Capacitor

From HvWiki

An electronic component that stores energy in an electric field.

The energy stored in a capacitor is given as $V 2 C / 2$ , where $V$ is the voltage stored in the capacitor and $C$ the capacitance of the capacitor (in Farads, not microFarads).

The charge in a capacitor, $Q$ , its stored voltage, $V$ , and its value of capacitance, $C$ , are related by $Q = V C$ . Eg, a 1000uF capacitor charged to 10V has .01C (coulombs) of charge stored in it.

Since an ampere is defined as one coulomb per second, manipulating the above equation tells us that the rate of change in voltage of the capacitor is given by the current flowing through it divided by the capacitance. In calculus terms, $\frac{dv}{dt}=\frac{I}{C}$ .

Make-up of Capacitor

Any two conductors seperated by an insulator has a capacitance. A basic capacitor is the parallel plate capacitor. This is just two flat conductors (like metal sheets) parallel to each other, separated by a dielectric. The capacitance of this capacitor is given as:

$C = \frac{\epsilon A}{d}$

where $C$ is the capacitance, $A$ the area of the plates in square metres, $d$ the distance between the plates in metres and $ε$ what is called the permittivity of the material filling the capacitor. For vacuum the value is $ε 0 = 8.854 x 10 - 12 C 2 / N . m 2$ . This is approximately the same value as of air.

Real-world capacitor equivalent

Leakage

Dielectrics being imperfect, the capacitor does leak some charge over time. This is like a resistor in parallel with the capacitor, $R p$ .

Equivalent series resistance

The conductors in the capacitor do have some resistance. This is modelled by a resistor in series with the capacitor, $R s$ , called the Equivalent series resistance or ESR. It's typically less than one Ohm.

ESR cannot be measured just by sticking an ohmeter across the capacitor leads.

This site shows an example of how to build a circuit to measure ESR as well as how to.

99 Cent ESR Test Adapter

Maximum ESR before it is recommended to replace a capacitor for voltage rating below 35 V:

Capacitance	Equivalent series resistance
1 µF	4 Ω
2.2 µF	3 Ω
4.7 µF	2 Ω
10 µF	1.5 Ω
22 µF	1 Ω
47 µF	1 Ω
100 µF	0.5 Ω
220 µF	0.3 Ω
470 µF	0.1 Ω
1000 µF	0.05 Ω

Equivalent series inductance

The conductors inside also have some inductance, $L s$ , called the Equivalent series inductance or ESL (L standing for inductance).

Capacitor analogues

Some analogues that might make it easier to comprehend the function of a capacitor.

Water analogue

A capacitor behaves like a vertical hollow tube (eg. measuring cylinder) of water. The height of water represents the voltage and the area of the base of the tube is the capacitance. If water is poured into the top of the cylinder (a current flow), the rate at which the height of the water (the voltage) goes up depends on how quickly the water is flowing in, and the area.

This analogy works for a lot of things. The formula for potential energy stored in the cylinder is analogous, if a hole is put in the cylinder it leaks away according to the same equations and if two cylinders are connected via a hose they equalise in the same way.

But be sure to check out the other water analogue of a capacitor, part of the Wikipedia:Hydraulic_analogy.

Mechanical analogues

A commonly given analogy for a capacitor is a spring, where the force from the spring is the voltage and the distanced stretched is the charge.

A fairly equivalent analogy is a blown up balloon, where the air pressure is the voltage, and the distance stretched the charge.

Types of Capacitors

Ceramic

Class 1

Highly stable over temperature, voltage, frequency and time.
Based on titanium dioxide but can have low levels of barium titanate, calcium titanate, and other things added to increase the dielectric constant, or to get a desired temperature slope.

Class 2

Nonlinear temperature,frequency and voltage response.
Unstable over time, the original capacity may in some cases be restored by heating over the Curie temperature at 150 °C.

Class 3

Same low performance as Class 2
Low dielectric strength (16-50 V)
Very high dielectric constant

Plastic Film

Polyester (Mylar) or Greencap

Small size
Low price
Use when performance is not important

Polystyrene

Low loss
High stability
Low dielectric absorption
Low dielectric strength
Use for critical filters

Metalised Film or MKT

Electrolytic

Electrolytics are cheap and have large capacitances for their sizes. They are polarised and must be used in a circuit the right way around. Electrolytics are not so good for higher frequencies.

Dry tantalum

Low ESR
Low leakage current
Good temperature stability

Have a tendency to fail short circuit at overvoltage or overtemperature, the fumes of burning capacitors are noxious. In some cases a series resistor is needed to reduce the possibility of fire. Tolerable peak reverse voltage is commonly between 15% at 25 °C and 5% at 85 °C.

Homemade Capacitors

It is impossible to make a capacitor at home that has the same low price and quality as one you can buy so it only makes sense in very special cases to make capacitors instead of buying them.

How to make a rolled foil capacitor - 60 kV, 3.5 nF

Charging Capacitors

Constant Voltage

When the switch is closed the capacitor $C$ will start charging via the resistor $R$ to the voltage $V$ . By Kirchoff's voltage law, at any given time the voltage across the resistor and the voltage across the capacitor must add up to $V$ . Initially the voltage across the capacitor is 0V so the voltage across the resistor must be the full $V$ so the maximum possible current flows through the circuit and the capacitor charges quickly. As the capacitor charges, however, its voltage increases and the voltage across the resistor must decrease, decreasing the current through the circuit and hence the rate at which the capacitor charges.

The value $R * C$ is in seconds and is often called the time constant. Every time constant a capacitor is charged by roughly %63. For example, a 100uF capacitor being charged on a 10V supply via a 10k resistor has a time constant of 1s. If it starts discharged, after 1s it will be about %63 charged, or charged to about 6.3V. After the 2s it will have added %63 of the remaining 3.7V, ie. about 2.3V, and will be charged to about 8.6V. After 3s it will be charged to about 9.5V.

Percentage Charged after N Time Constants
N	%
0	0
1	63.2
2	86.5
3	95.0
4	98.2
5	99.3
6	99.8

In theory, the rate at which the capacitor charges slows down forever and the capacitor is never quite 100% charged. After five time constants, however, the capacitor should be about 99.3% charged and this is often taken as close enough in practice.

The theoretically exact description of the voltage of the capacitor at any time is given by $V_c=V_s(1 - e^\frac{-t}{RC})$ , where $V c$ is the voltage across the capacitor at any time $t$ seconds for a given $V s$ supply voltage and $R$ ohms resistance and $C$ Farads capacitance. $e$ is a special number, Euler's constant, which can be approximated to 2.72.

Constant Current

Capacitor Banks

However which way that capacitors are arranged together, the total amount of energy they can store does not change.

Parallel Capacitor Banks

When capacitors are put in parallel, the capacitances simply add up. Eg, a 120uF, 16V capacitor in parallel with a 100uF, 50V capacitor is equivalent to a 220uF, 16V capacitor. (The voltage rating does not increase, it is equal to the lowest voltage rating in the parallel bank.)

Series Capacitor Banks

With capacitors in series things get a bit more complicated, however, the capacitance always goes down and the voltage rating up.

If $n$ equal capacitors are put in series, the capacitance is an $n t h$ of the capacitance of any one alone and the rating is $n$ times the rating of one.

In general, if any capacitors are put in series, the combined capacitance is found by

$\frac{1}{C_t} = \sum^n_{i=1} C_i =\frac{1}{C_1}+\frac{1}{C_2}+\cdots+\frac{1}{C_n}$

So the combined capacitance of a 100uF, a 22uF and a 47uF is about 13uF. Note that this capacitance is always less than the smallest capacitance.

Working out the maximum voltage of series capacitor banks isn't straightforward. It's best to work out the voltage that will be stored in each capacitor on an individual basis and check against each capacitor's ratings.

Mixed Capacitor Banks

Complicated capacitor banks containing both series and parallel capacitors can be reduced by grouping capacitors that are in series or parallel, replacing them with an equivalent capacitor and repeating until just one equivalent capacitor is found.

Charging Series Capacitor Banks

If capacitors were ideal and didn't leak, the voltage, $V c$ , across any capacitor in a series bank that had been charged to $V s$ would be given by $V_c=V_s \bullet \frac{C_t}{C_c}$ Where $C t$ is the total capacitance and $C c$ is the capacitance of the capacitor being looked at.

For very leaky capacitors, like elecrolytics, each capacitor can be considered to have a resistor in parallel with it and it's better to treat the voltages as being divided by these resistors.

Since it's unreliable to assume how the capacitors will leak, often resistors are deliberately put in parallel with the individual capacitors. The leakage through these resistors is designed to be much more than the leakage within the capacitor, so that the internal leakage can be ignored and the voltage across the capacitors will be set by this resistor network. The value of each resistor, $R c$ should be $R_c=R_t \bullet \frac{C_t}{C_c}$ Where $R t$ is the total resistance in parallel with the bank.

Impedance of a Capacitor

DC

Strictly speaking the term DC represents an electrical system that has been steady state for an eternity in the past and will be forever. An ideal capacitor blocks DC (acts like an infinite impedance) so when analysing a steady state circuit capacitors can be replaced by an open circuit. In practice the impedance of a capacitor to DC is the leakage resistance.

If an ohmeter is put across an ideal uncharged capacitor the resistance seems to start at a reading of $0Ω$ and rise to infinity. In practice it starts at the ESR of the capacitor and converges on the leakage resistance, but this is not a practical way to measure ESR.

See above, Real-World Capacitor Limitations.

AC

A capacitor has a reactance to sinusoidal AC flow given by $X_C=\frac{1}{\omega C}$ where $C$ is the capacitance in farads and $ω = 2π f$ , f the frequency in hertz. See the AC page when it gets put up. To see the effect of a capacitor on non-sinusoidal signals, use Fourier Analysis.

A capacitor has a lower impedance to higher frequencies.

Further reading

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Categories: Electronics

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