Introduction to Electrical MachinesErkan Mee

Coulombs Law

Remember Like charges repel one anotherOpposite charges attract one anotherThe force of repulsion/attraction get weaker as the charges are farther apart.

Charges and ForcesIn air, e= 8.85 x 10-12 Fm-1|| = 1, Fa = -Fb

Unit vector r?

These are all unit vectors, |i| = 1They have a direction, and a magnitude of 1 adds direction to a quantity without changing its magnitudee.g.... speed = 100m/s is a speed S100(1/2, 1/2, 0)m/s is a velocity v =S , 100m/s, North-East () = (1/2, 1/2, 0) in this case.

Charges and FieldsFa =+QaEbFb =+QbEaEb(r) is the electric field set up by charge b at distance r (point a)Ea(r) is the electric field set up by charge a at distance r (point b)

Charges and FieldsE = -V/dF = +q(-V/d)F = qE againWhere E is the field set up inside the capacitor

Charges and FieldsVE = -V/d

Several Charges?EaEbEcEdEe

Several Charges?EaEbEcEdEeETOTETOT

Charge Density : 3D3Dr(r) in C/mm31mm3 = r C

r(ra) > r(rb)

Charge Density : 2D2Dr(r) in C/mm21mm2 = r Cr(ra) > r(rb)

Charge Density : 1D1Dr(r) in C/mm1mm = r C

r(ra) > r(rb)

Gausss Law

Gausss Law : Crude AnalogyTry to measure the rain on a rainy dayMethod 1 : count the raindrops as they fall, and add them upcf Coulombs LawMethod 2 : Hold up an umbrella (a surface) and see how wet it gets.cf Gausss LawMethod 1 is a divide and-conquer or microscopic approachMethod 2 is a more gross or macroscopic approachThey must give the same answer.

Electric Field LinesThese are all correct as E-field lines are simply cartoonsFor now, adopt a drawing scheme such that 1C = 1 E-line.

Lines of Electric FieldHow many field lines cross out of the circle?8C 8 lines16C 16 lines32C 32 lines

Gausss Law : Cartoon VersionThe number of electric field lines leaving a closed surface is equal to the charge enclosed by that surfaceS(E-field-lines) a Charge EnclosedN Coulombs aN lines

Gausss Law Proper (L)S(E-lines) proportional to (Charge Enclosed)D = ED.ds = r(r)dv = r(r)dxdydzD.ds = charge enclosed= 0 = 8.85 x 10-12 in a vacuum

Digression/RevisionArea IntegralsThis area gets wetter!

Area Integrals whats happening?This area gets wetter!

Area Integrals whats happening?Clearly, as the areas are the same, the angle between the area and the rainfall matters

Area Integrals whats happening?dsdsExtreme cases at 180 - maximum rainfall at 90, no rainfall

Flux of rain (rainfall) through an area dsFluxrain = R.ds|R||ds|cos(q)Rds cos(q)Fluxrain = 0 for 90 cos(q) = 0Fluxrain = -Rds for 180 cos(q) = -1Generally, Fluxrain = Rds cos(q)-1 < cos(q) < +1

Potential

Potential Start Simply VE = -V/d

E=-(rate of change of V with distance)Remember the capacitor

E = -V/dShould really be E = -dV/dxAnd if V = Mx+c, dV/dx = M = constantThen E = -M as shownIn 3D, dV/dx becomes (dV/dx, dV/dy, dV/dz) = V, soE = -V = -(dV/dx, dV/dy, dV/dz)

E = -V

Potential : AnalogyThese contour lines are lines of equal gravitational potential energy mghWhere they are close together, the effect of the gravitational field is strongThe field acts in a direction perpendicular to the countours and it points in a negative direction (i.e. thats the way you will fall!)

Potential - commentsWalking around a contour expends no energyIn a perfect worldi.e. no-one moves the hill as you walk!Walking to the top of the hill and back again expends no energyIn a perfect worldi.e. the hill stays still and you recoup the energy you expend while climbing as you descend (using your internal generator!)

Electric Fields and Potentials are the SameVoltage contours

Potential Difference : Formal Definition (L)a xb xThe Potential Difference (Voltage) between a and b is the the work done to move a 1C charge from a to b

Potential Difference : Formal Definition (L)The Potential Difference (Voltage) between a and b is the the work done to move a 1C charge from a to bIn 1D, Work = -FdIn 3D, Work = -F.dlForce = F = QE =+1E = EWork done = -E.dlTotal Work done = -abE.dl

Line integral revision

E

Potential Difference = -abE.dl

ab is a line integralIn general mathematics, the value of a line integral depends upon the path dl takes from a to bIn this potential calculation, the path does not matterSo : choose a convenient path

Potential Difference : Worked Example Point charge Q(b)(a)Place a 1C charge at (a)Move it to (b)Work done in this movement is the potential difference (voltage) between (a) and (b)

Capacitance

Some Capacitorsinsulatorconductor

Capacitance : DefinitionTake two chunks of conductorSeparated by insulatorApply a potential V between themCharge will appear on the conductors, with Q+ = +CV on the higher-potential and Q- = -CV on the lower potential conductorC depends upon both the geometry and the nature of the material that is the insulatorQ+ = +CVQ- = -CV

Magnetic Fields

The Story so FarMaxwells 1st Equation Maxwells 2nd Equation or or

What creates a magnetic field?

What else creates amagnetic field B?Stationary charge no B-fieldStationary charge no B-fieldMoving charge non-zero B-field

Current = Moving Charges

Direction of B, H fields?Right hand : thumb = current,fingers = B-fieldCC

Magnitude of B, H fields?Take an (infinitesimally small) piece of wirePass a current I through itThe magnitude of the ring of field directly around it is given by dB = moIdl 4pr2So, for example, B1>B2>B3

If only it were that simple Unfortunately, dB = moIdl 4pr2 is a special caseThe element Idl creates B-fields elsewhere (i.e. everywhere) as shown and, for example, B4

The Biot-Savart LawLJ

x

Worked Example of Biot-Savart Law : Infinite Line of CurrentdBxB

Worked Example of Biot-Savart Law : Infinite Line of CurrentdBIdlfdfrdfr

Amperes Law

Try this Create a contour for integration (a circle seems to make sense here!)

H.dl = Current IenclosedThis is, as it turns out, Amperes Law and is the magnetic-field equivalent of Gausss lawIf we define H=B, B=H, then H.dl = Current enclosed = J.dsI3I2I1I4I5I6

H.dl = Current IenclosedI3I2I1I4

Amperes Law Worked ExampleCalculate the magnetic field H bothOutside (r>R) and Inside (r

Outside r>R, H.dl = Ienclosed Br

Inside r

Faradays Law

Changing Magnetic Field Current and Voltage

Faradays LawFaradays Law : Rate of change of magnetic flux through a loop = emf (voltage) around the loop

Lenzs LawB, HLenzs Law emf appears and current flows that creates a magnetic field that opposes the change in this case an increase hence the negative sign in Faradays Law.CV-, V+

Lenzs LawB, HLenzs Law emf appears and current flows that creates a magnetic field that opposes the change in this case an decrease hence the negative sign in Faradays Law.V+, V-C

Faradays LawRate of change of magnetic flux through a loop = emf around the loop

Maxwell so far Integral form Differential form Note :Maxwell#1, Maxwell#2 and Maxwell#4 are completeMaxwell#3 is still incomplete (just!)

Whats the point of Faraday?Take a circuitPass a current through itMagnetic field is created (Ampere)Put another circuit nearbyIf the induced magnetic field changes in time, Faradays Law causes an emf and current to appearThis is Magnetic Inductance and the Mutual Inductance between two circuits expresses the strength with which they couple inductively.It can be used to signal to/from (and provide power for) remote circuits, or circuits embedded in (say) the body.

InductanceTake a circuitPass a current through itMagnetic field is created (Ampere)This field passes through the circuit that created itIf the magnetic field is time-varying, it induces an emf and thus a current in the circuit.This emf opposes the change in magnetic field that caused it and thus induces a current in the opposite direction from the current that caused the magnetic field in the first place!This is (self-) inductanceIt depends upon the geometry of the circuit and what it contains (bits of iron?).

SON

Take it as experimental fact that like charges repel (etc) and that the force between two charges is proportional to the product of the two charges, the inverse square of the distance between them and has some funny constants,p and e in it! Charge Q1 pushes charge Q2 with the force given, and vice versa. Closer = stronger force.This force can be, and is, best seen as the effect of an (invisible) FIELD set up by one charge that causes a force to be exerted on a second charge. Here, Charge Q1 sets up a field E1 at the location of charge Q2 and the Force on Q2 is given by F = Q2E1. The field is called E, the Electric Field and is given by Coulomb's Law (above).Note that electric field (a) points away from a positive charge (b) is proportional to the charge and (c) is inversely proportional to the square of the distance from the charge.Of course, we can do it all backwards and find the field and force due to Q2 at the position of Q1 - and we get the same force, with the opposite sign. This makes complete physical sense - the two charges push each other apart with equal force. If this were not true, we'd be worried, as one charge would be pushing harder on the other than vice versa.Imagine a chest expander with one end pulling harder than the other. What would happen then?Unit vector? Here's what it means. Formally, it's a vector with a direction, but magnitude of 1. So multiplying by it gives a scalar quantity (e.g. speed) a direction (changing it into a velocity) but does NOT change the magnitude. there's a simple example at the foot.In Electromagntics, unit vectors are all over the place - giving electric and magnetic fields direction, without messing up their magnitudes.Electric field E, at a point, is simply a vector that is the force on a +1C test charge place at that point and we can view the force between these two charges as either:-

The force on charge a, of the electric field set up by charge b at distance r (i.e at point a),or The force on charge b, of the electric field set up by charge a at distance r (i.e at point b).

Looking at the simplest charge-creation structure I can imagine (parallel-plate capacitor of whoch more anon), we can see that the Electric Field is defined as a vector, E(x,y,z), or E(r), whose magnitude expresses the strength of the force on a 1C charge placed at a point in space and whose direction is the direction of that force at that point in space. Note that the parallel-plate capacitor gives rise to a particularly simple field pattern all right-left in this case, constant between the plates and zero outside them (you may guess that this ideal case is something of an idealisation and we will look at it more once we have some more tools for examining things).As described the field is constant in magnitude, points right-left and is therefore negative. The voltage rises from zero to V linearly between the plates and its worth noting here that the voltage is positive while the field is negative (E=-V/d). If we swopped the positive and negative charges on the plates, both would reverse in sign.Before electric field, scientists proposed several theories to explain the inexplicable - "action at a distance" ... how can one body affect another that is distant from it? The electric field explains it nicely and later we will see that the magnetic field expresses a different form of action-at-a-distance.By the way, there were some odd theories of this action-at-a-distance stuff before electric field was proposed - some involving an invisible "aether" that surrounded everything and carried the force (may the force be with you ...). If this were a Physics lecture, we'd spend time on these intriguing, but unsuccessful, theories. However ...More practically, multiple charges each produce a field at some distant point and the field simply add vectoriallyas you see here. Here we calculate the total field at the location of a "test charge", caused by three "field-creating" charges Qa..Qb Qe. The resultant field gives the resultant force:-

Heres a nice analogy. Clouds of smoke are illegal here (!) but if you had one, you would NOT try to count the smoke particles. What you would do is to measure the density of particles (in a box of size 1mm3, say) and then add up all of the mm3 of smoke to get a total. As mm3 tend to m3 and thence to infinitesimally small volume, the sum of little boxes of charge becomes an integral.Of course, charge can be contained in a plane (as it is on the plate of a capacitor) whereupon we measure it in C/mm2 (or m2, or whatever).Or in 1D, as it is on a wire Lets do this in cartoon form first. The ethos of Gauss is that we look, not at the number of lines of field that emanate from a point charge and then add them up (or integrate them) but that we effectively count the number of field lines leaving a closed surface (actually, we find the total electric flux more later). What we will then find is that if we choose our closed surface carefully, the maths becomes almost trivial. I know that the prospect of surface integrals is causing you a deep sinking feeling, but bear with me theyre OK and only very simple examples will come your way in this course!Lets stick to 2D for now and adopt a drawing scheme whereby each coulomb of charge is represented by (or is viewed as being able to generate) 1 electric field line (this is 100% arbitrary!). Well choose a circular surface with malice aforethought!Then, 8C leads to 8 lines of E-field poking through the surface. 16C leads to 16 E-lines and 32C to 32 E-lines and so on.The number of electric field lines leaving a closed surface is proportional to the charge enclosed by the surface.If we re-phrase this as The total electric flux leaving a closed surface is proportional to the charge enclosed by the surface, then we have actually stated Gausss law, and thus Maxwells first equation!

And, happily, all of this is true in 3D as well.So - biting the bullet, here is Gauss properly. D is just E multiplied by the dielectric constant , which is 0 = 8.85 x 10-12 for a vacuum and more for materials with some dielectric properties. The charge enclosed now becomes a volume integral of charge/volume in a closed surface and the number of E-field lines becomes the integral of E (actually D) over the same closed surface. Whats the dot product, D.E, all about, though?Flux density D and field E have a direction - and here's why.Look at the water-flux example once more. The same rainfall has dramatically different effects depending upon its direction. Look at the rectangular umbrellas. So - the direction, or orientation, is at least as important as the area. A huge umbrella held at 90 to the rainfall won't keep you very dry.Mathematically, this means that when we talk about an element of area (perversely, usually called ds ... "s" standing for surface) we actually have to make it a vector and give it direction - chosen to be PERPENDICULAR to the surface. With this convention, we call is ds. Get used to this - we will see it again.

And here they are in close-up.Here is the same thing in simple maths the dot product automatically take account of the foreshortening effect that makes the same area, in the same flux of rain, get either wetter or not, depending upon its angle to the flux.Lets start with the simplest, parallel-plate capacitor again here are the E-field and Voltage graphs as before.E = -V/d, everything is nocely one-dimensional. Wed expect that E= -V/d, V = -Ed would be a special case of the real, general expression (which will say something like E = f(V) and V = g(E)).

And thats what the truth turns out to be. In 1D, E = -V/d is a special case of E = -dV/dx, which is iself a special case of E = (-dV/dx, -dV/dy, -dV/dz).This thing in the brackets is called the GRADIENT (d/dx, d/dy, d/dz) and is, in math-speak, a VECTOR OPERATOR (gasp!). In this case, we are applying it to a scalar (V) and thus getting a vector (E) out of it.The shorthand for (d/dx, d/dy, d/dz) is that funny upside-down triangle thing = (d/dx, d/dy, d/dz). We will do other things with it later.Heres what I find to be a VERY useful analogy. Electric field is rather like gravitational field. E-field pulls a positive charge along its field lines, while gravitational field pulls an object (e.g. YOU) along its field lines (e.g. down, unless youre in outer space). We know about gravitational potential energy (mgh) and that climbing a hill (or even climbing on to a chair) increases your potential energy the extra energy comes, of course, from the cornflakes that you ate this morning and have since turned into kinetic and potential energy (PE). Lines that connect together all points that have the same gravitational PE are called contour lines, as they connect points that all have the same mgh i.e. HEIGHT!Electric Potential is very similar to gravitational PE and we might expect to find lines of equal potential (voltage to you!) useful as well.Some obvious remarks And here is how things look in an electric force field set up by a pair of (here curved) electrodes with a voltage, or potential, applied between them. The electrodes are clearly all at the same voltage as points on any bit of a good conductor must be (Why? Think about it and ask) so they are EQUIPOTENTIAL lines or surfaces in 3D. In between the conductors , the shape of the equipotentials is affected by the shape of the electrodes and also, potentially, by any stuff that we place between the electrodes. Note in passing and without proof here that E-field lines and equipotentials cross at 90 .. This is not a coincidence. For that same reason, E-field lines and electrodes meet at 90.Now lets get a little more precise. The gravitational potential difference between two points h1 and h2 is the energy expended (or got back) in moving a 1kg weight from h1 to h2.The potential difference between two points a and b is DEFINED as the energy expended (or got back) in moving a 1C charge from a to b. Just as in the case of gravity, the path taken doesnt matter the energy expended/gained will be the same and thus the potential difference is the same.Here is all of that, written down mathematically in1D and 3D unfortunately, the work done is now a line integral. Effectively, we have broken the path from a to be into an infinite number of elements dl , each of which is infinitesimally small and added them all up or integrated them. The dot product E.dl appears because moving perpendicular to the E-field is the same as walking along a countour line and in that case E and dl are at 90 to one another. In the absence of friction, no energy is either expended or gained. Most energy is expended/gained when E and dl are at 180and 0 to one another respectively. The dot product takes care of that very neatly. The negative sign simply says that work is done when forcing a movement against an opposing force and work is got back when the movement is with the force is with you(!).Line integral -E.dl means form lots of infinitesimally small -E.dls and then add them up just like this.Generally, the whole point of a line integral is to take account, mathematically, of the path taken. In the case of potential, the path does not matter so it makes sense to choose a path for integration that makes the maths easy. This is best seen in a worked example.Capacitors consist of two chunks of conductor, separated by a volume of dielectric (insulator). Any two such chunks form a capacitor. Here are some examples.Self-explanatory, I think. By geometry, we mean the shape and separation of the two chunks of conductor. The material material that is the insulator means that two identical capacitors with (say) ceramic between the plates of one and air netween the plates of the other will have different capacitances.We all know that a bar magnet can create a magnetic field, creating the magnetic flux density B and its associated magnetic field H. Fundamentally, however, magnetic field is associated with moving charges In this case, a static charge creates no B-field .. When it moves, a B-field appears, when it stops, the B-field disappears again.Or a current (which is simply a lot of moving charges!) creates a magnetic field when it flows, which reverses in sign when the current changes direction.Theres a neat reminder as to which way the field points. Point your right hand thumb along the current and unless you have a very weird right hand, your fingers will point along the direction of the field lines.Looking at the simplest situation first lets look at the magnetic field around an infinitesimally small lump of current I in a piece of wire dl.1) The B-field lines are circles wrapped around Idl as given by the right-hand rule.2) The field strength falls off as 1/r2 with distance.3) Field is, not surprisingly, porportional to current I.4) There are factors of 4 and to include, but not to worry about/discuss(!).

However the element Idl creates field elsewhere as well . In fact everywhere from + to -! What a nuisance.We have an expression for this, but it needs the use of a vector cross product (LMN) so well have to reveisit that first.Brace yourself and get your right hand ready for action

Here is the physical law that relates the element of magnetic field dB caused by an elemant of current Idl the Biot-Savart law.The magnitude is what we saw a few slides ago 1) The B-field lines are circles wrapped around Idl as given by the right-hand rule.2) The field strength falls off as 1/r2 with distance.3) Field is, not surprisingly, porportional to current I.4) There are factors of 4 and to include, but not to worry about/discuss(!).But now we also have the vector element that ensures that we point the resultant field in the correct direction and that is where the cross product comes in. As drawn here, we are calculating dl x r, which is a vector, pointing around the circle drawn, tangential to the circle, and anticlockwise, looking from the right of this diagram.

And, of course, we can integrate all the elements dB to get B the total magnetic field.

Some trigonometry and geometry.R = rsin(f) is easy. The rdf stuff is less so.First we have to view the triangle that has df as its acute angle as two radii of a cirle of radius r. The we can use the arc-of-a-circle formula to calculate the length of the arc, rdf. We can then use that in the magnified triangle to get the rdf/dl formula.Lets sneak up on Amperes Law.For no apparently good reason, lets create a circular contour and integrate B.dl around it(!). If we do that, we get I.In other words, if we integrate B around a closed contour that has a current I flowing through it, we get IB.dl = I.

Amperes Law as just discovered works with all current distributions and all contours we have just chosen a very symmetric example. Here is a general example of currents that are enclosed by a contour and not enclosed. The total enclosed current here is I1+I2+I3-I4.Note in passing also that when the current distribution is continuous (i.e. not confined to wires) we must write the current enclosed as the integral of current density J through an area that is bounded by whatever contour we have chosen.Note in passing that the shape of the surface chosen to calculate J.ds does not matter the same current, I1+I2+I3-I4, flows through both the blue-green flat surface shown here and the lilac bulgy surface.We will trip over this later in the context of DISPLACEMENT CURRRENT, as we try to put the finishing touches to Maxwells Equations.For a contour outside the wire, ALL of the current is enclosed and the result is exactly the same as that for an infinitely-narrow wire.Faraday found that by moving a magnet about near a loop of wire, he could cause a current to flow in the wire and by implication a voltage, or electromotive force (emf) to appear around the loop.He found that the emf was equal to the rate of change of the total magnetic flux through the loop bigger magnet, bigger emf, faster movement, bigger emf. Here is Faradays Law expressed as a line integral of electric field (which is a potential, voltage or emf) and the surface integral of the magnetic flux/unit area (B) the total magnetic flux throgh the loop. Note that the emf dsappears if the flux stops changing. There is a negative sign in Faradays Law and this is why.There is a negative sign in Faradays Law and this is why.Inductance (pedantically, self-inductance) explained. We can now see why inductors oppose a change in current as it gives rise to a change in magnetic field, which the inductor feels duty-bound, by Lenzs and Faradays Laws, to oppose!