11/6/2012
Overview
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•
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23-DC Generators Part 1
Induced emf Equation
Power
Torque
Armature Reaction
ECEGR 450
Electromechanical Energy Conversion
Dr. Louie
Induced emf
Induced emf
• Recall for a single coil with one turn that is
rotating in a uniform magnetic field the average
emf is:
ec
2
• emf at the terminals of the machine must
account for number of turns per coil and number
of coils arranged in series (or in parallel)
P
Em
P
Na
m
• Frequency of ec in Hz is:
P
4
m
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Induced emf Equation
ea
P
m
PCNc
a
P
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Example
• Therefore
PNa
C
Nc
a
 Na: number turns between positive and negative
terminals
 C: total number of coils
 a: number of parallel paths (lap winding: a = P)
 Nc: number of turns per coil
 Ec: average induced emf in a single coil (V)
 m: mechanical speed of the rotor (rad/s)
 p: flux per pole
f
2
Consider a 24-slot, lap wound, 2 pole dc
generator with 18 turns per coil. Find the average
induced emf if p = 0.05 Wb and the rotor rotates
at 183.2 rad/s.
m
 ea: average emf between the brushes (V)
let
PCNc
(machine constant)
a
therefore
ea K a P m
Ka
Dr. Louie
“slot” refers to the notches in the rotor for armature windings.
Number of slots = number of coils.
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11/6/2012
Example
Question
Consider a 24-slot, lap wound, 2 pole dc
generator with 18 turns per coil. Find the average
induced emf if p = 0.05 Wb and the rotor rotates
at 183.2 rad/s.
ea
PCNc
P
2 24 18 0.05 182.3
2
m
a
• For a lap-wound machine with constant C, Nc and
m, how does increasing the number of poles P
affect the terminal voltage?
1259.6V
7
Dr. Louie
Dr. Louie
Question
Induced emf Equation
• For a lap-wound machine with constant C, Nc and
m, how does increasing the number of poles P
affect the terminal voltage?
• It has no effect on induced voltage. For lapwound machines, a = P so that:
ea
PCNc
a
P
m
PCNc
P
P
m
CNc
P
• Electrical power to the load:
Pd
eaia
9
Ka Pia
m
Te
• The average torque on a single turn-coil is
m
• Next examine the torque developed using a fields
approach
Dr. Louie
2BiLr
 L : length of one conductor along the side of the
rotor (m)
 r: radius at which each conductor is located (m)
• For stable operation, the electrical power
developed must equal the mechanical power
Td
10
• Recall the torque developed by a single turn coil
rotating in a uniform magnetic field
 Td: torque(Nm)
Pd
i
ma
Developed Torque
• Developed mechanical torque:
Td
P
Dr. Louie
Induced emf Equation
Td
Ka
 Pd: power developed by the generator (W)
 ia: armature current, assumed to be constant (A)
m
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Pd
8
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Tc
2BicLr
 ic: average current in the coil (current is full wave
rectified)
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Developed Torque
Magnetization Characteristic
• The total torque developed by the machine is:
Td
2BLria
 from ea
• Let Ap be the area of each pole
AP
Td
• Induced emf is a function of flux per pole and
armature speed
CNc
a
2 rL
P
2CNcP
BApia
2 a
recall that: Ka
(m2),
then
This is exactly the
same as the torque
derived from
examining the
electrical power
Ka Pia
PCNc
a
k f if
• Is kf always the same value for a given machine?
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Magnetization Characteristic
Ka
 yields Ea
m
Ka
P
m
and
P
14
• Is kf always the same value for a given machine?
 kf: constant of proportionality
 if: field current (A)
• Combining Ea
Dr. Louie
Magnetization Characteristic
• Flux per pole is dependent on and proportional
with the mmf provided by the field current
• Mathematically
Dr. Louie
m
 Induced emf is then proportional to flux per pole
Magnetization Characteristic
P
P
• Flux per pole is dependent on the mmf provided
by the field current
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Dr. Louie
Ka
• Assume the armature is open-circuited and is
rotating at the rated speed of the machine
 kf relates flux to field current
 Since the machine is made of ferromagnetic
material, it can saturate
 When it saturates, it requires more current to
produce an incremental amount of flux
 So, kf is a function of field current
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Magnetization Characteristic
• The magnetization curve will reflect that:
k f if
 Magnetic circuit of the flux passes through a
ferromagnetic material and an air gap
 Ferromagnetic material (non-linear B-H curve)
 Air-gap (linear B-H curve)
 Residual magnetism in material will give rise to a
residual emf in the armature at zero field current
 Hysteresis effect
k f if
• Assuming speed is constant, the induced emf is
proportional to the field current
• Remember that kf is dependent on the field
current due to saturation
• We therefore expect a non-linear current-toinduced emf relationship
 This is the magnetization curve of the machine
Dr. Louie
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Magnetization Characteristic
Armature Reaction
• er: residual induced emf (volts)
• Flux in the armature is from two sources:
 field winding
 armature current (when connected to a load)
mmf of the air gap
• Fluxes are perpendicular to each other
• Resulting distortion can have a profound effect on
the operation of the machine
• Read 5.8 of the text
emf (no load)
Air-gap line
mmf of the machine
er
Field Current
(or mmf)
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Armature Reaction
Armature Reaction
N
x
x
S
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Dr. Louie
Armature Reaction
Armature Reaction
Interpole Windings
Relocated Brushes
N
N
N
x
x
Dr. Louie
x
x
x
x
b
S
x
x
N
x
x
x
x
Current-carrying
conductors are shorted.
Excessive sparking occurs.
a
S
S
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S
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Summary
• Terminal voltage for DC generator:
ea
PCNc
a
P
m
• Non-linear relationship between field current and
terminal voltage due to saturation
• Interaction between flux generated by field
winding and armature winding is “Armature
Reaction”, and decreases performance of machine
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