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Basics of Power circuits
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Basics of Power circuitry including basic electrical elements and basic laws of circuit theory and circuit reduction techniques.

### Transcript of Basics of Power Circuits

• Basics of Power circuits

• ContentsArticles

Electrical resistance and conductance 1Inductor 10Capacitor 24Ohm's law 42Kirchhoff's circuit laws 50Current divider 54Voltage divider 57Y- transform 61Nodal analysis 66Mesh analysis 69Superposition theorem 72Thvenin's theorem 73Norton's theorem 77Maximum power transfer theorem 80

ReferencesArticle Sources and Contributors 84Image Sources, Licenses and Contributors 86

Ramesh Singh, DTU (Formerly Delhi College of Engineering)[email protected]

• Electrical resistance and conductance 1

Electrical resistance and conductance

Electromagnetism

Electricity Magnetism

The electrical resistance of an electrical conductor is the opposition to the passage of an electric current throughthat conductor; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electricalresistance shares some conceptual parallels with the mechanical notion of friction. The SI unit of electrical resistanceis the ohm (), while electrical conductance is measured in siemens (S).An object of uniform cross section has a resistance proportional to its resistivity and length and inverselyproportional to its cross-sectional area. All materials show some resistance, except for superconductors, which havea resistance of zero.The resistance (R) of an object is defined as the ratio of voltage across it (V) to current through it (I), while theconductance (G) is the inverse:

For a wide variety of materials and conditions, V and I are directly proportional to each other, and therefore R and Gare constant (although they can depend on other factors like temperature or strain). This proportionality is calledOhm's law, and materials that satisfy it are called "Ohmic" materials.In other cases, such as a diode or battery, V and I are not directly proportional, or in other words the IV curve is nota straight line through the origin, and Ohm's law does not hold. In this case, resistance and conductance are lessuseful concepts, and more difficult to define. The ratio V/I is sometimes still useful, and is referred to as a "chordalresistance" or "static resistance",[][] as it corresponds to the inverse slope of a chord between the origin and an IVcurve. In other situations, the derivative may be most useful; this is called the "differential resistance".

Ramesh Singh, DTU (Formerly Delhi College of Engineering)[email protected]

• Electrical resistance and conductance 2

Introduction

The hydraulic analogy compares electric currentflowing through circuits to water flowing through

pipes. When a pipe (left) is filled with hair(right), it takes a larger pressure to achieve thesame flow of water. Pushing electric current

through a large resistance is like pushing waterthrough a pipe clogged with hair: It requires alarger push (electromotive force) to drive the

same flow (electric current).

In the hydraulic analogy, current flowing through a wire (or resistor) islike water flowing through a pipe, and the voltage drop across the wireis like the pressure drop which pushes water through the pipe.Conductance is proportional to how much flow occurs for a givenpressure, and resistance is proportional to how much pressure isrequired to achieve a given flow. (Conductance and resistance arereciprocals.)

The voltage drop (i.e., difference in voltage between one side of theresistor and the other), not the voltage itself, provides the driving forcepushing current through a resistor. In hydraulics, it is similar: Thepressure difference between two sides of a pipe, not the pressure itself,determines the flow through it. For example, there may be a largewater pressure above the pipe, which tries to push water down throughthe pipe. But there may be an equally large water pressure below thepipe, which tries to push water back up through the pipe. If thesepressures are equal, no water will flow. (In the image at right, the waterpressure below the pipe is zero.)

The resistance and conductance of a wire, resistor, or other element is generally determined by two factors: geometry(shape) and materials.Geometry is important because it is more difficult to push water through a long, narrow pipe than a wide, short pipe.In the same way, a long, thin copper wire has higher resistance (lower conductance) than a short, thick copper wire.Materials are important as well. A pipe filled with hair restricts the flow of water more than a clean pipe of the sameshape and size. In a similar way, electrons can flow freely and easily through a copper wire, but cannot as easily flowthrough a steel wire of the same shape and size, and they essentially cannot flow at all through an insulator likerubber, regardless of its shape. The difference between, copper, steel, and rubber is related to their microscopicstructure and electron configuration, and is quantified by a property called resistivity.

Conductors and resistors

A 65 resistor, as identified by its electronic color code(bluegreenblack-gold). An ohmmeter could be used to verify this

value.

Those substances through which electricity can floware called conductors. A piece of conducting materialof a particular resistance meant for use in a circuit iscalled a resistor. Conductors are made ofhigh-conductivity materials such as metals, inparticular copper and aluminium. Resistors, on theother hand, are made of a wide variety of materials depending on factors such as the desired resistance, amount ofenergy that it needs to dissipate, precision, and costs.

Ramesh Singh, DTU (Formerly Delhi College of Engineering)[email protected]

• Electrical resistance and conductance 3

Ohm's law

The current-voltage characteristics of four devices: Two resistors, a diode, and a battery.The horizontal axis is voltage drop, the vertical axis is current. Ohm's law is satisfiedwhen the graph is a straight line through the origin. Therefore, the two resistors are

"ohmic", but the diode and battery are not.

Ohm's law is an empirical law relatingthe voltage V across an element to thecurrent I through it:

(V is directly proportional to I). Thislaw is not always true: For example, itis false for diodes, batteries, etc.However, it is true to a very goodapproximation for wires and resistors(assuming that other conditions, including temperature, are held fixed). Materials or objects where Ohm's law is trueare called "ohmic", whereas objects which do not obey Ohm's law are '"non-ohmic".

Relation to resistivity and conductivity

A piece of resistive material with electricalcontacts on both ends.

The resistance of a given object depends primarily on two factors:What material it is made of, and its shape. For a given material, theresistance is inversely proportional to the cross-sectional area; forexample, a thick copper wire has lower resistance than anotherwise-identical thin copper wire. Also, for a given material, theresistance is proportional to the length; for example, a long copper wirehas higher resistance than an otherwise-identical short copper wire.The resistance R and conductance G of a conductor of uniform crosssection, therefore, can be computed as

where is the length of the conductor, measured in metres [m], A is the cross-section area of the conductormeasured in square metres [m], (sigma) is the electrical conductivity measured in siemens per meter (Sm1), and (rho) is the electrical resistivity (also called specific electrical resistance) of the material, measured in ohm-metres(m). The resistivity and conductivity are proportionality constants, and therefore depend only on the material thewire is made of, not the geometry of the wire. Resistivity and conductivity are reciprocals: . Resistivity isa measure of the material's ability to oppose electric current.This formula is not exact: It assumes the current density is totally uniform in the conductor, which is not always truein practical situations. However, this formula still provides a good approximation for long thin conductors such aswires.Another situation for which this formula is not exact is with alternating current (AC), because the skin effect inhibitscurrent flow near the center of the conductor. Then, the geometrical cross-section is different from the effectivecross-section in which current is actually flowing, so the resistance is higher than expected. Similarly, if twoconductors are near each other carrying AC current, their resistances will increase due to the proximity effect. Atcommercial power frequency, these effects are significant for large conductors carrying large currents, such asbusbars in an electrical substation,[1] or large power cables carrying more than a few hundred amperes.

Ramesh Singh, DTU (Formerly Delhi College of Engineering)[email protected]

• Electrical resistance and conductance 4

What determines resistivity?The resistivity of different materials varies by an enormous amount: For example, the conductivity of teflon is about1030 times lower than the conductivity of copper. Why is there such a difference? Loosely speaking, a metal haslarge numbers of "delocalized" electrons that are not stuck in any one place, but free to move across large distances,whereas in an insulator (like teflon), each electron is tightly bound to a single molecule, and a great force is requiredto pull it away. Semiconductors lie between these two extremes. More details can be found in the article: Electricalresistivity and conductivity. For the case of electrolyte solutions, see the article: Conductivity (electrolytic).Resistivity varies with temperature. In semiconductors, resistivity also changes when light is shining on it. These arediscussed below.

Measuring resistanceAn instrument for measuring resistance is called an ohmmeter. Simple ohmmeters cannot measure low resistancesaccurately because the resistance of their measuring leads causes a voltage drop that interferes with themeasurement, so more accurate devices use four-terminal sensing.

Typical resistances

Component Resistance ()

1 meter of copper wirewith 1mm diameter

0.02[2]

0.03[3]

AA battery (typicalinternal resistance)

0.1[4]

Incandescent light bulbfilament (typical)

200-1000[5]

Human body 1000 to 100,000[6]

Static and differential resistance

The IV curve of a non-ohmic device (purple). The static resistance at point A is the inverse slope of line B throughthe origin. The differential resistance at A is the inverse slope of tangent line C.

Ramesh Singh, DTU (Formerly Delhi College of Engineering)[email protected]

• Electrical resistance and conductance 5

The IV curve of a component with negative differential resistance, an unusual phenomenon where the IV curve isnon-monotonic.Many electrical elements, such as diodes and batteries do not satisfy Ohm's law. These are called non-ohmic ornonlinear, and are characterized by an IV curve which is not a straight line through the origin.Resistance and conductance can still be defined for non-ohmic elements. However, unlike ohmic resistance,nonlinear resistance is not constant but varies with the voltage or current through the device; its operating point.There are two types:[][]

Static resistance (also called chordal or DC resistance) - This corresponds to the usual definition of resistance;the voltage divided by the current

.

It is the slope of the line (chord} from the origin through the point on the curve. Static resistance determinesthe power dissipation in an electrical component. Points on the IV curve located in the 2nd or 4th quadrants,for which the slope of the chordal line is negative, have negative static resistance. Passive devices, which haveno source of energy, cannot have negative static resistance. However active devices such as transistors orop-amps can synthesize negative static resistance with feedback, and it is used in some circuits such asgyrators.

Differential resistance (also called dynamic, incremental or small signal resistance) - Differential resistance isthe derivative of the voltage with respect to the current; the slope of the IV curve at a point

.

If the IV curve is nonmonotonic (with peaks and troughs), the curve will have a negative slope in someregions; so in these regions the device has negative differential resistance. Devices with negative differentialresistance can amplify a signal applied to them, and are used to make amplifiers and oscillators. These includetunnel diodes, Gunn diodes, IMPATT diodes, magnetron tubes, and unijunction transistors.

Ramesh Singh, DTU (Formerly Delhi College of Engineering)[email protected]

• Electrical resistance and conductance 6

AC circuits

The voltage (red) and current (blue) versus time (horizontal axis) for a capacitor(top) and inductor (bottom). Since the amplitude of the current and voltage

sinusoids are the same, the absolute value of impedance is 1 for both the capacitorand the inductor (in whatever units the graph is using). On the other hand, the

phase difference between current and voltage is -90 for the capacitor; therefore,the complex phase of the impedance of the capacitor is -90. Similarly, the phase

difference between current and voltage is +90 for the inductor; therefore, thecomplex phase of the impedance of the inductor is +90.

When an alternating current flows through acircuit, the relation between current andvoltage across a circuit element ischaracterized not only by the ratio of theirmagnitudes, but also the difference in theirphases. For example, in an ideal resistor, themoment when the voltage reaches itsmaximum, the current also reaches itsmaximum (current and voltage areoscillating in phase). But for a capacitor orinductor, the maximum current flow occursas the voltage passes through zero andvice-versa (current and voltage areoscillating 90 out of phase, see image atright). Complex numbers are used to keeptrack of both the phase and magnitude ofcurrent and voltage:

where: t is time, V(t) and I(t) are, respectively, voltage and current as a function of time, V0, I0, Z, and Y are complex numbers, Z is called impedance, Y is called admittance, Re indicates real part, is the angular frequency of the AC current, is the imaginary unit.The impedance and admittance may be expressed as complex numbers which can be broken into real and imaginaryparts:

where R and G are resistance and conductance respectively, X is reactance, and B is susceptance. For ideal resistors,Z and Y reduce to R and G respectively, but for AC networks containing capacitors and inductors, X and B arenonzero.

for AC circuits, just as for DC circuits.

Frequency dependence of resistance

Another complication of AC circuits is that the resistance and conductance can be frequency-dependent. One reason,mentioned above is the skin effect (and the related proximity effect). Another reason is that the resistivity itself maydepend on frequency (see Drude model, deep-level traps, resonant frequency, KramersKronig relations, etc.)

Ramesh Singh, DTU (Formerly Delhi College of Engineering)[email protected]

• Electrical resistance and conductance 7

Energy dissipation and Joule heating

Running current through a resistance creates heat,in a phenomenon called Joule heating. In thispicture, a cartridge heater, warmed by Joule

heating, is glowing red hot.

Resistors (and other elements with resistance) oppose the flow ofelectric current; therefore, electrical energy is required to push currentthrough the resistance. This electrical energy is dissipated, heating theresistor in the process. This is called Joule heating (after JamesPrescott Joule), also called ohmic heating or resistive heating.

The dissipation of electrical energy is often undesired, particularly inthe case of transmission losses in power lines. High voltagetransmission helps reduce the losses by reducing the current for a givenpower.

On the other hand, Joule heating is sometimes useful, for example inelectric stoves and other electric heaters (also called resistive heaters).As another example, incandescent lamps rely on Joule heating: the filament is heated to such a high temperature thatit glows "white hot" with thermal radiation (also called incandescence).

The formula for Joule heating is:

where P is the power (energy per unit time) converted from electrical energy to thermal energy, R is the resistance,and I is the current through the resistor.

Dependence of resistance on other conditions

Temperature dependenceNear room temperature, the resistivity of metals typically increases as temperature is increased, while the resistivityof semiconductors typically decreases as temperature is increased. The resistivity of insulators and electrolytes mayincrease or decrease depending on the system. For the detailed behavior and explanation, see Electrical resistivityand conductivity.As a consequence, the resistance of wires, resistors, and other components often change with temperature. This effectmay be undesired, causing an electronic circuit to malfunction at extreme temperatures. In some cases, however, theeffect is put to good use. When temperature-dependent resistance of a component is used purposefully, thecomponent is called a resistance thermometer or thermistor. (A resistance thermometer is made of metal, usuallyplatinum, while a thermistor is made of ceramic or polymer.)Resistance thermometers and thermistors are generally used in two ways. First, they can be used as thermometers:By measuring the resistance, the temperature of the environment can be inferred. Second, they can be used inconjunction with Joule heating (also called self-heating): If a large current is running through the resistor, theresistor's temperature rises and therefore its resistance changes. Therefore, these components can be used in acircuit-protection role similar to fuses, or for feedback in circuits, or for many other purposes. In general,self-heating can turn a resistor into a nonlinear and hysteretic circuit element. For more details seeThermistor#Self-heating effects.If the temperature T does not vary too much, a linear approximation is typically used:

where is called the temperature coefficient of resistance, is a fixed reference temperature (usually room temperature), and is the resistance at temperature . The parameter is an empirical parameter fitted from measurement data. Because the linear approximation is only an approximation, is different for different reference temperatures. For this reason it is usual to specify the temperature that was measured at with a suffix, such as

Ramesh Singh, DTU (Formerly Delhi College of Engineering)[email protected]

• Electrical resistance and conductance 8

, and the relationship only holds in a range of temperatures around the reference.[7]

The temperature coefficient is typically +3103 K1 to +6103 K1 for metals near room temperature. It isusually negative for semiconductors and insulators, with highly variable magnitude.[8]

Strain dependenceJust as the resistance of a conductor depends upon temperature, the resistance of a conductor depends upon strain.By placing a conductor under tension (a form of stress that leads to strain in the form of stretching of the conductor),the length of the section of conductor under tension increases and its cross-sectional area decreases. Both theseeffects contribute to increasing the resistance of the strained section of conductor. Under compression (strain in theopposite direction), the resistance of the strained section of conductor decreases. See the discussion on strain gaugesfor details about devices constructed to take advantage of this effect.

Light illumination dependenceSome resistors, particularly those made from semiconductors, exhibit photoconductivity, meaning that theirresistance changes when light is shining on them. Therefore they are called photoresistors (or light dependentresistors). These are a common type of light detector.

SuperconductivitySuperconductors are materials that have exactly zero resistance and infinite conductance, because they can have V=0and I0. This also means there is no joule heating, or in other words no dissipation of electrical energy. Therefore, ifsuperconductive wire is made into a closed loop, current will keep flowing around the loop forever. Superconductorsrequire cooling to temperatures near 4 K with liquid helium for most metallic superconductors like NbSn alloys, orcooling to temperatures near 77K with liquid nitrogen for the expensive, brittle and delicate ceramic hightemperature superconductors. Nevertheless, there are many technological applications of superconductivity,including superconducting magnets.

References[1] Fink and Beaty, Standard Handbook for Electrical Engineers 11th Edition, page 17-19[2] The resistivity of copper is about 1.710-8m. See (http:/ / hypertextbook. com/ facts/ 2004/ BridgetRitter. shtml).[3] Electric power substations engineering by John Douglas McDonald, p 18-37, google books link (http:/ / books. google. com/

books?id=e__hltcUQIQC& pg=PT363)[4] (http:/ / data. energizer. com/ PDFs/ BatteryIR. pdf) For a fresh Energizer E91 AA alkaline battery, the internal resistance varies from 0.9 at

-40C, to 0.1 at +40C.[5] A 60W light bulb in the USA (120V mains electricity) draws RMS current 60W/120V=500mA, so its resistance is 120V/500mA=240 ohms.

The resistance of a 60W light bulb in Europe (230V mains) would be 900 ohms. The resistance of a filament is temperature-dependent; thesevalues are for when the filament is already heated up and the light is already glowing.

[6] 100,000 ohms for dry skin contact, 1000 ohms for wet or broken skin contact. Other factors and conditions are relevant as well. See electricshock article for more details. Also see:

[7] Ward, MR, Electrical Engineering Science, pp3640, McGraw-Hill, 1971.[8] See Electrical resistivity and conductivity for a table. The temperature coefficient of resistivity is similar but not identical to the temperature

coefficient of resistance. The small difference is due to thermal expansion changing the dimensions of the resistor.

Ramesh Singh, DTU (Formerly Delhi College of Engineering)[email protected]

• Electrical resistance and conductance 9

External links The Notion of Electrical Resistance (http:/ / independent. academia. edu/ Csoliverez/ Papers/ 1848699/

The_Notion_of_Electrical_Resistance_by_Soliverez). Review of the equations that determine the value ofelectrical resistance.

Vehicular Electronics Laboratory: Resistance Calculator (http:/ / www. cvel. clemson. edu/ emc/ calculators/Resistance_Calculator/ index. html''Clemson)

Ramesh Singh, DTU (Formerly Delhi College of Engineering)[email protected]

• Inductor 10

Inductor

Inductor

A selection of low-value inductors

Type Passive

Working principle Electromagnetic induction

Electronic symbol

An inductor, also called a coil or reactor, is a passive two-terminalelectrical component which resists changes in electric current passingthrough it. It consists of a conductor such as a wire, usually wound intoa coil. When a current flows through it, energy is stored temporarily ina magnetic field in the coil. When the current flowing through aninductor changes, the time-varying magnetic field induces a voltage inthe conductor, according to Faradays law of electromagneticinduction, which opposes the change in current that created it.

An inductor is characterized by its inductance, the ratio of the voltage to the rate of change of current, which hasunits of henries (H). Many inductors have a magnetic core made of iron or ferrite inside the coil, which serves toincrease the magnetic field and thus the inductance. Along with capacitors and resistors, inductors are one of thethree passive linear circuit elements that make up electric circuits. Inductors are widely used in alternating current(AC) electronic equipment, particularly in radio equipment. They are used to block the flow of AC current whileallowing DC to pass; inductors designed for this purpose are called chokes. They are also used in electronic filters toseparate signals of different frequencies, and in combination with capacitors to make tuned circuits, used to tuneradio and TV receivers.

Ramesh Singh, DTU (Formerly Delhi College of Engineering)[email protected]

• Inductor 11

OverviewInductance (L) results from the magnetic field around a current-carrying conductor; the electric current through theconductor creates a magnetic flux. Inductance is a geometrical property of a circuit which is determined by howmuch magnetic flux through the circuit is created by a given current i

(1)

Any wire or other conductor will generate a magnetic field when current flows through it, so every conductor hassome inductance. In inductors the conductor is shaped to increase the magnetic field. Winding the wire into a coilincreases the number of times the magnetic flux lines link the circuit, increasing the field and thus the inductance.The more turns, the higher the inductance. By winding the coil on a "magnetic core" made of a ferromagneticmaterial like iron, the magnetizing field from the coil will induce magnetization in the material, increasing themagnetic flux. The high permeability of a ferromagnetic core can increase the inductance of a coil by a factor ofseveral thousand over what it would be without it.

Constitutive equationAny change in the current through an inductor creates a changing flux, inducing a voltage across the inductor. ByFaraday's Law of Induction the voltage induced by any change in magnetic flux through the circuit is

From (1) above

(2)

So inductance is also a measure of the amount of electromotive force (voltage) generated per unit change in current.For example, an inductor with an inductance of 1 henry produces an EMF of 1 volt when the current through theinductor changes at the rate of 1 ampere per second. This is usually taken to be the constitutive relation (definingequation) of the inductor.

Lenz's lawThe polarity (direction) of the induced voltage is given by Lenz's law, which states that it will be such as to opposethe change in current. For example, if the current through an inductor is increasing, the induced voltage will bepositive at the terminal through which the current enters and negative at the terminal through which it leaves. Theenergy from the external circuit necessary to overcome this potential 'hill' is stored in the magnetic field of theinductor; the inductor is sometimes said to be "charging". If the current is decreasing, the induced voltage will benegative at the terminal through which the current enters. Energy from the magnetic field is being returned to thecircuit; the inductor is said to be "discharging".

Ideal and real inductorsIn circuit theory, inductors are idealized as obeying the mathematical relation (2) above precisely. An "idealinductor" has inductance, but no resistance or capacitance, and does not dissipate or radiate energy. However realinductors have side effects which cause their behavior to depart from this simple model. They have resistance (due tothe resistance of the wire and energy losses in core material), and parasitic capacitance (due to the electric fieldbetween the turns of wire which are at slightly different potentials). At high frequencies the capacitance begins toaffect the inductor's behavior; at some frequency, real inductors behave as resonant circuits, becoming self-resonant.Above the resonant frequency the capacitive reactance becomes the dominant part of the impedance. At higherfrequencies, resistive losses in the windings increase due to skin effect and proximity effect.

Ramesh Singh, DTU (Formerly Delhi College of Engineering)[email protected]

• Inductor 12

Inductors with ferromagnetic cores have additional energy losses due to hysteresis and eddy currents in the core,which increase with frequency. At high currents, iron core inductors also show gradual departure from ideal behaviordue to nonlinearity caused by magnetic saturation of the core. An inductor may radiate electromagnetic energy intosurrounding space and circuits, and may absorb electromagnetic emissions from other circuits, causingelectromagnetic interference (EMI). Real-world inductor applications may consider these parasitic parameters asimportant as the inductance.

Applications

A ferrite "bead" choke, consisting of anencircling ferrite cylinder, removes electronic

noise from a computer power cord.

Inductors are used extensively in analog circuits and signal processing.Inductors in conjunction with capacitors form tuned circuits which canemphasize or filter out specific signal frequencies. Applications rangefrom the use of large inductors in power supplies, which in conjunctionwith filter capacitors remove residual hums known as the mains hum orother fluctuations from the direct current output, to the smallinductance of the ferrite bead or torus installed around a cable toprevent radio frequency interference from being transmitted down thewire. Smaller inductor/capacitor combinations provide tuned circuitsused in radio reception and broadcasting, for instance. Inductors areused as the energy storage device in many switched-mode powersupplies to produce DC current. The inductor supplies energy to thecircuit to keep current flowing during the "off" switching periods.

Two (or more) inductors in proximity that have coupled magnetic flux(mutual inductance) form a transformer, which is a fundamentalcomponent of every electric utility power grid. The efficiency of atransformer may decrease as the frequency increases due to eddycurrents in the core material and skin effect on the windings. The sizeof the core can be decreased at higher frequencies. For this reason,aircraft use 400 hertz alternating current rather than the usual 50 or 60hertz, allowing a great saving in weight from the use of smallertransformers.[1]

Inductors are also employed in electrical transmission systems, wherethey are used to limit switching currents and fault currents. In this field, they are more commonly referred to asreactors.

Because inductors have complicated side effects (detailed below) which cause them to depart from ideal behavior,because they can radiate electromagnetic interference (EMI), and most of all because of their bulk which preventsthem from being integrated on semiconductor chips, the use of inductors is declining in modern electronic devices,particularly compact portable devices. Real inductors are increasingly being replaced by active circuits such as thegyrator which can synthesize inductance using capacitors.

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• Inductor 13

Inductor construction

A ferrite core inductor with two47mH windings.

An inductor usually consists of a coil of conducting material, typically insulatedcopper wire, wrapped around a core either of plastic or of a ferromagnetic (orferrimagnetic) material; the latter is called an "iron core" inductor. The highpermeability of the ferromagnetic core increases the magnetic field and confinesit closely to the inductor, thereby increasing the inductance. Low frequencyinductors are constructed like transformers, with cores of electrical steellaminated to prevent eddy currents. 'Soft' ferrites are widely used for cores aboveaudio frequencies, since they do not cause the large energy losses at highfrequencies that ordinary iron alloys do. Inductors come in many shapes. Mostare constructed as enamel coated wire (magnet wire) wrapped around a ferritebobbin with wire exposed on the outside, while some enclose the wirecompletely in ferrite and are referred to as "shielded". Some inductors have anadjustable core, which enables changing of the inductance. Inductors used toblock very high frequencies are sometimes made by stringing a ferrite bead on a

wire.

Small inductors can be etched directly onto a printed circuit board by laying out the trace in a spiral pattern. Somesuch planar inductors use a planar core.Small value inductors can also be built on integrated circuits using the same processes that are used to maketransistors. Aluminium interconnect is typically used, laid out in a spiral coil pattern. However, the small dimensionslimit the inductance, and it is far more common to use a circuit called a "gyrator" that uses a capacitor and activecomponents to behave similarly to an inductor.

Types of inductor

Air core inductor

Double helix oscillation transformer fora spark gap transmitter. Transformerconsists of two helical inductors. Theinner inductor is moved to adjust themutual inductance between the two

coils.

The term air core coil describes an inductor that does not use a magnetic coremade of a ferromagnetic material. The term refers to coils wound on plastic,ceramic, or other nonmagnetic forms, as well as those that have only air insidethe windings. Air core coils have lower inductance than ferromagnetic corecoils, but are often used at high frequencies because they are free from energylosses called core losses that occur in ferromagnetic cores, which increase withfrequency. A side effect that can occur in air core coils in which the winding isnot rigidly supported on a form is 'microphony': mechanical vibration of thewindings can cause variations in the inductance.

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• Inductor 14

Collection of RF inductors, showing techniques to reduce losses. The threetop right and the loopstick antenna, bottom, have basket windings.

At high frequencies, particularly radio frequencies(RF), inductors have higher resistance and otherlosses. In addition to causing power loss, inresonant circuits this can reduce the Q factor of thecircuit, broadening the bandwidth. In RF inductors,which are mostly air core types, specializedconstruction techniques are used to minimize theselosses. The losses are due to these effects:

Skin effect: The resistance of a wire to highfrequency current is higher than its resistance to direct current because of skin effect. Radio frequency alternatingcurrent does not penetrate far into the body of a conductor but travels along its surface. Therefore, in a solid wire,most of the cross sectional area of the wire is not used to conduct the current, which is in a narrow annulus on thesurface. This effect increases the resistance of the wire in the coil, which may already have a relatively highresistance due to its length and small diameter.

Proximity effect: Another similar effect that also increases the resistance of the wire at high frequencies isproximity effect, which occurs in parallel wires that lie close to each other. The individual magnetic field ofadjacent turns induces eddy currents in the wire of the coil, which causes the current in the conductor to beconcentrated in a thin strip on the side near the adjacent wire. Like skin effect, this reduces the effectivecross-sectional area of the wire conducting current, increasing its resistance.

Parasitic capacitance: The capacitance between individual wire turnsof the coil, called parasitic capacitance, does not cause energy losses butcan change the behavior of the coil. Each turn of the coil is at a slightlydifferent potential, so the electric field between neighboring turns storescharge on the wire, so the coil acts as if it has a capacitor in parallelwith it. At a high enough frequency this capacitance can resonate withthe inductance of the coil forming a tuned circuit, causing the coil tobecome self-resonant.

To reduce parasitic capacitance and proximity effect, RF coils areconstructed to avoid having many turns lying close together, parallel to oneanother. The windings of RF coils are often limited to a single layer, andthe turns are spaced apart. To reduce resistance due to skin effect, inhigh-power inductors such as those used in transmitters the windings aresometimes made of a metal strip or tubing which has a larger surface area,and the surface is silver-plated. Basket-weave coils: To reduce proximity effect and parasitic

capacitance, multilayer RF coils are wound in patterns in whichsuccessive turns are not parallel but crisscrossed at an angle; these areoften called honeycomb or basket-weave coils.

Spiderweb coils: Another construction technique with similaradvantages is flat spiral coils. These are often wound on a flat insulating support with radial spokes or slots, withthe wire weaving in and out through the slots; these are called spiderweb coils. The form has an odd number ofslots, so successive turns of the spiral lie on opposite sides of the form, increasing separation.

Litz wire: To reduce skin effect losses, some coils are wound with a special type of radio frequency wire called litz wire. Instead of a single solid conductor, litz wire consists of several smaller wire strands that carry the

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• Inductor 15

current. Unlike ordinary stranded wire, the strands are insulated from each other, to prevent skin effect fromforcing the current to the surface, and are braided together. The braid pattern ensures that each wire strand spendsthe same amount of its length on the outside of the braid, so skin effect distributes the current equally between thestrands, resulting in a larger cross-sectional conduction area than an equivalent single wire.

Ferromagnetic core inductor

A variety of types of ferrite core inductors andtransformers

Ferromagnetic-core or iron-core inductors use a magnetic core made ofa ferromagnetic or ferrimagnetic material such as iron or ferrite toincrease the inductance. A magnetic core can increase the inductanceof a coil by a factor of several thousand, by increasing the magneticfield due to its higher magnetic permeability. However the magneticproperties of the core material cause several side effects which alter thebehavior of the inductor and require special construction:

Core losses: A time-varying current in a ferromagnetic inductor,which causes a time-varying magnetic field in its core, causesenergy losses in the core material that are dissipated as heat, due totwo processes:

Eddy currents: From Faraday's law of induction, the changingmagnetic field can induce circulating loops of electric current inthe conductive metal core. The energy in these currents isdissipated as heat in the resistance of the core material. The amount of energy lost increases with the areainside the loop of current.

Hysteresis: Changing or reversing the magnetic field in the core also causes losses due to the motion of the tinymagnetic domains it is composed of. The energy loss is proportional to the area of the hysteresis loop in theBH graph of the core material. Materials with low coercivity have narrow hysteresis loops and so lowhysteresis losses.For both of these processes, the energy loss per cycle of alternating current is constant, so core losses increaselinearly with frequency. Online core loss calculators[2] are available to calculate the energy loss. Using inputssuch as input voltage, output voltage, output current, frequency, ambient temperature, and inductance thesecalculators can predict the losses of the inductors core and AC/DC based on the operating condition of thecircuit being used.[3]

Nonlinearity: If the current through a ferromagnetic core coil is high enough that the magnetic core saturates, theinductance will not remain constant but will change with the current through the device. This is callednonlinearity and results in distortion of the signal. For example, audio signals can suffer intermodulationdistortion in saturated inductors. To prevent this, in linear circuits the current through iron core inductors must belimited below the saturation level. Some laminated cores have a narrow air gap in them for this purpose, andpowdered iron cores have a distributed air gap. This allows higher levels of magnetic flux and thus highercurrents through the inductor before it saturates.[4]

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Laminated core inductor

Laminated iron core ballastinductor for a metal halide

lamp

Low-frequency inductors are often made with laminated cores to prevent eddycurrents, using construction similar to transformers. The core is made of stacks of thinsteel sheets or laminations oriented parallel to the field, with an insulating coating onthe surface. The insulation prevents eddy currents between the sheets, so any remainingcurrents must be within the cross sectional area of the individual laminations, reducingthe area of the loop and thus reducing the energy losses greatly. The laminations aremade of low-coercivity silicon steel, to reduce hysteresis losses.

Ferrite-core inductor

For higher frequencies, inductors are made with cores of ferrite. Ferrite is a ceramicferrimagnetic material that is nonconductive, so eddy currents cannot flow within it.The formulation of ferrite is xxFe2O4 where xx represents various metals. For inductor cores soft ferrites are used,which have low coercivity and thus low hysteresis losses. Another similar material is powdered iron cemented with abinder.

Toroidal core inductor

Toroidal inductor in the power supplyof a wireless router

In an inductor wound on a straight rod-shaped core, the magnetic field linesemerging from one end of the core must pass through the air to reenter the coreat the other end. This reduces the field, because much of the magnetic fieldpath is in air rather than the higher permeability core material. A highermagnetic field and inductance can be achieved by forming the core in a closedmagnetic circuit. The magnetic field lines form closed loops within the corewithout leaving the core material. The shape often used is a toroidal ordoughnut-shaped ferrite core. Because of their symmetry, toroidal cores allowa minimum of the magnetic flux to escape outside the core (called leakageflux), so they radiate less electromagnetic interference than other shapes. Toroidal core coils are manufactured ofvarious materials, primarily ferrite, powdered iron and laminated cores.[5]

Variable inductor

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(left) Inductor with a threaded ferrite slug (visible at top) that can be turned to move it into or out of the coil. 4.2 cmhigh. (right) A variometer used in radio receivers in the 1920sProbably the most common type of variable inductor today is one with a moveable ferrite magnetic core, which canbe slid or screwed in or out of the coil. Moving the core farther into the coil increases the permeability, increasingthe magnetic field and the inductance. Many inductors used in radio applications (usually less than 100MHz) useadjustable cores in order to tune such inductors to their desired value, since manufacturing processes have certaintolerances (inaccuracy). Sometimes such cores for frequencies above 100MHz are made from highly conductivenon-magnetic material such as aluminum.[citation needed] They decrease the inductance because the magnetic fieldmust bypass them.Air core inductors can use sliding contacts or multiple taps to increase or decrease the number of turns included inthe circuit, to change the inductance. A type much used in the past but mostly obsolete today has a spring contactthat can slide along the bare surface of the windings. The disadvantage of this type is that the contact usuallyshort-circuits one or more turns. These turns act like a single-turn short-circuited transformer secondary winding; thelarge currents induced in them cause power losses.A type of continuously variable air core inductor is the variometer. This consists of two coils with the same numberof turns connected in series, one inside the other. The inner coil is mounted on a shaft so its axis can be turned withrespect to the outer coil. When the two coils' axes are collinear, with the magnetic fields pointing in the samedirection, the fields add and the inductance is maximum. When the inner coil is turned so its axis is at an angle withthe outer, the mutual inductance between them is smaller so the total inductance is less. When the inner coil is turned180 so the coils are collinear with their magnetic fields opposing, the two fields cancel each other and theinductance is very small. This type has the advantage that it is continuously variable over a wide range. It is used inantenna tuners and matching circuits to match low frequency transmitters to their antennas.Another method to control the inductance without any moving parts requires an additional DC current bias windingwhich controls the permeability of an easily saturable core material. See Magnetic amplifier.

Circuit theoryThe effect of an inductor in a circuit is to oppose changes in current through it by developing a voltage across itproportional to the rate of change of the current. An ideal inductor would offer no resistance to a constant directcurrent; however, only superconducting inductors have truly zero electrical resistance.The relationship between the time-varying voltage v(t) across an inductor with inductance L and the time-varyingcurrent i(t) passing through it is described by the differential equation:

When there is a sinusoidal alternating current (AC) through an inductor, a sinusoidal voltage is induced. Theamplitude of the voltage is proportional to the product of the amplitude (IP) of the current and the frequency (f) of thecurrent.

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In this situation, the phase of the current lags that of the voltage by /2.If an inductor is connected to a direct current source with value I via a resistance R, and then the current source isshort-circuited, the differential relationship above shows that the current through the inductor will discharge with anexponential decay:

Laplace circuit analysis (s-domain)When using the Laplace transform in circuit analysis, the impedance of an ideal inductor with no initial current isrepresented in the s domain by:

whereis the inductance, and

is the complex frequency.If the inductor does have initial current, it can be represented by: adding a voltage source in series with the inductor, having the value:

whereis the inductance, andis the initial current in the inductor.

(Note that the source should have a polarity that is aligned with the initial current) or by adding a current source in parallel with the inductor, having the value:

whereis the initial current in the inductor.

is the complex frequency.

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Inductor networksInductors in a parallel configuration each have the same potential difference (voltage). To find their total equivalentinductance (Leq):

The current through inductors in series stays the same, but the voltage across each inductor can be different. The sumof the potential differences (voltage) is equal to the total voltage. To find their total inductance:

These simple relationships hold true only when there is no mutual coupling of magnetic fields between individualinductors.

Stored energyNeglecting losses, the energy (measured in joules, in SI) stored by an inductor is equal to the amount of workrequired to establish the current through the inductor, and therefore the magnetic field. This is given by:

where L is inductance and I is the current through the inductor.This relationship is only valid for linear (non-saturated) regions of the magnetic flux linkage and current relationship.In general if one decides to find the energy stored in a LTI inductor that has initial current in a specific time between

and can use this:

That we have

where

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Q factorAn ideal inductor would have no resistance or energy losses. However, real inductors have winding resistance fromthe metal wire forming the coils. Since the winding resistance appears as a resistance in series with the inductor, it isoften called the series resistance. The inductor's series resistance converts electric current through the coils into heat,thus causing a loss of inductive quality. The quality factor (or Q) of an inductor is the ratio of its inductive reactanceto its resistance at a given frequency, and is a measure of its efficiency. The higher the Q factor of the inductor, thecloser it approaches the behavior of an ideal, lossless, inductor. High Q inductors are used with capacitors to makeresonant circuits in radio transmitters and receivers. The higher the Q is, the narrower the bandwidth of the resonantcircuit.The Q factor of an inductor can be found through the following formula, where L is the inductance, R is theinductor's effective series resistance, is the radian operating frequency, and the product L is the inductivereactance:

Notice that Q increases linearly with frequency if L and R are constant. Although they are constant at lowfrequencies, the parameters vary with frequency. For example, skin effect, proximity effect, and core losses increaseR with frequency; winding capacitance and variations in permeability with frequency affect L.Qualitatively at low frequencies and within limits, increasing the number of turns N improves Q because L varies asN2 while R varies linearly with N. Similarly, increasing the radius r of an inductor improves Q because L varies as r2

while R varies linearly with r. So high Q air core inductors often have large diameters and many turns. Both of thoseexamples assume the diameter of the wire stays the same, so both examples use proportionally more wire (copper). Ifthe total mass of wire is held constant, then there would be no advantage to increasing the number of turns or theradius of the turns because the wire would have to be proportionally thinner.Using a high permeability ferromagnetic core can greatly increase the inductance for the same amount of copper, sothe core can also increase the Q. Cores however also introduce losses that increase with frequency. The core materialis chosen for best results for the frequency band. At VHF or higher frequencies an air core is likely to be used.Inductors wound around a ferromagnetic core may saturate at high currents, causing a dramatic decrease ininductance (and Q). This phenomenon can be avoided by using a (physically larger) air core inductor. A welldesigned air core inductor may have a Q of several hundred.

Inductance formulaeThe table below lists some common simplified formulas for calculating the approximate inductance of severalinductor constructions.

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Construction Formula Notes

Cylindrical air-core coil[]

L = inductance in henries (H) 0 = permeability of free space = 4 10

7 H/m K = Nagaoka coefficient[]

N = number of turns A = area of cross-section of the coil in square metres (m2) l = length of coil in metres (m)

Straight wire conductor[6]

L = inductance l = cylinder length c = cylinder radius 0 = permeability of free space = 4 10

7 H/m = conductor permeability p = resistivity = phase rate

Exact if = 0 or =

L = inductance (nH)[7][8]

l = length of conductor (mm) d = diameter of conductor (mm) f = frequency

Cu or Al (i.e., relativepermeability is one)

l > 100 d[9]

d2 f > 1mm2 MHz

L = inductance (nH)[10][8]

l = length of conductor (mm) d = diameter of conductor (mm) f = frequency

Cu or Al (i.e., relativepermeability is one)

l > 100 d[9]

d2 f < 1mm2 MHz

Short air-core cylindricalcoil

[11]

L = inductance (H) r = outer radius of coil (in) l = length of coil (in) N = number of turns

Multilayer air-core coil[citationneeded]

L = inductance (H) r = mean radius of coil (in) l = physical length of coil winding (in) N = number of turns d = depth of coil (outer radius minus inner radius) (in)

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Flat spiral air-corecoil

[12][citation needed]

L = inductance (H) r = mean radius of coil (cm) N = number of turns d = depth of coil (outer radius minus inner radius) (cm)

L = inductance (H) r = mean radius of coil (in) N = number of turns d = depth of coil (outer radius minus inner radius) (in)

accurate to within 5 percent ford > 0.2 r.[]

Toroidal core (circularcross-section)

[13] L = inductance (H) d = diameter of coil winding (in) N = number of turns D = 2 * radius of revolution (in)

L = inductance (H) d = diameter of coil winding (in) N = number of turns D = 2 * radius of revolution (in)

approximation when d < 0.1 D

Toroidal core (rectangularcross-section)

[]

L = inductance (H) d1 = inside diameter of toroid (in) d2 = outside diameter of toroid (in) N = number of turns h = height of toroid (in)

Notes[7][7] , subst. radius = d/2 and cgs units[8][8] , convert to natural logarithms and inches to mm.[9] states for l < 100 d, include d/2l within the parentheses.[10][10] , subst. radius = d/2 and cgs units[11][11] ARRL Handbook, 66th Ed. American Radio Relay League (1989).[12][12] For the second formula, which cites to .

References Terman, Frederick (1943). Radio Engineers' Handbook. McGraw-Hill Wheeler, H. A. (October, 1938). "Simple Inductance Formulae for Radio Coils". Proc. I. R. E. 16 (10): 1398. doi:

10.1109/JRPROC.1928.221309 (http:/ / dx. doi. org/ 10. 1109/ JRPROC. 1928. 221309)

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External linksGeneral How stuff works (http:/ / electronics. howstuffworks. com/ inductor1. htm) The initial concept, made very simple Capacitance and Inductance (http:/ / www. lightandmatter. com/ html_books/ 4em/ ch07/ ch07. html) A chapter

from an online textbook Spiral inductor models (http:/ / www. mpdigest. com/ issue/ Articles/ 2005/ aug2005/ agilent/ Default. asp).

Article on inductor characteristics and modeling. Online coil inductance calculator (http:/ / www. 66pacific. com/ calculators/ coil_calc. aspx). Online calculator

calculates the inductance of conventional and toroidal coils using formulas 3, 4, 5, and 6, above. AC circuits (http:/ / www. phys. unsw. edu. au/ ~jw/ AC. html) Understanding coils and transforms (http:/ / www. mikroe. com/ en/ books/ keu/ 03. htm) Bowley, Roger (2009). "Inductor" (http:/ / www. sixtysymbols. com/ videos/ inductor. htm). Sixty Symbols. Brady

Haran for the University of Nottingham.

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Capacitor

Capacitor

Miniature low-voltage capacitors (next to a cm ruler)

Electronic symbol

A typical electrolytic capacitor

4 electrolytic capacitors of different voltages andcapacitance

A capacitor (originally known as a condenser) is a passivetwo-terminal electrical component used to store energy electrostaticallyin an electric field. The forms of practical capacitors vary widely, butall contain at least two electrical conductors separated by a dielectric(insulator); for example, one common construction consists of metalfoils separated by a thin layer of insulating film. Capacitors are widelyused as parts of electrical circuits in many common electrical devices.

When there is a potential difference (voltage) across the conductors, astatic electric field develops across the dielectric, causing positivecharge to collect on one plate and negative charge on the other plate.Energy is stored in the electrostatic field. An ideal capacitor ischaracterized by a single constant value, capacitance. This is the ratioof the electric charge on each conductor to the potential differencebetween them. The SI unit of capacitance is the farad, which is equal toone coulomb per volt.

The capacitance is greatest when there is a narrow separation betweenlarge areas of conductor, hence capacitor conductors are often calledplates, referring to an early means of construction. In practice, thedielectric between the plates passes a small amount of leakage currentand also has an electric field strength limit, the breakdown voltage. Theconductors and leads introduce an undesired inductance and resistance.

Capacitors are widely used in electronic circuits for blocking directcurrent while allowing alternating current to pass. In analog filter networks, they smooth the output of powersupplies. In resonant circuits they tune radios to particular frequencies. In electric power transmission systems theystabilize voltage and power flow.[1]

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History

Battery of four Leyden jars inMuseum Boerhaave, Leiden, the

Netherlands.

In October 1745, Ewald Georg von Kleist of Pomerania in Germany found thatcharge could be stored by connecting a high-voltage electrostatic generator by awire to a volume of water in a hand-held glass jar.[2] Von Kleist's hand and thewater acted as conductors, and the jar as a dielectric (although details of themechanism were incorrectly identified at the time). Von Kleist found thattouching the wire resulted in a powerful spark, much more painful than thatobtained from an electrostatic machine. The following year, the Dutch physicistPieter van Musschenbroek invented a similar capacitor, which was named theLeyden jar, after the University of Leiden where he worked.[3] He also wasimpressed by the power of the shock he received, writing, "I would not take asecond shock for the kingdom of France."[4]

Daniel Gralath was the first to combine several jars in parallel into a "battery" toincrease the charge storage capacity. Benjamin Franklin investigated the Leydenjar and came to the conclusion that the charge was stored on the glass, not in thewater as others had assumed. He also adopted the term "battery",[5][6] (denoting

the increasing of power with a row of similar units as in a battery of cannon), subsequently applied to clusters ofelectrochemical cells.[7] Leyden jars were later made by coating the inside and outside of jars with metal foil, leavinga space at the mouth to prevent arcing between the foils.[citation needed] The earliest unit of capacitance was the jar,equivalent to about 1 nanofarad.[8]

Leyden jars or more powerful devices employing flat glass plates alternating with foil conductors were usedexclusively up until about 1900, when the invention of wireless (radio) created a demand for standard capacitors, andthe steady move to higher frequencies required capacitors with lower inductance. A more compact constructionbegan to be used of a flexible dielectric sheet such as oiled paper sandwiched between sheets of metal foil, rolled orfolded into a small package.Early capacitors were also known as condensers, a term that is still occasionally used today. The term was first usedfor this purpose by Alessandro Volta in 1782, with reference to the device's ability to store a higher density ofelectric charge than a normal isolated conductor.[9]

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Theory of operation

Overview

Charge separation in a parallel-plate capacitorcauses an internal electric field. A dielectric(orange) reduces the field and increases the

capacitance.

A simple demonstration of a parallel-platecapacitor

A capacitor consists of two conductors separated by a non-conductiveregion.[10] The non-conductive region is called the dielectric. Insimpler terms, the dielectric is just an electrical insulator. Examples ofdielectric media are glass, air, paper, vacuum, and even asemiconductor depletion region chemically identical to the conductors.A capacitor is assumed to be self-contained and isolated, with no netelectric charge and no influence from any external electric field. Theconductors thus hold equal and opposite charges on their facingsurfaces,[11] and the dielectric develops an electric field. In SI units, acapacitance of one farad means that one coulomb of charge on eachconductor causes a voltage of one volt across the device.[12]

An ideal capacitor is wholly characterized by a constant capacitance C,defined as the ratio of charge Q on each conductor to the voltage Vbetween them:[10]

Because the conductors (or plates) are close together, the oppositecharges on the conductors attract one another due to their electricfields, allowing the capacitor to store more charge for a given voltagethan if the conductors were separated, giving the capacitor a largecapacitance.Sometimes charge build-up affects the capacitor mechanically, causingits capacitance to vary. In this case, capacitance is defined in terms ofincremental changes:

Hydraulic analogy

In the hydraulic analogy, a capacitor is analogousto a rubber membrane sealed inside a pipe. This

animation illustrates a membrane beingrepeatedly stretched and un-stretched by the flowof water, which is analogous to a capacitor beingrepeatedly charged and discharged by the flow of

charge.

In the hydraulic analogy, charge carriers flowing through a wire areanalogous to water flowing through a pipe. A capacitor is like a rubbermembrane sealed inside a pipe. Water molecules cannot pass throughthe membrane, but some water can move by stretching the membrane.The analogy clarifies a few aspects of capacitors:

The current alters the charge on a capacitor, just as the flow ofwater changes the position of the membrane. More specifically, theeffect of an electric current is to increase the charge of one plate ofthe capacitor, and decrease the charge of the other plate by an equal

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amount. This is just like how, when water flow moves the rubber membrane, it increases the amount of water onone side of the membrane, and decreases the amount of water on the other side.

The more a capacitor is charged, the larger its voltage drop; i.e., the more it "pushes back" against the chargingcurrent. This is analogous to the fact that the more a membrane is stretched, the more it pushes back on the water.

Charge can flow "through" a capacitor even though no individual electron can get from one side to the other.This is analogous to the fact that water can flow through the pipe even though no water molecule can pass throughthe rubber membrane. Of course, the flow cannot continue the same direction forever; the capacitor willexperience dielectric breakdown, and analogously the membrane will eventually break.

The capacitance describes how much charge can be stored on one plate of a capacitor for a given "push" (voltagedrop). A very stretchy, flexible membrane corresponds to a higher capacitance than a stiff membrane.

A charged-up capacitor is storing potential energy, analogously to a stretched membrane.

Energy of electric fieldWork must be done by an external influence to "move" charge between the conductors in a capacitor. When theexternal influence is removed, the charge separation persists in the electric field and energy is stored to be releasedwhen the charge is allowed to return to its equilibrium position. The work done in establishing the electric field, andhence the amount of energy stored, is[13]

Here Q is the charge stored in the capacitor, V is the voltage across the capacitor, and C is the capacitance.In the case of a fluctuating voltage V(t), the stored energy also fluctuates and hence power must flow into or out ofthe capacitor. This power can be found by taking the time derivative of the stored energy:

Currentvoltage relationThe current I(t) through any component in an electric circuit is defined as the rate of flow of a charge Q(t) passingthrough it, but actual chargeselectronscannot pass through the dielectric layer of a capacitor. Rather, an electronaccumulates on the negative plate for each one that leaves the positive plate, resulting in an electron depletion andconsequent positive charge on one electrode that is equal and opposite to the accumulated negative charge on theother. Thus the charge on the electrodes is equal to the integral of the current as well as proportional to the voltage,as discussed above. As with any antiderivative, a constant of integration is added to represent the initial voltageV(t0). This is the integral form of the capacitor equation:

[14]

Taking the derivative of this and multiplying by C yields the derivative form:[15]

The dual of the capacitor is the inductor, which stores energy in a magnetic field rather than an electric field. Itscurrent-voltage relation is obtained by exchanging current and voltage in the capacitor equations and replacing Cwith the inductanceL.

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DC circuits

A simple resistor-capacitor circuit demonstratescharging of a capacitor.

A series circuit containing only a resistor, a capacitor, a switch and aconstant DC source of voltage V0 is known as a charging circuit.

[] Ifthe capacitor is initially uncharged while the switch is open, and theswitch is closed at t0, it follows from Kirchhoff's voltage law that

Taking the derivative and multiplying by C, gives a first-orderdifferential equation:

At t = 0, the voltage across the capacitor is zero and the voltage across the resistor is V0. The initial current is thenI(0)=V0/R. With this assumption, solving the differential equation yields

where 0= RC is the time constant of the system. As the capacitor reaches equilibrium with the source voltage, thevoltages across the resistor and the current through the entire circuit decay exponentially. The case of discharging acharged capacitor likewise demonstrates exponential decay, but with the initial capacitor voltage replacing V0 andthe final voltage being zero.

AC circuitsImpedance, the vector sum of reactance and resistance, describes the phase difference and the ratio of amplitudesbetween sinusoidally varying voltage and sinusoidally varying current at a given frequency. Fourier analysis allowsany signal to be constructed from a spectrum of frequencies, whence the circuit's reaction to the various frequenciesmay be found. The reactance and impedance of a capacitor are respectively

where j is the imaginary unit and is the angular frequency of the sinusoidal signal. The j phase indicates that theAC voltage V= ZI lags the AC current by 90: the positive current phase corresponds to increasing voltage as thecapacitor charges; zero current corresponds to instantaneous constant voltage, etc.Impedance decreases with increasing capacitance and increasing frequency. This implies that a higher-frequencysignal or a larger capacitor results in a lower voltage amplitude per current amplitudean AC "short circuit" or ACcoupling. Conversely, for very low frequencies, the reactance will be high, so that a capacitor is nearly an opencircuit in AC analysisthose frequencies have been "filtered out".Capacitors are different from resistors and inductors in that the impedance is inversely proportional to the definingcharacteristic; i.e., capacitance.

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Laplace circuit analysis (s-domain)When using the Laplace transform in circuit analysis, the capacitance of an ideal capacitor with no initial charge isrepresented in the s domain by:

where C is the capacitance, and s is the complex frequency.

Parallel-plate model

Dielectric is placed between two conductingplates, each of area A and with a separation of d

The simplest capacitor consists of two parallel conductive platesseparated by a dielectric with permittivity (such as air). The modelmay also be used to make qualitative predictions for other devicegeometries. The plates are considered to extend uniformly over an areaA and a charge density = Q/A exists on their surface. Assumingthat the width of the plates is much greater than their separation d, theelectric field near the centre of the device will be uniform with themagnitude E = /. The voltage is defined as the line integral of theelectric field between the plates

Solving this for C = Q/V reveals that capacitance increases with area and decreases with separation

The capacitance is therefore greatest in devices made from materials with a high permittivity, large plate area, andsmall distance between plates.A parallel plate capacitor can only store a finite amount of energy before dielectric breakdown occurs. Thecapacitor's dielectric material has a dielectric strength Ud which sets the capacitor's breakdown voltage at V= Vbd =Udd. The maximum energy that the capacitor can store is therefore

We see that the maximum energy is a function of dielectric volume, permittivity, and dielectric strength per distance.So increasing the plate area while decreasing the separation between the plates while maintaining the same volumehas no change on the amount of energy the capacitor can store. Care must be taken when increasing the plateseparation so that the above assumption of the distance between plates being much smaller than the area of the platesis still valid for these equations to be accurate. In addition, these equations assume that the electric field is entirelyconcentrated in the dielectric between the plates. In reality there are fringing fields outside the dielectric, for examplebetween the sides of the capacitor plates, which will increase the effective capacitance of the capacitor. This could beseen as a form of parasitic capacitance. For some simple capacitor geometries this additional capacitance term can becalculated analytically.[] It becomes negligibly small when the ratio of plate area to separation is large.

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Several capacitors in parallel.

Networks

For capacitors in parallelCapacitors in a parallel configuration each have the same appliedvoltage. Their capacitances add up. Charge is apportionedamong them by size. Using the schematic diagram to visualizeparallel plates, it is apparent that each capacitor contributes tothe total surface area.

For capacitors in series

Several capacitors in series.

Connected in series, the schematic diagram reveals that theseparation distance, not the plate area, adds up. The capacitorseach store instantaneous charge build-up equal to that of everyother capacitor in the series. The total voltage difference fromend to end is apportioned to each capacitor according to theinverse of its capacitance. The entire series acts as a capacitorsmaller than any of its components.

Capacitors are combined in series to achieve a higher working voltage, for example for smoothing a highvoltage power supply. The voltage ratings, which are based on plate separation, add up, if capacitance andleakage currents for each capacitor are identical. In such an application, on occasion series strings areconnected in parallel, forming a matrix. The goal is to maximize the energy storage of the network withoutoverloading any capacitor. For high-energy storage with capacitors in series, some safety considerations mustbe applied to ensure one capacitor failing and leaking current will not apply too much voltage to the otherseries capacitors.

Voltage distribution in parallel-to-series networks.

To model the distribution of voltages from a single charged capacitor connected in parallel to a chain ofcapacitors in series :

Note: This is only correct if all capacitance values are equal.The power transferred in this arrangement is:

Series connection is also sometimes used to adapt polarized electrolytic capacitors for bipolar AC use. Two identical polarized electrolytic capacitors are connected back to back to form a bipolar capacitor with half the nominal capacitance of either.[16] However, the anode film can only withstand a small reverse voltage.[17] This arrangement can lead to premature failure as the anode film is broken down during the reverse-conduction phase and partially rebuilt during the forward phase.[18] A factory-made non-polarized electrolytic capacitor has both plates anodized so that it can withstand rated voltage in both directions; such capacitors also have

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about half the capacitance per unit volume of polarized capacitors.

Non-ideal behaviorCapacitors deviate from the ideal capacitor equation in a number of ways. Some of these, such as leakage current andparasitic effects are linear, or can be assumed to be linear, and can be dealt with by adding virtual components to theequivalent circuit of the capacitor. The usual methods of network analysis can then be applied. In other cases, suchas with breakdown voltage, the effect is non-linear and normal (i.e., linear) network analysis cannot be used, theeffect must be dealt with separately. There is yet another group, which may be linear but invalidate the assumption inthe analysis that capacitance is a constant. Such an example is temperature dependence. Finally, combined parasiticeffects such as inherent inductance, resistance, or dielectric losses can exhibit non-uniform behavior at variablefrequencies of operation.

Breakdown voltageAbove a particular electric field, known as the dielectric strength Eds, the dielectric in a capacitor becomesconductive. The voltage at which this occurs is called the breakdown voltage of the device, and is given by theproduct of the dielectric strength and the separation between the conductors,[19]

The maximum energy that can be stored safely in a capacitor is limited by the breakdown voltage. Due to the scalingof capacitance and breakdown voltage with dielectric thickness, all capacitors made with a particular dielectric haveapproximately equal maximum energy density, to the extent that the dielectric dominates their volume.[20]

For air dielectric capacitors the breakdown field strength is of the order 2 to 5 MV/m; for mica the breakdown is 100to 300 MV/m, for oil 15 to 25 MV/m, and can be much less when other materials are used for the dielectric.[21] Thedielectric is used in very thin layers and so absolute breakdown voltage of capacitors is limited. Typical ratings forcapacitors used for general electronics applications range from a few volts to 1 kV. As the voltage increases, thedielectric must be thicker, making high-voltage capacitors larger per capacitance than those rated for lower voltages.The breakdown voltage is critically affected by factors such as the geometry of the capacitor conductive parts; sharpedges or points increase the electric field strength at that point and can lead to a local breakdown. Once this starts tohappen, the breakdown quickly tracks through the dielectric until it reaches the opposite plate, leaving carbon behindcausing a short circuit.[22]

The usual breakdown route is that the field strength becomes large enough to pull electrons in the dielectric fromtheir atoms thus causing conduction. Other scenarios are possible, such as impurities in the dielectric, and, if thedielectric is of a crystalline nature, imperfections in the crystal structure can result in an avalanche breakdown asseen in semi-conductor devices. Breakdown voltage is also affected by pressure, humidity and temperature.[23]

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• Capacitor 32

Equivalent circuit

Two different circuit models of a real capacitor

An ideal capacitor only stores and releases electrical energy, withoutdissipating any. In reality, all capacitors have imperfections within thecapacitor's material that create resistance. This is specified as theequivalent series resistance or ESR of a component. This adds a realcomponent to the impedance:

As frequency approaches infinity, the capacitive impedance (orreactance) approaches zero and the ESR becomes significant. As thereactance becomes negligible, power dissipation approaches PRMS =VRMS /RESR.

Similarly to ESR, the capacitor's leads add equivalent seriesinductance or ESL to the component. This is usually significant only atrelatively high frequencies. As inductive reactance is positive and increases with frequency, above a certainfrequency capacitance will be canceled by inductance. High-frequency engineering involves accounting for theinductance of all connections and components.

If the conductors are separated by a material with a small conductivity rather than a perfect dielectric, then a smallleakage current flows directly between them. The capacitor therefore has a finite parallel resistance,[12] and slowlydischarges over time (time may vary greatly depending on the capacitor material and quality).

Q factorThe quality factor (or Q) of a capacitor is the ratio of its reactance to its resistance at a given frequency, and is ameasure of its efficiency. The higher the Q factor of the capacitor, the closer it approaches the behavior of an ideal,lossless, capacitor.The Q factor of a capacitor can be found through the following formula:

Where: is frequency in radians per second, is the capacitance, is the capacitive reactance, and is the series resistance of the capacitor.

Ripple currentRipple current is the AC component of an applied source (often a switched-mode power supply) (whose frequencymay be constant or varying). Ripple current causes heat to be generated within the capacitor due to the dielectriclosses caused by the changing field strength together with the current flow across the slightly resistive supply lines orthe electrolyte in the capacitor. The equivalent series resistance (ESR) is the amount of internal series resistance onewould add to a perfect capacitor to model this. Some types of capacitors, primarily tantalum and aluminumelectrolytic capacitors, as well as some film capacitors have a specified rating value for maximum ripple current. Tantalum electrolytic capacitors with solid manganese dioxide electrolyte are limited by ripple current and

generally have the highest ESR ratings in the capacitor family. Exceeding their ripple limits can lead to shorts andburning parts.

Ramesh Singh, DTU (Formerly Delhi College of Engineering)[email protected]

• Capacitor 33

Aluminium electrolytic capacitors, the most common type of electrolytic, suffer a shortening of life expectancy athigher ripple currents. If ripple current exceeds the rated value of the capacitor, it tends to result in explosivefailure.

Ceramic capacitors generally have no ripple current limitation and have some of the lowest ESR ratings. Film capacitors have very low ESR ratings but exceeding rated ripple current may cause degradation failures.

Capacitance instabilityThe capacitance of certain capacitors decreases as the component ages. In ceramic capacitors, this is caused bydegradation of the dielectric. The type of dielectric, ambient operating and storage temperatures are the mostsignificant aging factors, while the operating voltage has a smaller effect. The aging process may be reversed byheating the component above the Curie point. Aging is fastest near the beginning of life of the component, and thedevice stabilizes over time.[24] Electrolytic capacitors age as the electrolyte evaporates. In contrast with ceramiccapacitors, this occurs towards the end of life of the component.Temperature dependence of capacitance is usually expressed in parts per million (ppm) per C. It can usually betaken as a broadly linear function but can be noticeably non-linear at the temperature extremes. The temperaturecoefficient can be either positive or negative, sometimes even amongst different samples of the same type. In otherwords, the spread in the range of temperature coefficients can encompass zero. See the data sheet in the leakagecurrent section above for an example.Capacitors, especially ceramic capacitors, and older designs such as paper capacitors, can absorb sound wavesresulting in a microphonic effect. Vibration moves the plates, causing the capacitance to vary, in turn inducing ACcurrent. Some dielectrics also generate piezoelectricity. The resulting interference is especially problematic in audioapplications, potentially causing feedback or unintended recording. In the reverse microphonic effect, the varyingelectric field between the capacitor plates exerts a physical force, moving them as a speaker. This can generateaudible sound, but drains energy and stresses the dielectric and the electrolyte, if any.

Current and voltage reversalCurrent reversal occurs when the current changes direction. Voltage reversal is the change of polarity in a circuit.Reversal is generally described as the percentage of the maximum rated voltage that reverses polarity. In DC circuits,this will usually be less than 100% (often in the range of 0 to 90%), whereas AC circuits experience 100% reversal.In DC circuits and pulsed circuits, current and voltage reversal are affected by the damping of the system. Voltagereversal is encountered in RLC circuits that are under-damped. The current and voltage reverse direction, forming aharmonic oscillator between the inductance and capacitance. The current and voltage will tend to oscillate and mayreverse direction several times, with each peak being lower than the previous, until the system reaches anequilibrium. This is often referred to as ringing. In comparison, critically damped or over-damped systems usually donot experience a voltage reversal. Reversal is also encountered in AC circuits, where the peak current will be equalin each direction.For maximum life, capacitors usually need to be able to handle the maximum amount of reversal that a system willexperience. An AC circuit will experience 100% voltage reversal, while under-damped DC circuits will experienceless than 100%. Reversal creates excess electric fields in the dielectric, causes excess heating of both the dielectricand the conductors, and can dramatically shorten the life expectancy of the capacitor. Reversal ratings will oftenaffect the design considerations for the capacitor, from the choice of dielectric materials and voltage ratings to thetypes of internal connections used.[25]

Ramesh Singh, DTU (Formerly Delhi College of Engineering)[email protected]

• Capacitor 34

Dielectric absorptionCapacitors made with some types of dielectric material show "dielectric absorption" or "soakage". On discharging acapacitor and disconnecting it, after a short time it may develop a voltage due to hysteresis in the dielectric. Thiseffect can be objectionable in applications such as precision sample and hold circuits.

LeakageLeakage is equivalent to a resistor in parallel with the capacitor. Constant exposure to heat can cause dielectricbreakdown and excessive leakage, a problem often seen in older vacuum tube circuits, particularly where oiled paperand foil capacitors were used. In many vacuum tube circuits, interstage coupling capacitors are used to conduct avarying signal from the plate of one tube to the grid circuit of the next stage. A leaky capacitor can cause the gridcircuit voltage to be raised from its normal bias setting, causing excessive current or signal distortion in thedownstream tube. In power amplifiers this can cause the plates to glow red, or current limiting resistors to overheat,even fail. Similar considerations apply to component fabricated solid-state (transistor) amplifiers, but owing to lowerheat production and the use of modern polyester dielectric barriers this once-common problem has become relativelyrare.

Electrolytic failure from disuseElectrolytic capacitors are conditioned when manufactured by applying a voltage sufficient to initiate the properinternal chemical state. This state is maintained by regular use of the equipment. If a system using electrolyticcapacitors is unused for a long period of time it can lose its conditioning, and will generally fail with a short circuitwhen next operated, permanently damaging the capacitor. To prevent this in tube equipment, the voltage can beslowly brought up using a variable transformer (variac) on the mains, over a twenty or thirty minute interval.Transistor equipment is more problematic as such equipment may be sensitive to low voltage ("brownout")conditions, with excessive currents due to improper bias in some circuits.[citation needed]

Capacitor typesPractical capacitors are available commercially in many different forms. The type of internal dielectric, the structureof the plates and the device packaging all strongly affect the characteristics of the capacitor, and its applications.Values available range from very low (picofarad range; while arbitrarily low values are in principle possible, stray(parasitic) capacitance in any circuit is the limiting factor) to about 5kF supercapacitors.Above approximately 1 microfarad electrolytic capacitors are usually used because of their small size and low costcompared with other technologies, unless their relatively poor stability, life and polarised nature make themunsuitable. Very high capacity supercapacitors use a porous carbon-based electrode material.

Dielectric materials

Capacitor materials. From left: multilayerceramic, ceramic disc, multilayer polyester film,tubular ceramic, polystyrene, metalized polyesterfilm, aluminum electrolytic. Major scale divisions

are in centimetres.

Most types of capacitor include a dielectric spacer, which increasestheir capacitance. These dielectrics are most often insulators. However,low capacitance devices are available with a vacuum between theirplates, which allows extremely high voltage operation and low losses.Variable capacitors with their plates open to the atmosphere werecommonly used in radio tuning circuits. Later designs use polymer foildielectric between the moving and stationary plates, with no significantair space between them.

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• Capacitor 35

In order to maximise the charge that a capacitor can hold, the dielectric material needs to have as high a permittivityas possible, while also having as high a breakdown voltage as possible.Several solid dielectrics are available, including paper, plastic, glass, mica and ceramic materials. Paper was usedextensively in older devices and offers relatively high voltage performance. However, it is susceptible to waterabsorption, and has been largely replaced by plastic film capacitors. Plastics offer better stability and agingperformance, which makes them useful in timer circuits, although they may be limited to low operating temperaturesand frequencies. Ceramic capacitors are generally small, cheap and useful for high frequency applications, althoughtheir capacitance varies strongly with voltage and they age poorly. They are broadly categorized as class 1dielectrics, which have predictable variation of capacitance with temperature or class 2 dielectrics, which can operateat higher voltage. Glass and mica capacitors are extremely reliable, stable and tolerant to high temperatures andvoltages, but are too expensive for most mainstream app