Capacitors
Discussion
history
Nearly everyone is familiar with the static charge generated by friction — a phenomenon formally known as triboelectricity. Walking across a carpeted floor, combing one's hair on a dry day, or pulling transparent tape off a roll all result in the separation of small amounts of positive and negative charge. The earliest known written account of charging by friction goes back as far as the 6th century BCE when the Greek scientistThales of Miletus (635–543 BCE) noted that amber rubbed with animal fur acquired the ability to pick up small bits of material. For roughly the next 2300 years, wherever electricity was studied, somebody had to take two different materials and rub them together to create separated islands of positive and negative charge.
Fast forward to 18th century Europe, an era known as the Enlightenment, a time and place characterized by the expansion of culture and the acquisition of knowledge. Among the empowered and educated classes of the Enlightenment, science was a fashionable pursuit and lectures on scientific subjects were well attended. Those given by electricians were among the most popular. (The word electrician originally referred to a person knowledgeable in the nature of static electricity.) Electricity was a hot topic in the 18th century and much exploration was being done with electrostatic machines that generated charge by friction.
While friction is an easy and inexpensive means to separate charge for use in electric experiments, the amounts of charge available are quite small. If electricity was going to be anything other than an irritating side effect of walking across a carpet, some means for increasing the amount of charge available for experiments had to be found.
The first device for storing charge was discovered in the winter of 1745–46 by two electricians working independently: Ewald von Kleist (1715–1759), dean of the cathedral at Kammin, Prussia (now Kamień, Poland), and Pieter van Musschenbroek (1692–1761), professor of mathematics and physics at the University of Leyden in Holland (now spelled Leiden). The device built by von Kleist consisted of a medicine bottle partly filled with water and sealed with a cork. A nail was pushed through the cork and into the water. Holding the bottle in one hand, the nail was then brought in contact with the terminal of an electrostatic machine allowed to acquire some charge. When von Kleist reached for the nail to remove it from the stopper while still holding the bottle the separated charges were able to reunite by flowing through his body. Van Musschenbroek's device and experiences with it were almost the same as von Kleist's, but with three major exceptions. First, a visiting student Andreas Cunæus (1712–1788) made the shocking discovery not van Musschenbroek himself; second, he made many improvements to the device (most importantly, removing the water and wrapping the inside and outside of the jar with metallic foil); and third, he wrote his colleagues to tell them all about it.
As I see that this sheet is not completely filled, I would like to tell you about a new but terrible experiment, which I advise you never to try yourself, nor would I, who experienced it and survived by the grace of God, do it again for all the kingdom of France. I was engaged in displaying the powers of electricity. An iron tube AB was suspended from blue-silk lines; a globe, rapidly spun and rubbed, was located near A, and communicated its electrical power to AB. From a point near the other end B a brass wire hung, in my right hand I held the globe D, partly filled with water, into which the wire dipped, with my left hand E I tried to draw the snapping sparks that jump from the iron tube to the finger, thereupon my right hand F was struck with such force that my hole body quivered just like someone hit by lightning. Generally the blow does not break the glass, no matter how thin it is, nor does it knock the hand away; but the arm and entire body are affected so terribly I can't describe it. I thought I was done for. But here are some peculiarities. When the globe D is made of English glass there is no effect, or almost none; German glass must be used, Dutch doesn't work either; D does not have to be a globe, a drinking glass will do; nor does it matter if it is large or small, thick or thin, tall or short, or of any particular shape; but it must be made of German or Bohemian glass. The globe D that almost killed me was of very thin white glass, five inches in diameter. Most other note-worthy phenomena I here omit. Suffice it that the man should stand directly on the ground; that the same one who holds the globe should draw the spark; the effect is small if two men participate, one grasping the globe and the other pulling the sparks. If the globe D rests on metal lying on a wooden table, and someone touches the metal with one hand and elicits sparks with the other, he also will be struck with an immense force. I've found out so much about electricity that I've reached the point where I understand nothing and can explain nothing. Well, I've filled this sheet up pretty well.
Pieter van Musschenbroek, 1746 (paid link)
Never say "never try" something — especially something "terrible" — because then everyone will want to try it. Soon scientists across the Continent (and Benjamin Franklin in America) were constructing their own new and improved electric charge storage devices.
Raw notes…
- Refinement of design: filled with water, ink, vinegar, melted butter, wine or beer, and finally nothing (foil). A glass vessel partly coated, inside and out, with metal foil and the Leyden jar (de Leidsche flesch) was born.
- Applications by Benjamin Franklin (1706–1790) United States
- Such a device is known today as a capacitor.
- 1746 French clergyman and physicist Jean Antoine Nollet demonstrated that electricity could be transmitted instantaneously over great distances suggesting that communications could be sent by electricity much faster than a human messenger could carry them. With the connivance of the Abbot of the Grand Convent of the Carthusians in Paris he assembled 200 monks in a long snaking line with each monk holding the ends of eight metre long wires to form a chain about one mile long. Without warning he connected a Leyden Jar to the ends of the line giving the unsuspecting monks a powerful electric shock and noted with satisfaction that all the monks started swearing and contorting, reacting simultaneously to the shock. A second demonstration was performed at Versailles for King Louis XV, this time by sending current through a chain of 180 Royal Guards since by now the monks were less than cooperative. The King was both impressed and amused as the soldiers all jumped simultaneously when the circuit was completed.
- Observations sur quelques nouveaux Phénomènes d'Electricité. Par M. l' Abbé NOLLET (20 Avril 1746). Mémoires de l'Académie royale des sciences.
theory
Informal definition of capacitance
Formal definition of capacitance. The capacitance (C) of an electrostatic system is the ratio of the quantity of charge separated (Q) to the potential difference applied (V).
| C = | Q |
| V |
The SI unit of capacitance is the farad [F], which is equivalent to the coulomb per volt [C/V].
| ⎡ ⎢ ⎣ |
F = | C | ⎤ ⎥ ⎦ |
| V |
One farad is generally considered a large capacitance.
Energy storage
| Q | Q | |||||||
| U = | ⌠ ⎮ ⌡ |
V dq = | ⌠ ⎮ ⌡ |
q | dq = | 1 | Q2 | |
| C | 2 | C | ||||||
| 0 | 0 |
Since Q = CV, and also since C = Q/V
| Uc = ½QV = ½CV2 = ½ | Q2 |
| C |
The English scientist Henry Cavendish (1731–1810) determined the factors affecting capacitance. The capacitance (C) of a parallel plate capacitor is…
- directly proportional to the area (A) of one plate
- inversely proportional to the separation (d) between the plates
- directly proportional to the dielectric constant (κ, the Greek letter kappa) of the material between the plates
| C = | κε0A |
| d |
Derivation
| C = | Q | = | σA | = | 1 | σA | = | ε0 | σA | = | ε0A | ||
| V | Ed | E | d | σ | d | d |
More advanced…
Cylindrical capacitor (e.g., coaxial cables)
| C = | 2πκε0ℓ |
| ln(r2/r1) |
Spherical capacitor
| C = | 4πκε0 |
| (1/r1) − (1/r2) |
If the two radii are nearly similar (e.g., positive troposphere h = 8.5 km above negative Earth), then the equation above can be approximated as …
| C ≈ | 4πκε0r2 |
| h |
Self-capacitance of a sphere (e.g., van de Graaff generator). Let r2 → ∞.
C = 4πε0r
More on dielectrics in the next section.
large capacitors
Two (three?) examples: in power supplies, the condenser microphone (and the Theremin?).
Typically, they are used for power-supply smoothing (or decoupling) to eliminate spikes or dropouts
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Just because an electrical device is unplugged doesn't mean it's safe to open it up and work inside. Heavy appliances, like this microwave oven, often contain capacitors capable of storing significant amounts of electric energy. An accidental and quick discharge could result in serious injury or death. (The capacitor is the oval shaped metal canister on the right.)
Condenser microphones. The word "condenser" is a now nearly obsolete term meaning "capacitor".
A backwards condenser microphone is a what?
A condenser microphone is basically a capacitor with one fixed plate and one light, thin, free plate called a diaphragm. This second plate is so light that sound waves are powerful enough to set it vibrating. This causes the distance between the fixed and stationary plate to change. When the plate separation changes, the capacitance changes. The plates are charged to a constant value when in use and the changing capacitance results in a changing voltage.
Sound, you will recall, is a longitudinal wave; a series of alternating high and low pressure regions called compressions and rarefactions that propagate through a medium such as air. A high pressure compression striking the microphone pushes the diaphragm inward, reducing the plate separation, increasing the capacitance, and decreasing the voltage. A low pressure rarefaction pulls the diaphragm back out, increasing the plate separation, decreasing the capacitance, and increasing the voltage. The voltage thus turns out to be inversely proportional to the air pressure. ?????? This doesn't work right. Pressure and voltage should be directly proportional.
Condenser microphone equations
| C = | Q | = | ε0A |
| V | d |
| Q = | ε0AVbias | = ε0AVbias [μP] |
| d | ||
| d | Q = ε0AVbias | d | P |
| dt | dt |
The voltage is small and the changes even smaller, so an amplifier circuit is needed to bring the signal up to a useable level.
| type | sounds produce changes in… |
which cause changes in… |
which result in changes in… |
|---|---|---|---|
| carbon | granule density | resistance | voltage |
| condenser | plate separation | capacitance | voltage |
| dynamic | coil location | flux | voltage |
| piezoelectric | compression | polarization | voltage |
More examples
- Theremin
- Capacitive proximity sensors
- Capacitance fuel gauging systems
small capacitors
We are surrounded by teeny, tiny capacitors. They're everywhere! Two examples: DRAM and the MEMS accelerometer.
dynamic random access memory (DRAM). The basis of a dynamic RAM cell is a capacitor. The first commercially available DRAM chip was the Intel 1103, introduced in 1970.
MEMS (micro electromechanical system) accelerometer. Acceleration is determined from the differential capacitance of adjacent capacitors.
| Parts of a typical low-g MEMS accelerometer | ||||||||||||||||||||
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| characteristic | value | |||
|---|---|---|---|---|
| beam: | proof mass | 0.1 | μg | |
| length | 280 | μm | ||
| thickness | 2 | μm | ||
| suspension height | 1.6 | μm | ||
| resonant frequency | 10–22 | kHz | ||
| plates: | length | 38 | μm | |
| separation | 1.3 | μm | ||
| minimum detectable displacement | 0.02 | nm | ||
| capacitance: | total | 100 | fF | per plate |
| minimum detectable change | 0.02 | fF | per plate | |
| maximum change | 10 | fF | per plate | |
| acceleration: | measurable range | ±5 | g | |
| minimum detectable change | 0.002 | g | ||
| maximum shock | 1000 | g | ||
MEMS accelerometers are used…
- to detect sudden automobile deceleration and then pretension seat belts or deploy air bags as needed
- to determine velocity, position, and orientation for inertial navigation and flight data recording systems
- to monitor vibrations in machines and detect unusual conditions such as asymmetric loading or potential failure
- to determine if a mechanical device has been dropped and then activate safe mode (like a portable hard drive)
- to detect forced entry and then activate an alarm (like a car alarm)
- to detect violent earthquakes and then shut off pipelines or water mains (as a seismic senor)
- to measure changes in inclination and warn of possible building collapse or just to determine if a surface is level or true (as a tilt sensor)
- to detect the orientation of a hand held display and enter landscape or portrait mode as appropriate
- to monitor motion in cameras and then stabilize the image recorded
- to control gaming systems
- to analyze motion in professional and collegiate sports
- to count the number of steps taken by a person walking or jogging (as a pedometer)
- to measure the strength of the local gravitational field for geologic studies