Basics of Power circuits

ContentsArticles

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

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

Article LicensesLicense 88

Ramesh Singh, DTU (Formerly Delhi College of Engineering)singhramesh2@gmail.com

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 (Ω), 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 I–V 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 I–V

curve. In other situations, the derivative may be most useful; this is called the "differential resistance".

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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 Ω resistor, as identified by its electronic color code(blue–green–black-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.

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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 [m²], σ (sigma) is the electrical conductivity measured in siemens per meter (S·m−1), andρ (rho) is the electrical resistivity (also called specific electrical resistance) of the material, measured in ohm-metres(Ω·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.

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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 (Ω)

1 meter of copper wirewith 1mm diameter

0.02[2]

1 km overhead powerline(typical)

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.

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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 I–V 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.

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Electrical resistance and conductance 6

AC circuits

Impedance and admittance

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, Kramers–Kronig relations, etc.)

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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

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Electrical resistance and conductance 8

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

The temperature coefficient is typically +3×10−3 K−1 to +6×10−3 K−1 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 I≠0. 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 1.7×10-8Ωm. 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 0.9Ω at

-40°C, to 0.1Ω at +40°C.[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, pp36–40, 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.

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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)

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Inductor 10

Inductor

Inductor

A selection of low-value inductors

Type Passive

Working principle Electromagnetic induction

First production Michael Faraday (1831)

Electronic symbol

Axial lead inductors (100uH)

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 Faraday’s 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.

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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 φ 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.

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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

Large 50 MVAR three-phase iron-core loadinginductor at German utility substation

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

Radio frequency inductor

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.

Spiderweb-wound coil for crystal radio

Adjustable ferrite slug RF coilusing basket winding and litz wire

• 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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Inductor 16

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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Inductor 17

(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 100 MHz) useadjustable cores in order to tune such inductors to their desired value, since manufacturing processes have certaintolerances (inaccuracy). Sometimes such cores for frequencies above 100 MHz 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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Inductor 18

In this situation, the phase of the current lags that of the voltage by π/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 19

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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Inductor 20

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, ω is the radian operating frequency, and the product ωL 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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Inductor 21

Construction Formula Notes

Cylindrical air-core coil[]

• L = inductance in henries (H)• μ0 = 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• μ0 = permeability of free space = 4 × 10−7 H/m• μ = conductor permeability• p = resistivity• ω = phase rate

Exact if ω = 0 or ω = ∞

• 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 > 1 mm2 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 < 1 mm2 MHz

Short air-core cylindricalcoil

[11]

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

Multilayer air-core coil[citation

needed]

• L = inductance (µH)• 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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Inductor 22

Flat spiral air-corecoil

[12][citation needed]

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

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

• L = inductance (µH)• 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 (µH)• 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 ρ = 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 ρ = 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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Inductor 23

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 24

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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Capacitor 25

Solid-body, resin-dipped 10 μF 35 V tantalumcapacitors. The + sign indicates the positive lead.

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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Capacitor 26

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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Capacitor 27

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:

Current–voltage 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 charges—electrons—cannot 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 inductance L.

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Capacitor 28

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 τ0 = 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 ω 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 amplitude—an 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 analysis—those 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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Capacitor 29

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 ε (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 ±ρ = ±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 = ρ/ε. 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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Capacitor 30

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 =VRMS² /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.

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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]

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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 5 kF 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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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 applications. Electrolytic capacitors and supercapacitors areused to store small and larger amounts of energy, respectively, ceramic capacitors are often used in resonators, andparasitic capacitance occurs in circuits wherever the simple conductor-insulator-conductor structure is formedunintentionally by the configuration of the circuit layout.Electrolytic capacitors use an aluminum or tantalum plate with an oxide dielectric layer. The second electrode is aliquid electrolyte, connected to the circuit by another foil plate. Electrolytic capacitors offer very high capacitancebut suffer from poor tolerances, high instability, gradual loss of capacitance especially when subjected to heat, andhigh leakage current. Poor quality capacitors may leak electrolyte, which is harmful to printed circuit boards. Theconductivity of the electrolyte drops at low temperatures, which increases equivalent series resistance. While widelyused for power-supply conditioning, poor high-frequency characteristics make them unsuitable for manyapplications. Electrolytic capacitors will self-degrade if unused for a period (around a year), and when full power isapplied may short circuit, permanently damaging the capacitor and usually blowing a fuse or causing failure ofrectifier diodes (for instance, in older equipment, arcing in rectifier tubes). They can be restored before use (anddamage) by gradually applying the operating voltage, often done on antique vacuum tube equipment over a period of30 minutes by using a variable transformer to supply AC power. Unfortunately, the use of this technique may be lesssatisfactory for some solid state equipment, which may be damaged by operation below its normal power range,requiring that the power supply first be isolated from the consuming circuits. Such remedies may not be applicable tomodern high-frequency power supplies as these produce full output voltage even with reduced input.Tantalum capacitors offer better frequency and temperature characteristics than aluminum, but higher dielectricabsorption and leakage.[26]

Polymer capacitors (OS-CON, OC-CON, KO, AO) use solid conductive polymer (or polymerized organicsemiconductor) as electrolyte and offer longer life and lower ESR at higher cost than standard electrolytic capacitors.A Feedthrough is a component that, while not serving as its main use, has capacitance and is used to conduct signalsthrough a circuit board.Several other types of capacitor are available for specialist applications. Supercapacitors store large amounts ofenergy. Supercapacitors made from carbon aerogel, carbon nanotubes, or highly porous electrode materials, offerextremely high capacitance (up to 5 kF as of 2010[27]) and can be used in some applications instead of rechargeablebatteries. Alternating current capacitors are specifically designed to work on line (mains) voltage AC power circuits.They are commonly used in electric motor circuits and are often designed to handle large currents, so they tend to bephysically large. They are usually ruggedly packaged, often in metal cases that can be easily grounded/earthed. Theyalso are designed with direct current breakdown voltages of at least five times the maximum AC voltage.

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Structure

Capacitor packages: SMD ceramic at top left;SMD tantalum at bottom left; through-hole

tantalum at top right; through-hole electrolytic atbottom right. Major scale divisions are cm.

The arrangement of plates and dielectric has many variationsdepending on the desired ratings of the capacitor. For small values ofcapacitance (microfarads and less), ceramic disks use metalliccoatings, with wire leads bonded to the coating. Larger values can bemade by multiple stacks of plates and disks. Larger value capacitorsusually use a metal foil or metal film layer deposited on the surface ofa dielectric film to make the plates, and a dielectric film ofimpregnated paper or plastic – these are rolled up to save space. Toreduce the series resistance and inductance for long plates, the platesand dielectric are staggered so that connection is made at the commonedge of the rolled-up plates, not at the ends of the foil or metalized filmstrips that comprise the plates.

The assembly is encased to prevent moisture entering the dielectric –early radio equipment used a cardboard tube sealed with wax. Modern paper or film dielectric capacitors are dippedin a hard thermoplastic. Large capacitors for high-voltage use may have the roll form compressed to fit into arectangular metal case, with bolted terminals and bushings for connections. The dielectric in larger capacitors isoften impregnated with a liquid to improve its properties.

Several axial-lead electrolytic capacitors

Capacitors may have their connecting leads arranged in manyconfigurations, for example axially or radially. "Axial" means that theleads are on a common axis, typically the axis of the capacitor'scylindrical body – the leads extend from opposite ends. Radial leadsmight more accurately be referred to as tandem; they are rarely actuallyaligned along radii of the body's circle, so the term is inexact, althoughuniversal. The leads (until bent) are usually in planes parallel to that ofthe flat body of the capacitor, and extend in the same direction; theyare often parallel as manufactured.

Small, cheap discoidal ceramic capacitors have existed since the 1930s,and remain in widespread use. Since the 1980s, surface mount packages for capacitors have been widely used. Thesepackages are extremely small and lack connecting leads, allowing them to be soldered directly onto the surface ofprinted circuit boards. Surface mount components avoid undesirable high-frequency effects due to the leads andsimplify automated assembly, although manual handling is made difficult due to their small size.

Mechanically controlled variable capacitors allow the plate spacing to be adjusted, for example by rotating or slidinga set of movable plates into alignment with a set of stationary plates. Low cost variable capacitors squeeze togetheralternating layers of aluminum and plastic with a screw. Electrical control of capacitance is achievable with varactors(or varicaps), which are reverse-biased semiconductor diodes whose depletion region width varies with appliedvoltage. They are used in phase-locked loops, amongst other applications.

Capacitor markingsMost capacitors have numbers printed on their bodies to indicate their electrical characteristics. Larger capacitorslike electrolytics usually display the actual capacitance together with the unit (for example, 220 μF). Smallercapacitors like ceramics, however, use a shorthand consisting of three numbers and a letter, where the numbers showthe capacitance in pF (calculated as XY × 10Z for the numbers XYZ) and the letter indicates the tolerance (J, K or Mfor ±5%, ±10% and ±20% respectively).

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Additionally, the capacitor may show its working voltage, temperature and other relevant characteristics.

ExampleA capacitor with the text 473K 330V on its body has a capacitance of 47 × 103 pF = 47 nF (±10%) with a workingvoltage of 330 V.

Applications

This mylar-film, oil-filled capacitor has very lowinductance and low resistance, to provide the

high-power (70 megawatt) and high speed (1.2microsecond) discharge needed to operate a dye

laser.

Energy storage

A capacitor can store electric energy when disconnected from itscharging circuit, so it can be used like a temporary battery. Capacitorsare commonly used in electronic devices to maintain power supplywhile batteries are being changed. (This prevents loss of information involatile memory.)

Conventional capacitors provide less than 360 joules per kilogram ofenergy density, whereas a conventional alkaline battery has a densityof 590 kJ/kg.

In car audio systems, large capacitors store energy for the amplifier touse on demand. Also for a flash tube a capacitor is used to hold thehigh voltage.

Pulsed power and weapons

Groups of large, specially constructed, low-inductance high-voltagecapacitors (capacitor banks) are used to supply huge pulses of currentfor many pulsed power applications. These include electromagneticforming, Marx generators, pulsed lasers (especially TEA lasers), pulseforming networks, radar, fusion research, and particle accelerators.

Large capacitor banks (reservoir) are used as energy sources for the exploding-bridgewire detonators or slapperdetonators in nuclear weapons and other specialty weapons. Experimental work is under way using banks ofcapacitors as power sources for electromagnetic armour and electromagnetic railguns and coilguns.

Power conditioning

A 10 millifarad capacitor in an amplifier powersupply

Reservoir capacitors are used in power supplies where they smooth theoutput of a full or half wave rectifier. They can also be used in chargepump circuits as the energy storage element in the generation of highervoltages than the input voltage.

Capacitors are connected in parallel with the power circuits of mostelectronic devices and larger systems (such as factories) to shunt awayand conceal current fluctuations from the primary power source toprovide a "clean" power supply for signal or control circuits. Audioequipment, for example, uses several capacitors in this way, to shuntaway power line hum before it gets into the signal circuitry. The

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Capacitor 38

capacitors act as a local reserve for the DC power source, and bypass AC currents from the power supply. This isused in car audio applications, when a stiffening capacitor compensates for the inductance and resistance of the leadsto the lead-acid car battery.

Power factor correction

A high-voltage capacitor bank usedfor power factor correction on a

power transmission system.

In electric power distribution, capacitors are used for power factor correction.Such capacitors often come as three capacitors connected as a three phase load.Usually, the values of these capacitors are given not in farads but rather as areactive power in volt-amperes reactive (var). The purpose is to counteractinductive loading from devices like electric motors and transmission lines tomake the load appear to be mostly resistive. Individual motor or lamp loads mayhave capacitors for power factor correction, or larger sets of capacitors (usuallywith automatic switching devices) may be installed at a load center within abuilding or in a large utility substation.

Suppression and coupling

Signal coupling

Polyester film capacitors are frequently used ascoupling capacitors.

Because capacitors pass AC but block DC signals (when charged up tothe applied dc voltage), they are often used to separate the AC and DCcomponents of a signal. This method is known as AC coupling or"capacitive coupling". Here, a large value of capacitance, whose valueneed not be accurately controlled, but whose reactance is small at thesignal frequency, is employed.

Decoupling

A decoupling capacitor is a capacitor used to protect one part of acircuit from the effect of another, for instance to suppress noise ortransients. Noise caused by other circuit elements is shunted through the capacitor, reducing the effect they have onthe rest of the circuit. It is most commonly used between the power supply and ground. An alternative name isbypass capacitor as it is used to bypass the power supply or other high impedance component of a circuit.

Noise filters and snubbers

When an inductive circuit is opened, the current through the inductance collapses quickly, creating a large voltage across the open circuit of the switch or relay. If the inductance is large enough, the energy will generate a spark, causing the contact points to oxidize, deteriorate, or sometimes weld together, or destroying a solid-state switch. A snubber capacitor across the newly opened circuit creates a path for this impulse to bypass the contact points, thereby preserving their life; these were commonly found in contact breaker ignition systems, for instance. Similarly, in smaller scale circuits, the spark may not be enough to damage the switch but will still radiate undesirable radio frequency interference (RFI), which a filter capacitor absorbs. Snubber capacitors are usually employed with a low-value resistor in series, to dissipate energy and minimize RFI. Such resistor-capacitor combinations are available

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Capacitor 39

in a single package.Capacitors are also used in parallel to interrupt units of a high-voltage circuit breaker in order to equally distributethe voltage between these units. In this case they are called grading capacitors.In schematic diagrams, a capacitor used primarily for DC charge storage is often drawn vertically in circuit diagramswith the lower, more negative, plate drawn as an arc. The straight plate indicates the positive terminal of the device,if it is polarized (see electrolytic capacitor).

Motor startersIn single phase squirrel cage motors, the primary winding within the motor housing is not capable of starting arotational motion on the rotor, but is capable of sustaining one. To start the motor, a secondary "start" winding has aseries non-polarized starting capacitor to introduce a lead in the sinusoidal current. When the secondary (start)winding is placed at an angle with respect to the primary (run) winding, a rotating electric field is created. The forceof the rotational field is not constant, but is sufficient to start the rotor spinning. When the rotor comes close tooperating speed, a centrifugal switch (or current-sensitive relay in series with the main winding) disconnects thecapacitor. The start capacitor is typically mounted to the side of the motor housing. These are called capacitor-startmotors, that have relatively high starting torque. Typically they can have up-to four times as much starting torquethan a split-phase motor and are used on applications such as compressors, pressure washers and any small devicerequiring high starting torques.Capacitor-run induction motors have a permanently connected phase-shifting capacitor in series with a secondwinding. The motor is much like a two-phase induction motor.Motor-starting capacitors are typically non-polarized electrolytic types, while running capacitors are conventionalpaper or plastic film dielectric types.

Signal processingThe energy stored in a capacitor can be used to represent information, either in binary form, as in DRAMs, or inanalogue form, as in analog sampled filters and CCDs. Capacitors can be used in analog circuits as components ofintegrators or more complex filters and in negative feedback loop stabilization. Signal processing circuits also usecapacitors to integrate a current signal.

Tuned circuits

Capacitors and inductors are applied together in tuned circuits to select information in particular frequency bands.For example, radio receivers rely on variable capacitors to tune the station frequency. Speakers use passive analogcrossovers, and analog equalizers use capacitors to select different audio bands.The resonant frequency f of a tuned circuit is a function of the inductance (L) and capacitance (C) in series, and isgiven by:

where L is in henries and C is in farads.

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Capacitor 40

SensingMost capacitors are designed to maintain a fixed physical structure. However, various factors can change thestructure of the capacitor, and the resulting change in capacitance can be used to sense those factors.Changing the dielectric:

The effects of varying the characteristics of the dielectric can be used for sensing purposes. Capacitors with anexposed and porous dielectric can be used to measure humidity in air. Capacitors are used to accuratelymeasure the fuel level in airplanes; as the fuel covers more of a pair of plates, the circuit capacitance increases.

Changing the distance between the plates:Capacitors with a flexible plate can be used to measure strain or pressure. Industrial pressure transmitters usedfor process control use pressure-sensing diaphragms, which form a capacitor plate of an oscillator circuit.Capacitors are used as the sensor in condenser microphones, where one plate is moved by air pressure, relativeto the fixed position of the other plate. Some accelerometers use MEMS capacitors etched on a chip tomeasure the magnitude and direction of the acceleration vector. They are used to detect changes inacceleration, in tilt sensors, or to detect free fall, as sensors triggering airbag deployment, and in many otherapplications. Some fingerprint sensors use capacitors. Additionally, a user can adjust the pitch of a thereminmusical instrument by moving his hand since this changes the effective capacitance between the user's handand the antenna.

Changing the effective area of the plates:Capacitive touch switches are now used on many consumer electronic products.

Hazards and safetyCapacitors may retain a charge long after power is removed from a circuit; this charge can cause dangerous or evenpotentially fatal shocks or damage connected equipment. For example, even a seemingly innocuous device such as adisposable camera flash unit powered by a 1.5 volt AA battery contains a capacitor which may be charged to over300 volts. This is easily capable of delivering a shock. Service procedures for electronic devices usually includeinstructions to discharge large or high-voltage capacitors, for instance using a Brinkley stick. Capacitors may alsohave built-in discharge resistors to dissipate stored energy to a safe level within a few seconds after power isremoved. High-voltage capacitors are stored with the terminals shorted, as protection from potentially dangerousvoltages due to dielectric absorption.Some old, large oil-filled paper or plastic film capacitors contain polychlorinated biphenyls (PCBs). It is known thatwaste PCBs can leak into groundwater under landfills. Capacitors containing PCB were labelled as containing"Askarel" and several other trade names. PCB-filled paper capacitors are found in very old (pre-1975) fluorescentlamp ballasts, and other applications.Capacitors may catastrophically fail when subjected to voltages or currents beyond their rating, or as they reach theirnormal end of life. Dielectric or metal interconnection failures may create arcing that vaporizes the dielectric fluid,resulting in case bulging, rupture, or even an explosion. Capacitors used in RF or sustained high-current applicationscan overheat, especially in the center of the capacitor rolls. Capacitors used within high-energy capacitor banks canviolently explode when a short in one capacitor causes sudden dumping of energy stored in the rest of the bank intothe failing unit. High voltage vacuum capacitors can generate soft X-rays even during normal operation. Propercontainment, fusing, and preventive maintenance can help to minimize these hazards.High-voltage capacitors can benefit from a pre-charge to limit in-rush currents at power-up of high voltage directcurrent (HVDC) circuits. This will extend the life of the component and may mitigate high-voltage hazards.

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Capacitor 41

Swollen caps of electrolytic capacitors –special design of semi-cut caps prevents

capacitors from bursting

This high-energycapacitor from a

defibrillator can deliverover 500 joules of

energy. A resistor isconnected between theterminals for safety, to

allow the stored energy tobe released.

Catastrophic failure

Notes[10][10] Ulaby, p.168[11][11] Ulaby, p.157[12][12] Ulaby, p.169[14][14] Dorf, p.263[15][15] Dorf, p.260[19][19] Ulaby, p.170[27] http:/ / en. wikipedia. org/ w/ index. php?title=Capacitor& action=edit

References• Dorf, Richard C.; Svoboda, James A. (2001). Introduction to Electric Circuits (http:/ / books. google. com/

books?id=l-weAQAAIAAJ) (5th ed.). New York: John Wiley & Sons. ISBN 9780471386896.• Ulaby, Fawwaz Tayssir (1999). Fundamentals of Applied Electromagnetics (http:/ / books. google. com/

books?id=a_C8QgAACAAJ). Upper Saddle River, New Jersey: Prentice Hall. ISBN 9780130115546.• Zorpette, Glenn (2005). "Super Charged: A Tiny South Korean Company is Out to Make Capacitors Powerful

enough to Propel the Next Generation of Hybrid-Electric Cars" (http:/ / www. spectrum. ieee. org/ jan05/ 2777).IEEE Spectrum (North American ed.) 42 (1): 32. doi: 10.1109/MSPEC.2005.1377872 (http:/ / dx. doi. org/ 10.1109/ MSPEC. 2005. 1377872). ISSN  0018-9235 (http:/ / www. worldcat. org/ issn/ 0018-9235).

• The ARRL Handbook for Radio Amateurs (68th ed.). Newington, CT: The Amateur Radio Relay League. 1991.• Huelsman, Lawrence P. (1972). Basic Circuit Theory: With Digital Computations (http:/ / books. google. com/

books?id=hnkjAAAAMAAJ). Series in computer applications in electrical engineering. Englewood Cliffs:Prentice-Hall. ISBN 9780130574305.

• Philosophical Transactions of the Royal Society LXXII, Appendix 8, 1782 (Volta coins the word condenser)• Maini, A. K. (1997). Electronic Projects for Beginners (http:/ / books. google. com/ books?id=Dx3Mdx_oDHsC)

(2nd ed.). India: Pustak Mahal. ISBN 9788122301526.• "The First Condenser - A Beer Glass" (http:/ / www. sparkmuseum. com/ BOOK_LEYDEN. HTM).

SparkMuseum.• Currier, Dean P. (2000). "Adventures in Cybersound - Ewald Christian von Kleist" (http:/ / web. archive. org/

web/ 20080625014024/ http:/ / www. acmi. net. au/ AIC/ VON_KLEIST_BIO. html). Archived from the original(http:/ / www. acmi. net. au/ AIC/ VON_KLEIST_BIO. html) on 2008-06-25.

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Capacitor 42

External links• Howstuffworks.com: How Capacitors Work (http:/ / electronics. howstuffworks. com/ capacitor. htm/ printable)• CapSite 2009: Introduction to Capacitors (http:/ / my. execpc. com/ ~endlr/ )• Capacitor Tutorial (http:/ / www. sentex. ca/ ~mec1995/ gadgets/ caps/ caps. html) – Includes how to read

capacitor temperature codes• Introduction to Capacitor and Capacitor codes (http:/ / www. robotplatform. com/ electronics/ capacitor/

capacitor. html)• Low ESR Capacitor Manufacturers (http:/ / www. capacitorlab. com/ low-esr-capacitor-manufacturers/ )• How Capacitor Works – Capacitor Markings and Color Codes (http:/ / freecircuits. org/ 2012/ 01/

capacitors-basics-working/ )

Ohm's law

V, I, and R, the parameters of Ohm's law.

Electromagnetism

•• Electricity•• Magnetism

Ohm's law states that the current through a conductor between two points is directly proportional to the potentialdifference across the two points. Introducing the constant of proportionality, the resistance,[1] one arrives at the usualmathematical equation that describes this relationship:[]

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Ohm's law 43

where I is the current through the conductor in units of amperes, V is the potential difference measured across theconductor in units of volts, and R is the resistance of the conductor in units of ohms. More specifically, Ohm's lawstates that the R in this relation is constant, independent of the current.[2]

The law was named after the German physicist Georg Ohm, who, in a treatise published in 1827, describedmeasurements of applied voltage and current through simple electrical circuits containing various lengths of wire. Hepresented a slightly more complex equation than the one above (see History section below) to explain hisexperimental results. The above equation is the modern form of Ohm's law.In physics, the term Ohm's law is also used to refer to various generalizations of the law originally formulated byOhm. The simplest example of this is:

where J is the current density at a given location in a resistive material, E is the electric field at that location, and σ isa material dependent parameter called the conductivity. This reformulation of Ohm's law is due to GustavKirchhoff.[3]

HistoryIn January 1781, before Georg Ohm's work, Henry Cavendish experimented with Leyden jars and glass tubes ofvarying diameter and length filled with salt solution. He measured the current by noting how strong a shock he felt ashe completed the circuit with his body. Cavendish wrote that the "velocity" (current) varied directly as the "degree ofelectrification" (voltage). He did not communicate his results to other scientists at the time,[] and his results wereunknown until Maxwell published them in 1879.[4]

Ohm did his work on resistance in the years 1825 and 1826, and published his results in 1827 as the book Diegalvanische Kette, mathematisch bearbeitet (The galvanic circuit investigated mathematically).[5] He drewconsiderable inspiration from Fourier's work on heat conduction in the theoretical explanation of his work. Forexperiments, he initially used voltaic piles, but later used a thermocouple as this provided a more stable voltagesource in terms of internal resistance and constant potential difference. He used a galvanometer to measure current,and knew that the voltage between the thermocouple terminals was proportional to the junction temperature. He thenadded test wires of varying length, diameter, and material to complete the circuit. He found that his data could bemodeled through the equation

where x was the reading from the galvanometer, l was the length of the test conductor, a depended only on thethermocouple junction temperature, and b was a constant of the entire setup. From this, Ohm determined his law ofproportionality and published his results.Ohm's law was probably the most important of the early quantitative descriptions of the physics of electricity. Weconsider it almost obvious today. When Ohm first published his work, this was not the case; critics reacted to histreatment of the subject with hostility. They called his work a "web of naked fancies"[6] and the German Minister ofEducation proclaimed that "a professor who preached such heresies was unworthy to teach science."[7] Theprevailing scientific philosophy in Germany at the time asserted that experiments need not be performed to developan understanding of nature because nature is so well ordered, and that scientific truths may be deduced throughreasoning alone.[8] Also, Ohm's brother Martin, a mathematician, was battling the German educational system. Thesefactors hindered the acceptance of Ohm's work, and his work did not become widely accepted until the 1840s.Fortunately, Ohm received recognition for his contributions to science well before he died.

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Ohm's law 44

In the 1850s, Ohm's law was known as such and was widely considered proved, and alternatives, such as "Barlow'slaw", were discredited, in terms of real applications to telegraph system design, as discussed by Samuel F. B. Morsein 1855.[9]

While the old term for electrical conductance, the mho (the inverse of the resistance unit ohm), is still used, a newname, the siemens, was adopted in 1971, honoring Ernst Werner von Siemens. The siemens is preferred in formalpapers.In the 1920s, it was discovered that the current through an ideal resistor actually has statistical fluctuations, whichdepend on temperature, even when voltage and resistance are exactly constant; this fluctuation, now known asJohnson–Nyquist noise, is due to the discrete nature of charge. This thermal effect implies that measurements ofcurrent and voltage that are taken over sufficiently short periods of time will yield ratios of V/I that fluctuate fromthe value of R implied by the time average or ensemble average of the measured current; Ohm's law remains correctfor the average current, in the case of ordinary resistive materials.Ohm's work long preceded Maxwell's equations and any understanding of frequency-dependent effects in ACcircuits. Modern developments in electromagnetic theory and circuit theory do not contradict Ohm's law when theyare evaluated within the appropriate limits.

ScopeOhm's law is an empirical law, a generalization from many experiments that have shown that current isapproximately proportional to electric field for most materials. It is less fundamental than Maxwell's equations and isnot always obeyed. Any given material will break down under a strong-enough electric field, and some materials ofinterest in electrical engineering are "non-ohmic" under weak fields.[10][11]

Ohm's law has been observed on a wide range of length scales. In the early 20th century, it was thought that Ohm'slaw would fail at the atomic scale, but experiments have not borne out this expectation. As of 2012, researchers havedemonstrated that Ohm's law works for silicon wires as small as four atoms wide and one atom high.[12]

Microscopic origins

Drude Model electrons (shown here in blue) constantlybounce among heavier, stationary crystal ions (shown

in red).

The dependence of the current density on the applied electric fieldis essentially quantum mechanical in nature; (see Classical andquantum conductivity.) A qualitative description leading to Ohm'slaw can be based upon classical mechanics using the Drude modeldeveloped by Paul Drude in 1900.[13][14]

The Drude model treats electrons (or other charge carriers) likepinballs bouncing among the ions that make up the structure of thematerial. Electrons will be accelerated in the opposite direction tothe electric field by the average electric field at their location.With each collision, though, the electron is deflected in a randomdirection with a velocity that is much larger than the velocitygained by the electric field. The net result is that electrons take azigzag path due to the collisions, but generally drift in a directionopposing the electric field.

The drift velocity then determines the electric current density and its relationship to E and is independent of the collisions. Drude calculated the average drift velocity from p = −eEτ where p is the average momentum, −e is the charge of the electron and τ is the average time between the collisions. Since both the momentum and the current

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Ohm's law 45

density are proportional to the drift velocity, the current density becomes proportional to the applied electric field;this leads to Ohm's law.

Hydraulic analogyA hydraulic analogy is sometimes used to describe Ohm's law. Water pressure, measured by pascals (or PSI), is theanalog of voltage because establishing a water pressure difference between two points along a (horizontal) pipecauses water to flow. Water flow rate, as in liters per second, is the analog of current, as in coulombs per second.Finally, flow restrictors—such as apertures placed in pipes between points where the water pressure ismeasured—are the analog of resistors. We say that the rate of water flow through an aperture restrictor isproportional to the difference in water pressure across the restrictor. Similarly, the rate of flow of electrical charge,that is, the electric current, through an electrical resistor is proportional to the difference in voltage measured acrossthe resistor.Flow and pressure variables can be calculated in fluid flow network with the use of the hydraulic ohmanalogy.[15][16] The method can be applied to both steady and transient flow situations. In the linear laminar flowregion, Poiseuille's law describes the hydraulic resistance of a pipe, but in the turbulent flow region thepressure–flow relations become nonlinear.The hydraulic analogy to Ohm's law has been used, for example, to approximate blood flow through the circulatorysystem.[17]

Circuit analysis

Ohm's law triangle

In circuit analysis, three equivalent expressions of Ohm's law are usedinterchangeably:

Each equation is quoted by some sources as the defining relationship of Ohm'slaw,[][18][19] or all three are quoted,[20] or derived from a proportional form,[21] oreven just the two that do not correspond to Ohm's original statement maysometimes be given.[22][23]

The interchangeability of the equation may be represented by a triangle, where V (voltage) is placed on the topsection, the I (current) is placed to the left section, and the R (resistance) is placed to the right. The line that dividesthe left and right sections indicate multiplication, and the divider between the top and bottom sections indicatesdivision (hence the division bar).

Resistive circuitsResistors are circuit elements that impede the passage of electric charge in agreement with Ohm's law, and aredesigned to have a specific resistance value R. In a schematic diagram the resistor is shown as a zig-zag symbol. Anelement (resistor or conductor) that behaves according to Ohm's law over some operating range is referred to as anohmic device (or an ohmic resistor) because Ohm's law and a single value for the resistance suffice to describe thebehavior of the device over that range.Ohm's law holds for circuits containing only resistive elements (no capacitances or inductances) for all forms ofdriving voltage or current, regardless of whether the driving voltage or current is constant (DC) or time-varying suchas AC. At any instant of time Ohm's law is valid for such circuits.Resistors which are in series or in parallel may be grouped together into a single "equivalent resistance" in order to apply Ohm's law in analyzing the circuit. This application of Ohm's law is illustrated with examples in "How To

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Ohm's law 46

Analyze Resistive Circuits Using Ohm's Law" on wikiHow.

Reactive circuits with time-varying signalsWhen reactive elements such as capacitors, inductors, or transmission lines are involved in a circuit to which AC ortime-varying voltage or current is applied, the relationship between voltage and current becomes the solution to adifferential equation, so Ohm's law (as defined above) does not directly apply since that form contains onlyresistances having value R, not complex impedances which may contain capacitance ("C") or inductance ("L").Equations for time-invariant AC circuits take the same form as Ohm's law, however, the variables are generalized tocomplex numbers and the current and voltage waveforms are complex exponentials.[24]

In this approach, a voltage or current waveform takes the form , where t is time, s is a complex parameter, andA is a complex scalar. In any linear time-invariant system, all of the currents and voltages can be expressed with thesame s parameter as the input to the system, allowing the time-varying complex exponential term to be canceled outand the system described algebraically in terms of the complex scalars in the current and voltage waveforms.The complex generalization of resistance is impedance, usually denoted Z; it can be shown that for an inductor,

and for a capacitor,

We can now write,

where V and I are the complex scalars in the voltage and current respectively and Z is the complex impedance.This form of Ohm's law, with Z taking the place of R, generalizes the simpler form. When Z is complex, only the realpart is responsible for dissipating heat.In the general AC circuit, Z varies strongly with the frequency parameter s, and so also will the relationship betweenvoltage and current.

For the common case of a steady sinusoid, the s parameter is taken to be , corresponding to a complex sinusoid. The real parts of such complex current and voltage waveforms describe the actual sinusoidal currents and

voltages in a circuit, which can be in different phases due to the different complex scalars.

Linear approximationsOhm's law is one of the basic equations used in the analysis of electrical circuits. It applies to both metal conductorsand circuit components (resistors) specifically made for this behaviour. Both are ubiquitous in electrical engineering.Materials and components that obey Ohm's law are described as "ohmic" [25] which means they produce the samevalue for resistance (R = V/I) regardless of the value of V or I which is applied and whether the applied voltage orcurrent is DC (direct current) of either positive or negative polarity or AC (alternating current).In a true ohmic device, the same value of resistance will be calculated from R = V/I regardless of the value of theapplied voltage V. That is, the ratio of V/I is constant, and when current is plotted as a function of voltage the curveis linear (a straight line). If voltage is forced to some value V, then that voltage V divided by measured current I willequal R. Or if the current is forced to some value I, then the measured voltage V divided by that current I is also R.Since the plot of I versus V is a straight line, then it is also true that for any set of two different voltages V1 and V2applied across a given device of resistance R, producing currents I1 = V1/R and I2 = V2/R, that the ratio(V1-V2)/(I1-I2) is also a constant equal to R. The operator "delta" (Δ) is used to represent a difference in a quantity,so we can write ΔV = V1-V2 and ΔI = I1-I2. Summarizing, for any truly ohmic device having resistance R, V/I =ΔV/ΔI = R for any applied voltage or current or for the difference between any set of applied voltages or currents.

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Ohm's law 47

The I–V curves of four devices: Two resistors, a diode, and a battery. The two resistorsfollow Ohm's law: The plot is a straight line through the origin. The other two devices do

not follow Ohm's law.

There are, however, components ofelectrical circuits which do not obeyOhm's law; that is, their relationshipbetween current and voltage (their I–Vcurve) is nonlinear (or non-ohmic). Anexample is the p-n junction diode(curve at right). As seen in the figure,the current does not increase linearlywith applied voltage for a diode. One can determine a value of current (I) for a given value of applied voltage (V)from the curve, but not from Ohm's law, since the value of "resistance" is not constant as a function of appliedvoltage. Further, the current only increases significantly if the applied voltage is positive, not negative. The ratio V/Ifor some point along the nonlinear curve is sometimes called the static, or chordal, or DC, resistance,[26][27] but asseen in the figure the value of total V over total I varies depending on the particular point along the nonlinear curvewhich is chosen. This means the "DC resistance" V/I at some point on the curve is not the same as what would bedetermined by applying an AC signal having peak amplitude ΔV volts or ΔI amps centered at that same point alongthe curve and measuring ΔV/ΔI. However, in some diode applications, the AC signal applied to the device is smalland it is possible to analyze the circuit in terms of the dynamic, small-signal, or incremental resistance, defined asthe one over the slope of the V–I curve at the average value (DC operating point) of the voltage (that is, one over thederivative of current with respect to voltage). For sufficiently small signals, the dynamic resistance allows the Ohm'slaw small signal resistance to be calculated as approximately one over the slope of a line drawn tangentially to theV-I curve at the DC operating point.[]

Temperature effectsOhm's law has sometimes been stated as, "for a conductor in a given state, the electromotive force is proportional tothe current produced." That is, that the resistance, the ratio of the applied electromotive force (or voltage) to thecurrent, "does not vary with the current strength ." The qualifier "in a given state" is usually interpreted as meaning"at a constant temperature," since the resistivity of materials is usually temperature dependent. Because theconduction of current is related to Joule heating of the conducting body, according to Joule's first law, thetemperature of a conducting body may change when it carries a current. The dependence of resistance ontemperature therefore makes resistance depend upon the current in a typical experimental setup, making the law inthis form difficult to directly verify. Maxwell and others worked out several methods to test the law experimentallyin 1876, controlling for heating effects.[28]

Relation to heat conductionsOhm's principle predicts the flow of electrical charge (i.e. current) in electrical conductors when subjected to theinfluence of voltage differences; Jean-Baptiste-Joseph Fourier's principle predicts the flow of heat in heat conductorswhen subjected to the influence of temperature differences.The same equation describes both phenomena, the equation's variables taking on different meanings in the two cases.Specifically, solving a heat conduction (Fourier) problem with temperature (the driving "force") and flux of heat (therate of flow of the driven "quantity", i.e. heat energy) variables also solves an analogous electrical conduction (Ohm)problem having electric potential (the driving "force") and electric current (the rate of flow of the driven "quantity",i.e. charge) variables.The basis of Fourier's work was his clear conception and definition of thermal conductivity. He assumed that, all else being the same, the flux of heat is strictly proportional to the gradient of temperature. Although undoubtedly true for small temperature gradients, strictly proportional behavior will be lost when real materials (e.g. ones having a

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Ohm's law 48

thermal conductivity that is a function of temperature) are subjected to large temperature gradients.A similar assumption is made in the statement of Ohm's law: other things being alike, the strength of the current ateach point is proportional to the gradient of electric potential. The accuracy of the assumption that flow isproportional to the gradient is more readily tested, using modern measurement methods, for the electrical case thanfor the heat case.

Other versionsOhm's law, in the form above, is an extremely useful equation in the field of electrical/electronic engineeringbecause it describes how voltage, current and resistance are interrelated on a "macroscopic" level, that is, commonly,as circuit elements in an electrical circuit. Physicists who study the electrical properties of matter at the microscopiclevel use a closely related and more general vector equation, sometimes also referred to as Ohm's law, havingvariables that are closely related to the V, I, and R scalar variables of Ohm's law, but which are each functions ofposition within the conductor. Physicists often use this continuum form of Ohm's Law:[29]

where "E" is the electric field vector with units of volts per meter (analogous to "V" of Ohm's law which has units ofvolts), "J" is the current density vector with units of amperes per unit area (analogous to "I" of Ohm's law which hasunits of amperes), and "ρ" (Greek "rho") is the resistivity with units of ohm·meters (analogous to "R" of Ohm's lawwhich has units of ohms). The above equation is sometimes written[30] as J = E where "σ" (Greek "sigma") is theconductivity which is the reciprocal of ρ.

Current flowing through a uniform cylindrical conductor (such as a round wire)with a uniform field applied.

The potential difference between two points isdefined as:[31]

with the element of path along theintegration of electric field vector E. If theapplied E field is uniform and oriented alongthe length of the conductor as shown in thefigure, then defining the voltage V in the usualconvention of being opposite in direction tothe field (see figure), and with theunderstanding that the voltage V is measureddifferentially across the length of theconductor allowing us to drop the Δ symbol,the above vector equation reduces to the scalar equation:

Since the E field is uniform in the direction of wire length, for a conductor having uniformly consistent resistivity ρ,the current density J will also be uniform in any cross-sectional area and oriented in the direction of wire length, sowe may write:[32]

Substituting the above 2 results (for E and J respectively) into the continuum form shown at the beginning of thissection:

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Ohm's law 49

The electrical resistance of a uniform conductor is given in terms of resistivity by:[32]

where l is the length of the conductor in SI units of meters, a is the cross-sectional area (for a round wire a = πr2 if ris radius) in units of meters squared, and ρ is the resistivity in units of ohm·meters.After substitution of R from the above equation into the equation preceding it, the continuum form of Ohm's law fora uniform field (and uniform current density) oriented along the length of the conductor reduces to the more familiarform:

A perfect crystal lattice, with low enough thermal motion and no deviations from periodic structure, would have noresistivity,[33] but a real metal has crystallographic defects, impurities, multiple isotopes, and thermal motion of theatoms. Electrons scatter from all of these, resulting in resistance to their flow.The more complex generalized forms of Ohm's law are important to condensed matter physics, which studies theproperties of matter and, in particular, its electronic structure. In broad terms, they fall under the topic of constitutiveequations and the theory of transport coefficients.

Magnetic effectsIf an external B-field is present and the conductor is not at rest but moving at velocity v, then an extra term must beadded to account for the current induced by the Lorentz force on the charge carriers.

In the rest frame of the moving conductor this term drops out because v= 0. There is no contradiction because theelectric field in the rest frame differs from the E-field in the lab frame: E ' = E + v×B. Electric and magnetic fieldsare relative, see Lorentz transform.If the current J is alternating because the applied voltage or E-field varies in time, then reactance must be added toresistance to account for self-inductance, see electrical impedance. The reactance may be strong if the frequency ishigh or the conductor is coiled.See Hall effect for some other implication of a magnetic field.

References[3] Olivier Darrigol, Electrodynamics from Ampère to Einstein (http:/ / books. google. com/ books?id=ZzeYSbqITWkC& pg=PA70&

dq="alternative+ formulation+ of+ Ohm's+ law"+ isbn:0198505949& lr=& as_drrb_is=q& as_minm_is=0& as_miny_is=& as_maxm_is=0&as_maxy_is=& as_brr=0#v=onepage& q="alternative formulation of Ohm's law" isbn:0198505949& f=false), p.70, Oxford University Press,2000 ISBN 0-19-850594-9.

[4] Sanford P. Bordeau (1982) Volts to Hertz...the Rise of Electricity. Burgess Publishing Company, Minneapolis, MN. pp.86–107, ISBN0-8087-4908-0

[6] Davies, B, "A web of naked fancies?", Physics Education 15 57–61, Institute of Physics, Issue 1, Jan 1980 (http:/ / www. iop. org/ EJ/ S/UNREG/ an0VsEw7ynSY0UzIRNaNVQ/ abstract/ 0031-9120/ 15/ 1/ 314)

[7] Hart, IB, Makers of Science, London, Oxford University Press, 1923. p. 243. (http:/ / www. foresight. org/ news/ negativeComments.html#loc037)

[8] Herbert Schnädelbach, Philosophy in Germany 1831-1933, pages 78-79, Cambridge University Press, 1984 ISBN 0521296463.[12] B. Weber, S. Mahapatra, H. Ryu, S. Lee, A. Fuhrer, T. C. G. Reusch, D. L. Thompson, W. C. T. Lee, G. Klimeck, L. C. L. Hollenberg, M.

Y. Simmons. "Ohm’s Law Survives to the Atomic Scale" (http:/ / www. sciencemag. org/ content/ 335/ 6064/ 64) Science 6 January 2012:Vol. 335 no. 6064 pp. 64-67 accessdate=2012-1-6

[16] A. Esposito, "A Simplified Method for Analyzing Circuits by Analogy", Machine Design, October 1969, pp. 173–177.[25] Hughes, E, Electrical Technology, pp10, Longmans, 1969.[30] Seymour J, Physical Electronics, Pitman, 1972, pp 53–54[31] Lerner L, Physics for scientists and engineers, Jones & Bartlett, 1997, pp. 685–686 (http:/ / books. google. com/

books?id=Nv5GAyAdijoC& pg=PA685)

Ramesh Singh, DTU (Formerly Delhi College of Engineering)singhramesh2@gmail.com

Ohm's law 50

[32] Lerner L, Physics for scientists and engineers, Jones & Bartlett, 1997, pp. 732–733 (http:/ / books. google. com/books?id=Nv5GAyAdijoC& pg=PA732)

[33] Seymour J, Physical Electronics, pp 48–49, Pitman, 1972

External links• John C. Shedd and Mayo D. Hershey, "The History of Ohm's Law" (http:/ / books. google. com/

books?id=8CQDAAAAMBAJ& pg=PA599& dq="Popular+ Science"+ "Ohm's+ law"& hl=en&ei=stULTZfxDMbKhAfxlr3-Cw& sa=X& oi=book_result& ct=result& resnum=1&ved=0CCMQ6AEwAA#v=onepage& q& f=false), Popular Science, December 1913, pages 599-614, BonnierCorporation ISSN 0161-7370, gives the history of Ohm's investigations, prior work, Ohm's false equation in thefirst paper, illustration of Ohm's experimental apparatus.

• Morton L. Schagrin, "Resistance to Ohm's Law" (http:/ / dx. doi. org/ 10. 1119/ 1. 1969620), American Journal ofPhysics, July 1963, Volume 31, Issue 7, pp. 536–47. Explores the conceptual change underlying Ohm'sexperimental work.

• Kenneth L. Caneva, "Ohm, Georg Simon." (http:/ / www. encyclopedia. com/ topic/ Georg_Simon_Ohm. aspx#1)Complete Dictionary of Scientific Biography. 2008

Kirchhoff's circuit lawsKirchhoff's circuit laws are two approximate equalities that deal with the current and voltage in electrical circuits.They were first described in 1845 by Gustav Kirchhoff.[1] This generalized the work of Georg Ohm and preceded thework of Maxwell. Widely used in electrical engineering, they are also called Kirchhoff's rules or simply Kirchhoff'slaws (see also Kirchhoff's laws for other meanings of that term).Both of Kirchhoff's laws can be understood as corollaries of the Maxwell equations in the low-frequency limit --conventionally called "DC" circuits. They serve as first approximations for AC circuits.[2]

Kirchhoff's current law (KCL)

The current entering any junction is equal to thecurrent leaving that junction. i2 + i3 = i1 + i4

This law is also called Kirchhoff's first law, Kirchhoff's point rule,or Kirchhoff's junction rule (or nodal rule).

The principle of conservation of electric charge implies that:At any node (junction) in an electrical circuit, the sum ofcurrents flowing into that node is equal to the sum of currentsflowing out of that node, or:

The algebraic sum of currents in a network of conductorsmeeting at a point is zero.

Recalling that current is a signed (positive or negative) quantityreflecting direction towards or away from a node, this principle can bestated as:

n is the total number of branches with currents flowing towards or away from the node.This formula is valid for complex currents:

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Kirchhoff's circuit laws 51

The law is based on the conservation of charge whereby the charge (measured in coulombs) is the product of thecurrent (in amperes) and the time (in seconds).

LimitationsKCL, in its usual form, is dependent on the assumption that current flows only in conductors, and that whenevercurrent flows into one end of a conductor it immediately flows out the other end. This is not a safe assumption forAC circuits.[2] It may be possible to salvage the form of KCL by considering "parasitic capacitances" distributedalong the conductors.[2] However, this greatly detracts from the simplicity of KCL and invalidates the notion oftopological circuit diagram as discussed below. Significant violations of KCL can occur[3][4] even at 60Hz, which isnot a very high frequency.In other words, KCL is valid only if the total electric charge, , remains constant in the region being considered. Inpractical cases this is always so when KCL is applied at a geometric point. When investigating a finite region,however, it is possible that the charge density within the region may change. Since charge is conserved, this can onlycome about by a flow of charge across the region boundary. This flow represents a net current, and KCL is violated.Formally, from the volume integral of the current continuity equation,

where is the current density vector and is the volume of the regionConverting the volume integral to a surface integral using the divergence theorem

Hence,

The right-hand side vanishes if is independent of time. If practically all of is contained within small regions,conducting wires for instance, then the left-hand side can be interpreted as a sum of discrete currents and KCL isrecovered, providing that .A particular case where KCL does not hold is the current entering a single plate of a capacitor. To illustrate thispoint, imagine a closed surface around that single plate, current enters through the surface, but does not exit, thusviolating KCL. Certainly, the currents through a closed surface around the entire capacitor will meet KCL, since thecurrent entering one plate is balanced by the current exiting the other plate, and that is usually all that is important incircuit analysis, but there is a problem when considering a single plate. Another common example, the current in anantenna enters the antenna from the transmitter feeder, but no current exits from the other end (Johnson and Graham,pp. 36–37).Maxwell introduced the concept of displacement currents to describe these questionable situations. The currentflowing into a capacitor plate is equal to the rate of charge accumulation, and similarly, equal to the rate of change ofelectric flux (SI system unit of measurement for both, electric flux and electric charge, is Coulombs). This rate ofchange of flux, , is what Maxwell called displacement current ;

If displacement currents are included, Kirchhoff's current law is once again validated. Displacement currents are notreal currents, in that they do not consist of moving charges; they should be viewed more as a correction factor tokeep KCL valid. In the case of the capacitor plate, the real current entering the plate is exactly cancelled by adisplacement current leaving the plate, and transitioning towards the opposite plate.

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Kirchhoff's circuit laws 52

This can also be expressed in terms of vector field quantities through divergence of Ampère's law with Maxwell'scorrection, and combining Gauss's law, yielding:

This is simply the charge conservation equation (in integral form, it says that the current flowing out of a closedsurface is equal to the rate of loss of charge within the enclosed volume (divergence theorem). Kirchhoff's currentlaw is equivalent to the statement that the divergence of the current is zero, true for time-invariant ρ, or always true ifthe displacement current is included with J.

UsesA matrix version of Kirchhoff's current law is the basis of most circuit simulation software, such as SPICE.Kirchhoff's current law combined with Ohm's Law is used in nodal analysis.

Kirchhoff's voltage law (KVL)

The sum of all the voltages around the loop is equal tozero. v1 + v2 + v3 - v4 = 0

This law is also called Kirchhoff's second law, Kirchhoff's loop(or mesh) rule, and Kirchhoff's second rule.

Similarly to KCL, it can be stated as:

Here, n is the total number of voltages measured. The voltagesmay also be complex:

This law is based on one of the Maxwell equations, namely theMaxwell-Faraday law of induction, which states that the voltagedrop around any closed loop is equal to the rate-of-change of theflux threading the loop. The amount of flux depends on the area ofthe loop and on the magnetic field strength. KVL states the loop voltage is zero. The Maxwell equations tell us thatthe loop voltage will be small if the area of the loop is small, the magnetic field is weak, and/or the magnetic field isslowly changing.

Routine engineering techniques -- such as the use of coaxial cable and twisted pairs -- can be used to minimize straymagnetic fields and minimize the area of vulnerable loops. Utilization of these techniques creates an arrangement,whereby KVL becomes a useful approximation for situations where its application was imprecise.

LimitationsKVL is based on the assumption that there is no fluctuating magnetic field linking the closed loop. This is not a safeassumption for AC circuits.[2] In the presence of a changing magnetic field the electric field is not a conservativevector field. Therefore the electric field can not be the gradient of any potential. That is to say, the line integral of theelectric field around the loop is not zero, directly contradicting KVL.It may be possible to salvage the form of KVL by considering "parasitic inductances" (including mutual inductances)distributed along the conductors.[2] These are treated as imaginary circuit elements that produce a voltage drop equalto the rate-of-change of the flux. However, this greatly detracts from the simplicity of KVL and invalidates thenotion of topological circuit diagram.

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Kirchhoff's circuit laws 53

GeneralizationIn the DC limit, the voltage drop around any loop is zero. This includes imaginary loops arranged arbitrarily in space-- not limited to the loops delineated by the circuit elements and conductors. In the low-frequency limit, this is acorollary of Faraday's law of induction (which is one of the Maxwell equations).This has practical application in situations involving "static electricity".

Topological circuit diagramsThe approximations that lead to Kirchhoff's circuit laws are part of a package that also leads to topological circuitdiagrams, i.e. the idea that the physical and geometrical layout of the circuit does not matter; the only thing thatmatters is the topology as determined by the conductors and circuit elements connected to the nodes. These can betreated as the arcs and nodes of formal graph theory. In other words, Kirchhoff's laws say it suffices to use a circuitdiagram that is purely schematic. This is a very useful, powerful simplification.This works fine in the DC limit, but it is only a first approximation for AC circuits.[2] For high-power,high-precision, and/or high-frequency work, the deviations from Kirchhoff's laws cannot be neglected.[2] Thephysical and geometrical layout of the circuit matters, because it determines the magnitude of the parasiticcapacitances and inductances.[2][3][4]

ExampleAssume an electric network consisting of two voltage sources and three resistors:

According to the first law we have

The second law applied to the closed circuit s1 gives

The second law applied to the closed circuit s2 gives

Thus we get a linear system of equations in :

Assuming

the solution is

has a negative sign, which means that the direction of is opposite to the assumed direction (the directiondefined in the picture).

Ramesh Singh, DTU (Formerly Delhi College of Engineering)singhramesh2@gmail.com

Kirchhoff's circuit laws 54

References[1][1] Oldham, p.52[2] Ralph Morrison, Grounding and Shielding Techniques in Instrumentation Wiley-Interscience (1986) ISBN 0471838055[4][4] Non-contact voltage detector

• Paul, Clayton R. (2001). Fundamentals of Electric Circuit Analysis. John Wiley & Sons. ISBN 0-471-37195-5.• Kalil T. Swain Oldham, The doctrine of description: Gustav Kirchhoff, classical physics, and the "purpose of all

science" in 19th-century Germany, ProQuest, 2008, ISBN 0-549-83131-2.• Serway, Raymond A.; Jewett, John W. (2004). Physics for Scientists and Engineers (6th ed.). Brooks/Cole.

ISBN 0-534-40842-7.• Tipler, Paul (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern

Physics (5th ed.). W. H. Freeman. ISBN 0-7167-0810-8.• Howard W. Johnson, Martin Graham, High-speed signal propagation: advanced black magic, Prentice Hall

Professional, 2003 ISBN 0-13-084408-X.

External links• MIT video lecture (http:/ / academicearth. org/ lectures/ basic-circuit-analysis-method-kvl-and-kcl-mmethod) on

the KVL and KCL methods

Current divider

Figure 1: Schematic of an electrical circuit illustrating currentdivision. Notation RT. refers to the total resistance of the circuit to

the right of resistor RX.

In electronics, a current divider is a simple linearcircuit that produces an output current (IX) that is afraction of its input current (IT). Current divisionrefers to the splitting of current between the branches ofthe divider. The currents in the various branches ofsuch a circuit will always divide in such a way as tominimize the total energy expended.

The formula describing a current divider is similar inform to that for the voltage divider. However, the ratiodescribing current division places the impedance of theunconsidered branches in the numerator, unlike voltagedivision where the considered impedance is in thenumerator. This is because in current dividers, totalenergy expended is minimized, resulting in currentsthat go through paths of least impedance, therefore the inverse relationship with impedance. On the other hand,voltage divider is used to satisfy Kirchhoff's Voltage Law. The voltage around a loop must sum up to zero, so thevoltage drops must be divided evenly in a direct relationship with the impedance.

To be specific, if two or more impedances are in parallel, the current that enters the combination will be splitbetween them in inverse proportion to their impedances (according to Ohm's law). It also follows that if theimpedances have the same value the current is split equally.

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Current divider 55

Resistive dividerA general formula for the current IX in a resistor RX that is in parallel with a combination of other resistors of totalresistance RT is (see Figure 1):

where IT is the total current entering the combined network of RX in parallel with RT. Notice that when RT iscomposed of a parallel combination of resistors, say R1, R2, ... etc., then the reciprocal of each resistor must be addedto find the total resistance RT:

General caseAlthough the resistive divider is most common, the current divider may be made of frequency dependentimpedances. In the general case the current IX is given by:

Using AdmittanceInstead of using impedances, the current divider rule can be applied just like the voltage divider rule if admittance(the inverse of impedance) is used.

Take care to note that YTotal is a straightforward addition, not the sum of the inverses inverted (as you would do for astandard parallel resistive network). For Figure 1, the current IX would be

Example: RC combination

Figure 2: A low pass RC current divider

Figure 2 shows a simple current divider made up of acapacitor and a resistor. Using the formula above, the currentin the resistor is given by:

where ZC = 1/(jωC) is the impedance of the capacitor and jis the imaginary unit.

The product τ = CR is known as the time constant of thecircuit, and the frequency for which ωCR = 1 is called thecorner frequency of the circuit. Because the capacitor has zero impedance at high frequencies and infinite impedanceat low frequencies, the current in the resistor remains at its DC value IT for frequencies up to the corner frequency,whereupon it drops toward zero for higher frequencies as the capacitor effectively short-circuits the resistor. In otherwords, the current divider is a low pass filter for current in the resistor.

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Current divider 56

Loading effect

Figure 3: A current amplifier (gray box) driven by a Norton source (iS, RS) andwith a resistor load RL. Current divider in blue box at input (RS,Rin) reduces the

current gain, as does the current divider in green box at the output (Rout,RL)

The gain of an amplifier generally dependson its source and load terminations. Currentamplifiers and transconductance amplifiersare characterized by a short-circuit outputcondition, and current amplifiers andtransresistance amplifiers are characterizedusing ideal infinite impedance currentsources. When an amplifier is terminated bya finite, non-zero termination, and/or drivenby a non-ideal source, the effective gain isreduced due to the loading effect at theoutput and/or the input, which can be understood in terms of current division.

Figure 3 shows a current amplifier example. The amplifier (gray box) has input resistance Rin and output resistanceRout and an ideal current gain Ai. With an ideal current driver (infinite Norton resistance) all the source current iSbecomes input current to the amplifier. However, for a Norton driver a current divider is formed at the input thatreduces the input current to

which clearly is less than iS. Likewise, for a short circuit at the output, the amplifier delivers an output current io = Aiii to the short-circuit. However, when the load is a non-zero resistor RL, the current delivered to the load is reducedby current division to the value:

Combining these results, the ideal current gain Ai realized with an ideal driver and a short-circuit load is reduced tothe loaded gain Aloaded:

The resistor ratios in the above expression are called the loading factors. For more discussion of loading in otheramplifier types, see loading effect.

Unilateral versus bilateral amplifiers

Figure 4: Current amplifier as a bilateral two-port network; feedback throughdependent voltage source of gain β V/V

Figure 3 and the associated discussion refersto a unilateral amplifier. In a more generalcase where the amplifier is represented by atwo port, the input resistance of theamplifier depends on its load, and the outputresistance on the source impedance. Theloading factors in these cases must employthe true amplifier impedances includingthese bilateral effects. For example, takingthe unilateral current amplifier of Figure 3,the corresponding bilateral two-port network is shown in Figure 4 based upon h-parameters.[1] Carrying out theanalysis for this circuit, the current gain with feedback Afb is found to be

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Current divider 57

That is, the ideal current gain Ai is reduced not only by the loading factors, but due to the bilateral nature of thetwo-port by an additional factor[2] ( 1 + β (RL / RS ) Aloaded ), which is typical of negative feedback amplifiercircuits. The factor β (RL / RS ) is the current feedback provided by the voltage feedback source of voltage gain βV/V. For instance, for an ideal current source with RS = ∞ Ω, the voltage feedback has no influence, and for RL = 0Ω, there is zero load voltage, again disabling the feedback.

References and notes[1] The h-parameter two port is the only two-port among the four standard choices that has a current-controlled current source on the output side.[2] Often called the improvement factor or the desensitivity factor.

External links• University of Texas: Notes on electronic circuit theory (http:/ / utwired. engr. utexas. edu/ rgd1/ lesson05. cfm)

Voltage divider

Figure 1: Voltage divider

In electronics or EET, a voltage divider (also known as a potentialdivider) is a linear circuit that produces an output voltage (Vout) thatis a fraction of its input voltage (Vin). Voltage division refers to thepartitioning of a voltage among the components of the divider.

An example of a voltage divider consists of two resistors in series ora potentiometer. It is commonly used to create a reference voltage,or to get a low voltage signal proportional to the voltage to bemeasured, and may also be used as a signal attenuator at lowfrequencies. For direct current and relatively low frequencies, avoltage divider may be sufficiently accurate if made only ofresistors; where frequency response over a wide range is required,(such as in an oscilloscope probe), the voltage divider may havecapacitive elements added to allow compensation for loadcapacitance. In electric power transmission, a capacitive voltage divider is used for measurement of high voltage.

General caseA voltage divider referenced to ground is created by connecting two electrical impedances in series, as shown inFigure 1. The input voltage is applied across the series impedances Z1 and Z2 and the output is the voltage across Z2.Z1 and Z2 may be composed of any combination of elements such as resistors, inductors and capacitors.Applying Ohm's Law, the relationship between the input voltage, Vin, and the output voltage, Vout, can be found:

Proof:

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Voltage divider 58

The transfer function (also known as the divider's voltage ratio) of this circuit is simply:

In general this transfer function is a complex, rational function of frequency.

Examples

Resistive divider

Figure 2: Simple resistive voltage divider

A resistive divider is the case where both impedances, Z1 and Z2, arepurely resistive (Figure 2).

Substituting Z1 = R1 and Z2 = R2 into the previous expression gives:

If R1 = R2 then

If Vout=6V and Vin=9V (both commonly used voltages), then:

and by solving using algebra, R2 must be twice the value of R1.

To solve for R1:

To solve for R2:

Any ratio greater than 1 is possible. That is, using resistors alone it is not possible to either invert the voltage orincrease Vout above Vin.

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Voltage divider 59

Low-pass RC filter

Figure 3: Resistor/capacitor voltage divider

Consider a divider consisting of a resistor and capacitor as shownin Figure 3.

Comparing with the general case, we see Z1 = R and Z2 is theimpedance of the capacitor, given by

where XC is the reactance of the capacitor, C is the capacitance ofthe capacitor, j is the imaginary unit, and ω (omega) is the radianfrequency of the input voltage.

This divider will then have the voltage ratio:

.

The product τ (tau) = RC is called the time constant of the circuit.The ratio then depends on frequency, in this case decreasing as frequency increases. This circuit is, in fact, a basic(first-order) lowpass filter. The ratio contains an imaginary number, and actually contains both the amplitude andphase shift information of the filter. To extract just the amplitude ratio, calculate the magnitude of the ratio, that is:

Inductive dividerInductive dividers split AC input according to inductance:

The above equation is for non-interacting inductors; mutual inductance (as in an autotransformer) will alter theresults.Inductive dividers split DC input according to the resistance of the elements as for the resistive divider above.

Capacitive dividerCapacitive dividers do not pass DC input.For an AC input a simple capacitive equation is:

Any leakage current in the capactive elements requires use of the generalized expression with two impedances. Byselection of parallel R and C elements in the proper proportions, the same division ratio can be maintained over auseful range of frequencies. This is the principle applied in compensated oscilloscope probes to increasemeasurement bandwidth.

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Voltage divider 60

Loading effectThe voltage output of a voltage divider is not fixed but varies according to the load. To obtain a reasonably stableoutput voltage the output current should be a small fraction of the input current. The drawback of this is that most ofthe input current is wasted as heat in the divider. An alternative is to use a voltage regulator.

ApplicationsVoltage dividers are used for adjusting the level of a signal, for bias of active devices in amplifiers, and formeasurement of voltages. A Wheatstone bridge and a multimeter both include voltage dividers. A potentiometer isused as a variable voltage divider in the volume control of a radio.

References

Further reading• Horowitz, Paul; Hill, Winfield (1989). The Art of Electronics. Cambridge University Press.

External links• Voltage divider or potentiometer calculations (http:/ / www. sengpielaudio. com/ calculator-voltagedivider. htm)• Voltage divider tutorial video in HD (http:/ / afrotechmods. com/ tutorials/ 2011/ 11/ 28/ voltage-divider-tutorial/

)• Online calculator to choose the values by series E24, E96 (http:/ / www. magic-worlld. narod. ru)• Online voltage divider calculator: chooses the best pair from a given series and also gives the color code (http:/ /

www. cl-projects. de/ projects/ tools/ resmatch-en. phtml)• Voltage divider theory (http:/ / www. tedpavlic. com/ teaching/ osu/ ece209/ support/ circuits_sys_review. pdf) -

RC low-pass filter example and voltage divider using Thévenin's theorem

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Y-Δ transform 61

Y-Δ transformThe Y-Δ transform, also written wye-delta and also known by many other names, is a mathematical technique tosimplify the analysis of an electrical network. The name derives from the shapes of the circuit diagrams, which lookrespectively like the letter Y and the Greek capital letter Δ. This circuit transformation theory was published byArthur Edwin Kennelly in 1899.[1] It is widely used in analysis of three-phase electric power circuits.The Y-Δ transform can be considered a special case of the star-mesh transform for three resistors.

Names

Illustration of the transform in its T-Π representation,

The Y-Δ transform is known by avariety of other names, mostly basedupon the two shapes involved, listed ineither order. The Y, spelled out as wye,can also be called T or star; the Δ,spelled out as delta, can also be calledtriangle, Π (spelled out as pi), ormesh. Thus, common names for thetransformation include wye-delta ordelta-wye, star-delta, star-mesh, or T-Π.

Basic Y-Δ transformation

Δ and Y circuits with the labels which are used in this article.

The transformation is used to establishequivalence for networks with threeterminals. Where three elements terminate ata common node and none are sources, thenode is eliminated by transforming theimpedances. For equivalence, the impedancebetween any pair of terminals must be thesame for both networks. The equationsgiven here are valid for complex as well asreal impedances.

Equations for the transformation from Δ-load to Y-load 3-phase circuit

The general idea is to compute the impedance at a terminal node of the Y circuit with impedances , toadjacent nodes in the Δ circuit by

where are all impedances in the Δ circuit. This yields the specific formulae

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Y-Δ transform 62

Equations for the transformation from Y-load to Δ-load 3-phase circuit

The general idea is to compute an impedance in the Δ circuit by

where is the sum of the products of all pairs of impedances in the Y circuit andis the impedance of the node in the Y circuit which is opposite the edge with . The formula for the

individual edges are thus

A proof of the existence and uniqueness of the transformationThe feasibility of the transformation can be shown as a consequence of superposition theorem in electric circuit. Ashort proof, rather than derived as a corollary of the more general star-mesh transform, can be given as follows. Theequivalence lies in the statement that for any external voltages ( , and ) applying at the three nodes ( ,

and ), the corresponding currents ( , and ) are exactly the same for both the Y and Δ circuit, andvice versa. In this proof, we start with given external currents at the nodes. According to superposition theorem, thevoltages can be obtained by studying the linear summation of the resulting voltages at the nodes of following threeproblems: apply at the three nodes with current (1) , , , (2) , ,

and (3) , , . It can be readily shown that due to Kirchhoff's circuitlaws, one has . One notes that now each problem is relatively simple, since it only involves onesingle ideal current source. To obtain exactly the same outcome voltages at the nodes for each problem, theequivalent resistances in two circuits must be the same, this can be easily found by using the basic rules of series andparallel circuits:

Though usually six equations are more than enough to express three variables ( ) in term of the otherthree variables( ), here it is straightforward to show that these equations indeed lead to the abovedesigned expressions. In fact, the superposition theorem not only establishes the relation between the values of theresistances, but also guarantees the uniqueness of such solution.

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Y-Δ transform 63

Simplification of networksResistive networks between two terminals can theoretically be simplified to a single equivalent resistor (moregenerally, the same is true of impedance). Series and parallel transforms are basic tools for doing so, but for complexnetworks such as the bridge illustrated here, they do not suffice.The Y-Δ transform can be used to eliminate one node at a time and produce a network that can be further simplified,as shown.

Transformation of a bridge resistor network, using the Y-Δ transform to eliminate node D, yields an equivalent network that mayreadily be simplified further.

The reverse transformation, Δ-Y, which adds a node, is often handy to pave the way for further simplification aswell.

Transformation of a bridge resistor network, using the Δ-Y transform, also yields anequivalent network that may readily be simplified further.

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Y-Δ transform 64

Graph theoryIn graph theory, the Y-Δ transform means replacing a Y subgraph of a graph with the equivalent Δ subgraph. Thetransform preserves the number of edges in a graph, but not the number of vertices or the number of cycles. Twographs are said to be Y-Δ equivalent if one can be obtained from the other by a series of Y-Δ transforms in eitherdirection. For example, the Petersen family is a Y-Δ equivalence class.

Demonstration

Δ-load to Y-load transformation equations

Δ and Y circuits with the labels that are used in this article.

To relate from Δ tofrom Y, the

impedance between two correspondingnodes is compared. The impedance ineither configuration is determined as ifone of the nodes is disconnected fromthe circuit.The impedance between N1 and N2with N3 disconnected in Δ:

To simplify, let be the sum of .

Thus,

The corresponding impedance between N1 and N2 in Y is simple:

hence:

(1)

Repeating for :

(2)

and for :

(3)

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Y-Δ transform 65

From here, the values of can be determined by linear combination (addition and/or subtraction).For example, adding (1) and (3), then subtracting (2) yields

thus,

where For completeness:

(4)

(5)

(6)

Y-load to Δ-load transformation equationsLet

.We can write the Δ to Y equations as

(1)

(2)

(3)

Multiplying the pairs of equations yields

(4)

(5)

(6)

and the sum of these equations is

(7)

Factor from the right side, leaving in the numerator, canceling with an in the denominator.

(8)

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Y-Δ transform 66

Note the similarity between (8) and {(1),(2),(3)}Divide (8) by (1)

which is the equation for . Dividing (8) by (2) or (3) (expressions for or ) gives the remaining equations.

Notes[1] A.E. Kennelly, Equivalence of triangles and stars in conducting networks, Electrical World and Engineer, vol. 34, pp. 413–414, 1899.

References• William Stevenson, Elements of Power System Analysis 3rd ed., McGraw Hill, New York, 1975, ISBN

0-07-061285-4

External links• Star-Triangle Conversion (http:/ / www. designcabana. com/ knowledge/ electrical/ basics/ resistors): Knowledge

on resistive networks and resistors• Calculator of Star-Triangle transform (http:/ / www. elektro-energetika. cz/ calculations/ transfigurace.

php?language=english)

Nodal analysis

Kirchhoff's current law is the basis of nodalanalysis.

In electric circuits analysis, nodal analysis, node-voltage analysis, orthe branch current method is a method of determining the voltage(potential difference) between "nodes" (points where elements orbranches connect) in an electrical circuit in terms of the branchcurrents.

In analyzing a circuit using Kirchhoff's circuit laws, one can either donodal analysis using Kirchhoff's current law (KCL) or mesh analysisusing Kirchhoff's voltage law (KVL). Nodal analysis writes anequation at each electrical node, requiring that the branch currentsincident at a node must sum to zero. The branch currents are written interms of the circuit node voltages. As a consequence, each branchconstitutive relation must give current as a function of voltage; anadmittance representation. For instance, for a resistor, Ibranch = Vbranch* G, where G (=1/R) is the admittance (conductance) of the resistor.

Nodal analysis is possible when all the circuit elements' branch constitutive relations have an admittancerepresentation. Nodal analysis produces a compact set of equations for the network, which can be solved by hand ifsmall, or can be quickly solved using linear algebra by computer. Because of the compact system of equations, manycircuit simulation programs (e.g. SPICE) use nodal analysis as a basis. When elements do not have admittancerepresentations, a more general extension of nodal analysis, modified nodal analysis, can be used.

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Nodal analysis 67

While simple examples of nodal analysis focus on linear elements, more complex nonlinear networks can also besolved with nodal analysis by using Newton's method to turn the nonlinear problem into a sequence of linearproblems.

Method1. Note all connected wire segments in the circuit. These are the nodes of nodal analysis.2.2. Select one node as the ground reference. The choice does not affect the result and is just a matter of convention.

Choosing the node with the most connections can simplify the analysis.3.3. Assign a variable for each node whose voltage is unknown. If the voltage is already known, it is not necessary to

assign a variable.4.4. For each unknown voltage, form an equation based on Kirchhoff's current law. Basically, add together all

currents leaving from the node and mark the sum equal to zero. Finding the current between two nodes is nothingmore than "the node you're on, minus the node you're going to, divided by the resistance between the two nodes."

5. If there are voltage sources between two unknown voltages, join the two nodes as a supernode. The currents ofthe two nodes are combined in a single equation, and a new equation for the voltages is formed.

6. Solve the system of simultaneous equations for each unknown voltage.

Examples

Basic case

Basic example circuit with one unknown voltage, V1.

The only unknown voltage in this circuit is V1.There are three connections to this node andconsequently three currents to consider. Thedirection of the currents in calculations is chosento be away from the node.

1. Current through resistor R1: (V1 - VS) / R12. Current through resistor R2: V1 / R23. Current through current source IS: -ISWith Kirchhoff's current law, we get:

This equation can be solved in respect to V1:

Finally, the unknown voltage can be solved by substituting numerical values for the symbols. Any unknown currentsare easy to calculate after all the voltages in the circuit are known.

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Nodal analysis 68

Supernodes

In this circuit, VA is between two unknownvoltages, and is therefore a supernode.

In this circuit, we initially have two unknown voltages, V1 and V2.The voltage at V3 is already known to be VB because the otherterminal of the voltage source is at ground potential.

The current going through voltage source VA cannot be directlycalculated. Therefore we can not write the current equations for eitherV1 or V2. However, we know that the same current leaving node V2must enter node V1. Even though the nodes can not be individuallysolved, we know that the combined current of these two nodes is zero.This combining of the two nodes is called the supernode technique,and it requires one additional equation: V1 = V2 + VA.

The complete set of equations for this circuit is:

By substituting V1 to the first equation and solving in respect to V2,we get:

References• P. Dimo Nodal Analysis of Power Systems Abacus Press Kent 1975

External links• Branch current method [1]

• Online four-node problem solver [2]

• Simple Nodal Analysis Example [3]

References[1] http:/ / www. allaboutcircuits. com/ vol_1/ chpt_10/ 2. html[2] http:/ / www. catc. ac. ir/ mazlumi/ node. php[3] http:/ / wiki. syncleus. com/ index. php/ Nodal_Analysis

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Mesh analysis 69

Mesh analysis

Figure 1: Essential meshes of the planar circuit labeled 1, 2, and 3. R1, R2, R3, 1/sc,and Ls represent the impedance of the resistors, capacitor, and inductor values inthe s-domain. Vs and is are the values of the voltage source and current source,

respectively.

Mesh analysis (or the mesh currentmethod) is a method that is used to solveplanar circuits for the currents (andindirectly the voltages) at any place in thecircuit. Planar circuits are circuits that canbe drawn on a plane surface with no wirescrossing each other. A more generaltechnique, called loop analysis (with thecorresponding network variables called loopcurrents) can be applied to any circuit,planar or not. Mesh analysis and loopanalysis both make use of Kirchhoff’svoltage law to arrive at a set of equationsguaranteed to be solvable if the circuit has asolution.[1] Mesh analysis is usually easierto use when the circuit is planar, comparedto loop analysis.[2]

Mesh currents and essential meshes

Figure 2: Circuit with mesh currents labeled as i1, i2, and i3. The arrows show thedirection of the mesh current.

Mesh analysis works by arbitrarily assigningmesh currents in the essential meshes (alsoreferred to as independent meshes). Anessential mesh is a loop in the circuit thatdoes not contain any other loop. Figure 1labels the essential meshes with one, two,and three.[3]

A mesh current is a current that loopsaround the essential mesh and the equationsare set solved in terms of them. A meshcurrent may not correspond to anyphysically flowing current, but the physicalcurrents are easily found from them.[2] It isusual practice to have all the mesh currentsloop in the same direction. This helpsprevent errors when writing out theequations. The convention is to have all the mesh currents looping in a clockwise direction.[3] Figure 2 shows thesame circuit from Figure 1 with the mesh currents labeled.

Solving for mesh currents instead of directly applying Kirchhoff's current law and Kirchhoff's voltage law cangreatly reduce the amount of calculation required. This is because there are fewer mesh currents than there arephysical branch currents. In figure 2 for example, there are six branch currents but only three mesh currents.

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Mesh analysis 70

Setting up the equationsEach mesh produces one equation. These equations are the sum of the voltage drops in a complete loop of the meshcurrent.[3] For problems more general than those including current and voltage sources, the voltage drops will be theimpedance of the electronic component multiplied by the mesh current in that loop.[4]

If a voltage source is present within the mesh loop, the voltage at the source is either added or subtracted dependingon if it is a voltage drop or a voltage rise in the direction of the mesh current. For a current source that is notcontained between two meshes, the mesh current will take the positive or negative value of the current sourcedepending on if the mesh current is in the same or opposite direction of the current source.[3] The following is thesame circuit from above with the equations needed to solve for all the currents in the circuit.

Once the equations are found, the system of linear equations can be solved by using any technique to solve linearequations.

Special casesThere are two special cases in mesh current: currents containing a supermesh and currents containing dependentsources.

Supermesh

Figure 3: Circuit with a supermesh. Supermesh occurs because the current sourceis in between the essential meshes.

A supermesh occurs when a current sourceis contained between two essential meshes.The circuit is first treated as if the currentsource is not there. This leads to oneequation that incorporates two meshcurrents. Once this equation is formed, anequation is needed that relates the two meshcurrents with the current source. This will bean equation where the current source isequal to one of the mesh currents minus theother. The following is a simple example ofdealing with a supermesh.[2]

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Mesh analysis 71

Dependent sources

Figure 4: Circuit with dependent source. ix is the current upon which the dependentsource depends.

A dependent source is a current source orvoltage source that depends on the voltageor current of another element in the circuit.When a dependent source is containedwithin an essential mesh, the dependentsource should be treated like an independentsource. After the mesh equation is formed, adependent source equation is needed. Thisequation is generally called a constraintequation. This is an equation that relates thedependent source’s variable to the voltage orcurrent that the source depends on in the circuit. The following is a simple example of a dependent source.[2]

External links• Mesh current method [5]

• Online three-mesh problem solver [6]

References[1] Hayt, William H., & Kemmerly, Jack E. (1993). Engineering Circuit Analysis (5th ed.), New York: McGraw Hill.[2] Nilsson, James W., & Riedel, Susan A. (2002). Introductory Circuits for Electrical and Computer Engineering. New Jersey: Prentice Hall.[3] Lueg, Russell E., & Reinhard, Erwin A. (1972). Basic Electronics for Engineers and Scientists (2nd ed.). New York: International Textbook

Company.[4] Puckett, Russell E., & Romanowitz, Harry A. (1976). Introduction to Electronics (2nd ed.). San Francisco: John Wiley and Sons, Inc.[5] http:/ / www. allaboutcircuits. com/ vol_1/ chpt_10/ 3. html[6] http:/ / www. catc. ac. ir/ mazlumi/ mesh. php

Ramesh Singh, DTU (Formerly Delhi College of Engineering)singhramesh2@gmail.com

Superposition theorem 72

Superposition theoremThe superposition theorem for electrical circuits states that for a linear system the response (Voltage or Current) inany branch of a bilateral linear circuit having more than one independent source equals the algebraic sum of theresponses caused by each independent source acting alone, while all other independent sources are replaced by theirinternal impedances.To ascertain the contribution of each individual source, all of the other sources first must be "turned off" (set to zero)by:1. Replacing all other independent voltage sources with a short circuit (thereby eliminating difference of potential.

i.e. V=0, internal impedance of ideal voltage source is ZERO (short circuit)).2. Replacing all other independent current sources with an open circuit (thereby eliminating current. i.e. I=0,

internal impedance of ideal current source is infinite (open circuit).This procedure is followed for each source in turn, then the resultant responses are added to determine the trueoperation of the circuit. The resultant circuit operation is the superposition of the various voltage and current sources.The superposition theorem is very important in circuit analysis. It is used in converting any circuit into its Nortonequivalent or Thevenin equivalent.Applicable to linear networks (time varying or time invariant) consisting of independent sources, linear dependentsources, linear passive elements Resistors, Inductors, Capacitors and linear transformers.Another point that should be considered is that superposition only works for voltage and current but not power. Inother words the sum of the powers is not the real consumed power. To calculate power we should first usesuperposition to find both current and voltage of that linear element and then calculate sum of the multiplied voltagesand currents respectively.

References•• Electronic Devices and Circuit Theory 9th ed. by Boylestad and Nashelsky•• Basic Circuit Theory , By: C. A. Desoer and E. H. Kuh

External links• On the Application of Superposition to Dependent Sources in Circuit Analysis [1] - proves superposition of

dependent sources is valid

References[1] http:/ / users. ece. gatech. edu/ mleach/ papers/ superpos. pdf

Ramesh Singh, DTU (Formerly Delhi College of Engineering)singhramesh2@gmail.com

Thévenin's theorem 73

Thévenin's theoremThévenin's theorem holds, to illustrate in DC circuit theory terms, that (see image):

• Any linear electrical network with voltage and current sources and only resistances can be replaced atterminals A-B by an equivalent voltage source Vth in series connection with an equivalent resistance Rth.

• This equivalent voltage Vth is the voltage obtained at terminals A-B of the network with terminals A-B opencircuited.

• This equivalent resistance Rth is the resistance obtained at terminals A-B of the network with all itsindependent current sources open circuited and all its independent voltage sources short circuited.

For AC systems, the theorem can be applied to reactive impedances as well as resistances.The theorem was independently derived in 1853 by the German scientist Hermann von Helmholtz and in 1883 byLéon Charles Thévenin (1857–1926), an electrical engineer with France's national Postes et Télégraphestelecommunications organization.[1][2][3][4][5][6]

Thévenin's theorem and its dual, Norton's theorem, are widely used for circuit analysis simplification and to studycircuit's initial-condition and steady-state response.[7][8] Thévenin's theorem can be used to convert any circuit'ssources and impedances to a Thévenin equivalent; use of the theorem may in some cases be more convenient thanuse of Kirchhoff's circuit laws.[9][6]

Any black box containing resistances only andvoltage and current sources can be replaced to a

Thévenin equivalent circuit consisting of anequivalent voltage source in series connection

with an equivalent resistance.

Calculating the Thévenin equivalent

To calculate the equivalent circuit, the resistance and voltage areneeded, so two equations are required. These two equations are usuallyobtained by using the following steps, but any conditions placed on theterminals of the circuit should also work:

1. Calculate the output voltage, VAB, when in open circuit condition(no load resistor—meaning infinite resistance). This is VTh.

2. Calculate the output current, IAB, when the output terminals areshort circuited (load resistance is 0). RTh equals VTh divided by thisIAB.

The equivalent circuit is a voltage source with voltage VTh in series with a resistance RTh.Step 2 could also be thought of as:

2a. Replace voltage sources with short circuits, and current sources with open circuits.2b. Calculate the resistance between terminals A and B. This is RTh.

The Thévenin-equivalent voltage is the voltage at the output terminals of the original circuit. When calculating aThévenin-equivalent voltage, the voltage divider principle is often useful, by declaring one terminal to be Vout andthe other terminal to be at the ground point.The Thévenin-equivalent resistance is the resistance measured across points A and B "looking back" into the circuit.It is important to first replace all voltage- and current-sources with their internal resistances. For an ideal voltagesource, this means replace the voltage source with a short circuit. For an ideal current source, this means replace thecurrent source with an open circuit. Resistance can then be calculated across the terminals using the formulae forseries and parallel circuits. This method is valid only for circuits with independent sources. If there are dependentsources in the circuit, another method must be used such as connecting a test source across A and B and calculatingthe voltage across or current through the test source.

Ramesh Singh, DTU (Formerly Delhi College of Engineering)singhramesh2@gmail.com

Thévenin's theorem 74

Example

Step 0: The original circuit

Step 1: Calculating the equivalent outputvoltage

Step 2: Calculating the equivalentresistance

Step 3: The equivalent circuit

In the example, calculating the equivalent voltage:

(notice that R1 is not taken into consideration, as above calculations are done in an open circuit condition between Aand B, therefore no current flows through this part, which means there is no current through R1 and therefore novoltage drop along this part)Calculating equivalent resistance:

Conversion to a Norton equivalent

A Norton equivalent circuit is related to the Thévenin equivalent by thefollowing:

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Thévenin's theorem 75

Practical limitations•• Many, if not most circuits are only linear over a certain range of values, thus the Thévenin equivalent is valid only

within this linear range and may not be valid outside the range.• The Thévenin equivalent has an equivalent I–V characteristic only from the point of view of the load.•• The power dissipation of the Thévenin equivalent is not necessarily identical to the power dissipation of the real

system. However, the power dissipated by an external resistor between the two output terminals is the sameregardless of how the internal circuit is represented.

A proof of the theoremThe proof involves two steps. First use superposition theorem to construct a solution, and then use uniquenesstheorem to show the solution is unique. The second step is usually implied. Firstly, using the superposition theorem,in general for any linear "black box" circuit which contains voltage sources and resistors, one can always write downits voltage as a linear function of the corresponding current as follows

where the first term reflects the linear summation of contributions from each voltage source, while the second termmeasures the contribution from all the resistors. The above argument is due to the fact that the voltage of the blackbox for a given current is identical to the linear superposition of the solutions of the following problems: (1) toleave the black box open circuited but activate individual voltage source one at a time and, (2) to short circuit all thevoltage sources but feed the circuit with a certain ideal voltage source so that the resulting current exactly reads (or an ideal current source of current ). Once the above expression is established, it is straightforward to show that

and are the single voltage source and the single series resistor in question.

References[1][1] Helmholtz[2][2] Thévenin (1883a)[3][3] Thévenin (1883b)[4][4] Johnson (2003a)[5][5] Brittain[6][6] Dorf[7][7] Brenner[8][8] Elgerd[9][9] Dwight

Bibliography• Brenner, Egon; Javid, Mansour (1959). "Chapter 12 - Network Functions" (http:/ / books. google. ca/ books/

about/ Analysis_of_electric_circuits. html?id=V4FrAAAAMAAJ& redir_esc=y). Analysis of Electric Circuits.McGraw-Hill. pp. 268–269.

• Brittain, J.E. (March 1990). "Thevenin's theorem" (http:/ / ieeexplore. ieee. org/ search/ searchresult.jsp?newsearch=true& queryText=James+ E. + Brittain+ Thevenin's+ theorem& . x=41& . y=17). IEEE Spectrum27 (3): 42. doi: 10.1109/6.48845 (http:/ / dx. doi. org/ 10. 1109/ 6. 48845). Retrieved 1 February 2013.

• Dorf, Richard C.; Svoboda, James A. (2010). "Chapter 5 - Circuit Theorems" (http:/ / ca. wiley. com/ WileyCDA/WileyTitle/ productCd-EHEP000347. html). Introduction to Electric Circuits (8th ed.). Hoboken, NJ: John Wiley& Sons. pp. 162–207. ISBN 978-0-470-52157-1.

• Dwight, Herbert B. (1949). "Sec. 2 - Electric and Magnetic Circuits". In Knowlton, A.E. Standard Handbook forElectrical Engineers (8th ed.). McGraw-Hill. p. 26.

• Elgerd, Olle I. (2007). "Chapter 10, Energy System Transients - Surge Phenomena and Symmetrical Fault Analysis" (http:/ / books. google. ca/ books/ about/ Electric_Energy_Systems_Theory. html?id=AKTi3UxfhlgC&

Ramesh Singh, DTU (Formerly Delhi College of Engineering)singhramesh2@gmail.com

Thévenin's theorem 76

redir_esc=y). Electric Energy Systems Theory: An Introduction. Tata McGraw-Hill. pp. 402–429.ISBN 978-0070192300.

• Helmhotz, H. (1853). "Über einige Gesetze der Vertheilung elektrischer Ströme in körperlichen Leitern mitAnwendung auf die thierisch-elektrischen Versuche (Some laws concerning the distribution of electrical currentsin conductors with applications to experiments on animal electricity)" (http:/ / gallica. bnf. fr/ ark:/ 12148/bpt6k151746. image. f225. langFR). Annalen der Physik und Chemie 89 (6): 211–233.

• Johnson, D.H. (2003a). "Origins of the equivalent circuit concept: the voltage-source equivalent" (http:/ / www.ece. rice. edu/ ~dhj/ paper1. pdf). Proceedings of the IEEE 91 (4): 636–640. doi: 10.1109/JPROC.2003.811716(http:/ / dx. doi. org/ 10. 1109/ JPROC. 2003. 811716).

• Johnson, D.H. (2003b). "Origins of the equivalent circuit concept: the current-source equivalent" (http:/ / www.ece. rice. edu/ ~dhj/ paper2. pdf). Proceedings of the IEEE 91 (5): 817–821. doi: 10.1109/JPROC.2003.811795(http:/ / dx. doi. org/ 10. 1109/ JPROC. 2003. 811795).

• Thévenin, L. (1883a). "Extension de la loi d’Ohm aux circuits électromoteurs complexes (Extension of Ohm’s lawto complex electromotive circuits)" (http:/ / books. google. com/ ?id=shUAAAAAMAAJ& pg=PA222&lpg=PA222& dq=Extension+ de+ la+ loi+ dâ��Ohm+ aux+ circuits+ électromoteurs+ complexes#v=onepage&q=Extension de la loi dâ��Ohm aux circuits électromoteurs complexes& f=false). Annales Télégraphiques. 3e

series 10: 222–224.• Thévenin, L. (1883b). "Sur un nouveau théorème d'électricité dynamique (On a new theorem of dynamic

electricity)". Comptes rendus hebdomadaires des séances de l'Académie des Sciences 97: 159–161.• Wenner, F. (1926). "Sci. Paper S531, A principle governing the distribution of current in systems of linear

conductors". Washington, D.C.: Bureau of Standards.

External links• Thevenin's theorem at allaboutcircuits.com (http:/ / www. allaboutcircuits. com/ vol_1/ chpt_10/ 8. html)• Filter-Order Filters: Shortcut via Thévenin Equivalent Source (http:/ / www. tedpavlic. com/ teaching/ osu/

ece209/ support/ circuits_sys_review. pdf) — showing on p. 4 complex circuit's Thévenin's theorem simplicationto first-order low-pass filter and associated voltage divider, time constant and gain.

Ramesh Singh, DTU (Formerly Delhi College of Engineering)singhramesh2@gmail.com

Norton's theorem 77

Norton's theoremKnown in Europe as the Mayer–Norton theorem, Norton's theorem holds, to illustrate in DC circuit theory terms,that (see image):

• Any linear electrical network with voltage and current sources and only resistances can be replaced atterminals A-B by an equivalent current source INO in parallel connection with an equivalent resistance RNO.

• This equivalent current INO is the current obtained at terminals A-B of the network with terminals A-B shortcircuited.

• This equivalent resistance RNO is the resistance obtained at terminals A-B of the network with all its voltagesources short circuited and all its current sources open circuited.

For AC systems the theorem can be applied to reactive impedances as well as resistances.The Norton equivalent circuit is used to represent any network of linear sources and impedances at a givenfrequency.

Any black box containing resistances only andvoltage and current sources can be replaced by an

equivalent circuit consisting of an equivalentcurrent source in parallel connection with an

equivalent resistance.

Norton's theorem and its dual, Thévenin's theorem, are widely used forcircuit analysis simplification and to study circuit's initial-conditionand steady-state response.

Norton's theorem was independently derived in 1926 by Siemens &Halske researcher Hans Ferdinand Mayer (1895–1980) and Bell Labsengineer Edward Lawry Norton (1898–1983).[1][2][3][4][5]

To find the equivalent,1. Find the Norton current INo. Calculate the output current, IAB, with

a short circuit as the load (meaning 0 resistance between A and B).This is INo.

2. Find the Norton resistance RNo. When there are no dependent sources (all current and voltage sources areindependent), there are two methods of determining the Norton impedance RNo.

• Calculate the output voltage, VAB, when in open circuit condition (i.e., no load resistor — meaning infiniteload resistance). RNo equals this VAB divided by INo.

or• Replace independent voltage sources with short circuits and independent current sources with open circuits.

The total resistance across the output port is the Norton impedance RNo.This is equivalent to calculating the Thevenin resistance.

However, when there are dependent sources, the more general method must be used. This method is not shownbelow in the diagrams.• Connect a constant current source at the output terminals of the circuit with a value of 1 Ampere and

calculate the voltage at its terminals. This voltage divided by the 1 A current is the Norton impedance RNo.This method must be used if the circuit contains dependent sources, but it can be used in all cases evenwhen there are no dependent sources.

Ramesh Singh, DTU (Formerly Delhi College of Engineering)singhramesh2@gmail.com

Norton's theorem 78

Example of a Norton equivalent circuit

Step 0: The original circuit Step 1: Calculating the equivalent output current Step 2: Calculating the equivalent resistance

Step 3: The equivalent circuit

In the example, the total current Itotal is given by:

The current through the load is then, using the current divider rule:

And the equivalent resistance looking back into the circuit is:

So the equivalent circuit is a 3.75 mA current source in parallel with a 2 kΩ resistor.

Conversion to a Thévenin equivalent

A Norton equivalent circuit is related to the Thévenin equivalent by thefollowing equations:

Ramesh Singh, DTU (Formerly Delhi College of Engineering)singhramesh2@gmail.com

Norton's theorem 79

Queueing theoryThe passive circuit equivalent of "Norton's theorem" in queuing theory is called the Chandy Herzog Wootheorem.[6][7] In a reversible queueing system, it is often possible to replace an uninteresting subset of queues by asingle (FCFS or PS) queue with an appropriately chosen service rate.[8]

References[1][1] Mayer[2][2] Norton[3][3] Johnson (2003b)[4][4] Brittain[5][5] Dorf[6][6] Johnson (2003a)[7][7] Gunther[8][8] Chandy et al.

Bibliography• Brittain, J.E. (March 1990). "Thevenin's theorem" (http:/ / ieeexplore. ieee. org/ search/ searchresult.

jsp?newsearch=true& queryText=James+ E. + Brittain+ Thevenin's+ theorem& . x=41& . y=17). IEEE Spectrum27 (3): 42. doi: 10.1109/6.48845 (http:/ / dx. doi. org/ 10. 1109/ 6. 48845). Retrieved 1 February 2013.

• Chandy, K. M.; Herzog, U.; Woo, L. (Jan 1975). "Parametric Analysis of Queuing Networks" (http:/ / scholar.google. ca/ scholar_url?hl=en& q=http:/ / citeseerx. ist. psu. edu/ viewdoc/ download?doi=10. 1. 1. 93. 9312&rep=rep1& type=pdf& sa=X& scisig=AAGBfm1HEBU-rSFYLTIePQWPitczchOopA& oi=scholarr&ei=L3wQUfP9DOHWiwKYtICQAQ& ved=0CC4QgAMoADAA). IBM Journal of Research and Development19 (1): 36–42. doi: 10.1147/rd.191.0036 (http:/ / dx. doi. org/ 10. 1147/ rd. 191. 0036).

• Dorf, Richard C.; Svoboda, James A. (2010). "Chapter 5 – Circuit Theorems" (http:/ / ca. wiley. com/WileyCDA/ WileyTitle/ productCd-EHEP000347. html). Introduction to Electric Circuits (8th ed.). Hoboken, NJ:John Wiley & Sons. pp. 162–207. ISBN 978-0-470-52157-1.

• Gunther, N.J. (2004). Analyzing computer systems performance : with PERL::PDQ (http:/ / books. google. com/?id=rp1EZKnr48kC& pg=PA83#v=onepage& q& f=false) (Online-Ausg. ed.). Berlin: Springer. p. 281.ISBN 3-540-20865-8.

• Johnson, D.H. (2003). "Origins of the equivalent circuit concept: the voltage-source equivalent" (http:/ / www.ece. rice. edu/ ~dhj/ paper1. pdf). Proceedings of the IEEE 91 (4): 636–640. doi: 10.1109/JPROC.2003.811716(http:/ / dx. doi. org/ 10. 1109/ JPROC. 2003. 811716).

• Johnson, D.H. (2003). "Origins of the equivalent circuit concept: the current-source equivalent" (http:/ / www.ece. rice. edu/ ~dhj/ paper2. pdf). Proceedings of the IEEE 91 (5): 817–821. doi: 10.1109/JPROC.2003.811795(http:/ / dx. doi. org/ 10. 1109/ JPROC. 2003. 811795).

• Mayer, H. F. (1926). "Ueber das Ersatzschema der Verstärkerröhre (On equivalent circuits for electronicamplifiers]". Telegraphen- und Fernsprech-Technik 15: 335–337.

• Norton, E. L. (1926). Technical Report TM26–0–1860 – Design of finite networks for uniform frequencycharacteristic. Bell Laboratories.

External links• Norton's theorem at allaboutcircuits.com (http:/ / www. allaboutcircuits. com/ vol_1/ chpt_10/ 9. html)

Ramesh Singh, DTU (Formerly Delhi College of Engineering)singhramesh2@gmail.com

Maximum power transfer theorem 80

Maximum power transfer theoremIn electrical engineering, the maximum power transfer theorem states that, to obtain maximum external powerfrom a source with a finite internal resistance, the resistance of the load must equal the resistance of the source asviewed from its output terminals. Moritz von Jacobi published the maximum power (transfer) theorem around 1840;it is also referred to as "Jacobi's law".[1]

The theorem results in maximum power transfer, and not maximum efficiency. If the resistance of the load is madelarger than the resistance of the source, then efficiency is higher, since a higher percentage of the source power istransferred to the load, but the magnitude of the load power is lower since the total circuit resistance goes up.If the load resistance is smaller than the source resistance, then most of the power ends up being dissipated in thesource, and although the total power dissipated is higher, due to a lower total resistance, it turns out that the amountdissipated in the load is reduced.The theorem states how to choose (so as to maximize power transfer) the load resistance, once the source resistanceis given. It is a common misconception to apply the theorem in the opposite scenario. It does not say how to choosethe source resistance for a given load resistance. In fact, the source resistance that maximizes power transfer isalways zero, regardless of the value of the load resistance.The theorem can be extended to AC circuits that include reactance, and states that maximum power transfer occurswhen the load impedance is equal to the complex conjugate of the source impedance.

Maximizing power transfer versus power efficiencyThe theorem was originally misunderstood (notably by Joule) to imply that a system consisting of an electric motordriven by a battery could not be more than 50% efficient since, when the impedances were matched, the power lostas heat in the battery would always be equal to the power delivered to the motor. In 1880 this assumption was shownto be false by either Edison or his colleague Francis Robbins Upton, who realized that maximum efficiency was notthe same as maximum power transfer. To achieve maximum efficiency, the resistance of the source (whether abattery or a dynamo) could be made close to zero. Using this new understanding, they obtained an efficiency ofabout 90%, and proved that the electric motor was a practical alternative to the heat engine.

The condition of maximum power transfer does not result in maximum efficiency. If we define the efficiency asthe ratio of power dissipated by the load to power developed by the source, then it is straightforward to calculatefrom the above circuit diagram that

Consider three particular cases:

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Maximum power transfer theorem 81

• If , then • If or then • If , then The efficiency is only 50% when maximum power transfer is achieved, but approaches 100% as the load resistanceapproaches infinity, though the total power level tends towards zero. Efficiency also approaches 100% if the sourceresistance approaches zero, and 0% if the load resistance approaches zero. In the latter case, all the power isconsumed inside the source (unless the source also has no resistance), so the power dissipated in a short circuit iszero.

Impedance matchingA related concept is reflectionless impedance matching. In radio, transmission lines, and other electronics, there isoften a requirement to match the source impedance (such as a transmitter) to the load impedance (such as anantenna) to avoid reflections in the transmission line.

Calculus-based proof for purely resistive circuits(See Cartwright[2] for a non-calculus-based proof)

In the diagram opposite, power is being transferred from the source,with voltage and fixed source resistance , to a load withresistance , resulting in a current . By Ohm's law, is simplythe source voltage divided by the total circuit resistance:

The power dissipated in the load is the square of the currentmultiplied by the resistance:

The value of for which this expression is a maximum could be calculated by differentiating it, but it is easier tocalculate the value of for which the denominator

is a minimum. The result will be the same in either case. Differentiating the denominator with respect to :

For a maximum or minimum, the first derivative is zero, so

or

In practical resistive circuits, and are both positive, so the positive sign in the above is the correct solution.To find out whether this solution is a minimum or a maximum, the denominator expression is differentiated again:

This is always positive for positive values of and , showing that the denominator is a minimum, and thepower is therefore a maximum, when

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Maximum power transfer theorem 82

A note of caution is in order here. This last statement, as written, implies to many people that for a given load, thesource resistance must be set equal to the load resistance for maximum power transfer. However, this equation onlyapplies if the source resistance cannot be adjusted, e.g., with antennas (see the first line in the proof stating "fixedsource resistance"). For any given load resistance a source resistance of zero is the way to transfer maximum powerto the load. As an example, a 100 volt source with an internal resistance of 10 ohms connected to a 10 ohm load willdeliver 250 watts to that load. Make the source resistance zero ohms and the load power jumps to 1000 watts.

In reactive circuitsThe theorem also applies where the source and/or load are not totally resistive. This invokes a refinement of themaximum power theorem, which says that any reactive components of source and load should be of equal magnitudebut opposite phase. (See below for a derivation.) This means that the source and load impedances should be complexconjugates of each other. In the case of purely resistive circuits, the two concepts are identical. However, physicallyrealizable sources and loads are not usually totally resistive, having some inductive or capacitive components, and sopractical applications of this theorem, under the name of complex conjugate impedance matching, do, in fact, exist.If the source is totally inductive (capacitive), then a totally capacitive (inductive) load, in the absence of resistivelosses, would receive 100% of the energy from the source but send it back after a quarter cycle. The resultant circuitis nothing other than a resonant LC circuit in which the energy continues to oscillate to and fro. This is calledreactive power. Power factor correction (where an inductive reactance is used to "balance out" a capacitive one), isessentially the same idea as complex conjugate impedance matching although it is done for entirely different reasons.For a fixed reactive source, the maximum power theorem maximizes the real power (P) delivered to the load bycomplex conjugate matching the load to the source.For a fixed reactive load, power factor correction minimizes the apparent power (S) (and unnecessary current)conducted by the transmission lines, while maintaining the same amount of real power transfer. This is done byadding a reactance to the load to balance out the load's own reactance, changing the reactive load impedance into aresistive load impedance.

Proof

In this diagram, AC power is being transferred from the source, withphasor magnitude voltage (peak voltage) and fixed sourceimpedance , to a load with impedance , resulting in a phasormagnitude current . is simply the source voltage divided bythe total circuit impedance:

The average power dissipated in the load is the square of thecurrent multiplied by the resistive portion (the real part) of the

load impedance:

where the resistance and reactance are the real and imaginary parts of , and is the imaginary part of.

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Maximum power transfer theorem 83

To determine the values of and (since , , and are fixed) for which this expression is amaximum, we first find, for each fixed positive value of , the value of the reactive term for which thedenominator

is a minimum. Since reactances can be negative, this denominator is easily minimized by making

The power equation is now reduced to:

and it remains to find the value of which maximizes this expression. However, this maximization problem hasexactly the same form as in the purely resistive case, and the maximizing condition can be found in thesame way.The combination of conditions••can be concisely written with a complex conjugate (the *) as:

Notes

References•• H.W. Jackson (1959) Introduction to Electronic Circuits, Prentice-Hall.

External links• The complex conjugate matching false idol (http:/ / www. analog-rf. com/ match. shtml)• Conjugate matching versus reflectionless matching (http:/ / www. ece. rutgers. edu/ ~orfanidi/ ewa/ ch12. pdf)

(PDF) taken from Electromagnetic Waves and Antennas (http:/ / www. ece. rutgers. edu/ ~orfanidi/ ewa/ )• The Spark Transmitter. 2. Maximising Power, part 1. (http:/ / home. freeuk. net/ dunckx/ wireless/ maxpower1/

maxpower1. html)• Jacobi's theorem (http:/ / www. du. edu/ ~jcalvert/ tech/ jacobi. htm) - unconfirmed claim that theorem was

discovered by Moritz Jacobi• (http:/ / chem. ch. huji. ac. il/ ~eugeniik/ history/ jacobi. html) MH Jacobi Biographical notes• Google Docs Spreadsheet (http:/ / spreadsheets. google. com/ pub?key=pANWpjmy2L4xC96TaLF6MEA)

calculating max power transfer efficiencies by Sholto Maud and Dino Cevolatti.

Ramesh Singh, DTU (Formerly Delhi College of Engineering)singhramesh2@gmail.com

Article Sources and Contributors 84

Article Sources and ContributorsElectrical resistance and conductance  Source: http://en.wikipedia.org/w/index.php?oldid=569574128  Contributors: 2602:306:32D7:D870:CCB:816D:2FCB:B55C, A. B., A8UDI, AC+793888, Adamtester, Ahoerstemeier, Alan Liefting, AlanD, Alansohn, Alfred Centauri, Allstarecho, Aloplop, Alphachimp, Alzo55, Amaury, Amigadave, Ams627, Ancheta Wis, Andyjsmith, AnnaFrodesiak, Anwar saadat, Arc de Ciel, ArnoldReinhold, Artelius, Arthena, Ashishbhatnagar72, Ateş, Aurumpotestasest, Avihu, AxelBoldt, BD2412, Bar0n, Barticus88, Beatnik8983,BenFrantzDale, Bgruber, Blues-harp, Bm gub, Bowlhover, Brockert, Bryan Derksen, Can't sleep, clown will eat me, CanOfWorms, Canderson7, Cantaloupe2, CapitalLetterBeginning, Ccrrccrr,Chaojoker, Chetvorno, Chovain, Christian75, CinchBug, Coasterlover1994, CosineKitty, Courcelles, Cpl Syx, Csoliverez, CyrilB, DHN, DJIndica, DV8 2XL, DVdm, DabMachine, Damonjackson, DanielCD, Danny Rathjens, Darkgecko, David Eppstein, Dbenbenn, Dead avengers, Deipnosopher, Delirium, Delldot, Delta G, Denisarona, DerHexer, Destroyer000, DexDor,Dhruv17singhal, Dicklyon, Dingabat111, Discospinster, Dkasak, Dlohcierekim, Dmitry sychov, Dnas, Dogears, Dominik, Donarreiskoffer, Doubledoubletripletrouble, Download, Dpiddy1337,Drjem3, Drmies, E71, ESkog, EWikist, Editor at Large, Editzu, Electron9, Elikrieg, Elliskev, Embrittled, Enochlau, Enormousdude, Epbr123, Estnyboer, Eumolpo, Evercat, Everyking, Evilsaltine, Filipporso, Finalnight, Foant, FreeScientificInformation, Frodood, FrozenMan, Gaffydantane, Gaius Cornelius, Gaurav.pal, Gene Nygaard, George Makepeace, Giftlite, Gil Dawson,Gilderien, Gilliam, Girjesh.dubey, Glasseyes24, Glen, Glenn, Gran2, Gtg204y, Guptaankit89, Hadal, Handuka, Harley peters, Hazel77, Headbomb, Heron, Hibbleton, Hut 8.5, Hylian loach, Ialready forgot, Icairns, Incompetence, InsanityBringer, Isomerise, Ixfd64, J36miles, JHMM13, JaGa, Jatkins, JenVan, Jim1138, John of Reading, Johndburger, Jolly Janner, Joshkurien, Joy,Jrockley, Jrousseau, Jrpresto, Jusdafax, JustAGal, JustUser, Jwonder, Kaleal92, Katieh5584, Kostisl, Koyn, Kunal3224, Kurzon, L Kensington, L3lackEyedAngels, Lanthanum-138, Lenko,Lewispb, Light current, LizardJr8, Loodog, Looscan, Looxix, Lradrama, Lseixas, Lupo, M7, Malcolm Farmer, Mark Chung, Mark viking, Matusz, Meawoppl, Mebden, Michael Hardy, MicroBioHawk, Mike.lifeguard, Mild Bill Hiccup, MisterSheik, Mjrice, Mkill, MrRadioGuy, Nascar1996, Nathparkling, Nikai, No such user, NorwegianBlue, NovaDog, Nrets, Nunh-huh, Ojs,OldakQuill, Omegatron, Orphan Wiki, Oshwah, Oskard, P.asgaripour, Patrick, Paverider, Pedro, Perl, Peterlin, Philip Trueman, Phillipsacp, PhySusie, Physchim62, Pi, Piano non troppo, Pichote,Plugwash, Possum, Quantpole, Quistnix, Quiyst, RR, Randallash, Raymondwinn, Razorflame, Reedy, ResearchRave, Rich Farmbrough, Richj, RickK, Rjstott, Rob Hurt, Robert K S, RobertG,RockMagnetist, RogierBrussee, Roy da Vinci, Royboycrashfan, Rwestafer, SCEhardt, Salsb, Satori Son, Sax Russell, Sbyrnes321, SchreiberBike, SchreyP, Seaphoto, Searchme, Shipmaster,Shoefly, Sillybilly, SimDarthMaul, SimonD, Simone, SirEditALot, Sirtrebuchet, SoCalSuperEagle, SomeFajitaSomewhere, SpaceFlight89, SpecialPiggy, Spinningspark, Stephen Burnett, StevenWeston, StradivariusTV, Supremeknowledge, SutharsanJIsles, Syed Wamiq Ahmed Hashmi, TDogg310, TStein, Teamfruit, Template namespace initialisation script, Texture, The RamblingMan, TheGrimReaper®™©, Theresa knott, Thubing, Tide rolls, Tim Starling, Tommy2010, TomyDuby, Tonsofpcs, Tonywalton, Traxs7, Trevor MacInnis, Trurle, Unbitwise, Vaxquis,VeejayJameson, Velella, Venny85, Vilerage, Voidxor, Vssun, Wayward, Wbm1058, Webclient101, Welsh, WikiLeon, Wikipelli, Wisibility, Wtshymanski, Wwoods, XJaM, Yahia.barie,Yakeyglee, Yansa, Yjwong, Youssefsan, Yuje, Zharradan.angelfire, Zhida, کاشف عقیل ,عبد المؤمن ,טוקיוני, 老 陳, 699 anonymous edits

Inductor  Source: http://en.wikipedia.org/w/index.php?oldid=570112839  Contributors: 2607:F140:400:1036:E981:A517:26E5:5DBE, A. Carty, AbJ32, Acronymsical, Aditya Cholan, Adpete,Aempirei, Aidanlister, Aitias, Akendall, Alertjean, Alfred Centauri, Amaraiel, Amp71, Andonic, Antikon, Apolkhanov, Arch dude, Armstrong1113149, Army1987, ArnoldReinhold, ArthurRubin, Arunsingh16, Ascidian, Atlant, Austin RS, BD2412, BabyBatter, Baseball Bugs, Bdieseldorff, Bemoeial, Ben-Zin, Benjah-bmm27, Berean Hunter, Bernard François, Berrinam, BertHickman, BhavdipGadhiya, BillC, Bobblewik, Bogdangiusca, Boulaur, Breakeydown, CAkira, Cbdorsett, Cfallin, Chairboy, Chan siuman, CharlesC, Chetvorno, Chowbok, Christian75,Christopher Mahan, Congruence, Conversion script, Copeland.James.H, CosineKitty, Cpl Syx, CyrilB, Cyrius, DV8 2XL, Dalstadt, DeadEyeArrow, Defrector, Dendodge, Dgrant, Dicklyon,Dlrohrer2003, Download, Dratman, Drhlajos, Ebraminio, Eclecticology, Elcap, Elspec, Enviromet, Even stevenson, FearXtheXfro, Frankie1969, Fresheneesz, FrozenMan, Fumitol, Funandtrvl,G-W, Gail, Gaius Cornelius, Galoubet, Gene Nygaard, GeoGreg, Gerben49, Giftlite, Glenn, Glrx, Gobonobo, Grafen, Grandfatherclok, GreenSpigot, Guy Macon, Gzuckier, Götz, Haham hanuka,Harland1, Harriv, HazardX21, Hefo, Hephaestos, Heron, Hgrosser, Highsand, Hmo, Hooperbloob, Hu12, Immibis, InvertRect, Iwsh, J. W. Love, Jauhienij, JayC, Jiang, Jlg4104, Joel D. Reid,John of Reading, Junglecat, KNfLrPnKNsT, Kar.ma, Karthik262399, Katherine, Kbwikipedia, Keenan Pepper, Khalid Mahmood, Kingpin13, Krishnavedala, Lexicon, Light current, Lightmouse,Lindosland, Lindseyrose, Little Mountain 5, LizGere, Lmatt, Loggie, Lommer, Looxix, Lornova, Lovecz, LucasVB, LukeB 11, M jurrens, Manuel Trujillo Berges, Maralia, MarsRover,Matthiaspaul, Maxzimet, Mean as custard, Meggar, Meisongbei, MetaNest, Mhims, Michael Hardy, Mike Dill, Mike Rosoft, Mike1024, Mild Bill Hiccup, Mintguy, MisterSheik, Mkill,Mondebleu, Mormegil, Munozdj, Myanw, NawlinWiki, Nczempin, Nedim Ardoğa, Nemu, Neonil, Niceguyedc, Nickptar, Night Goblin, Nikai, Nk, No such user, Ohnoitsjamie, Oli Filth,Omegatron, Oooh.oooh, Pandamonia, Papa November, Patrick, Paul Foxworthy, Penterwast, Petedarnell, Pewahl, Philip Trueman, Pinethicket, Pirateer, Pjacobi, Pol098, Prari, Prateekgoyl, RTC,RandomXYZb, RaseaC, Reddi, Rich Farmbrough, RickK, Rickcwalker, Rivertorch, Rjwilmsi, Rogper, Romanm, Ronhjones, Rtdrury, Salgueiro, Sankalpdravid, ScAvenger lv, Seaphoto,Searchme, SebastianHelm, ServAce85, Shaddack, ShiftyDave, Smack, Snafflekid, Sparkie82, Spinningspark, Srich32977, Srleffler, Ssd, Starsong, Stephenb, Steve Quinn, Steve carlson,Superherogirl7, TDogg310, THEN WHO WAS PHONE?, TYLER, Teles, TenPoundHammer, The Original Wildbear, Theo10011, Thorseth, Thumperward, Tide rolls, Tt801, Ulfbastel,Unforgiven24, UninvitedCompany, Utcursch, Vahid alpha, Velle, Vipinhari, Vssun, Wagino 20100516, Webclient101, Wiebelfrotzer, WikiWebbie, Wikigi, Wolfkeeper, Wtshymanski, Yekrats,Youandme, Your Lord and Master, Zangar, Zhinker, Zoicon5, 575 ,ساجد امجد ساجد anonymous edits

Capacitor  Source: http://en.wikipedia.org/w/index.php?oldid=569576958  Contributors: 10v1walsha, 124Nick, 2001:660:330F:A4:D6BE:D9FF:FE15:81A1,2001:67C:10EC:3F42:8000:0:0:1F6, 2602:304:47EB:97A9:21B:77FF:FEAD:46DE, 2602:306:C434:6980:BD19:FD5B:5ADC:C7C3, 31.6, 4.35.185.xxx, A. B., A. Carty, A. Parrot,Abhishekchavan79, Adam1213, Addshore, Ahoerstemeier, AhsanAli408, Airplaneman, Akamad, Alai, Aldie, Alex Khimich, Alex Sims, Alex.muller, Alexander Bell, Alfred Centauri,Aloysius314, Altered Walter, Altermike, Am088, Amcbride, Andonic, Andre Engels, Andy.Cowley, Anon lynx, AntonioSajonia, Ap, Arch dude, Arnero, ArnoldReinhold, Asher196, Atlant, AudiO Phile, Avakar, AxelBoldt, Axlq, B137, BAxelrod, BMunage, Banime, Bayezit.dirim, Bemoeial, Benfriesen12, Bentogoa, BeowulfNode, Berean Hunter, Bert Hickman, BertSen, BinaryFrog,Binksternet, Bissinger, Blackcloak, Blue520, Bobo192, Bobstay, Bogdangiusca, Bread2u, Brews ohare, Bryan Derksen, Bullet train, Calltech, Cameracut, Can't sleep, clown will eat me, Canley,Capricorn42, CaseInPoint, Catgut, Ccrrccrr, CecilWard, Chan siuman, Chetvorno, Chongkian, Circuit13, Clienthopeless, Cloudswrest, Cogniac, Coleycole, Conversion script, Coolhandscot,Corny131, Craig Pemberton, Crazytales, Crocodealer, Crunchy Numbers, Cst17, CyborgTosser, 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S.Örvarr.S, S3000, SCEhardt, SDC, Sadi Carnot, Saeed,Salako1999, Samkline, SamuelRiv, Sanalks, Sangak, Sangwine, Sankalpdravid, Sbyrnes321, Schneelocke, Schuym1, Scohoust, ScottSteiner, Scottfisher, SeanMack, Searchme, SebastianBreier,Sertrel, Seven of Nine, Shaddack, ShakataGaNai, ShakingSpirit, Shen, Shenme, Shirt58, Sietse Snel, Simdude2u, Sjakkalle, Sleepsfortheweak, Slicky, Smack, Snafflekid, Snigbrook, Snowolf,Soam Vasani, Sodium, Sole Soul, Solipsist, Sonett72, Sonia, SpectrumAnalyser, Spike Wilbury, Spinningspark, Starsong, Stephenb, Steve carlson, Steven Zhang, Stevenj, StoneCold89,StradivariusTV, StuartH, Supspirit, Suruena, Symane, Syrthiss, Szzuk, THEN WHO WAS PHONE?, TREXJET, TYelliot, Tabby, Tangotango, Tarret, Tawker, TenOfAllTrades, Terraguy,Texture, Tezcatlipoca, Tgabor, Thaflinger, The Original Wildbear, Thedatastream, Theneokid, Theo10011, Theoneintraining, Theresa knott, Thorseth, Thue, Tide rolls, Tim Starling, Timwi,Tjfulopp, Tleave2000, Tom harrison, Tom.Reding, Tommy2010, TorQue Astur, TreeSmiler, Treekids, Trigaranus, Tubular, Turk oğlan, Tylerdmace, Tylerni7, Ulflund, Uncle Dick, Urhixidur,User A1, Vahid alpha, Vanished user 47736712, Vegaswikian, Veinor, Ventusa, Vin300, Vividvilla, Vladislav Pogorelov, Vokesk, Vortex112, W0lfie, Wanderer57, Warut, Waveguy,Wavelength, Wayward, Wdfarmer, Widr, Wiki alf, WikiDao, Wikiborg, Wikid77, Wingstarsoft, Wjbeaty, Wknight94, Wolfkeeper, Woohookitty, Wstorr, Wtshymanski, Xochimec, Xspartachris,Yekrats, Yiyi303, Yk Yk Yk, Young Pioneer, Yuiwii, Yyy, Zaereth, Zaphraud, Zbxgscqf, Zelikazi, ZeroOne, Zginder, Zondran, Zueignung, Zundark, Zyxoas, Zzzzzzus, Милан Јелисавчић,1448 anonymous edits

Ohm's law  Source: http://en.wikipedia.org/w/index.php?oldid=553410448  Contributors: 0612, AC+79 3888, AbJ32, AbigailAbernathy, Acroterion, Adamarthurryan, Adi4094, AdjustShift, Ahammer92, Ahoerstemeier, AirBa, Aleksa Lukic, Alex.muller, Alfie66, Anaxial, Ancheta Wis, Andre Engels, AndrewHowse, Ann Stouter, Ap, Arakunem, ArchStanton69, ArnoldReinhold, Art Carlson, Arthena, Arvindn, AshishG, AttoRenato, Av99, Avicennasis, AxelBoldt, BD2412, Babybackbullcrap, Bakabaka, BenKovitz, Benjak, Bentogoa, Bevo, BillC, Bkell, Blackcloak,

Ramesh Singh, DTU (Formerly Delhi College of Engineering)singhramesh2@gmail.com

Article Sources and Contributors 85

Bloomingdedalus, Boby01, BonBonRaj, Boobloverandfanylicker, Booyabazooka, Borgx, Bowlhover, Breakeydown, Brews ohare, Bri09154642, Brianann MacAmhlaidh, Brighterorange,Bumm13, Bushcarrot, Callumm12, Can't sleep, clown will eat me, Canadian-Bacon, Capecodeph, Capricorn42, Careflite, CaseInPoint, Casper2k3, Cbrown1023, Cengime, Challisrussia, Chenzw,ChevyS1953, ChrisHodgesUK, Chrislk02, Christian List, Circuit dreamer, Closedmouth, Cmdrjameson, Cobaltbluetony, Cocytus, Conversion script, Cool3, Corvus cornix, Cpl Syx,Crazycomputers, Csoliverez, DJ Clayworth, DS1000, DV8 2XL, DabMachine, Dabomb87, Danno uk, Danski14, Darakom, DarkAudit, Dcoetzee, Delldot, Denisarona, Dicklyon, Discospinster,Dolphin51, DrBob, DragonHawk, Dto, Dust Filter, Editor at Large, El-mister911, Electron9, Eloc Jcg, Elpiades, Epbr123, Erik9, Everyking, Excirial, Fabartus, Falcon8765, Favonian, Festorian,FinallyEditingWithAUsername, FlowRate, GB fan, GRBerry, Gaius Cornelius, Gdo01, Gene Nygaard, Gianluigi, Giftlite, Gilliam, Glenn, Glowing Star, Gogo Dodo, Goodnightmush,GorillaWarfare, Goudzovski, Graham87, Gregbard, Gurch, Hadal, Haljolad, Hammertime, Hasankhan101, Helenabella, Hemanshu, Heron, Hhhippo, Hooperbloob, Hotfuzzydice, Huussi, IIMusLiM HyBRiD II, Imzjustplayin, Iorek The Lost, Isarra (HG), J.delanoy, J00tel, J04n, JCraw, JLaTondre, JSquish, JV Smithy, JaGa, James-Chin, Jamesday, Jaybuoy, Jazza93, Je.rrt, Jeff G.,Jeh, Jfdwolff, Jmlk17, Jnersasi, John254, Johnbarry3434, Johndarrington, Johnringson, Jonmwang, Jpkotta, Jschnur, Karenjc, Kelpin, Kenny90655, Ket, Kjkolb, Kkchang, Kotecky, Krawi,Kukini, KurtRaschke, Kuru, KyraVixen, Kzollman, LFaraone, Lectonar, LedgendGamer, Lenehey, Liftarn, Light current, LilHelpa, Lindosland, LizardJr8, Looxix, Lyla1205, MKar, Mac Davis,Mackdaddy7, MagicHalo123, Mandarax, Manyirons, Marek69, Markedguy, Marx0728, Masudr, Mauledbytigers, Mccaskey, Mebden, MegX, Meggar, Melchoir, Mets501, Michael Hardy,Michael93555, MightyWarrior, Miguel33-NJITWILL, Mike Rosoft, Mild Bill Hiccup, Mirams, Miranda, Mkdw, Mlewis000, Moe Epsilon, Moeye, Montgomery '39, Moogee, Mortense,Mostafa.Hassan, Mrball25, Mspraveen, Nabla, NawlinWiki, Nczempin, Ndyguy, NerdyScienceDude, Nicholas2124, Nickpowerz, Ninjalemming, Nivix, No-Bullet, Oli Filth, Omegatron,Onionmon, Oo7565, Optical1000091, Orangemike, OwenX, Oxymoron83, Parthian Scribe, Parthpatel1903, Patrickthe8, Pb30, Pedro, Peteo666, Peter12220, Pharaoh of the Wizards, PhilHibbs,Philip Trueman, Phillipsacp, PierreAbbat, Pikiwyn, Pinethicket, Piszkosfred, Plugwash, PsiEpsilon, Psiphiorg, R. A. Wilson, Ran4, Rbakker99, Rdrosson, RedHillian, René Vápeník, RichFarmbrough, Richiekisatwiki, Rico402, Rifleman 82, Rjstott, Rjwilmsi, Rob.derosa, Robert K S, Rogerbrent, Rplasson, Rspanton, Rwbest, Ryulong, SGBailey, ST47, Sabih omar, Sai2020, Salsb,Sandover, Sbyrnes321, Scalable Vector Raccoon, Schizobullet, Scriber, Searchme, Seb, Senor Purple, Seven of Nine, Skezz, Skäpperöd, Slon02, Slyfoxx, Smalljim, Smb1001, Snowolf,SomeFajitaSomewhere, Someone else 90, Soraora, SpaceFlight89, Spacehippy, Spiff, Spinningspark, SpyMagician, Squids and Chips, Stariki, Stovl, StuartH, Stuwert Patterson, Stwalkerster,Surajt88, Suresh jeevanandam, THEN WHO WAS PHONE?, TStein, Tagishsimon, Tantalate, Tex23, Texture, The Rambling Man, The Thing That Should Not Be, The wub, The1337gamer,Theeltone, Throwaway85, Tide rolls, Timl, Toffile, Tom Cann 1974, TomyDuby, Tonywalton, Tordek ar, TotNoob102, Treandafilia, Tsi43318, UNIXCOFFEE928, Ulflund, Ultimate Wombat,UltraBibendum, Useight, Vaelor, Velella, Versus22, Vertium, Victuallers, Vipinhari, Vortexrealm, Vssun, W0lfie, WP137, Walesgal92, Waveguide2, Waveguy, WaysToEscape, WeeGee,Werdan7, Why Not A Duck, Wikidudeman, William Avery, WillowW, Windharp, Wjbeaty, Wolfmankurd, Wstorr, Wtshymanski, X201, XJaM, Xaonon, Xaven, Xezbeth, Ybenharim, Yintan,Yk Yk Yk, Zachary.nichols, Zeeman5469, Zetawoof, Zotix, Zundark, Zxmaster, Zzyzx11, பரிதிமதி, 1000 anonymous edits

Kirchhoff's circuit laws  Source: http://en.wikipedia.org/w/index.php?oldid=570297375  Contributors: 124Nick, 2001:6B0:1:1350:41A2:6DB9:3A6F:7EFC, AirBa, Alansohn, AliRajabi,Andres, Ashokreddy2, Atulgogtay, Aulis Eskola, Awickert, Banaticus, Baruneju, Berserkerus, Betterusername, Bigbug21, BillC, Billybobjoethethird, Bowlhover, CambridgeBayWeather, Can'tsleep, clown will eat me, Chrislk02, Clee845, Coasterlover1994, CutOffTies, Daniel.Cardenas, Danmichaelo, Dicklyon, DigitalPhase, Dirkbb, Doczilla, Drilnoth, Drmies, ENeville, Editor atLarge, Enigmaman, Enok.cc, Epbr123, Fox2k11, Gene Nygaard, Giftlite, Glen, Glenn, Gnomeza, Grebaldar, Gruzd, Gurch, Haitao32668011, Heron, Hhhippo, Hirak 99, Hooperbloob,Hovhannest, Hydrogen Iodide, Ideal gas equation, Ipatrol, Isarra (HG), JabberWok, Jdpipe, Jerry, Jionpedia, Jon Stockton, Jsd, Julesd, Karada, Kmarinas86, Knuckles, Kristolane, Kwinkunks,LouriePieterse, Lsmll, Maximus Rex, Merosonox, Mets501, Michael Hardy, Mormegil, Mrball25, Msadaghd, Myasuda, Narendran95, Nczempin, Nfwu, NickW557, Nonick, Noommos, Oliviosu,Omegatron, Ortho, Ot, OwenX, Ozhiker, Paolous, Paul August, Paverider, Pflodo, Pgadfor, Pharaoh of the Wizards, Philip Trueman, Piet Delport, Plesna, PoqVaUSA, Purgatory Fubar,RGForbes, Rich Farmbrough, Ripepette, Robert K S, Rogerbrent, Romanm, SMC, Sabih omar, Salvador85, Sdschulze, Searchme, Sikory, Skittleys, Skizzik, Spinningspark, Stevenj, Suffusion ofYellow, Suppiesman123, Svjo-2, Terraflorin, Texture, Thane, That Guy, From That Show!, The Original Wildbear, The Thing That Should Not Be, The wub, Thehakimboy, Trinibones,Tsi43318, User A1, Vampikay, Victorjimi, Vonkje, Vrenator, Waqasb, Wdwd, Widr, Wood Thrush, Wstorr, Xianyang, Yahia.barie, Yhljjang, Yintan, Zeroparallax, Zzyzx11, Περίεργος, 老 陳,312 anonymous edits

Current divider  Source: http://en.wikipedia.org/w/index.php?oldid=557760215  Contributors: Aaron Nitro Danielson, Balthamos, Boris Barowski, Brews ohare, Charco.gtr, Denisarona,Developer38, DexDor, Djbaniel, Gareth Griffith-Jones, Giraffedata, Hooperbloob, J04n, Jatkins, Just zis Guy, you know?, Karol Langner, Kephir, Kingdanielj, Knycz, Kyle1278, Light current,LiuyuanChen, Mlewis000, Mo7amedsalim, Mr. Accident, Myasuda, Oleander, Pearle, Postdlf, Rogerbrent, Starwiz, Theo10011, Toffile, UnionAttack, Zahical, 61 anonymous edits

Voltage divider  Source: http://en.wikipedia.org/w/index.php?oldid=569967165  Contributors: 2001:18E8:3:10B0:2C34:E72B:5CD:4B9E, Aaronsharpe, Ancheta Wis, Andreas Rejbrand,Andres, Arja36, Arnaud Dessein, Audriusa, Bento00, Binksternet, Biscuittin, Bobblehead, Brews ohare, Can't sleep, clown will eat me, CharlotteWebb, Chggr, Choihei, Circuit dreamer,Clementina, Colonies Chris, CyrilB, D4g0thur, DMCer, Daniel Musto, Danielw46, DexDor, Dgsmith84, Doodle77, Euancreid, Excirial, Feureau, Flying fish, Giraffedata, Glenn, Gnorkel,Hooperbloob, ICE77, Jackfork, Jbarchuk, Joldy, Jschuler16, Karol Langner, Kenyon, Kilva, Kolano, Laurascudder, Lhovius, Manequito, Mark Blackburn, MelbourneStar, Mike Segal, Mikm,MinorContributor, Mlewis000, Mnmngb, MrOllie, Mxn, N3rd4i, N5iln, Nan Je Ye, Nanite, Neundorfer, NewEnglandYankee, Nish p88, Oli Filth, Omegatron, PenguiN42, Pinethicket, Poulpy,RainbowOfLight, Rjwilmsi, Rogerbrent, Rowersimon, SGBailey, SaaHc2B, Sedwick2, Shyam, Steve carlson, Susts, TedPavlic, Tgeairn, Toffile, Verkhovensky, Woodshed, Wtshymanski,XxFranzxX, ZooFari, 182 anonymous edits

Y-Δ transform  Source: http://en.wikipedia.org/w/index.php?oldid=569585398  Contributors: A. Carty, Abdull, Alansohn, Alejo2083, Ap, Apparition11, Bkell, Blotwell, Bodanger, CBM,Cbdorsett, Charles Matthews, Chetvorno, DMahalko, DVdm, Damian Yerrick, DavidCary, Davr, Dicklyon, Dionyziz, Draksis314, Farhan678, Fresheneesz, Gamebm, Giftlite, Greudin,HMSSolent, Heron, Ideal gas equation, J36miles, JHunterJ, Jamelan, Karada, L'Aquatique, Linas, Lmdemasi, Michael Hardy, MonoAV, MonteChristof, Mwarren us, NameIsRon, OlegAlexandrov, PV=nRT, Pebkac, Phil Boswell, R'n'B, R.e.b., Reddi, René Vápeník, Saippuakauppias, Shyam, Slandete, SlothMcCarty, Tbhotch, Tejashs, The Anome, Thenoflyzone, Thryduulf,TutterMouse, Twri, Unraveled, Wdl1961n, Wtshymanski, Xyzzy n, Zzyzx11, 124 anonymous edits

Nodal analysis  Source: http://en.wikipedia.org/w/index.php?oldid=567529382  Contributors: 8r455, Alexmailbox100, Andrei Stroe, Arthur Rubin, Bo102010, Christian75, David Nemati, DebeoMorium, Dicklyon, Envy0, FatimahMaj, Heron, Highsand, HyperFlexed, Itaharesay, JJ Harrison, Jmorgan, Komal singh b, Lyctc, Miterdale, Mohsinjibran, Mrbobjrsrv, Omegatron, PeteX, PetteriAimonen, Pinethicket, Rogerbrent, Ronit.parikh, Slightsmile, Spinningspark, Tentinator, The Thing That Should Not Be, Umansky, Wilson SK Jin, Y2H, Yahia.barie, YoungGeezer, Пика Пика,老 陳, 57 anonymous edits

Mesh analysis  Source: http://en.wikipedia.org/w/index.php?oldid=565843134  Contributors: 92kenneth, Akilaa, Alan Liefting, Alansohn, Alexmailbox100, Alfred Centauri, Amalas,Andrewman327, Btyner, Cedars, Chatlanin, Dfdeboer, Dicklyon, Download, Ferengi, GorillaWarfare, Heron, Itaharesay, Izno, J04n, KD5TVI, Lyctc, Materialscientist, Michael Hardy, Mion,Mrball25, Nosnibor80, Omegatron, Paverider, Potatoswatter, QUILzhunter931, Rocketrod1960, Ronit.parikh, S3000, Somnath gaikwad (maharashtra), Spinningspark, TBrandley, Toffile,Tsi43318, Ubardak, Vik-Thor, Widr, Windharp, Wtshymanski, Zanypoetboy45, 53 anonymous edits

Superposition theorem  Source: http://en.wikipedia.org/w/index.php?oldid=557148933  Contributors: Abdull, Alfred Centauri, Amalas, Chzz, Deepesh.sh92, Donniewherman, Faigl.ladislav,Fresheneesz, GregorB, Gwen-chan, Jaymie94, Masolis, Matt j, Mild Bill Hiccup, Muchness, No such user, Petr Kopač, Poiselius, R'n'B, R-skin, RainbowOfLight, Rich Farmbrough, Rogerbrent,Ronit.parikh, SailorfromNH, Sidman62, Sonett72, Spinningspark, Tabletop, Venu Muriki, Wtshymanski, Y2H, Zzyzx11, 37 anonymous edits

Thévenin's theorem  Source: http://en.wikipedia.org/w/index.php?oldid=567681884  Contributors: Abdull, AdjustShift, Alejo2083, Alexius08, Alfred Centauri, Americanhero, AmigoDoPaulo,Ampre, Analogguru, AndrewHowse, André Neves, Animum, Arabani, Atlant, BeteNoir, Betsumei, Boneless555, Capmaster, Cbdorsett, Cblambert, Charles Matthews, Choihei, Cikicdragan,Constant314, Cwkmail, Cwtiyar, Dashingjose, Dicklyon, Finlux, Flyguy649, Fresheneesz, Gamebm, Giftlite, Greudin, Heron, Hhhippo, Hooperbloob, I love Laura very much, Int21h, Jeff G., JoeDecker, JoeSmack, KALYAN T.V., Larryisgood, Lihwc, Lingwitt, Lionelbrits, Lyla1205, Michael Hardy, Mikhail Ryazanov, Mild Bill Hiccup, MisterSheik, Mlewis000, NTox, Noishe,Nukerebel, Ojan, Omegatron, Plugwash, Positronium, Rdrosson, Rich Farmbrough, Rogerbrent, Saud678, Senanayake, Shadowlynk, Sim, Simon12, Smack, Spinningspark, Stimpak, Sudeepshenoy2007, Super-Magician, Surfer43, TParis, TedPavlic, Tempodivalse, Tide rolls, TomyDuby, Topbanana, Tsi43318, Unyoyega, VonWoland, WeeGee, WikiUserPedia, Wstorr,Wtshymanski, YCM Interista, Yatloong, 114 anonymous edits

Norton's theorem  Source: http://en.wikipedia.org/w/index.php?oldid=566281491  Contributors: Abdull, Alexius08, Allankk, Arabani, Atlant, Banaticus, BeteNoir, Bulwersator, CDN99,Cbdorsett, Cblambert, Charles Matthews, Clarince63, Closedmouth, CommonsDelinker, Cst17, Dicklyon, DragonHawk, Drumkid, Ejuyoung, Eras-mus, Fresheneesz, Gareth Jones, Giftlite,Gilderien, Gregbard, Greudin, Haymaker, Heatherawalls, Heron, Hooperbloob, Ixfd64, JJ Harrison, Jamars, JoeSmack, Jojhutton, Kingpin13, LachlanA, Lanny's, Lingwitt, Looxix, Melonkelon,Michael Hardy, Mlewis000, MonteChristof, Nozog, Omegatron, Paranoid Android1208, PhilKnight, Plugwash, Positronium, Qxz, R'n'B, Ranjitbbsr2, RayKiddy, Rdrosson, Richard D. LeCour,Rino Su, Rogerbrent, TedPavlic, Tjmayerinsf, Tomas e, TomyDuby, Tsi43318, VonWoland, Wtshymanski, ZZ9pluralZalpha, Zoicon5, Σ, 66 anonymous edits

Maximum power transfer theorem  Source: http://en.wikipedia.org/w/index.php?oldid=555179742  Contributors: AxelBoldt, BD2412, BenKovitz, BeteNoir, Bobo192, Captain Quirk,Catslash, Charles Matthews, Chetvorno, Chilidog69, Colonel Warden, Drbreznjev, Eeekster, Everyking, FMasic, Geometry guy, Giftlite, Heron, Hufwiki, Ignacioerrico, JJ Harrison, Jerzy,KD5TVI, Laocmo, Light current, Loren.wilton, Michael Hardy, Mitcherator, Nocal, Oleg Alexandrov, Omegatron, Overjive, Positronium, R'n'B, Rhombus, Rlove, Rogerbrent, Rtdrury, Saud678,Sciurinæ, Sholto Maud, Steve Sheehy, Steven Weston, Telaclavo, The Anome, Timo Honkasalo, Toffile, Wdwd, Wolfkeeper, Wtshymanski, Yurik.kas, Zzyzx11, 58 anonymous edits

Ramesh Singh, DTU (Formerly Delhi College of Engineering)singhramesh2@gmail.com

Image Sources, Licenses and Contributors 86

Image Sources, Licenses and ContributorsFile:VFPt Solenoid correct2.svg  Source: http://en.wikipedia.org/w/index.php?title=File:VFPt_Solenoid_correct2.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors:Geek3File:ResistanceHydraulicAnalogy.svg  Source: http://en.wikipedia.org/w/index.php?title=File:ResistanceHydraulicAnalogy.svg  License: Creative Commons Zero  Contributors:User:Sbyrnes321File:register.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Register.jpg  License: GNU Free Documentation License  Contributors: Haragayato, cropd by hidro.File:FourIVcurves.svg  Source: http://en.wikipedia.org/w/index.php?title=File:FourIVcurves.svg  License: Creative Commons Zero  Contributors: User:Sbyrnes321File:Resistivity geometry.png  Source: http://en.wikipedia.org/w/index.php?title=File:Resistivity_geometry.png  License: GNU Free Documentation License  Contributors: User:Omegatronfile:DifferentialChordalResistance.svg  Source: http://en.wikipedia.org/w/index.php?title=File:DifferentialChordalResistance.svg  License: Creative Commons Zero  Contributors:User:Sbyrnes321file:Negative_differential_resistance.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Negative_differential_resistance.svg  License: Creative Commons Attribution-ShareAlike 3.0Unported  Contributors: Omegatron and Alessio DamatoFile:VI phase.png  Source: http://en.wikipedia.org/w/index.php?title=File:VI_phase.png  License: Public Domain  Contributors: Jeffrey PhilippsonFile:Cartridge-heater-hot.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Cartridge-heater-hot.jpg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: MaxellatorImage:Electronic component inductors.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Electronic_component_inductors.jpg  License: Creative Commons Attribution-ShareAlike3.0 Unported  Contributors: meFile:Inductor.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Inductor.svg  License: Public Domain  Contributors: JjbeardFile:Resistor shaped Inductors.png  Source: http://en.wikipedia.org/w/index.php?title=File:Resistor_shaped_Inductors.png  License: Creative Commons Attribution 3.0  Contributors: VahidalphaImage:Drosselspule im Umspannwerk Bisamberg.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Drosselspule_im_Umspannwerk_Bisamberg.jpg  License: Creative CommonsAttribution-Sharealike 3.0  Contributors: Mario Sedlak (talk)Image:Ferrite bead no shell.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Ferrite_bead_no_shell.jpg  License: GNU Free Documentation License  Contributors: OmegatronImage:Choke electronic component Epcos 2x47mH 600mA common mode.jpg  Source:http://en.wikipedia.org/w/index.php?title=File:Choke_electronic_component_Epcos_2x47mH_600mA_common_mode.jpg  License: Public Domain  Contributors: Mike1024Image:Transmitter tank coil.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Transmitter_tank_coil.jpg  License: Public Domain  Contributors: Chetvorno, 1 anonymous editsImage:Hf spoler og transformatorer.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Hf_spoler_og_transformatorer.jpg  License: GNU Free Documentation License  Contributors:Original uploader was Glenn at da.wikipediaImage:Inductor radio (crystal).JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Inductor_radio_(crystal).JPG  License: Creative Commons Attribution-Sharealike 3.0,2.5,2.0,1.0 Contributors: F1jmmImage:Kreuzwickelspule.png  Source: http://en.wikipedia.org/w/index.php?title=File:Kreuzwickelspule.png  License: Public Domain  Contributors: de:User:PeterFrankfurtImage:Aplikimi i feriteve.png  Source: http://en.wikipedia.org/w/index.php?title=File:Aplikimi_i_feriteve.png  License: Creative Commons Attribution-Sharealike 3.0  Contributors:User:FIEK-KompjuterikeImage:Vorschaltdrossel Kvg.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Vorschaltdrossel_Kvg.jpg  License: Public domain  Contributors: Ulfbastel at de.wikipediaFile:3Com OfficeConnect ADSL Wireless 11g Firewall Router 2012-10-28-0869.jpg  Source:http://en.wikipedia.org/w/index.php?title=File:3Com_OfficeConnect_ADSL_Wireless_11g_Firewall_Router_2012-10-28-0869.jpg  License: Creative Commons Zero  Contributors: User:Slickfile:Ferrite slug tuned inductor with pot core.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Ferrite_slug_tuned_inductor_with_pot_core.JPG  License: Creative Commons Zero Contributors: User:Chetvornofile:Variometer.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Variometer.jpg  License: Public Domain  Contributors: ChetvornoImage:inductors in parallel.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Inductors_in_parallel.svg  License: GNU Free Documentation License  Contributors: OmegatronImage:inductors in series.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Inductors_in_series.svg  License: GNU Free Documentation License  Contributors: OmegatronFile:Photo-SMDcapacitors.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Photo-SMDcapacitors.jpg  License: Public Domain  Contributors: Shaddack, WhymFile:Capacitor Symbol alternative.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Capacitor_Symbol_alternative.svg  License: Public Domain  Contributors: M. Mehdi SalemNaraghiFile:Condensador electrolitico 150 microF 400V.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Condensador_electrolitico_150_microF_400V.jpg  License: unknown Contributors: WilltronFile:Electrolytic capacitor.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Electrolytic_capacitor.jpg  License: Creative Commons Attribution 3.0  Contributors: Vahid alphaFile:Tantalum capacitors.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Tantalum_capacitors.jpg  License: Creative Commons Attribution 3.0  Contributors: MataresephotosFile:Leidse flessen Museum Boerhave december 2003 2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Leidse_flessen_Museum_Boerhave_december_2003_2.jpg  License: GNUFree Documentation License  Contributors: Original uploader was Alvinrune at en.wikipediaFile:Capacitor schematic with dielectric.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Capacitor_schematic_with_dielectric.svg  License: Creative CommonsAttribution-Sharealike 3.0  Contributors: Papa NovemberFile:Plattenkondensator hg.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Plattenkondensator_hg.jpg  License: Creative Commons Attribution 3.0  Contributors: Hannes Grobe(talk)File:CapacitorHydraulicAnalogyAnimation.gif  Source: http://en.wikipedia.org/w/index.php?title=File:CapacitorHydraulicAnalogyAnimation.gif  License: Creative Commons Zero Contributors: User:Sbyrnes321File:RC switch.svg  Source: http://en.wikipedia.org/w/index.php?title=File:RC_switch.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: PureCoreFile:Parallel plate capacitor.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Parallel_plate_capacitor.svg  License: Public Domain  Contributors: inductiveloadFile:capacitors in parallel.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Capacitors_in_parallel.svg  License: GNU Free Documentation License  Contributors: OmegatronFile:capacitors in series.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Capacitors_in_series.svg  License: GNU Free Documentation License  Contributors: OmegatronFile:Capacitor equivalent circuits.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Capacitor_equivalent_circuits.svg  License: Creative Commons Zero  Contributors:User:GorillaWarfareFile:Condensators.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Condensators.JPG  License: GNU Free Documentation License  Contributors: de:Benutzer:AkaFile:Axial electrolytic capacitors.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Axial_electrolytic_capacitors.jpg  License: Creative Commons Attribution 3.0  Contributors:MataresephotosFile:Mylar-film oil-filled low-inductance capacitor 6.5 MFD @ 5000 VDC.jpg  Source:http://en.wikipedia.org/w/index.php?title=File:Mylar-film_oil-filled_low-inductance_capacitor_6.5_MFD_@_5000_VDC.jpg  License: Creative Commons Zero  Contributors: User:ZaerethFile:Capacitor.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Capacitor.jpg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Daniel Christensen aten.wikipediaFile:Condensor bank 150kV - 75MVAR.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Condensor_bank_150kV_-_75MVAR.jpg  License: Public Domain  Contributors: PhilippeMertensFile:Polyester film capacitor.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Polyester_film_capacitor.jpg  License: GNU Free Documentation License  Contributors:MataresephotosFile:Defekte Kondensatoren.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Defekte_Kondensatoren.jpg  License: Creative Commons Attribution-Sharealike 2.0  Contributors: UserSmial on de.wikipedia

Ramesh Singh, DTU (Formerly Delhi College of Engineering)singhramesh2@gmail.com

Image Sources, Licenses and Contributors 87

File:High-energy capacitor from a defibrillator 42 MFD @ 5000 VDC.jpg  Source:http://en.wikipedia.org/w/index.php?title=File:High-energy_capacitor_from_a_defibrillator_42_MFD_@_5000_VDC.jpg  License: Creative Commons Zero  Contributors: User:ZaerethFile:Exploded Electrolytic Capacitor.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Exploded_Electrolytic_Capacitor.jpg  License: Creative Commons Attribution-Sharealike 3.0 Contributors: User:Frizb99File:OhmsLaw.svg  Source: http://en.wikipedia.org/w/index.php?title=File:OhmsLaw.svg  License: Public Domain  Contributors: Waveguide2 (talk) (Transferred by Nk/Originally uploaded byWaveguide2)File:Electrona in crystallo fluentia.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Electrona_in_crystallo_fluentia.svg  License: GNU Free Documentation License  Contributors:RafaelgarciaFile:Ohm's law triangle.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Ohm's_law_triangle.svg  License: Public Domain 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Omegatron, SteveZodiac, WikipediaMasterImage:Thevenin to Norton2.PNG  Source: http://en.wikipedia.org/w/index.php?title=File:Thevenin_to_Norton2.PNG  License: GNU Free Documentation License  Contributors: Originaluploader was Yatloong at en.wikipediaFile:Norton equivelant.png  Source: http://en.wikipedia.org/w/index.php?title=File:Norton_equivelant.png  License: Creative Commons Attribution-ShareAlike 3.0 Unported  Contributors:DrumkidImage:Norton step 2.png  Source: http://en.wikipedia.org/w/index.php?title=File:Norton_step_2.png  License: GNU Free Documentation License  Contributors: EugeneZelenko, Ilmari Karonen,Inductiveload, Ma-Lik, Omegatron, WikipediaMasterImage:Norton step 4.png  Source: http://en.wikipedia.org/w/index.php?title=File:Norton_step_4.png  License: GNU Free Documentation License  Contributors: Althiphika, EugeneZelenko,Ilmari Karonen, Inductiveload, Ma-Lik, Omegatron, WikipediaMaster, 1 anonymous editsFile:Source and load circuit.png  Source: http://en.wikipedia.org/w/index.php?title=File:Source_and_load_circuit.png  License: GNU Free Documentation License  Contributors:EugeneZelenko, Ilmari Karonen, Ma-Lik, OmegatronFile:Maximum Power Transfer Graph.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Maximum_Power_Transfer_Graph.svg  License: Public Domain  Contributors: InductiveloadFile:maxpowertheorem.png  Source: http://en.wikipedia.org/w/index.php?title=File:Maxpowertheorem.png  License: Public Domain  Contributors: Maksim, WikipediaMasterFile:Source and load boxes.png  Source: http://en.wikipedia.org/w/index.php?title=File:Source_and_load_boxes.png  License: GNU Free Documentation License  Contributors: EugeneZelenko,Glenn, Ilmari Karonen, Omegatron, 1 anonymous edits

Ramesh Singh, DTU (Formerly Delhi College of Engineering)singhramesh2@gmail.com

License 88

LicenseCreative Commons Attribution-Share Alike 3.0 Unported//creativecommons.org/licenses/by-sa/3.0/

Ramesh Singh, DTU (Formerly Delhi College of Engineering)singhramesh2@gmail.com