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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 "how-to" aspects that aren't covered in an encyclopedia. The book will focus on how to use the components to build useful circuits.


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Prerequisite Topics


Most important / Required knowledge

Moderately Important / Aids in comprehension

Slightly Important / Related or helpful


Multivariable Calculus


Physics in Electronics
Electricity and Magnetism

Other Useful Topics

International System of Units


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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 all-time 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 college-level 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

Materials which contain movable charges that can flow with minimal resistance.
Materials with few or no movable charges, or with charges which flow with extremely high resistance.
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.

  1. 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".
  2. 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 non-metals and are good at blocking the flow of electrons.
  3. 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.



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 (Electro-Magnetic 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.



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 built-up 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 multi-trillions 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 6.2415×1018 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 q1 and q2 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 = 8.9875×109 N·m2/C2 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, Q1, Q2, Q3, through Qn, on the charge q in question. Each of the Q1 through Qn 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 Qi charge would give the magnitude of the electric force exerted by each of the Qi 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.


  • College Physics Volume 2 by Doug Davis, Saunders College Publishing, Orlando, FL, 1994

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Voltage, Current, and Power

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Basic Understanding


Experiments show us that electric point charges attract or repel as calculated by Coulomb's law. Integrating (summing) over a distribution of points charges as they are assembled into a specific system configuration allows us to determine a scalar value defined as the electrical potential or electric field of a specific point. This mathematical definition is very useful in electronics circuit theory.


The potential difference between two test points resulting from the distribution of charge in the circuit, usually measured in volts.


Net amount of charge (coulombs) (number of electrons x electron charge) flowing past a specified point during a time interval (seconds), usually measured in Amps (1 Amp = 1 coulomb / 1 second). In typical components and systems the quantity of electrons is quite large and the aggregate charge flow is referred to as electricity.


Energy given in a certain amount of time, usually measured in watts.

Law of Charges

Opposite charges attract while similar charges repel.


When electricity passes through a wire it creates a moving magnetic field around the wire. The typical unit of measure is Henrys.


When electric fields or charge distributions are created in a physical system that stores recoverable energy, characteristics of the physical components which affect calculation of the electrical quantities are defined as capacitance. The base unit of measure is Farads, however microfarads (μF), are used much more often.


When potential difference creates movement of electrons between two points, some of the potential energy formerly available in the system is irreversibly transferred from the electric field or the electrons moving through the component via collisions with atoms and molecules within the material. Ohm's Law, V=IR, defines resistance as R=V/I where V is the voltage difference applied across the component, I is the resulting current flow in Amps, and R is a constant created by characteristics of the component which is calculated from the measured voltage loss of the measured current passing through the component.

Electric Field


A charged particle such as a proton or electron may "feel" an electrical force on it in a certain environment. This force is typically due to the presence of other charges nearby. The force will have a direction and magnitude, and can be represented by a vector. (A vector is simply a quantity that represents the direction and magnitude of something.) The magnitude of the force depends on the charge of the particle, the charge on the particles around it, and how close or far away they are: 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. The direction of the force depends on the location of the surrounding charges.

In describing the electrical environment at that location , it is said there is an electric field at that location. The electric field is defined as the force that a single unit of charge would feel at that location. In some systems of measurement, the unit of charge is the charge of a single proton; in others it is the coulomb. A coulomb is the charge of 6.24×1018 protons

The relationship between force and electric field for a single charged particle is given by the following equation:

The bold letters indicate vector quantities. This means that a charge q, in an electric field E, having a certain direction and a magnitude E, would have a force F on it, in the same direction and with a magnitude F. Considering only the magnitudes, the following would result from the definition.

    E = F/q    these are all magnitudes or numerical quantities

The net electric field E, at a location is due to the presence of all other charges nearby, similar to the net electric force F, if there was a charge q at that location. The contribution of one of these other charges to the total (or net) electric field is a vector E contribution, which for a point charge can be derived from Coulomb's Law. Distributions of charge density in various shapes may also yield vector E contributions to the total electric field, to be added in as vector quantities. Practically speaking, most electricians, electrical engineers, and other electrical circuit builders and hobbyists seldom do these sorts of electric field calculations. Electric field calculations of this sort are more of a theoretical physics or special applications problem, so these calculations are omitted here in favor of more applicable material. See Electric Field for such information on electric field formulas.

There is an electrical force on a charge only if there is a charge subject to the force at a location in an electric field. However, even if there is no such charge subject to the force, there could still be an electric field at a point. This means that an electric field is a property of a location or point in space and its electrical environment, which would determine what a charge q would "feel" if it were there.



Now, a micro-physics review: Work is causing displacement (or movement) of an object or matter against a force. Energy is the ability to perform work like this. Energy can be kinetic energy or potential energy. Kinetic energy is the energy a mass has because it is moving. Potential energy in an object, in matter, in a charge or other situation has the ability to perform work or to be converted into kinetic energy or a different kind of potential energy.

A reason why a particle or a charge may have potential energy could be because it is located at a point in a force field, such as a gravitational field, electric field, or magnetic field. In the presence of such a field, gravity or electric or magnetic forces could cause the particle or charge to move faster or move against resistive forces, representing a conversion of potential energy to kinetic energy or work. The amount of potential energy it has would depend on its location. Moving from one location to another could cause a change in its potential energy.
For example, an object near the surface of the earth placed high would have a certain amount of gravitational potential energy based on its mass, location (height or altitude) in and strength of the Earth's gravitational field. If the object were to drop from this location (height) to a new lower location, at least some of its gravitational potential energy would be converted to kinetic energy, resulting in the object moving down. The difference in gravitational potential energy could be calculated from one location to another, but determining the absolute potential energy of the object is arbitrary, so ground level is chosen arbitrarily as the height where its gravitational potential energy equals zero. The potential energy at all other heights is determined from the mass of the object, location relative to the ground level, and strength of the gravitational field.

All energy values are numerical or scalar quantities, not vectors.

Electric Potential Energy


Somewhat similarly, a charged particle at a certain point or location in an electrical environment (i. e. an electric field) would have a certain amount of electric potential energy based on its charge, location, and the electric field there, which could be based on quantity and locations of all other charges nearby. If the charge were to move from this location to a new location or point, it could cause a change in its electric potential energy. This difference in electric potential energy in the charge particle would be proportional to its charge and it could be an increase or a decrease. From measurements and calculations, one may be able to determine this difference in electric potential energy, but coming up with an absolute figure for its potential is difficult and typically not necessary. Therefore, in a manner somewhat similar to gravitational potential energy, an arbitrary location or point nearby, often somewhere in the electric circuit in question, is chosen to be the point where the electric potential energy would be zero, if the charge were there. Often the wiring, circuit, or appliance will be connected to the ground, so this ground point is often chosen to be the zero point. The electric potential energy at all other points is determined relative to the ground level. The SI unit of electric potential energy is the joule.

Electric Potential


Because the electric potential energy of a charged particle (or object) is proportional to its charge and otherwise simply dependent on its location (point where it's at), a useful value to use is electric potential. Electric potential (symbolized by V) at a point is defined as the electric potential energy (PE) per unit positive charge (q) that a charge would have at that given point (location). At a point a, the electric potential at a is given by:

                     Va = (PE of charge at a)/q

Somewhat analogously to an electric field, electrical potential is a property of a location and the electrical conditions there, whether or not there is a charge present there subject to these conditions. On the other hand, electric potential energy is more analogous to electric force in that for it to be present, there should be a subject charged particle or object which has that energy. Electric potential is often simply called potential by physicists. Because the SI unit of electric potential energy is the joule and because the SI unit of charge is the coulomb, the SI unit for electric potential, the volt (symbolized by V), is defined as a joule per coulomb (J/C).

Because electric potential energy is based on an arbitrary point where its value is set at as zero, the value of electric potential at a given point is also based on this same arbitrary zero point (reference point where the potential is set at zero). The potential at a given point a is then the difference between potentials from point a to the zero point, often called a ground node (or just ground).

Calculations of electric potential energy or electric potential based on Coulomb's Law are sometimes theoretically possible, such as might be for electric field calculations, but again these are of mostly theoretical interest and not often done in practical applications. Therefore, such calculations are also omitted here in favour of more applicable material.

Often it is of interest to compare the potentials at two different points, which we may call point a and point b. Then the electric potential difference between points a and b (Vab) would be defined as the electric potential at b minus the electric potential at a.

                        Vab = Vb - Va

The unit for electric potential difference is the volt, the same as for electric potential. Electric potential difference is often simply called potential difference by physicists. Under direct current (DC) conditions and at any one instant in time under alternating current (AC), potential and potential difference are numerical or scalar quantities, not vectors, and they can have positive or negative values.



Voltage is electric potential expressed in volts. Similarly, potential difference expressed in volts is often called voltage difference or often referred to as voltage across two points or across an electrical component. The terms electric potential, potential, and potential difference are terms more often used by physicists. Since these quantities are almost always expressed in volts (or some related unit such as millivolts), engineers, electricians, hobbyists, and common people usually use the term voltage instead of potential. Furthermore, in practical applications, electrical force, electric field, and electrical potential energy of charged particles are not discussed nearly as often as voltage, power, and energy in a macroscopic sense.

Additional note: The following explains why voltage is "analogous" to the pressure of a fluid in a pipe (although, of course, it is only an analogy, not exactly the same thing), and it also explains the strange-sounding "dimensions" of voltage. Consider the potential energy of compressed air being pumped into a tank. The energy increases with each new increment of air. Pressure is that energy divided by the volume, which we can understand intuitively. Now consider the energy of electric charge (measured in coulombs) being forced into a capacitor. Voltage is that energy per charge, so voltage is analogous to a pressure-like sort of forcefulness. Also, dimensional analysis tells us that voltage ("energy per charge") is charge per distance, the distance being between the plates of the capacitor. (More discussion is on page 16 of "Industrial Electronics," by D. J. Shanefield, Noyes Publications, Boston, 2001.)



When an electric circuit is operating in Direct Current (DC) mode, all voltages and voltage differences in the circuit are typically constant (do not vary) with time. When a circuit is operating under Alternating Current (AC) conditions, the voltages in the circuit vary periodically with time; the voltages are a sinusoidal function of time, such as V(t) = a sin (b t) with constant a and b, or some similar function. The number of times the period repeats (or "cycles") per unit time is called the frequency of V(t). Under DC conditions or at any one instant in time under AC, potential (or voltage) and potential difference (or voltage difference) are numerical or scalar quantities, not vectors, and they can have positive or negative values. However, in AC mode, the overall function of voltage with time V(t), can be expressed as a complex number or a phasor for a given frequency. The frequency can be expressed in cycles per second or simply sec-1, which is called Hertz (Hz) in SI units. Typical commercial electric power provided in the United States is AC at a frequency of 60 Hz (50 Hz for India).



Ground is shown on electronics diagrams, but it isn't really a component. It is simply the node which has been assigned a voltage of zero. It is represented by one of the symbols below. Technically, any single node can be assigned as ground, and other voltages are measured relative to it. However, the convention is to only assign it in one of two ways, related to the type of power supply. In a single supply situation, such as a circuit powered by a single battery, the ground point is usually defined as the more negative of the power source's terminals. This makes all voltages in the circuit positive with respect to ground (usually), and is a common convention. For a split-supply device, such as a circuit driven by a center-tapped transformer, usually the center voltage is defined as ground, and there are equal and roughly symmetrical positive and negative voltages in the circuit.

Signal ground
Ground for a signal. Since wires have a certain amount of resistance to them, ground points in a circuit aren't all at exactly the same voltage. It is important in practical circuit design to separate the power supply ground from the signal ground from the shielding ground, etc. In circuits where minimum noise is especially important, power regulator circuitry should have thick wires or traces connecting the grounds, in a sequence from the power supply to the "cleanest" ground at the output of the filters of the power supply, which will then be a "star point" for the grounds of the signal circuitry.
Chassis ground
A direct connection to the chassis of the device. This is used for EMI shielding and also for safety ground in line AC powered devices.
Earth ground
Used in radio or power distribution systems, a connection to the earth itself. Also the other end of the connection for the safety ground, since the power line voltage will seek a path through the earth back to the power line supply station. This was the original usage of the word "ground", and the more modern meaning of the word would have been called a "floating ground".

The earth ground symbol and signal ground symbol are often interchanged without regard to their original meanings. As far as signal-level electronics (and this book) is concerned, ground almost always means a signal ground or floating ground, not connected to the earth itself.

(see: http://www.prosoundweb.com/studyhall/rane/grounding/ground2.php)



Electric current, often called just current, is the movement of charge in a conductor (such as a wire) or into, out of, or through an electrical component. Current is quantified as a rate of positive charge movement past a certain point or through a cross-sectional area. Simply put, current is quantified as positive charge per unit time. However, since current is a vector quantity, the direction in which the current flows is still important. Current flow in a given direction can be positive or negative; the negative sign means that positive charges move opposite of the given direction. The quantity of current at a certain point is typically symbolized by a capital or small letter I with a designation which direction the current I is moving. The SI unit of current is the ampere (A), one of the fundamental units of physics. See ampere for the definition of ampere. Sometimes, ampere is informally abbreviated to amp. The definition of a coulomb (C), the SI unit of charge, is based on an ampere. A coulomb is the amount of positive charge passing a point when a constant one ampere current flows by the point for one second. The second is the SI unit of time. In other words, a coulomb equals an ampere-second (A·s). An ampere is a coulomb per second (C/s).

Conventional Current


Typically, current is in a metal and constitutes movement of electrons which have negative charge; however, people initially thought that current had a positive charge. The result is that even though current is the flow of negative electrons and flows from the negative to the positive terminal of a battery, when people do circuit analysis they pretend that current is a flow of positive particles and flows from the positive to the negative terminal of a battery (or other power source). Actually, it is more complicated than this, since current can be made up of electrons, holes, ions, protons, or any charged particle. Since the actual charge carriers are usually ignored when analyzing a circuit, current is simplified and thought of as flowing from positive to negative, and is known as conventional current.

Analogy to pebble tossing: I have pebbles and I am throwing them into a basket. In doing this the basket gains pebbles and I lose pebbles. So there is a negative current of pebbles to the basket because it is gaining pebbles, and there is a positive current of pebbles to me because I am losing pebbles. In pebble tossing the currents have equal strength but in opposite directions.

Current I is represented in amperes (A) and equals x number of y



Power is energy per unit of time. The SI unit for power is the watt (W) which equals a joule per second (J/s), with joule being the SI unit for energy and second being the SI unit for time. When somebody plugs an appliance into a receptacle to use electricity to make that appliance function, that person provides electrical energy for the appliance. The appliance usually functions by turning that electrical energy into heat, light, or work — or perhaps converts it into electrical energy again in a different form. If this situation is ongoing, it is said that the receptacle or electric power company delivers power to the appliance. The current from the receptacle going in and out of the appliance effectively carries the power and the appliance absorbs the power.

Multiplying a unit of power by a unit of time would result in a unit that represents a quantity of energy. Therefore, multiplying a kilowatt by an hour gives a kilowatt-hour (kW·h), a unit often used by electrical power companies to represent an amount of electrical energy generated or provided to consumers. Oddly, in marketing and retail packaging, portable batteries are often rated in "mAh", which are normally units of power, but do not identify how much power the battery can supply. Rather, the mAh rating is always at the implied voltage defined by the battery's chemistry. For example, a common, premium quality 18650 Li-ion battery would be advertised/rated 3400 mAh, at most. Even though "3400 mAh" is in power units, Li-ion chemistry battery voltage is typically around 3.7 volts, and this does identify the battery as one that should be able to store about 3400 mAh x 3.7 volts = 12.6 mWh (milliwatt hours) of energy per charge.) Also oddly, in marketing and retail packaging, a larger (e.g. vehicle) battery is often advertised based on the "Amps" or "Cold Cranking Amps" (CCA) the battery can provide to indicate the maximum power it can deliver. Although Amps are not power units, these ratings do represent the amount of power that can be output.

For direct current (DC), power P can be calculated by multiplying the voltage and current, when they are known.

                     P = V I

Note that energy/charge is multiplied by charge/time to give energy/time. At any one point in time t in alternating current (AC) circuitry, power P(t) equals voltage V(t) times current I(t).

         P(t) = V(t) I(t) at any one time t

Calculations of AC power averaged over time will be discussed under AC power.



An electronic circuit is a system in which conventional current flows from the positive terminal of a source, through a load, to the negative terminal of the source. But the current will only flow when there is a closed path from the positive to negative terminal. If there is a discontinuity or an open circuit in its path, the current will not flow and hence the circuit will be non functional. The current does not flow since the open circuit acts like an infinite resistance.

Short Circuit


A short circuit is another name for a node, although it usually means an unintentional node. Has current through it but no voltage across it.

Open Circuit


Has potential across it but no current through it.

Properties of wires


Theoretical circuit connection (wire) has no resistance or inductance. Real wires always have voltage over them if there is current flowing through them (resistance). On high frequencies there are measurable voltage potentials over wire links if there is flowing alternating current through wires (inductance like in inductors).

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.



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. Non-Metals like Wood and Rubber do not conduct electricity so easily. Therefore, they are used to make Non-Conductors.

Semi Conductor

Matter that conducts electricity in a manner between that of Conductors and Non-Conductors. For example, Silicon (Si) and Germanium (Ge) conduct electricity better than non-conductors 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


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


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 is defined as the ratio of current over voltage denoted as Y measured in mho
or siemens (S)


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 enT

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 PV and the electric loss energy is PR Then the electric energy delivered:

P = PV - PR

Black Body Radiation


Further experience with conductors that conduct electricity . It is observed that all conductors that conduct electricity exhibit

  1. Change in Temperature
  2. Release Radiant Heat Energy into the surrounding



Connect a conductor with an electric source in a closed loop . Plot the value I at different f to have a I - f diagram



for f<fo

Current increasing with increasing f .
Radiant heat is a wave travels at velocity v = λf carries energy E = m v2 .

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


  1. All conductor that conducts Electricity has a threshold frequency fo
  2. The Radiant Heat Energy is a Light Wave of dual Wave Particle characteristic. Sometimes it behaves like Particle, sometimes it behaves like Wave
  3. 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


Electronics | Foreword | Basic Electronics | Complex Electronics | Electricity | Machines | History of Electronics | Appendix | edit



A cell, or electrochemical cell, Galvanic cell or Voltaic cell, is a component used to store and provide electrical energy. The energy in a cell is stored as a chemical potential energy, and electrical energy can be extracted when the chemicals undergo a chemical reaction. A battery is the term used to refer to multiple cells that are used together as one unit, and can be connected in series, parallel or in a combination of series and parallel connections.

The properties of the cell can be determined by the chemistry of the cell (i.e. what it is made up of), and its physical design. Different chemistries will affect cell voltage, energy density, internal resistivity, and other factors. Physical design will affect the total cell energy capacity, total internal resistance and the shape and size of the cell. Generally, increasing the size of a cell, while maintaining the cell chemistry, will result in a greater energy capacity and lower internal resistance, and consequently a greater current supplying capability, and vice versa.

Cells are categorised as Primary (Dry) cells, and Secondary (Wet) cells, based on their chemistry. Primary cells are single-use and disposable, for example alkaline cells. Secondary cells are rechargeable, for example lead-acid 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.


Electronics | Foreword | Basic Electronics | Complex Electronics | Electricity | Machines | History of Electronics | Appendix | edit



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 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 op-amp can have a resistance near 1012 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 Pv and Electric Energy Loss is Pr 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.

2003 means 200×103 = 200kΩ
600 means 60×100 = 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:


ρ is the electrical resistivity (resistance to electricity) of the material,
L is the length of the material
A is the cross-sectional 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 ohm-meter 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 cross-sectional 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 cross-sectional area decreases; the resistance, multiplied by the area, will be a constant. If the cross-sectional 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 (Ω-m).

Conductors Ω-m (Ohm-meter)
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 nickel-chromium alloy commonly used in heating elements)
Graphite ~10-6
Carbon 3.5×10-5
Pure Germanium 0.6
Pure Silicon 640
Common purified water ~103
Ultra-pure water ~105
Pure Gallium Arsenide ~106
Diamond ~1010
Glass 1010 to 1014
Mica 9×1013
Rubber 1013 to 1016
Organic polymers ~1014
Sulfur ~1015
Quartz (fused) 5 to 75×1016
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 non-linear 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.



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 cross-sectional area remains 1. However, if you were to make connections on the sides, the exact opposite would be true: the cross-sectional area would be 4 and the length 1, resulting in 0.25 ohms total resistance.

Cross-Sectional 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 cross-sectional 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 cross-sectional 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 cross-sectional 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.)



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 Ω-meters. Lastly, we calculate the cross-sectional area of the conductor, which is 1 cm × 1 cm = 1 cm2 (0.0001 m2). Then, we put it all in the formula, converting cm to m:

units m2 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
Temperature-sensitive 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.


  • 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 RA is placed between the input voltage and the output node, and the resistor RB 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 ROUT = RA||RB. 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
  • Pull-up / Pull-down: 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 side-effects 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 (pull-down) or a high voltage (pull-up) 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 R1 and R2 of an op-amp inverting amplifier) supplied by two voltages (VIN and -VOUT) having opposite polarities. The two voltage sources "pull" the output point in opposite directions; as a result, if R2/R1 = -VOUT/VIN, the point becomes a virtual ground. Placement of a pull-up or pull-down 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 current-to-voltage 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 current-to-voltage converter where the resistor may have a significant resistance as an op-amp compensates the "undesired" voltage drop across it (unfortunately, this remedy may be applied only in low-current 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.



Resistors are available as pre-fabricated, real-world components. The behavior of such components deviates from an ideal resistor in certain ways. Therefore, real-world 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.



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 E-series 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 E-series are also available, e.g. for high-precision 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 E-series, 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 E-series standard. Although typically at a lower price.

Series: Resistors that combine in series add the nominal tolerances together.

Example: For two resistors in series RA = 1.5kΩ±130Ω and RB = 500Ω±25Ω, the tolerance is 130Ω + 25Ω, resulting in a final resistor value RT = 2kΩ±155Ω.

Parallel: Resistors that combine in parallel have a combined tolerance that is slightly more complex.

Example: For two resistors in parallel RA = 1.5kΩ±130Ω and RB = 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.



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 point-to-point 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.






Further Readings



Electronics | Foreword | Basic Electronics | Complex Electronics | Electricity | Machines | History of Electronics | Appendix | edit

General remarks

Some different capacitors for electronic equipment

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.



Theory of conventional construction

Model of a capacitor

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

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

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

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

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

Theory of electrochemical construction

Scheme on double layer on electrode (BMD model).
1. IHP Inner Helmholtz Layer
2. OHP Outer Helmholtz Layer
3. Diffuse layer
4. Solvated ions
5. Specifically adsorptive ions (Pseudocapacitance)
6. Solvent molecule.

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 so-called electrochemical capacitors. In contrast to ceramic, film and electrolytic capacitors, supercapacitors, also known as electrical double-layer capacitors (EDLC) or ultracapacitors do not have a conventional dielectric. The capacitance value of an electrochemical capacitor is determined by two high-capacity 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 (double-layer capacitance) and the
  • electrochemical storage achieved by a faradaic electron charge-transfer 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 double-layer by itself.[2]



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)








Reactance is defined as the ratio of Voltage over Current



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




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



Below is a simple RC Circuit:

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

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

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

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

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

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

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

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

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

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

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:

So this means that 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 most-used 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:
    • Double-layer capacitors were named for the physical phenomenon of the Helmholtz double-layer
    • Pseudocapacitors were named for their ability to store electric energy electro-chemically with reversible faradaic charge-transfer
    • Hybrid capacitors combine double-layer and pseudocapacitors to increase power density
  • Seldom-used Silver mica, glass, silicon, air-gap 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.

Overview over the most commonly used fixed capacitors in electronic equipment
Overview over the most commonly used fixed capacitors in electronic equipment

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, DC-link 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 built-in capacitors with metal conductive areas in different layers of a multi-layer printed circuit board and kludges such as twisting together two pieces of insulated wire also exist.


Principle charge storage of different capacitor types and their inherent voltage progression

The most common dielectrics are:

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

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. Helmholtz-layer thickness are given in the table below.

Key parameters[3][4][5][6][7]
Capacitor style Dielectric Permittivity
at 1 kHz
dielectric strength
Minimum thickness
of the dielectric
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
9,6[8] 710 < 0.01 (6.3 V)
< 0.8 (450 V)
Tantalum electrolytic capacitors Tantalum pentoxide
26[8] 625 < 0.01 (6.3 V)
< 0.08 (40 V)
Niobium electrolytic capacitors Niobium pentoxide,
42 455 < 0.01 (6.3 V)
< 0.10 (40 V)
Double-layer capacitors
Helmholtz double-layer - - < 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. Razor-thin 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 vs. voltage ranges of different capacitor types
Capacitance ranges vs. voltage ranges of different capacitor types

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.


Capacitor volumetric efficiency increased from 1970 to 2005 (click image to enlarge)

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.

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.

Comparing the three main capacitor types it shows, that a broad range of overlapping functions for many general-purpose and industrial applications exists in electronic equipment.
Comparing the three main capacitor types it shows, that a broad range of overlapping functions for many general-purpose and industrial applications exists in electronic equipment.

Capacitor - types and styles


Ceramic capacitors

Construction of a Multi-Layer Ceramic Capacitor (MLCC)

A ceramic capacitor is a non-polarized 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, by-pass 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 well-suited 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] feed-through capacitors for noise suppression by low-pass filters[12] and ceramic power capacitors for transmitters and HF applications.[13][14]

Features and applications as well as disadvantages of ceramic capacitors
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, by-pass 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

Three examples of different film capacitor configurations for increasing surge current ratings

Film capacitors or plastic film capacitors are non-polarized 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 self-healing 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]

Characteristics of plastic film materials for film capacitors
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 RInsul•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/mm3) 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 motor-run applications[19]

Features and applications as well as disadvantages of film capacitors
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 self-healing 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 self-healing properties: internal short may be disabling. Larger dimensions than metallized alternative.
Polypropylene (PP) film capacitors Polypropylene
Most popular film capacitor dielectric. Predictable linear and low capacitance change with operating temperature. Suitable for applications in Class-1 frequency-determining 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 over-voltages or voltage reversals than oil-impregnated MKV-capacitors 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 semi-critical 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 non-critical 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 lead-free 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 mission-critical 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

MKV power capacitor, double-sided metallized paper (field-free mechanical carrier of the electrodes), polypropylene film (dielectric), windings impregnated with insulating oil

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 volt-amps.

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 field-free 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]

Features and applications as well as disadvantages of film-based power capacitors
Capacitor type Dielectric Features/applications Disadvantages
Metallized paper power capacitors Paper impregnated with insulating oil or epoxy resin Self-healing properties. Originally impregnated with wax, oil or epoxy. Oil-Kraft 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 self-healing. Potential catastrophic failure due to high stored energy.
PP dielectric,
field-free paper
power capacitors
(MKV power capacitors)
Double-sided (field-free) metallized paper as electrode carrier. PP as dielectic, impregnated with insulating oil, epoxy resin or insulating gas Self-healing. 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 double-sided
metallized PP power capacitors
PP as dielectric, impregnated with insulating oil, epoxy resin or insulating gas Highest capacitance per volume power capacitor. Self-healing. Broad range of applications such as general-purpose, 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 self-healing.

Electrolytic capacitors


Electrolytic capacitors have a metallic anode covered with an oxidized layer used as dielectric. The second electrode is a non-solid (wet) or solid electrolyte. Electrolytic capacitors are polarized. Three families are available, categorized according to their 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.

Some important values of the different electrolytic capacitors
Anode material Electrolyte Capacitance
Max. rated
at 85 °C
ripple current
(roughned foil)
non solid,
e.g. Ethylene glycol,
0.1–2,700,000 600 150 0.05–2.0
Manganese dioxide
0.1–1,500 40 175 0.5–2.5
conductive polymere
(e.g. Polypyrrole)
10–1,500 25 125 10–30
(roughned foil)
non solid
Sulfuric acid
0.1–1,000 630 125
non solid
sulfuric acid
0.1–15,000 150 200
Manganese dioxide
0.1–3,300 125 150 1.5–15
conductive polymere
(e.g. Polypyrrole)
10–1,500 35 125 10–30
Manganese dioxide
1–1,500 10 125 5–20
conductive polymere
(e.g. Polypyrrole)
2.2–1,000 25 105 10–30
1) Ripple current at 100 kHz and 85 °C / volumen (nominal dimensions)

The large capacitance per unit volume of electrolytic capacitors make them valuable in relatively high-current and low-frequency 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 low-frequency 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 Non-Polarized 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]

Features and applications as well as disadvantages of electrolytic capacitors
Capacitor type Dielectric Features/applications Disadvantages
Electrolytic capacitors
with non solid
(wet, liquid)
Aluminum dioxide
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 by-pass, coupling, smoothing and buffer applications in power supplies and DC-links. 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 end-of-life, showing failures like "capacitor plague"
Tantalum pentoxide
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
Tantalum pentoxide
Niobium pentoxide
Tantalum and niobium with smaller dimensions for a given capacitance/voltage vs aluminum. Stable electrical parameters. Good long-term high temperature performance. Lower ESR lower than non-solid (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
Tantalum pentoxide
Niobium pentoxide
Greatly reduced ESR compared with manganese or non-solid (wet) elelectrolytics. Higher ripple current ratings. Extended operational life. Stable electrical parameters. Self-healing.[26] Used for smoothing and buffering in smaller power supplies especially in SMD. Polarized. Highest leakage current among electrolytics. Higher prices than non-solid or manganese dioxide. Voltage limited to about 100 V. Explodes when voltage, current, or slew rates are exceeded or under reverse voltage.


Hierarchical classification of supercapacitors and related types
Ragone chart showing power density vs. energy density of various capacitors and batteries
Classification of supercapacitors into classes regarding to IEC 62391-1, IEC 62567and DIN EN 61881-3 standards

Supercapacitors (SC),[27] comprise a family of electrochemical capacitors. Supercapacitor, sometimes called ultracapacitor is a generic term for electric double-layer 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]

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 double-layer by itself.[27]

Supercapacitors are divided into three families, based on the design of the electrodes:

  • Double-layer capacitors – with carbon electrodes or derivates with much higher static double-layer 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 double-layer capacitance and pseudocapacitance, such as lithium-ion 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 well-suited 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 62391-1 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, CAP-XX, DLCAP, EneCapTen, EVerCAP, DynaCap, Faradcap, GreenCap, Goldcap, HY-CAP, Kapton capacitor, Super capacitor, SuperCap, PAS Capacitor, PowerStor, PseudoCap, Ultracapacitor making it difficult for users to classify these capacitors.

Features and applications as well as disadvantages of supercapacitors
Capacitor type Dielectric Features/applications Disadvantages
Helmholtz double-layer plus faradaic pseudo-capacitance 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
Helmholtz double-layer plus faradaic pseudo-capacitance. 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, nano-scale 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, As of 2012 glass capacitors were in use as SMD version for applications requiring ultra-reliable and ultra-stable service.

Power capacitors

  • Vacuum capacitors—used in high power RF transmitters
  • SF6 gas filled capacitors—used as capacitance standard in measuring bridge circuits

Special capacitors

  • Printed circuit boards—metal conductive areas in different layers of a multi-layer 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

Features and applications as well as disadvantages of miscellaneous capacitors
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. Self-healing if arc-over current is limited. Very high cost. Fragile. Large. Relatively low capacitance.
SF6-gas filled capacitors SF6 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. Ultra-reliable. Ultra-stable. 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 oxide-nitride-oxide (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 one-time 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.

Features and applications as well as disadvantages of variable 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. Self-healing if arc-over current is limited. Very high cost. Fragile. Large dimensions.
SF6 gas filled tuning capacitor SF6 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



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 × 1012) pieces.[35] This market is dominated by ceramic capacitors with estimate of approximately one trillion (1 × 1012) items per year.[1]

Detailed estimated figures in value for the main capacitor families are:

All other capacitor types are negligible in terms of value and quantity compared with the above types.

Capacitor - Electrical characteristics


Series-equivalent circuit

Series-equivalent circuit model of a capacitor

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 non-linear, voltage-dependent 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 series-equivalent circuit the electrical characteristics are defined by:

  • C, the capacitance of the capacitor
  • Rinsul, the insulation resistance of the dielectric, not to be confused with the insulation of the housing
  • Rleak, the resistance representing the leakage current of the capacitor
  • RESR, the equivalent series resistance which summarizes all ohmic losses of the capacitor, usually abbreviated as "ESR"
  • LESL, the equivalent series inductance which is the effective self-inductance of the capacitor, usually abbreviated as "ESL".

Using a series equivalent circuit instead of a parallel equivalent circuit is specified by IEC/EN 60384-1.

Standard values and tolerances


The "rated capacitance" CR or "nominal capacitance" CN 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 low-voltage AC measuring method at a temperature of 20 °C with frequencies of

  • 100 kHz, 1 MHz (preferred) or 10 MHz for non-electrolytic capacitors with CR ≤ 1 nF:
  • 1 kHz or 10 kHz for non-electrolytic capacitors with 1 nF < CR ≤ 10 μF
  • 100/120 Hz for electrolytic capacitors
  • 50/60 Hz or 100/120 Hz for non-electrolytic capacitors with CR > 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.

Tolerances of capacitors and their letter codes
E series Tolerance
CR > 10 pF Letter code CR < 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 high-quality circuits such as precision oscillators and timers. General applications such as non-critical 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.

Temperature coefficients of some common capacitors
Type of capacitor,
dielectric material
Temperature coefficient
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
±20 % −40 to +85/105/125 °C
Tantalum electrolytic capacitor
±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 non-solid 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 charge-carrying ions. As higher the anode structure is roughned as more the capacitance value decreases with increasing frequency. Low voltage types with highly-roughened 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 non-linearity in low-distortion 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

Relation between rated and category temperature range and applied 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 60384-1 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.


Simplified series-equivalent circuit of a capacitor for higher frequencies (above); vector diagram with electrical reactances XESL and XC and resistance ESR and for illustration the impedance Z and dissipation factor tan δ

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 series-equivalent 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


have the same value (), then the impedance will only be determined by .

Typical impedance curves for different capacitance values over frequency showing the typical form with a decreasing impedance values below resonance and increasing values above resonance. As higher the capacitance as lower the resonance.

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 switched-mode 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 end-surface 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 "face-down" construction associated with multi-anode technology in tantalum electrolytic capacitors further reduced ESL. Capacitor families such as the so-called MOS capacitor or silicon capacitors offer solutions when capacitors at frequencies up to the GHz range are needed.

Inductance (ESL) and self-resonant 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 self-resonant frequency. The self-resonant frequency is the lowest frequency at which the impedance passes through a minimum. For any AC application the self-resonant frequency is the highest frequency at which capacitors can be used as a capacitive component.

This is critically important for decoupling high-speed 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 non-linear, 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 60384-1 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 non-electrolytic capacitors with CR ≤ 1 nF:
  • 1 kHz or 10 kHz for non-electrolytic capacitors with 1 nF < CR ≤ 10 μF
  • 100/120 Hz for electrolytic capacitors
  • 50/60 Hz or 100/120 Hz for non-electrolytic capacitors with CR > 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.

Comparization of ohmic losses for different capacitor types
for resonant circuits (Reference frequency 1 MHz)
Capacitor type Capacitance
at 100 kHz
at 1 MHz
tan δ
at 1 MHz
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 switched-mode 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, so-called 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 vendor-related 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

Limiting conditions for capacitors operating with AC loads

An AC load only can be applied to a non-polarized 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.
Typical rms AC voltage curves as a function of frequency, for 4 different capacitance values of a 63 V DC film capacitor series

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 self-discharge 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 Rins". This resistance is represented by the resistor Rins in parallel with the capacitor in the series-equivalent 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 self-discharge over insulation resistance with decreasing capacitor voltage follows the formula

With stored DC voltage  and self-discharge constant

Thus, after  voltage  drops to 37% of the initial value.

The self-discharge 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 time-determining 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 self-discharge 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 self-discharge constant of about 2000–4000 s.[42]

Insulation resistance respectively the self-discharge 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

The general leakage current behavior of electrolytic capacitors depend on the kind of electrolyte

For electrolytic capacitors the insulation resistance of the dielectric is termed "leakage current". This DC current is represented by the resistor Rleak in parallel with the capacitor in the series-equivalent circuit of electrolytic capacitors. This resistance between the terminals of a capacitor is also finite. Rleak 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 self-repair weaknesses by building up new layers. The time required depends generally on the electrolyte. Solid electrolytes drop faster than non-solid electrolytes but remain at a slightly higher level.

The leakage current in non-solid electrolytic capacitors as well as in manganese oxide solid tantalum capacitors decreases with voltage-connected time due to self-healing effects. Although electrolytics leakage current is higher than current flow over insulation resistance in ceramic or film capacitors, the self-discharge 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.



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 time-delayed dipole discharging, a phenomenon that is also called dielectric relaxation, "soakage" or "battery action".

Values of dielectric absorption for some often used capacitors
Type of capacitor Dielectric Absorption
Air and vacuum capacitors Not measurable
Class-1 ceramic capacitors, NP0 0.6%
Class-2 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%
Double-layer capacitor or super capacitors data not available

In many applications of capacitors dielectric absorption is not a problem but in some applications, such as long-time-constant integrators, sample-and-hold circuits, switched-capacitor analog-to-digital converters, and very low-distortion 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 parallel-plate 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.

Comparison of energy stored in capacitors with the same dimensions but with different rated voltages and capacitance values
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 sponge-like 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 drying-out effects for non-solid materials.


Aging of different Class 2 ceramic capacitors compared with NP0-Class 1 ceramic capacitor

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 ferroelectric-related 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 self-healing 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

The electrical values of electrolytic capacitors with non-solid electrolyte changes over the time due to evaporation of electrolyte. Reaching specified limits of the parameters the capacitors will be count as "wear out failure".

Electrolytic capacitors with non-solid 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 so-called "10-degree-law". 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

The life time (load life) of capacitors correspondents with the time of constant random failure rate shown in the bathtub curve. For electrolytic capacitors with non-solid electrolyte and supercapacitors ends this time with the beginning of wear out failures due to evaporation of electrolyte

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 post-shipment 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 (109) component-hours 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 UR). 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




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 non-solid 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 non-solid electrolytes, too. Special series for 125 °C with organic solvents like GBL are specified up to 10 years storage time ensure without pre-condition the proper electrical behavior of the capacitors.[61]

For antique radio equipment, "pre-conditioning" 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 60384-1 in the following sections.[62]

Ceramic capacitors

  • IEC/EN 60384-8—Fixed capacitors of ceramic dielectric, Class 1
  • IEC/EN 60384-9—Fixed capacitors of ceramic dielectric, Class 2
  • IEC/EN 60384-21—Fixed surface mount multilayer capacitors of ceramic dielectric, Class 1
  • IEC/EN 60384-22—Fixed surface mount multilayer capacitors of ceramic dielectric, Class 2

Film capacitors

  • IEC/EN 60384-2—Fixed metallized polyethylene-terephthalate film dielectric d.c. capacitors
  • IEC/EN 60384-11—Fixed polyethylene-terephthalate film dielectric metal foil d.c. capacitors
  • IEC/EN 60384-13—Fixed polypropylene film dielectric metal foil d.c. capacitors
  • IEC/EN 60384-16—Fixed metallized polypropylene film dielectric d.c. capacitors
  • IEC/EN 60384-17—Fixed metallized polypropylene film dielectric a.c. and pulse
  • IEC/EN 60384-19—Fixed metallized polyethylene-terephthalate film dielectric surface mount d.c. capacitors
  • IEC/EN 60384-20—Fixed metalized polyphenylene sulfide film dielectric surface mount d.c. capacitors
  • IEC/EN 60384-23—Fixed metallized polyethylene naphthalate film dielectric chip d.c. capacitors

Electrolytic capacitors

  • IEC/EN 60384-3—Surface mount fixed tantalum electrolytic capacitors with manganese dioxide solid electrolyte
  • IEC/EN 60384-4—Aluminium electrolytic capacitors with solid (MnO2) and non-solid electrolyte
  • IEC/EN 60384-15—fixed tantalum capacitors with non-solid and solid electrolyte
  • IEC/EN 60384-18—Fixed aluminium electrolytic surface mount capacitors with solid (MnO2) and non-solid electrolyte
  • IEC/EN 60384-24—Surface mount fixed tantalum electrolytic capacitors with conductive polymer solid electrolyte
  • IEC/EN 60384-25—Surface mount fixed aluminium electrolytic capacitors with conductive polymer solid electrolyte


  • IEC/EN 62391-1—Fixed electric double-layer capacitors for use in electric and electronic equipment - Part 1: Generic specification
  • IEC/EN 62391-2—Fixed electric double-layer capacitors for use in electronic equipment - Part 2: Sectional specification - Electric double-layer capacitors for power application

Capacitor symbols

Capacitor symbols
Capacitor Polarized





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" e-caps 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 × 10Z pF), the letters J, K or M indicate the tolerance (±5%, ±10% and ±20% respectively) and VOLTS V represents the working voltage.


  • 105K 330V implies a capacitance of 10 × 105 pF = 1 µF (K = ±10%) with a working voltage of 330 V.
  • 473M 100V implies a capacitance of 47 × 103 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 short-marking 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


As of 2013 Capacitors do not use color coding.

Polarity marking


Aluminum e-caps with non-solid electrolyte have a polarity marking at the cathode (minus) side. Aluminum, tantalum, and niobium e-caps with solid electrolyte have a polarity marking at the anode (plus) side. Supercapacitor are marked at the minus side.


  1. a b J. Ho, T. R. Jow, S. Boggs, Historical Introduction to Capacitor Technology, PDF [1]
  2. a b c Adam Marcus Namisnyk (2003-06-23). "A Survey of Electrochemical Supercapacitor Technology" (PDF). Retrieved 2011-06-24.
  3. WIMA, Characteristics of Metallized Film Capacitors in Comparison with Other Dielectrics [2]
  4. Film Capacitors, TDK Epcos, General technical information
  5. AVX, Dielectric Comparison Chart
  6. Holystone, Capacitor Dielectric Comparison, Technical Note 3
  7. Power Film Capacitors for Industrial Applications, P. Bettacchi, D. Montanari, D. Zanarini, D. Orioli, G. Rondelli, A. Sanua, KEMET Electronics [3]
  8. a b Template:Literatur
  9. General technical information of (RFI/EMI)Noise suppression capacitors on AC mains [4]
  10. Vishay, Capacitors - RFI Class X/Y
  11. X2Y® Technology
  12. Murata, Three-terminal Capacitor Structure, No.TE04EA-1.pdf 98.3.20
  13. Vishay, Ceramic RF-Power Capacitors
  14. Vishay. "Capacitors - RF Power". Vishay. Retrieved 2013-03-09.
  15. Passive component magazine, Nov./Dec. 2005, F. Jacobs, Polypropylene Capacitor Film Resin, p. 29 ff [5]
  16. Paumanok Publications, PCInewsletterOct2007cmp Paumanok Publications, Inc.
  17. WIMA, RFI Capacitors
  18. WIMA Snubber Capacitors
  19. Amrad Engeneering Inc., Motor run capacitors
  20. Epcos, Capacitors for power electronics, General technical information
  21. Sanyo, Capacitor lecture POSCAP (Ta) (Basic), Polymerized electrolyte
  22. CDE, Motor Start Capacitors
  23. Rubycon, Aluminum Electrolytic Capacitors for Strobe Flash
  24. Fischer & Tausche, Electrolytic capacitor for audio frequency
  25. Vishay, Wet Electrolyte Tantalum Capacitors, Introduction
  26. Self-healing Characteristics of Solid Electrolytic Capacitor with Polypyrrole Electrolyte, Yamamoto Hideo [6]
  27. a b c B. E. Conway (1999). Electrochemical Supercapacitors: Scientific Fundamentals and Technological Applications. Berlin: Springer. ISBN 0306457369. Retrieved 2013-05-02. see also Brian E. Conway in Electrochemistry Encyclopedia: Electrochemical Capacitors — Their Nature, Function and Applications
  28. Template:Cite techreport
  29. Elzbieta Frackowiak, Francois Beguin, PERGAMON, Carbon 39 (2001) 937–950, Carbon materials for the electrochemical storage of energy in Capacitors
  30. 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 978-953-307-830-4
  31. Elton
  32. IPDiA, 3D Silicon Capacitors
  33. Tettex instruments, SF6 Gas insulated Standard Capacitors
  34. AVX, Performance Characteristics of Multilayer Glass Capacitors
  35. highbeam business, Electronic Capacitors SIC 3675, Industry report
  36. 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
  37. Siliziumkondensator, Vishay, HPC0603A
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  40. WIMA, Selection of Capacitors for Pulse Applications
  41. Kemet, General information DC Film Capacitors
  42. Wima,Insulation Resistance
  43. [8], Capacitors for reduced sound emissions.
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  45. Kemet, Polymer Tantalum Chip Capacitors
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  49. NCC, KME series
  50. KEMET, series PHE450
  51. Metallized Polypropylene Film Energy Storage Capacitors For Low Pulse Duty, Ralph M. Kerrigan, NWL Capacitor Division [11]
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  54. Takaaki Tsurumi & Motohiro Shono & Hirofumi Kakemoto & Satoshi Wada & Kenji Saito & Hirokazu Chazono, Mechanism of capacitance aging under DC-bias field in X7R-MLCCs Published online: 23 March 2007, # Springer Science + Business Media, LLC 2007 [12]
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  59. 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]
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Electronics | Foreword | Basic Electronics | Complex Electronics | Electricity | Machines | History of Electronics | Appendix | edit



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 is the characteristic of the Inductor to generate 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, μ0, 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
μr = relative permeability of core material
N = number of turns
A = area of cross-section of the coil / m2
l = length of coil / m

The self-inductance of a straight, round wire in free space

Lself = 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 self-inductance 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

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)

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)

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 cross-section)

L = inductance / H
μr = relative permeability of core material
N = number of turns
r = radius of coil winding / m
D = overall diameter of toroid / m

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 Q-factor, 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, μ0 multiplied by a factor, the relative permeability, μr
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;


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 time-invariant 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 a coil of several turns is connected to an electricity source in a closed loop, the current in the circuit induces a magnetic field that has the same properties as a Magnetic Field of a Magnet.

When the current is turned off, the Magnetic Field does not exist.

Conducting Coil is called ElectroMagnet

Inductor and Alternating Current Voltage (AC)


Inductor's Voltage


Inductor's Current



, where .



Angle Difference Between Voltage and Current


For Lossless Inductor

The angle difference between Voltage and Current is 90

For Lossy Inductor

Changing the value of L and RL will change the value of Angle of Difference, Angular Frequency, Frequency 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



A electron moving through space creates a magnetic field that spins around the charge according to the right hand rule. The magnetic field is created by the spin of the moving electron. If the wire is bent in the shape of a ring, when its current is flowing it magnetic field will resemble water flowing through a hose. In order for the ring to have a magnetic field, its magnetic field must first displace the magnetic field that is already there. This is why inductors initially resist any changes in current when a voltage is applied. Over time the magnetic field changes to reflect the magnetic field of the ring and current starts flowing.

Inductors resist changes in current and take time to adjust.

A popular example of inductance is an electromagnet. It is essentially an inductor connected to dc with a piece of metal in its core. The flow of current creates a magnetic flow that mimicks a magnet. The direction of current determines the polarity of the magnet.

The nice thing about electromagnets is the strength of the current determines the strength of the magnetic field, so the more current the more magnetic field. Also reversing the direction of the current switches the polarity of the electromagnet.

This property allows electromagnets to be used as switches. As the current increases the magnet becomes more repulsive to other magnets.

Electromagnets are also used in loudspeakers. You have a voltage that is dependent on distance so as the distance decreases the voltage increases and as the distance increases the voltage decreases. The result is the ability to program the loudspeaker according to a vibration pattern.




Consider n inductors in series, as above. The voltage across the whole arrangement (i.e. across the two terminals) must be equal to the sum of the voltages across the individual inductors:

from our definition of inductance, where the voltage is the inductance multiplied by the rate of change of current, we get

where i1 is the current in element 1, and so on. Since the current in every element the series must always be the same (by Kirchhoff's Current Law), we can see that

where i is the current in the network. Factorising, we get:


If we now call all the element in the series a single, equivalent inductance, Leq, we see that


This means that when in series, the total inductance is just the sum of all the constituent inductances.



When inductors are in parallel, then each inductor has the same voltage across it, that is, the voltage present at the terminals of the network. This can just be called v. Now, our equation describing the equivalent inductance, Leq, is:

where ieq is the current through the network.

By Kirchhoff's Current Law, we have

Differentiating with respect to time gives

Now by rearranging the general equation describing inductance in the i'th element, we can obtain, for each term in the above,

Substituting into the equation before, we get


Rearranging, we get

So, therefore:

This is identical to the rule for combining resistors.

Inductor Construction

An inductor is the electrical equivalent of a flywheel. When you introduce energy to an inductor, the current flows slowly at first, just like a flywheel doesn't instantly get up to speed. And if you try to stop a flywheel instantly, something will break. If you let current through an inductor, then open the switch, the energy of the inductor must go somewhere - it will arc back across the switch, quickly ruining it - thus the condensor used in coil-type automotive ignition systems: it stores the energy until the inefficiency of the flywheel can turn it into heat.

Inductors are formed of a coil of conductive material. Normally they are made of copper wire, but not always (Example: aluminum wire, or spiral pattern etched on circuit board). The material around and within the coil affects its properties; common types are air-core (only a coil of wire), iron-core, and ferrite core. Iron and ferrite types are more efficient because they conduct the magnetic field much better than air; of the two, ferrite is more efficient because stray electricity cannot flow through it. Ferrite is more expensive but operates at much higher frequencies than iron cores.

Some inductors have more than a core, which is just a rod the coil is formed about. Some are formed like transformers, using two E-shaped pieces facing each other, the wires wound about the central leg of the E's. The E's are made of laminated iron/steel or ferrite.

Toroidal inductors are most efficient of all, they are wound around a donut shape which is made of ferrite. They are more difficult to make, because the formed coil cannot be manufactured directly on the toroid - it must be wound onto it.

Important qualities of angf inductor

There are several important properties for an inductor.

  • Current carrying capacity is determined by wire thickness.
  • Q, or quality, is determined by the uniformity of the windings, as well as the core material and how thoroughly it surrounds the coil.
  • Last but not least, the inductance of the coil.

The inductance is determined by several factors.

  • coil shape: short and squat is best
  • core material
  • windings: winding in opposite directions will cancel out the inductance effect, and you will have only a resistor.

(someone add formulas etc. for determining inductance, please.)

Inductors/Real Inductors

In the real world, various factors contribute to non-idealities within inductors.

Nonzero Resistance


Nearly all common components have wire leads. Inductors specifically have this drawback in addition to the fact that they are typically made from a length of wire which is coiled around some form of material. This wire typically does not have zero resistance. While small lengths of wire made of typical conductive materials, such as copper, larger coils will have a relatively large amount of this wire, leading to an added resistance which may not be neglected. This also becomes true when the potential applied across the inductor increases.

To account for this added resistivity, a non-ideal inductor may be modeled as an ideal inductor in series with a resistor. This resistance may be measured under DC steady-state conditions.

High-frequency considerations


Although quite small, lead wires do have some amount of self-inductance. This will usually be neglectable except in the case of very small values for inductive components as well as for high frequency design. To account for the inductance added by these wires, one may model such a non-ideal inductor by including series inductors on for each of the two (or more) leading wires coming from the component.

Simplifying Capacitor/Inductor Networks

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Capacitors and inductors follow similar laws as resistors when it comes to simplification.

Simplifying Capacitor Configurations


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.

File:Electronics Capacitorsparallel.png

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

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 100V capacitors in series results in the same as having one capacitor with the total maximum voltage of 200V. 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 the a different voltage over them.

File:Electronics Capacitorsseries.png

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

Simplifying Inductor Configurations


An inductor works by opposing current change

Inductors in Parallel


Each inductor has a decreased amount of current flowing through it.
Take two inductors of the same strength that are in parallel. This divides the current so half the current is flowing through each inductor.

Inductors in Series


Inductors in series are just like resistors in series. Simply add them up.


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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.

Voltage-controlled 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 Vcis a control voltage.
An example of a VCVS may be an idealized amplifier, where A is the gain of the amplifier.

Current-controlled 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 Ic is a control current.

Voltage-controlled 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 Vc is a control voltage.

Current-controlled 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 Ic 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, Ib is the control and the collector current Ic is the output.


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 push-button 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.

(single|double) pole ((single|double) throw|changeover)

  • SPST = single pole single throw, a simple on-off switch.
  • SPDT = single pole double throw, a simple changeover or on-off-on 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 "on-off" 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 push-button 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


DC Voltage and Current

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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

  1. Voltage is the potential difference between two charged objects.
  2. Potentials can be added or subtracted in series to make larger or smaller potentials as is commonly done in batteries.
  3. Positive charge flow from areas of high potential to lower potential.
  4. 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


-I1 + I2 + I3 = 0 ↔ I1 = I2 + I3

I1 - I2 - I3 - I4 = 0 ↔ I2 + I3 + I4 = I1

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.




Similary if is the voltage across then

In general for n series resistors the voltage dropped across one of them say is


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


The current through according to Ohms Law is


Dividing equation (2) by (1)


In general with n Resistors the current is

Or possibly more simply


Nodal Analysis

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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 opposite of each other, this signifies 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.


  1. Identify all nodes.
  2. 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).
  3. Assign voltage variables to the other nodes (these are node voltages.)
  4. 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.)
  5. 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

  1. 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.
  2. 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 'super-node', consisting of the source and the nodes it is connected to. Write a KCL equation for all current entering and leaving the super-node. 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 (V2-V1=source value). This new system of equations can be solved as in Step 5 above.
  3. Dependent Voltage Source
    Solution: Same as an independent voltage source, with an extra step. First write a super-node 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.



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;


Mesh Analysis

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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.)


  1. Draw circuit in planar form (if possible.)
  2. 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.
  3. Write a KVL equation in terms of mesh currents for each mesh.
  4. 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 'super-mesh' 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.



Given the Circuit below, find the currents , .

The circuit has 2 loops indicated on the diagram. Using KVL we get:
Simplifying we get the simultaneous equations:

solving to get:

Thevenin/Norton Equivalents

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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 allow a voltage source in series with a resistor to be replaced by a current source in parallel with that exact same 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 (and also Norton) equivalent consists of RTh alone

Thevenin Equivalents


The Thevenin equivalent circuit of a (two-terminal) 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 open-circuited. Find the current from terminal A to terminal B when they are short-circuited. Then

The Thevenin voltage source value is equivalent to the open-circuit 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

Figure 1: Circuit for the determination of Equivalents

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.

Figure 2: Circuit for the Equivalent Resistance

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.

Figure 3: Equivalent Thevenin Circuit

The Norton Equivalent is created by doing a source transformation using .(2)

Figure 4: Equivalent Norton Circuit

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.


If the value of form equation 1 is substituted into equation 3.


Now look at Figure 1 and calculate 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.


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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, V1, and resistor, R, a current i1 flows. The current i1 has a value according to Ohm’s Law. Similarly in Figure 1 (b) with independent voltage source, V2, and resistor, R, a current i2 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 V1/R has a value i1 and the other term is i2 this gives equation 3. This is basically what the Superposition Theorem states.


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.


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 R1.

Step 1: Short circuit V2 and solve for v1. By voltage divider rule.


Short circuit V1 and solve for v2. By voltage divider rule.


Step 2: Sum the effects.

Using equations 5 and 6.

If and then