Introduction to
Electrical Machines
Erkan Meşe
Coulomb’s 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
â
Qa
r
Qb
Fa
Fb
Fa =-QaâQb
Fb =+QbâQa
4per2
In air, e= 8.85 x 10-12 Fm-1
|â| = 1, Fa = -Fb
4per2
Unit vector âr?
â3
1 unit
â1
â4
â2
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/2, 1/2, 0)m/s is a velocity v =Sâ , 100m/s, North-East ()
â = (1/2, 1/2, 0) in this case.
Charges and Fields
Fb =+QbâQa
Fa =-QaâQb
4per2
4per2
Fa =+QaEb
Fb =+QbEa
Where Eb =-Qbâ
Where Ea = +Qaâ
4per2
4per2
Eb(r)
is the electric field
Ea(r)
is the electric field
set up by charge b at
set up by charge a at
distance r (point a)
distance r (point b)
Charges and Fields
d
0
+q
F
E
Voltage V
+
+
+
+
+
+
V
E = -V/d
F = +q(-V/d)
F = qE again
Where E is the
field set up
inside the
capacitor
Charges and Fields
V
V
0
|E|
0
E = -V/d
Several Charges?
+Qc
+Qd
-Qe
+Qa
-Qb
Ea
Eb
Ec
Ed
Ee
Several Charges?
+Qc
+Qd
-Qe
+Qa
-Qb
ETOT
Ea
Eb
Ec
Ed
Ee
ETOT
Charge Density : 3D
3D
r(r) in C/mm3
1mm3 = r C
1mm3
r(ra) > r(rb)
Charge Density : 2D
r(ra) > r(rb)
2D
r(r) in C/mm2
1mm2 = r C
1mm2
Charge Density : 1D
r(ra) > r(rb)
1D
r(r) in C/mm
1mm = r C
1mm
Gauss’s Law
Gauss’s 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 Coulomb’s Law
Method 2 : Hold up an umbrella (a “surface”) and see how
wet it gets.




cf Gauss’s 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
1C
1C
These are all “correct” as E-field lines are simply cartoons
For now, adopt a drawing scheme such that 1C = 1 E-line.
1C
Lines of Electric Field
How many field lines
cross out of the circle?
8C  8 lines
16C  16 lines
32C
16C
8C
32C  32 lines
Gauss’s 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  aN lines
Gauss’s Law Proper (L)





S(E-lines) proportional to (Charge Enclosed)
D = εE
D.ds
= r(r)dv
= r(r)dxdydz
D.ds = charge enclosed
ε= ε0 = 8.85 x 10-12 in a vacuum
Digression/Revision
Area Integrals
This area gets
wetter!
Area Integrals – what’s happening?
Rainfall
Rainfall
ds
This area gets
wetter!
Area Integrals – what’s happening?
Rainfall
Rainfall
ds
Clearly, as the areas are the same, the angle between the
area and the rainfall matters …
Area Integrals – what’s happening?
Rainfall, R
ds
Rainfall, R
ds
Extreme cases
at 180° - maximum rainfall
at 90°, no rainfall
Flux of rain (rainfall) through an area
ds

Fluxrain = R.ds
|R||ds|cos(q)
 Rds cos(q)




Fluxrain = 0 for 90° … cos(q) = 0
Fluxrain = -Rds for 180° … cos(q) = -1
Generally, Fluxrain = Rds cos(q)

-1 < cos(q) < +1
Potential
Potential … Start Simply …
V
Remember the capacitor
V
0
|E|
0
E = -V/d
E=-(rate of change
of V with distance)
E = -V/d

Should really be E = -dV/dx




And if V = Mx+c, dV/dx = M = constant
Then E = -M as shown
In 3D, dV/dx becomes
(dV/dx, dV/dy, dV/dz) = V, so
E = -V = -(dV/dx, dV/dy, dV/dz)
E = -V
Potential : Analogy
100M
150M
250M
200M
300M
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. that’s 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
5V
5V
1V
2V
0V
3V
E-Field lines
4V
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
5V
bx
5V
E-Field lines
0V
a x 1C
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 = QE =+1E = E
Work done = -E.dl
Total Work done = -abE.dl
Line integral …revision
E
E
dl
Potential Difference = -abE.dl
 ab



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)
1C
Q
E
Capacitance
Some Capacitors
conductor
insulator
Capacitance : Definition

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
+++++++++++
+++++++++++
+++++++++++
V
V
0
Magnetic Fields
The Story so Far
Maxwell’s 1st Equation …
 D.ds = charge enclosed =  rv dv
or … .D = rv
Maxwell’s 2nd Equation …
 B.ds = 0
or … .B = 0
What creates a magnetic field?
S
B = mH
N
What else creates a
magnetic field B?
B
Stationary charge
 no B-field
Moving charge
 non-zero B-field
Stationary charge
 no B-field
Current = Moving Charges
B
B
I
I
Direction of B, H fields?
Right hand : thumb = current,
fingers = B-field
B
B
I
I
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, B1>B2>B3
r3
r2
I
dl
r1
B1
B2
B3
I
If only it were that simple …


Unfortunately,
dB = moIdl
4pr2
is a special case
The element Idl creates Bfields elsewhere (i.e.
everywhere) as shown …
and, for example, B4<B1,
B5<B2, B6<B3
as the Idl  B distance
increases
r3
r6
r2
dl
r1
I
r5
r4
B4
B1
B5
B2
B3
B6
The Biot-Savart Law
m0I dl  a r
dB =
4p r 2
dB
âr
r
I
dl
L
J
x
Worked Example of Biot-Savart Law
: Infinite Line of Current
m0I dl  a r
dB =
4p r 2
dB →→B
âr
r
I
dl
x
Worked Example of Biot-Savart Law
: Infinite Line of Current
m0I dl  a r
dB =
4p r 2
df
dB
.
R = r sin f 
r
f
dl
df
f
dl
I
rdf
rdf
sin f  =
, dl =
dl
sin f 
Ampere’s Law
Try this …
Create a contour for integration (a circle seems to make sense here!)
mI
B=
a from the Biot-Savart law (a is tangential to circle)
2p R
  mI 
Calculate  B.dl =  
 a.dl note : B and dl are parallel
2
p
R


 B.dl =  Bdl = B  dl , ( B = constant)
 mI 
 mI 
 B.dl = 
  dl = 
  2p R = mI
 2p R 
 2p R 
dl
B
 B.dl = mI
 B.dl = mI
. =I
 Hdl
B
dl
dl
B
I
H.dl = Current Ienclosed


This is, as it turns out, Ampere’s Law and is the
magnetic-field equivalent of Gauss’s law
If we define H=B÷μ, B=μH, then
H.dl = Current “enclosed” = J.ds
I4
I1
I3
I2
I6
I5
Take a closed contour
These currents are “enclosed”
And these currents are not!
H.dl = Current Ienclosed
I4
I1
I3
I2
Faraday’s Law
Changing Magnetic Field  Current
and Voltage
B, H
N
Current
S
Faraday’s Law
B, H
N
S
Fmagnetic = total magnetic “flux” = B.ds
VLOOP = -E.dl
Faraday’s Law : Rate of change of magnetic flux through a loop
= emf (voltage) around the loop
 E.dl = 
loop
d F mag
dt
dB
=  
.ds
surface dt
Lenz’s Law
B, H
N
S
Iinduced
V-, V+
Lenz’s 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 Faraday’s Law.
Lenz’s Law
B, H
N
S
Iinduced
V+, V-
Lenz’s 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 Faraday’s Law.
Faraday’s Law
Rate of change of magnetic flux through a loop = emf around the loop
dB
.ds
 E.dl =  
loop
surface dt
dB
or ...   E = 
in differential form
dt
Maxwell so far …
Integral form …
Maxwell#1 :  D.ds =  r dV
Maxwell#2 :  B.ds = 0
Maxwell#3 :  H.dl =  Jds
.
dB
Maxwell#4 :  E.dl =  
.ds
dt
Differential form …
.D = r
.B = 0
 H = J
dB
 E = 
dt
Note :
Maxwell#1, Maxwell#2 and Maxwell#4 are complete
Maxwell#3 is still incomplete (just!)
What’s 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, Faraday’s 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.
B
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!
inductance
This is (self-)
It depends upon the geometry of the
circuit and what it contains (bits of
iron?).
B
SON