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Introduction to Electrical Machines

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Title: Introduction to Electrical Machines


1
Introduction to Electrical Machines
  • Erkan Mese

2
Coulombs Law
3
Remember
  • Like charges repel one another
  • Opposite charges attract one another
  • The force of repulsion/attraction get weaker as
    the charges are farther apart.

4
Charges and Forces
In air, e 8.85 x 10-12 Fm-1 â 1, Fa -Fb
5
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.
6
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)
7
Charges and Fields
E -V/d F q(-V/d) F qE again Where E is the
field set up inside the capacitor
8
Charges and Fields
V
E -V/d
9
Several Charges?
Ea Eb Ec Ed Ee
10
Several Charges?
Ea Eb Ec Ed Ee ETOT
ETOT
11
Charge Density 3D
3D r(r) in C/mm3 1mm3 r C
r(ra) gt r(rb)
12
Charge Density 2D
r(ra) gt r(rb)
2D r(r) in C/mm2 1mm2 r C
13
Charge Density 1D
r(ra) gt r(rb)
1D r(r) in C/mm 1mm r C
14
Gausss Law
15
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.

16
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.
17
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
18
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
19
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

20
Digression/Revision Area Integrals
This area gets wetter!
21
Area Integrals whats happening?
This area gets wetter!
22
Area Integrals whats happening?
Clearly, as the areas are the same, the angle
between the area and the rainfall matters
23
Area Integrals whats happening?
ds
ds
Extreme cases at 180 - maximum rainfall at 90,
no rainfall
24
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

25
Potential
26
Potential Start Simply
V
Remember the capacitor
E -V/d E-(rate of change of V with distance)
27
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
28
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!)
29
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!)

30
Electric Fields and Potentials are the Same
Voltage contours
31
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
32
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

33
Line integral revision
E


34
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

35
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)
36
Capacitance
37
Some Capacitors
conductor
insulator
38
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
39
Magnetic Fields
40
The Story so Far
Maxwells 1st Equation
or
Maxwells 2nd Equation
or
41
What creates a magnetic field?
42
What else creates a magnetic field B?
Stationary charge no B-field
Stationary charge no B-field
Moving charge non-zero B-field
43
Current Moving Charges
44
Direction of B, H fields? Right hand thumb
current, fingers B-field
C
C
45
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

46
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

47
The Biot-Savart Law
x
L
J
48
Worked Example of Biot-Savart Law Infinite Line
of Current
ò
dB
??B
x
49
Worked Example of Biot-Savart Law Infinite Line
of Current
dB
r
I
f
dl
df
rdf
50
Amperes Law
51
Try this
Create a contour for integration (a circle seems
to make sense here!)
52
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
53
òH.dl Current Ienclosed
I4
I1
I3
I2
54
Amperes Law Worked Example
  • Calculate the magnetic field H both
  • Outside (rgtR)
  • and
  • Inside (rltR)
  • A wire with uniformly-distributed current
    I, current density

55
Outside rgtR, òH.dl Ienclosed
B
r
56
Inside rltR, òH.dl Ienclosed
B?
R
r
57
Faradays Law
58
Changing Magnetic Field Current and Voltage
59
Faradays Law
Faradays Law Rate of change of magnetic flux
through a loop emf (voltage) around the loop
60
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.
61
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.
62
Faradays Law
Rate of change of magnetic flux through a loop
emf around the loop
63
Maxwell so far
Integral form
Differential form
Note Maxwell1, Maxwell2 and Maxwell4 are
complete Maxwell3 is still incomplete (just!)
64
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.
65
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?).
66
SON
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