these, and so is the capacitance. Capacitance values can range between 1μF and 1mF (sometimes up into the farad range). These are compact capacitors -- higher (best) energy density of any material other than tantalums. Electrolytic capacitors also very lossy -- at high frequencies they have the highest (worst) ESR of any capacitor material. They are useful for smoothing power supplies because of the high capacitance.

Air-gap

Aerogel

These capacitors are more compact than normal electrolytic capacitors, giving capacitance values in the farad range, but normally have an extremely low breakdown voltage.

Super capacitors 2500 F to 5000 F

## Capacitor Connection

### Capacitors in Series

Capacitors in series are the same as increasing the distance between two capacitor plates. As well, it should be noted that placing two 100 V capacitors in series results in the same as having one capacitor with the total maximum voltage of 200 V. This, however, is not recommended to be done in practice. Especially with capacitors of different values. In a capacitor network in series, all capacitors can have a different voltage over them.

In a series configuration, the capacitance of all the capacitors combined is the reciprocal of the sum of the reciprocals of the capacitance of all the capacitors.

$\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots + \frac{1}{C_n}$

### Capacitors in Parallel

Capacitors in parallel are the same as increasing the total surface area of the capacitors to create a larger capacitor with more capacitance. In a capacitor network in parallel, all capacitors have the same voltage over them.

In a parallel configuration, the capacitance of the capacitors in parallel is the sum of the capacitance of all the capacitors.

$C_{eq} = C_1 + C_2 + \cdots + C_n \,\!$

## RC Circuit

### Introduction

An RC circuit is short for 'Resistor-Capacitor' circuit. A capacitor takes a finite amount of time to discharge through a resistor, which varies with the values of the resistor and capacitor. A capacitor acts interestingly in an electronic circuit, practically speaking as a combination of a voltage source and a variable resistor.

### Basics

Below is a simple RC Circuit:

There is a capacitor in parallel with the resistor and current probe. The way the capacitor functions is by acting as a very low resistance load when the circuit is initially turned on. This is illustrated below:

Initially, the capacitor has a very low resistance, almost 0. Since electricity takes the path of least resistance, almost all the electricity flows through the capacitor, not the resistor, as the resistor has considerably higher resistance.

As a capacitor charges, its resistance increases as it gains more and more charge. As the resistance of the capacitor climbs, electricity begins to flow not only to the capacitor, but through the resistor as well:

Once the capacitor's voltage equals that of the battery, meaning it is fully charged, it will not allow any current to pass through it. As a capacitor charges its resistance increases and becomes effectively infinite (open connection) and all the electricity flows through the resistor.

Once the voltage source is disconnected, however, the capacitor acts as a voltage source itself:

As time goes on, the capacitor's charge begins to drop, and so does its voltage. This means less current flowing through the resistor:

Once the capacitor is fully discharged, you are back to square one:

If one were to do this with a light and a capacitor connected to a battery, what you would see is the following:

1. Switch is closed. Light does not light up.
2. Light gradually becomes brighter and brighter...
3. Light is at full luminosity.
4. Switch is released. Light continues to shine.
6. Light is off.

This is how a capacitor acts. However, what if you changed the values of R1? C1? The voltage of the battery? We will examine the mathematical relationship between the resistor, capacitor, and charging rate below.

### The Time Constant

In order to find out how long it takes for a capacitor to fully charge or discharge, or how long it takes for the capacitor to reach a certain voltage, you must know a few things. First, you must know the starting and finishing voltages. Secondly, you must know the time constant of the circuit you have. Time constant is denoted by the Greek letter 'tau' or τ. The formula to calculate this time constant is:

$\tau={R}{C} \,\!$

Great, so what does this mean? The time constant is how long it takes for a capacitor to charge to 63% of its full charge. This time, in seconds, is found by multiplying the resistance in ohms and the capacitance in farads.

According to the formula above, there are two ways to lengthen the amount of time it takes to discharge. One would be to increase the resistance, and the other would be to increase the capacitance of the capacitor. This should make sense. It should be noted that the formula compounds, such that in the second time constant, it charges another 63%, based on the original 63%. This gives you about 86.5% charge in the second time constant. Below is a table.

Time Constant Charge
1 63%
2 87%
3 95%
4 98%
5 99+%

For all practicality, by the 5th time constant it is considered that the capacitor is fully charged or discharged.

put some stuff in here about how discharging works the same way, and the function for voltage based on time

$v(t)=\frac{1}{C} \int i(t)dt$

Where i(t) is the current flowing through the capacitor as a function of time.

This equation is often used in another form. By differentiating with respect to time:

$i(t)=C\frac{dv(t)}{dt}$

Substituting v/r for i(t) and integrating the above equation gives you an equation used to describe the charging and discharging characteristics of RC circuits. A charging characteristic curve exponentially increases from 0% (0 volts) and approaches 100% full (maximum voltage), similarly, a discharge curve starts at the theoretical 100% (maximum voltage) and exponentially falls back to 0% (0 volts).

## Capacitor Specifications

When a capacitor is being discussed, it is referred to with certain "specifications" or characteristics. Capacitors are usually "specified" in the following manner-

• they are specified by type (tantalum, electrolytic, etc.)
• they are specified by package (axial, radial, as discussed above).
• they are specified by how to connect to them, their connection type (such as "snap in" or leaded, or threaded screw holes, or surface mount).
• they are specified by capacitance value, e.g. in microfarads (µF).
• they are specified by voltage rating (i.e., 30 V). This indicates the maximum voltage under which it is safe to use the referenced capacitor.
• some types, such as electrolytic capacitors, are specified by operating temperature (usually 80 or 120 °C), which reflects the maximum temperature that the capacitor can reach before failing. Note- common practice is to use capacitors well below their maximum operating voltage and temperature in order to ensure longevity.
• they can be specified by other parameters, including ESR or "equivalent series resistance" (explained above). Also, some capacitors can be specified by UL or other safety rating. A "X" type capacitor indicates that the capacitor meets certain standards one of which is that it is appropriate to be used with line-level voltages (such as 117 or 220 V) typically found from the wall outlet, as well as that it can withstand surges typically found in power distribution systems.
• they are specified in percentage accuracy, i.e., how much they are likely to deviate from their rated capacitance. Common ratings are + or - 20%.

## Capacitor Variations

• Variable Capacitor: