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Basics of Power circuits Contents

Articles Electrical resistance and conductance 1 10 24 Ohm's law 42 Kirchhoff's circuit laws 50 Current divider 54 divider 57 Y-Δ transform 61 Nodal analysis 66 69 72 Thévenin's theorem 73 Norton's theorem 77 Maximum power transfer theorem 80 References Article Sources and Contributors 84 Image Sources, Licenses and Contributors 86 Article Licenses License 88

Ramesh Singh, DTU (Formerly Delhi College of ) [email protected] Electrical resistance and conductance 1 Electrical resistance and conductance

Electromagnetism

• •

The electrical resistance of an is the opposition to the passage of an through that conductor; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical notion of friction. The SI unit of electrical resistance is 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 inversely proportional to its cross-sectional area. All materials show some resistance, except for superconductors, which have a 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 the conductance (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 G are constant (although they can depend on other factors like or strain). This proportionality is called Ohm's law, and materials that satisfy it are called "Ohmic" materials. In other cases, such as a or battery, V and I are not directly proportional, or in other words the I–V curve is not a straight line through the origin, and Ohm's law does not hold. In this case, resistance and conductance are less useful concepts, and more difficult to define. The ratio V/I is sometimes still useful, and is referred to as a "chordal resistance" 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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Introduction

In the , current flowing through a (or ) is like water flowing through a pipe, and the across the wire is like the drop which pushes water through the pipe. Conductance is proportional to how much flow occurs for a given pressure, and resistance is proportional to how much pressure is required to achieve a given flow. (Conductance and resistance are reciprocals.)

The voltage drop (i.e., difference in voltage between one side of the The hydraulic analogy compares electric current resistor and the other), not the voltage itself, provides the driving flowing through circuits to water flowing through pushing current through a resistor. In , it is similar: The pipes. When a pipe (left) is filled with hair pressure difference between two sides of a pipe, not the pressure itself, (right), it takes a larger pressure to achieve the determines the flow through it. For example, there may be a large same flow of water. Pushing electric current through a large resistance is like pushing water water pressure above the pipe, which tries to push water down through through a pipe clogged with hair: It requires a the pipe. But there may be an equally large water pressure below the larger push () to drive the pipe, which tries to push water back up through the pipe. If these same flow (electric current). are equal, no water will flow. (In the image at right, the water pressure 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 same shape and size. In a similar way, electrons can flow freely and easily through a copper wire, but cannot as easily flow through a steel wire of the same shape and size, and they essentially cannot flow at all through an like rubber, regardless of its shape. The difference between, copper, steel, and rubber is related to their microscopic structure and electron configuration, and is quantified by a property called resistivity.

Conductors and

Those substances through which electricity can flow are called conductors. A piece of conducting material

of a particular resistance meant for use in a circuit is A 65 Ω resistor, as identified by its electronic color code called a resistor. Conductors are made of (blue–green–black-gold). An could be used to verify this high-conductivity materials such as metals, in value. particular copper and aluminium. Resistors, on the other hand, are made of a wide variety of materials depending on factors such as the desired resistance, amount of energy that it needs to dissipate, precision, and costs.

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Ohm's law

Ohm's law is an empirical law relating the voltage V across an element to the current I through it:

(V is directly proportional to I). This The current-voltage characteristics of four devices: Two resistors, a diode, and a battery. law is not always true: For example, it The horizontal axis is voltage drop, the vertical axis is current. Ohm's law is satisfied is false for , batteries, etc. when the graph is a straight line through the origin. Therefore, the two resistors are "ohmic", but the diode and battery are not. However, it is true to a very good approximation for and resistors (assuming that other conditions, including temperature, are held fixed). Materials or objects where Ohm's law is true are called "ohmic", whereas objects which do not obey Ohm's law are '"non-ohmic".

Relation to resistivity and conductivity

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

A piece of resistive material with electrical contacts on both ends.

where is the length of the conductor, measured in metres [m], A is the cross-section area of the conductor measured in square metres [m²], σ (sigma) is the electrical conductivity measured in siemens per meter (S·m−1), and ρ (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 the wire is made of, not the geometry of the wire. Resistivity and conductivity are reciprocals: . Resistivity is a measure of the material's ability to oppose electric current. This formula is not exact: It assumes the is totally uniform in the conductor, which is not always true in practical situations. However, this formula still provides a good approximation for long thin conductors such as wires. Another situation for which this formula is not exact is with (AC), because the skin effect inhibits current flow near the center of the conductor. Then, the geometrical cross-section is different from the effective cross-section in which current is actually flowing, so the resistance is higher than expected. Similarly, if two conductors are near each other carrying AC current, their resistances will increase due to the proximity effect. At commercial power frequency, these effects are significant for large conductors carrying large currents, such as busbars in an electrical substation,[1] or large power cables carrying more than a few hundred amperes.

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What determines resistivity? The resistivity of different materials varies by an enormous amount: For example, the conductivity of teflon is about 1030 times lower than the conductivity of copper. Why is there such a difference? Loosely speaking, a metal has large 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 required to pull it away. Semiconductors lie between these two extremes. More details can be found in the article: Electrical resistivity and conductivity. For the case of solutions, see the article: Conductivity (electrolytic). Resistivity varies with temperature. In semiconductors, resistivity also changes when light is shining on it. These are discussed below.

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

Typical resistances

Component Resistance (Ω)

[2] 1 meter of copper wire 0.02 with 1mm diameter

[3] 1 km overhead power 0.03 line (typical)

[4] AA battery (typical 0.1 internal resistance)

[5] Incandescent light bulb 200-1000 filament (typical)

[6] Human body 1000 to 100,000

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 through the origin. The differential resistance at A is the inverse slope of tangent line C.

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The IV curve of a component with negative differential resistance, an unusual phenomenon where the IV curve is non-monotonic. Many electrical elements, such as diodes and batteries do not satisfy Ohm's law. These are called non-ohmic or nonlinear, 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 determines the 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 have no source of energy, cannot have negative static resistance. However active devices such as or op-amps can synthesize negative static resistance with feedback, and it is used in some circuits such as . • Differential resistance (also called dynamic, incremental or small signal resistance) - Differential resistance is the 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 some regions; so in these regions the device has negative differential resistance. Devices with negative differential resistance can amplify a signal applied to them, and are used to make and oscillators. These include tunnel diodes, Gunn diodes, IMPATT diodes, magnetron tubes, and unijunction transistors.

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AC circuits

Impedance and admittance

When an alternating current flows through a circuit, the relation between current and voltage across a circuit element is characterized not only by the ratio of their magnitudes, but also the difference in their phases. For example, in an ideal resistor, the moment when the voltage reaches its maximum, the current also reaches its maximum (current and voltage are oscillating in phase). But for a capacitor or inductor, the maximum current flow occurs as the voltage passes through zero and vice-versa (current and voltage are oscillating 90° out of phase, see image at right). Complex numbers are used to keep The voltage (red) and current (blue) versus time (horizontal axis) for a capacitor track of both the phase and magnitude of (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 capacitor current and voltage: and 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, the where: complex phase of the impedance of the inductor is +90°. • t is time, • V(t) and I(t) are, respectively, voltage and current as a function of time, • V , I , Z, and Y are complex numbers, 0 0 • 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 imaginary parts:

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 and , X and B are nonzero.

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 may depend on frequency (see Drude model, deep-level traps, resonant frequency, Kramers–Kronig relations, etc.)

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Energy dissipation and

Resistors (and other elements with resistance) oppose the flow of electric current; therefore, is required to push current through the resistance. This electrical energy is dissipated, heating the resistor in the process. This is called Joule heating (after ), also called ohmic heating or resistive heating. The dissipation of electrical energy is often undesired, particularly in the case of transmission losses in power lines. High voltage Running current through a resistance creates , transmission helps reduce the losses by reducing the current for a given in a phenomenon called Joule heating. In this power. picture, a cartridge heater, warmed by Joule heating, is glowing red hot. On the other hand, Joule heating is sometimes useful, for example in electric 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 that it 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 dependence Near room temperature, the resistivity of metals typically increases as temperature is increased, while the resistivity of semiconductors typically decreases as temperature is increased. The resistivity of insulators and may increase or decrease depending on the system. For the detailed behavior and explanation, see Electrical resistivity and conductivity. As a consequence, the resistance of wires, resistors, and other components often change with temperature. This effect may be undesired, causing an to malfunction at extreme . In some cases, however, the effect is put to good use. When temperature-dependent resistance of a component is used purposefully, the component is called a resistance thermometer or thermistor. (A resistance thermometer is made of metal, usually platinum, 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 in conjunction with Joule heating (also called self-heating): If a large current is running through the resistor, the resistor's temperature rises and therefore its resistance changes. Therefore, these components can be used in a circuit-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 see Thermistor#Self-heating effects. If the temperature T does not vary too much, a linear approximation is typically used:

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

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] Electrical resistance and conductance 8

, and the relationship only holds in a range of temperatures around the reference.[7] The temperature coefficient is typically +3×10−3 K−1 to +6×10−3 K−1 for metals near room temperature. It is usually negative for semiconductors and insulators, with highly variable magnitude.[8]

Strain dependence Just 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 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 these effects contribute to increasing the resistance of the strained section of conductor. Under compression (strain in the opposite direction), the resistance of the strained section of conductor decreases. See the discussion on strain gauges for details about devices constructed to take advantage of this effect.

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

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

References [1] Fink and Beaty, Standard Handbook for Electrical Engineers 11th Edition, page 17-19 -8 [2] The resistivity of copper is about 1.7×10 Ωm. See (http:/ / hypertextbook. com/ facts/ 2004/ BridgetRitter. shtml).

[3] 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; these values 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 electric shock article for more details. Also see: [7] Ward, MR, 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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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 of electrical resistance.

• Vehicular Laboratory: Resistance Calculator (http:/ / www. cvel. clemson. edu/ emc/ calculators/

Resistance_Calculator/ index. html''Clemson)

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] Inductor 10 Inductor

Inductor

A selection of low-value inductors

Type Passive

Working principle Electromagnetic induction

First production (1831)

Electronic symbol

An inductor, also called a coil or reactor, is a passive two-terminal electrical component which resists changes in electric current passing through it. It consists of a conductor such as a wire, usually wound into a coil. When a current flows through it, energy is stored temporarily in a in the coil. When the current flowing through an inductor changes, the time-varying magnetic field induces a voltage in Axial lead inductors (100uH) the conductor, according to Faraday’s law of electromagnetic induction, which opposes the change in current that created it.

An inductor is characterized by its , the ratio of the voltage to the rate of change of current, which has units of henries (H). Many inductors have a magnetic core made of iron or ferrite inside the coil, which serves to increase the magnetic field and thus the inductance. Along with capacitors and resistors, inductors are one of the three passive 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 while allowing DC to pass; inductors designed for this purpose are called chokes. They are also used in electronic filters to separate signals of different frequencies, and in combination with capacitors to make tuned circuits, used to tune radio and TV receivers.

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Overview Inductance (L) results from the magnetic field around a current-carrying conductor; the electric current through the conductor creates a . Inductance is a geometrical property of a circuit which is determined by how much 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 has some inductance. In inductors the conductor is shaped to increase the magnetic field. Winding the wire into a coil increases 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 ferromagnetic material like iron, the magnetizing field from the coil will induce in the material, increasing the magnetic flux. The high permeability of a ferromagnetic core can increase the inductance of a coil by a factor of several thousand over what it would be without it.

Constitutive equation Any change in the current through an inductor creates a changing flux, inducing a voltage across the inductor. By Faraday'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 when the current through the inductor changes at the rate of 1 ampere per second. This is usually taken to be the constitutive relation (defining equation) of the inductor.

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

Ideal and real inductors In circuit theory, inductors are idealized as obeying the mathematical relation (2) above precisely. An "ideal inductor" has inductance, but no resistance or , and does not dissipate or radiate energy. However real inductors have side effects which cause their behavior to depart from this simple model. They have resistance (due to the resistance of the wire and energy losses in core material), and (due to the between the turns of wire which are at slightly different potentials). At high frequencies the capacitance begins to affect 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 higher frequencies, resistive losses in the windings increase due to skin effect and proximity effect.

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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 behavior due to nonlinearity caused by magnetic of the core. An inductor may radiate electromagnetic energy into surrounding space and circuits, and may absorb electromagnetic emissions from other circuits, causing electromagnetic interference (EMI). Real-world inductor applications may consider these parasitic parameters as important as the inductance.

Applications

Inductors are used extensively in analog circuits and signal processing. Inductors in conjunction with capacitors form tuned circuits which can emphasize or filter out specific signal frequencies. Applications range from the use of large inductors in power supplies, which in conjunction with filter capacitors remove residual hums known as the mains hum or other fluctuations from the output, to the small inductance of the ferrite bead or torus installed around a cable to prevent radio frequency interference from being transmitted down the wire. Smaller inductor/capacitor combinations provide tuned circuits used in radio reception and broadcasting, for instance. Inductors are Large 50 MVAR three-phase iron-core loading inductor at German utility substation used as the energy storage device in many switched-mode power supplies to produce DC current. The inductor supplies energy to the circuit to keep current flowing during the "off" switching periods.

Two (or more) inductors in proximity that have coupled magnetic flux (mutual inductance) form a , which is a fundamental component of every electric utility power grid. The efficiency of a transformer may decrease as the frequency increases due to eddy currents in the core material and skin effect on the windings. The size of the core can be decreased at higher frequencies. For this reason, aircraft use 400 hertz alternating current rather than the usual 50 or 60 hertz, allowing a great saving in weight from the use of smaller A ferrite "bead" , consisting of an [1] encircling ferrite cylinder, removes electronic . noise from a power cord. Inductors are also employed in electrical transmission systems, where they are used to limit switching currents and fault currents. In this field, they are more commonly referred to as reactors. 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 prevents them 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 the which can synthesize inductance using capacitors.

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Inductor construction

An inductor usually consists of a coil of conducting material, typically insulated copper wire, wrapped around a core either of or of a ferromagnetic (or ferrimagnetic) material; the latter is called an "iron core" inductor. The high permeability of the ferromagnetic core increases the magnetic field and confines it closely to the inductor, thereby increasing the inductance. Low frequency inductors are constructed like transformers, with cores of electrical steel laminated to prevent eddy currents. 'Soft' ferrites are widely used for cores above audio frequencies, since they do not cause the large energy losses at high frequencies that ordinary iron alloys do. Inductors come in many shapes. Most are constructed as enamel coated wire ( wire) wrapped around a ferrite bobbin with wire exposed on the outside, while some enclose the wire A ferrite core inductor with two completely in ferrite and are referred to as "shielded". Some inductors have an 47mH windings. adjustable core, which enables changing of the inductance. Inductors used to block 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. Some such planar inductors use a planar core. Small value inductors can also be built on integrated circuits using the same processes that are used to make transistors. Aluminium interconnect is typically used, laid out in a spiral coil pattern. However, the small dimensions limit the inductance, and it is far more common to use a circuit called a "gyrator" that uses a capacitor and active components to behave similarly to an inductor.

Types of inductor

Air core inductor

The term air core coil describes an inductor that does not use a magnetic core made 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 inside the windings. Air core coils have lower inductance than ferromagnetic core coils, but are often used at high frequencies because they are free from energy losses called core losses that occur in ferromagnetic cores, which increase with frequency. A side effect that can occur in air core coils in which the winding is not rigidly supported on a form is 'microphony': mechanical vibration of the windings can cause variations in the inductance.

Double helix oscillation transformer for a spark gap . Transformer consists of two helical inductors. The inner inductor is moved to adjust the mutual inductance between the two coils.

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Radio frequency inductor

At high frequencies, particularly radio frequencies (RF), inductors have higher resistance and other losses. In addition to causing power loss, in resonant circuits this can reduce the of the circuit, broadening the bandwidth. In RF inductors, which are mostly air core types, specialized construction techniques are used to minimize these Collection of RF inductors, showing techniques to reduce losses. The three losses. The losses are due to these effects: top right and the loopstick antenna, bottom, have basket windings.

• Skin effect: The resistance of a wire to high frequency current is higher than its resistance to direct current because of skin effect. Radio frequency alternating current 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 the surface. This effect increases the resistance of the wire in the coil, which may already have a relatively high resistance due to its length and small diameter. • Proximity effect: Another similar effect that also increases the resistance of the wire at high frequencies is proximity effect, which occurs in parallel wires that lie close to each other. The individual magnetic field of adjacent turns induces eddy currents in the wire of the coil, which causes the current in the conductor to be concentrated in a thin strip on the side near the adjacent wire. Like skin effect, this reduces the effective cross-sectional area of the wire conducting current, increasing its resistance. • Parasitic capacitance: The capacitance between individual wire turns of the coil, called parasitic capacitance, does not cause energy losses but can change the behavior of the coil. Each turn of the coil is at a slightly different potential, so the electric field between neighboring turns stores charge on the wire, so the coil acts as if it has a capacitor in parallel with it. At a high enough frequency this capacitance can resonate with the inductance of the coil forming a tuned circuit, causing the coil to become self-resonant. Spiderweb-wound coil for crystal radio

To reduce parasitic capacitance and proximity effect, RF coils are constructed to avoid having many turns lying close together, parallel to one another. The windings of RF coils are often limited to a single layer, and the turns are spaced apart. To reduce resistance due to skin effect, in high-power inductors such as those used in the windings are sometimes 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 which successive turns are not parallel but crisscrossed at an angle; these are often called honeycomb or basket-weave coils. Adjustable ferrite slug RF coil using basket winding and • Spiderweb coils: Another construction technique with similar advantages is flat spiral coils. These are often wound on a flat insulating support with radial spokes or slots, with the wire weaving in and out through the slots; these are called spiderweb coils. The form has an odd number of slots, 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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current. Unlike ordinary stranded wire, the strands are insulated from each other, to prevent skin effect from forcing the current to the surface, and are braided together. The braid pattern ensures that each wire strand spends the same amount of its length on the outside of the braid, so skin effect distributes the current equally between the strands, resulting in a larger cross-sectional conduction area than an equivalent single wire.

Ferromagnetic core inductor

Ferromagnetic-core or iron-core inductors use a magnetic core made of a ferromagnetic or ferrimagnetic material such as iron or ferrite to increase the inductance. A magnetic core can increase the inductance of a coil by a factor of several thousand, by increasing the magnetic field due to its higher magnetic permeability. However the magnetic properties of the core material cause several side effects which alter the behavior 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, causes energy losses in the core material that are dissipated as heat, due to two processes: • Eddy currents: From Faraday's law of induction, the changing A variety of types of ferrite core inductors and transformers magnetic field can induce circulating loops of electric current in the conductive metal core. The energy in these currents is dissipated as heat in the resistance of the core material. The amount of energy lost increases with the area inside the loop of current. • Hysteresis: Changing or reversing the magnetic field in the core also causes losses due to the motion of the tiny magnetic domains it is composed of. The energy loss is proportional to the area of the hysteresis loop in the BH graph of the core material. Materials with low coercivity have narrow hysteresis loops and so low hysteresis losses. For both of these processes, the energy loss per cycle of alternating current is constant, so core losses increase linearly with frequency. Online core loss calculators[2] are available to calculate the energy loss. Using inputs such as input voltage, output voltage, output current, frequency, ambient temperature, and inductance these calculators can predict the losses of the inductors core and AC/DC based on the operating condition of the circuit being used.[3] • Nonlinearity: If the current through a ferromagnetic core coil is high enough that the magnetic core saturates, the inductance will not remain constant but will change with the current through the device. This is called nonlinearity and results in distortion of the signal. For example, audio signals can suffer intermodulation distortion in saturated inductors. To prevent this, in linear circuits the current through iron core inductors must be limited below the saturation level. Some laminated cores have a narrow air gap in them for this purpose, and powdered iron cores have a distributed air gap. This allows higher levels of magnetic flux and thus higher currents through the inductor before it saturates.[4]

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Laminated core inductor

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

Ferrite-core inductor Laminated iron core ballast inductor for a metal halide For higher frequencies, inductors are made with cores of ferrite. Ferrite is a ceramic lamp ferrimagnetic material that is nonconductive, so eddy currents cannot flow within it. The formulation of ferrite is xxFe O where xx represents various metals. For inductor cores soft ferrites are used, 2 4 which have low coercivity and thus low hysteresis losses. Another similar material is powdered iron cemented with a binder.

Toroidal core inductor

In an inductor wound on a straight rod-shaped core, the magnetic field lines emerging from one end of the core must pass through the air to reenter the core at the other end. This reduces the field, because much of the magnetic field path is in air rather than the higher permeability core material. A higher magnetic field and inductance can be achieved by forming the core in a closed . The magnetic field lines form closed loops within the core without leaving the core material. The shape often used is a toroidal or Toroidal inductor in the power supply doughnut-shaped ferrite core. Because of their symmetry, toroidal cores allow of a wireless router a minimum of the magnetic flux to escape outside the core (called leakage flux), so they radiate less electromagnetic interference than other shapes. Toroidal core coils are manufactured of various materials, primarily ferrite, powdered iron and laminated cores.[5]

Variable inductor

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(left) Inductor with a threaded ferrite slug (visible at top) that can be turned to move it into or out of the coil. 4.2 cm high. (right) A variometer used in radio receivers in the 1920s Probably the most common type of variable inductor today is one with a moveable ferrite magnetic core, which can be slid or screwed in or out of the coil. Moving the core farther into the coil increases the permeability, increasing the magnetic field and the inductance. Many inductors used in radio applications (usually less than 100 MHz) use adjustable cores in order to tune such inductors to their desired value, since manufacturing processes have certain tolerances (inaccuracy). Sometimes such cores for frequencies above 100 MHz are made from highly conductive non-magnetic material such as aluminum.[citation needed] They decrease the inductance because the magnetic field must bypass them. Air core inductors can use sliding contacts or multiple taps to increase or decrease the number of turns included in the circuit, to change the inductance. A type much used in the past but mostly obsolete today has a spring contact that can slide along the bare surface of the windings. The disadvantage of this type is that the contact usually short-circuits one or more turns. These turns act like a single-turn short-circuited transformer secondary winding; the large 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 number of turns connected in series, one inside the other. The inner coil is mounted on a shaft so its axis can be turned with respect to the outer coil. When the two coils' axes are collinear, with the magnetic fields pointing in the same direction, the fields add and the inductance is maximum. When the inner coil is turned so its axis is at an angle with the outer, the mutual inductance between them is smaller so the total inductance is less. When the inner coil is turned 180° so the coils are collinear with their magnetic fields opposing, the two fields cancel each other and the inductance is very small. This type has the advantage that it is continuously variable over a wide range. It is used in antenna 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 winding which controls the permeability of an easily saturable core material. See Magnetic .

Circuit theory The effect of an inductor in a circuit is to oppose changes in current through it by developing a voltage across it proportional to the rate of change of the current. An ideal inductor would offer no resistance to a constant direct current; 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-varying current 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. The amplitude of the voltage is proportional to the product of the amplitude (I ) of the current and the frequency (f) of the P current.

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

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

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

where is the inductance, and is 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:

where is the initial current in the inductor. is the complex frequency.

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Inductor networks Inductors in a parallel configuration each have the same potential difference (voltage). To find their total equivalent inductance (L ): eq

The current through inductors in series stays the same, but the voltage across each inductor can be different. The sum of 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 individual inductors.

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

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

That we have

where

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Q factor An ideal inductor would have no resistance or energy losses. However, real inductors have winding resistance from the metal wire forming the coils. Since the winding resistance appears as a resistance in series with the inductor, it is often 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 reactance to its resistance at a given frequency, and is a measure of its efficiency. The higher the Q factor of the inductor, the closer it approaches the behavior of an ideal, lossless, inductor. High Q inductors are used with capacitors to make resonant circuits in radio transmitters and receivers. The higher the Q is, the narrower the bandwidth of the resonant circuit. The Q factor of an inductor can be found through the following formula, where L is the inductance, R is the inductor's effective series resistance, ω is the radian operating frequency, and the product ωL is the inductive reactance:

Notice that Q increases linearly with frequency if L and R are constant. Although they are constant at low frequencies, the parameters vary with frequency. For example, skin effect, proximity effect, and core losses increase R 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 as N2 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 those examples assume the diameter of the wire stays the same, so both examples use proportionally more wire (copper). If the total of wire is held constant, then there would be no advantage to increasing the number of turns or the radius 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, so the core can also increase the Q. Cores however also introduce losses that increase with frequency. The core material is 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 in inductance (and Q). This phenomenon can be avoided by using a (physically larger) air core inductor. A well designed air core inductor may have a Q of several hundred.

Inductance formulae The table below lists some common simplified formulas for calculating the approximate inductance of several inductor constructions.

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Construction Formula Notes

[] Cylindrical air-core coil

• L = inductance in henries (H) • μ = permeability of free space = 4 × 10−7 H/m 0 [] • 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)

[6] Straight wire conductor Exact if ω = 0 or ω = ∞

• L = inductance • l = cylinder length • c = cylinder radius • μ = permeability of free space = 4 × 10−7 H/m 0 • μ = conductor permeability • p = resistivity • ω = phase rate

• Cu or Al (i.e., relative permeability is one) [9] [7][8] • l > 100 d • L = inductance (nH) • d2 f > 1 mm2 MHz • l = length of conductor (mm) • d = diameter of conductor (mm) • f = frequency

• Cu or Al (i.e., relative permeability is one) [9] [10][8] • l > 100 d • L = inductance (nH) • d2 f < 1 mm2 MHz • l = length of conductor (mm) • d = diameter of conductor (mm) • f = frequency

Short air-core cylindrical [11] coil • 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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Flat spiral air-core [12] coil [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)

accurate to within 5 percent for [] d > 0.2 r. • L = inductance (µH) • r = mean radius of coil (in) • N = number of turns • d = depth of coil (outer radius minus inner radius) (in)

Toroidal core (circular [13] cross-section) • 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

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

Toroidal core (rectangular [] cross-section) • L = inductance (µH) • d = inside diameter of toroid (in) 1 • d = outside diameter of toroid (in) 2 • N = number of turns • h = height of toroid (in)

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

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

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

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External links General

• 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

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

Electronic symbol

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

When there is a potential difference (voltage) across the conductors, a

static electric field develops across the dielectric, causing positive A typical electrolytic capacitor charge to collect on one plate and negative charge on the other plate. Energy is stored in the electrostatic field. An ideal capacitor is characterized by a single constant value, capacitance. This is the ratio of the on each conductor to the potential difference between them. The SI unit of capacitance is the farad, which is equal to one coulomb per volt.

The capacitance is greatest when there is a narrow separation between large areas of conductor, hence capacitor conductors are often called plates, referring to an early means of construction. In practice, the dielectric between the plates passes a small amount of leakage current and also has an electric field strength limit, the breakdown voltage. The 4 electrolytic capacitors of different and conductors and leads introduce an undesired inductance and resistance. capacitance Capacitors are widely used in electronic circuits for blocking direct current while allowing alternating current to pass. In analog filter networks, they smooth the output of power supplies. In resonant circuits they tune radios to particular frequencies. In electric power transmission systems they stabilize voltage and power flow.[1]

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

History

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

Battery of four Leyden jars in Daniel Gralath was the first to combine several jars in parallel into a "battery" to Museum Boerhaave, Leiden, the increase the charge storage capacity. Benjamin Franklin investigated the Leyden Netherlands. jar and came to the conclusion that the charge was stored on the glass, not in the water 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 of electrochemical cells.[7] Leyden jars were later made by coating the inside and outside of jars with metal foil, leaving a 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 used exclusively up until about 1900, when the invention of wireless (radio) created a demand for standard capacitors, and the steady move to higher frequencies required capacitors with lower inductance. A more compact construction began to be used of a flexible dielectric sheet such as oiled paper sandwiched between sheets of metal foil, rolled or folded into a small package. Early capacitors were also known as condensers, a term that is still occasionally used today. The term was first used for this purpose by in 1782, with reference to the device's ability to store a higher density of electric charge than a normal isolated conductor.[9]

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Theory of operation

Overview

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

An ideal capacitor is wholly characterized by a constant capacitance C, Charge separation in a parallel-plate capacitor defined as the ratio of charge ±Q on each conductor to the voltage V causes an internal electric field. A dielectric between them:[10] (orange) reduces the field and increases the capacitance.

Because the conductors (or plates) are close together, the opposite charges on the conductors attract one another due to their electric fields, allowing the capacitor to store more charge for a given voltage than if the conductors were separated, giving the capacitor a large capacitance. Sometimes charge build-up affects the capacitor mechanically, causing its capacitance to vary. In this case, capacitance is defined in terms of incremental changes:

A simple demonstration of a parallel-plate capacitor

Hydraulic analogy

In the hydraulic analogy, charge carriers flowing through a wire are analogous to water flowing through a pipe. A capacitor is like a rubber membrane sealed inside a pipe. Water molecules cannot pass through the membrane, but some water can move by stretching the membrane. In the hydraulic analogy, a capacitor is analogous The analogy clarifies a few aspects of capacitors: to a rubber membrane sealed inside a pipe. This animation illustrates a membrane being • The current alters the charge on a capacitor, just as the flow of repeatedly stretched and un-stretched by the flow water changes the position of the membrane. More specifically, the of water, which is analogous to a capacitor being effect of an electric current is to increase the charge of one plate of repeatedly charged and discharged by the flow of charge. the capacitor, and decrease the charge of the other plate by an equal

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amount. This is just like how, when water flow moves the rubber membrane, it increases the amount of water on one 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 charging current. 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 through the rubber membrane. Of course, the flow cannot continue the same direction forever; the capacitor will experience 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" (voltage drop). 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 field Work must be done by an external influence to "move" charge between the conductors in a capacitor. When the external influence is removed, the charge separation persists in the electric field and energy is stored to be released when the charge is allowed to return to its equilibrium position. The work done in establishing the electric field, and hence 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 of the capacitor. This power can be found by taking the time derivative of the stored energy:

Current–voltage relation The current I(t) through any component in an electric circuit is defined as the rate of flow of a charge Q(t) passing through it, but actual charges—electrons—cannot pass through the dielectric layer of a capacitor. Rather, an electron accumulates on the negative plate for each one that leaves the positive plate, resulting in an electron depletion and consequent positive charge on one electrode that is equal and opposite to the accumulated negative charge on the other. 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 voltage V(t ). This is the integral form of the capacitor equation:[14] 0

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. Its current-voltage relation is obtained by exchanging current and voltage in the capacitor equations and replacing C with the inductance L.

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DC circuits

A series circuit containing only a resistor, a capacitor, a and a constant DC source of voltage V is known as a charging circuit.[] If 0 the capacitor is initially uncharged while the switch is open, and the switch is closed at t , it follows from Kirchhoff's voltage law that 0

Taking the derivative and multiplying by C, gives a first-order A simple resistor-capacitor circuit demonstrates differential equation: charging of a capacitor.

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

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

AC circuits Impedance, the vector sum of reactance and resistance, describes the phase difference and the ratio of amplitudes between sinusoidally varying voltage and sinusoidally varying current at a given frequency. Fourier analysis allows any signal to be constructed from a spectrum of frequencies, whence the circuit's reaction to the various frequencies may 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 the AC voltage V = ZI lags the AC current by 90°: the positive current phase corresponds to increasing voltage as the capacitor charges; zero current corresponds to instantaneous constant voltage, etc. Impedance decreases with increasing capacitance and increasing frequency. This implies that a higher-frequency signal or a larger capacitor results in a lower voltage amplitude per current amplitude—an AC "" or AC coupling. Conversely, for very low frequencies, the reactance will be high, so that a capacitor is nearly an open circuit 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 defining characteristic; i.e., capacitance.

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

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

Parallel-plate model

The simplest capacitor consists of two parallel conductive plates separated by a dielectric with ε (such as air). The model may also be used to make qualitative predictions for other device geometries. The plates are considered to extend uniformly over an area A and a ±ρ = ±Q/A exists on their surface. Assuming that the width of the plates is much greater than their separation d, the electric field near the centre of the device will be uniform with the magnitude E = ρ/ε. The voltage is defined as the line integral of the electric field between the plates

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

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, and small distance between plates. A parallel plate capacitor can only store a finite amount of energy before dielectric breakdown occurs. The capacitor's dielectric material has a U which sets the capacitor's breakdown voltage at V = V = d bd U d. The maximum energy that the capacitor can store is therefore d

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 volume has no change on the amount of energy the capacitor can store. Care must be taken when increasing the plate separation so that the above assumption of the distance between plates being much smaller than the area of the plates is still valid for these equations to be accurate. In addition, these equations assume that the electric field is entirely concentrated in the dielectric between the plates. In reality there are fringing fields outside the dielectric, for example between the sides of the capacitor plates, which will increase the effective capacitance of the capacitor. This could be seen as a form of parasitic capacitance. For some simple capacitor geometries this additional capacitance term can be calculated analytically.[] It becomes negligibly small when the ratio of plate area to separation is large.

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Networks

For capacitors in parallel Capacitors in a parallel configuration each have the same applied voltage. Their add up. Charge is apportioned among them by size. Using the schematic diagram to visualize parallel plates, it is apparent that each capacitor contributes to

the total surface area. Several capacitors in parallel.

For capacitors in series Connected in series, the schematic diagram reveals that the separation distance, not the plate area, adds up. The capacitors each store instantaneous charge build-up equal to that of every other capacitor in the series. The total voltage difference from Several capacitors in series. end to end is apportioned to each capacitor according to the inverse of its capacitance. The entire series acts as a capacitor smaller than any of its components.

Capacitors are combined in series to achieve a higher working voltage, for example for smoothing a high voltage power supply. The voltage ratings, which are based on plate separation, add up, if capacitance and leakage currents for each capacitor are identical. In such an application, on occasion series strings are connected in parallel, forming a matrix. The goal is to maximize the energy storage of the network without overloading any capacitor. For high-energy storage with capacitors in series, some safety considerations must be applied to ensure one capacitor failing and leaking current will not apply too much voltage to the other series 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 of capacitors 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 behavior Capacitors deviate from the ideal capacitor equation in a number of ways. Some of these, such as leakage current and parasitic effects are linear, or can be assumed to be linear, and can be dealt with by adding virtual components to the equivalent circuit of the capacitor. The usual methods of network analysis can then be applied. In other cases, such as with breakdown voltage, the effect is non-linear and normal (i.e., linear) network analysis cannot be used, the effect must be dealt with separately. There is yet another group, which may be linear but invalidate the assumption in the analysis that capacitance is a constant. Such an example is temperature dependence. Finally, combined parasitic effects such as inherent inductance, resistance, or dielectric losses can exhibit non-uniform behavior at variable frequencies of operation.

Breakdown voltage Above a particular electric field, known as the dielectric strength E , the dielectric in a capacitor becomes ds conductive. The voltage at which this occurs is called the breakdown voltage of the device, and is given by the product 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 scaling of capacitance and breakdown voltage with dielectric thickness, all capacitors made with a particular dielectric have approximately 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 100 to 300 MV/m, for oil 15 to 25 MV/m, and can be much less when other materials are used for the dielectric.[21] The dielectric is used in very thin layers and so absolute breakdown voltage of capacitors is limited. Typical ratings for capacitors used for general electronics applications range from a few to 1 kV. As the voltage increases, the dielectric 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; sharp edges or points increase the electric field strength at that point and can lead to a local breakdown. Once this starts to happen, the breakdown quickly tracks through the dielectric until it reaches the opposite plate, leaving carbon behind causing a short circuit.[22] The usual breakdown route is that the field strength becomes large enough to pull electrons in the dielectric from their atoms thus causing conduction. Other scenarios are possible, such as impurities in the dielectric, and, if the dielectric is of a crystalline nature, imperfections in the crystal structure can result in an avalanche breakdown as seen in semi-conductor devices. Breakdown voltage is also affected by pressure, humidity and temperature.[23]

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] Capacitor 32

Equivalent circuit

An ideal capacitor only stores and releases electrical energy, without dissipating any. In reality, all capacitors have imperfections within the capacitor's material that create resistance. This is specified as the equivalent series resistance or ESR of a component. This adds a real component to the impedance:

As frequency approaches infinity, the capacitive impedance (or reactance) approaches zero and the ESR becomes significant. As the reactance becomes negligible, power dissipation approaches P = RMS V ² /R . RMS ESR Similarly to ESR, the capacitor's leads add equivalent series Two different circuit models of a real capacitor inductance or ESL to the component. This is usually significant only at relatively high frequencies. As inductive reactance is positive and increases with frequency, above a certain frequency capacitance will be canceled by inductance. High-frequency engineering involves accounting for the inductance of all connections and components. If the conductors are separated by a material with a small conductivity rather than a perfect dielectric, then a small leakage current flows directly between them. The capacitor therefore has a finite parallel resistance,[12] and slowly discharges over time (time may vary greatly depending on the capacitor material and quality).

Q factor The quality factor (or Q) of a capacitor is the ratio of its reactance to its resistance at a given frequency, and is a measure 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 current Ripple current is the AC component of an applied source (often a switched-mode power supply) (whose frequency may be constant or varying). Ripple current causes heat to be generated within the capacitor due to the dielectric losses caused by the changing field strength together with the current flow across the slightly resistive supply lines or the electrolyte in the capacitor. The equivalent series resistance (ESR) is the amount of internal series resistance one would add to a perfect capacitor to model this. Some types of capacitors, primarily tantalum and aluminum electrolytic 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 and burning parts.

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] Capacitor 33

• Aluminium electrolytic capacitors, the most common type of electrolytic, suffer a shortening of life expectancy at higher ripple currents. If ripple current exceeds the rated value of the capacitor, it tends to result in explosive failure. • 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 instability The capacitance of certain capacitors decreases as the component ages. In ceramic capacitors, this is caused by degradation of the dielectric. The type of dielectric, ambient operating and storage temperatures are the most significant aging factors, while the operating voltage has a smaller effect. The aging process may be reversed by heating the component above the Curie point. Aging is fastest near the beginning of life of the component, and the device stabilizes over time.[24] Electrolytic capacitors age as the electrolyte evaporates. In contrast with ceramic capacitors, 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 be taken as a broadly linear function but can be noticeably non-linear at the temperature extremes. The temperature coefficient can be either positive or negative, sometimes even amongst different samples of the same type. In other words, the spread in the range of temperature coefficients can encompass zero. See the data sheet in the leakage current section above for an example. Capacitors, especially ceramic capacitors, and older designs such as paper capacitors, can absorb sound waves resulting in a microphonic effect. Vibration moves the plates, causing the capacitance to vary, in turn inducing AC current. Some also generate piezoelectricity. The resulting interference is especially problematic in audio applications, potentially causing feedback or unintended recording. In the reverse microphonic effect, the varying electric field between the capacitor plates exerts a physical force, moving them as a speaker. This can generate audible sound, but drains energy and stresses the dielectric and the electrolyte, if any.

Current and voltage reversal Current 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. Voltage reversal is encountered in RLC circuits that are under-damped. The current and voltage reverse direction, forming a harmonic oscillator between the inductance and capacitance. The current and voltage will tend to oscillate and may reverse direction several times, with each peak being lower than the previous, until the system reaches an equilibrium. This is often referred to as ringing. In comparison, critically damped or over-damped systems usually do not experience a voltage reversal. Reversal is also encountered in AC circuits, where the peak current will be equal in each direction. For maximum life, capacitors usually need to be able to handle the maximum amount of reversal that a system will experience. An AC circuit will experience 100% voltage reversal, while under-damped DC circuits will experience less than 100%. Reversal creates excess electric fields in the dielectric, causes excess heating of both the dielectric and the conductors, and can dramatically shorten the life expectancy of the capacitor. Reversal ratings will often affect the design considerations for the capacitor, from the choice of dielectric materials and voltage ratings to the types of internal connections used.[25]

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] Capacitor 34

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

Leakage Leakage is equivalent to a resistor in parallel with the capacitor. Constant exposure to heat can cause dielectric breakdown and excessive leakage, a problem often seen in older circuits, particularly where oiled paper and foil capacitors were used. In many vacuum tube circuits, interstage coupling capacitors are used to conduct a varying signal from the plate of one tube to the grid circuit of the next stage. A leaky capacitor can cause the grid circuit voltage to be raised from its normal bias setting, causing excessive current or signal distortion in the downstream 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 () amplifiers, but owing to lower heat production and the use of modern polyester dielectric barriers this once-common problem has become relatively rare.

Electrolytic failure from disuse Electrolytic capacitors are conditioned when manufactured by applying a voltage sufficient to initiate the proper internal chemical state. This state is maintained by regular use of the equipment. If a system using electrolytic capacitors is unused for a long period of time it can lose its conditioning, and will generally fail with a short circuit when next operated, permanently damaging the capacitor. To prevent this in tube equipment, the voltage can be slowly 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 types Practical capacitors are available commercially in many different forms. The type of internal dielectric, the structure of 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 cost compared with other , unless their relatively poor stability, life and polarised nature make them unsuitable. Very high capacity supercapacitors use a porous carbon-based electrode material.

Dielectric materials

Most types of capacitor include a dielectric spacer, which increases their capacitance. These dielectrics are most often insulators. However, low capacitance devices are available with a vacuum between their plates, which allows extremely high voltage operation and low losses. Variable capacitors with their plates open to the atmosphere were Capacitor materials. From left: multilayer commonly used in radio tuning circuits. Later designs use polymer foil ceramic, ceramic disc, multilayer polyester film, dielectric between the moving and stationary plates, with no significant tubular ceramic, polystyrene, metalized polyester air space between them. film, aluminum electrolytic. Major scale divisions are in centimetres.

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In order to maximise the charge that a capacitor can hold, the dielectric material needs to have as high a permittivity as 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 used extensively in older devices and offers relatively high voltage performance. However, it is susceptible to water absorption, and has been largely replaced by plastic film capacitors. offer better stability and aging performance, which makes them useful in timer circuits, although they may be limited to low operating temperatures and frequencies. Ceramic capacitors are generally small, cheap and useful for high frequency applications, although their capacitance varies strongly with voltage and they age poorly. They are broadly categorized as class 1 dielectrics, which have predictable variation of capacitance with temperature or class 2 dielectrics, which can operate at higher voltage. Glass and mica capacitors are extremely reliable, stable and tolerant to high temperatures and voltages, but are too expensive for most mainstream applications. Electrolytic capacitors and supercapacitors are used to store small and larger amounts of energy, respectively, ceramic capacitors are often used in , and parasitic capacitance occurs in circuits wherever the simple conductor-insulator-conductor structure is formed unintentionally by the configuration of the circuit layout. Electrolytic capacitors use an aluminum or tantalum plate with an oxide dielectric layer. The second electrode is a liquid electrolyte, connected to the circuit by another foil plate. Electrolytic capacitors offer very high capacitance but suffer from poor tolerances, high instability, gradual loss of capacitance especially when subjected to heat, and high leakage current. Poor quality capacitors may leak electrolyte, which is harmful to printed circuit boards. The conductivity of the electrolyte drops at low temperatures, which increases equivalent series resistance. While widely used for power-supply conditioning, poor high-frequency characteristics make them unsuitable for many applications. Electrolytic capacitors will self-degrade if unused for a period (around a year), and when full power is applied may short circuit, permanently damaging the capacitor and usually blowing a fuse or causing failure of rectifier diodes (for instance, in older equipment, arcing in rectifier tubes). They can be restored before use (and damage) by gradually applying the operating voltage, often done on antique vacuum tube equipment over a period of 30 minutes by using a variable transformer to supply AC power. Unfortunately, the use of this technique may be less satisfactory 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 to modern 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 dielectric absorption and leakage.[26] Polymer capacitors (OS-CON, OC-CON, KO, AO) use solid conductive polymer (or polymerized organic semiconductor) 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 signals through a circuit board. Several other types of capacitor are available for specialist applications. Supercapacitors store large amounts of energy. Supercapacitors made from carbon aerogel, carbon nanotubes, or highly porous electrode materials, offer extremely high capacitance (up to 5 kF as of 2010[27]) and can be used in some applications instead of rechargeable batteries. 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 be physically large. They are usually ruggedly packaged, often in metal cases that can be easily grounded/earthed. They also are designed with direct current breakdown voltages of at least five times the maximum AC voltage.

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] Capacitor 36

Structure

The arrangement of plates and dielectric has many variations depending on the desired ratings of the capacitor. For small values of capacitance (microfarads and less), ceramic disks use metallic coatings, with wire leads bonded to the coating. Larger values can be made by multiple stacks of plates and disks. Larger value capacitors usually use a metal foil or metal film layer deposited on the surface of a dielectric film to make the plates, and a dielectric film of impregnated paper or plastic – these are rolled up to save space. To reduce the series resistance and inductance for long plates, the plates Capacitor packages: SMD ceramic at top left; and dielectric are staggered so that connection is made at the common SMD tantalum at bottom left; through-hole tantalum at top right; through-hole electrolytic at edge of the rolled-up plates, not at the ends of the foil or metalized film bottom right. Major scale divisions are cm. strips 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 dipped in a hard thermoplastic. Large capacitors for high-voltage use may have the roll form compressed to fit into a rectangular metal case, with bolted terminals and bushings for connections. The dielectric in larger capacitors is often impregnated with a liquid to improve its properties. Capacitors may have their connecting leads arranged in many configurations, for example axially or radially. "Axial" means that the leads are on a common axis, typically the axis of the capacitor's cylindrical body – the leads extend from opposite ends. Radial leads might more accurately be referred to as tandem; they are rarely actually aligned along radii of the body's circle, so the term is inexact, although universal. The leads (until bent) are usually in planes parallel to that of the flat body of the capacitor, and extend in the same direction; they are often parallel as manufactured. Several axial-lead electrolytic capacitors

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. These packages are extremely small and lack connecting leads, allowing them to be soldered directly onto the surface of printed circuit boards. Surface mount components avoid undesirable high-frequency effects due to the leads and simplify 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 sliding a set of movable plates into alignment with a set of stationary plates. Low cost variable capacitors squeeze together alternating 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 applied voltage. They are used in phase-locked loops, amongst other applications.

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

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

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

Applications

Energy storage

A capacitor can store electric energy when disconnected from its charging circuit, so it can be used like a temporary battery. Capacitors are commonly used in electronic devices to maintain power supply while batteries are being changed. (This prevents loss of information in volatile memory.) Conventional capacitors provide less than 360 joules per kilogram of energy density, whereas a conventional alkaline battery has a density of 590 kJ/kg. In car audio systems, large capacitors store energy for the amplifier to use on demand. Also for a flash tube a capacitor is used to hold the high voltage.

Pulsed power and weapons This mylar-film, oil-filled capacitor has very low Groups of large, specially constructed, low-inductance high-voltage inductance and low resistance, to provide the capacitors (capacitor banks) are used to supply huge pulses of current high-power (70 megawatt) and high speed (1.2 microsecond) discharge needed to operate a dye for many pulsed power applications. These include electromagnetic laser. forming, Marx generators, pulsed lasers (especially TEA lasers), pulse forming networks, radar, fusion research, and particle accelerators. Large capacitor banks (reservoir) are used as energy sources for the exploding-bridgewire detonators or slapper detonators in nuclear weapons and other specialty weapons. Experimental work is under way using banks of capacitors as power sources for electromagnetic armour and electromagnetic railguns and coilguns.

Power conditioning

Reservoir capacitors are used in power supplies where they smooth the output of a full or half wave rectifier. They can also be used in charge circuits as the energy storage element in the generation of higher voltages than the input voltage. Capacitors are connected in parallel with the power circuits of most electronic devices and larger systems (such as factories) to away and conceal current fluctuations from the primary power source to provide a "clean" power supply for signal or control circuits. Audio equipment, for example, uses several capacitors in this way, to shunt A 10 millifarad capacitor in an amplifier power away power line hum before it gets into the signal circuitry. The supply

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] Capacitor 38

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

Power factor correction

In electric power distribution, capacitors are used for 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 a reactive power in volt-amperes reactive (var). The purpose is to counteract inductive loading from devices like electric motors and transmission lines to make the load appear to be mostly resistive. Individual motor or lamp loads may have capacitors for power factor correction, or larger sets of capacitors (usually with automatic switching devices) may be installed at a load center within a building or in a large utility substation.

Suppression and coupling

A high-voltage capacitor bank used for power factor correction on a power transmission system.

Signal coupling

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

Decoupling Polyester film capacitors are frequently used as A decoupling capacitor is a capacitor used to protect one part of a coupling capacitors. circuit from the effect of another, for instance to suppress noise or transients. Noise caused by other circuit elements is shunted through the capacitor, reducing the effect they have on the rest of the circuit. It is most commonly used between the power supply and . An alternative name is bypass 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

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] Capacitor 39

in a single package. Capacitors are also used in parallel to interrupt units of a high-voltage in order to equally distribute the 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 diagrams with 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 starters In single phase squirrel cage motors, the primary winding within the motor housing is not capable of starting a rotational motion on the rotor, but is capable of sustaining one. To start the motor, a secondary "start" winding has a series 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 force of the rotational field is not constant, but is sufficient to start the rotor spinning. When the rotor comes close to operating speed, a centrifugal switch (or current-sensitive relay in series with the main winding) disconnects the capacitor. The start capacitor is typically mounted to the side of the motor housing. These are called capacitor-start motors, that have relatively high starting torque. Typically they can have up-to four times as much starting torque than a split-phase motor and are used on applications such as , pressure washers and any small device requiring high starting torques. Capacitor-run induction motors have a permanently connected phase-shifting capacitor in series with a second winding. The motor is much like a two-phase induction motor. Motor-starting capacitors are typically non-polarized electrolytic types, while running capacitors are conventional paper or plastic film dielectric types.

Signal processing The energy stored in a capacitor can be used to represent information, either in binary form, as in DRAMs, or in analogue form, as in analog sampled filters and CCDs. Capacitors can be used in analog circuits as components of integrators or more complex filters and in negative feedback loop stabilization. Signal processing circuits also use capacitors 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 analog crossovers, 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 is given by:

where L is in henries and C is in farads.

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Sensing Most capacitors are designed to maintain a fixed physical structure. However, various factors can change the structure 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 an exposed and porous dielectric can be used to measure humidity in air. Capacitors are used to accurately measure 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 used for 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, relative to the fixed position of the other plate. Some accelerometers use MEMS capacitors etched on a chip to measure the magnitude and direction of the acceleration vector. They are used to detect changes in acceleration, in tilt sensors, or to detect free fall, as sensors triggering airbag deployment, and in many other applications. Some fingerprint sensors use capacitors. Additionally, a user can adjust the pitch of a theremin musical instrument by moving his hand since this changes the effective capacitance between the user's hand and the antenna. Changing the effective area of the plates: Capacitive touch are now used on many consumer electronic products.

Hazards and safety Capacitors may retain a charge long after power is removed from a circuit; this charge can cause dangerous or even potentially fatal shocks or damage connected equipment. For example, even a seemingly innocuous device such as a disposable camera flash unit powered by a 1.5 volt AA battery contains a capacitor which may be charged to over 300 volts. This is easily capable of delivering a shock. Service procedures for electronic devices usually include instructions to discharge large or high-voltage capacitors, for instance using a Brinkley stick. Capacitors may also have built-in discharge resistors to dissipate stored energy to a safe level within a few seconds after power is removed. High-voltage capacitors are stored with the terminals shorted, as protection from potentially dangerous voltages due to dielectric absorption. Some old, large oil-filled paper or plastic film capacitors contain polychlorinated biphenyls (PCBs). It is known that waste 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) fluorescent lamp ballasts, and other applications. Capacitors may catastrophically fail when subjected to voltages or currents beyond their rating, or as they reach their normal end of life. Dielectric or metal interconnection failures may create arcing that vaporizes the dielectric , resulting in case bulging, rupture, or even an explosion. Capacitors used in RF or sustained high-current applications can overheat, especially in the center of the capacitor rolls. Capacitors used within high-energy capacitor banks can violently explode when a short in one capacitor causes sudden dumping of energy stored in the rest of the bank into the failing unit. High voltage vacuum capacitors can generate soft X-rays even during normal operation. Proper containment, 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 direct current (HVDC) circuits. This will extend the life of the component and may mitigate high-voltage hazards.

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Swollen caps of electrolytic capacitors – This high-energy Catastrophic failure special design of semi-cut caps prevents capacitor from a capacitors from bursting defibrillator can deliver over 500 joules of energy. A resistor is connected between the terminals for safety, to allow the stored energy to be released.

Notes [10] Ulaby, p.168 [11] Ulaby, p.157 [12] Ulaby, p.169 [14] Dorf, p.263 [15] Dorf, p.260 [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.

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] 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 potential difference across the two points. Introducing the constant of proportionality, the resistance,[1] one arrives at the usual mathematical equation that describes this relationship:[]

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where I is the current through the conductor in units of amperes, V is the potential difference measured across the conductor in units of volts, and R is the resistance of the conductor in units of ohms. More specifically, Ohm's law states that the R in this relation is constant, independent of the current.[2] The law was named after the German physicist , who, in a treatise published in 1827, described measurements of applied voltage and current through simple electrical circuits containing various lengths of wire. He presented a slightly more complex equation than the one above (see History section below) to explain his experimental 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 by Ohm. 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 is a material dependent parameter called the conductivity. This reformulation of Ohm's law is due to Gustav Kirchhoff.[3]

History In January 1781, before Georg Ohm's work, Henry Cavendish experimented with Leyden jars and glass tubes of varying diameter and length filled with salt solution. He measured the current by noting how strong a shock he felt as he completed the circuit with his body. Cavendish wrote that the "velocity" (current) varied directly as the "degree of " (voltage). He did not communicate his results to other scientists at the time,[] and his results were unknown 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 Die galvanische Kette, mathematisch bearbeitet (The galvanic circuit investigated mathematically).[5] He drew considerable inspiration from Fourier's work on heat conduction in the theoretical explanation of his work. For experiments, he initially used voltaic piles, but later used a thermocouple as this provided a more stable voltage source 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 then added test wires of varying length, diameter, and material to complete the circuit. He found that his data could be modeled through the equation

where x was the reading from the galvanometer, l was the length of the test conductor, a depended only on the thermocouple junction temperature, and b was a constant of the entire setup. From this, Ohm determined his law of proportionality and published his results. Ohm's law was probably the most important of the early quantitative descriptions of the physics of electricity. We consider it almost obvious today. When Ohm first published his work, this was not the case; critics reacted to his treatment of the subject with hostility. They called his work a "web of naked fancies"[6] and the German Minister of Education proclaimed that "a professor who preached such heresies was unworthy to teach science."[7] The prevailing scientific philosophy in Germany at the time asserted that experiments need not be performed to develop an understanding of nature because nature is so well ordered, and that scientific truths may be deduced through reasoning alone.[8] Also, Ohm's brother Martin, a mathematician, was battling the German educational system. These factors 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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In the 1850s, Ohm's law was known as such and was widely considered proved, and alternatives, such as "Barlow's law", were discredited, in terms of real applications to telegraph system design, as discussed by Samuel F. B. Morse in 1855.[9] While the old term for electrical conductance, the mho (the inverse of the resistance unit ohm), is still used, a new name, the siemens, was adopted in 1971, honoring Ernst Werner von Siemens. The siemens is preferred in formal papers. In the 1920s, it was discovered that the current through an ideal resistor actually has statistical fluctuations, which depend on temperature, even when voltage and resistance are exactly constant; this fluctuation, now known as Johnson–Nyquist noise, is due to the discrete nature of charge. This thermal effect implies that measurements of current and voltage that are taken over sufficiently short periods of time will yield ratios of V/I that fluctuate from the value of R implied by the time average or ensemble average of the measured current; Ohm's law remains correct for 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 AC circuits. Modern developments in electromagnetic theory and circuit theory do not contradict Ohm's law when they are evaluated within the appropriate limits.

Scope Ohm's law is an empirical law, a generalization from many experiments that have shown that current is approximately proportional to electric field for most materials. It is less fundamental than Maxwell's equations and is not always obeyed. Any given material will break down under a strong-enough electric field, and some materials of interest 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's law would fail at the atomic scale, but experiments have not borne out this expectation. As of 2012, researchers have demonstrated that Ohm's law works for silicon wires as small as four atoms wide and one atom high.[12]

Microscopic origins

The dependence of the current density on the applied electric field is essentially quantum mechanical in nature; (see Classical and quantum conductivity.) A qualitative description leading to Ohm's law can be based upon classical using the Drude model developed by Paul Drude in 1900.[13][14] The Drude model treats electrons (or other charge carriers) like pinballs bouncing among the ions that make up the structure of the material. Electrons will be accelerated in the opposite direction to the electric field by the average electric field at their location. With each collision, though, the electron is deflected in a random direction with a velocity that is much larger than the velocity

gained by the electric field. The net result is that electrons take a Drude Model electrons (shown here in blue) constantly zigzag path due to the collisions, but generally drift in a direction bounce among heavier, stationary crystal ions (shown opposing the electric field. in red).

The 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 , −e is the charge of the electron and τ is the average time between the collisions. Since both the momentum and the current

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] 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 analogy A hydraulic analogy is sometimes used to describe Ohm's law. Water pressure, measured by pascals (or PSI), is the analog of voltage because establishing a water pressure difference between two points along a (horizontal) pipe causes 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 is measured—are the analog of resistors. We say that the rate of water flow through an aperture restrictor is proportional 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 across the resistor. Flow and pressure variables can be calculated in fluid flow network with the use of the hydraulic ohm analogy.[15][16] The method can be applied to both steady and transient flow situations. In the linear laminar flow region, Poiseuille's law describes the hydraulic resistance of a pipe, but in the turbulent flow region the pressure–flow relations become nonlinear. The hydraulic analogy to Ohm's law has been used, for example, to approximate blood flow through the circulatory system.[17]

Circuit analysis

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

Each equation is quoted by some sources as the defining relationship of Ohm's law,[][18][19] or all three are quoted,[20] or derived from a proportional form,[21] or

even just the two that do not correspond to Ohm's original statement may Ohm's law triangle sometimes be given.[22][23] The interchangeability of the equation may be represented by a triangle, where V (voltage) is placed on the top section, the I (current) is placed to the left section, and the R (resistance) is placed to the right. The line that divides the left and right sections indicate multiplication, and the divider between the top and bottom sections indicates division (hence the division bar).

Resistive circuits Resistors are circuit elements that impede the passage of electric charge in agreement with Ohm's law, and are designed to have a specific resistance value R. In a schematic diagram the resistor is shown as a zig-zag symbol. An element (resistor or conductor) that behaves according to Ohm's law over some operating range is referred to as an ohmic device (or an ohmic resistor) because Ohm's law and a single value for the resistance suffice to describe the behavior of the device over that range. Ohm's law holds for circuits containing only resistive elements (no capacitances or ) for all forms of driving voltage or current, regardless of whether the driving voltage or current is constant (DC) or time-varying such as 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

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] Ohm's law 46

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

Reactive circuits with time-varying signals When reactive elements such as capacitors, inductors, or transmission lines are involved in a circuit to which AC or time-varying voltage or current is applied, the relationship between voltage and current becomes the solution to a differential equation, so Ohm's law (as defined above) does not directly apply since that form contains only resistances 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 to complex 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, and A is a complex scalar. In any linear time-invariant system, all of the currents and voltages can be expressed with the same s parameter as the input to the system, allowing the time-varying complex exponential term to be canceled out and 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 real part is responsible for dissipating heat. In the general AC circuit, Z varies strongly with the frequency parameter s, and so also will the relationship between voltage 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 approximations Ohm's law is one of the basic equations used in the analysis of electrical circuits. It applies to both metal conductors and 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 same value for resistance (R = V/I) regardless of the value of V or I which is applied and whether the applied voltage or current 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 the applied voltage V. That is, the ratio of V/I is constant, and when current is plotted as a function of voltage the curve is linear (a straight line). If voltage is forced to some value V, then that voltage V divided by measured current I will equal 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 V and V 1 2 applied across a given device of resistance R, producing currents I = V /R and I = V /R, that the ratio 1 1 2 2 (V -V )/(I -I ) is also a constant equal to R. The operator "delta" (Δ) is used to represent a difference in a quantity, 1 2 1 2 so we can write ΔV = V -V and ΔI = I -I . Summarizing, for any truly ohmic device having resistance R, V/I = 1 2 1 2 ΔV/ΔI = R for any applied voltage or current or for the difference between any set of applied voltages or currents.

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] Ohm's law 47

There are, however, components of electrical circuits which do not obey Ohm's law; that is, their relationship between current and voltage (their I–V curve) is nonlinear (or non-ohmic). An The I–V curves of four devices: Two resistors, a diode, and a battery. The two resistors example is the p-n junction diode follow Ohm's law: The plot is a straight line through the origin. The other two devices do (curve at right). As seen in the figure, not follow Ohm's law. the current does not increase linearly with 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 applied voltage. Further, the current only increases significantly if the applied voltage is positive, not negative. The ratio V/I for some point along the nonlinear curve is sometimes called the static, or chordal, or DC, resistance,[26][27] but as seen in the figure the value of total V over total I varies depending on the particular point along the nonlinear curve which is chosen. This means the "DC resistance" V/I at some point on the curve is not the same as what would be determined by applying an AC signal having peak amplitude ΔV volts or ΔI amps centered at that same point along the curve and measuring ΔV/ΔI. However, in some diode applications, the AC signal applied to the device is small and it is possible to analyze the circuit in terms of the dynamic, small-signal, or incremental resistance, defined as the one over the slope of the V–I curve at the average value (DC operating point) of the voltage (that is, one over the derivative of current with respect to voltage). For sufficiently small signals, the dynamic resistance allows the Ohm's law small signal resistance to be calculated as approximately one over the slope of a line drawn tangentially to the V-I curve at the DC operating point.[]

Temperature effects Ohm's law has sometimes been stated as, "for a conductor in a given state, the electromotive force is proportional to the current produced." That is, that the resistance, the ratio of the applied electromotive force (or voltage) to the current, "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 the conduction of current is related to Joule heating of the conducting body, according to Joule's first law, the temperature of a conducting body may change when it carries a current. The dependence of resistance on temperature therefore makes resistance depend upon the current in a typical experimental setup, making the law in this form difficult to directly verify. Maxwell and others worked out several methods to test the law experimentally in 1876, controlling for heating effects.[28]

Relation to heat conductions Ohm's principle predicts the flow of electrical charge (i.e. current) in electrical conductors when subjected to the influence of voltage differences; Jean-Baptiste-Joseph Fourier's principle predicts the flow of heat in heat conductors when 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 (the rate of flow of the driven "quantity", i.e. heat energy) variables also solves an analogous electrical conduction (Ohm) problem having (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

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] 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 at each point is proportional to the gradient of electric potential. The accuracy of the assumption that flow is proportional to the gradient is more readily tested, using modern measurement methods, for the electrical case than for the heat case.

Other versions Ohm's law, in the form above, is an extremely useful equation in the field of electrical/electronic engineering because 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 microscopic level use a closely related and more general vector equation, sometimes also referred to as Ohm's law, having variables that are closely related to the V, I, and R scalar variables of Ohm's law, but which are each functions of position 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 of volts), "J" is the current density vector with units of amperes per unit area (analogous to "I" of Ohm's law which has units of amperes), and "ρ" (Greek "rho") is the resistivity with units of ohm·meters (analogous to "R" of Ohm's law which has units of ohms). The above equation is sometimes written[30] as J = E where "σ" (Greek "sigma") is the conductivity which is the reciprocal of ρ. The potential difference between two points is defined as:[31]

with the element of path along the integration of electric field vector E. If the applied E field is uniform and oriented along the length of the conductor as shown in the figure, then defining the voltage V in the usual convention of being opposite in direction to the field (see figure), and with the understanding that the voltage V is measured Current flowing through a uniform cylindrical conductor (such as a round wire) differentially across the length of the with a uniform field applied. conductor 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, so we may write:[32]

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

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] 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 r is 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 for a uniform field (and uniform current density) oriented along the length of the conductor reduces to the more familiar form:

A perfect crystal lattice, with low enough thermal motion and no deviations from periodic structure, would have no resistivity,[33] but a real metal has crystallographic defects, impurities, multiple isotopes, and thermal motion of the atoms. 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 the properties of matter and, in particular, its electronic structure. In broad terms, they fall under the topic of constitutive equations and the theory of transport coefficients.

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

In the rest frame of the moving conductor this term drops out because v= 0. There is no contradiction because the electric field in the rest frame differs from the E-field in the lab frame: E ' = E + v×B. Electric and magnetic fields are 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 to resistance to account for self-inductance, see . The reactance may be strong if the frequency is high or the conductor is coiled. See 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, ISBN 0-8087-4908-0

[6] Davies, B, "A web of naked fancies?", 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 , 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) [email protected] 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, Bonnier Corporation ISSN 0161-7370, gives the history of Ohm's investigations, prior work, Ohm's false equation in the first 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 of Physics, July 1963, Volume 31, Issue 7, pp. 536–47. Explores the conceptual change underlying Ohm's experimental 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 laws

Kirchhoff'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 the work of Maxwell. Widely used in electrical engineering, they are also called Kirchhoff's rules or simply Kirchhoff's laws (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)

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 of currents flowing into that node is equal to the sum of currents flowing out of that node, or: The algebraic sum of currents in a network of conductors meeting at a point is zero. Recalling that current is a signed (positive or negative) quantity reflecting direction towards or away from a node, this principle can be stated as: The current entering any junction is equal to the current leaving that junction. i + i = i + i 2 3 1 4

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

Limitations KCL, in its usual form, is dependent on the assumption that current flows only in conductors, and that whenever current flows into one end of a conductor it immediately flows out the other end. This is not a safe assumption for AC circuits.[2] It may be possible to salvage the form of KCL by considering "parasitic capacitances" distributed along the conductors.[2] However, this greatly detracts from the simplicity of KCL and invalidates the notion of topological as discussed below. Significant violations of KCL can occur[3][4] even at 60Hz, which is not a very high frequency. In other words, KCL is valid only if the total electric charge, , remains constant in the region being considered. In practical 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 only come 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 region Converting 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 is recovered, providing that . A particular case where KCL does not hold is the current entering a single plate of a capacitor. To illustrate this point, imagine a closed surface around that single plate, current enters through the surface, but does not exit, thus violating KCL. Certainly, the currents through a closed surface around the entire capacitor will meet KCL, since the current entering one plate is balanced by the current exiting the other plate, and that is usually all that is important in circuit analysis, but there is a problem when considering a single plate. Another common example, the current in an antenna 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 current flowing into a capacitor plate is equal to the rate of charge accumulation, and similarly, equal to the rate of change of (SI system unit of measurement for both, electric flux and electric charge, is Coulombs). This rate of change of flux, , is what Maxwell called ;

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

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This can also be expressed in terms of vector field quantities through divergence of Ampère's law with Maxwell's correction, 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 closed surface is equal to the rate of loss of charge within the enclosed volume (divergence theorem). Kirchhoff's current law is equivalent to the statement that the divergence of the current is zero, true for time-invariant ρ, or always true if the displacement current is included with J.

Uses A 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)

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 voltages may also be complex:

This law is based on one of the Maxwell equations, namely the Maxwell-Faraday law of induction, which states that the voltage The sum of all the voltages around the loop is equal to zero. v + v + v - v = 0 drop around any closed loop is equal to the rate-of-change of the 1 2 3 4 flux threading the loop. The amount of flux depends on the area of the loop and on the magnetic field strength. KVL states the loop voltage is zero. The Maxwell equations tell us that the loop voltage will be small if the area of the loop is small, the magnetic field is weak, and/or the magnetic field is slowly changing.

Routine engineering techniques -- such as the use of coaxial cable and twisted pairs -- can be used to minimize stray magnetic 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.

Limitations KVL is based on the assumption that there is no fluctuating magnetic field linking the closed loop. This is not a safe assumption for AC circuits.[2] In the presence of a changing magnetic field the electric field is not a conservative vector field. Therefore the electric field can not be the gradient of any potential. That is to say, the line integral of the electric 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 equal to the rate-of-change of the flux. However, this greatly detracts from the simplicity of KVL and invalidates the notion of topological circuit diagram.

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Generalization In 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 a corollary of Faraday's law of induction (which is one of the Maxwell equations). This has practical application in situations involving "".

Topological circuit diagrams The approximations that lead to Kirchhoff's circuit laws are part of a package that also leads to topological circuit diagrams, i.e. the idea that the physical and geometrical layout of the circuit does not matter; the only thing that matters is the topology as determined by the conductors and circuit elements connected to the nodes. These can be treated as the arcs and nodes of formal graph theory. In other words, Kirchhoff's laws say it suffices to use a circuit diagram 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] The physical and geometrical layout of the circuit matters, because it determines the magnitude of the parasitic capacitances and inductances.[2][3][4]

Example Assume 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 s gives 1

The second law applied to the closed circuit s gives 2

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 direction defined in the picture).

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] Kirchhoff's circuit laws 54

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

In electronics, a current divider is a simple linear circuit that produces an output current (I ) that is a X fraction of its input current (I ). Current division T refers to the splitting of current between the branches of the divider. The currents in the various branches of such a circuit will always divide in such a way as to minimize the total energy expended.

The formula describing a current divider is similar in form to that for the . However, the ratio describing current division places the impedance of the unconsidered branches in the numerator, unlike voltage Figure 1: Schematic of an electrical circuit illustrating current division where the considered impedance is in the division. Notation R . refers to the total resistance of the circuit to T numerator. This is because in current dividers, total the right of resistor R . X energy expended is minimized, resulting in currents that 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 the voltage 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 split between them in inverse proportion to their impedances (according to Ohm's law). It also follows that if the impedances have the same value the current is split equally.

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] Current divider 55

Resistive divider A general formula for the current I in a resistor R that is in parallel with a combination of other resistors of total X X resistance R is (see Figure 1): T

where I is the total current entering the combined network of R in parallel with R . Notice that when R is T X T T composed of a parallel combination of resistors, say R , R , ... etc., then the reciprocal of each resistor must be added 1 2 to find the total resistance R : T

General case Although the resistive divider is most common, the current divider may be made of frequency dependent impedances. In the general case the current I is given by: X

Using Admittance Instead 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 Y is a straightforward addition, not the sum of the inverses inverted (as you would do for a Total standard parallel resistive network). For Figure 1, the current I would be X

Example: RC combination

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

where Z = 1/(jωC) is the impedance of the capacitor and j C is the imaginary unit.

The product τ = CR is known as the time constant of the Figure 2: A low pass RC current divider circuit, and the frequency for which ωCR = 1 is called the corner frequency of the circuit. Because the capacitor has zero impedance at high frequencies and infinite impedance at low frequencies, the current in the resistor remains at its DC value I for frequencies up to the corner frequency, T whereupon it drops toward zero for higher frequencies as the capacitor effectively short-circuits the resistor. In other words, the current divider is a low pass filter for current in the resistor.

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] Current divider 56

Loading effect

The of an amplifier generally depends on its source and load terminations. Current amplifiers and transconductance amplifiers are characterized by a short-circuit output condition, and current amplifiers and transresistance amplifiers are characterized using ideal infinite impedance current Figure 3: A current amplifier (gray box) driven by a Norton source (i , R ) and sources. When an amplifier is terminated by S S with a resistor load R . Current divider in blue box at input (R ,R ) reduces the a finite, non-zero termination, and/or driven L S in current gain, as does the current divider in green box at the output (R ,R ) by a non-ideal source, the effective gain is out L reduced due to the loading effect at the output 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 R and output resistance in R and an ideal current gain A . With an ideal current driver (infinite Norton resistance) all the source current i out i S becomes input current to the amplifier. However, for a Norton driver a current divider is formed at the input that reduces the input current to

which clearly is less than i . Likewise, for a short circuit at the output, the amplifier delivers an output current i = A S o i i to the short-circuit. However, when the load is a non-zero resistor R , the current delivered to the load is reduced i L by current division to the value:

Combining these results, the ideal current gain A realized with an ideal driver and a short-circuit load is reduced to i the loaded gain A : loaded

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

Unilateral versus bilateral amplifiers

Figure 3 and the associated discussion refers to a unilateral amplifier. In a more general case where the amplifier is represented by a two , the input resistance of the amplifier depends on its load, and the output resistance on the source impedance. The loading factors in these cases must employ the true amplifier impedances including Figure 4: Current amplifier as a bilateral two-port network; feedback through these bilateral effects. For example, taking dependent voltage source of gain β V/V the unilateral current amplifier of Figure 3, the corresponding bilateral two-port network is shown in Figure 4 based upon h-parameters.[1] Carrying out the analysis for this circuit, the current gain with feedback A is found to be fb

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] Current divider 57

That is, the ideal current gain A is reduced not only by the loading factors, but due to the bilateral nature of the i two-port by an additional factor[2] ( 1 + β (R / R ) A ), which is typical of negative feedback amplifier L S loaded circuits. The factor β (R / R ) is the current feedback provided by the voltage feedback source of voltage gain β L S V/V. For instance, for an ideal current source with R = ∞ Ω, the voltage feedback has no influence, and for R = 0 S L Ω, 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

In electronics or EET, a voltage divider (also known as a potential divider) is a linear circuit that produces an output voltage (V ) that out is a fraction of its input voltage (V ). Voltage division refers to the in partitioning of a voltage among the components of the divider. An example of a voltage divider consists of two resistors in series or a potentiometer. It is commonly used to create a reference voltage, or to get a low voltage signal proportional to the voltage to be measured, and may also be used as a signal attenuator at low frequencies. For direct current and relatively low frequencies, a voltage divider may be sufficiently accurate if made only of resistors; where frequency response over a wide range is required, (such as in an oscilloscope probe), the voltage divider may have Figure 1: Voltage divider capacitive elements added to allow compensation for load capacitance. In electric power transmission, a capacitive voltage divider is used for measurement of high voltage.

General case A voltage divider referenced to ground is created by connecting two electrical impedances in series, as shown in Figure 1. The input voltage is applied across the series impedances Z and Z and the output is the voltage across Z . 1 2 2 Z and Z may be composed of any combination of elements such as resistors, inductors and capacitors. 1 2 Applying Ohm's Law, the relationship between the input voltage, V , and the output voltage, V , can be found: in out

Proof:

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

A resistive divider is the case where both impedances, Z and Z , are 1 2 purely resistive (Figure 2). Substituting Z = R and Z = R into the previous expression gives: 1 1 2 2

If R = R then 1 2

If V =6V and V =9V (both commonly used voltages), then: out in

Figure 2: Simple resistive voltage divider and by solving using algebra, R must be twice the value of R . 2 1

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 or increase V above V . out in

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] Voltage divider 59

Low-pass RC filter

Consider a divider consisting of a resistor and capacitor as shown in Figure 3. Comparing with the general case, we see Z = R and Z is the 1 2 impedance of the capacitor, given by

where X is the reactance of the capacitor, C is the capacitance of C the capacitor, j is the imaginary unit, and ω (omega) is the radian frequency of the input voltage. This divider will then have the voltage ratio:

Figure 3: Resistor/capacitor voltage divider

.

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 and phase shift information of the filter. To extract just the amplitude ratio, calculate the magnitude of the ratio, that is:

Inductive divider Inductive dividers split AC input according to inductance:

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

Capacitive divider Capacitive 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. By selection of parallel R and C elements in the proper proportions, the same division ratio can be maintained over a useful range of frequencies. This is the principle applied in compensated oscilloscope probes to increase measurement bandwidth.

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] Voltage divider 60

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

Applications Voltage dividers are used for adjusting the level of a signal, for bias of active devices in amplifiers, and for measurement of voltages. A and a multimeter both include voltage dividers. A potentiometer is used 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

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] Y-Δ transform 61 Y-Δ transform

The Y-Δ transform, also written wye-delta and also known by many other names, is a mathematical technique to simplify the analysis of an . The name derives from the shapes of the circuit diagrams, which look respectively like the letter Y and the Greek capital letter Δ. This circuit transformation theory was published by Arthur 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

The Y-Δ transform is known by a variety of other names, mostly based upon the two shapes involved, listed in either order. The Y, spelled out as wye, can also be called T or star; the Δ, spelled out as delta, can also be called triangle, Π (spelled out as pi), or mesh. Thus, common names for the Illustration of the transform in its T-Π representation, transformation include wye-delta or delta-wye, star-delta, star-mesh, or T-Π.

Basic Y-Δ transformation

The transformation is used to establish equivalence for networks with three terminals. Where three elements terminate at a common node and none are sources, the node is eliminated by transforming the impedances. For equivalence, the impedance between any pair of terminals must be the same for both networks. The equations given here are valid for complex as well as real impedances. Δ and Y circuits with the labels which are used in this article.

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 , to adjacent nodes in the Δ circuit by

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

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] 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 and is 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 transformation The feasibility of the transformation can be shown as a consequence of superposition theorem in electric circuit. A short proof, rather than derived as a corollary of the more general star-mesh transform, can be given as follows. The equivalence 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, and vice versa. In this proof, we start with given external currents at the nodes. According to superposition theorem, the voltages can be obtained by studying the linear summation of the resulting voltages at the nodes of following three problems: apply at the three nodes with current (1) , , , (2) , , and (3) , , . It can be readily shown that due to Kirchhoff's circuit laws, one has . One notes that now each problem is relatively simple, since it only involves one single ideal current source. To obtain exactly the same outcome voltages at the nodes for each problem, the equivalent resistances in two circuits must be the same, this can be easily found by using the basic rules of series and parallel circuits:

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

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] Y-Δ transform 63

Simplification of networks Resistive networks between two terminals can theoretically be simplified to a single equivalent resistor (more generally, the same is true of impedance). Series and parallel transforms are basic tools for doing so, but for complex networks 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 may readily be simplified further.

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

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

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] Y-Δ transform 64

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

Demonstration

Δ-load to Y-load transformation equations

To relate from Δ to from Y, the impedance between two corresponding nodes is compared. The impedance in either configuration is determined as if one of the nodes is disconnected from the circuit. The impedance between N and N 1 2 with N disconnected in Δ: 3 Δ and Y circuits with the labels that are used in this article.

To simplify, let be the sum of .

Thus,

The corresponding impedance between N and N in Y is simple: 1 2

hence:

(1)

Repeating for :

(2)

and for :

(3)

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

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

In electric circuits analysis, nodal analysis, node-voltage analysis, or the branch current method is a method of determining the voltage (potential difference) between "nodes" (points where elements or branches connect) in an electrical circuit in terms of the branch currents. In analyzing a circuit using Kirchhoff's circuit laws, one can either do nodal analysis using Kirchhoff's current law (KCL) or mesh analysis using Kirchhoff's voltage law (KVL). Nodal analysis writes an equation at each electrical node, requiring that the branch currents incident at a node must sum to zero. The branch currents are written in terms of the circuit node voltages. As a consequence, each branch constitutive relation must give current as a function of voltage; an Kirchhoff's current law is the basis of nodal admittance representation. For instance, for a resistor, I = V analysis. branch branch * 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 admittance representation. Nodal analysis produces a compact set of equations for the network, which can be solved by hand if small, or can be quickly solved using linear algebra by computer. Because of the compact system of equations, many circuit simulation programs (e.g. SPICE) use nodal analysis as a basis. When elements do not have admittance representations, a more general extension of nodal analysis, modified nodal analysis, can be used.

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] Nodal analysis 67

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

Method 1. Note all connected wire segments in the circuit. These are the nodes of nodal analysis. 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. 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. 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 nothing more 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 of the 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

The only unknown voltage in this circuit is V . 1 There are three connections to this node and consequently three currents to consider. The direction of the currents in calculations is chosen to be away from the node. 1. Current through resistor R : (V - V ) / R 1 1 S 1 2. Current through resistor R : V / R 2 1 2 3. Current through current source I : -I S S Basic example circuit with one unknown voltage, V . 1 With Kirchhoff's current law, we get:

This equation can be solved in respect to V : 1

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

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] Nodal analysis 68

Supernodes

In this circuit, we initially have two unknown voltages, V and V . 1 2 The voltage at V is already known to be V because the other 3 B terminal of the voltage source is at ground potential. The current going through voltage source V cannot be directly A calculated. Therefore we can not write the current equations for either V or V . However, we know that the same current leaving node V 1 2 2 must enter node V . Even though the nodes can not be individually 1 solved, 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: V = V + V . 1 2 A The complete set of equations for this circuit is:

By substituting V to the first equation and solving in respect to V2, 1 In this circuit, V is between two unknown A we get: voltages, and is therefore a supernode.

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

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] Mesh analysis 69 Mesh analysis

Mesh analysis (or the mesh current method) is a method that is used to solve planar circuits for the currents (and indirectly the voltages) at any place in the circuit. Planar circuits are circuits that can be drawn on a plane surface with no wires crossing each other. A more general technique, called loop analysis (with the corresponding network variables called loop currents) can be applied to any circuit, planar or not. Mesh analysis and loop analysis both make use of Kirchhoff’s voltage law to arrive at a set of equations guaranteed to be solvable if the circuit has a Figure 1: Essential meshes of the planar circuit labeled 1, 2, and 3. R , R , R , 1/sc, [1] 1 2 3 solution. Mesh analysis is usually easier and Ls represent the impedance of the resistors, capacitor, and inductor values in to use when the circuit is planar, compared the s-domain. V and i are the values of the voltage source and current source, s s to loop analysis.[2] respectively.

Mesh currents and essential meshes

Mesh analysis works by arbitrarily assigning mesh currents in the essential meshes (also referred to as independent meshes). An essential mesh is a loop in the circuit that does not contain any other loop. Figure 1 labels the essential meshes with one, two, and three.[3]

A mesh current is a current that loops around the essential mesh and the equations are set solved in terms of them. A mesh current may not correspond to any physically flowing current, but the physical currents are easily found from them.[2] It is usual practice to have all the mesh currents Figure 2: Circuit with mesh currents labeled as i , i , and i . The arrows show the 1 2 3 loop in the same direction. This helps direction of the mesh current. prevent errors when writing out the equations. The convention is to have all the mesh currents looping in a clockwise direction.[3] Figure 2 shows the same 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 can greatly reduce the amount of calculation required. This is because there are fewer mesh currents than there are physical branch currents. In figure 2 for example, there are six branch currents but only three mesh currents.

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] Mesh analysis 70

Setting up the equations Each mesh produces one equation. These equations are the sum of the voltage drops in a complete loop of the mesh current.[3] For problems more general than those including current and voltage sources, the voltage drops will be the impedance of the 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 depending on if it is a voltage drop or a voltage rise in the direction of the mesh current. For a current source that is not contained between two meshes, the mesh current will take the positive or negative value of the current source depending on if the mesh current is in the same or opposite direction of the current source.[3] The following is the same 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 linear equations.

Special cases There are two special cases in mesh current: currents containing a supermesh and currents containing dependent sources.

Supermesh

A supermesh occurs when a current source is contained between two essential meshes. The circuit is first treated as if the current source is not there. This leads to one equation that incorporates two mesh currents. Once this equation is formed, an equation is needed that relates the two mesh currents with the current source. This will be an equation where the current source is equal to one of the mesh currents minus the Figure 3: Circuit with a supermesh. Supermesh occurs because the current source other. The following is a simple example of is in between the essential meshes. dealing with a supermesh.[2]

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] Mesh analysis 71

Dependent sources

A is a current source or voltage source that depends on the voltage or current of another element in the circuit. When a dependent source is contained within an essential mesh, the dependent source should be treated like an independent source. After the mesh equation is formed, a dependent source equation is needed. This

equation is generally called a constraint Figure 4: Circuit with dependent source. i is the current upon which the dependent x equation. This is an equation that relates the source depends. dependent source’s variable to the voltage or current 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) [email protected] Superposition theorem 72 Superposition theorem

The superposition theorem for electrical circuits states that for a linear system the response (Voltage or Current) in any branch of a bilateral linear circuit having more than one independent source equals the algebraic sum of the responses caused by each independent source acting alone, while all other independent sources are replaced by their internal 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 true operation 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 Norton equivalent or Thevenin equivalent. Applicable to linear networks (time varying or time invariant) consisting of independent sources, linear dependent sources, 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. In other words the sum of the powers is not the real consumed power. To calculate power we should first use superposition to find both current and voltage of that linear element and then calculate sum of the multiplied voltages and 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) [email protected] Thévenin's theorem 73 Thévenin's theorem

Thévenin'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 at terminals A-B by an equivalent voltage source V in series connection with an equivalent resistance R . th th • This equivalent voltage V is the voltage obtained at terminals A-B of the network with terminals A-B open th circuited. • This equivalent resistance R is the resistance obtained at terminals A-B of the network with all its th independent 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 by Léon Charles Thévenin (1857–1926), an electrical engineer with France's national Postes et Télégraphes telecommunications 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 study circuit's initial-condition and steady-state response.[7][8] Thévenin's theorem can be used to convert any circuit's sources and impedances to a Thévenin equivalent; use of the theorem may in some cases be more convenient than use of Kirchhoff's circuit laws.[9][6]

Calculating the Thévenin equivalent

To calculate the equivalent circuit, the resistance and voltage are needed, so two equations are required. These two equations are usually obtained by using the following steps, but any conditions placed on the terminals of the circuit should also work: Any black box containing resistances only and 1. Calculate the output voltage, V , when in open circuit condition AB voltage and current sources can be replaced to a (no load resistor—meaning infinite resistance). This is V . Thévenin equivalent circuit consisting of an Th 2. Calculate the output current, I , when the output terminals are equivalent voltage source in series connection AB with an equivalent resistance. short circuited (load resistance is 0). R equals V divided by this Th Th I . AB The equivalent circuit is a voltage source with voltage V in series with a resistance R . Th Th 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 R . Th The Thévenin-equivalent voltage is the voltage at the output terminals of the original circuit. When calculating a Thévenin-equivalent voltage, the voltage divider principle is often useful, by declaring one terminal to be V and out the 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 voltage source, this means replace the voltage source with a short circuit. For an ideal current source, this means replace the current source with an open circuit. Resistance can then be calculated across the terminals using the formulae for series and parallel circuits. This method is valid only for circuits with independent sources. If there are dependent sources in the circuit, another method must be used such as connecting a test source across A and B and calculating the voltage across or current through the test source.

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] Thévenin's theorem 74

Example

Step 1: Calculating the equivalent output voltage Step 3: The equivalent circuit Step 0: The original circuit Step 2: Calculating the equivalent resistance

In the example, calculating the equivalent voltage:

(notice that R is not taken into consideration, as above calculations are done in an open circuit condition between A 1 and B, therefore no current flows through this part, which means there is no current through R and therefore no 1 voltage drop along this part) Calculating equivalent resistance:

Conversion to a Norton equivalent

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

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] 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 same regardless of how the internal circuit is represented.

A proof of the theorem The proof involves two steps. First use superposition theorem to construct a solution, and then use uniqueness theorem 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 down its 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 term measures the contribution from all the resistors. The above argument is due to the fact that the voltage of the black box for a given current is identical to the linear superposition of the solutions of the following problems: (1) to leave the black box open circuited but activate individual voltage source one at a time and, (2) to short circuit all the voltage 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] Helmholtz [2] Thévenin (1883a) [3] Thévenin (1883b) [4] Johnson (2003a) [5] Brittain [6] Dorf [7] Brenner [8] Elgerd [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 Spectrum

27 (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 for Electrical 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) [email protected] 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 mit Anwendung auf die thierisch-elektrischen Versuche (Some laws concerning the distribution of electrical currents

in 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 law

to complex electromotive circuits)" (http:/ / books. google. com/ ?id=shUAAAAAMAAJ& pg=PA222&

lpg=PA222& dq=Extension+ de+ la+ loi+ d’Ohm+ aux+ circuits+ électromoteurs+ complexes#v=onepage& e q=Extension de la loi d’Ohm aux circuits électromoteurs complexes& f=false). Annales Télégraphiques. 3 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 simplication to first-order low-pass filter and associated voltage divider, time constant and gain.

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] Norton's theorem 77 Norton's theorem

Known 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 at terminals A-B by an equivalent current source I in parallel connection with an equivalent resistance R . NO NO • This equivalent current I is the current obtained at terminals A-B of the network with terminals A-B short NO circuited. • This equivalent resistance R is the resistance obtained at terminals A-B of the network with all its voltage NO sources 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 given frequency. Norton's theorem and its dual, Thévenin's theorem, are widely used for circuit analysis simplification and to study circuit's initial-condition and steady-state response. Norton's theorem was independently derived in 1926 by Siemens & Halske researcher Hans Ferdinand Mayer (1895–1980) and Bell Labs [1][2][3][4][5] engineer Edward Lawry Norton (1898–1983). Any black box containing resistances only and To find the equivalent, voltage and current sources can be replaced by an equivalent circuit consisting of an equivalent 1. Find the Norton current I . Calculate the output current, I , with current source in parallel connection with an No AB a short circuit as the load (meaning 0 resistance between A and B). equivalent resistance. This is I . No 2. Find the Norton resistance R . When there are no dependent sources (all current and voltage sources are No independent), there are two methods of determining the Norton impedance R . No • Calculate the output voltage, V , when in open circuit condition (i.e., no load resistor — meaning infinite AB load resistance). R equals this V divided by I . No AB No 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 R . No 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 shown below 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 R . No This method must be used if the circuit contains dependent sources, but it can be used in all cases even when there are no dependent sources.

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

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

Step 3: The equivalent circuit

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 the following equations:

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] Norton's theorem 79

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

References [1] Mayer [2] Norton [3] Johnson (2003b) [4] Brittain [5] Dorf [6] Johnson (2003a) [7] Gunther [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 Spectrum

27 (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 Development

19 (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 electronic amplifiers]". Telegraphen- und Fernsprech-Technik 15: 335–337. • Norton, E. L. (1926). Technical Report TM26–0–1860 – Design of finite networks for uniform frequency characteristic. 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) [email protected] Maximum power transfer theorem 80 Maximum power transfer theorem

In electrical engineering, the maximum power transfer theorem states that, to obtain maximum external power from a source with a finite internal resistance, the resistance of the load must equal the resistance of the source as viewed 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 made larger than the resistance of the source, then efficiency is higher, since a higher percentage of the source power is transferred 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 the source, and although the total power dissipated is higher, due to a lower total resistance, it turns out that the amount dissipated in the load is reduced. The theorem states how to choose (so as to maximize power transfer) the load resistance, once the source resistance is given. It is a common misconception to apply the theorem in the opposite scenario. It does not say how to choose the source resistance for a given load resistance. In fact, the source resistance that maximizes power transfer is always 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 occurs when the load impedance is equal to the complex conjugate of the source impedance.

Maximizing power transfer versus power efficiency The theorem was originally misunderstood (notably by Joule) to imply that a system consisting of an electric motor driven by a battery could not be more than 50% efficient since, when the impedances were matched, the power lost as heat in the battery would always be equal to the power delivered to the motor. In 1880 this assumption was shown to be false by either Edison or his colleague Francis Robbins Upton, who realized that maximum efficiency was not the same as maximum power transfer. To achieve maximum efficiency, the resistance of the source (whether a battery or a dynamo) could be made close to zero. Using this new understanding, they obtained an efficiency of about 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 as the ratio of power dissipated by the load to power developed by the source, then it is straightforward to calculate from the above circuit diagram that

Consider three particular cases:

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] 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 resistance approaches infinity, though the total power level tends towards zero. Efficiency also approaches 100% if the source resistance approaches zero, and 0% if the load resistance approaches zero. In the latter case, all the power is consumed inside the source (unless the source also has no resistance), so the power dissipated in a short circuit is zero.

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

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 with resistance , resulting in a current . By Ohm's law, is simply the source voltage divided by the total circuit resistance:

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

The value of for which this expression is a maximum could be calculated by differentiating it, but it is easier to calculate 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 the power is therefore a maximum, when

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] 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, the source resistance must be set equal to the load resistance for maximum power transfer. However, this equation only applies if the source resistance cannot be adjusted, e.g., with antennas (see the first line in the proof stating "fixed source resistance"). For any given load resistance a source resistance of zero is the way to transfer maximum power to the load. As an example, a 100 volt source with an internal resistance of 10 ohms connected to a 10 ohm load will deliver 250 to that load. Make the source resistance zero ohms and the load power jumps to 1000 watts.

In reactive circuits The theorem also applies where the source and/or load are not totally resistive. This invokes a refinement of the maximum power theorem, which says that any reactive components of source and load should be of equal magnitude but opposite phase. (See below for a derivation.) This means that the source and load impedances should be complex conjugates of each other. In the case of purely resistive circuits, the two concepts are identical. However, physically realizable sources and loads are not usually totally resistive, having some inductive or capacitive components, and so practical 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 resistive losses, would receive 100% of the energy from the source but send it back after a quarter cycle. The resultant circuit is nothing other than a resonant LC circuit in which the energy continues to oscillate to and fro. This is called reactive power. Power factor correction (where an inductive reactance is used to "balance out" a capacitive one), is essentially 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 by complex 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 by adding a reactance to the load to balance out the load's own reactance, changing the reactive load impedance into a resistive load impedance.

Proof

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

The average power dissipated in the load is the square of the current 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 .

Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] Maximum power transfer theorem 83

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

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 has exactly the same form as in the purely resistive case, and the maximizing condition can be found in the same 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) [email protected] Article Sources and Contributors 84 Article Sources and Contributors

Electrical 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+79 3888, Adamtester, Ahoerstemeier, Alan Liefting, AlanD, Alansohn, Alfred Centauri, Allstarecho, Aloplop, Alphachimp, Alzo55, Amaury, Amigadave, Ams627, Ancheta Wis, Andyjsmith, Anna Frodesiak, 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, Damon jackson, 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, Evil saltine, 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, I already 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, MicroBio Hawk, 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, Steven Weston, StradivariusTV, Supremeknowledge, SutharsanJIsles, Syed Wamiq Ahmed Hashmi, TDogg310, TStein, Teamfruit, Template namespace initialisation script, Texture, The Rambling Man, 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,

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Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] Article Sources and Contributors 85

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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't sleep, clown will eat me, Chrislk02, Clee845, Coasterlover1994, CutOffTies, Daniel.Cardenas, Danmichaelo, Dicklyon, DigitalPhase, Dirkbb, Doczilla, Drilnoth, Drmies, ENeville, Editor at Large, 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 of Yellow, Suppiesman123, Svjo-2, Terraflorin, Texture, Thane, That Guy, From That Show!, The Original Wildbear, The Thing That Should Not Be, The wub, Thehakimboy, Trinibones,

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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, Oleg Alexandrov, 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, Debeo Morium, Dicklyon, Envy0, FatimahMaj, Heron, Highsand, HyperFlexed, Itaharesay, JJ Harrison, Jmorgan, Komal singh b, Lyctc, Miterdale, Mohsinjibran, Mrbobjrsrv, Omegatron, PeteX, Petteri Aimonen, 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., Joe Decker, 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, Sudeep shenoy2007, 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) [email protected] Image Sources, Licenses and Contributors 86 Image Sources, Licenses and Contributors

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Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] Image Sources, Licenses and Contributors 87

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Ramesh Singh, DTU (Formerly Delhi College of Engineering) [email protected] License 88 License

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