Fundamentals of Engineering Electromagnetics - Chapter 4 potx

Some examples include: The electric field of electromagnetic waves e.g., radio waves or light is basically theinduced electric field; In electrical transformers, the induced electric fie

In 1831 Michael Faraday performed experiments to check whether current is produced in

a closed wire loop placed near a magnet, in analogy to dc currents producing magneticfields His experiment showed that this could not be done, but Faraday realized that atime-varying current in the loop was obtained while the magnet was being moved toward it or

induction It is perhaps the most important law of electromagnetism Without it therewould be no electricity from rotating generators, no telephone, no radio and television, nomagnetic memories, to mention but a few applications

The phenomenon of electromagnetic induction has a simple physical interpretation.Two charged particles (‘‘charges’’) at rest act on each other with a force given byCoulomb’s law Two charges moving with uniform velocities act on each other with anadditional force, the magnetic force If a particle is accelerated, there is another additionalforce that it exerts on other charged particles, stationary or moving As in the case of themagnetic force, if only a pair of charges is considered, this additional force is much smallerthan Coulomb’s force However, time-varying currents in conductors involve a vastnumber of accelerated charges, and produce effects significant enough to be easilymeasurable

This additional force is of the same form as the electric force (F ¼ QE) However,other properties of the electric field vector, E in this case, are different from those of the

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Deceased

McGill University, Montre´al, Quebec

University of Colorado, Boulder, Colorado

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electric field vector of static charges When we wish to stress this difference, we use aslightly different name: the induced electric field strength.

The induced electric field and electromagnetic induction have immense practicalconsequences Some examples include:

The electric field of electromagnetic waves (e.g., radio waves or light) is basically theinduced electric field;

In electrical transformers, the induced electric field is responsible for obtaininghigher or lower voltage than the input voltage;

The skin effect in conductors with ac currents is due to induced electric field;Electromagnetic induction is also the cause of ‘‘magnetic coupling’’ that may result

in undesired interference between wires (or metal traces) in any system withtime-varying current, an effect that increases with frequency

The goal of this chapter is to present:

Fundamental theoretical foundations for electromagnetic induction, most tantly Faraday’s law;

impor-Important consequences of electromagnetic induction, such as Lentz’s law and theskin effect;

Some simple and commonly encountered examples, such as calculation of theinductance of a solenoid and coaxial cable;

A few common applications, such as generators, transformers, electromagnets, etc

The practical sources of the induced electric field are time-varying currents in a broadersense If we have, for example, a stationary and rigid wire loop with a time-varyingcurrent, it produces an induced electric field However, a wire loop that changes shapeand/or is moving, carrying a time-constant current, also produces a time-varying current inspace and therefore induces an electric field Currents equivalent to Ampe`re’s currents in

a moving magnet have the same effect and therefore also produce an induced electric field.Note that in both of these cases there exists, in addition, a time-varying magneticfield Consequently, a time-varying (induced) electric field is always accompanied by

a time-varying magnetic field, and conversely, a time-varying magnetic field is alwaysaccompanied by a time-varying (induced) electric field

components of the electric field differ in the work done by the field in moving a pointcharge around a closed contour For the static electric field this work is always zero, butfor the induced electric field it is not Precisely this property of the induced electric fieldgives rise to a very wide range of consequences and applications Of course, a charge can

be situated simultaneously in both a static (Coulomb-type) and an induced field, thusbeing subjected to a total force

We know how to calculate the static electric field of a given distribution of charges,but how can we determine the induced electric field strength? When a charged particle

is moving with a velocity v with respect to the source of the magnetic field, the answerfollows from the magnetic force on the charge:

ð

V

J  dVr

In this equation, r is the distance of the point where the induced electric field is beingdetermined from the volume element dV In the case of currents over surfaces, JðtÞ  dV

If we know the distribution of time-varying currents, Eq (4.3) enables thedetermination of the induced electric field at any point of interest Most often it is notpossible to obtain the induced electric field strength in analytical form, but it can always

be evaluated numerically

Faraday’s law is an equation for the total electromotive force (emf ) induced in a closedloop due to the induced electric field This electromotive force is distributed along the loop(not concentrated at a single point of the loop), but we are rarely interested in thisdistribution Thus, Faraday’s law gives us what is relevant only from the circuit-theorypoint of view—the emf of the Thevenin generator equivalent to all the elementalgenerators acting in the loop

Consider a closed conductive contour C, either moving arbitrarily in a time-constantmagnetic field or stationary with respect to a system of time-varying currents producing aninduced electric field If the wire segments are moving in a magnetic field, there is aninduced field acting along them of the form in Eq (4.2), and if stationary, the inducedelectric field is given in Eq (4.3) In both cases, a segment of the wire loop behaves as anelemental generator of an emf

It can be shown that, whatever the cause of the induced electric field (the contourmotion, time-varying currents, or the combination of the two), the total emf induced in thecontour can be expressed in terms of time variation of the magnetic flux through thecontour:

The possibility of expressing the induced emf in terms of the magnetic flux alone is notsurprising We know that the induced electric field is always accompanied by a magneticfield, and the above equation only reflects the relationship that exists between the twofields (although the relationship itself is not seen from the equation) Finally, this equation

is valid only if the time variation of the magnetic flux through the contour is due either

to motion of the contour in the magnetic field or to time variation of the magnetic field

in which the contour is situated (or a combination of the two) No other cause of timevariation of the magnetic flux will result in an induced emf

Electric and Magnetic Field

The voltage between two points is defined as the line integral of the total electric fieldstrength, given in Eq (4.1), from one point to the other In electrostatics, the inducedelectric field does not exist, and voltage does not depend on the path between these points.This is not the case in a time-varying electric and magnetic field

Consider arbitrary time-varying currents and charges producing a time-varyingelectric and magnetic field, Fig 4.1 Consider two points, A and B, in this field, and twopaths, a and b, between them, as indicated in the figure The voltage between these twopoints along the two paths is given by

The integral between A and B of the static part is simply the potential difference between

VAB along a or b¼VAVBþ

ðB

Aalong a or b

the integral in this equation is different for paths a and b These paths form a closedcontour Applying Faraday’s law to that contour, we have

einduced in closed contour AaBbA ¼

along a and along b are different Consequently, the voltage between two points in a varying electric and magnetic field depends on the choice of integration path between thesetwo points

time-This is a very important practical conclusion for time-varying electrical circuits

It implies that, contrary to circuit theory, the voltage measured across a circuit by

a voltmeter depends on the shape of the leads connected to the voltmeter terminals Sincethe measured voltage depends on the rate of change of magnetic flux through the surfacedefined by the voltmeter leads and the circuit, this effect is particularly pronounced at highfrequencies

A time-varying current in one current loop induces an emf in another loop In linearmedia, an electromagnetic parameter that enables simple determination of this emf is themutual inductance

A wire loop with time-varying current creates a time-varying induced electricfield not only in the space around it but also along the loop itself As a consequence,there is a feedback—the current produces an effect which affects itself The parameterknown as inductance, or self-inductance, of the loop enables simple evaluation of thiseffect

1

the first contour, it creates a time-varying magnetic field, as well as a time-varying

where the first index denotes the source of the field (contour 1 in this case)

It is usually much easier to find the induced emf using Faraday’s law than in anyother way The magnetic flux density vector in linear media is proportional to the current

that causes the magnetic field It follows that the flux 12ðtÞ through C2 caused by thecurrent i1ðtÞin C1 is also proportional to i1ðtÞ:

constant depends only on the geometry of the system and the properties of the (linear)

sometimes in circuit theory by M

expression for the induced electric field in Eqs (4.3) and (4.5).] So, we can write

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inductance, which is a useful result since in some instances one of these is much simpler tocalculate than the other

Figure 4.2 Two coupled conductive contours

Note that mutual inductance can be negative as well as positive The sign depends onthe actual geometry of the system and the adopted reference directions along the twoloops: if the current in the reference direction of one loop produces a positive flux in theother loop, then mutual inductance is positive, and vice versa For calculating the flux,the normal to the loop surface is determined by the right-hand rule with respect to itsreference direction.

As mentioned, when a current in a contour varies in time, the induced electricfield exists everywhere around it and therefore also along its entire length Consequently,there is an induced emf in the contour itself This process is known as self-induction.The simplest (even if possibly not physically the clearest) way of expressing this emf is touse Faraday’s law:

eðtÞ ¼ dselfðtÞ

If the contour is in a linear medium (i.e., the flux through the contour is proportional

through the contour due to current iðtÞ in it and iðtÞ,

The constant L depends only on the geometry of the system, and its unit is again

a henry (H) In the case of a dc current, L ¼ =I , which can be used for determining theself-inductance in some cases in a simple manner

The self-inductances of two contours and their mutual inductance satisfy thefollowing condition:

The dimensionless coefficient k is called the coupling coefficient

There are many devices that make use of electric or magnetic forces Although this is notcommonly thought of, almost any such device can be made in an ‘‘electric version’’ and in

a ‘‘magnetic version.’’ We shall see that the magnetic forces are several orders ofmagnitude stronger than electric forces Consequently, devices based on magnetic forces

are much smaller in size, and are used more often when force is required For example,electric motors in your household and in industry, large cranes for lifting ferromagneticobjects, home bells, electromagnetic relays, etc., all use magnetic, not electric, forces.

A powerful method for determining magnetic forces is based on energy contained inthe magnetic field While establishing a dc current, the current through a contour has tochange from zero to its final dc value During this process, there is a changing magneticflux through the contour due to the changing current, and an emf is induced in theestablish the final static magnetic field, the sources have to overcome this emf, i.e., to spendsome energy A part (or all) of this energy is stored in the magnetic field and is known asmagnetic energy

Let n contours, with currents i1ðtÞ, i2ðtÞ, , inðtÞbe the sources of a magnetic field.Assume that the contours are connected to generators of electromotive forces

e1ðtÞ, e2ðtÞ, , enðtÞ Finally, let the contours be stationary and rigid (i.e., they cannot

contained in the magnetic field of such currents is

some value B, the volume density of magnetic energy is given by

The energy in a linear medium can now be found by integrating this expression overthe entire volume of the field:

In many cases, however, this is quite complicated

The magnetic force can also be evaluated as a derivative of the magnetic energy Thiscan be done assuming either (1) the fluxes through all the contours are kept constant or (2)the currents in all the contours are kept constant In some instances this enables verysimple evaluation of magnetic forces

Assume first that during a displacement dx of a body in the magnetic field along the

the currents in the contours appropriately The x component of the magnetic force acting

on the body is then obtained as

The signs in the two expressions for the force determine the direction of the force

In Eq (4.28), the positive sign means that when current sources are producing all thecurrents in the system (I ¼ const), the magnetic field energy increases, as the generators arethe ones that add energy to the system and produce the force

to Eq (4.3), lines of the induced electric field around the loop are circles centered at theloop axis normal to it, so that the line integral of the induced electric field around a

a wire loop, this field acts as a distributed generator along the entire loop length, and acurrent is induced in that loop

reached an extremely important conclusion: The induced electric field of time-varyingcurrents in one wire loop produces a time-varying current in an adjacent closed wire loop.Note that the other loop need not (and usually does not) have any physical contact withthe first loop This means that the induced electric field enables transport of energy fromone loop to the other through vacuum Although this coupling is actually obtained bymeans of the induced electric field, it is known as magnetic coupling

Note that if the wire loop C2 is not closed, the induced field nevertheless inducesdistributed generators along it The loop behaves as an open-circuited equivalent(The´venin) generator.

Figure 4.4 shows a permanent magnet approaching a stationary loop The permanentmagnet is equivalent to a system of macroscopic currents Since it is moving, the magneticflux created by these currents through the contour varies in time According to thereference direction of the contour shown in the figure, the change of flux is positive,

a current through the closed loop, which in turn produces its own magnetic field, shown inthe figure in dashed line As a result, the change of the magnetic flux, caused initially by themagnet motion, is reduced This is Lentz’s law: the induced current in a conductivecontour tends to decrease the change in magnetic flux through the contour Lentz’s lawdescribes a feedback property of electromagnetic induction

Figure 4.3 A circular loop C1with a time-varying current iðtÞ The induced electric field of thiscurrent is tangential to the circular loop C2indicated in dashed line, so that it results in a distributedemf around the loop

Figure 4.4 Illustration of Lentz’s law

4.3.3 Eddy Currents

A very important consequence of the induced electric field are eddy currents These arecurrents induced throughout a solid metal body when the body is situated in a time-varying magnetic (i.e., induced electric) field

As the first consequence of eddy currents, there is power lost to heat according

to Joule’s law Since the magnitude of eddy currents is proportional to the magnitude

of the induced electric field, eddy-current losses are proportional to the square offrequency

As the second consequence, there is a secondary magnetic field due to the inducedcurrents which, following Lentz’s law, reduces the magnetic field inside the body Both

of these effects are usually not desirable For example, in a ferromagnetic core shown inFig 4.5, Lentz’s law tells us that eddy currents tend to decrease the flux in the core, and themagnetic circuit of the core will not be used efficiently The flux density vector is thesmallest at the center of the core, because there the B field of all the induced currents adds

up The total magnetic field distribution in the core is thus nonuniform

To reduce these two undesirable effects, ferromagnetic cores are made of mutuallymuch smaller loops, the emf induced in these loops is consequently much smaller, and sothe eddy currents are also reduced significantly Of course, this only works if vector B isparallel to the sheets

In some instances, eddy currents are created on purpose For example, in inductionfurnaces for melting metals, eddy currents are used to heat solid metal pieces to meltingtemperatures

A time-invariant current in a homogeneous cylindrical conductor is distributed uniformlyover the conductor cross section If the conductor is not cylindrical, the time-invariantcurrent in it is not distributed uniformly, but it exists in the entire conductor A time-varying current has a tendency to concentrate near the surfaces of conductors At veryhigh frequencies, the current is restricted to a very thin layer near the conductor surface,practically on the surfaces themselves Because of this extreme case, the entirephenomenon of nonuniform distribution of time-varying currents in conductors isknown as the skin effect

Figure 4.5 Eddy currents in a piece of ferromagnetic core Note that the total B field in the core isreduced due to the opposite field created by eddy currents

The cause of skin effect is electromagnetic induction A time-varying magnetic field isaccompanied by a time-varying induced electric field, which in turn creates secondarytime-varying currents (induced currents) and a secondary magnetic field The inducedcurrents produce a magnetic flux which opposes the external flux (the same flux that

‘‘produced’’ the induced currents) As a consequence, the total flux is reduced The largerthe conductivity, the larger the induced currents are, and the larger the permeability, themore pronounced the flux reduction is Consequently, both the total time-varyingmagnetic field and induced currents inside conductors are reduced when compared withthe dc case

The skin effect is of considerable practical importance For example, at very highfrequencies a very thin layer of conductor carries most of the current Any conductor (orfor that matter, any other material), can be coated with silver (the best availableconductor) and practically the entire current will flow through this thin silver coating.Even at power frequencies in the case of high currents, the use of thick solid conductors isnot efficient, and bundled conductors are used instead

The skin effect exists in all conductors, but, as mentioned, the tendency of current andmagnetic flux to be restricted to a thin layer on the conductor surface is much morepronounced for a ferromagnetic conductor than for a nonferromagnetic conductor of thesame conductivity For example, for iron at 60 Hz the thickness of this layer is on the order

Figure 4.6 A ferromagnetic core for transformers and ac machines consists of thin insulatedsheets: (a) sketch of core and (b) photograph of a typical transformer core

of only 0.5 mm Consequently, solid ferromagnetic cores for alternating current electricmotors, generators, transformers, etc., would result in poor use of the ferromagneticmaterial and high losses Therefore, laminated cores made of thin, mutually insulatedsheets are used instead At very high frequencies, ferrites (ferrimagnetic ceramic materials)are used, because they have very low conductivity when compared to metallicferromagnetic materials.

Consider a body with a sinusoidal current of angular frequency ! and let thematerial of the body have a conductivity  and permeability  If the frequency is highenough, the current will be distributed over a very thin layer over the body surface, thecurrent density being maximal at the surface (and parallel to it), and decreasing rapidlywith the distance z from it:

The intensity of the current density vector decreases exponentially with increasing z At

at the boundary surface This distance is known as the skin depth For example, for

in some common materials at a few characteristic frequencies

The result for skin depth for iron at power frequencies (50 Hz or 60 Hz), ffi 5 mm,tells us something important Iron has a conductivity that is only about six times lessthan that of copper On the other hand, copper is much more expensive than iron Why

do we then not use iron wires for the distribution of electric power in our homes? Notingthat there are millions of kilometers of such wires, the savings would be very large.Unfortunately, due to a large relative permeability—iron has very small power-frequencyskin depth (a fraction of a millimeter)—the losses in iron wire are large, outweighing thesavings, so copper or aluminum are used instead

Keeping the current intensity the same, Joule losses increase with frequency due toincreased resistance in conductors resulting from the skin effect It can be shown thatJoule’s losses per unit area are given by

where H0 is the complex rms value of the tangential component of the vector H on the

to the current in the inner conductor This is an example of the proximity effect If

in addition there is normal cable current in the outer conductor, it is the same but opposite

to the current on the conductor outer surface, so the two cancel out We are left with

a current over the inner conductor and a current over the inside surface of the outerconductor This combination of the skin and proximity effects is what is usually actuallyencountered in practice

Redistribution of Current on Parallel Wires and Printed Traces

Consider as the next example three long parallel wires a certain distance apart lying inone plane The three ends are connected together at one and at the other end of the wires,and these common ends are connected by a large loop to a generator of sinusoidal emf.Are the currents in the three wires the same? At first glance we should expect them to

be the same, but due to the induced electric field they are not: the current intensity in themiddle wire will always be smaller than in the other two

The above example is useful for understanding the distribution of ac current acrossthe cross section of a printed metal strip, such as a trace on a printed-circuit board Thedistribution of current across the strip will not be uniform (which it is at zero frequency).The current amplitude will be much greater along the strip edges than along its center.This effect is sometimes referred to as the edge effect, but it is, in fact, the skin effect in stripconductors Note that for a strip line (consisting of two close parallel strips) this effect isvery small because the induced electric fields due to opposite currents in the two stripspractically cancel out

Circuit theory is the basic tool of electrical engineers, but it is approximate and thereforehas limitations These limitations can be understood only using electromagnetic-fieldtheory We consider here the approximations implicit in Kirchhoff ’s voltage law (KVL).This law states that the sum of voltages across circuit branches along any closed path iszero and that voltages and currents in circuit branches do not depend on the circuit actualgeometrical shape Basically, this means that this law neglects the induced electric field

produced by currents in the circuit branches This field increases with frequency, sothat at a certain frequency (depending on circuit properties and its actual size) theinfluence of the induced electric field on circuit behavior becomes of the same order ofmagnitude as that due to generators in the circuit The analysis of circuit behavior in suchcases needs to be performed by electromagnetic analysis, usually requiring numericalsolutions.

As a simple example, consider the circuit in Fig 4.7, consisting of several printedtraces and two lumped (pointlike, or much smaller than a wavelength) surface-mountcomponents For a simple two-loop circuit 10 cm
20 cm in size, already at a frequency of

10 MHz circuit analysis gives results with errors exceeding 20% The tabulated values

in Fig 4.7 show the calculated and measured complex impedance seen by the generator

at different frequencies

Several useful practical conclusions can be drawn The first is that for circuits thatcontain wires or traces and low-valued resistors, this effect will become pronounced atlower frequencies The second is that the behavior of an ac circuit always depends on thecircuit shape, although in some cases this effect might be negligible (A completeelectromagnetic numerical solution of this circuit would give exact agreement with theory.)This directly applies to measurements of ac voltages (and currents), since the leads of themeter are also a part of the circuit Sometimes, there is an emf induced in the meter leadsdue to flux through loops formed by parts of the circuit and the leads This can lead toerrors in voltage measurements, and the loops that give rise to the error emf are oftenreferred to as ground loops

Some substances have zero resistivity at very low temperatures For example, lead haszero resistivity below about 7.3 K (just a little bit warmer than liquid helium) Thisphenomenon is known as superconductivity, and such conductors are said to be

Frequency Calculated Re(Z) Measured Re(Z) Calculated Im(Z) Measured Im(Z)

superconductors Some ceramic materials (e.g., yttrium barium oxide) become conductors at temperatures as ‘‘high’’ as about 70 K (corresponding to the temperature ofliquid nitrogen) Superconducting loops have an interesting property when placed in atime-varying magnetic field The Kirchhoff voltage law for such a loop has the form

since the emf in the loop is d=dt and the loop has zero resistance From this equation,

it is seen that the flux through a superconducting loop remains constant Thus, it isnot possible to change the magnetic flux through a superconducting loop by means

of electromagnetic induction The physical meaning of this behavior is the following:

If a superconducting loop is situated in a time-varying induced electric field, thecurrent induced in the loop must vary in time so as to produce exactly the same inducedelectric field in the loop, but in the opposite direction If this were not so, infinite currentwould result

AND FARADAY’S LAW

An ac generator, such as the one sketched in Fig 4.8, can be explained using Faraday’slaw A rectangular wire loop is rotating in a uniform magnetic field (for example, betweenthe poles of a magnet) We can measure the induced voltage in the wire by connecting

The loop is rotating about this axis with an angular velocity ! If we assume that at t ¼ 0vector B is parallel to vector n normal to the surface of the loop, the induced emf in theloop is given by

In practice, the coil has many turns of wire instead of a single loop, to obtain a largerinduced emf Also, usually the coil is not rotating, but instead the magnetic field is rotatingaround it, which avoids sliding contacts of the generator

Figure 4.8 A simple ac generator

4.4.2 Induction Motors

Motors transform electric to mechanical power through interaction of magnetic fluxand electric current [1,20,26] Electric motors are broadly categorized as ac and dc motors,with a number of subclassifications in each category This section describes the basicoperation of induction motors, which are most often encountered in industrial use.The principles of the polyphase induction motor are here explained on the example

of the most commonly used three-phase version In essence, an induction motor is

a transformer Its magnetic circuit is separated by an air gap into two portions Thefixed stator carries the primary winding, and the movable rotor the secondary winding,

as shown in Fig 4.9a An electric power system supplies alternating current to the primary

Figure 4.9 (a) Cross section of a three-phase induction motor 1-10, 2-20, and 3-30 mark theprimary stator windings, which are connected to an external three-phase power supply (b) Time-domain waveforms in the windings of the stator and resulting magnetic field vector rotation as afunction of time

winding, which induces currents in the secondary (short-circuited or closed through

an external impedance) and thus causes the motion of the rotor The key distinguishablefeature of this machine with respect to other motors is that the current in the secondary

is produced only by electromagnetic induction, i.e., not by an external power source.The primary windings are supplied by a three-phase system currents, which producethree stationary alternating magnetic fields Their superposition yields a sinusoidallydistributed magnetic field in the air gap of the stator, revolving synchronously with thepower-supply frequency The field completes one revolution in one cycle of the statorcurrents with the shown angular arrangement in the stator, results in a rotating magneticfield with a constant magnitude and a mechanical angular speed that depends on thefrequency of the electric supply

Two main types of induction motors differ in the configuration of the secondarywindings In squirrel-cage motors, the secondary windings of the rotor are constructedfrom conductor bars, which are short-circuited by end rings In the wound-rotor motors,the secondary consists of windings of discrete conductors with the same number of poles

as in the primary stator windings

The velocity of flowing liquids that have a small, but finite, conductivity can be measuredusing electromagnetic induction In Fig 4.10, the liquid is flowing through a flat insulatingpipe with an unknown velocity v The velocity of the fluid is roughly uniform over thecross section of the pipe To measure the fluid velocity, the pipe is in a magnetic field with

a flux density vector B normal to the pipe Two small electrodes are in contact with thefluid at the two ends of the pipe cross section A voltmeter with large input impedanceshows a voltage V when connected to the electrodes The velocity of the fluid is then given

and angular frequency ! flowing through it The conductor is encircled by a flexible thin

per unit length We show that if we measure the amplitude of the voltage between the

Figure 4.10 Measurement of fluid velocity

There are dN ¼ N0dl turns of wire on a length dl of the strip The magnetic flux

As an example of the measurement of ac voltage, consider a straight copper wire of radius

points 1 and 2, with leads as shown in Fig 4.12 If b ¼ 50 cm and c ¼ 20 cm, we will

Figure 4.11 A method for measuring ac current in a conductor without inserting an ammeter intothe circuit

Figure 4.12 Measurement of ac voltage

and (3) ! ¼ 106rad=s We assume that the resistance of the copper conductor per unit

The voltage measured by the voltmeter (i.e., the voltage between its very terminals,and not between points 1 and 2) is

Vvoltmeter¼ ðV1V2Þ e ¼ R0bi  e ð4:37Þ

contour containing the voltmeter and the wire segment between points 1 and 2 (we neglectthe size of the voltmeter) This emf is approximately given by

depend on frequency The difference between this potential difference and the voltageindicated by the voltmeter for the three specified frequencies is (1) 117.8 mV, (2) 3.74 mV,and (3) 3.74 V This difference represents an error in measuring the potential differenceusing the voltmeter with such leads We see that in case (2) the relative error is as large as189%, and that in case (3) such a measurement is meaningless

When a magnetized disk with small permanent magnets (created in the writing process)moves in the vicinity of the air gap of a magnetic head, it will produce time-variable flux inthe head magnetic core and the read-and-write coil wound around the core As a result, an

positive and negative pulses This is sketched in Fig 4.13 (For the description of the

Figure 4.13 A hard disk magnetized through the write process induces a emf in the read process:when the recorded magnetic domains change from south to north pole or vice versa, a voltage pulseproportional to the remanent magnetic flux density is produced The pulse can be negative orpositive

Figure 4. 9 (a) Cross section of a three-phase induction motor 1-1 0, 2-2 0, and 3-3 0... sliding contacts of the generator

Figure 4. 8 A simple ac generator

4. 4.2 Induction Motors

Motors... class="text_page_counter">Trang 14< /span>

where H0 is the complex rms value of the tangential component of the vector H on the

to