Capacitors

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© 1998-2008 by Glenn Elert -- A Work in Progress
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Discussion

history

Nearly everyone is familiar with the static charge generated by friction — a phenomena 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 Sixth Century BCE when the Greek scientist Thales of Miletus a.k.a. Θαλής ο Μιλήσιος (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 Eighteenth 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 Eighteenth 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 Georg 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 (now spelled Leiden) in Holland. 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 Cunaeus (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.

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.

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 …

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/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

U =  1  CV2 =  1   Q2  =  1  QV
2 2 C 2

Henry Cavendish (1731-1810) determined the factors affecting capacitance. The capacitance (C) of a parallel plate capacitor is …

C =  κε0A
d

Derivation

C =  Q  =  σA  =  1   σA  =  ε0   σA  =  ε0A
V Ed E   d σ d d

More advanced: cylindrical (coaxial cables)

C =   
2πε0ℓ
ln ⎛
⎝		 b ⎞
⎠
a  
   

and spherical (e.g. positive ionosphere 300 km above negative earth)

C =   
4πε0
   1  −  1   
  a b  
   

and self-capacitance of a sphere (e,g, van de Graaf generator)

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

Large Capacitors — Family Portrait
   
 
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 electrical 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 very 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.


[magnify]

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.

Microphones and How They Work
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

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 [magnify]

  suspended parts     fixed-plate half-capacitors     other fixed parts
  proof mass (beam)     high capacitance     tether anchors
  folded tethers (springs)     medium capacitance     polysilicon substrate
        low capacitance      

Acceleration is determined from the differential capacitance of adjacent capacitors.

Characteristics of a Typical Low-g MEMS Accelerometer
  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 …

Summary

Problems

practice

  1. Write something.
    • Answer it.
  2. Write something.
    • Answer it.
  3. Write something.
    • Answer it.
  4. Write something completely different.
    • Answer it.

numerical

  1. problems

Resources

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