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Foreword
Aim of This Textbook
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The aim of this textbook is to explain the design and function of electronic circuits and components. The text covers electronic circuit components, DC analysis, and AC analysis.
It should be useful to beginner hobbyists as well as beginner engineering students, teaching both theory and practical applications.
It should be thought of as a companion project to the Wikipedia articles about electronics. While Wikipedia covers many details about the technology used in electronics components and related fields, the Electronics Wikibook covers a lot of the "howto" aspects that aren't covered in an encyclopedia. The book will focus on how to use the components to build useful circuits.
Prerequisites
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Prerequisite Topics
Most important / Required knowledge
Moderately Important / Aids in comprehension
Slightly Important / Related or helpful
Other Useful Topics
Preface
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Importance of Electronics
Electronics is the study and use of devices that control the flow of electrons (or other charged particles). These devices can be used to process information or perform tasks using electromagnetic power.
Electronic circuits can be found in numerous household products, including such items as telephones, computers, and CD players. Electronic devices have also allowed greatly increased precision in scientific measurements.
Interest in the field of electronics increased around 1900 and the advent of radio. Interest reached an alltime high in the 1940s, 50s, 60s, with the invention of transistor radios, the launch of Sputnik, and the science and math educational push to win the space race. Interest in electronics as a hobby in the 1970s led to the advent of the personal computer (PC).
Electronics have since seen a decline in hobbyist interest. Electronics is now generally studied as part of a collegelevel program in electrical engineering.
This book is an attempt at reviving the hobbyist mentality that made electronics so big in the first place, by making electronics concepts more accessible and giving practical knowledge, as well as providing technical information for the student.
Charge and Coulomb's Law
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Two atoms are walking down the street. The first atom says to the second atom "I think I lost an electron!" The second says "Are you sure?" To which the first states "I'm positive!"
Basic Understanding
 Conductors
 Materials which contain movable charges that can flow with minimal resistance.
 Insulators
 Materials with few or no movable charges, or with charges which flow with extremely high resistance.
 Semiconductors
 Materials whose behavior ranges between that of a conductor and that of an insulator under different conditions. Their conducting behavior may be heavily dependent on temperature. They are useful because we are able to change their conducting behavior to be dependent on many other factors.
 The Atom
 An atom contains a positively charged nucleus and one or more negatively charged electrons. The atom exists in three states: neutral, positively charged, and negatively charged. A neutral atom has the same number of electrons and protons, a positively charged atom has more protons than electrons and a negatively charged atom has more electrons than protons.
 (+) and () Ions
 An ion is an atom that has an unequal number of electrons and protons. Ions are formed when a neutral atom gains or loses electrons during a chemical reaction. In a battery, the positive side has + ions, means there are fewer electrons than protons, giving it an overall positive charge, and ve side, more electrons than protons, giving it an overall negative charge. +ve and ve charge will attract each other, and it is the use of such an attractive force that allows the battery to do work.
Note: Electric current is not the same as electron flow as is widely mistaken. Firstly, the total current has the opposite direction compared to electron flow. This is a lucky "mistake" on our forefathers' part to put it this way. It is also because of this lucky legacy that we are reminded that electricity can flow in materials other than metals alone. For example, in water, it is not electrons that flow, it is ions, and the +ve ions and ve ions flow in opposite directions, contributing half of the total current each.
Balance of Charge
Atoms, the smallest particles of matter that retain the properties of the matter, are made of protons, electrons, and neutrons. Protons have a positive charge, Electrons have a negative charge that cancels the proton's positive charge. Neutrons are particles that are similar to a proton but have a neutral charge. There are no differences between positive and negative charges except that particles with the same charge repel each other and particles with opposite charges attract each other. If a solitary positive proton and negative electron are placed near each other they will come together to form a hydrogen atom. This repulsion and attraction (force between stationary charged particles) is known as the Electrostatic Force and extends theoretically to infinity, but is diluted as the distance between particles increases.
When an atom has one or more missing electrons it is left with a positive charge, and when an atom has at least one extra electron it has a negative charge. Having a positive or a negative charge makes an atom an ion. Atoms only gain and lose protons and neutrons through fusion, fission, and radioactive decay. Although atoms are made of many particles and objects are made of many atoms, they behave similarly to charged particles in terms of how they repel and attract.
In an atom the protons and neutrons combine to form a tightly bound nucleus. This nucleus is surrounded by a vast cloud of electrons circling it at a distance but held near the protons by electromagnetic attraction (the electrostatic force discussed earlier). The cloud exists as a series of overlapping shells / bands in which the inner valence bands are filled with electrons and are tightly bound to the atom. The outer conduction bands contain no electrons except those that have accelerated to the conduction bands by gaining energy. With enough energy an electron will escape an atom (compare with the escape velocity of a space rocket). When an electron in the conduction band decelerates and falls to another conduction band or the valence band a photon is emitted. This is known as the photoelectric effect.
A laser is formed when electrons travel back and forth between conduction bands emitting synchronized photons.
 When the conduction and valence bands overlap, the atom is a conductor and allows for the free movement of electrons. Conductors are metals and can be thought of as a bunch of atomic nuclei surrounded by a churning "sea of electrons".
 When there is a large energy level gap between the conduction and valence bands, the atom is an insulator; it traps electrons. Many insulators are nonmetals and are good at blocking the flow of electrons.
 When there is a small energy level gap between the conduction and valence bands, the atom is a semiconductor. Semiconductors behave like conductors and insulators, and work using the conduction and valence bands. The electrons in the outer valence band are known as holes. They behave like positive charges because of how they flow. In semiconductors electrons collide with the material and their progress is halted. This makes the electrons have an effective mass that is less than their normal mass. In some semiconductors holes have a larger effective mass than the conduction electrons.
Electronic devices are based on the idea of exploiting the differences between conductors, insulators, and semiconductors but also exploit known physical phenomena such as electromagnetism and phosphorescence.
Conductors
In a metal the electrons of an object are free to move from atom to atom. Due to their mutual repulsion (calculable via Coulomb's Law ), the valence electrons are forced from the centre of the object and spread out evenly across its surface in order to be as far apart as possible. This cavity of empty space is known as a Faraday Cage and stops electromagnetic radiation, such as charge, radio waves, and EMPs (ElectroMagnetic Pulses) from entering and leaving the object. If there are holes in the Faraday Cage then radiation can pass.
One of the interesting things to do with conductors is demonstrate the transfer of charge between metal spheres. Start by taking two identical and uncharged metal spheres which are each suspended by insulators (such as a pieces of string). The first step involves putting sphere 1 next to but not touching sphere 2. This causes all the electrons in sphere 2 to travel away from sphere 1 to the far end of sphere 2. So sphere 2 now has a negative end filled with electrons and a positive end lacking electrons. Next sphere 2 is grounded by contact with a conductor connected with the earth and the earth takes its electrons leaving sphere 2 with a positive charge. The positive charge (absence of electrons) spreads evenly across the surface due to its lack of electrons. If suspended by strings, the relatively negatively charged sphere 1 will attract the relatively positively charged sphere 2.
Insulators
In an insulator the charges of a material are stuck and cannot flow. This allows an imbalance of charge to build up on the surface of the object by way of the triboelectric effect. The triboelectric effect (rubbing electricity effect) involves the exchange of electrons when two different insulators such as glass, hard rubber, amber, or even the seat of one's trousers, come into contact. The polarity and strength of the charges produced differ according to the material composition and its surface smoothness. For example, glass rubbed with silk will build up a charge, as will hard rubber rubbed with fur. The effect is greatly enhanced by rubbing materials together.
 Van de Graaff Generator: A charge pump (pump for electrons) that generates static electricity. In a Van de Graaff generator, a conveyor belt uses rubbing to pick up electrons, which are then deposited on metal brushes. The end result is a charge difference.
Because the material being rubbed is now charged, contact with an uncharged object or an object with the opposite charge may cause a discharge of the builtup static electricity by way of a spark. A person simply walking across a carpet may build up enough charge to cause a spark to travel over a centimetre. The spark is powerful enough to attract dust particles to cloth, destroy electrical equipment, ignite gas fumes, and create lightning. In extreme cases the spark can destroy factories that deal with gunpowder and explosives. The best way to remove static electricity is by discharging it through grounding. Humid air will also slowly discharge static electricity. This is one reason cells and capacitors lose charge over time.
Note: The concept of an insulator changes depending on the applied voltage. Air looks like an insulator when a low voltage is applied. But it breaks down as an insulator, becomes ionised, at about ten kilovolts per centimetre. A person could put their shoe across the terminals of a car battery and it would look like an insulator. But putting a shoe across a ten kilovolt powerline will cause a short.
Quantity of Charge
Protons and electrons have opposite but equal charge. Because in almost all cases, the charge on protons or electrons is the smallest amount of charge commonly discussed, the quantity of charge of one proton is considered one positive elementary charge and the charge of one electron is one negative elementary charge. Because atoms and such particles are so small, and charge in amounts of multitrillions of elementary charges are usually discussed, a much larger unit of charge is typically used. The coulomb is a unit of charge, which can be expressed as a positive or negative number, which is equal to approximately 6.2415×10^{18} elementary charges. Accordingly, an elementary charge is equal to approximately 1.602×10^{19} coulombs. The commonly used abbreviation for the coulomb is a capital C. The SI definition of a coulomb is the quantity of charge which passes a point over a period of 1 second ( s ) when a current of 1 ampere (A) flows past that point, i.e., C = A·s or A = C/s. You may find it helpful during later lessons to retain this picture in your mind (even though you may not recall the exact number). An ampere is one of the fundamental units in physics from which various other units are defined, such as the coulomb.
Force between Charges: Coulomb's Law
The repulsive or attractive electrostatic force between charges decreases as the charges are located further from each other by the square of the distance between them. An equation called Coulomb's law determines the electrostatic force between two charged objects. The following picture shows a charge q at a certain point with another charge Q at a distance of r away from it. The presence of Q causes an electrostatic force to be exerted on q.
The magnitude of the electrostatic force F, on a charge q, due to another charge Q, equals Coulomb's constant multiplied by the product of the two charges (in coulombs) divided by the square of the distance r, between the charges q and Q. Here a capital Q and small q are scalar quantities used for symbolizing the two charges, but other symbols such as q_{1} and q_{2} have been used in other sources. These symbols for charge were used for consistency with the electric field article in Wikipedia and are consistent with the Reference below.
F = magnitude of electrostatic force on charge q due to another charge Q
r = distance (magnitude quantity in above equation) between q and Q
k = Coulomb's constant = 8.9875×10^{9} N·m^{2}/C^{2} in free space
The value of Coulomb's constant given here is such that the preceding Coulomb's Law equation will work if both q and Q are given in units of coulombs, r in metres, and F in newtons and there is no dielectric material between the charges. A dielectric material is one that reduces the electrostatic force when placed between charges. Furthermore, Coulomb's constant can be given by:
where = permittivity. When there is no dielectric material between the charges (for example, in free space or a vacuum),
Air is only very weakly dielectric and the value above for will work well enough with air between the charges. If a dielectric material is present, then
where is the dielectric constant which depends on the dielectric material. In a vacuum (free space), and thus . For air, . Typically, solid insulating materials have values of and will reduce electric force between charges. The dielectric constant can also be called relative permittivity, symbolized as in Wikipedia.
Highly charged particles close to each other exert heavy forces on each other; if the charges are less or they are farther apart, the force is less. As the charges move far enough apart, their effect on each other becomes negligible.
Any force on an object is a vector quantity. Vector quantities such as forces are characterized by a numerical magnitude (i. e. basically the size of the force) and a direction. A vector is often pictured by an arrow pointing in the direction. In a force vector, the direction is the one in which the force pulls the object. The symbol is used here for the electric force vector. If charges q and Q are either both positive or both negative, then they will repel each other. This means the direction of the electric force on q due to Q is away from Q in exactly the opposite direction, as shown by the red arrow in the preceding diagram. If one of the charges is positive and the other negative, then they will attract each other. This means that the direction of on q due to Q is exactly in the direction towards Q, as shown by the blue arrow in the preceding diagram. The Coulomb's equation shown above will give a magnitude for a repulsive force away from the Q charge. A property of a vector is that if its magnitude is negative, the vector will be equal to a vector with an equivalent but positive magnitude and exactly the opposite direction. So, if the magnitude given by the above equation is negative due to opposite charges, the direction of the resulting force will be directly opposite of away from Q, meaning the force will be towards Q, an attractive force. In other sources, different variations of Coulombs' Law are given, including vector formulas in some cases (see Wikipedia link and reference(s) below).
In many situations, there may be many charges, Q_{1}, Q_{2}, Q_{3}, through Q_{n}, on the charge q in question. Each of the Q_{1} through Q_{n} charges will exert an electric force on q. The direction of the force depends on the location of the surrounding charges. A Coulomb's Law calculation between q and a corresponding Q_{i} charge would give the magnitude of the electric force exerted by each of the Q_{i} charges for i = 1 through n, but the direction of each of the component forces must also be used to determine the individual force vectors, . To determine the total electric force on q, the electric force contributions from each of these charges add up as vector quantities, not just like ordinary (or scalar) numbers.
The total electric force on q is additive to any other forces affecting it, but all of the forces are to be added together as vectors to obtain the total force on the charged object q. In many cases, there are billions of electrons or other charges present, so that geometrical distributions of charges are used with equations stemming from Coulomb's Law. Practically speaking, such calculations are usually of more interest to a physicist than an electrician, electrical engineer, or electronics hobbyist, so they will not be discussed much more in this book, except in the section on capacitors.
In addition to the electrostatic forces described here, electromagnetic forces are created when the charges are moving. These will be described later.
Reference(s):
 College Physics Volume 2 by Doug Davis, Saunders College Publishing, Orlando, FL, 1994
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Basic Concepts
What is Electronics?
Electronics is the study of flow of electrons in various materials or space subjected to various conditions. In the past, electronics dealt with the study of Vacuum Tubes or Thermionic valves, today it mainly deals with flow of electrons in semiconductors. However, despite these technological differences, the main focus of electronics remains the controlled flow of electrons through a medium. By controlling the flow of electrons, we can make them perform special tasks, such as power an induction motor or heat a resistive coil.
Plumbing Analogy: A simple way to understand electrical circuits is to think of them as pipes. Let's say you have a simple circuit with a voltage source and a resistor between the positive and negative terminals on the source. When the circuit is powered, electrons will move from the negative terminal, through the resistor, and into the positive terminal. The resistor is basically a path of conduction that resists the movement of electrons. This circuit could also be represented as a plumbing network. In the plumbing network, the resistor would be equivalent to a section of pipe, where the water is forced to move around several barriers to pass through, effectively slowing its flow. If the pipe is level, no water will flow in an organized fashion, since the pressure is equal throughout the pipe. However, if we tilt the pipe to a vertical position (similar to turning on a voltage source), a pressure difference is created (similar to a voltage difference) and the water begins flowing through the pipe. This flow of water is similar to the flow of electrons in a circuit.
Electricity
To understand electronics, you need to understand electricity and what it is. Basically, electricity is the flow of electrons due to a difference in electrical charge between two points. This difference in charge is created due to a difference in electron density. If you have a point where the electron density is higher than the electron density at another point, the electrons in the area of higher density will want to balance the charge by migrating towards the area with lower density. This migration is referred to as electrical current. Thus, flow in an electrical circuit is induced by putting more electrons on one side of the circuit than the other, forcing them to move through the circuit to balance the charge density.
Electric Charge
Observations tell us that matter can either be electrically neutral (that is, have no net charge), or carry a positive or negative charge. On a microscopic level, a negative charge corresponds to an excess of electrons in the material (which each carry a 'unit' of negative charge), and a positive charge a shortage of electrons. We denote the charge of an object by , which is measured in Coulombs.
Coulomb's Law
Two objects that have the same type of charge are known to repel, whereas objects with opposite charges attract. The force between charged objects is given by Coulomb's Law:
where and are the charges of the two objects (positive or negative), is the distance between them, and a universal constant.
Electric field
Suppose we have a fixed particle (or a collection of fixed particles), then we can calculate the force on any given (test) particle with charge by applying Coulomb's Law. There is a force at any point in space; we say that there is an electric field due to the (fixed) charges. The field at any point is equal to the net force on a charged particle divided by its charge, or:
Lorentz's Law
When a charge in motion passes through a magnetic field, the magnetic field will push a positive charge upward and negative charge downward in the direction perpendicular to the initial direction traveled. The magnitude of the magnetic force on the charge is given by Lorentz's Law:
ElectroMagnetic Force
The sum of the Coulomb and Lorentz's Forces is called the ElectroMagnetic Force:
Electricity and Matter
All matter interacts with Electricity, and are divided into three categories: Conductors, Semi Conductors, and Non Conductors.
 Matter that conducts Electricity easily. Metals like Zinc (Zn) and Copper (Cu) conduct electricity very easily. Therefore, they are used to make Conductors.
 Matter that does not conduct Electricity at all. NonMetals like Wood and Rubber do not conduct electricity so easily. Therefore, they are used to make NonConductors.
 Matter that conducts electricity in a manner between that of Conductors and NonConductors. For example, Silicon (Si) and Germanium (Ge) conduct electricity better than nonconductors but worse than conductors. Therefore, they are used to make Semi Conductors.
Electricity and Conductors
Normally, all conductors have a zero net charge . If there is an electric force that exerts a pressure on the charges in the conductor to force charges to move in a straight line result in a stream of electric charge moving in a straight line
Voltage
 The pressure the electric force exert on the charges is called voltage denoted as V measured in Volt (V) and defined as the ratio of Work Done on Charge
Current
 The moving of straight lines of electric charges in the conductor is called current denoted as I measured in Ampere (A) and defined as Charge flow through an area in a unit of time
Conductance
 Conductance is defined as the ratio of current over voltage denoted as Y measured in mho
Resistance
 Resistance is defined as the ratio of voltage over current denoted as R measured in Ohm
Generally, resistance of any conductor is found to increase with increasing temperature
For Conductor
 R = Ro(1 + nT)
For Semi Conductor
 R = Ro e^{nT}
When a conductor conducts electricity, it dissipates heat energy into the surrounding . This results in a loss of electric energy transmitted . If the electric supply energy is P_{V} and the electric loss energy is P_{R} Then the electric energy delivered:
 P = P_{V}  P_{R}
Black Body Radiation
Further experience with conductors that conduct electricity . It is observed that all conductors that conduct electricity exhibit
 Change in Temperature
 Release Radiant Heat Energy into the surrounding
Experiment
Connect a conductor with an electric source in a closed loop . Plot the value I at different f to have a I  f diagram
Observation
for f<fo
 Current increasing with increasing f .
 Radiant heat is a wave travels at velocity v = λf carries energy E = m v^{2} .
for f=fo,
 Current stops increasing .
 Radiant heat is a wave travels at velocity v = c (speed of Light) carries energy E = hfo .
for f>fo,
 Current remains at the value of current at fo .
 Radiant heat is a wave travels at velocity v = c (speed of Light) carries energy E = h nfo
Conclusion
 All conductor that conducts Electricity has a threshold frequency fo
 The Radiant Heat Energy is a Light Wave of dual Wave Particle characteristic. Sometimes it behaves like Particle, sometimes it behaves like Wave
 At Frequency f > fo . The energy of the Light is quantized . it can only have the value of multiple integer of fo . E = hf = h nfo
Cells
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Cells
 Cell: Two materials with a voltage difference between them. This causes current to flow, which does work. Electrons travel from the cathode, do some work, and are absorbed by the anode.
 Anode: Destination of electrons.
 Cathode: Source of electrons.
 ions: An atom with an imbalance of electrons.
 cell operation: The cell runs and electrons are depleted at the cathode and accumulate at the anode. This creates a reverse voltage which stops the flow of electrons.
 irreversible: At some point the voltage difference reactions between the cathode and anode will decrease to a point that the cell in unusable. At this point, in an irreversible cell, the voltage difference is irreplaceably lost, and the cell is of no further use.
 reversible: Able to run the cell backwards.
 rechargeable: In a rechargeable cell, when the voltage difference between the cathode and anode decreases, the cell can be recharged, thereby increasing the voltage difference to a suitable level to allow continued use.
 humid air will discharge cells.
 cells are usually made of toxic or corrosive substances, for example lead and sulphuric acid. Such substances have been known to explode.
 Electronegativity [16]
What is the relationship between voltage and electronegativity?
 Electronegativity is a concept in chemistry used to measure and predict the relative likelihood of a chemical reaction causing electrons to shift from one chemical to another resulting in ions and molecular bonds. A battery cell operates by allowing two chemicals to react and supply ions to the anode and cathode. When the supply of a reactant is consumed, the battery is dead. It no longer produces different electrical potential at the anode and cathode driven by the chemical reaction.
 Voltage is the electrical potential of a point due to surrounding measurable electric charge distributions and points as calculated by application Coulomb's Law. Voltage difference between two points connected by a conductor results in electron flow.
Resistors
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Resistor
A resistor is a block or material that limits the flow of current. The greater the resistance, the lower the current will be, assuming the same voltage imposed on the resistor. The hydraulic analogy of a resistor would be the pipe with water flowing through it. The wider the diameter of a pipe, the higher the water flow through the pipe, assuming the same pressure difference on the terminals of a pipe.
Resistor's Symbol
Resistors have two leads (points of contact) to which the resistor can be connected to an electrical circuit. A symbol for a resistor used in electrical circuit diagrams is shown below.
The endpoints at the left and right sides of the symbol indicate the points of contact for the resistor. The ratio of the voltage to current will always be positive, since a higher voltage on one side of a resistor is a positive voltage, and a current will flow from the positive side to the negative side, resulting in a positive current. If the voltage is reversed, the current is reversed, leading again to a positive resistance.
Resistance
Resistance is a characteristic of Resistor indicates the measurement of current opposition . Resistance has a symbol R measured in Ohm (Ω) . The ratio of voltage to current is referred to as Ohm's Law, and is one of the most basic laws that govern electronics.
An ohm is the amount of resistance which passes one ampere of current when a one volt potential is placed across it. (The ohm is actually defined as the resistance which dissipates one watt of power when one ampere of current is passed through it.)
Resistance can vary from very small to very large. A superconductor has zero resistance, while something like the input to an opamp can have a resistance near 10^{12} Ω, and even higher resistances are possible.
Resistance and Temperature
For most materials, resistance increases with increasing Temperature
 For Conductor .
 For Semi Conductor .
Resistance and Electric Power Loss
Resistance converts Electrical Energy into Heat this causes Electric Energy Loss.
NOTE : Resistors which dissipate large amounts of power are cooled so that they are not destroyed, typically with finned heatsinks.
If Electric Energy Supply is P_{v} and Electric Energy Loss is P_{r} Then, Electric Energy Delivered is
The ratio of Electric Energy Delivered over Electric Energy Supplied indicates the Efficiency of Electric Power Supply
Resistor's Labeling (See also Identification)
A manufactured resistor is usually labeled with the nominal value (value to be manufactured to) and sometimes a tolerance. Rectangular resistors will usually contain numbers that indicate a resistance and a multiplier. If there are three or four numbers on the resistor, the first numbers are a resistance value, and the last number refers to the number of zeroes in the multiplier. If there is an R in the value, the R takes the place of the decimal point.
 Examples
 2003 means 200×10^{3} = 200kΩ
 600 means 60×10^{0} = 60Ω
 2R5 means 2.5Ω
 R01 means 0.01Ω
Cylindrical resistors (axial) usually have colored bands that indicate a number and a multiplier. Resistance bands are next to each other, with a tolerance band slightly farther away from the resistance bands. Starting from the resistance band side of the resistor, each colour represents a number in the same fashion as the number system shown above.
Colour System

Black Brown Red Orange Yellow Green Blue Violet Grey White 0 1 2 3 4 5 6 7 8 9
Clue : B.B.ROY of Great Britain was a Very Good Worker. Additional Colours: A gold band in the multiplier position means 0.1, but means a 5% tolerance in the tolerance position. A silver band in the multiplier position means 0.01, but means 10% in the tolerance position.
Resistor's Construction
The resistance R of a component is dependent on its physical dimension and can be calculated using:
where
 ρ is the electrical resistivity (resistance to electricity) of the material,
 L is the length of the material
 A is the crosssectional area of the material.
If you increase ρ or L you increase the resistance of the material, but if you increase A you decrease the resistance of the material.
Resistivity of the Material
Every material has its own resistivity, depending on its physical makeup. Most metals are conductors and have very low resistivity; whereas, insulators such as rubber, wood, and air all have very high resistivity. The inverse of resistivity is conductivity, which is measured in units of Siemens/metre (S/m) or, equivalently. mhos/metre.
In the following chart, it is not immediately obvious how the unit ohmmeter is selected. Considering a solid block of the material to be tested, one can readily see that the resistance of the block will decrease as its crosssectional area increases (thus widening the conceptual "pipe"), and will increase as the length of the block increases (lengthening the "pipe"). Given a fixed length, the resistance will increase as the crosssectional area decreases; the resistance, multiplied by the area, will be a constant. If the crosssectional area is held constant, as the length is increased, the resistance increases in proportion, so the resistance divided by the length is similarly a constant. Thus the bulk resistance of a material is typically measured in ohm meters squared per meter, which simplifies to ohm  meter (Ωm).

Conductors Ωm (Ohmmeter) Silver 1.59×10^{8} Copper 1.72×10^{8} Gold 2.44×10^{8} Aluminum 2.83×10^{8} Tungsten 5.6×10^{8} Iron 10×10^{8} Platinum 11×10^{8} Lead 22×10^{8} Nichrome 1.50×10^{6} (A nickelchromium alloy commonly used in heating elements) Graphite ~10^{6} Carbon 3.5×10^{5} Semiconductors Pure Germanium 0.6 Pure Silicon 640 Common purified water ~10^{3} Ultrapure water ~10^{5} Pure Gallium Arsenide ~10^{6} Insulators Diamond ~10^{10} Glass 10^{10} to 10^{14} Mica 9×10^{13} Rubber 10^{13} to 10^{16} Organic polymers ~10^{14} Sulfur ~10^{15} Quartz (fused) 5 to 75×10^{16} Air very high
Silver, copper, gold, and aluminum are popular materials for wires, due to low resistivity. Silicon and germanium are used as semiconductors. Glass, rubber, quartz crystal, and air are popular dielectrics, due to high resistivity.
Many materials, such as air, have a nonlinear resistance curve. Normal undisturbed air has a high resistance, but air with a high enough voltage applied will become ionized and conduct very easily.
The resistivity of a material also depends on its temperature. Normally, the hotter an object is, the more resistance it has. At high temperatures, the resistance is proportional to the absolute temperature. At low temperatures, the formula is more complicated, and what counts as a high or low temperature depends on what the resistor is made from. In some materials the resistivity drops to zero below a certain temperature. This is known as superconductivity, and has many useful applications.
(Some materials, such as silicon, have less resistance at higher temperatures.)
For all resistors, the change in resistance for a small increase in temperature is directly proportional to the change in temperature.
Current passing through a resistor will warm it up. Many components have heat sinks to dissipate that heat. The heatsink keeps the component from melting or setting something on fire.
Length
The length of an object is directly proportional to its resistance. As shown in the diagram below, 1 unit cubed of material has 1 ohm of resistance. However, when 4 units are stacked lengthwise and a connection is made to the front and back sides respectively, the total resistance is 4 ohms. This is because the length of the unit is 4, whereas the crosssectional area remains 1. However, if you were to make connections on the sides, the exact opposite would be true: the crosssectional area would be 4 and the length 1, resulting in 0.25 ohms total resistance.
CrossSectional Area
Increasing area is the same as having resistors in parallel, so as you increase the area you add more paths for current to take.
The resistance of a material is inversely proportional to its crosssectional area. This is shown in the diagram below, where 1 unit cubed has one ohm of resistance. However, if 4 units cubed are stacked on top of each other in the fashion such that there is 4 units squared of crosssectional area, and the electrical connections are made to the front and back such that the connections are on the largest sides, the resultant resistance would be 0.25 ohms.
Additional note: There are two reasons why a small crosssectional area tends to raise resistance. One is that the electrons, all having the same negative charge, repel each other. Thus there is resistance to many being forced into a small space. The other reason is that they collide, causing "scattering," and therefore they are diverted from their original directions. (More discussion is on page 27 of "Industrial Electronics," by D. J. Shanefield, Noyes Publications, Boston, 2001.)
Example
For instance, if you wanted to calculate the resistance of a 1 cm high, 1 cm wide, 5 cm deep block of copper, as shown in the diagram below:
You would first need to decide how it's oriented. Suppose you want to use it from front to back (lengthwise), like a piece of wire, with electrical contacts on the front and rear faces. Next you need to find the length, L. As shown, it is 5 cm long (0.05 m). Then, we look up the resistivity of copper on the table, 1.6×10^{8} Ωmeters. Lastly, we calculate the crosssectional area of the conductor, which is 1 cm × 1 cm = 1 cm^{2} (0.0001 m^{2}). Then, we put it all in the formula, converting cm to m:
units m^{2} cancel:
Which, after evaluating, gives you a final value of 8.0×10^{6} Ω, or 8 microohms, a very small resistance. The method shown above included the units to demonstrate how the units cancel out, but the calculation will work as long as you use consistent units.
 Internet Hint: Google calculator can do calculations like this for you, automatically converting units. This example can be calculated with this link: [17]
Properties of the material
 Wirewound: Used for power resistors, since the power per volume ratio is highest. These usually have the lowest noise.
 Carbon Film: These are easy to produce, but usually have lots of noise because of the properties of the material.
 Metal Film: These resistors have thermal and voltage noise attributes that are between carbon and wirewound.
 Ceramic: Useful for high frequency applications.
Resistor Connection
Resistors in Series
Resistors in series are equivalent to having one long resistor. If the properties of two resistors are equivalent, except the length, the final resistance will be the sum of the two construction methods:
This means that the resistors add when in series.
 Christmas tree lights are usually connected in series, with the unfortunate effect that if one light blows, the others will all go out (This happens because the circuit is not complete, if a circuit is not complete then the current cannot flow, hence the light bulbs all go out). However, most modern Christmas light strings have built in shunt resistors in parallel to the bulb, so that current will flow past the blown light bulb.
Resistors in Parallel
In a parallel circuit, current is divided among multiple paths. This means that two resistors in parallel have a lower equivalent resistance than either of the parallel resistors, since both resistors allow current to pass. Two resistors in parallel will be equivalent to a resistor that is twice as wide:
Since conductances (the inverse of resistance) add in parallel, you get the following equation:
For example, two 4 Ω resistors in parallel have an equivalent resistance of only 2 Ω.
To simplify mathematical equations, resistances in parallel can be represented with two vertical lines "" (as in geometry). For two resistors the parallel formula simplifies to:
Combinations of series and parallel
Resistors in parallel are evaluated as if in a mathematical set of "parentheses." The most basic group of resistors in parallel is evaluated first, then the group in series with the new equivalent resistor, then the next group of resistors in parallel, and so on. For example, the above portion would be evaluated as follows:
Resistor variations
 Variable Resistor or Potentiometer
 Variable resistors are tunable, meaning you can turn a dial or slide a contact and change the resistance. They are used as knobs to control the volume of a stereo, or as a dimmer for a lamp. The term Potentiometer is often abbreviated as 'pot'. It is constructed like a resistor, but has a sliding tap contact. Potentiometers are used as Voltage Dividers. It is rare to find a variable resistor with only two leads. Most are potentiometers with three leads, even if one is not connected to anything.
 Rheostat
 A variation of the potentiostat with a high current rating, which is used to control the amount of power going through a load, such as a motor.
 Thermistor
 Temperaturesensitive resistor, in which the resistance decreases as the temperature rises. They are used in fire alarms, so if things get too hot the current rises and trips a switch that sounds an alarm.
 LDR (Light Dependent Resistor) or Photoresistor
 A resistor which changes values depending on the amount of light shining on its surface. The resistance decreases as the amount of light increases. They are used in street lamps, so when it gets dark the current decreases and turns on the street lamp.
Applications
 Voltage division / Attenuation: Sometimes a voltage will be too large to measure, so a means to linearly reduce the voltage is required. Placing two resistors in series to ground will provide a point in the middle to tap. Resistor R_{A} is placed between the input voltage and the output node, and the resistor R_{B} is placed between the output node and ground. This creates a voltage divider to lessen the output voltage. Typically, the resistors are near the value of ~10kΩ. The Thevenin model of the circuit gives an output resistance R_{OUT} = R_{A}R_{B}. A larger output resistance will more likely be affected by the input resistance of the measuring circuit (this is a desired effect in the transistor biasing circuits). Placement of the voltage divider should be close to the measuring circuit, to minimize noise (in this arrangement, it will be also lessened Rb/(Ra + Rb) times). The output voltage of the voltage divider is
 Pullup / Pulldown: If there is nothing to drive a signal node, the node will be left "floating" (for example, such a situation occurs at the trigger input of a car alarm system when the driver has switched off the internal lamp). This may lead to unintended values being measured, or causing sideeffects when the voltage is propagated down the remainder of the circuit. To prevent this, a relatively high value resistor (usually ~10kΩ to ~1MΩ) is placed between the node and ground (pulldown) or a high voltage (pullup) to bring the voltage of the "floating" node near to the voltage it is being pulled. A resistive voltage divider is another example where the upper resistor "pulls" the output point up toward the input voltage while the lower resistor "pulls" the output point toward the ground. This idea is evolved in the circuit of a resistive voltage summer (for example, the resistors R_{1} and R_{2} of an opamp inverting amplifier) supplied by two voltages (V_{IN} and V_{OUT}) having opposite polarities. The two voltage sources "pull" the output point in opposite directions; as a result, if R_{2}/R_{1} = V_{OUT}/V_{IN}, the point becomes a virtual ground. Placement of a pullup or pulldown resistor does not have a significant effect on the performance of the circuit, if they have high resistances.
 Current limiting / Isolation: In order to protect circuits from conditions that may cause too much current in a device, a current limiting resistor is inserted in the middle of the circuit. A digital input to a microcontroller may benefit from a current limiting resistor. The inputs to modern microcontrollers have protection circuitry built in that will protect the input from an overvoltage condition, provided that the current is small enough. For instance, a common microcontroller will be capable of withstanding 20mA. If there are 12V nets on a circuit board, or in a system, the digital input will benefit from a 350Ω resistor (refer to calculation below). Usually a slightly larger resistance is used in practice, but too large of a resistor will cause noise, and may prevent the input from being able to read the voltage. It is good practice to place the resistor as close as possible to the microcontroller input, so that an accidental short on the board will mean that the microcontroller input is likely still protected.
 Line termination / Impedance matching: The properties of an electric wave propagating through a conductor (such as a wire) create a reflection, which can be viewed as unwanted noise. A reflection can be eliminated by maximizing the power transfer between the conductor and the termination resistor. By matching the resistance (more importantly the impedance ), the wave will not cause a reflection. The voltage of the echo V_r is calculated below in reference to the original signal V_o as a function of the conductor impedance Z_C and the terminator impedance Z_T. As the name implies, the termination resistor goes at the end of the conductor.
 Current sensing: Measurement of a current cannot be done directly. There are three major ways to measure a current: a resistor, a hall sensor, and an inductor. The hall sensor and inductor use a property of the magnetic field to sense the current through a nearby conductor. According to Ohm's law, if a current I flows through a resistor R, a voltage V = R.I appears across the resistor. Therefore, the resistor can act as a passive currenttovoltage converter. In this arrangement, the resistor should have a very low value (sometimes on the order of ~0.01Ω), so it does not affect the current flow or heat up; however, a smaller value has a lower voltage to read, which means more noise may be introduced. This contradiction is solved in the circuit of an active currenttovoltage converter where the resistor may have a significant resistance as an opamp compensates the "undesired" voltage drop across it (unfortunately, this remedy may be applied only in lowcurrent measurements). The current sense resistor should be placed as close as possible to where the measurement occurs, in order not to disturb the circuit.
 Filtering: Filtering is discussed later, after an introduction to capacitors and inductors. Filters are best placed close to where measurement takes place.
Specifications
Resistors are available as prefabricated, realworld components. The behavior of such components deviates from an ideal resistor in certain ways. Therefore, realworld resistors are not only specified by their resistance, but also by other parameters. In order to select a manufactured resistance, the entire range of specifications should be considered. Usually, exact values do not need to be known, but ranges should be determined.
Nominal Resistance
The nominal resistance is the resistance that can be expected when ordering a resistor. Finding a range for the resistance is necessary, especially when operating on signals. Resistors do not come in all of the values that will be necessary. Sometimes resistor values can be manipulated by shaving off parts of a resistor (in industrial environments this is sometimes done with a LASER to adjust a circuit), or by combining several resistors in series and parallel.
Available resistor values typically come with a resistance value from a so called resistor series. Resistor series are sets of standard, predefined resistance values. The values are actually made up from a geometric sequence within each decade. In every decade there are supposed to be resistance values, with a constant step factor. The standard resistor values within a decade are derived by using the step factor
rounded to a two digit precision. Resistor series are named E, according to the used value of in the above formula.
n Values/Decade Step factor i Series  6 1.47 E6 12 1.21 E12 24 1.10 E24 48 1.05 E48
For example, in the E12 series for , the resistance steps in a decade are, after rounding the following 12 values:
1.00, 1.20, 1.50, 1.80, 2.20, 2.70, 3.30, 3.90, 4.70, 5.60, 6.80, and 8.20
and actually available resistors from the E12 series are for example resistors with a nominal value of 120Ω or 4.7kΩ.
Tolerances
A manufactured resistor has a certain tolerance to which the resistance may differ from the nominal value. For example, a 2kΩ resistor may have a tolerance of ±5%, leaving a resistor with a value between 1.9kΩ and 2.1kΩ (i.e. 2kΩ±100Ω). The tolerance must be accounted for when designing circuits. A circuit with an absolute voltage of 5V±0.0V in a voltage divider network with two resistors of 2kΩ±5% will have a resultant voltage of 5V±10% (i.e. 5V±0.1V). The final resistor tolerances are found by taking the derivative of the resistor values, and plugging the absolute deviations into the resulting equation.
The above mentioned Eseries which are used to provide standardized nominal resistance values, are also coupled to standardized nominal tolerances. The fewer steps within a decade there are, the larger the allowed tolerance of a resistor from such a series is. More precises resistors, outside of the mentioned Eseries are also available, e.g. for highprecision measurement equipment. Common tolerances, colors and key characters used to identify them are for example:
Series Values/Decade Tolerance Color Code Character Code  E6 6 ±20% [none] [none] E12 12 ±10% silver K E24 24 ±5% gold J E48 48 ±2% red G   ±1% brown F   ±0.5%  D   ±0.25%  C   ±0.1%  B
Resistor manufacturers can benefit from this standardization. They manufacture resistors first, and afterwards they measure them. If a resistor does not meet the nominal value within the defined tolerance of one Eseries, it might still fit into a lower series, and doesn't have to be thrown away, but can be sold as being compliant to that lower Eseries standard. Although typically at a lower price.
Series: Resistors that combine in series add the nominal tolerances together.
 Derivation:
 Example: For two resistors in series R_{A} = 1.5kΩ±130Ω and R_{B} = 500Ω±25Ω, the tolerance is 130Ω + 25Ω, resulting in a final resistor value R_{T} = 2kΩ±155Ω.
Parallel: Resistors that combine in parallel have a combined tolerance that is slightly more complex.
 Derivation:
 Example: For two resistors in parallel R_{A} = 1.5kΩ±130Ω and R_{B} = 500Ω±25Ω.
Power Rating
Because the purpose of a resistor is to dissipate power in the form of heat, the resistor has a rating (in watts) at which the resistor can continue to dissipate before the temperature overwhelms the resistor and causes it to overheat. When a resistor overheats, the material begins to melt away, which will cause the resistance to increase (usually), until the resistor breaks.
Operating Temperature
Related to power rating, the operating temperature is the temperature that the resistor can continue to operate before being destroyed.
Maximum Voltage
In order to avoid sparkovers or material breakdown a certain maximum voltage over a resistor must not be exceeded. The maximum voltage is part of a resistor's specification, and typically a function of the resistor's physical length, distance of the leads, material and coating.
For example, a resistor with a maximum operating voltage of 1kV can have a length in the area of 2", while a 0.3" resistor can operate under up to several tens of volts, probably up to a hundred volts. When working with dangerous voltages it is essential to check the actual specification of a resistor, instead of only trusting it because of the length.
Temperature Coefficient
This parameter refers to the constant in which the resistance changes per degree Celsius (units in C^{1}). The change in temperature is not linear over the entire range of temperatures, but can usually be thought of as linear around a certain range (usually around room temperature). However, the resistance should be characterized over a large range if the resistor is to be used as a thermistor in those ranges. The simplified linearized formula for the affect on temperature to a resistor is expressed in an equation:
Capacity and Inductance
Real world resistors not only show the physical property of resistance, but also have a certain capacity and inductance. These properties start to become important, if a resistor is used in some high frequency circuitry. Wire wound resistors, for example, show an inductance which typically make them unusable above 1kHz.
Packaging
Resistors can be packaged in any way possible, but are divided into surface mount, through hole, soldering tag and a few more forms. Surface mount is connected to the same side that the resistor is on. Through hole resistors have leads (wires) that typically go through the circuit board and are soldered to the board on the side opposite the resistor, hence the name. Resistors with leads are also used in pointtopoint circuits without circuit boards. Soldering tag resistors have lugs to solder wires or high current connectors onto.
Usual packages for surface mount resistors are rectangular, referenced by a length and a width in mils (thousands of an inch). For instance, an 0805 resistor is a rectangle with length .08" x .05", with contacts (metal that connects to the resistor) on either side. Typical through hole resistors are cylindrical, referenced either by the length (such as 0.300") or by a typical power rating that is common to the length (a 1/4W resistor is typically 0.300"). This length does not include the length of the leads.
Related Wikimedia resources
Wikibooks
 Circuit Idea: Passive voltagetocurrent converter shows how the bare resistor can act as a simple voltagetocurrent converter.
Wikipedia
 Voltagetocurrent converter builds consecutively the passive and active versions of the voltagetocurrent converter.
 Currenttovoltage converter is dedicated to the passive and active versions of the inverse currenttovoltage converter.
Further Readings
 Resistors in Series and Parallel: Electronics for Beginners
Capacitors
Electronics  Foreword  Basic Electronics  Complex Electronics  Electricity  Machines  History of Electronics  Appendix  edit
General remarks
Capacitors are a good example of the fact that even the simplest device can become complicated given 250 years of evolution. (Citation J. Ho, T. R. Jow, St. Boggs, Historical Introduction to Capacitor Technology)^{[1]}
Capacitors, together with resistors, inductors and memristors, belong to the group of "passive components" for electronic equipment. Although in absolute figures the most common capacitors are integrated capacitors, e.g. in DRAMs or in flash memory structures, this article is concentrated on discrete components.
Capacitors
Theory of conventional construction
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.
Theory of electrochemical construction
Besides the conventional static storage of electric energy in an electric field, two other storage principles to store electric energy in a capacitor exist. They are socalled electrochemical capacitors. In contrast to ceramic, film and electrolytic capacitors, supercapacitors, also known as electrical doublelayer capacitors (EDLC) or ultracapacitors do not have a conventional dielectric. The capacitance value of an electrochemical capacitor is determined by two highcapacity storage principles. These principles are:
 electrostatic storage within Helmholtz double layers achieved on the phase interface between the surface of the electrodes and the electrolyte (doublelayer capacitance) and the
 electrochemical storage achieved by a faradaic electron chargetransfer by specifically adsorpted ions with redox reactions (pseudocapacitance). Unlike batteries, in the faradaic redox reactions, the ions simply cling to the atomic structure of an electrode without making or braking chemical bonds, and no or negligibly small chemical modifications are involved in charge/discharge.
The ratio of the storage resulting from each principle can vary greatly, depending on electrode design and electrolyte composition. Pseudocapacitance can increase the capacitance value by as much as an order of magnitude over that of the doublelayer by itself.^{[2]}
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.
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:
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
Current
Reactance
Reactance is defined as the ratio of Voltage over Current
Impedance
Impedance is defined as the sum of Capacitor's Resistance and Reactance
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 =
Changing the value of R and C will change the value of Phase Angle, Angular Frequency, Frequency and Time
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.
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.
RC Circuit
Introduction
An RC circuit is short for 'ResistorCapacitor' 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:
 Switch is closed. Light does not light up.
 Light gradually becomes brighter and brighter...
 Light is at full luminosity.
 Switch is released. Light continues to shine.
 Light begins to fade...
 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.
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:
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
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:
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).
Capacitors  general remarks
Common capacitors and their names
Capacitors are divided into two mechanical groups: Fixed capacitors with fixed capacitance values and variable capacitors with variable (trimmer) or adjustable (tunable) capacitance values.
The most important group is the fixed capacitors. Many got their names from the dielectric. For a systematic classification these characteristics can't be used, because one of the oldest, the electrolytic capacitor, is named instead by its cathode construction. So the mostused names are simply historical.
The most common kinds of capacitors are:
 Ceramic capacitors have a ceramic dialectric.
 Film and paper capacitors are named for their dielectrics.
 Aluminum, tantalum and niobium electrolytic capacitors are named after the material used as the anode and the construction of the cathode
 Supercapacitor is the family name for:
 Doublelayer capacitors were named for the physical phenomenon of the Helmholtz doublelayer
 Pseudocapacitors were named for their ability to store electric energy electrochemically with reversible faradaic chargetransfer
 Hybrid capacitors combine doublelayer and pseudocapacitors to increase power density
 Seldomused Silver mica, glass, silicon, airgap and vacuum capacitors were named for their dielectric.
Capacitors in each family have similar physical design features, but vary, for example, in the form of the terminals.
In addition to the above shown capacitor types, which derived their name from historical development, there are many individual capacitors that have been named based on their application. They include:
 Power capacitors, motor capacitors, DClink capacitors, suppression capacitors, audio crossover capacitors, lighting ballast capacitors, snubber capacitors, coupling, decoupling or bypassing capacitors.
Often, more than one capacitor family is employed for these applications, e.g. interference suppression can use ceramic capacitors or film capacitors.
Specialized devices such as builtin capacitors with metal conductive areas in different layers of a multilayer printed circuit board and kludges such as twisting together two pieces of insulated wire also exist.
Dielectrics
The most common dielectrics are:
 Ceramics
 Plastic films
 Oxide layer on metal (Aluminum, Tantalum, Niobium)
 Natural materials like mica, glass, paper, air, vacuum
All of them store their electrical charge statically within an electric field between two (parallel) electrodes.
Beneath this conventional capacitors a family of electrochemical capacitors called Supercapacitors was developed. Supercapacitors don't have a conventional dielectric. They store their electrical charge statically in
 Helmholtz doublelayers (Doublelayer capacitors)
and additional electrochemical with faradaic charge transfer
 with a pseudocapacitance (Pseudocapacitors)
 or with both storage principles together (Hybrid capacitors).
The most important material parameters of the different dielectrics used and the appr. Helmholtzlayer thickness are given in the table below.
Capacitor style  Dielectric  Permittivity at 1 kHz 
Maximum/realized. dielectric strength V/µm 
Minimum thickness of the dielectric µm 

Ceramic capacitors, Class 1 
paraelectric  12–40  < 100(?)  1 
Ceramic capacitors, Class 2 
ferroelectric  200–14,000  < 25(?)  0.5 
Film capacitors  Polypropylene ( PP)  2.2  650/450  1.9 – 3.0 
Film capacitors  Polyethylen terephthalate, Polyester (PET) 
3.3  580/280  0.7–0.9 
Film capacitors  Polyphenylene sulfide (PPS)  3.0  470/220  1.2 
Film capacitors  Polyethylene naphthalate (PEN)  3.0  500/300  0.9–1.4 
Film capacitors  Polytetrafluoroethylene (PTFE)  2.0  450(?)/250  5.5 
Paper capacitors  Paper  3.5–5.5  60  5–10 
Aluminium electrolytic capacitors  Aluminium oxide Al_{2}O_{3} 
9,6^{[8]}  710  < 0.01 (6.3 V) < 0.8 (450 V) 
Tantalum electrolytic capacitors  Tantalum pentoxide Ta_{2}O_{5} 
26^{[8]}  625  < 0.01 (6.3 V) < 0.08 (40 V) 
Niobium electrolytic capacitors  Niobium pentoxide, Nb_{2}O_{5} 
42  455  < 0.01 (6.3 V) < 0.10 (40 V) 
Supercapacitors Doublelayer capacitors 
Helmholtz doublelayer      < 0.001 (2.7 V) 
Vacuum capacitors  Vacuum  1  40   
Air gap capacitors  Air  1  3.3   
Glass capacitors  Glass  5–10  450   
Mica capacitors  Mica  5–8  118  4–50 
The capacitor's plate area can be adapted to the wanted capacitance value. The permittivity and the dielectric thickness are the determining parameter for capacitors. Ease of processing is also crucial. Thin, mechanically flexible sheets can be wrapped or stacked easily, yielding large designs with high capacitance values. Razorthin metallized sintered ceramic layers covered with metallized electrodes however, offer the best conditions for the miniaturization of circuits with SMD styles.
A short view to the figures in the table above gives the explanation for some simple facts:
 Supercapacitors have the highest capacitance density because of its special charge storage principles
 Electrolytic capacitors have lesser capacitance density than supercapacitors but the highest capacitance density of conventional capacitors because its thin dielectric.
 Ceramic capacitors class 2 have much higher capacitance values in a given case than class 1 capacitors because of their much higher permittivity.
 Film capacitors with their different plastic film material do have a small spread in the dimensions for a given capacitance/voltage value of a film capacitor because the minimum dielectric film thickness differs between the different film materials.
Capacitance and voltage range
Capacitance ranges from picofarad to more than hundreds of farad. Voltage ratings can reach 100 kilovolts. In general, capacitance and voltage correlates with physical size and cost.
Miniaturization
As in other areas of electronics, volumetric efficiency measures the performance of electronic function per unit volume. For capacitors, the volumetric efficiency is measured with the "CV product", calculated by multiplying the capacitance (C) by the maximum voltage rating (V), divided by the volume. From 1970 to 2005, volumetric efficiencies have improved dramatically.

Wound metallized paper capacitor from the early 1930s in hardpaper case, capacitance value specified in "cm" in the cgs system; 5,000 cm corresponds to 28 nF
Overlapping range of applications
These individual capacitors can perform their application independent of their affiliation to an above shown capacitor type, so that an overlapping range of applications between the different capacitor types exists.
Capacitor  types and styles
Ceramic capacitors
A ceramic capacitor is a nonpolarized fixed capacitor made out of two or more alternating layers of ceramic and metal in which the ceramic material acts as the dielectric and the metal acts as the electrodes. The ceramic material is a mixture of finely ground granules of paraelectric or ferroelectric materials, modified by mixed oxides that are necessary to achieve the capacitor's desired characteristics. The electrical behavior of the ceramic material is divided into two stability classes:
 Class 1 ceramic capacitors with high stability and low losses compensating the influence of temperature in resonant circuit application. Common EIA/IEC code abbreviations are C0G/NP0, P2G/N150, R2G/N220, U2J/N750 etc.
 Class 2 ceramic capacitors with high volumetric efficiency for buffer, bypass and coupling applications Common EIA/IEC code abbreviations are: X7R/2XI, Z5U/E26, Y5V/2F4, X7S/2C1, etc.
The great plasticity of ceramic raw material works well for many special applications and enables an enormous diversity of styles, shapes and great dimensional spread of ceramic capacitors. The smallest discrete capacitor, for instance, is a "01005" chip capacitor with the dimension of only 0.4 mm × 0.2 mm.
The construction of ceramic multilayer capacitors with mostly alternating layers results in single capacitors connected in parallel. This configuration increases capacitance and decreases all losses and parasitic inductances. Ceramic capacitors are wellsuited for high frequencies and high current pulse loads.
Because the thickness of the ceramic dielectric layer can be easily controlled and produced by the desired application voltage, ceramic capacitors are available with rated voltages up to the 30 kV range.
Some ceramic capacitors of special shapes and styles are used as capacitors for special applications, including RFI/EMI suppression capacitors for connection to supply mains, also known as safety capacitors,^{[9]}^{[10]} X2Y® capacitors for bypassing and decoupling applications,^{[11]} feedthrough capacitors for noise suppression by lowpass filters^{[12]} and ceramic power capacitors for transmitters and HF applications.^{[13]}^{[14]}
Capacitor type  Dielectric  Features/applications  Disadvantages 

Ceramic Class 1 capacitors  paraelectric ceramic mixture of Titanium dioxide modified by additives  Predictable linear and low capacitance change with operating temperature. Excellent high frequency characteristics with low losses. For temperature compensation in resonant circuit application. Available in voltages up to 15,000 V  Low permittivity ceramic, capacitors with low volumetric efficiency, larger dimensions than Class 2 capacitors 
Ceramic Class 2 capacitors  ferroelectric ceramic mixture of barium titanate and suitable additives  High permittivity, high volumetric efficiency, smaller dimensions than Class 1 capacitors. For buffer, bypass and coupling applications. Available in voltages up to 50,000 V.  Lower stability and higher losses than Class 1. Capacitance changes with change in applied voltage, with frequency and with aging effects. Slightly microphonic 
Film capacitors
Film capacitors or plastic film capacitors are nonpolarized capacitors with an insulating plastic film as the dielectric. The dielectric films are drawn to a thin layer, provided with metallic electrodes and wound into a cylindrical winding. The electrodes of film capacitors may be metallized aluminum or zinc, applied on one or both sides of the plastic film, resulting in metallized film capacitors or a separate metallic foil overlying the film, called film/foil capacitors.
Metallized film capacitors offer selfhealing properties. Dielectric breakdowns or shorts between the electrodes do not destroy the component. The metallized construction makes it possible to produce wound capacitors with larger capacitance values (up to 100 µF and larger) in smaller cases than within film/foil construction.
Film/foil capacitors or metal foil capacitors use two plastic films as the dielectric. Each film is covered with a thin metal foil, mostly aluminium, to form the electrodes. The advantage of this construction is the ease of connecting the metal foil electrodes, along with an excellent current pulse strength.
A key advantage of every film capacitor's internal construction is direct contact to the electrodes on both ends of the winding. This contact keeps all current paths very short. The design behaves like a large number of individual capacitors connected in parallel, thus reducing the internal ohmic losses (ESR) and parasitic inductance (ESL). The inherent geometry of film capacitor structure results in low ohmic losses and a low parasitic inductance, which makes them suitable for applications with high surge currents (snubbers) and for AC power applications, or for applications at higher frequencies.
The plastic films used as the dielectric for film capacitors are Polypropylene (PP), Polyester (PET), Polyphenylene sulfide (PPS), Polyethylene naphthalate (PEN), and Polytetrafluoroethylene or Teflon (PTFE). Polypropylene film material with a market share of something about 50% and Polyester film with something about 40% are the most used film materials. The rest of something about 10% will be used by all other materials including PPS and paper with roughly 3%, each.^{[15]}^{[16]}
Film material, abbreviated codes  

Film characteristics  PET  PEN  PPS  PP  
Relative permittivity at 1 kHz  3.3  3.0  3.0  2.2  
Minimum film thickness (µm)  0.7–0.9  0.9–1.4  1.2  2.4–3.0  
Moisture absorption (%)  low  0.4  0.05  <0.1  
Dielectric strength (V/µm)  580  500  470  650  
Commercial realized voltage proof (V/µm) 
280  300  220  400  
DC voltage range (V)  50–1,000  16–250  16–100  40–2,000  
Capacitance range  100 pF–22 µF  100 pF–1 µF  100 pF–0.47 µF  100 pF–10 µF  
Application temperature range (°C)  −55 to +125 /+150  −55 to +150  −55 to +150  −55 to +105  
ΔC/C versus temperature range (%)  ±5  ±5  ±1.5  ±2.5  
Dissipation factor (•10^{−4})  
at 1 kHz  50–200  42–80  2–15  0.5–5  
at 10 kHz  110–150  54–150  2.5–25  2–8  
at 100 kHz  170–300  120–300  12–60  2–25  
at 1 MHz  200–350  –  18–70  4–40  
Time constant R_{Insul}•C (s)  at 25 °C  ≥10,000  ≥10,000  ≥10,000  ≥100,000 
at 85 °C  1,000  1,000  1,000  10,000  
Dielectric absorption (%)  0.2–0.5  1–1.2  0.05–0.1  0.01–0.1  
Specific capacitance (nF•V/mm^{3})  400  250  140  50 
Some film capacitors of special shapes and styles are used as capacitors for special applications, including RFI/EMI suppression capacitors for connection to the supply mains, also known as safety capacitors,^{[17]} Snubber capacitors for very high surge currents,^{[18]} Motor run capacitors, AC capacitors for motorrun applications^{[19]}
Capacitor type  Dielectric  Features/applications  Disadvantages 

Metallized film capacitors  PP, PET, PEN, PPS, (PTFE)  Metallized film capacitors are significantly smaller in size than film/foil versions and have selfhealing properties.  Thin metallized electrodes limit the maximum current carrying capability respectively the maximum possible pulse voltage. 
Film/foil film capacitors  PP, PET, PTFE  Film/foil film capacitors have the highest surge ratings/pulse voltage, respectively. Peak currents are higher than for metallized types.  No selfhealing properties: internal short may be disabling. Larger dimensions than metallized alternative. 
Polypropylene (PP) film capacitors  Polypropylene (Treofan®) 
Most popular film capacitor dielectric. Predictable linear and low capacitance change with operating temperature. Suitable for applications in Class1 frequencydetermining circuits and precision analog applications. Very narrow capacitances. Extremely low dissipation factor. Low moisture absorption, therefore suitable for "naked" designs with no coating. High insulation resistance. Usable in high power applications such as snubber or IGBT. Used also in AC power applications, such as in motors or power factor correction. Very low dielectric losses. High frequency and high power applications such as induction heating. Widely used for safety/EMI suppression, including connection to power supply mains.  Maximum operating temperature of 105 °C. Relatively low permittivity of 2.2. PP film capacitors tend to be larger than other film capacitors. More susceptible to damage from transient overvoltages or voltage reversals than oilimpregnated MKVcapacitors for pulsed power applications. 
Polyester (PET) film (Mylar) capacitors 
Polyethylene terephthalate, Polyester (Hostaphan®, Mylar®)  Smaller in size than functionally comparable polypropylene film capacitors. Low moisture absorption. Have almost completely replaced metallized paper and polystyrene film for most DC applications. Mainly used for general purpose applications or semicritical circuits with operating temperatures up to 125 °C. Operating voltages up to 60,000 V DC.  Usable at low (AC power) frequencies. Limited use in power electronics due to higher losses with increasing temperature and frequency. 
Polyethylene naphthalate (PEN) film capacitors 
Polyethylene naphthalate (Kaladex®)  Better stability at high temperatures than PET. More suitable for high temperature applications and for SMD packaging. Mainly used for noncritical filtering, coupling and decoupling, because temperature dependencies are not significant.  Lower relative permittivity and lower dielectric strength imply larger dimensions for a given capacitance and rated voltage than PET. 
Polyphenylene Sulfide (PPS) film capacitors 
Polyphenylene (Torelina®)  Small temperature dependence over the entire temperature range and a narrow frequency dependence in a wide frequency range. Dissipation factor is quite small and stable. Operating emperatures up to 270 °C. Suitable for SMD. Tolerate increased reflow soldering temperatures for leadfree soldering mandated by the RoHS 2002/95/European Union directive  Above 100 °C, the dissipation factor increases, increasing component temperature, but can operate without degradation. Cost is usually higher than PP. 
Polytetrafluoroethylene (PTFE) (Teflon film) capacitors 
Polytetrafluoroethylene (Teflon®)  Lowest loss solid dielectric. Operating temperatures up to 250 °C. Extremely high insulation resistance. Good stability. Used in missioncritical applications.  Large size (due to low dielectric constant). Higher cost than other film capacitors. 
Polycarbonate (PC) film capacitors 
Polycarbonate  Almost completely replaced by PP  Limited manufacturers 
Polystyrene (PS) film capacitors 
Polystyrene (Styroflex)  Almost completely replaced by PET  Limited manufacturers 
Polysulphone film capacitors  Polysulfone  Similar to polycarbonate. Withstand full voltage at comparatively higher temperatures.  Only development, no series found (2012) 
Polyamide film capacitors  Polyamide  Operating temperatures of up to 200 °C. High insulation resistance. Good stability. Low dissipation factor.  Only development, no series found (2012) 
Polyimide film (Kapton) capacitors 
Polyimide (Kapton)  Highest dielectric strength of any known plastic film dielectric.  Only development, no series found (2012) 
Film power capacitors
A related type is the power film capacitor. The materials and construction techniques used for large power film capacitors mostly are similar to those of ordinary film capacitors. However, capacitors with high to very high power ratings for applications in power systems and electrical installations are often classified separately, for historical reasons. The standardization of ordinary film capacitors is oriented on electrical and mechanical parameters. The standardization of power capacitors by contrast emphasizes the safety of personnel and equipment, as given by the local regulating authority.
As modern electronic equipment gained the capacity to handle power levels that were previously the exclusive domain of "electrical power" components, the distinction between the "electronic" and "electrical" power ratings blurred. Historically, the boundary between these two families was approximately at a reactive power of 200 voltamps.
Film power capacitors mostly use polypropylene film as the dielectric. Other types include metallized paper capacitors (MP capacitors) and mixed dielectric film capacitors with polypropylene dielectrics. MP capacitors serve for cost applications and as fieldfree carrier electrodes (soggy foil capacitors) for high AC or high current pulse loads. Windings can be filled with an insulating oil or with epoxy resin to reduce air bubbles, thereby preventing short circuits.
They find use as converters to change voltage, current or frequency, to store or deliver abruptly electric energy or to improve the power factor. The rated voltage range of these capacitors is from approximately120 V AC (capacitive lighting ballasts) to 100 kV.^{[20]}

Power film capacitor for AC Power factor correction (PFC), packaged in a cylindrical metal can
Capacitor type  Dielectric  Features/applications  Disadvantages 

Metallized paper power capacitors  Paper impregnated with insulating oil or epoxy resin  Selfhealing properties. Originally impregnated with wax, oil or epoxy. OilKraft paper version used in certain high voltage applications. Mostly replaced by PP.  Large size. Highly hygroscopic, absorbing moisture from the atmosphere despite plastic enclosures and impregnates. Moisture increases dielectric losses and decreases insulation resistance. 
Paper film/foil power capacitors  Kraft paper impregnated with oil  Paper covered with metal foils as electrodes. Low cost. Intermittent duty, high discharge applications.  Physically large and heavy. Significantly lower energy density than PP dielectric. Not selfhealing. Potential catastrophic failure due to high stored energy. 
PP dielectric, fieldfree paper power capacitors (MKV power capacitors) 
Doublesided (fieldfree) metallized paper as electrode carrier. PP as dielectic, impregnated with insulating oil, epoxy resin or insulating gas  Selfhealing. Very low losses. High insulation resistance. High inrush current strength. High thermal stability. Heavy duty applications such as commutating with high reactive power, high frequencies and a high peak current load and other AC applications.  Physically larger than PP power capacitors. 
Single or doublesided metallized PP power capacitors 
PP as dielectric, impregnated with insulating oil, epoxy resin or insulating gas  Highest capacitance per volume power capacitor. Selfhealing. Broad range of applications such as generalpurpose, AC capacitors, motor capacitors, smoothing or filtering, DC links, snubbing or clamping, damping AC, series resonant DC circuits, DC discharge, AC commutation, AC power factor correction.  critical for reliable high voltage operation and very high inrush current loads, limited heat resistance (105 °C) 
PP film/foil power capacitors  Impregnated PP or insulating gas, insulating oil, epoxy resin or insulating gas  Highest inrush current strength  Larger than the PP metallized versions. Not selfhealing. 
Electrolytic capacitors
Electrolytic capacitors have a metallic anode covered with an oxidized layer used as dielectric. The second electrode is a nonsolid (wet) or solid electrolyte. Electrolytic capacitors are polarized. Three families are available, categorized according to their dielectric.
 Aluminum electrolytic capacitors with aluminum oxide as dielectric
 Tantalum electrolytic capacitors with tantalum pentoxide as dielectric
 Niobium electrolytic capacitors with niobium pentoxide as dielectric.
The anode is highly roughened to increase the surface area. This and the relatively high permittivity of the oxide layer gives these capacitors very high capacitance per unit volume compared with film or ceramic capacitors.
The permittivity of tantalum pentoxide is approximately three times higher than aluminium dioxide, producing significantly smaller components. However, permittivity determines only the dimensions. Electrical parameters, especially conductivity, are established by the electrolyte's material and composition. Three general types of electrolytes are used:
 non solid (wet, liquid)—conductivity approximately 10 mS/cm and are the lowest cost
 solid manganese oxide—conductivity approximately 100 mS/cm offer high quality and stability
 solid conductive polymer (Polypyrrole)—conductivity approximately 10,000 mS/cm,^{[21]} offer ESR values as low as <10 mΩ
Internal losses of electrolytic capacitors, prevailing used for decoupling and buffering applications, are determined by the kind of electrolyte.
Anode material  Electrolyte  Capacitance range (µF) 
Max. rated voltage at 85 °C (V) 
Upper categorie temperature (°C) 
Specific ripple current (mA/mm^{3}) ^{1)} 

Aluminum (roughned foil) 
non solid, e.g. Ethylene glycol, DMF, DMA, GBL 
0.1–2,700,000  600  150  0.05–2.0 
solid, Manganese dioxide (MnO_{2} 
0.1–1,500  40  175  0.5–2.5  
solid conductive polymere (e.g. Polypyrrole) 
10–1,500  25  125  10–30  
Tantalum (roughned foil) 
non solid Sulfuric acid 
0.1–1,000  630  125  – 
Tantalum (sintered) 
non solid sulfuric acid 
0.1–15,000  150  200  – 
solid Manganese dioxide (MnO_{2} 
0.1–3,300  125  150  1.5–15  
solid conductive polymere (e.g. Polypyrrole) 
10–1,500  35  125  10–30  
Niobium (sintered) 
solid Manganese dioxide (MnO_{2} 
1–1,500  10  125  5–20 
solid conductive polymere (e.g. Polypyrrole) 
2.2–1,000  25  105  10–30  

The large capacitance per unit volume of electrolytic capacitors make them valuable in relatively highcurrent and lowfrequency electrical circuits, e.g. in power supply filters for decoupling unwanted AC components from DC power connections or as coupling capacitors in audio amplifiers, for passing or bypassing lowfrequency signals and storing large amounts of energy. The relatively high capacitance value of an electrolytic capacitor combined with the very low ESR of the polymer electrolyte of polymer capacitors, especially in SMD styles, makes them a competitor to MLC chip capacitors in personal computer power supplies.
Bipolar electrolytics (also called NonPolarized capacitors) contain two anodized aluminium foils, behaving like two capacitors connected in series opposition.
Electolytic capacitors for special applications include motor start capacitors,^{[22]} flashlight capacitors^{[23]} and audio frequency capacitors.^{[24]}
Capacitor type  Dielectric  Features/applications  Disadvantages 

Electrolytic capacitors with non solid (wet, liquid) electrolyte 
Aluminum dioxide Al_{2}O_{3} 
Very large capacitance to volume ratio. Capacitance values up to 2,700,000 µF/6.3 V. Voltage up to 550 V. Lowest cost per capacitance/voltage values. Used where low losses and high capacitance stability are not of major importance, especially for lower frequencies, such as bypass, coupling, smoothing and buffer applications in power supplies and DClinks.  Polarized. Significant leakage. Relatively high ESRTemplate:Dn and ESL values, limiting high ripple current and high frequency applications. Lifetime calculation required because drying out phenomenon. Vent or burst when overloaded, overheated or connected wrong polarized. Water based electrolyte may vent at endoflife, showing failures like "capacitor plague" 
Tantalum pentoxide Ta_{2}O_{5} 
Wet tantalum electrolytic capacitors (wet slug)^{[25]} Lowest leakage among electrolytics. Voltage up to 630 V (tantalum film) or 125 V (tantalum sinter body). Hermetically sealed. Stable and reliable. Military and space applications.  Polarized. Violent explosion when voltage, ripple current or slew rates are exceeded, or under reverse voltage. Expensive.  
[Electrolytic capacitors with solid [Manganese dioxide]] electrolyte 
Aluminum dioxide Al_{2}O_{3} Tantalum pentoxide Ta_{2}O_{5}, Niobium pentoxide Nb_{2}O_{5} 
Tantalum and niobium with smaller dimensions for a given capacitance/voltage vs aluminum. Stable electrical parameters. Good longterm high temperature performance. Lower ESR lower than nonsolid (wet) electrolytics.  Polarized. About 125 V. Low voltage and limited, transient, reverse or surge voltage tolerance. Possible combustion upon failure. ESR much higher than conductive polymer electrolytics. Manganese expected to be replaced by polymer. 
Electrolytic capacitors with solid Polymer electrolyte (Polymer capacitors) 
Aluminum dioxide Al_{2}O_{3}, Tantalum pentoxide Ta_{2}O_{5}, Niobium pentoxide Nb_{2}O_{5} 
Greatly reduced ESR compared with manganese or nonsolid (wet) elelectrolytics. Higher ripple current ratings. Extended operational life. Stable electrical parameters. Selfhealing.^{[26]} Used for smoothing and buffering in smaller power supplies especially in SMD.  Polarized. Highest leakage current among electrolytics. Higher prices than nonsolid or manganese dioxide. Voltage limited to about 100 V. Explodes when voltage, current, or slew rates are exceeded or under reverse voltage. 
Supercapacitors
Supercapacitors (SC),^{[27]} comprise a family of electrochemical capacitors. Supercapacitor, sometimes called ultracapacitor is a generic term for electric doublelayer capacitors (EDLC), pseudocapacitors and hybrid capacitors. They don't have a conventional solid dielectric. The capacitance value of an electrochemical capacitor is determined by two storage principles, both of which contribute to the total capacitance of the capacitor:^{[28]}^{[29]}^{[30]}
 Doublelayer capacitance – Storage is achieved by separation of charge in a Helmholtz double layer at the interface between the surface of a conductor and an electrolytic solution. The distance of separation of charge in a doublelayer is on the order of a few Angstroms (0.3–0.8 nm). This storage is electrostatic in origin.^{[2]}
 Pseudocapacitance – Storage is achieved by redox reactions, electrosorbtion or intercalation on the surface of the electrode or by specifically adsorpted ions that results in a reversible faradaic chargetransfer. The pseudocapacitance is faradaic in origin.^{[2]}
The ratio of the storage resulting from each principle can vary greatly, depending on electrode design and electrolyte composition. Pseudocapacitance can increase the capacitance value by as much as an order of magnitude over that of the doublelayer by itself.^{[27]}
Supercapacitors are divided into three families, based on the design of the electrodes:
 Doublelayer capacitors – with carbon electrodes or derivates with much higher static doublelayer capacitance than the faradaic pseudocapacitance
 Pseudocapacitors – with electrodes out of metal oxides or conducting polymers with a high amount of faradaic pseudocapacitance
 Hybrid capacitors – capacitors with special and asymmetric electrodes that exhibit both significant doublelayer capacitance and pseudocapacitance, such as lithiumion capacitors
Supercapacitors bridge the gap between conventional capacitors and rechargeable batteries. They have the highest available capacitance values per unit volume and the greatest energy density of all capacitors. They support up to 12,000 Farads/1.2 Volt,^{[31]} with capacitance values up to 10,000 times that of electrolytic capacitors.^{[27]} While existing supercapacitors have energy densities that are approximately 10% of a conventional battery, their power density is generally 10 to 100 times greater. Power density is defined as the product of energy density, multiplied by the speed at which the energy is delivered to the load. The greater power density results in much shorter charge/discharge cycles than a battery is capable, and a greater tolerance for numerous charge/discharge cycles. This makes them wellsuited for parallel connection with batteries, and may improve battery performance in terms of power density.
Within electrochemical capacitors, the electrolyte is the conductive connection between the two electrodes, distinguishing them from electrolytic capacitors, in which the electrolyte only forms the cathode, the second electrode.
Supercapacitors are polarized and must operate with correct polarity. Polarity is controlled by design with asymmetric electrodes, or, for symmetric electrodes, by a potential applied during the manufacturing process.
Supercapacitors support a broad spectrum of applications for power and energy requirements, including:
 Low supply current during longer times for memory backup in (SRAMs) in electronic equipment
 Power electronics that require very short, high current, as in the KERSsystem in Formula 1 cars
 Recovery of braking energy for vehicles such as buses and trains
Supercapacitors are rarely interchangeable, especially those with higher energy densities. IEC standard 623911 Fixed electric double layer capacitors for use in electronic equipment identifies four application classes:
 Class 1, Memory backup, discharge current in mA = 1 • C (F)
 Class 2, Energy storage, discharge current in mA = 0.4 • C (F) • V (V)
 Class 3, Power, discharge current in mA = 4 • C (F) • V (V)
 Class 4, Instantaneous power, discharge current in mA = 40 • C (F) • V (V)
Exceptional for electronic components like capacitors are the manifold different trade or series names used for supercapacitors like: APowerCap, BestCap, BoostCap, CAPXX, DLCAP, EneCapTen, EVerCAP, DynaCap, Faradcap, GreenCap, Goldcap, HYCAP, Kapton capacitor, Super capacitor, SuperCap, PAS Capacitor, PowerStor, PseudoCap, Ultracapacitor making it difficult for users to classify these capacitors.

Maxwell MC and BC ultracapacitor cells and modules.jpg
Supercapacitor/Ultracapacitor cells and modules for high current loads
Capacitor type  Dielectric  Features/applications  Disadvantages 

Supercapacitors Pseudocapacitors 
Helmholtz doublelayer plus faradaic pseudocapacitance  Energy density typically tens to hundreds of times greater than conventional electrolytics. More comparable to batteries than to other capacitors. Large capacitance/volume ratio. Relatively low ESR. Thousands of farads. RAM memory backup. Temporary power during battery replacement. Rapidly absorbs/delivers much larger currents than batteries. Hundreds of thousands of charge/discharge cycles. Hybrid vehicles. Recuperation  Polarized. Low operating voltage per cell. (Stacked cells provide higher operating voltage.) Relatively high cost. 
Hybrid capacitors Lithium ion capacitors (LIC) 
Helmholtz doublelayer plus faradaic pseudocapacitance. Anode doped with lithium ions.  Higher operating voltage. Higher energy density than common EDLCs, but smaller than lithium ion batteries (LIB). No thermal runaway reactions.  Polarized. Low operating voltage per cell. (Stacked cells provide higher operating voltage.) Relatively high cost. 
Miscellaneous capacitors
Beneath the above described capacitors covering more or less nearly the total market of discrete capacitors some new developments or very special capacitor types as well as older types can be found in electronics.
Integrated capacitors
 Integrated capacitors—in integrated circuits, nanoscale capacitors can be formed by appropriate patterns of metallization on an isolating substrate. They may be packaged in multiple capacitor arrays with no other semiconductive parts as discrete components.^{[32]}
 Glass capacitors—First Leyden jar capacitor was made of glass, Template:As of glass capacitors were in use as SMD version for applications requiring ultrareliable and ultrastable service.
Power capacitors
 Vacuum capacitors—used in high power RF transmitters
 SF_{6} gas filled capacitors—used as capacitance standard in measuring bridge circuits
Special capacitors
 Printed circuit boards—metal conductive areas in different layers of a multilayer printed circuit board can act as a highly stable capacitor. It is common industry practice to fill unused areas of one PCB layer with the ground conductor and another layer with the power conductor, forming a large distributed capacitor between the layers.
 Wire—2 pieces of insulated wire twisted together. Capacitance alues usually range from 3 pF to 15 pF. Used in homemade VHF circuits for oscillation feedback.
Obsolete capacitors
 Mica capacitors—the first capacitors with stable frequency behavior and low losses, used for military radio applications during World War II
 Airgap capacitors—used by the first sparkgap transmitters
Capacitor type  Dielectric  Features/applications  Disadvantages 

Air gap capacitors  Air  Low dielectric loss. Used for resonating HF circuits for high power HF welding.  Physically large. Relatively low capacitance. 
Vacuum capacitors  Vacuum  Extremely low losses. Used for high voltage, high power RF applications, such as transmitters and induction heating. Selfhealing if arcover current is limited.  Very high cost. Fragile. Large. Relatively low capacitance. 
SF_{6}gas filled capacitors  SF_{6} gas  High precision.^{[33]} Extremely low losses. Very high stability. Up to 1600 kV rated voltage. Used as capacitance standard in measuring bridge circuits.  Very high cost 
Metallized mica (Silver mica) capacitors  Mica  Very high stability. No aging. Low losses. Used for HF and low VHF RF circuits and as capacitance standard in measuring bridge circuits. Mostly replaced by Class 1 ceramic capacitors  Higher cost than class 1 ceramic capacitors 
Glass capacitors  Glass  Better stability and frequency than silver mica. Ultrareliable. Ultrastable. Resistant to nuclear radiation. Operating temperature: −75 °C to +200 °C and even short overexposure to +250 °C.^{[34]}  Higher cost than class 1 ceramic 
Integrated capacitors  oxidenitrideoxide (ONO)  Thin (down to 100 µm). Smaller footprint than most MLCC. Low ESL. Very high stability up to 200 °C. High reliability  Customized production 
Variable capacitors
Variable capacitors may have their capacitance changed by mechanical motion. Generally two versions of variable capacitors has to be to distinguished
 Tuning capacitor – variable capacitor for intentionally and repeatedly tuning an oscillator circuit in a radio or another tuned circuit
 Trimmer capacitor – small variable capacitor usually for onetime oscillator circuit internal adjustment
Variable capacitors include capacitors that use a mechanical construction to change the distance between the plates, or the amount of plate surface area which overlaps. They mostly use air as dielectric medium.
Semiconductive variable capacitance diodes are not capacitors in the sense of passive components but can change their capacitance as a function of the applied reverse bias voltage and are used like a variable capacitor. They have replaced much of the tuning and trimmer capacitors.
Capacitor type  Dielectric  Features/applications  Disadvantages 

Air gap tuning capacitors  Air  Circular or various logarithmic cuts of the rotor electrode for different capacitance curves. Split rotor or stator cut for symmetric adjustment. Ball bearing axis for noise reduced adjustment. For high professional devices.  Large dimensions. High cost. 
Vacuum tuning capacitors  Vacuum  Extremely low losses. Used for high voltage, high power RF applications, such as transmitters and induction heating. Selfhealing if arcover current is limited.  Very high cost. Fragile. Large dimensions. 
SF_{6} gas filled tuning capacitor  SF_{6}  Extremely low losses. Used for very high voltage high power RF applications.  Very high cost, fragile, large dimensions 
Air gap trimmer capacitors  Air  Mostly replaced by semiconductive variable capacitance diodes  High cost 
Ceramic trimmer capacitors  Class 1 ceramic  Linear and stable frequency behavior over wide temperature range  High cost 
Market
Discrete capacitors today are industrial products produced in very large quantities for use in electronic and in electrical equipment. Globally, the market for fixed capacitors was estimated at approximately US$18 billion in 2008 for 1,400 billion (1.4 × 10^{12}) pieces.^{[35]} This market is dominated by ceramic capacitors with estimate of approximately one trillion (1 × 10^{12}) items per year.^{[1]}
Detailed estimated figures in value for the main capacitor families are:
 Ceramic capacitors—US$8.3 billion (46%);
 Aluminum electrolytic capacitors—US$ 3.9 billion (22%);
 Film capacitors and Paper capacitors—US$ 2.6 billion, (15%);
 Tantalum electrolytic capacitors—US$ 2.2 billion (12%);
 Super capacitors (Doublelayer capacitors)—US$ 0.3 billion (2%); and
 Others like silver mica and vacuum capacitors—US$ 0.7 billion (3%).
All other capacitor types are negligible in terms of value and quantity compared with the above types.
Capacitor  Electrical characteristics
Seriesequivalent circuit
Discrete capacitors deviate from the ideal capacitor. An ideal capacitor only stores and releases electrical energy, with no dissipation. Capacitor components have losses and parasitic inductive parts. These imperfections in material and construction can have positive implications such as linear frequency and temperature behavior in class 1 ceramic capacitors. Conversely, negative implications include the nonlinear, voltagedependent capacitance in class 2 ceramic capacitors or the insufficient dielectric insulation of capacitors leading to leakage currents.
All properties can be defined and specified by a series equivalent circuit composed out of an idealized capacitance and additional electrical components which model all losses and inductive parameters of a capacitor. In this seriesequivalent circuit the electrical characteristics are defined by:
 C, the capacitance of the capacitor
 R_{insul}, the insulation resistance of the dielectric, not to be confused with the insulation of the housing
 R_{leak}, the resistance representing the leakage current of the capacitor
 R_{ESR}, the equivalent series resistance which summarizes all ohmic losses of the capacitor, usually abbreviated as "ESR"
 L_{ESL}, the equivalent series inductance which is the effective selfinductance of the capacitor, usually abbreviated as "ESL".
Using a series equivalent circuit instead of a parallel equivalent circuit is specified by IEC/EN 603841.
Standard values and tolerances
The "rated capacitance" C_{R} or "nominal capacitance" C_{N} is the value for which the capacitor has been designed. Actual capacitance depends on the measured frequency and ambient temperature. Standard measuring conditions are a lowvoltage AC measuring method at a temperature of 20 °C with frequencies of
 100 kHz, 1 MHz (preferred) or 10 MHz for nonelectrolytic capacitors with C_{R} ≤ 1 nF:
 1 kHz or 10 kHz for nonelectrolytic capacitors with 1 nF < C_{R} ≤ 10 μF
 100/120 Hz for electrolytic capacitors
 50/60 Hz or 100/120 Hz for nonelectrolytic capacitors with C_{R} > 10 μF
For supercapacitors a voltage drop method is applied for measuring the capacitance value. .
Capacitors are available in geometrically increasing preferred values (E series standards) specified in IEC/EN 60063. According to the number of values per decade, these were called the E3, E6, E12, E24 etc. series. The range of units used to specify capacitor values has expanded to include everything from pico (pF), nano (nF) and microfarad (µF) to farad (F). Millifarad and kilofarad are uncommon.
The percentage of allowed deviation from the rated value is called tolerance. The actual capacitance value should be within its tolerance limits, or it is out of specification. IEC/EN 60062 specifies a letter code for each tolerance.
E series  Tolerance  

C_{R} > 10 pF  Letter code  C_{R} < 10 pF  Letter code  
E 96  1%  F  0.1 pF  B 
E 48  2%  G  0.25 pF  C 
E 24  5%  J  0.5 pF  D 
E 12  10%  K  1 pF  F 
E 6  20%  M  2 pF  G 
E3  −20/+50%  S     
−20/+80%  Z     
The required tolerance is determined by the particular application. The narrow tolerances of E24 to E96 are used for highquality circuits such as precision oscillators and timers. General applications such as noncritical filtering or coupling circuits employ E12 or E6. Electrolytic capacitors, which are often used for filtering and bypassing capacitors mostly have a tolerance range of ±20% and need to conform to E6 (or E3) series values.
Temperature dependence
Capacitance typically varies with temperature. The different dielectrics express great differences in temperature sensitivity. The temperature coefficient is expressed in parts per million (ppm) per degree Celsius for class 1 ceramic capacitors or in % over the total temperature range for all others.
Type of capacitor, dielectric material 
Temperature coefficient ΔC/C 
Application temperature range 

Ceramic capacitor class 1 paraelectric NP0 
± 30 ppm/K (±0.5 %)  −55 to +125 °C 
Ceramic capacitor class 2 ferroelectric X7R 
±15 %  −55 to +125 °C 
Ceramic capacitor class 2, ferroelectric Y5V 
+22 % / −82 %  −30 to +85 °C 
Film capacitor Polypropylene ( PP) 
±2.5 %  −55 to +85/105 °C 
Film capacitor Polyethylen terephthalate, Polyester (PET) 
+5 %  −55 to +125/150 °C 
Film capacitor Polyphenylene sulfide (PPS) 
±1.5 %  −55 to +150 °C 
Film capacitor Polyethylene naphthalate (PEN) 
±5 %  −40 to +125/150 °C 
Film capacitor Polytetrafluoroethylene (PTFE) 
?  −40 to +130 °C 
Metallized paper capacitor (impregnated)  ±10 %  −25 to +85 °C 
Aluminum electrolytic capacitor Al_{2}O_{3} 
±20 %  −40 to +85/105/125 °C 
Tantalum electrolytic capacitor Ta_{2}O_{5} 
±20 %  −40 to +125 °C 
Frequency dependence
Most discrete capacitor types have more or less capacitance changes with increasing frequencies. The dielectric strength of class 2 ceramic and plastic film diminishes with rising frequency. Therefore their capacitance value decreases with increasing frequency. This phenomenon for ceramic class 2 and plastic film dielectrics is related to dielectric relaxation in which the time constant of the electrical dipoles is the reason for the frequency dependence of permittivity. The graphs below show typical frequency behavior of the capacitance for ceramic and film capacitors.
For electrolytic capacitors with nonsolid electrolyte, mechanical motion of the ions occurs. Their movability is limited so that at higher frequencies not all areas of the roughened anode structure are covered with chargecarrying ions. As higher the anode structure is roughned as more the capacitance value decreases with increasing frequency. Low voltage types with highlyroughened anodes display capacitance at 100 kHz approximately 10 to 20% of the value measured at 100 Hz.
Voltage dependence
Capacitance may also change with applied voltage. This effect is more prevalent in class 2 ceramic capacitors. The permittivity of ferroelectric class 2 material depends on the applied voltage. Higher applied voltage lowers permittivity. The change of capacitance can drop to 80% of the value measured with the standardized measuring voltage of 0.5 or 1.0 V. This behavior is a small source of nonlinearity in lowdistortion filters and other analog applications. In audio applications this can be the reason for harmonic distortion.
Film capacitors and electrolytic capacitors have no significant voltage dependence.
Rated and category voltage
The voltage at which the dielectric becomes conductive is called the breakdown voltage, and is given by the product of the dielectric strength and the separation between the electrodes. The dielectric strength depends on temperature, frequency, shape of the electrodes, etc. Because a breakdown in a capacitor normally is a short circuit and destroys the component, the operating voltage is lower than the breakdown voltage. The operating voltage is specified such that the voltage may be applied continuously throughout the life of the capacitor.
In IEC/EN 603841 the allowed operating voltage is called "rated voltage" or "nominal voltage". The rated voltage (UR) is the maximum DC voltage or peak pulse voltage that may be applied continuously at any temperature within the rated temperature range.
The voltage proof of nearly all capacitors decreases with increasing temperature. For some applications it is important to use a higher temperature range. Lowering the voltage applied at a higher temperature maintains safety margins. For some capacitor types therefore the IEC standard specify a second "temperature derated voltage" for a higher temperature range, the "category voltage". The category voltage (UC) is the maximum DC voltage or peak pulse voltage that may be applied continuously to a capacitor at any temperature within the category temperature range.
The relation between both voltages and temperatures is given in the picture right.
Impedance
In general, a capacitor is seen as a storage component for electric energy. But this is only one capacitor function. A capacitor can also act as an AC resistor. In many cases the capacitor is used as a decoupling capacitor to filter or bypass undesired biased AC frequencies to the ground. Other applications use capacitors for capacitive coupling of AC signals; the dielectric is used only for blocking DC. For such applications the AC resistance is as important as the capacitance value.
The frequency dependent AC resistance is called impedance and is the complex ratio of the voltage to the current in an AC circuit. Impedance extends the concept of resistance to AC circuits and possesses both magnitude and phase at a particular frequency. This is unlike resistance, which has only magnitude.
The magnitude represents the ratio of the voltage difference amplitude to the current amplitude, is the imaginary unit, while the argument gives the phase difference between voltage and current.
In capacitor data sheets, only the impedance magnitude Z is specified, and simply written as "Z" so that the formula for the impedance can be written in Cartesian form
where the real part of impedance is the resistance (for capacitors ) and the imaginary part is the reactance .
As shown in a capacitor's seriesequivalent circuit, the real component includes an ideal capacitor , an inductance and a resistor . The total reactance at the angular frequency therefore is given by the geometric (complex) addition of a capacitive reactance (Capacitance) and an inductive reactance (Inductance): .
To calculate the impedance the resistance has to be added geometrically and then is given by
 . The impedance is a measure of the capacitor's ability to pass alternating currents. In this sense the impedance can be used like Ohms law
to calculate either the peak or the effective value of the current or the voltage.
In the special case of resonance, in which the both reactive resistances
 and
have the same value (), then the impedance will only be determined by .
The impedance specified in the datasheets often show typical curves for the different capacitance values. With increasing frequency as the impedance decreases down to a minimum. The lower the impedance, the more easily alternating currents can be passed through the capacitor. At the apex, the point of resonance, where XC has the same value than XL, the capacitor has the lowest impedance value. Here only the ESR determines the impedance. With frequencies above the resonance the impedance increases again due to the ESL of the capacitor. The capacitor becomes to an inductance.
As shown in the graph, the higher capacitance values can fit the lower frequencies better while the lower capacitance values can fit better the higher frequencies.
Aluminum electrolytic capacitors have relatively good decoupling properties in the lower frequency range up to about 1 MHz due to their large capacitance values. This is the reason for using electrolytic capacitors in standard or switchedmode power supplies behind the rectifier for smoothing application.
Ceramic and film capacitors are already out of their smaller capacitance values suitable for higher frequencies up to several 100 MHz. They also have significantly lower parasitic inductance, making them suitable for higher frequency applications, due to their construction with endsurface contacting of the electrodes. To increase the range of frequencies, often an electrolytic capacitor is connected in parallel with a ceramic or film capacitor.^{[36]}
Many new developments are targeted at reducing parasitic inductance (ESL). This increases the resonance frequency of the capacitor and, for example, can follow the constantly increasing switching speed of digital circuits. Miniaturization, especially in the SMD multilayer ceramic chip capacitors (MLCC), increases the resonance frequency. Parasitic inductance is further lowered by placing the electrodes on the longitudinal side of the chip instead of the lateral side. The "facedown" construction associated with multianode technology in tantalum electrolytic capacitors further reduced ESL. Capacitor families such as the socalled MOS capacitor or silicon capacitors offer solutions when capacitors at frequencies up to the GHz range are needed.
Inductance (ESL) and selfresonant frequency
ESL in industrial capacitors is mainly caused by the leads and internal connections used to connect the capacitor plates to the outside world. Large capacitors tend to have higher ESL than small ones because the distances to the plate are longer and every mm counts as an inductance.
For any discrete capacitor, there is a frequency above DC at which it ceases to behave as a pure capacitor. This frequency, where is as high as , is called the selfresonant frequency. The selfresonant frequency is the lowest frequency at which the impedance passes through a minimum. For any AC application the selfresonant frequency is the highest frequency at which capacitors can be used as a capacitive component.
This is critically important for decoupling highspeed logic circuits from the power supply. The decoupling capacitor supplies transient current to the chip. Without decouplers, the IC demands current faster than the connection to the power supply can supply it, as parts of the circuit rapidly switch on and off. To counter this potential problem, circuits frequently use multiple bypass capacitors—small (100 nF or less) capacitors rated for high frequencies, a large electrolytic capacitor rated for lower frequencies and occasionally, an intermediate value capacitor.
Ohmic losses, ESR, dissipation factor, and quality factor
The summarized losses in discrete capacitors are ohmic AC losses. DC losses are specified as "leakage current" or "insulating resistance" and are negligible for an AC specification. AC losses are nonlinear, possibly depending on frequency, temperature, age or humidity. The losses result from two physical conditions:
 line losses including internal supply line resistances, the contact resistance of the electrode contact, line resistance of the electrodes, and in "wet" aluminum electrolytic capacitors and especially supercapacitors, the limited conductivity of liquid electrolytes and
 dielectric losses from dielectric polarization.
The largest share of these losses in larger capacitors is usually the frequency dependent ohmic dielectric losses. For smaller components, especially for wet electrolytic capacitors, conductivity of liquid electrolytes may exceed dielectric losses. To measure these losses, the measurement frequency must be set. Since commercially available components offer capacitance values cover 15 orders of magnitude, ranging from pF (10^{−12} F) to some 1000 F in supercapacitors, it is not possible to capture the entire range with only one frequency. IEC 603841 states that ohmic losses should be measured at the same frequency used to measure capacitance. These are:
 100 kHz, 1 MHz (preferred) or 10 MHz for nonelectrolytic capacitors with C_{R} ≤ 1 nF:
 1 kHz or 10 kHz for nonelectrolytic capacitors with 1 nF < C_{R} ≤ 10 μF
 100/120 Hz for electrolytic capacitors
 50/60 Hz or 100/120 Hz for nonelectrolytic capacitors with C_{R} > 10 μF
A capacitor's summarized resistive losses may be specified either as ESR, as a dissipation factor(DF, tan δ), or as quality factor (Q), depending on application requirements.
Capacitors with higher ripple current loads, such as electrolytic capacitors, are specified with equivalent series resistance ESR. ESR can be shown as an ohmic part in the above vector diagram. ESR values are specified in datasheets per individual type.
The losses of film capacitors and some class 2 ceramic capacitors are mostly specified with the dissipation factor tan δ. These capacitors have smaller losses than electrolytic capacitors and mostly are used at higher frequencies up to some hundred MHz. However the numeric value of the dissipation factor, measured at the same frequency, is independent on the capacitance value and can be specified for a capacitor series with a range of capacitance. The dissipation factor is determined as the tangent of the reactance () and the ESR, and can be shown as the angle δ between imaginary and the impedance axis.
If the inductance is small, the dissipation factor can be approximated as:
Capacitors with very low losses, such as ceramic Class 1 and Class 2 capacitors, specify resistive losses with a quality factor (Q). Ceramic Class 1 capacitors are especially suitable for LC resonant circuits with frequencies up to the GHz range, and precise high and low pass filters. For an electrically resonant system, Q represents the effect of electrical resistance and characterizes a resonator's bandwidth relative to its center or resonant frequency . Q is defined as the reciprocal value of the dissipation factor.
A high Q value is for resonant circuits a mark of the quality of the resonance.
Capacitor type  Capacitance (pF) 
ESR at 100 kHz (mΩ) 
ESR at 1 MHz (mΩ) 
tan δ at 1 MHz (10^{−4}) 
Quality factor 

Silicon capacitor^{[37]}  560  400  —  2,5  4000 
Mica capacitor^{[38]}  1000  650  65  4  2500 
Class 1 ceramic capacitor (NP0)^{[39]} 
1000  1600  160  10  1000 
Limiting current loads
A capacitor can act as an AC resistor, coupling AC voltage and AC current between two points. Every AC current flow through a capacitor generates heat inside the capacitor body. These dissipation power loss is caused by and is the squared value of the effective (RMS) current
The same power loss can be written with the dissipation factor as
The internal generated heat has to be distributed to the ambient. The temperature of the capacitor, which is established on the balance between heat produced and distributed, shall not exceed the capacitors maximum specified temperature. Hence, the ESR or dissipation factor is a mark for the maximum power (AC load, ripple current, pulse load, etc.) a capacitor is specified for.
AC currents may be a:
 ripple current—an effective (RMS) AC current, coming from an AC voltage superimposed of an DC bias, a
 pulse current—an AC peak current, coming from an voltage peak, or an
 AC current—an effective (RMS) sinusoidal current
Ripple and AC currents mainly warms the capacitor body. By this currents internal generated temperature influences the breakdown voltage of the dielectric. Higher temperature lower the voltage proof of all capacitors. In wet electrolytic capacitors higher temperatures force the evaporation of electrolytes, shortening the life time of the capacitors. In film capacitors higher temperatures may shrink the plastic film changing the capacitor's properties.
Pulse currents, especially in metallized film capacitors, heat the contact areas between end spray (schoopage) and metallized electrodes. This may reduce the contact to the electrodes, heightening the dissipation factor.
For safe operation, the maximal temperature generated by any AC current flow through the capacitor is a limiting factor, which in turn limits AC load, ripple current, pulse load, etc.
Ripple current
A "ripple current" is the RMS value of a superimposed AC current of any frequency and any waveform of the current curve for continuous operation at a specified temperature. It arises mainly in power supplies (including switchedmode power supplies) after rectifying an AC voltage and flows as charge and discharge current through the decoupling or smoothing capacitor. The "rated ripple current" shall not exceed a temperature rise of 3, 5 or 10 °C, depending on the capacitor type, at the specified maximum ambient temperature.
Ripple current generates heat within the capacitor body due to the ESR of the capacitor. The ESR, composed out of the dielectric losses caused by the changing field strength in the dielectric and the losses resulting out of the slightly resistive supply lines or the electrolyte depends on frequency and temperature. Higher frequencies heighten the ESR and higher temperatures lower the ESR slightly.
The types of capacitors used for power applications have a specified rated value for maximum ripple current. These are primarily aluminum electrolytic capacitors, and tantalum as well as some film capacitors and Class 2 ceramic capacitors.
Aluminium electrolytic capacitors, the most common type for power supplies, experience shorter life expectancy at higher ripple currents. Exceeding the limit tends to result in explosive failure.
Tantalum electrolytic capacitors with solid manganese dioxide electrolyte are also limited by ripple current. Exceeding their ripple limits tends to shorts and burning components.
For film and ceramic capacitors, normally specified with a loss factor tan δ, the ripple current limit is determined by temperature rise in the body of approximately 10 °C. Exceeding this limit may destroy the internal structure and cause shorts.
Pulse current
The rated pulse load for a certain capacitor is limited by the rated voltage, the pulse repetition frequency, temperature range and pulse rise time. The "pulse rise time" , represents the steepest voltage gradient of the pulse (rise or fall time) and is expressed in volts per μs (V/μs).
The rated pulse rise time is also indirectly the maximum capacity of an applicable peak current . The peak current is defined as:
where: is in A; in µF; in V/µs
The permissible pulse current capacity of a metallized film capacitor generally allows an internal temperature rise of 8 to 10 °K.
In the case of metallized film capacitors, pulse load depends on the properties of the dielectric material, the thickness of the metallization and the capacitor's construction, especially the construction of the contact areas between the end spray and metallized electrodes. High peak currents may lead to selective overheating of local contacts between end spray and metallized electrodes which may destroy some of the contacts, leading to increasing ESR.
For metallized film capacitors, socalled pulse tests simulate the pulse load that might occur during an application, according to a standard specification. IEC 60384 part 1, specifies that the test circuit is charged and discharged intermittently. The test voltage corresponds to the rated DC voltage and the test comprises 10000 pulses with a repetition frequency of 1 Hz. The pulse stress capacity is the pulse rise time. The rated pulse rise time is specified as 1/10 of the test pulse rise time.
The pulse load must be calculated for each application. A general rule for calculating the power handling of film capacitors is not available because of vendorrelated internal construction details. To prevent the capacitor from overheating the following operating parameters have to be considered:
 peak current per µF
 Pulse rise or fall time dv/dt in V/µs
 relative duration of charge and discharge periods (pulse shape)
 maximum pulse voltage (peak voltage)
 peak reverse voltage;
 Repetition frequency of the pulse
 Ambient temperature
 Heat dissipation (cooling)
Higher pulse rise times are permitted for pulse voltage lower than the rated voltage.
Examples for calculations of individual pulse loads are given by many manufactures, e.g. WIMA^{[40]} and Kemet.^{[41]}
AC current
An AC load only can be applied to a nonpolarized capacitor. Capacitors for AC applications are primarily film capacitors, metallized paper capacitors, ceramic capacitors and bipolar electrolytic capacitors.
The rated AC load for an AC capacitor is the maximum sinusoidal effective AC current (rms) which may be applied continuously to a capacitor within the specified temperature range. In the datasheets the AC load may be expressed as
 rated AC voltage at low frequencies,
 rated reactive power at intermediate frequencies,
 reduced AC voltage or rated AC current at high frequencies.
The rated AC voltage for film capacitors is generally calculated so that an internal temperature rise of 8 to 10 °K is the allowed limit for safe operation. Because dielectric losses increase with increasing frequency, the specified AC voltage has to be derated at higher frequencies. Datasheets for film capacitors specify special curves for derating AC voltages at higher frequencies.
If film capacitors or ceramic capacitors only have a DC specification, the peak value of the AC voltage applied has to be lower than the specified DC voltage.
AC loads can occur in AC Motor run capacitors, for voltage doubling, in snubbers, lighting ballast and for power factor correction PFC for phase shifting to improve transmission network stability and efficiency, which is one of the most important applications for large power capacitors. These mostly large PP film or metallized paper capacitors are limited by the rated reactive power VAr.
Bipolar electrolytic capacitors, to which an AC voltage may be applicable, are specified with a rated ripple current.
Insulation resistance and selfdischarge constant
The resistance of the dielectric is finite, leading to some level of DC "leakage current" that causes a charged capacitor to lose charge over time. For ceramic and film capacitors, this resistance is called "insulation resistance R_{ins}". This resistance is represented by the resistor R_{ins} in parallel with the capacitor in the seriesequivalent circuit of capacitors. Insulation resistance must not be confused with the outer isolation of the component with respect to the environment.
The time curve of selfdischarge over insulation resistance with decreasing capacitor voltage follows the formula
With stored DC voltage and selfdischarge constant
Thus, after voltage drops to 37% of the initial value.
The selfdischarge constant is an important parameter for the insulation of the dielectric between the electrodes of ceramic and film capacitors. For example, a capacitor can be used as the timedetermining component for time relays or for storing a voltage value as in a sample and hold circuits or operational amplifiers.
Class 1 ceramic capacitors have an insulation resistance of at least 10 GΩ, while class 2 capacitors have at least 4 GΩ or a selfdischarge constant of at least 100 s. Plastic film capacitors typically have an insulation resistance of 6 to 12 GΩ. This corresponds to capacitors in the uF range of a selfdiscarge constant of about 2000–4000 s.^{[42]}
Insulation resistance respectively the selfdischarge constant can be reduced if humidity penetrates into the winding. It is partially strongly temperature dependent and decreases with increasing temperature. Both decrease with increasing temperature.
In electrolytic capacitors, the insulation resistance is defined as leakage current.
Leakage current
For electrolytic capacitors the insulation resistance of the dielectric is termed "leakage current". This DC current is represented by the resistor R_{leak} in parallel with the capacitor in the seriesequivalent circuit of electrolytic capacitors. This resistance between the terminals of a capacitor is also finite. R_{leak} is lower for electrolytics than for ceramic or film capacitors.
The leakage current includes all weak imperfections of the dielectric caused by unwanted chemical processes and mechanical damage. It is also the DC current that can pass through the dielectric after applying a voltage. It depends on the interval without voltage applied (storage time), the thermic stress from soldering, on voltage applied, on temperature of the capacitor, and on measuring time.
The leakage current drops in the first minutes after applying DC voltage. In this period the dielectric oxide layer can selfrepair weaknesses by building up new layers. The time required depends generally on the electrolyte. Solid electrolytes drop faster than nonsolid electrolytes but remain at a slightly higher level.
The leakage current in nonsolid electrolytic capacitors as well as in manganese oxide solid tantalum capacitors decreases with voltageconnected time due to selfhealing effects. Although electrolytics leakage current is higher than current flow over insulation resistance in ceramic or film capacitors, the selfdischarge of modern non solid electrolytic capacitors takes several weeks.
A particular problem with electrolytic capacitors is storage time. Higher leakage current can be the result of longer storage times. These behaviors are limited to electrolytes with a high percentage of water. Organic solvents such as GBL do not have high leakage with longer storage times.
Leakage current is normally measured 2 or 5 minutes after applying rated voltage.
Microphonics
All ferroelectric materials exhibit piezoelectricity a piezoelectric effect. Because Class 2 ceramic capacitors use ferroelectric ceramics dielectric, these types of capacitors may have electrical effects called microphonics. Microphonics (microphony) describes how electronic components transform mechanical vibrations into an undesired electrical signal (noise).^{[43]} The dielectric may absorb mechanical forces from shock or vibration by changing thickness and changing the electrode separation, affecting the capacitance, which in turn induces an AC current. The resulting interference is especially problematic in audio applications, potentially causing feedback or unintended recording.
In the reverse microphonic effect, varying the electric field between the capacitor plates exerts a physical force, turning them into an audio speaker. High current impulse loads or high ripple currents can generate audible sound from the capacitor itself, draining energy and stressing the dielectric.^{[44]}
Dielectric absorption (soakage)
Dielectric absorption occurs when a capacitor that has remained charged for a long time discharges only incompletely when briefly discharged. Although an ideal capacitor would reach zero volts after discharge, real capacitors develop a small voltage from timedelayed dipole discharging, a phenomenon that is also called dielectric relaxation, "soakage" or "battery action".
Type of capacitor  Dielectric Absorption 

Air and vacuum capacitors  Not measurable 
Class1 ceramic capacitors, NP0  0.6% 
Class2 ceramic capacitors, X7R  2.5% 
Polypropylene film capacitors (PP)  0.05 to 0.1% 
Polyester film capacitors (PET)  0.2 to 0.5% 
Polyphenylene sulfide film capacitors (PPS)  0.05 to 0.1% 
Polyethylene naphthalate film capacitors (PEN)  1.0 to 1.2% 
Tantalum electrolytic capacitors with solid electrolyte  2 to 3%,^{[45]} 10%^{[46]} 
Aluminium electrolytic capacitor with non solid electrolyte  10 to 15% 
Doublelayer capacitor or super capacitors  data not available 
In many applications of capacitors dielectric absorption is not a problem but in some applications, such as longtimeconstant integrators, sampleandhold circuits, switchedcapacitor analogtodigital converters, and very lowdistortion filters, it is important that the capacitor does not recover a residual charge after full discharge, and capacitors with low absorption are specified.^{[47]} The voltage at the terminals generated by the dielectric absorption may in some cases possibly cause problems in the function of an electronic circuit or can be a safety risk to personnel. In order to prevent shocks most very large capacitors are shipped with shorting wires that need to be removed before they are used.^{[48]}
Energy density
The capacitance value depends on the dielectric material (ε), the surface of the electrodes (A) and the distance (d) separating the electrodes and is given by the formula of a plate capacitor:
The separation of the electrodes and the voltage proof of the dielectric material defines the breakdown voltage of the capacitor. The breakdown voltage is proportional to the thickness of the dielectric.
Theoretically, given two capacitors with the same mechanical dimensions and dielectric, but one of them have half the thickness of the dielectric. With the same dimensions this one could place twice the parallelplate area inside. This capacitor has theoretically 4 times the capacitance as the first capacitor but half of the voltage proof.
Since the energy density stored in a capacitor is given by:
thus a capacitor having a dielectric half as thick as another has 4 times higher capacitance but ½ voltage proof, yielding an equal maximum energy density.
Therefore, dielectric thickness does not affect energy density within a capacitor of fixed overall dimensions. Using a few thick layers of dielectric can support a high voltage, but low capacitance, while thin layers of dielectric produce a low breakdown voltage, but a higher capacitance.
This assumes that neither the electrode surfaces nor the permittivity of the dielectric change with the voltage proof. A simple comparison with two existing capacitor series can show whether reality matches theory. The comparison is easy, because the manufacturers use standardized case sizes or boxes for different capacitance/voltage values within a series.
Electrolytic capacitors NCC, KME series Ǿ D × H = 16.5 mm × 25 mm^{[49]} 
Metallized PP film capacitors KEMET; PHE 450 series W × H × L = 10.5 mm × 20.5 mm × 31.5 mm^{[50]} 

Capacitance/Voltage  Stored Energy  Capacitance/Voltage  Stored Energy 
4700 µF/10 V  235 mWs  1.2 µF/250 V  37.5 mWs 
2200 µF/25 V  688 mWs  0.68 µF/400 V  54.4 mWs 
220 µF/100 V  1100 mWs  0.39 µF/630 V  77.4 mWs 
22 µF/400 V  1760 mWs  0.27 µF/1000 V  135 mWs 
In reality modern capacitor series do not fit the theory. For electrolytic capacitors the spongelike rough surface of the anode foil gets smoother with higher voltages, decreasing the surface area of the anode. But because the energy increases squared with the voltage, and the surface of the anode decreases lesser than the voltage proof, the energy density increases clearly. For film capacitors the permittivity changes with dielectric thickness and other mechanical parameters so that the deviation from the theory has other reasons.^{[51]}
Comparing the capacitors from the table with a supercapacitor, the highest energy density capacitor family. For this, the capacitor 25 F/2.3 V in dimensions D × H = 16 mm × 26 mm from Maxwell HC Series, compared with the electrolytic capacitor of approximately equal size in the table. This supercapacitor has roughly 5000 times higher capacitance than the 4700/10 electrolytic capacitor but ¼ of the voltage and has about 66,000 mWs (0.018 Wh) stored electrical energy,^{[52]} approximately 100 times higher energy density (40 to 280 times) than the electrolytic capacitor.
Long time behavior, aging
Electrical parameters of capacitors may change over time during storage and application. The reasons for parameter changings are different, it may be a property of the dielectric, environmental influences, chemical processes or dryingout effects for nonsolid materials.
Aging
In ferroelectric Class 2 ceramic capacitors, capacitance decreases over time. This behavior is called "aging". This aging occurs in ferroelectric dielectrics, where domains of polarization in the dielectric contribute to the total polarization. Degradation of polarized domains in the dielectric decreases permittivity and therefore capacitance over time.^{[53]}^{[54]} The aging follows a logarithmic law. This defines the decrease of capacitance as constant percentage for a time decade after the soldering recovery time at a defined temperature, for example, in the period from 1 to 10 hours at 20 °C. As the law is logarithmic, the percentage loss of capacitance will twice between 1 h and 100 h and 3 times between 1 h and 1,000 h and so on. Aging is fastest near the beginning, and the absolute capacitance value stabilizes over time.
The rate of aging of Class 2 ceramic capacitors depends mainly on its materials. Generally, the higher the temperature dependence of the ceramic, the higher the aging percentage. The typical aging of X7R ceramic capacitors is about 2.5&nbs;% per decade.^{[55]} The aging rate of Z5U ceramic capacitors is significantly higher and can be up to 7% per decade.
The aging process of Class 2 ceramic capacitors may be reversed by heating the component above the Curie point.
Class 1 ceramic capacitors and film capacitors do not have ferroelectricrelated aging. Environmental influences such as higher temperature, high humidity and mechanical stress can, over a longer period, lead to a small irreversible change in the capacitance value sometimes called aging, too.
The change of capacitance for P 100 and N 470 Class 1 ceramic capacitors is lower than 1%, for capacitors with N 750 to N 1500 ceramics it is ≤ 2%. Film capacitors may lose capacitance due to selfhealing processes or gain it due to humidity influences. Typical changes over 2 years at 40 °C are, for example, ±3 % for PE film capacitors and ±1 % PP film capacitors.
Life time
Electrolytic capacitors with nonsolid electrolyte age as the electrolyte evaporates. This evaporation depends on temperature and the current load the capacitors experience. Electrolyte escape influences capacitance and ESR. Capacitance decreases and the ESR increases over time. In contrast to ceramic, film and electrolytic capacitors with solid electrolytes, "wet" electrolytic capacitors reach a specified "end of life" reaching a specified maximum change of capacitance or ESR. End of life, "load life" or "lifetime" can be estimated either by formula or diagrams^{[56]} or roughly by a socalled "10degreelaw". A typical specification for an electrolytic capacitor states a lifetime of 2,000 hours at 85 °C, doubling for every 10 degrees lower temperature, achieving lifespan of approximately 15 years at room temperature.
Supercapacitors also experience electrolyte evaporation over time. Estimation is similar to wet electrolytic capacitors. Additional to temperature the voltage and current load influence the life time. Lower voltage than rated voltage and lower current loads as well as lower temperature extend the life time.
Failure rate
Capacitors are reliable components with low failure rates, achieving life expectancies of decades under normal conditions. Most capacitors pass a test at the end of production similar to a "burn in", so that early failures are found during production, reducing the number of postshipment failures.
Reliability for capacitors is usually specified in numbers of Failures In Time (FIT) during the period of constant random failures. FIT is the number of failures that can be expected in one billion (10^{9}) componenthours of operation at fixed working conditions (e.g. 1000 devices for 1 million hours, or 1 million devices for 1000 hours each, at 40 °C and 0.5 U_{R}). For other conditions of applied voltage, current load, temperature, mechanical influences and humidity the FIT can recalculated with terms standardized for industrial^{[57]} or military^{[58]} contexts.
Additional information
Soldering
Capacitors may experience changes to electrical parameters due to environmental influences like soldering, mechanical stress factors (vibration, shock) and humidity. The greatest stress factor is soldering. The heat of the solder bath, especially for SMD capacitors, can cause ceramic capacitors to change contact resistance between terminals and electrodes; in film capacitors, the film may shrink, and in wet electrolytic capacitors the electrolyte may boil. A recovery period enables characteristics to stabilize after soldering; some types may require up to 24 hours. Some properties may change irreversibly by a few per cent from soldering.
Electrolytic behavior from storage or disuse
Electrolytic capacitors with nonsolid electrolyte are "aged" during manufacturing by applying rated voltage at high temperature for a sufficient time to repair all cracks and weaknesses that may have occurred during production. Some electrolytes with a high water content react quite aggressively or even violently with unprotected aluminum. This leads to a "storage" or "disuse" problem of electrolytic capacitors manufactured before the 1980s. Chemical processes weaken the oxide layer when these capacitors are not used for too long. New electrolytes with "inhibitors" or "passivators" were developed during the 1980s to solve this problem.^{[59]}^{[60]} As of 2012 the standard storage time for electronic components of two years at room temperature substantiates (cased) by the oxidation of the terminals will be specified for electrolytic capacitors with nonsolid electrolytes, too. Special series for 125 °C with organic solvents like GBL are specified up to 10 years storage time ensure without precondition the proper electrical behavior of the capacitors.^{[61]}
For antique radio equipment, "preconditioning" of older electrolytic capacitors may be recommended. This involves applying the operating voltage for some 10 minutes over a current limiting resistor to the terminals of the capacitor. Applying a voltage through a safety resistor repairs the oxide layers.
IEC/EN standards
The tests and requirements to be met by capacitors for use in electronic equipment for approval as standardized types are set out in the generic specification IEC/EN 603841 in the following sections.^{[62]}
Ceramic capacitors
 IEC/EN 603848—Fixed capacitors of ceramic dielectric, Class 1
 IEC/EN 603849—Fixed capacitors of ceramic dielectric, Class 2
 IEC/EN 6038421—Fixed surface mount multilayer capacitors of ceramic dielectric, Class 1
 IEC/EN 6038422—Fixed surface mount multilayer capacitors of ceramic dielectric, Class 2
Film capacitors
 IEC/EN 603842—Fixed metallized polyethyleneterephthalate film dielectric d.c. capacitors
 IEC/EN 6038411—Fixed polyethyleneterephthalate film dielectric metal foil d.c. capacitors
 IEC/EN 6038413—Fixed polypropylene film dielectric metal foil d.c. capacitors
 IEC/EN 6038416—Fixed metallized polypropylene film dielectric d.c. capacitors
 IEC/EN 6038417—Fixed metallized polypropylene film dielectric a.c. and pulse
 IEC/EN 6038419—Fixed metallized polyethyleneterephthalate film dielectric surface mount d.c. capacitors
 IEC/EN 6038420—Fixed metalized polyphenylene sulfide film dielectric surface mount d.c. capacitors
 IEC/EN 6038423—Fixed metallized polyethylene naphthalate film dielectric chip d.c. capacitors
Electrolytic capacitors
 IEC/EN 603843—Surface mount fixed tantalum electrolytic capacitors with manganese dioxide solid electrolyte
 IEC/EN 603844—Aluminium electrolytic capacitors with solid (MnO2) and nonsolid electrolyte
 IEC/EN 6038415—fixed tantalum capacitors with nonsolid and solid electrolyte
 IEC/EN 6038418—Fixed aluminium electrolytic surface mount capacitors with solid (MnO2) and nonsolid electrolyte
 IEC/EN 6038424—Surface mount fixed tantalum electrolytic capacitors with conductive polymer solid electrolyte
 IEC/EN 6038425—Surface mount fixed aluminium electrolytic capacitors with conductive polymer solid electrolyte
Supercapacitors
 IEC/EN 623911—Fixed electric doublelayer capacitors for use in electric and electronic equipment  Part 1: Generic specification
 IEC/EN 623912—Fixed electric doublelayer capacitors for use in electronic equipment  Part 2: Sectional specification  Electric doublelayer capacitors for power application
Capacitor symbols
Capacitor  Polarized capacitor Electrolytic capacitor 
Bipolar electrolytic capacitor 
Feed through capacitor 
Tuning variable capacitor 
Trimmer variable capacitor 
Markings
Imprinted
Capacitors, like most other electronic components and if enough space is available, have imprinted markings to indicate manufacturer, type, electrical and thermal characteristics, and date of manufacture. If they are large enough the capacitor is marked with:
 manufacturer's name or trademark;
 manufacturer's type designation;
 polarity of the terminations (for polarized capacitors)
 rated capacitance;
 tolerance on rated capacitance
 rated voltage and nature of supply (AC or DC)
 climatic category or rated temperature;
 year and month (or week) of manufacture;
 certification marks of safety standards (for safety EMI/RFI suppression capacitors)
Polarized capacitors have polarity markings, usually "" (minus) sign on the side of the negative electrode for electrolytic capacitors or a stripe or "+" (plus) sign, see #Polarity marking. Also, the negative lead for leaded "wet" ecaps is usually shorter.
Smaller capacitors use a shorthand notation. The most commonly used format is: XYZ J/K/M VOLTS V, where XYZ represents the capacitance (calculated as XY × 10^{Z} pF), the letters J, K or M indicate the tolerance (±5%, ±10% and ±20% respectively) and VOLTS V represents the working voltage.
Examples:
 105K 330V implies a capacitance of 10 × 10^{5} pF = 1 µF (K = ±10%) with a working voltage of 330 V.
 473M 100V implies a capacitance of 47 × 10^{3} pF = 47 nF (M = ±20%) with a working voltage of 100 V.
Capacitance, tolerance and date of manufacture can be indicated with a short code specified in IEC/EN 60062. Examples of shortmarking of the rated capacitance (microfarads): µ47 = 0,47 µF, 4µ7 = 4,7 µF, 47µ = 47 µF
The date of manufacture is often printed in accordance with international standards.
 Version 1: coding with year/week numeral code, "1208" is "2012, week number 8".
 Version 2: coding with year code/month code. The year codes are: "R" = 2003, "S"= 2004, "T" = 2005, "U" = 2006, "V" = 2007, "W" = 2008, "X" = 2009, "A" = 2010, "B" = 2011, "C" = 2012, "D" = 2013, etc. Month codes are: "1" to "9" = Jan. to Sept., "O" = October, "N" = November, "D" = December. "X5" is then "2009, May"
For very small capacitors like MLCC chips no marking is possible. Here only the traceability of the manufacturers can ensure the identification of a type.
Colour coding
Template:As of Capacitors do not use color coding.
Polarity marking
Aluminum ecaps with nonsolid electrolyte have a polarity marking at the cathode (minus) side. Aluminum, tantalum, and niobium ecaps with solid electrolyte have a polarity marking at the anode (plus) side. Supercapacitor are marked at the minus side.
Footnotes
 ↑ ^{a} ^{b} J. Ho, T. R. Jow, S. Boggs, Historical Introduction to Capacitor Technology, PDF [1]
 ↑ ^{a} ^{b} ^{c} Adam Marcus Namisnyk (23 June 2003). "A Survey of Electrochemical Supercapacitor Technology". http://services.eng.uts.edu.au/cempe/subjects_JGZ/eet/Capstone%20thesis_AN.pdf. Retrieved 20110624.
 ↑ WIMA, Characteristics of Metallized Film Capacitors in Comparison with Other Dielectrics [2]
 ↑ Film Capacitors, TDK Epcos, General technical information
 ↑ AVX, Dielectric Comparison Chart
 ↑ Holystone, Capacitor Dielectric Comparison, Tecnical Note 3
 ↑ Power Film Capacitors for Industrial Applications, P. Bettacchi, D. Montanari, D. Zanarini, D. Orioli, G. Rondelli, A. Sanua, KEMET Electronics [3]
 ↑ ^{a} ^{b} Template:Literatur
 ↑ General technical information of (RFI/EMI)Noise suppression capacitors on AC mains [4]
 ↑ Vishay, Capacitors  RFI Class X/Y
 ↑ X2Y® Technology
 ↑ Murata, Threeterminal Capacitor Structure, No.TE04EA1.pdf 98.3.20
 ↑ Vishay, Ceramic RFPower Capacitors
 ↑ Vishay. "Capacitors  RF Power". Vishay. http://www.vishay.com/capacitors/ceramicrfpower/. Retrieved 20130309.
 ↑ Passive component magazine, Nov./Dec. 2005, F. Jacobs, Polypropylene Capacitor Film Resin, p. 29 ff [5]
 ↑ Paumanok Publications, PCInewsletterOct2007cmp Paumanok Publications, Inc.
 ↑ WIMA, RFI Capacitors
 ↑ WIMA Snubber Capacitors
 ↑ Amrad Engeneering Inc., Motor run capacitors
 ↑ Epcos, Capacitors for power electronics, General technical information
 ↑ Sanyo, Capacitor lecture POSCAP (Ta) (Basic), Polymerized electrolyte
 ↑ CDE, Motor Start Capacitors
 ↑ Rubycon, Aluminum Electrolytic Capacitors for Strobe Flash
 ↑ Fischer & Tausche, Electrolytic capacitor for audio frequency
 ↑ Vishay, Wet Electrolyte Tantalum Capacitors, Introduction
 ↑ Selfhealing Characteristics of Solid Electrolytic Capacitor with Polypyrrole Electrolyte, Yamamoto Hideo [6]
 ↑ ^{a} ^{b} ^{c} B. E. Conway (1999). Electrochemical Supercapacitors: Scientific Fundamentals and Technological Applications. Berlin: Springer. ISBN 0306457369. http://books.google.de/books?id=8yvzlr9TqI0C&pg=PA1&redir_esc=y. Retrieved 2013, Mai 02. see also Brian E. Conway in Electrochemistry Encyclopedia: Electrochemical Capacitors — Their Nature, Function and Applications
 ↑ Template:Cite techreport
 ↑ Elzbieta Frackowiak, Francois Beguin, PERGAMON, Carbon 39 (2001) 937–950, Carbon materials for the electrochemical storage of energy in Capacitors
 ↑ Yu.M. Volfkovich, A.A. Mikhailin, D.A. Bograchev, V.E. Sosenkin and V.S. Bagotsky, Studies of Supercapacitor Carbon Electrodes with High Pseudocapacitance, A. N. Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences, Moscow, Russia, Dr. Ujjal Kumar Sur (Ed.), ISBN 9789533078304
 ↑ Elton
 ↑ IPDiA, 3D Silicon Capacitors
 ↑ Tettex instruments, SF_{6} Gas insulated Standard Capacitors
 ↑ AVX, Performance Characteristics of Multilayer Glass Capacitors
 ↑ highbeam business, Electronic Capacitors SIC 3675, Industry report
 ↑ Murata: Basics of capacitors, lesson 2 Includes graph showing impedance as a function of frequency for different capacitor types; electrolytics are the only ones with a large component due to ESR
 ↑ Siliziumkondensator, Vishay, HPC0603A
 ↑ Simic Electronics, Chip Mica Capacitors
 ↑ AVX, NP0, 1000 pF 100 V, 0805, Q >= 1000 (1 MHz), [7]
 ↑ WIMA, Selection of Capacitors for Pulse Applications
 ↑ Kemet, General information DC Film Capacitors
 ↑ Wima,Insulation Resistance
 ↑ [8], Capacitors for reduced sound emissions.
 ↑ Kemet, Are your military ceramic capacitors subject to the piezoelectric effect? [9]
 ↑ Kemet, Polymer Tantalum Chip Capacitors
 ↑ AVX, ANALYSIS OF SOLID TANTALUM CAPACITOR LEAKAGE CURRENT
 ↑ "Understand Capacitor Soakage to Optimize Analog Systems" by Bob Pease 1982 [10]
 ↑ * "Modeling Dielectric Absorption in Capacitors", by Ken Kundert
 ↑ NCC, KME series
 ↑ KEMET, series PHE450
 ↑ Metallized Polypropylene Film Energy Storage Capacitors For Low Pulse Duty, Ralph M. Kerrigan, NWL Capacitor Division [11]
 ↑ Maxwell HC Series / docs/datasheet_hc_series_1013793.pdf
 ↑ Template:Literatur
 ↑ Takaaki Tsurumi & Motohiro Shono & Hirofumi Kakemoto & Satoshi Wada & Kenji Saito & Hirokazu Chazono, Mechanism of capacitance aging under DCbias field in X7RMLCCs Published online: 23 March 2007, # Springer Science + Business Media, LLC 2007 [12]
 ↑ Christopher England, Johanson dielectrics, Ceramic Capacitor Aging Made Simple [13]
 ↑ Electrolytic Capacitor Lifetime Estimation, Dr. Arne Albertsen, Jianghai Europe, [14]
 ↑ IEC/EN 61709, Electric components. Reliability. Reference conditions for failure rates and stress models for conversion
 ↑ MILHDBK217F Reliability Prediction of Electronic Equipment
 ↑ J. L. Stevens, T. R. Marshall, A. C. Geiculescu M., C. R. Feger, T. F. Strange, Carts USA 2006, The Effects of Electrolyte Composition on the Deformation Characteristics of Wet Aluminum ICD Capacitors, [15]
 ↑ Alfonso Berduque, Zongli Dou, Rong Xu, BHC Components Ltd (KEMET), Electrochemical Studies for Aluminium Electrolytic Capacitor Applications: Corrosion Analysis of Aluminium in Ethylene GlycolBased Electrolytes pdf
 ↑ Vishay BCcomponents, Introduction Aluminum Capacitors, paragraph "Storage", Revision: 10May12, Document Number: 28356, pdf
 ↑ IEC/EN/DIN Standards, BeuthVerlag
Inductors
Electronics  Foreword  Basic Electronics  Complex Electronics  Electricity  Machines  History of Electronics  Appendix  edit
Inductor
An inductor is a passive electronic component dependent on frequency used to store electric energy in the form of a magnetic field. An inductor has the symbol
Inductance
Inductance is the characteristic of the Inductor to generates a magnetic field for a given current. Inductance has a letter symbol L and measured in units of Henry (H).
This section list formulas for inductances in specific situations. Beware that some of the equations are in Imperial units.
The permeability of free space, μ_{0}, is constant and is defined to be exactly equal to 4π×10^{7} H m^{1}.
Basic inductance formula for a cylindrical coil
 L = inductance / H
 μ_{r} = relative permeability of core material
 N = number of turns
 A = area of crosssection of the coil / m^{2}
 l = length of coil / m
The selfinductance of a straight, round wire in free space

 L_{self} = self inductance / H(?)

 b = wire length /m
 a = wire radius /m
 = relative permeability of wire
If you make the assumption that b >> a and that the wire is nonmagnetic (), then this equation can be approximated to
 (for low frequencies)
 (for high frequencies due to the skin effect)
 L = inductance / H
 b = wire length / m
 a = wire radius / m
The inductance of a straight wire is usually so small that it is neglected in most practical problems. If the problem deals with very high frequencies (f > 20 GHz), the calculation may become necessary. For the rest of this book, we will assume that this selfinductance is negligible.
Inductance of a short air core cylindrical coil in terms of geometric parameters:
 L = inductance in μH
 r = outer radius of coil in inches
 l = length of coil in inches
 N = number of turns
 L = inductance in μH
Multilayer air core coil
 L = inductance in μH
 r = mean radius of coil in inches
 l = physical length of coil winding in inches
 N = number of turns
 d = depth of coil in inches (i.e., outer radius minus inner radius)
 L = inductance in μH
Flat spiral air core coil

 L = inductance / H
 r = mean radius of coil / m
 N = number of turns
 d = depth of coil / m (i.e. outer radius minus inner radius)
 L = inductance / H
Hence a spiral coil with 8 turns at a mean radius of 25 mm and a depth of 10 mm would have an inductance of 5.13µH.
Winding around a toroidal core (circular crosssection)

 L = inductance / H
 μ_{r} = relative permeability of core material
 N = number of turns
 r = radius of coil winding / m
 D = overall diameter of toroid / m
 L = inductance / H
Quality of good inductor
There are several important properties for an inductor that may need to be considered when choosing one for use in an electronic circuit. The following are the basic properties of a coil inductor. Other factors may be important for other kinds of inductor, but these are outside the scope of this article.
 Current carrying capacity is determined by wire thickness and resistivity.
 The quality factor, or Qfactor, describes the energy loss in an inductor due to imperfection in the manufacturing.
 The inductance of the coil is probably most important, as it is what makes the inductor useful. The inductance is the response of the inductor to a changing current.
The inductance is determined by several factors.
 Coil shape: short and squat is best
 Core material
 The number of turns in the coil. These must be in the same direction, or they will cancel out, and you will have a resistor.
 Coil diameter. The larger the diameter (core area) the larger the induction.
Coil's Characteristics
For a Coil that has the following dimension
 Area enclosed by each turn of the coil is A
 Length of the coil is 'l'
 Number of turns in the coil is N
 Permeability of the core is μ. μ is given by the permeability of free space, μ_{0} multiplied by a factor, the relative permeability, μ_{r}
 The current in the coil is 'i'
The magnetic flux density, B, inside the coil is given by:
We know that the flux linkage in the coil, λ, is given by;
Thus,
The flux linkage in an inductor is therefore proportional to the current, assuming that A, N, l and μ all stay constant. The constant of proportionality is given the name inductance (measured in Henries) and the symbol L:
Taking the derivative with respect to time, we get:
Since L is timeinvariant in nearly all cases, we can write:
Now, Faraday's Law of Induction states that:
We call the electromotive force (emf) of the coil, and this is opposite to the voltage v across the inductor, giving:
This means that the voltage across an inductor is equal to the rate of change of the current in the inductor multiplied by a factor, the inductance. note that for a constant current, the voltage is zero, and for an instantaneous change in current, the voltage is infinite (or rather, undefined). This applies only to ideal inductors which do not exist in the real world.
This equation implies that
 The voltage across an inductor is proportional to the derivative of the current through the inductor.
 In inductors, voltage leads current.
 Inductors have a high resistance to high frequencies, and a low resistance to low frequencies. This property allows their use in filtering signals.
An inductor works by opposing current change. Whenever an electron is accelerated, some of the energy that goes into "pushing" that electron goes into the electron's kinetic energy, but much of that energy is stored in the magnetic field. Later when that or some other electron is decelerated (or accelerated the opposite direction), energy is pulled back out of the magnetic field.
Inductor and Direct Current Voltage (DC)
When connect a Coil of several turns to an Electricity source in a closed loop . The current in the circuit set up a Magnetic Field that has the same properties like a Magnetic Field of a Magnet
 B = L I
When the current is turned off , the Magnetic Field does not exist .
 B = 0
Conducting Coil is called ElectroMagnet
Inductor and Alternating Current Voltage (AC)
Inductor's Voltage
Inductor's Current
Reactance
Impedance
Angle Difference Between Voltage and Current
For Lossless Inductor
 The angle difference between Voltage and Current is 90
For Lossy Inductor
Change the value of L and R_{L} will change the value of Angle of Difference , Angular Frequency, requenct and Time
Time Constant
Quality factor
Quality factor denoted as Q is defined as the ability to store energy to the sum total of all energy losses within the component
Inductor's Connection
Series Connection
Parallel Connection
See Also
Other Components
Electronics  Foreword  Basic Electronics  Complex Electronics  Electricity  Machines  History of Electronics  Appendix  edit
Ideal voltage sources
 An ideal voltage source is a fundamental electronics component that creates a constant voltage between two points regardless of whatever else is connected to it. Since it is ideal, some circuit configurations are not allowed, such as short circuits, which would create infinite current. (I = V / 0)
 A water analogy would be a pump with pressure sensors on both sides. The difference in pressure between the in port and out port is constantly measured, regardless of the absolute pressure of each side, and the pump speed is adjusted so that the pressure difference stays constant.
 Real voltage sources, such as batteries, power supplies, piezoelectric disks, generators, steam turbines, wall outlets, etc. have an internal source impedance (in series with the ideal voltage source), which is very important to understand.
Ideal current sources
 An ideal current source is an electronics component that creates a constant current through a section of circuit, regardless of whatever else is connected to it. Since it is ideal, some circuit configurations are not allowed, such as open circuits, which would create an infinite voltage.
 A water analogy would be a pump with a flow meter. It measures the amount of water flowing by per unit time and changes the speed of the pump so that the current flow is constant.
 Real current sources, such as batteries, power supplies, piezoelectric disks, generators, etc. have an internal source impedance (in parallel with the source), which is very important to understand.
 Real sources generally behave more like voltage sources than current sources, because the internal impedance in series is very low. A current source can be created from a voltage source with a circuit such as a current mirror.
Dependent Sources
 A dependent source is either a voltage or a current source which is dependent upon another value within the circuit, usually another voltage or current. Typically, these are used in circuit modeling and analysis.
There are four main types of such sources.
Voltagecontrolled voltage source (VCVS)
 This is a voltage source whose value is controlled by another voltage elsewhere in the circuit. Its output will typically be given as , where A is a gain term and V_{c}is a control voltage.
 An example of a VCVS may be an idealized amplifier, where A is the gain of the amplifier.
Currentcontrolled voltage source (CCVS)
 This is a voltage source whose value is controlled by a current elsewhere in the circuit. Its output is typically given as , where A is a gain term and I_{c} is a control current.
Voltagecontrolled current source (VCCS)
 This is a current source whose value is controlled by a voltage elsewhere in the circuit. Its output is typically given as , where A is a gain term and V_{c} is a control voltage.
Currentcontrolled current source (CCCS)
 This is a current source whose value is controlled by a current elsewhere in the circuit. Its output is typically given as , where A is a gain term and I_{c} is a control current.
 An example of a CCCS is an idealized bipolar junction transistor, which may be thought of as a small current controlling a larger one. Specifically the base current, I_{b} is the control and the collector current I_{c} is the output.
Switch
 A switch is a mechanical device that connects or disconnects two parts of a circuit.
 A switch is a short circuit when it is on.
And it is a open circuit when it is off.
 When you turn a switch on it completes a circuit that allows current to flow. When you turn the switch off it creates an air gap (depending on the type of switch), and since air is an insulator no current flows.
 A switch is a device for making or breaking an electric circuit.
 Usually the switch has two pieces of metal called contacts that touch to make a circuit, and separate to break the circuit. The contact material is chosen for its resistance to corrosion, because most metals form insulating oxides that would prevent the switch from working. Sometimes the contacts are plated with noble metals. They may be designed to wipe against each other to clean off any contamination. Nonmetallic conductors, such as conductive plastic, are sometimes used. The moving part that applies the operating force to the contacts is called the actuator, and may be a rocker, a toggle or dolly, a pushbutton or any type of mechanical linkage.
Contact Arrangements
 Switches can be classified according to the arrangement of their contacts. Some contacts are normally open until closed by operation of the switch, while normally closed contacts are opened by the switch action. A switch with both types of contact is called a changeover switch.
 The terms pole and throw are used to describe switch contacts. A pole is a set of contacts that belong to a single circuit. A throw is one of two or more positions that the switch can adopt. These terms give rise to the following abbreviations.
 S (single), D (double).
 T (throw), CO (changeover).
 CO = DT.
(singledouble) pole ((singledouble) throwchangeover)
 SPST = single pole single throw, a simple onoff switch.
 SPDT = single pole double throw, a simple changeover or onoffon switch.
 SPCO = single pole changeover, equivalent to SPDT.
 DPST = double pole single throw, equivalent to two SPST switches controlled by a single mechanism.
 DPDT = double pole double throw, equivalent to two SPDT switches controlled by a single mechanism.
 DPCO = double pole changeover, equivalent to DPDT.
 Switches with larger numbers of poles or throws can be described by replacing the "S" or "D" with a number.
Biased Switches
 A biased switch is one containing a spring that returns the actuator to a certain position. The "onoff" notation can be modified by placing parentheses around all positions other than the resting position. For example, an (on)off(on) switch can be switched on by moving the actuator in either direction away from the centre, but returns to the central off position when the actuator is released.
 The momentary pushbutton switch is a type of biased switch. This device makes contact when the button is pressed and breaks when the button is released.
Special Types
 Switches can be designed to respond to any type of mechanical stimulus: for example, vibration (the trembler switch), tilt, air pressure, fluid level (the float switch), the turning of a key (key switch), linear or rotary movement (the limit switch or microswitch).
 The mercury tilt switch consists of a blob of mercury inside a glass bulb. The two contacts pass through the glass, and are shorted together when the bulb is tilted to make the mercury roll on to them. The advantage of this type of switch is that the liquid metal flows around particles of dirt and debris that might otherwise prevent the contacts of a conventional switch from closing.
See also
 Audio for robotics
 Diodes
 Identification
 Integrated circuits
 Loudspeakers
 * Loudspeakers, Enclosures
 Measuring instruments
 Mechanical components
 Motors
 Power sources
 Software development
 Resistors
 * Resistors, Light dependent
 * Resistors, Physics
 Software components
 Thermistors
 Vacuum tubes
DC Voltage and Current Laws
Electronics  Foreword  Basic Electronics  Complex Electronics  Electricity  Machines  History of Electronics  Appendix  edit
Ohm's Law
Ohm's law describes the relationship between voltage, current, and resistance.Voltage and current are proportional to the potential difference and inversely proportional to the resistance of the circuit
 Voltage (V) is measured in volts (V); Current (I) in amperes (A); and resistance (R) in ohms (Ω).
In this example, the current going through any point in the circuit, I, will be equal to the voltage V divided by the resistance R.
In this example, the voltage across the resistor, V, will be equal to the supplied current, I, times the resistance R.
If two of the values (V, I, or R) are known, the other can be calculated using this formula.
Any more complicated circuit has an equivalent resistance that will allow us to calculate the current draw from the voltage source. Equivalent resistance is worked out using the fact that all resistors are either in parallel or series. Similarly, if the circuit only has a current source, the equivalent resistance can be used to calculate the voltage dropped across the current source.
Kirchoff's Voltage Law
Kirchoff's Voltage Law (KVL):
 The sum of voltage drops around any loop in the circuit that starts and ends at the same place must be zero.
Voltage as a Physical Quantity
 Voltage is the potential difference between two charged objects.
 Potentials can be added or subtracted in series to make larger or smaller potentials as is commonly done in batteries.
 Positive charge flow from areas of high potential to lower potential.
 All the components of a circuit have resistance that acts as a potential drop.
Kirchoff's Current Law
Kirchoff's Current Law (KCL):
 The sum of all current entering a node must equal the sum of all currents leaving the node.
KCL Example
I_{1} + I_{2} + I_{3} = 0 ↔ I_{1} = I_{2} + I_{3}
I_{1}  I_{2}  I_{3}  I_{4} = 0 ↔ I_{2} + I_{3} + I_{4} = I_{1}
Here is more about Kirchhoff's laws, which can be integrated here
Consequences of KVL and KCL
Voltage Dividers
If two circuit elements are in series, there is a voltage drop across each element, but the current through both must be the same. The voltage at any point in the chain divides according to the resistances. A simple circuit with two (or more) resistors in series with a source is called a voltage divider.
Figure A: Voltage Divider circuit.
Consider the circuit in Figure A. According to KVL the voltage is dropped across resistors and . If a current i flows through the two series resistors then by Ohm's Law.
 .
So
Therefore
Similary if is the voltage across then
In general for n series resistors the voltage dropped across one of them say is
Where
Voltage Dividers as References
Clearly voltage dividers can be used as references. If you have a 9 volt battery and you want 4.5 volts, then connect two equal valued resistors in series and take the reference across the second and ground. There are clearly other concerns though, the first concern is current draw and the effect of the source impedance. Clearly connecting two 100 ohm resistors is a bad idea if the source impedance is, say, 50 ohms. Then the current draw would be 0.036 mA which is quite large if the battery is rated, say, 200 milliampere hours. The loading is more annoying with that source impedance too, the reference voltage with that source impedance is . So clearly, increasing the order of the resistor to at least 1 k is the way to go to reduce the current draw and the effect of loading. The other problem with these voltage divider references is that the reference cannot be loaded if we put a 100 Ω resistor in parallel with a 10 kΩ resistor. When the voltage divider is made of two 10 kΩ resistors, then the resistance of the reference resistor becomes somewhere near 100 Ω. This clearly means a terrible reference. If a 10 MΩ resistor is used for the reference resistor will still be some where around 10 kΩ but still probably less. The effect of tolerances is also a problem; if the resistors are rated 5% then the resistance of 10 kΩ resistors can vary by ±500 Ω. This means more inaccuracy with this sort of reference.
Current Dividers
If two elements are in parallel, the voltage across them must be the same, but the current divides according to the resistances. A simple circuit with two (or more) resistors in parallel with a source is called a current divider.
Figure B: Parallel Resistors.
If a voltage V appears across the resistors in Figure B with only and for the moment then the current flowing in the circuit, before the division, i is according to Ohms Law.
Using the equivalent resistance for a parallel combination of resistors is
 (1)
The current through according to Ohms Law is
 (2)
Dividing equation (2) by (1)
Similarly
In general with n Resistors the current is
Or possibly more simply
Where
Nodal Analysis
Electronics  Foreword  Basic Electronics  Complex Electronics  Electricity  Machines  History of Electronics  Appendix  edit
Nodes
A node is a section of a circuit which connects components to each other. All of the current entering a node must leave a node, according to Kirchoff's Current Law. Every point on the node is at the same voltage, no matter how close it is to each component, because the connections between components are perfect conductors. This voltage is called the node voltage, and is the voltage difference between the node and an arbitrary reference, the ground point. The ground point is a node which is defined as having zero voltage. The ground node should be chosen carefully for convenience. Note that the ground node does not necessarily represent an actual connection to ground, it is just a device to make the analysis simpler. For example, if a node has a voltage of 5 Volts, then the voltage drop between that node and the ground node will be 5 Volts.
Note that in real circuits, nodes are made up of wires, which are not perfect conductors, and so the voltage is not perfectly the same everywhere on the node. This distinction is only important in demanding applications, such as low noise audio, high speed digital circuits (like modern computers), etc. If we look at how a particular circuit functions an engineer might be able to select check points that are diametrically apposite of each other, this signafies two points of current crossing over to another point this can be another method in testing a circuit to determine how nodes work.
Nodal Analysis
Nodal analysis is a formalized procedure based on KCL equations.
Steps:
 Identify all nodes.
 Choose a reference node. Identify it with reference (ground) symbol. A good choice is the node with the most branches, or a node which can immediately give you another node voltage (e.g., below a voltage source).
 Assign voltage variables to the other nodes (these are node voltages.)
 Write a KCL equation for each node (sum the currents leaving the node and set equal to zero). Rearrange these equations into the form A*V1+B*V2=C (or similar for equations with more voltage variables.)
 Solve the system of equations from step 4. There are a number of techniques that can be used: simple substitution, Cramer's rule, the adjoint matrix method, etc.
Complications in Nodal Analysis
 Dependent Current Source
 Solution: Write KVL equations for each node. Then express the extra variable (whatever the current source depends on) in terms of node voltages. Rearrange into the form from step 4 above. Solve as in step 5.
 Independent Voltage Source
 Problem: We know nothing about the current through the voltage source. We cannot write KCL equations for the nodes the voltage source is connected to.
 Solution: If the voltage source is between the reference node and any other node, we have been given a 'free' node voltage: the node voltage must be equal to the voltage source value! Otherwise, use a 'supernode', consisting of the source and the nodes it is connected to. Write a KCL equation for all current entering and leaving the supernode. Now we have one equation and two unknowns (the node voltages). Another equation that relates these voltages is the equation provided by the voltage source (V2V1=source value). This new system of equations can be solved as in Step 5 above.
 Dependent Voltage Source
 Solution: Same as an independent voltage source, with an extra step. First write a supernode KCL equation. Then write the source controlling quantity (dependence quantity?) in terms of the node voltages. Rearrange the equation to be in the A*V1+B*V2=C form. Solve the system as above.
Example
Given the Circuit below, find the voltages at all nodes.
node 0: (defined as ground node)
node 1: (free node voltage)
node 2:
node 3:
which results in the following system of linear equations:
therefore, the solution is:
Another solution with KCL would be to solve node in terms of node 2;
External Links
Mesh Analysis
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Meshes
A 'mesh' (also called a loop) is simply a path through a circuit that starts and ends at the same place. For the purpose of mesh analysis, a mesh is a loop that does not enclose other loops.
Mesh Analysis
Similar to nodal analysis, mesh analysis is a formalized procedure based on KVL equations. A caveat: mesh analysis can only be used on 'planar' circuits (i.e. there are no crossed, but unconnected, wires in the circuit diagram.)
Steps:
 Draw circuit in planar form (if possible.)
 Identify meshes and name mesh currents. Mesh currents should be in the clockwise direction. The current in a branch shared by two meshes is the difference of the two mesh currents.
 Write a KVL equation in terms of mesh currents for each mesh.
 Solve the resulting system of equations.
Complication in Mesh Analysis
1. Dependent Voltage Sources
Solution: Same procedure, but write the dependency variable in terms of mesh currents.
2. Independent Current Sources
Solution: If current source is not on a shared branch, then we have been given one of the mesh currents! If it is on a shared branch, then use a 'supermesh' that encircles the problem branch. To make up for the mesh equation you lose by doing this, use the mesh current relationship implied by the current source (i.e. ).
3. Dependent Current Sources
Solution: Same procedure as for an independent current source, but with an extra step to eliminate the dependency variable. Write the dependency variable in terms of mesh currents.
Example
Given the Circuit below, find the currents , .
The circuit has 2 loops indicated on the diagram. Using KVL we get:
Loop1:
Loop2:
Simplifying we get the simultaneous equations:
solving to get:
Thevenin and Norton Equivalent Circuits
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Source Transformation
Any linear time invariant network of impedances can be reduced to one equivalent impedance. In particular, any network of sources and resistors can be reduced to one ideal source and one resistor, in either the Thevenin or Norton configurations. In this way, a complicated network attached to a load resistor can be reduced to a single voltage divider (Thevenin) or current divider (Norton).
Thevenin and Norton equivalents let you replace a Voltage source in series with a resistor by a current source in parallel with a resistor, or vice versa. This is called a source transformation.
The point to be noted is that the block that is replaced with such an equivalent should be linear and time invariant, i.e. a linear change in the electrical source in that block produces a linear change in the equivalent source, and the behavior can be replicated if the initial conditions are replicated. The above shown transformation figures are true only if the circuit contains at least one independent voltage or current source. If the circuit comprises only dependent sources then Thevenin equivalent consists of R_{Th} alone
Thevenin Equivalents
The Thevenin equivalent circuit of a (twoterminal) network consists of a voltage source in series with a resistor. The Thevenin equivalent will have the same output voltage and current regardless of what is attached to the terminals.
Techniques For Finding Thevenin Equivalents
 Network contains no sources (only resistors): The Thevenin resistance is equal to the equivalent resistance of the network. The Thevenin voltage is zero.
 Basic: Works for any network except one with no independent sources. Find the voltage across the terminals (with positive reference at terminal A) when they are opencircuited. Find the current from terminal A to terminal B when they are shortcircuited. Then
The Thevenin voltage source value is equivalent to the opencircuit voltage.
If the network has no dependent sources, the independent sources can be zeroed, and the Thevenin resistance is equal to the equivalent resistance of the network with zeroed sources. Then, find .
 Only Dependent Sources:
If the network has only dependent sources, either attach a test voltage source to the terminal points and measure the current that passes from the positive terminal, or attach a test current source to the terminal points and measure the voltage difference across the terminals. In both cases you will have values for and , allowing you to use the relation to find the Thevenin resistance.
Norton Equivalents
Norton equivalents can be found by performing a source transformation on the Thevenin equivalent. The Norton Equivalent of a Thevenin Equivalent consists of a current source, in parallel with .
Thevenin and Norton Equivalent
The steps for creating the Equivalent are:
 1. Remove the load circuit.
 2. Calculate the voltage, V, at the output from the original sources.
 3. Now replace voltage sources with shorts and current sources with open circuits.
 4. Replace the load circuit with an imaginary ohm meter and measure the total resistance, R, looking back into the circuit, with the sources removed.
 5. The equivalent circuit is a voltage source with voltage V in series with a resistance R in series with the load.
The Thevenin Equivalent is determined with as the load as shown in Figure 1. The first step is to open circuit . Then the voltage v is calculated with open circuited must be calculated. The voltage across is this is because no current flows in the circuit so the voltage across must be by KVL.
Since this circuit does not contain any dependent sources, all that needs to be done is for all the Independent Voltage sources to be shorted and for all Independent Current Sources to be open circuited. This results in the circuit shown in Figure 2.
Now the Thevenin Resistance is calculated looking into the two nodes. The Thevenin resistance is clearly . The Thevenin Equivalent is shown in Figure 3 and and have the values shown below.
 (1)
The Norton Equivalent is created by doing a source transformation using .(2)
If and and then
As a final note if the voltage across is calculate by Voltage Divider Rule using the Thevenin Equivalent circuit in Figure 3.
 (3)
If the value of form equation 1 is substituted into equation 3.
 (4)
Now look at Figure 1 and calcute by voltage divider rule it has the same value as equation 4. If the current through is calculated in Figure 4 by current divider rule.
Substituting equation 2 into 5.
If equation 4 and Ohm's Law are used to get the voltage across equation 3 is reached.
Please note: The "", a symbol that is used as an operator here, holds higher precedence than the "+" operator. As such, it is evaluated before a sum.
See Norton's theorem and Thevenin's theorem for more examples.
Superposition
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Superposition Principle
 Figure 1: The circuits showing the linearity of resistors.
Most basic electronic circuits are composed of linear elements. Linear elements are circuit elements which follow Ohm’s Law. In Figure 1 (a) with independent voltage source, V_{1}, and resistor, R, a current i_{1} flows. The current i_{1} has a value according to Ohm’s Law. Similarly in Figure 1 (b) with independent voltage source, V2, and resistor, R, a current i_{2} flows. In Figure 1 (c) with independent voltage sources, V1 and V2, and resistor, R, a current i flows. Using Ohm’s Law equation 1 is reached. If some simple algebra is used then equation 2 is reached. But V_{1}/R has a value i_{1} and the other term is i_{2} this gives equation 3. This is basically what the Superposition Theorem states.
 (1)
 (2)
 (3)
The Superposition Theorem states that the effect of all the sources with corresponding stimuli on a circuit of linear elements is equal to the algebraic sum of each individual effect. Each individual effect is calculated by removing all other stimuli by replacing voltage sources with short circuits and current sources with open circuits. Dependent sources can be removed as long as the controlling stimuli is not set to zero. The process of calculating each effect with one stimulus connected at a time is continued until all the effects are calculated. If kth stimulus is denoted sk and the effect created by sk denoted ek.
 (4)
The steps for using superposition are as follows:
 1. Calculate the effect of each source in turn with all other independent voltage sources short circuited and independent current sources open circuited.
 2. Sum these effects to get the complete effect.
Note: the removal of each source is often stated differently as: replace each voltage source with its internal resistance and each current with its internal resistance. This is identical to what has been stated above. This is because a real voltage source consists of an independent voltage source in series with its internal resistance and a real current source consists of an independent current source in parallel with its internal resistance.
Superposition Example
 Figure 2: The circuit for the example.
Problem: Calculate the voltage, v, across resistor R_{1}.
Step 1: Short circuit V_{2} and solve for v_{1}. By voltage divider rule.
 (5)
Short circuit V_{1} and solve for v_{2}. By voltage divider rule.
 (6)
Step 2: Sum the effects.
Using equations 5 and 6.
If and then
Diagnostic Equipment
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Diagnostic and Testing Equipment
There is a wide array of devices used to test and diagnose electronic equipment. This chapter will attempt to explain the differences and different types of equipment used by electronics technicians and engineers.
Ammeter
An ammeter measures current.Current in electronics is usually measured in mA which are called milliamperes, which are 1/1000s of an ampere.
. The ammeter's terminals must be in series with the current being measured. Ammeters have a small resistance (typically 50 ohms) so that they only have a small effect on the current.
Basically an ammeter consists of a coil that can rotate inside a magnet, but a spring is trying to push the coil back to zero. The larger the current that flows through the coil, the larger the angle of rotation, the torque (= a rotary force) created by the current being counteracted by the return torque of the spring.
. Usually ammeters are connected in parallel with various switched resistors that can extend the range of currents that can be measured. Assume, for example, that the basic ammeter is "1000 ohms per volt", which means that to get the fullscale deflection of the pointer a current of 1 mA is needed (1 volt divided by 1000 ohms is 1 mA  see "Ohm's Law").
. To use that ammeter to read 10 mA fullscale it is shunted with another resistance, so that when 10 mA flows, 9 mA will flow through the shunt, and only 1 mA will flow through the meter. Similarly, to extend the range of the ammeter to 100 mA the shunt will carry 99 mA, and the meter only 1 mA.
Ohmmeter
An ohmmeter measures resistance.The two terminals of ohmmeter are each placed on a terminal of the resistance being measured. This resistance should be isolated from other effects. (It should be taken out of a circuit, if it is in one.)
Ohmmeters are basically ammeters that are connected to an internal battery, with a suitable resistance in series. Assume that the basic ammeter is "1000 ohms per volt", meaning that 1 mA is needed for fullscale deflection. When the external resistance that is connected to its terminals is zero (the leads are connected together at first for calibration), then the internal, variable, resistor in series with the ammeter is adjusted so that 1 mA will flow; that will depend on the voltage of the battery, and as the battery runs down that setting will change. The full scale point is marked as zero resistance. If an external resistance is then connected to the terminals that causes only half of the current to flow (0.5 mA in this example), then the external resistance will equal the internal resistance, and the scale is marked accordingly. When no current flows, the scale will read infinity resistance. The scale of an ohmmeter is NOT linear.Ohmmeters are usually usuful in cheking the short circuit and open circuit in boards.
Voltmeter
A voltmeter measures voltage.The voltmeter's terminals must be in parallel with the voltage being measured. Voltmeters have a large resistance (typically 1 megaohm), so that they only have a small effect on the voltage.
Multimeter
A multimeter is a combination device, (usually) capable of measuring current, resistance, or voltage. Most modern models measure all three, and include other features such as a diode tester, which can be used to measure continuity in circuits (emitting a loud 'beep' if there is a short).
Oscilloscope
An oscilloscope, commonly called a 'scope' by technicians, is used to display a voltage waveform on a screen, usually graphing voltage as a function of time.
Spectrum Analyzer
Spectrum analyzer shows voltage (or power) densities as function of frequency on radio frequency spectrum. Spectrum analyzer can use analog frequency scanning principle (like radio receiver always changing frequency and measuring receiving amplitude) or digital sampling and FFT (Fast Fourier Transformation).
Logic analyzer
A logic analyzer is, in effect, a specialised oscilloscope. The key difference between an analyzer and an oscilloscope is that the analyzer can only display a digital (on/off) waveform, whereas an oscilloscope can display any voltage (depending on the type of probe connected). The other difference is that logic analyzers tend to have many more signal inputs than oscilloscopes  usually 32 or 64, versus the two channels most oscilloscopes provide. Logic analyzers can be very useful for debugging complex logic circuits, where one signal's state may be affected by many other signals.
Frequency counter
A frequency counter is a relatively simple instrument used to measure the frequency of a signal in Hertz (cycles per second). Most counters work by counting the number of signal cycles that occur in a given time period (usually one second). This count is the frequency of the signal in Hertz, which is displayed on the counter's display.
Electrometer
A voltmeter with extremely high input resistance capable of measuring electrical charge with minimal influence to that charge. Ubiquitous in nucleonics, physics and biomedical disciplines. Enables the direct verification of charge measured in coulombs according to Q=CV. Additionally, electrometers can generally measure current flows in the femtoampere range, i.e. .000000000000001 ampere.
Signal Generator
DC Circuit Analysis
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DC Circuit Analysis
In this chapter, capacitors and inductors will be introduced (without considering the effects of AC current.) The big thing to understand about Capacitors and Inductors in DC Circuits is that they have a transient (temporary) response. During the transient period, capacitors build up charge and stop the flow of current (eventually acting like infinite resistors.) Inductors build up energy in the form of magnetic fields, and become more conductive. In other words, in the steadystate (long term behavior), capacitors become open circuits and inductors become short circuits. Thus, for DC analysis, you can replace a capacitor with an empty space and an inductor with a wire. The only circuit components that remain are voltage sources, current sources, and resistors.
Capacitors and Inductors at DC
DC steadystate (meaning the circuit has been in the same state for a long time), we've seen that capacitors act like open circuits and inductors act like shorts. The above figures show the process of replacing these circuit devices with their DC equivalents. In this case, all that remains is a voltage source and a lone resistor. (An AC analysis of this circuit can be found in the AC section.)
Resistors
If a circuits contains only resistors possibly in a combination of parallel and series connections then an equivalent resistance is determined. Then Ohm's Law is used to determine the current flowing in the main circuit. A combination of voltage and current divider rules are then used to solve for other required currents and voltages.
Simplify the following:
(a) (b) (c)
 Figure 1: Simple circuits series circuits.
The circuit in Figure 1 (a) is very simple if we are given R and V, the voltage of the source, then we use Ohm's Law to solve for the current. In Figure 1 (b) if we are given R1, R2 and V then we combine the resistor into an equivalent resistors noting that are in series. Then we solve for the current as before using Ohm's Law. In Figure 1 (c) if the resistors are labeled clockwise from the top resistor R1, R2 and R3 and the voltage source has the value: V. The analysis proceeds as follows.
This is the formula for calculating the equivalent resistance of series resistor. The current is now calculated using Ohm's Law.
If the voltage is required across the third resistor then we can use voltage divider rule.
Or alternatively one could use Ohm's Law together with the current just calculated.
 Figure 2: Simple parallel circuits.
In Figure 2 (a) if the Resistor nearest the voltage source is R1 and the other resistor R2. If we need to solve for the current i. then we proceed as before. First we calculate the equivalent resistance then use Ohm's Law to solve for the current. The resistance of a parallel combination is:
So the current, i, flowing in the circuit is, by Ohm's Law:
If we need to solve for current through R2 then we can use current divider rule.
 (1)
But it would probably have been simpler to have used the fact that V most be dropped across R2. This means that we can simply use Ohm's Law to calculate the current through R2. The equation is just equation 1. In Figure 2 (b) we do exactly the same thing except this time there are three resistors this means that the equivalent resistance will be:
Using this fact we do exactly the same thing.
 Figure 3: Combined parallel and series circuits
In Figure 3 (a), if the three resistors in the outer loop of the circuit are R1, R2 and R3 and the other resistor is R4. It is simpler to see what is going on if we combine R2 and R3 into their series equivalent resistance . It is clear now that the equivalent resistance is R1 in series with the parallel combination of and R4. If we want to calculate the voltage across the parallel combination of R4 and then we just use voltage divider.
If we want to calculate the current through R2 and R3 then we can use the voltage across and Ohm's law.
Or we could calculate the current in the main circuit and then use current divider rule to get the current.
In Figure 3 (b) we take the same approach simplifying parallel combinations and series combinations of resistors until we get the equivalent resistance.
In Figure 3 (c) this process doesn't work then because there are resistors connected in a delta this means that there is no way to simplify this beyond transforming them to a star or wye connection.
Note: To calculate the current draw from the source the equivalent resistance always must be calculated. But if we just need the voltage across a series resistor this may be necessary. If we want to calculate the current in parallel combination then we must use either current divider rule or calculate the voltage across the resistor and then use Ohm's law to get the current. The second method will often require less work since the current flowing from the source is required for the use of current divider rule. The use of current divider rule is much simpler in the case when the source is a current source because the value of the current is set by the current source.
The above image shows three points 1, 2, and 3 connected with resistors R_{1}, R_{2}, and R_{3} with a common point. Such a configuration is called a star network or a Yconnection.
The above image shows three points 1, 2, and 3 connected with resistor R_{12}, R_{23}, and R_{31}. The configuration is called a delta network or delta connection.
We have seen that the series and parallel networks can be reduced by the use of simple equations. Now we will derive similar relations to convert a star network to delta and vice versa. Consider the points 1 and 2. The resistance between them in the star case is simply
R_{1} + R_{2}
For the delta case, we have
R_{12}  (R_{31} + R_{23})
We have similar relations for the points 2, 3 and 3, 1. Making the substitution r_{1}= R_{23} etc., we have, simplifying,
in the most general case. If all the resistances are equal, then R = r/3.
Measuring Instruments
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Measuring Instruments
Ammeters
Ammeters are devices that measure current. Current in electronics is usually measured in mA which are called milliamperes, which are 1/1000s of an ampere.
..... Basically an ammeter consists of a coil that can rotate inside a magnet, but a spring is trying to push the coil back to zero. The larger the current that flows through the coil, the larger the angle of rotation, the torque (= a rotary force) created by the current being counteracted by the return torque of the spring.
..... Usually ammeters are connected in parallel with various switched resistors that can extend the range of currents that can be measured. Assume, for example, that the basic ammeter is "1000 ohms per volt", which means that to get the fullscale deflection of the pointer a current of 1 mA is needed (1 volt divided by 1000 ohms is 1 mA  see "Ohm's Law").
..... To use that ammeter to read 10 mA fullscale it is shunted with another resistance, so that when 10 mA flows, 9 mA will flow through the shunt, and only 1 mA will flow through the meter. Similarly, to extend the range of the ammeter to 100 mA the shunt will carry 99 mA, and the meter only 1 mA.
Ohmmeters
Ohmmeters are basically ammeters that are connected to an internal battery, with a suitable resistance in series. Assume that the basic ammeter is "1000 ohms per volt", meaning that 1 mA is needed for fullscale deflection. When the external resistance that is connected to its terminals is zero (the leads are connected together at first for calibration), then the internal, variable, resistor in series with the ammeter is adjusted so that 1 mA will flow; that will depend on the voltage of the battery, and as the battery runs down that setting will change. The full scale point is marked as zero resistance. If an external resistance is then connected to the terminals that causes only half of the current to flow (0.5 mA in this example), then the external resistance will equal the internal resistance, and the scale is marked accordingly. When no current flows, the scale will read infinity resistance. The scale of an ohmmeter is NOT linear.Ohmmeters are usually useful in checking the short circuit and open circuit in boards.
Multimeters
Multimeters contain Ohmeters, Voltmeters, Ammeters and a variety of capabilities to measure other quantities. AC and DC voltages are most often measurable. Frequency of AC voltages. Multimeters also feature a continuity detector, basically an Ohmmeter with a beeper if the multimeter sees less than 100 Ω then it beeps otherwise it is silent. This is very useful for finding whether components are connected when debugging or testing circuits. Multimeters are also often able to measure capacitance and inductance. This may be achieved using a Wien bridge. A diode tester is also generally onboard, this allows one to determine the anode and cathode of an unknown diode. A LCD display is also provided for easily reading of results.
Electronics Laboratory Instruments
Oscilloscope
The instrument is used to view AC waveforms. For better explanation of the oscilloscope.
Spectrum Analyzer
Signal Generator
This instrument is used to generate low voltage AC signals. Most common signal generators can create sinusoidal(sine), triangular and square waves of various frequencies. They are used in conjunction with the oscilloscope to test analogue circuits.
Logic Probe
This instrument generates high and low logic states to test digital circuits. If a logic probe is not available a square wave through a signal generator can be used. Square waves can also be used to test the response time of a digital circuits.
Noise in electronic circuits
 Electrical Noise
 any unwanted form of energy tending to interfere with the proper and easy reception and reproduction of wanted signals.
Classification
Based on Origin
 External noise
 Atmospheric
 Extraterrestrial
 solar
 Cosmic
 Industrial
 Internal noise
 Thermal Agitation Noise
 Shot Noise
 Transit Time Noise
 Flicker Noise
 Miscellaneous Sources
Thermal noise
 Thermal Agitation Noise
 Also known as Johnson noise or White noise.
where k = Boltzmann's constant = 1.38x10^{23}J/K

 T = absolute temperature, K = 273 + °C
 δ f = bandwidth of interest
 P_{n} = maximum noise power output of a resistor
Shot Noise
where i_{n} = r.m.s. shotnoise current
 e = charge of an electron = 1.6x10^{19}C
 i_{p} = direct diode current
 δ f = bandwidth of system
Noise Calculations
Addition due to several sources
noise voltages:
where R_{tot} = R_{1}+R_{2}+...
Addition due to Cascaded Amplifier stages
Analog Noise Models
CMOS
BJT
Noise in digital circuits:
Methods of reducing noise
Differential signaling
Differential signaling is a method of transmitting information electrically by means of two complementary signals sent on two separate wires. The technique can be used for both analogue signaling, as in some audio systems, and digital signaling, as in RS422, RS485, PCI Express and USB.
Good grounding
An ideal signal ground maintains zero voltage regardless of how much electrical current flows into ground or out of ground.
When lowlevel signals travel near high currents, their return currents shouldn't be allowed to flow in the same conductor. Otherwise, noise such as AC ripple on the high current will modulate the lowlevel signal.
References
Kennedy, George 'Electronic Communication Systems' , 3^{rd} Ed. ISBN 0070340544
Chapter 2: AC Circuits
AC Circuits
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Relationship between Voltage and Current
Resistor
In a resistor, the current is in phase with the voltage always. This means that the peaks and valleys of the two waveforms occur at the same times. Resistors can simply be defined as devices that perform the sole function of inhibiting the flow of current through an electrical circuit. Resistors are commercially available having various standard values, nevertheless variable resistors are also made called potentiometers, or pots for short. In theory electricity is a method used to harness myriad numbers including symbols to give the notion on how circuits function on a schematic drawing
Capacitor
The capacitor is different from the resistor in several ways. First, it consumes no real power. It does however, supply reactive power to the circuit. In a capacitor, as voltage is increasing the capacitor is charging. Thus a large initial current. As the voltage peaks the capacitor is saturated and the current falls to zero. Following the peak the circuit reverses and the charge leaves the capacitor. The next half of the cycle the circuit runs mirroring the first half.
The relationship between voltage and current in a capacitor is: . This is valid not only in AC but for any function v(t). As a direct consequence we can state that in the real world, the voltage across a capacitor is always a continuous function of the time.
If we apply the above formula to a AC voltage (i.e. ), we get for the current a 90° phase shift: .
In an AC circuit, current leads voltage by a quarter phase or 90 degrees. Note that while in DC circuits after the initial charge or discharge no current can flow, in AC circuits a current flows all the time into and out of the capacitor, depending on the impedance in the circuit. This is similar to the resistance in DC circuits, except that the impedance has 2 parts; the resistance included in the circuit, and also the reactance of the capacitor, which depends not only on the size of the capacitor, but also on the frequency of the applied voltage. In a circuit that has DC applied plus a signal, a capacitor can be used to block the DC, while letting the signal continue.
Inductor
In inductors, current is the negative derivative of voltage, meaning that however the voltage changes the current tries to oppose that change. When the voltage is not changing there is no current and no magnetic field.
In an AC Circuit, voltage leads current by a quarter phase or 90 degrees.
Voltage Defined as the derivative of the flux linkage:
Resonance
A circuit containing resistors, capacitors, and inductors is said to be in resonance when the reactance of the inductor cancels that of the capacitor to leave the resulting total resistance of the circuit to be equal to the value of the component resistor. The resonance state is achieved by fine tuning the frequency of the circuit to a value where the resulting impedance of the capacitor cancels that of the inductor, resulting in a circuit that appears entirely resistive.
See also
AC Voltage:
When the Voltage between two points is changing continuously with time then the voltage is called AC Voltage. For AC Voltages there will be no constant value so we define it by the average value called RMS value RMS means Root Means Square. In retrospect on AC the power that is used to theoretically determine by an unkown which we assume to be a steady flow, this is also what technicians label the fluctuation of current that determines how electricity moves in a diagram.
Phasors
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Phasors
Phasors provide a simple means of analyzing linear circuits. At the heart of phasor analysis lies Euler's formula:
 ^{[1]}
A complex exponential can also be expressed as
is called a phasor. It contains information about the magnitude and phase of a sinusoidal signal, but not the frequency or time. This simplifies use in circuit analysis, since most of the time, all quantities in the circuit will have the same frequency. (For circuits with sources at different frequencies, the principle of superposition must be used.)
A shorthand phasor notation is:
Note that this is simply a polar form, and can be converted to rectangular notation by (see figure one):
and back again by (see figure two):
 . Let us understand how this process works, even though a phasor causes circuits to heat up steady flow in current will explain on a basis that disallows overheating. Since according to the formula the outcome will be there is a time in duration which the current usage stays the same.
Sinusoidal Signals
To begin, we must first understand what sinusoidal signals are. Sinusoidal signals can be represented as
where A is the amplitude, is the frequency in radians per second, and is the phase angle in degrees(phase shift). We can return to the sinusoidal signal by taking the real part of Euler's formula:
For the moment, consider singlefrequency circuits. Every steady state current and voltage will have the same basic form:
 where is a phasor. So we can "divide through" by to get phasor circuit equations. We can solve these equations for some phasor circuit quantity , multiply by , and convert back to the sinusoidal form to find the timedomain sinusoidal steadystate solution.
Example
We have three sinusoidal signals with the same frequency added together:
In phasor notation, this is:
We can combine these terms to get one phasor notation. This is done first by separating the real and imaginary components:
The phasor notation can be written as:
Back to the time domain, we get the answer:
Footnotes
Impedance
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Definition
Impedance, , is the quantity that relates voltage and current in the frequency domain. (The tilde indicates a phasor. An overscore or arrow may also be used.)
 .
In rectangular form,
where R is the resistance and X is the reactance. Impedance is generally a function of frequency, i.e.
NOTE: ω = 2 π f
 where f is the frequency in cycles per second (f=50 or 60 Hertz usually, depending on the country concerned. Aircraft systems often use 400 Hertz.)
Reactance
Reactance (symbol ) is the resistance to current flow of a circuit element that can store energy (i.e. a capacitor or an inductor), and is measured in ohms.
The reactance of an inductor of inductance (in Henries), through which an alternating current of angular frequency flows is given by:
The reactance of a capacitor of capacitance (in Farads) is given similarly:
The two formulae for inductive reactance and capacitive reactance create interesting counterpoints. Notice that for inductive reactance, as the frequency of the AC increases, so does the reactance. Hence, higher frequencies result in lower current. The opposite is true of capacitive reactance: The higher the frequency of AC, the less reactance a capacitor will present.
Similarly, a more inductive inductor will present more reactance, while a capacitor with more capacitance will yield less reactance.
Resistors
Resistors have zero reactance, since they do not store energy, so their impedance is simply
 .
Capacitors
Capacitors have zero resistance, but do have reactance. it is stored in the power of a circuit or it can be also stored in a motor running off a AC current. Their impedance is
where C is the capacitance in farads. The reactance of one microfarad at 50 Hz is 3183 ohms, and at 60 Hz it is 2653 ohms. In much more basic terms storing energy would be equivalent to a battery the power source is active and stays in this manner without any loss of power, this is what is regarded as a means for reversing electricity in the process of keeping that current.
.
Inductors
Like capacitors, inductors have zero resistance, but have reactance. Their impedance is
where L is the inductance in henries. The reactance of one henry at 50 Hz is 314 ohms, and at 60 Hz it is 377 ohms.
Circuit Analysis Using Impedance
Analysis in the frequency domain proceeds exactly like DC analysis, but all currents and voltages are now phasors (and so have an angle). Impedance is treated exactly like a resistance, but is also a phasor (has an imaginary component/angle depending on the representation.)
(In the case that a circuit contains sources with different frequencies, the principle of superposition must be applied.)
Note that this analysis only applies to the steady state response of circuits. For circuits with transient characteristics, circuits must be analyzed in the Laplace domain, also known as sdomain analysis.
Steady State
Steady State
That can be said to be the condition of "rest", after all the changes/alterations were made. This may imply, for examples, that nothing at all happens, or that a "steady" current flows, or that a circuit has "settled down" to final values  that is until the next disturbance occurs.
If the input signal is not time invariant, say if is a sinosoid, the steady state will not be invariant either. The response of a systematic convergence can be considered to be composed of a transient response: the response to a disturbance, and the steady state response, in the absence of disturbance.
The transient part of the response tends to zero as time since a disturbance tends to infinity, so the steady state can be considered to be the response remaining as T > infinity.. In this case we make an assumption for error we can determine by calculation the extent of statistics how much power there is according to the formulae. Deducing the theory of certain parts of a sinuous making plausible solution in understanding the intricate mechanism of a steady state solution.
Chapter 3: Transient Analysis
RC Circuits
RL Circuits
LC Circuits
RCL Circuits
Chapter 4: Analog Circuits
Analog Circuits
As explained later, digital signals can only take one of two values at any one point. Analog signals, however, can take any value within a range. In modern electronics, many traditionally analog circuits are being replaced with digital ICs for various reasons:
 Reduced cost:
 Digital ICs can be programmed, meaning many different analog circuits can each be replaced with the same IC, reducing cost. For example, the same DSP chip might replace many different analog filters.
 Digital ICs can be easily expanded to put multiple functions on the same chip. For example, a DSP chip replacing multiple analog filters at the same time.
 Digital ICs can be reprogrammed without modifying the circuit, simplifying prototyping and field upgrades.
 Increased versatility:
 A digital IC can have its settings changed arbitrarily while it is being operated, these settings can even be changed by software. The equivalent in an analog circuit might require expensive and complicated switching techniques.
 Numeric precision: once a signal is converted to digital data, distortion and circuit noise are no longer a significant issue. A cheap modern watch driven by a digital circuit (such as a quartz oscillator and a counter) is more precise than any analog or mechanical clock.
 Ease of storage: digital data, once recorded, does not degrade as easily as analog data.
Digital circuits are also becoming more prevalent where there are no analog analogues, such as computers. Although analogue computers exist, their utility is severely limited by comparison.
Analog circuits are still more suitable for many functions. One important category is interfaces. Digital circuits are usually inferior to analog circuits for receiving and transmitting signals.
 A common digital interface, the differential line, needs an analog circuit to function well. The receiver must filter out common mode interference by computing the analog difference between the two lines. Most digital interfaces are designed as analog, only a few legacy interfaces (which have poor performance) use entirely digital circuits.
 Many radio signals are simply too high frequency to work with existing digital circuits. Radio modulators, demodulators, mixers, transmitters, and receivers are still analog. Some signals are even too high frequency for transistor circuits of any kind to amplify efficiently, such as microwave signals, which are still transmitted using vacuum tubes.
 Signals must be conditioned before conversion to digital to avoid aliasing: analog filters remove unwanted parts of signals.
 Many devices, such as monitors, require analog control. Even LCD monitors require analog circuits, although they need not be as sophisticated as CRT analog circuits.
 Audio equipment requires analog circuits. Speakers must be driven by analog signals, microphones produce analog signals. Signal of a microphone must be conditioned before conversion to digital, this usually entails sophisticated analog circuits such as preamplifers, compressors, and filters.
Vacuum Tubes
Passive Versus Active Components
Passive components have no gain and are not valves
 Voltage Regulator: an active component which accepts a range of voltages and outputs one constant voltage.
Vacuum Tube equals Thermonic Valve
A Vacuum Tube is a container (usually of glass) from which the air is removed. Inside the tube are two or more "Elements".
 Cathode: (electron emitter) has an electrically heated filament (which you can usually see glow red ) which spits out electrons that travel through the vacuum to the Anode (electron acceptor).
 Anode: Is a conductive (usually metal) plate that is connected to a positive voltage. The negative electrons flow from the Cathode to the Anode.
A vacuum tube with just Cathode and Anode elements is a DIODE. Current will flow only when the Anode has a positive voltage relative to the Cathode.
 Grid(s): metal gratings or grids are placed between the Cathode and Anode to produce devices that can amplify signals.
NOTE: A tube with 3 elements (one grid) is a TRIODE, with 2 grids a TETRODE, with 3 grids a PENTODE.
A Grid between the cathode and anode controls the flow of electrons. By applying a negative voltage to the grid it is possible to control the flow of electrons. This is the basis of the Vacuum Tube amplifier.
A problem with Vacuum Tubes is they are big, bulky, and often meant for applications involving a lot of power. They are very powerful which usually being a Vacuum. However it is possible to scale up vacuum tubes to very high powers, and many highpowered transmitters for broadcast FM and TV use vacuum power tubes.
Vacuum tube based amplifiers have largely been replaced by semiconductor transistor based amplifiers, as they are low power and much more compact.
The voltage difference in the direction from the cathode to the anode is known as the forward bias and is the normal operating mode. If the voltage applied to the Anode becomes negative relative to the Cathode, no electrons will flow. In electronics vacuum electron tube or valve is a device that controls current. Through a vacuum in sealed container. Vacuum tubes rely on thermionic emission of electrons from hot filament.
Klystron
A klystron is a vacuum tube used for production of microwave energy. This device is related to but not the same as a magnetron. The klystron was invented after the magnetron.
Klystrons work using a principle known as velocity modulation.
The klystron is a long narrow vacuum tube. There is an electron gun (heater, cathode, beam former) at one end and an anode at the other. In between is a series of donut shaped resonant cavity structures positioned so that the electron beam passes through the hole.
The first and last of the resonant cavities are electrically wired together.
At the cathode the electron beam is relatively smooth. There are natural slight increases and decreases in the electron density of the beam. As the beam passes through the holes of the resonant cavities, any changes in the electron beam cause some changes in the resting electro magnetic (EM) field of the cavities. The EM fields of the cavities begin to oscillate. The oscillating EM field of the cavities then has an effect on the electrons passing through, either slowing down or speeding up their passage.
As electrons are affected by the EM field of the first cavity they change their speed. This change in speed is called velocity modulation. By the time the electrons arrive at the last cavity there are definite groups in the beam. The groups interact strongly with the last cavity causing it to oscillate in a more pronounced way. Some of the last cavity's energy is tapped off and fed back to the first cavity to increase its oscillations. The stronger first cavity oscillations produce even stronger grouping of the electrons in the beam causing stronger oscillations in that last cavity and so on. This is positive feedback.
The output microwave energy is tapped of for use in high power microwave devices such as long range primary RADAR systems.
The klystron is a coherent microwave source in that it is possible to produce an output with a constant phase. This is a useful attribute when combined with signal processing to measure RADAR target attributes like Doppler shift.
Related microwave vacuum tubes are the Travelling Wave Tube (TWT) and the Travelling Wave Amplifier (TWA). A Hybrid device, which combines some aspects of these devices and the klystron, is a device called a Twicetron.
Magnetron
Magnetrons are used to produce microwaves.
This is the original device used for production of microwaves and was invented during the Second World War for use in RADAR equipment.
Magnetrons work using a principle known as velocity modulation.
A circular chamber, containing the cathode, is surrounded by and connected to a number of resonant cavities. The walls of the chamber are the anode. The cavity dimensions determine the frequency of the output signal. A strong magnetic field is passed through the chamber, produced by a powerful magnet. The cathode is similar to most thermionic valves, except heavy, rigid construction is necessary for power levels used in most Magnetrons. Early, experimental designs used directly heated cathodes. Modern, high powered designs use a rigid, tubular cathode enclosing a heater element.
Naturally excited electrons on the surface of the cathode are drawn off, into the chamber, toward the outer walls or anode. As the electrons move out, they pass through a magnetic field that produces a force perpendicular to the direction of motion and direction of the magnetic field. The faster the electrons move, the more sideways force is produced. The result is that the electrons rotate around the central cathode as they move toward the outside of the chamber.
As electrons move past the entrances to the resonant cavities a disturbance is made to the electro magnetic (EM) field that is at rest in the cavities. The cavity begins to oscillate. When another electron moves past the cavities, it also interacts with the internal EM field. The motion of the electron can be slowed or sped up by the cavity field. As more electrons interact with the cavity EM fields, the internal cavity oscillations increase and the effect on the passing electrons is more pronounced.
Eventually, bands of electrons rotating together develop within the central chamber. Any electrons that fall behind a band are given a kick by the resonant cavity fields. Any electrons going too fast have their excess energy absorbed by the cavities. This is the velocity modulation effect. The frequency of the resonance and electron interaction is in the order of GHz. (10^9 cycles per second)
In order to have a signal output from the magnetron, one of the cavities is taped with a slot or a probe to direct energy out into a waveguide for distribution.
Magnetrons for RADARs are pulsed with short duration and high current. Magnetrons for microwave ovens are driven with a continuous lower current.
The magnetrons for WWII bombers, operated by the RAF, were sometimes taped into a shielded box so that the aircrew could heat their inflight meals, hence the first microwave ovens.
Cathode Ray Tube
A cathode ray tube or CRT is a specialized vacuum tube in which images are produced when an electron beam strikes a phosphorescent surface. Television sets, computers, automated teller machines, video game machines, video cameras, monitors, oscilloscopes and radar displays all contain cathoderay tubes. Phosphor screens using multiple beams of electrons have allowed CRTs to display millions of colors.
The first cathode ray tube scanning device was invented by the German scientist Karl Ferdinand Braun in 1897. Braun introduced a CRT with a fluorescent screen, known as the cathode ray oscilloscope. The screen would emit a visible light when struck by a beam of electrons.
TV Tubes
TV tubes are basically cathode ray tubes. An electron beam is produced by an electricallyheated filament, and that beam is guided by two magnetic fields to a particular spot on the screen. The beam is moved so very quickly, that the eye can see not just one particular spot, but all the spots on the screen at once, forming a variable picture.
 Colours are produced by having 3 or more differently coloured screen spots activated at once to a variable degree.
The 2 magnetic fields are one for the vertical deflection, one for the horizontal deflection, and they are provided to the beam by external coils.
Oscilloscope Tubes
Oscilloscope tubes are basically the same as TV tubes, but the beam is guided by two electrostatic fields provided by internal pieces of metal. It is a necessity because an oscilloscope uses a very large range of synchronisation frequencies for the deflection, when a TV set uses fixed frequencies, and it would be too difficult to drive large coils on a so large band of frequency. They are much deeper for the same screen size as a TV tube because the deflection angle is little. A TV tube has a deflection angle of 90° for the be
External Links
Diodes
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Diode
Theoretically, a diode allows current flow in only one direction. An ideal diode acts as a perfect insulator for currents flowing in one direction and as a perfect conductor for currents flowing through it in the other direction. The direction in which the diode allows current to flow is called the forward bias direction and that in which current is resisted is called reverse bias direction. Diode has a symbol as shown
Construction
The modern semiconductor diode consists of two regions of semiconductor each having impurities of different types such that one side has excess holes (pregion) and the other has excess electrons (nregion). Such a junction of p and n regions is called a pn junction diode. The pregion has about twice as much area as the nregion to compensate for the lesser mobility of holes compared to electrons.
 + o [P  N]o  .Theoretically, a diode allows current flow in only one direction the most common type today is a crystalline piece of material with a p junction connected to two electrical terminals.
Operation
I V Curve
As seen in the graph above the diode actually works in both the forward region and the backward region. In the forward region the value of I and V are positive and in the backward region I and V are negative.
Forward Region
 Current and Voltage are positive
 When V < V_{d} . I = 0 . Diode does not conduct
 Khi V = V_{d} . I = 1mA . Diode starts to conduct . V_{d} = 0.3vGe , 0.6vSi
 V_{d} is called Forward Break Over Voltage
 Khi V > V_{d} . Diode conducts current . Current is calculated by ]
Backward Region
 Current and Voltage are Negative
 When the value of voltage is more negative than the Peak Inverse Voltage (PIV) Voltage the Diode will be destroyed
Ideal Diode
The real diode approaches the ideal diode in the sense that the reverse current is extremely small (less than 1fA) at least for a significant part of the characteristic, and the forward current is very high (on the order of 1mA). Although a real diode does not have the characteristics as the ideal diode, in theory it is possible to make an ideal diode if the concentrations of dopants in both the regions are infinite. However, there is no way of actually doing this and experiments do not agree.
The Shockley equation
The diode reverse (saturation) current is governed by the doping concentration. The current flowing through the device varies as the voltage applied across it changes as given by the Shockley diode equation (not to be confused with Schottky):
In the equation above is defined as , where is Boltzmann's constant, is the temperature in Kelvin, and is the magnitude of the charge on an electron.
In the forward bias direction, current flows with low voltage. If one draws a characteristic for this equation, a sharp increase in current can be seen at a particular voltage called the cutin voltage or the onvoltage.
In the reverse bias mode, the diode current is approximately . This is called the reverse saturation current because it looks like the diode is saturated with charge and cannot allow more current in the reverse bias direction than this.
Breakdown
However, a break from the above equation takes place at a point called breakdown voltage. One could think of it as the point where the Shockley equation breaks down and is no longer valid. There are two reasons for breakdown to occur.
Avalanche Breakdown
 This occurs as a result of excess minority carriers in a region. Minority carriers are those carriers that are in the wrong region. For example, electrons will be minority carriers in the pregion.
Zener Breakdown
 This is basically due to a size difference or dopant concentration difference. One of the regions has a greater region of depletion (Reverse bias voltage induces a depletion region, which is sparse in a densely doped region and dense in a sparsely doped region. )
See also Zener diodes
Summary
So basically, there are three modes in which a diode operates:
Forward
 No current flows until a small forward voltage is reached called cutin voltage.
Reverse
 The diode prevents current from flowing in the opposite direction. Current is small, and voltage can be large (but not exceeding the Zener voltage.)
Breakdown
 Once the diode voltage is more negative than the Zener voltage, the diode allows current to flow in the reverse direction.
When there is no voltage applied, the excess electrons of the N type semiconductor flow into the holes of the P type semiconductor. This creates a depletion region that acts as a voltage.
Diode Variations
Bridge Rectifier:
 A diode circuit that 'rectifies' alternating current (AC) into direct current (DC). The bridge rectifier is a fullwave rectifier, meaning that both the positive and negative portions of the wave become positive. (In a halfwave rectifier, positive stays positive, and negative becomes zero.) The bridge rectifier has advantages over other fullwave rectifier designs, because it reduces peakinverse voltage (PIV), the largest negative voltage across a single junction diode. By reducing the PIV, it becomes possible to use diodes with lower breakdown (Zener) voltages. This allows the use of cheaper diodes to perform the same function.
LED (Light Emitting Diode)
Schottky:
 A diode made from a metalsemiconductor junction, rather than an ptype/ntype silicon junction. These diodes typically have a much lower forward voltage drop than standard diodes (around 0.2V versus 0.6V).
Zener:
 A diode that is meant to be operated in the breakdown region. These diodes have lower Zener (breakdown) voltages, so that they can achieve the breakdown mode without melting. Unlike other diodes, these have very specific breakdown voltages, typically between 2 and 200 volts. See also Zener diodes
Varicap:
 A diode that is build to use the capacity in dependance of the cross voltage. Although they rectify, they will be used in tuning circuits as a replacement of manual operated variable capacitors. They permit the use of electronic equipment to be controlled, this permits a phaselocked loop and other circuits which needs stability and electronic control. Semiconductor diodes begin conducting electricity under certain conditons.
Transistors
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Transistor
A transistor is a solid state device made by joining three positivetype and negativetype semiconductors together. In general, all transistors have three pins: base, collector, and emitter. Transistor is a bipolar device that is a transistor with two junctions namely BE and CE DE EE FE. In theory we take a specified formulae incorporate this with using any type of meter in figuring the mathematical solution.
Construction
A lightly doped region called base is sandwiched between two regions called the emitter and collector respectively. The collector handles large quantities of current, hence its dopant concentration is the highest. The emitter's dopant concentration is slightly lesser, but its area is larger to provide for more current than the collector. The collector region should be heavily doped because electronhole pairs recombine in that region, while the emitter is not such a region. We can have two varieties in this kind of transistor.
NPN Transistor
An NPN transistor is made by joining one positivetype semiconductor in between two negativetype semiconductors. Here a lightly doped ptype semiconductor (semiconductor with more holes than electrons) is sandwiched between two welldoped ntype regions. It is like two pnjunctions facing away. An IEEE symbol for the NPN transistor is shown here. The arrow between the base and emitter is in the same direction as current flowing between the baseemitter junction. Power dissipated in the transistor is
 , where is the voltage between the collector and the emitter and is the collector current.
PNP Transistor
A PNP transistor is made by sandwiching a negativetype semiconductor in between two positivetype semiconductors.
Transistor Operation
Amplifier
A transistor conducts current when the base voltage is greater than BE junction's voltage. Current is non zero.
Switch
A transistor conducts current when the base voltage is greater than BE junction's voltage. Current is non zero. This corresponds to a closed switch. A transistor does not conduct current when the base voltage is less than BE junction's voltage. Current is zero. This correspond to an open switch.
External links
FET
Field Effect Transistor
The most common transistors today are FETs (Field Effect Transistor). These transistors are characterized as having a conductance between source and drain dependent on the voltage applied between the gate and the source terminals. The dependence is linear if the gate to drain voltage is also high along with the gate to source voltage. It turns into a squarelaw relationship if the gate to drain voltage is not enough.
Current Voltage Characteristics
Disadvantages
One of the issues that comes up in circuit design is that as chips get smaller the insulator gets thinner and it starts to look like Swiss cheese. As a result the insulator starts acting like a conductor. This is known as leakage current. One solution is to replace the insulator by a material with a higher dielectric coefficent.
Two types: enhancement and depletion. Enhancement is the standard MOSFET, in which a channel must be induced by applying voltage. Depletion MOSFETs have the channel implanted, and applying voltage causes the channel to cease being conductive.
FET transistors respond to the difference in voltage bias between the gate and the source.
MOSFET (MetalOxideSemiconductor FET): standard FET
JFET (Junction Fet): When voltage is applied between the source and drain current flows. Current only stops flowing when a voltage is applied to the gate.
MESFET (MEtalSemiconductor FET): pn junction is replaced with Scottky junction. Not made with Silicon.
HEMT (High Electron Mobility Transfer): A MESFET
PHEMT (Pseudomorphic HEMT):
MOSFET
Complementary Metal Oxide Semiconductor
CMOS , Complementary Metal Oxide Semiconductor is not a type of transistor. It is a logic family, based on MOS transistors.
Construction & Operation
CMOS is made of two FETs blocking the positive and negative voltages. Since only one FET can be on at a time, CMOS consumes negligible power during any of the logic states. But when a transition between states occurs, power is consumed by the device. This power consumed is of two types.
Shortcircuit power
 For a very short duration, both transistors are on and a very huge current flows through the device for that duration. This current accounts for about 10% of the total power consumed by the CMOS.
Dynamic power
 This is due to charge stored on the parasitic capacitance of the output node of the device. This parasitic capacitance depends on the wire's area, and closeness to other layers of metal in the IC, besides the relative permittivity of the quartz layer separating consecutive metal layers. It also depends (to a much smaller extent) upon the input capacitance of the next logic gate. This capacitance delays the rise in the output voltage and hence the rise or fall in the output of a gate is more like a that in a resistorcapacitor (RC) network. Thus the dynamic power consumed due to switching action in one gate is given by:
Amplifiers
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Circuit Models for Amplifiers
Amplifier are generally put in four categories
 voltage amplifier,
 current amplifier,
 transresistance amplifier and
 transconductance amplifier.
These models apply irrespective of the complexity of the internal circuit of the amplifier. The values of the model parameters can be found either by analyzing the amplifier circuit or by performing measurements at the amplifier temlinals.
The model for a voltage amplifier is shown in figure 1. Real amplifiers have input and output resistance. This is reflected in the model.
 Figure 1: A Voltage Amplifier.
Gain
Gain is the increase in the strength of a signal and is often expressed in decibels (dB). An increase of 3 dB is about equal to doubling in a linear scale. A gain of more than 1 is called amplification, while a gain of less than 1 is called Attenuation.
Gain is given different symbols depending of the type. For No load gains
 Voltage gain is A_{vo},
 Current gain A_{io},
 Transconductance G_{m}
 Transresistance R_{m}.
Using the model, the gain with a load can be calculated.
Transistor amplifiers
Common Emitter
QUALITATIVE CHARACTERISTICS
 Current Gain: ..... HIGH
 Voltage Gain: ..... HIGH
 Power Gain: ..... HIGH
 Input Impedance: ... AVERAGE
 Output Impedance:... AVERAGE
QUANTITATIVE CHARACTERISTICS Input Resistance(base): Zb=β×re' > β: Current Gain (Ic/Ib), where 'Ic' is Colector DC current and 'Ib' is DC Base current; > re': BaseEmitter dynamic resistor (Ut/Ie), where Ut is thermal voltage(≈25mV at 25°C) and 'Ie' is DC emitter current; Input Resistance(general): Zg= Zb  R1  R2, where R1 and R2 are the same as the picture above.
Common Collector
QUALITATIVE CHARACTERISTICS Current Gain: ..... HIGH Voltage Gain: ..... ≈1 Power Gain: ..... LOW Input Resistance: ... HIGH Output Resistance:... LOW
Common Base
QUALITATIVE CHARACTERISTICS Current Gain: ..... ≈1 Voltage Gain: ..... HIGH Power Gain: ..... AVERAGE Input Resistance: ... LOW Output Resistance:... HIGH
FET Configurations
As with BJT configurations, there are three FET configurations, each one corresponding to one of the terminals of the transistor.
Common Source
Common Drain
Common Gate
Biased Voltage Class
Transistors may be biased in a variety of classes. A trade off of linearity and power consumption is usually made where a Class A
Class A
The transistor is "on" all the time. We say 360 degrees of conduction, representing an entire period of the sine waveform. Ideally, this class produces very little distortion, however consumes a lot of power and is also least preferred.
This is the most linear of the classes, meaning the output signal is a truer representation of what was imputed. Here are the characteristics of the class: 1.The output device (transistor) conducts electricity for the entire cycle of input signal. In other words, they reproduce the entire waveform in its entirety. 2.These amps run hot, as the transistors in the power amp are on and running at full power all the time. 3.There is no condition where the transistor(s) is/are turned off. That doesn't mean that the amplifier is never or can never be turned off; it means the transistors doing the work inside the amplifier have a constant flow of electricity through them. This constant signal is called "bias". 4.Class A is the most inefficient of all power amplifier designs, averaging only around 20%.
Because of the way they are usually designed, Class A amplifiers are very inefficient. They do not need to be built in this way, but for every watt of output power, they can typically waste 45 watts as heat. They are usually very large, heavy and because of the 45 watts of heat energy released per watt of output, they run very hot, needing lots of ventilation (not at all ideal for a car, and rarely acceptable in a home). This is not due to the amplifier type, but the fact that even many design engineers do not fully understand how they work and often copy their designs from previous ones. The upside is that these amps are the most enjoyed of all amplifiers. When properly designed, the Class A amp gives the best representation of musical detail, with no clipping of the waveform. As a result the sound is cleaner and more linear; that is, it contains much lower levels of distortion.
They are the most accurate of all amps available, but at greater cost to manufacture, calling for tight tolerances and additional components for cooling and heat regulation.
Class AB
The transistor is "on" for slightly more than half the cycle (>180 degrees) of a sine wave and is the most common configuration used in pushpull audio power amplifiers. In pushpull amplifiers, Class AB produces mostly odd order distortion, however it is far more power efficient than Class A. Odd order distortion is not considered pleasing to hear in audio power amplifiers. This distortion can easily be removed with the addition of a simple negative feedback loop into the system as shown by the diagram below:
This type of amplifier is extremely easy to build and is the industry standard for audio amplifiers.
Issues with this type of amplifier include poor efficiency, size and cost. A typical class ab amplifier will have a power efficiency of 4080%. Because of this they require large heat sinks to cool the transistors, this also increases the cost of the amplifier due to the extra material to create these heat sinks.
This is the compromise of the bunch. Class AB operation has some of the best advantages of both Class A and Class B builtin. Its main benefits are sound quality comparable to that of Class A and efficiency similar to that of Class B. Most modern amp designs employ this topology.
Its main characteristics are: In fact, many Class AB amps operate in Class A at lower output levels, again giving the best of both worlds The output bias is set so that current flows in a specific output device for more than a half the signal cycle but less than the entire cycle. There is enough current flowing through each device to keep it operating so they respond instantly to input voltage demands. In the pushpull output stage, there is some overlap as each output device assists the other during the short transition, or crossover period from the positive to the negative half of the signal.
There are many implementations of the Class AB design. A benefit is that the inherent nonlinearity of Class B designs is almost totally eliminated, while avoiding the heatgenerating and wasteful inefficiencies of the Class A design. And as stated before, at some output levels, Class AB amps operate in Class A. It is this combination of good efficiency (around 50) with excellent linearity that makes class AB the most popular audio amplifier design.
There are quite a few excellent Class AB amps available. This is the design I recommended for most generaluse applications in home and car. Usually, parts choice rivals that of Class A amps, and dollar for dollar these are some of the best values in stereo amplification. There can be some variation in design principle, but generally these are welldesigned amps since their function is very wellunderstood by audio designers.
Class B
The transistor is "on" for only half the cycle (exactly 180 degrees) of a sine wave and is also very typically used in pushpull amplifier circuits. Ideally this class produces mostly odd order distortion. In audio applications it is believed that odd order distortion is not pleasing to hear. It is difficult to build a low distortion Class B amplifier and hence Class AB is almost universal.
In this amp, the positive and negative halves of the signal are dealt with by different parts of the circuit. The output devices continually switch on and off. Class B operation has the following characteristics: The input signal has to be a lot larger in order to drive the transistor appropriately. This is almost the opposite of Class A operation There have to be at least two output devices with this type of amp. This output stage employs two output devices so that each side amplifies each half of the waveform. [li Either both output devices are never allowed to be on at the same time, or the bias (remember, that trickle of electricity?) for each device is set so that current flow in one output device is zero when not presented with an input signal. Each output device is on for exactly one half of a complete signal cycle.
These amps run cooler than Class A amps, but the sound quality is not as pure, as there is a lot of "crossover" distortion, as one output device turns off and the other turns on over each signal cycle.
This type of amplifier design, or topology, gives us the term "pushpull," as this describes the tandem of output devices that deliver the audio signal to your speakers: one device pushes the signal, the other pulls the signal. They can be less expensive, because one can use two cheap output devices instead of one highquality one in the design.
As I mentioned before, the input signal has to be lot larger, meaning that from the amplifier input, it needs to be "stepped up" in a gain stage, so that the signal will allow the output transistors to operate more efficiently within their designed specifications. This means more circuitry in the path of your signal, degrading sound even before it gets to the output stage
Class C
The transistor is "on" less than half the cycle of a sine wave. We say <180 of conduction. This class produces both even and odd order distortion, however is very efficient.
Class D
The class D amp has been developed after the shortcomings of past generations, including classes A, B, AB, and C. Many people mistake the D as standing for digital. Although it is a “switching” amp, meaning it turns “on” and “off” at a specific frequency, it is a wrong assumption. D was simply the next letter in the alphabet. Consuming the least power out of its previous generations, the class D amps are generally smaller, making them ideal for mobile devices. It is because of their power efficiency, small size, and cheaper costs that the class D amps are quickly becoming the new industry standard for audio electronic. Companies such as Advanced Analog, Texas Instruments as well as other companies have released 50W stereo class d amplifiers that are the size of a penny and did not require any sort of heat sinking, something that was not possible with other types of amplifiers.
The basic design includes two MOSFET transistors in series, one pFET and one nFET being driven by a pulse width modulated (PWM) signal. Because of the properties of MOSFET transistors they are either fully on or fully off. When the transistor is off and the current is zero (so the amount of power wasted heating up the transistor is zero), or the transistor is fully on and the voltage across it is very close to zero (so the amount of power wasted heating up the transistor is again, very close to zero).
Because an analog signal needs to be transformed into a PWM signal a certain amount of distortion can occur, but the amount of distortion can be minimized. Because a PWM signal is very much like a digital signal, the sampling theorem states that if the sampling frequency is more than half the maximum frequency of the source it can be reproduced exactly. For audio signals the maximum frequency heard by humans is roughly 20kHz, so a PWM generator would only need to provide a minimum switching frequency of 40kHz. Because of the availability of faster components many class d amplifier designers will use switching frequencies closer to 400kHz to further reduce the distortion.
Issues of concern with a class d amplifier include electromagnetic emissions. Due to the presence of a medium frequency signal in the circuit, steps must be taken to reduce the emission of these signals that could interfere with other electronic devices.
==== Class E ==== (oops, class D again?) Switching amplifier These amplifiers are erroneously called "digital" amplifiers by the press and many audio "experts." Here's the skinny on Class D: While some Class D amps do run in true digital mode, using coherent binary data, most do not. They are better termed "switching" amplifiers, because here the output devices are rapidly switched on and off at least twice for each cycle. Depending on their switching frequency, they may be "switched on" or "off" millions of times a second. Class D operation is theoretically 100% efficient, but in practice, they are closer to 8090% efficiency. This efficiency gain is at the cost of highfidelity.
Think of Class D amps as being similar to a switchable power supply, but with audio signals controlling, or modulating, the switching action. To do this, you use a technology called Pulse Width Modulation (or PWM, a technology found in many CD players).
According to experts, audio signals can be used to modulate a PWM system to create a high power audio amplifier at fairly low voltages using very small components. Class D audio uses a fixed, high frequency signal having pulses that vary in width based on input signal amplitude. So, for example, a deep bass note creates a large pulse in the carrier signal. This can be translated into a musical signal by the on/off nature of the output devices.
Class D amplifiers are generally used for nonhighfidelity, or subwoofer applications.
There is a fifth (and, nominally, a sixth) class of amplifier, but they are rarely seen in practice in the consumer market. One is the Class G and the other Class H. These are similar in design to Class AB topologies, but both feature two power supplies that switch on or off, depending on the musical signal imputed. Using two power supplies improves efficiency enough to allow significantly more power for a given size and weight. Class G is becoming common for pro audio designs. Class H amps are designed to use the same topology as Class G, but it provides just enough voltage for optimum operation of the output devices. Again, its an attempt to increase efficiency, but at the expense of fidelity ultimately.
Class F
Class S
Amplifier
Operational amplifiers
Electronics/Operational amplifiers
Analog multipliers
Analog multipliers
An analog multiplier is a circuit with an output that is proportional to the product of two inputs:
where K is a constant value whose dimension is the inverse of a voltage. In general we might expect that the two inputs can be both positive or negative, and so can be the output. Anyway, most of the implementations work only if both inputs are strictly positive: this is not such a limit because we can shift the input and the output in order to have a core working only with positive signals but external interfaces working with any polarity (within certain limits according to the particular configuration).
Two possible implementations will be shown. Both will be using operational amplifiers, but the first one will use diodes to get the needed relationships, the second one MOSFET transistors.
Diode Implementations
As known, using operational amplifiers and diodes it's quite easy to obtain the logarithm and the exponential of a certain input. Remembering the property of logarithms:
we can multiply two signals first calculating their logarithm, then summing them and finally calculating the exponential of such a sum. From the point of view of mathematics, such an approach works as long as the two inputs are positive, because the logarithm of a negative number does not exist (in the real domain). We'll see that this limit is valid for the actual circuit as well, even if the reason will be more "physical". The block diagram of this implementation is the following:
If we simply append the circuits for logarithm, sum and exponential we get the following configuration:
for a quick overview on the behavior of the circuit, we'll assume that all the resistors R have the same value. It is obviously possible to use different values to get different results, but we will not consider it here. Let us use the following notation for the relationship between current and voltage on a diode:
where is the threshold voltage and I_{s} is the current flowing through the diode if it's inversepolarized. If we analyze the circuit without introducing any approximation we get:
so the final output is:
as it is clear, in the output there is the multiplication we were looking for, but there is another term we don't want. It can't be simply considered an error because it might be as great as the multiplication element, so it has to be removed. Anyway this is an easy task, since it is necessary only to add another stage to sum exactly , so we will have no error. The complete multiplier circuit is the following:
where the output voltage is given by:
that's exactly what we wanted. The circuit works as long as the following relationship is verified:
so the inputs can be zero or sightly negative but, since will be a small voltage, we are allowed to rewrite the relation simply as . From the mathematical point of view this is due to the fact that we can't calculate the logarithm of a negative number, from a physical point of view the limit is due to the fact that we can obtain only very small currents (almost zero) inversepolarizing the diodes.
In practical applications, the diodes are replaced with BJTs connected so to work like a diode.
MOS implementation
Since it is possible to use a MOSFET transistor as a voltage controlled resistor, it is possible to use this feature to create an analog multiplier. Referring to the picture on the right  the letters indicate the different pins: Drain, Source and Gate. MOS devices are symmetrical, so the drain and source can be exchanged without affecting the behavior of the device. Use source as the lowest voltage terminal and drain as the higher voltage terminal. When the voltage between gate and drain exceeds the threshold voltage, i.e. , the relationship between current and voltage is the following:
assuming we can always use this relationship, the analog multiplier configuration is the following:
where source and drain of both devices are pointed out. If and are positive, then the sources will be at virtual ground by the operational amplifier. The current flowing through is clear: one side of the resistor has the voltage , the other one is at (virtual) ground. That same current will flow through the MOS , thus defining the voltage . The current is given by:
but and . replacing and calculating we get:
considering the other MOS we have:
where and . Replacing we get:
from which the output voltage is:
which is what is wanted. The difference between the previous configurations are:
 the MOS implementation is simpler and requires fewer devices
 in the calculations for the diode configuration we did not introduce any approximation, while the MOS configuration we did.
In other words, the diode implementation is more complicated but it works fine for a wider range on inputs.
Chapter 4: Digital Circuits
Digital Circuits
Overview
In Digital circuits there are two parts one is Digital the other is circuits.
Digital
 means a description of system which operates on digits or numbers.
 The numbers that a digital system operates upon are binary numbers.One binary digit just have two values '0' and '1'. This '0' or '1' are analogous to 'off' and 'on', 'false' and 'true', etc. The Binary digIT is known as 'bit'. In day to day activities we use numbers and every digit of that number can be between 0  9. We use combination of such digital to for numbers bigger than 9. Similarly we can use combination of bits, i.e. combination of 1 & 0, to form bigger binary numbers.
Circuit can be any electric circuit and Digital circuit is a circuit which is designed to implement the description of digital circuit.
Digital circuits are developed using a special type of mathematics called 'Boolean Algebra'.
Boolean Algebra
Boolean Algebra
Boolean Algebra was created by George Boole (1815  1864) in his paper An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities, published in 1854. It had few applications at the time, but eventually scientists and engineers realized that his system could be used to create efficient computer logic. The Boolean system has two states: True (T) or False (F). This can be represented in several different ways as on or off, one or zero, yes or no, etc. These states are manipulated by three fundamental operations called logical operators: AND, OR and NOT. These operators take certain inputs and produce an output based on a predetermined table of results. For example, the AND operator takes two (or more) inputs and returns an 'on' result only when both (or all) inputs are 'on'.
 In these tables T means "True", or "Yes", or 1 (in electronics), and
 F means "False", or "No" or 0 (in electronics).
 For example, the 1st line of the AND table says that if A is F and B is F then the result also is F. This is also said often as If A is 0 and B is also 0, then A AND B = 0 times 0 = 0
 Also at the bottom of the AND table 1 times 1 is 1.
 Looking at the OR table, use plus instead of times: 0 plus 0 is 0, where again 0 is F and 1 is T. Only 0 and 1 exist, therefore if the result of the addition is more than 1, then change it to 1.
 The 3rd table shows NEGATION: whatever it is, change it to the other; 0 is changed into 1, and 1 is changed into 0.



These simple operators are good because they allow us to create very simple logic circuits: if user put in a quarter AND the Coke button is pressed, drop a Coke
There are ways, however, to combine these expressions to make much more complex but useful digital circuits. By using multiple operators on the same inputs, it is possible to create much more complex outputs. An expression like: A and B or C would have a truth table of the following:
A  B  C  X 
0  0  0  0 
0  0  1  1 
0  1  0  0 
0  1  1  1 
1  0  0  0 
1  0  1  1 
1  1  0  1 
1  1  1  1 
This truth table follows the rules If A and B are true, or C is true, then X is true
Exercise: Go through each case in the table and check it using the above statement.
Formal Mathematical Operators
AND is represented by or that is A AND B would be or .
OR is represented by or that is A OR B would be and .
NOT is represented by or that is NOT A is or .
If these three operators are combined then the NOR and NAND can be created. So A NOR B is or . NAND is or . Other notations are valid as well, and many different combinations of them arise in mathematics and engineering. No universal standard has been agreed upon. However, most sources will make explicit the particular notation used.
Boolean Algebra Laws
Boolean Algebra, like regular algebra, has certain rules. These rules are Associativity, Distributivity, Commutativity and De Morgan's Laws. Associativity, Commutativity and Distributivity only apply to the AND and OR operators. Some of these laws may seem trivial in normal Algebra but in other algebras they are not.
Associativity
Associativity is the property of algebra that the order of evaluation of the terms is immaterial.
Or
Distributivity
Distibutivity is the property that an operator can be applied to the terms within the brackets.
Or
Commutativity
Commutativity is the property that order of application of an operator is immaterial.
Or
De Morgan's Law
De Morgan's Law is a consequence of the fact that the not or negation operator is not distributive.
Or
Notes
It is important to note that
Or
This can be seen as either AND having a higher precedence or the fact that Associativity does not hold between AND and OR or that it is an invalid application of distributivity.
Another way of looking at this is the possible application of our understanding of normal algebra rules using the second notation. Where clearly the analogy between OR being addition and AND being multiplication is made. We would never make this error if this were high school algebra.
Rules
All of these Laws Result in a number of rules used for the reduction of Boolean Algebra.