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Introduction to Electrical Machines

- Erkan Mese

Coulombs Law

Remember

- Like charges repel one another
- Opposite charges attract one another
- The force of repulsion/attraction get weaker as

the charges are farther apart.

Charges and Forces

In air, e 8.85 x 10-12 Fm-1 â 1, Fa -Fb

Unit vector âr?

These are all unit vectors, âi 1 They have a

direction, and a magnitude of 1 â adds direction

to a quantity without changing its

magnitude e.g.... speed 100m/s is a speed

S 100(1/Ö2, 1/Ö2, 0)m/s is a velocity v Sâ ,

100m/s, North-East (ì) â (1/Ö2, 1/Ö2, 0) in

this case.

Charges and Fields

Fa QaEb

Fb QbEa

Eb(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 Fields

E -V/d F q(-V/d) F qE again Where E is the

field set up inside the capacitor

Charges and Fields

V

E -V/d

Several Charges?

Ea Eb Ec Ed Ee

Several Charges?

Ea Eb Ec Ed Ee ETOT

ETOT

Charge Density 3D

3D r(r) in C/mm3 1mm3 r C

r(ra) gt r(rb)

Charge Density 2D

r(ra) gt r(rb)

2D r(r) in C/mm2 1mm2 r C

Charge Density 1D

r(ra) gt r(rb)

1D r(r) in C/mm 1mm r C

Gausss Law

Gausss Law Crude Analogy

- Try to measure the rain on a rainy day
- Method 1 count the raindrops as they fall, and

add them up - cf Coulombs Law
- Method 2 Hold up an umbrella (a surface) and

see how wet it gets. - cf Gausss Law
- Method 1 is a divide and-conquer or

microscopic approach - Method 2 is a more gross or macroscopic

approach - They must give the same answer.

Electric Field Lines

These are all correct as E-field lines are

simply cartoons For now, adopt a drawing scheme

such that 1C 1 E-line.

Lines of Electric Field

How many field lines cross out of the circle?

8C Þ 8 lines

16C Þ 16 lines

32C Þ 32 lines

Gausss Law Cartoon Version

- The number of electric field lines leaving a

closed surface is equal to the charge enclosed by

that surface - S(E-field-lines) a Charge Enclosed

N Coulombs Þ aN lines

Gausss Law Proper (L)

- S(E-lines) proportional to (Charge Enclosed)
- D eE
- òòD.ds òòòr(r)dv òòòr(r)dxdydz
- òòD.ds charge enclosed
- e e0 8.85 x 10-12 in a vacuum

Digression/Revision Area Integrals

This 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?

ds

ds

Extreme cases at 180 - maximum rainfall at 90,

no rainfall

Flux of rain (rainfall) through an area ds

- Fluxrain R.ds
- Rdscos(q)
- Rds cos(q)
- Fluxrain 0 for 90 cos(q) 0
- Fluxrain -Rds for 180 cos(q) -1
- Generally, Fluxrain Rds cos(q)
- -1 lt cos(q) lt 1

Potential

Potential Start Simply

V

Remember the capacitor

E -V/d E-(rate of change of V with distance)

E -V/d

- Should really be E -dV/dx
- And if V Mxc, dV/dx M constant
- Then E -M as shown
- In 3D, dV/dx becomes (dV/dx, dV/dy, dV/dz) ÑV,

so - E -ÑV -(dV/dx, dV/dy, dV/dz)

E -ÑV

Potential Analogy

These contour lines are lines of equal

gravitational potential energy mgh Where they are

close together, the effect of the gravitational

field is strong The 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 - comments

- Walking around a contour expends no energy
- In a perfect world
- i.e. no-one moves the hill as you walk!
- Walking to the top of the hill and back again

expends no energy - In a perfect world
- i.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 Same

Voltage contours

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 b

b x

a x

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 b - In 1D, Work -Fd
- In 3D, Work -F.dl
- Force F QE 1E E
- Work done -E.dl
- Total Work done -òabE.dl

Line integral revision

E

Potential Difference -òabE.dl

- òab is a line integral
- In general mathematics, the value of a line

integral depends upon the path dl takes from a to

b - In this potential calculation, the path does not

matter - So choose a convenient path

Potential Difference Worked Example Point

charge Q

(b)

Place a 1C charge at (a)

Move it to (b)

Work done in this movement is the potential

difference (voltage) between (a) and (b)

(a)

Capacitance

Some Capacitors

conductor

insulator

Capacitance Definition

Q CV

- Take two chunks of conductor
- Separated by insulator
- Apply a potential V between them
- Charge will appear on the conductors, with Q

CV on the higher-potential and Q- -CV on the

lower potential conductor - C depends upon both the geometry and the nature

of the material that is the insulator

Q- -CV

Magnetic Fields

The Story so Far

Maxwells 1st Equation

or

Maxwells 2nd Equation

or

What creates a magnetic field?

What else creates a magnetic field B?

Stationary charge no B-field

Stationary charge no B-field

Moving charge non-zero B-field

Current Moving Charges

Direction of B, H fields? Right hand thumb

current, fingers B-field

C

C

Magnitude of B, H fields?

- Take an (infinitesimally small) piece of wire
- Pass a current I through it
- The magnitude of the ring of field directly

around it is given by dB moIdl 4pr2 - So, for example, B1gtB2gtB3

If only it were that simple

- Unfortunately, dB moIdl 4pr2 is a special

case - The element Idl creates B-fields elsewhere (i.e.

everywhere) as shown and, for example, B4ltB1,

B5ltB2, B6ltB3 as the Idl B distance increases

The Biot-Savart Law

x

L

J

Worked Example of Biot-Savart Law Infinite Line

of Current

ò

dB

??B

x

Worked Example of Biot-Savart Law Infinite Line

of Current

dB

r

I

f

dl

df

rdf

Amperes Law

Try this

Create a contour for integration (a circle seems

to make sense here!)

òH.dl Current Ienclosed

- This is, as it turns out, Amperes Law and is the

magnetic-field equivalent of Gausss law - If we define HBµ, BµH, then òH.dl Current

enclosed òòòJ.ds

I4

I5

I1

I3

I2

I6

òH.dl Current Ienclosed

I4

I1

I3

I2

Amperes Law Worked Example

- Calculate the magnetic field H both
- Outside (rgtR)
- and
- Inside (rltR)
- A wire with uniformly-distributed current

I, current density

Outside rgtR, òH.dl Ienclosed

B

r

Inside rltR, òH.dl Ienclosed

B?

R

r

Faradays Law

Changing Magnetic Field Current and Voltage

Faradays Law

Faradays Law Rate of change of magnetic flux

through a loop emf (voltage) around the loop

Lenzs Law

C

B, H

V-, V

Lenzs 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.

Lenzs Law

B, H

C

V, V-

Lenzs 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.

Faradays Law

Rate of change of magnetic flux through a loop

emf around the loop

Maxwell so far

Integral form

Differential form

Note Maxwell1, Maxwell2 and Maxwell4 are

complete Maxwell3 is still incomplete (just!)

Whats the point of Faraday?

Take a circuit Pass a current through it Magnetic

field is created (Ampere) Put another circuit

nearby If the induced magnetic field

changes in time, Faradays Law causes an emf

and current to appear This 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.

Inductance

Take a circuit Pass a current through it Magnetic

field is created (Ampere) This field passes

through the circuit that created it If 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-) inductance It depends upon

the geometry of the circuit and what it

contains (bits of iron?).

SON