Inductance And Inductors

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Inductance and inductorsiSCConsider a short circuit that is carrying acurrent iSC — we have seen shorts often andknow that the no voltage developed,regardless of current level.However, we also recall from basicelectromagnetic that a current in a wire willhave a magnetic field circulating around it.The field pattern is circular and the strengthof the field diminishes with increasingdistance from the wire.μoiscB 2πrWhenever charges are moving in a conductor, there will be magneticfields present. Since is takes some energy to generate the field, it canviewed as having some energy stored in it.In a typical wire in a typical EE 201 circuit, the magnetic field andassociated energy are very small and can probably be neglected in mostinstances.EE 201inductors – 1

Magnetic fields — a very short reviewMagnetic fields are characterized by two different field quantities,H magnetic field strength, units of amperes /meter (A/m)B magnetic flux density, units of tesla (T) [ V·s /m2 }The two are related and in common materials, either can be used as theprimary description of a magnetic field. In general, both are vectorquantities. (We will ignore the vector nature of the magnetic fields inour simplified discussion of inductors.) The relationship between B andH is given by the constitutive equation:B μHEE 201where µ is the permeability of the material where the magnetic fieldexists. The units of permeability are henries/meter (H/m). The freespace value of permeability is µo 4π 10–7 H/m. Most materials havethe free-space value of permeability, but some materials — usuallyalloys that contain iron, nickel, cobalt, or gadolinium — have higherpermeabilities. A higher permeability is characterized by a relativepermeability, µr, which is the factor by which the permeability isincreased over the free-space value.inductors –2

Magnetic flux densityIn discussing inductors, we will focus on B, the magnetic flux density.Visualizing EM fields and making sense of the different quantities isalways difficult. To develop a more intuitive feel for B, we can thinkabout the field in terms of field lines. The magnetic flux density indicateshow tightly packed the magnetic field lines are in a region of space.The more tightly packed they are (i.e. the higher the density), thestronger the field.B1B2 B1B3 B2end viewsEE 201inductors – 3

Magnetic fluxGiven a magnetic flux density, we can define an area in the spacewhere the magnetic field exists. Within that area, there will be somenumber of magnetic field lines. The number within the area is calledthe magnetic flux.ΦB B AΦB1Units of webers, Wb, ( V·s)ΦB2 ΦB1ΦB3 ΦB2ΦB1 ΦB2 !!EE 201inductors – 4

Faraday’s Law of InductionConsider a magnetic field with magnetic flux density, B. A loopof wire defines an area A that has lines of magnetic field crossingthrough it. The loop defines a magnetic flux.ΦB B A –vFFaraday discovered that if a wire loop has magnetic field lines cutting throughi — defining a magnetic flux in within the loop and if the magnetic flux ischanging, there will be voltage a induced across the terminals of the wire.(More rigorously, an electric field is produced between the open ends of thewire. Of course, we know that if there is an electric field, there will be acorresponding voltage difference.)dΦBvF dtThe negative sign comes from “Lenz’s Law”, whichsays that the magnetic field created by the current inthe loop caused by the induced vF must be oppositethe original magnetic field.The induced voltage can be the result of the magnetic field itself changing (aswith inductors — to be seen shortly) or by having the area change by movingthe loop within the field (as in the case of electric generators). Or both.EE 201inductors – 5

CoilWrapping N turns of wire around a core creates a coil. If the turns of thecoil are packed tightly together, the magnetic fields curling around eachindividual wire will merge to create an approximately uniform magneticfield extending down the length of the interior core. The interiormagnetic flux density will beμ NBcore icoil( l )where l is the length of the core, µ is permeability of the core materialand icoil is the current in wire.If the core has a cross-sectional area of A, the corresponding magneticflux isμN 2 AΦB N Bcore A icoil.( l )(The N turns increases the effective area intersecting the magnetic fluxdensity.)EE 201inductors – 6

InductanceμN 2 AΦB icoil( l )The magnetic flux is directly proportional to thecurrent. We can define the proportionality constantas the self-inductance.ΦB L icoilμN 2 Awhere L .lLInductor circuitsymbol.For a given amount of current, the field can be increased by: increasingthe number of turns, increasing the diameter of the coil (increasing A),decreasing the length of the coil, or by having the core made of amaterial with higher permeability.The units of inductance are henries or H (after American scientist JosephHenry.) 1 H 1 V·s/A. Typical values used in circuits range from about1 µH upwards to as much 0.1 H. (In power systems, much biggerinductors are possible.)EE 201inductors – 7

Inductor voltageA straight-forward application of Faraday’s Law tothe magnetic flux of an inductor gives:ΦB L icoildΦBdiL vL Ldtdtleading to the all-important current-voltage relationship for an inductor:diLvL LdtA consideration of Lenz’s Law leads to the indicated relationshipbetween current direction and voltage polarity — identical torelationship in a resistor or capacitor.iL vL–If the current is not changing, then there is no voltage and the inductorbehaves like a simple (if somewhat lengthy) short circuit. When thecurrent is changing, the changing field causes a voltage to bedeveloped that is proportional to the time derivative of the current.Conversely, if there is a voltage on the inductor, then current must bechanging. The current can be calculated by turning the aboveequation around:EE 2011 tiL (t) iL (0) vL (τ) dτL 0inductors – 8

ExampleiL ILLvL5A4VvL–1At1 ms 2 msFor t 1ms, diL /dt 0 vL 0.0t1 ms 2 msFor 1 ms t 2 ms, diL /dt 4000 A/s vL (0.001 H)(4000 A/s) 4 V.For t 2 ms, diL /dt 0 vL 0.Due to the derivative in theinductor equation, the currentcannot change instantaneously.This would require an infinitelylarge voltage and an infinitely largeamount of power.iL5A1AdiL dtvL !tHowever, the inductor voltage can change instantaneously, — the slopechange abruptly, as long as the current is continuous.EE 201inductors – 9

Inductor energyThe induced magnetic field requires energy in order for it to build up —the magnetic field represents energy stored in the inductor.To determine the stored energy, start with power. When the inductor is“amping up”, the power isdiLPL (t) iL (t) vL (t) LiLdtThe change in energy due to a change in current can be found byintegrating the power over time.iL(t)iL(t)1ΔE PL (t) dt iLdiL L [iL2 (t) iL2 (0)] i (0) i (0)2LLIf we choose iL(0) 0, which corresponds to no stored magnetic energy,the inductor energy corresponding to given a current can written as1 2EL LiL2EE 201Typical value: For L 1 mH and iL 10 A, E 0.05 J — not a whole lot.inductors – 10

Series and parallel inductor combinationsParallelieqLeqis i1 i2 i3 iL1L1iL2 L2 iL3 L3 vL– SeriesiL vL1 – LeqL1veq–L2L3– vL3 EE 201 vL2– ( )Inductor combinations are identicalto those for resistors.inductors – 11

EE 201 inductors – 2 Magnetic fields — a very short review Magnetic fields are characterized by two different field quantities, H magnetic field strength, units of amperes /meter (A/m) B magnetic flux density,

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