An Introduction
to Electrical Machines
P. Di Barba, University of Pavia, Italy
Academic year 2011-2012
Contents
Transformer
• 1. An overview of the device
• 2. Principle of operation of a single-phase transformer
• 3. Power (three-phase) transformer, signal transformer, autotransformer
Rotating electrical machines (REM)
• 1. Ampère law, Faraday principle, Lorentz force
• 2. An overview of REM
• 3. Rotating magnetic field
Asynchronous machine: induction motor
Synchronous machine: alternator
DC machine: commutator motor, dynamo
Permanent magnet motor
Stepping motor
Transformer: static electrical machine operating in AC regime.
It converts electric power into electric power, by varying the
power factors (V,I) and keeping the power (approximately)
constant.
1 : primary circuit (power in)
2 : secondary circuit (power out)
Single-phase
Three-phase
Applications
Power transformer
• Production systems operate at a reduced voltage
in order to decrease dielectric material size
• Transmission systems operate at a low current
in order to decrease conducting material size
• User systems operate at a low voltage
in order to increase safety
Signal (and power) transformer
• Adapts the voltage of two systems to be connected
• Adapts the impedance of two systems to be connected
• Couples two systems, keeping them electrically disconnected
Step up transformer:
increases V and decreases I
Step down transformer:
decreases V and increases I
Power range:
from 1 VA to 1000 MVA
Voltage range:
from 1 V to 400 kV
Basically, a transformer is composed of:
• a magnetic circuit;
• (at least) two magnetically-coupled electric circuits.
Single-phase power transformer (25 Hz < f < 400 Hz)
1
2
limb
yoke
1 2
windings
high-voltage terminal
lamination package (core)
low-voltage terminal
A<25 kVA
natural cooling
A>25 kVA
forced cooling
Three-phase
power transformer
(f=50 Hz) :
three pairs of windings
a
b
c
Three primary windings 1a,1b,1c
and three secondary windings 2a,2b,2c
Used connections:
star-delta, delta-delta, delta-star.
closed magnetic circuit
Signal transformer
single phase
20 Hz < f < 20 kHz
1
2
open magnetic circuit (f>20 kHz, core made of Fe2O3)
Operating principle (ideal case)
Model assumptions: infinite core permeability,
zero losses in the magnetic circuit and in the two electric circuits.
F
I2
2
Z
1
I1
1: N1 turns
2: N2 turns
Transformer equations
1
N1

V

V
,
k

2
1

k
N2

1

I

I2
1

k

A1  A 2  V1I1  V2 I 2


Transformer laws (ideal model) I
No-load operation (I2=0)
Let a sinusoidal voltage source v1(t) supply winding 1
Then, an infinitesimal magnetizing current I10 flows in
winding 1 and gives rise to a magnetomotive force (mmf)
Hopkinson law: magnetic flux
N1 I10  F
infinitesimal mmf
infinitesimal reluctance
non-infinitesimal sinusoidal flux
Transformer laws (ideal model) II
Flux F links both windings 1 and 2
Faraday-Lenz law: induced emf
dF
e1   N1
dt
KVL
e2   N 2
dF
dt
v1=-e1 , v2=-e2
In phasor notation:
V 1  jN1 F
V 2  jN 2 F
In terms of RMS values, it turns out to be:
V1 N1

k
V2 N 2
transformer voltage law
Primary circuit: supplied by V1
Secondary circuit: open circuited
F
2
1
2
1
I10
Kirchhoff voltage law
V1  E1  0
V2  E2  0
I10 magnetizing current
Terminal marking: currents entering the marked
terminals originate like fluxes in the magnetic core.
Transformer laws (ideal model) III
On-load operation
Current I2 is delivered by winding 2 to load Z
mmf N2I2 originates a secondary flux F2
opposing the primary flux F1 = FM , but
V1
FM 
4.44 N1 f
cannot vary due to the applied primary voltage V1
Transformer laws (ideal model) IV
Consequently, voltage source V1 is forced to deliver an
additional current I1 to winding 1 in order to compensate
the flux mismatch:
N1I1-N2I2=N1I10=0
It turns out to be:
I1 N 2 1


I 2 N1 k
transformer current law
Finally, the complex power conservation holds:
V1 I1  V2 I 2
Vector diagram
(Kapp diagram)
A resistive-inductive load is assumed
A more realistic model - 1
In a real transformer, power losses are not zero:
P0=GV2 active power in the magnetic core (eddy current and hysteresis)
Pc=RI2 active power in the electric circuits due to the Joule effect
Moreover, the finite magnetic permeability of the core causes a non-zero
magnetizing power:
Q0=BV2 reactive power in the magnetic circuit of main flux (inside the core)
QC=XI2 reactive power in the magnetic circuit of stray flux (outside the core)
P0 and Q0 dominate in the no-load operation
(measured in the open-circuit test)
Pc and Qc dominate in the on-load operation
(measured in the short-circuit test)
A more realistic model - 2
As a consequence, in a real transformer:
V1 is different from –E1 due to voltage drops caused by R and X of the
windings
F is not constant with respect to E1 because E1 varies with I1 for a given V1
V1 is different from kV2 due to on-load voltage drops
I1 is different from I2/k due to no-load primary current
A1 is different from A2 due to active and reactive power losses
Transformer equations based on the ideal model
are valid as a first approximation only
A more realistic model - 3
main flux
power loss (eddy
currents, hysteresis)
stray flux
power loss
(Joule effect)
A more realistic model - 4
1

V 2  k V1  ZI 2

I1  1 I 2  Y V1

k

2
2
P

P

RI

GV
 1
2
2
1

Q1  Q 2  XI22  BV12
Z  R  jX
Y  G  jB

Primary winding admittance
Secondary winding impedance
A more realistic model - 5
Modified Kapp diagram
Impedance adaption - 1
Problem: how to modify the value of an impedance Z
to be connected at a terminal pair A-B ?
V1
I1 
Zk 2
Impedance adaption - 2
1 V 2 1 V1 1 1 V1
I1  I 2 


k
k Z k k2 Z
Z k
Signal transformer: frequency response - 1
Signal transformer: frequency response - 2
Xm  Xc

Voltage drop - 1
V  V20 V2

1
V 1  V 20  V 2  ( R  jX ) I 2  V2  R  jX I 2 cos   jI 2 sen 
k
Voltage drop - 2
V  V20  V2  RI 2 cos   XI 2 sin 
Insulation transformer - 1
Measurement
Insulation transformer - 2
Safety
Insulation transformer - 3
Damping
Auto-transformer
V1 N 2  N1

k
N2
V2
I1
N2
1


I 2 N 2  N1 k
A1  V1I1  V2I2  A2  A
Ad  (V1  V2 ) I1  ( I 2  I1 )V2
Ad
1
A
Rotating Electrical Machine
Basically, it consists of:
• an electric circuit (inductor) originating the magnetic field;
• a magnetic circuit concentrating the field lines;
• an electric circuit (armature) experiencing electrodynamical
force or electromotive force.
Ampère law
ib
e
Faraday law
transformer effect
b(t)
e(t), i(t)
Faraday law
(motional effect)
bv e
Lorentz force
bi  F
stator
inductor circuit
air-gap
rotor
armature circuit
Magnetic field of a circular coil
dl
r
n
dB
B
i
 0i
B
n
2r
j axis
1
Rotating magnetic
field (two-phase) I
B1
2
r axis

B1 
0 I
2r
0 I
cos t n1 
0 I
2r
cos t
B2
B
0 I
0 I


B2 
cos t  n 2 
sent n 2   j
sent
2r
2
2r
2r

j axis
Rotating magnetic
field (two-phase) II
1
B1
2
r axis

B
B2
B  B1  B 2 
0 I
2r
cos t  j sent  
0 I
2r
e  jt
Rotating magnetic
field (three-phase):
j axis
1
Ferraris field
B1 
0 I
cos t n1
2r
0 I
2

B2 
cos t 
2r
3

0 I
r axis
3

 n2

4 

B3 
cos t 
 n3
2r
3 


B
2
3 0 I  jt
B
e
2 2r
Three-phase asynchronous motor (induction motor)
graphic symbol
As a motor, it converts electric power into mechanical power
air gap
shaft
ventilating fan
laminated stator and rotor
three-phase
inductor circuit
Squirrel-cage rotor
Wound rotor
W
bar
slip-ring commutator
(single-phase)
end ring
Induction motor: operating principle
• Inductor terminals are supplied by a three-phase symmetric voltage V1
• A three-phase balanced current I1 is delivered to inductor windings.
• An induction field B1 rotating at speed N0 = 120 f p-1 is originated
(f current frequency, p number of poles, speed in rpm).
• In stator and rotor windings, subject to a sinusoidal magnetic flux F,
electromotive forces
E1=2.22 m1 f F
and
E2=2.22 m2 f F
are induced, respectively.
F max value of flux, E1,2 rms value of electromotive force
m1,2 number of conductors in a winding
E2
three-phase balanced current I2 in the rotor windings
Magnetic effect of I2 :
• a synchronous rotating field B2 (speed N0) is originated,
which tends to decrease the inductor rotating field B1
• B1 depends on the applied voltage V1 and cannot vary
• voltage source V1 is forced to deliver an additional current I3 to the
inductor, such that rotating field B3 due to I3 compensates field B2
Mechanical effect of I2 :
the axially-directed conductors
placed in the radially-directed field B1
experience tangentially-directed forces Ft
The rotor experiences a torque C=2RFt: if unconstrained, it rotates.
Having defined the slip factor s=(N0-N)/N0
as the speed of the rotating field with respect to the rotor,
the frequency of E2 and I2 is f2=sf
If the opposing torque is equal to zero
then, the running torque CM=0 and
f≠0
I2=0
E2=0
f2=0
s=0
N=N0
the rotor is synchronous wrt the rotating field
If the opposing torque is different from zero
then, the running torque CM≠0 and
f≠0
I2 ≠ 0
E2 ≠ 0
f2 ≠ 0
0<s<1
N<N0
the rotor is asynchronous wrt the rotating field
(the rotor follows the field)
Torque-speed curve
Torque C is proportional to both flux F1 of the rotating field
and real part of the armature current I2cos f2 (active component):
C  kF1I 2 cos 2
Current I2 and power factor cos f2 depend on emf E2
and armature circuit impedance:
Z 2  R22  sL2 
2
resistive-inductive impedance R2+jsL2
On the other hand, electromotive force E2 depends
on both relative rotor speed N0-N and flux F1
E2  k ' N0  N F1
It turns out to be:
N 02 N 0  N R2
E2 R2
2
2


C  kF1
 k1F1

C

C
N
,
C

V
1
Z2 Z2
N 0 R2 2  N 0  N 2 L2 2
Torque-speed curve
Max torque
CM = k1F12 N0/(2L2)
when
N = N0(1-R2/L2)
C
C
M
self-starting
Cs = k1F12 N0R2/(R22+2L22)
operating point
CS
CR
0
N0
N
synchronous speed (rpm)
mechanical power Pm = CN
electric power Pe = √3 VI cosf
N0 
120 f
p
Speed control
decreasing V
decreasing f
C
increasing R2
(wound rotor only)
C
1
C
2
2
1
1
2
N
N
N
Synchronous generator
turbine
Three-phase output
alternator
Alternator structure
Three-phase
armature circuit
DC inductor circuit
(two-pole)
In low-power alternators
it can be replaced by a
permanent magnet.
As a generator, it converts mechanical power into electric power.
Alternator operating principle
• The inductor circuit is supplied by a DC current IE
• A magnetic flux FR sinusoidally distributed in the air-gap is originated
• A speed W is applied to the rotor shaft
• In the armature circuit a three-phase symmetric electromotive force
E = kWFR at an angular frequency  = Wp/2 is induced
• If the armature circuit is on load, a three-phase (balanced) current I
is delivered, so providing an active power P = 3EI cos f
• An equal amount of mechanical power must be supplied to the shaft
• Current I generates a synchronous rotating field (speed ),
which originates a flux FS opposing flux FR
• In order to keep E constant, excitation current IE must be increased
electric frequency
p
W
2
number of poles
rotor speed
I
I
Two-pole DC winding
Four-pole DC winding
DC machine
graphic symbol
From electric to mechanical power: motor
From mechanical to electric power: dynamo
air gap
salient
pole
ventilating fan
shaft
sliding brushes
with commutator
laminated stator and rotor
armature
inductor
DC machine - Principle of operation I
MOTOR
Inductor circuit is supplied by DC voltage VE
Current IE is originated
Magnetic field B, and so flux F, take place
Armature circuit is supplied by DC current IA
Armature conductors, carrying axial current IA and placed in radial field B,
are subject to tangential force F
Main effect
F
C
N
Pm = CN
Secondary effect
Due to the rotor motion, in the armature conductors an AC emf is induced,
which is subsequently rectified by the multi-sector commutator.
Therefore, a back emf E = kNF appears at the commutator terminals.
It turns out to be:
E=V
(V voltage across armature terminals)
Pe = EIA = CN = Pm (power balance)
DC machine - Principle of operation II
DYNAMO
Inductor circuit is supplied by DC voltage VE
Current IE is originated
Magnetic field B, and so flux F, take place
Rotor is given speed N, externally applied to the shaft
In the armature conductors, an AC emf is induced
At the commutator terminals a rectified emf E = kNF is available
Main effect
If armature terminals are connected to a resistive load
E
IA
Pe = EIA
Secondary effect
Armature conductors, carrying axial current IA and placed in radial field B,
are subject to tangential force F opposing the motion
F
C
N
Pm = CN
To maintain the motion, power Pm must be supplied to the shaft (Pm = Pe) .
Multi-sector commutator:
principle of operation
A
b
b
a
A
b
+
t1
B
a
t

a
t2
VAB
B
Each armature conductor is
electrically connected to a sector.
The polarity of each sector is reversed
when crossing the A-B line joining the
brushes (orthogonal to the pole axis).
The voltage VAB between brushes
exhibits always the same polarity.
with two conductors/sectors
t
VAB
t
with several conductors/sectors
As a result, the original AC signal is mechanically transformed
into a DC signal, and viceversa.
DC motor excitation schemes
I
Series excitation (motors for traction)
M
VE
IE
When Cres varies, both N and C vary
in such a way that CN  const
(constant power motor)
IA
IE
VE
M
VA
Independent excitation
V
IA
M
Derived excitation
When Cres varies, excitation current IE varies
 N  const (constant speed motor)
Equivalent circuit of the DC motor (with independent excitation)
inductor circuit
Torque
armature circuit
(Ra resistance
of armature
conductor)
Excitation flux
F  FI E 
Induced emf
(rectified)
E  kNF
k voltage constant
VE
C  k ' FI A  k ' F
RA
k’ torque constant
Torque-speed curve (independent-excitation motor)
F  F(IE )
self-starting
V
N0 
kF

k'F
V  kNF   C N 
C
RA
CS  k ' F
V
RA
Speed control
AC to DC conversion - 1
Ideal diode: current-voltage curve
AC to DC conversion - 2
iR  i1  i2
 R  RiR

Transformer + single phase rectifier
AC to DC conversion - 3
Single-phase rectifier (full-wave):
principle of operation
A rectified voltage vR is originated:
non-zero average value
Wound synchronous motor
Permanent magnet
synchronous motor
Magnets
Excitation
windings
SPM
IPM
Magnets
Synchronous motor: torque-speed curve
Three-phase inverter
Control system
Reference signal
feedback
Rotor position sensor
(Hall probe)
Brushless DC motor: principle of operation
Stepping motor: principle of operation
Permanent-magnet
stepping motor
Variable-reluctance
stepping motor