Lecture Notes EEE 360 - Warsaw University of Technology
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Transcript Lecture Notes EEE 360 - Warsaw University of Technology
EEE 360
Energy Conversion and
Transport
George G. Karady & Keith Holbert
Chapter 5
Transformers
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Lecture 9
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5.1 Construction
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Transformers
Primary winding
Supply
NP
Secondary winding
NS
Load
Laminated iron core
5.1 Basic components of single phase transformer
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Transformers
Primary
Winding
Multi-layer
Laminated
Iron Core
Secondary
Winding
H1 H2
X1
X2
Winding
Terminals
5.2 Single phase transformer arrangement
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Transformers
H2
X2
H2
X1
Vp
Vs
Vp
Vs
H1
X1
H1
X2
100
100
Vp ( t )
Vs( t )
Vp ( t )
0
100
Vs( t )
0
5
10
15
20
0
100
0
5
10
t
t
milli s
milli s
15
20
(b)
(a)
5.3 Polarity for transformer
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Transformers
Iron core
Insulation
Secondary
winding
Terminals
5.4 Small transformer construction a) Lamination, b) Iron core
with winding
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Transformers
Wires
Primary
Winding
Holder
Secondary
Winding
Insulation
5.5 Winding Construction
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Transformers
Figure 5.6
Dry-type
three-phase
transformer
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Transformers
Bushing
Steel
tank
Iron core
behind the steel
bar
Winding
Insulation
Radiator
Figure 5.7 Oil Insulated and cooled transformer
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Transformers
Metal bar
Exterior
porcelain tube
Flange
Interior
porcelain tube
Figure 5.8 Porcelain transformer
bushing
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Transformers
High voltage
bushing
Oil tank
Low voltage
bushing
Figure 5.9
Large Oil
cooled high
voltage
transformer
Cooling
radiators
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5.2 Magnetic circuit
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Transformers
1.8
Quality Sheet Steel
Magnetic Flux Density, B (T)
1.6
1.4
Ordinary Sheet Steel
1.2
Cast Steel
1
0.8
Cast Iron
0.6
0.4
0.2
0
0
500
1000
1500
2000
2500
Magnetic Field Intensity, H (A-turn/m)
Figure 5.10 B(H) Magnetization curves
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Transformers
Figure 5.11 B(H) Magnetization loss curves
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Transformers
• Ampere law
A
I m Nm H
Im
• Permeability and flux density
E
Nm
B H
• Flux
BA
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Fig 5.12 Magnetic
circuit
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Transformers
• Induced voltage
E Nm
d
dt
• The substitution of the above equations give
d B A
d H
d I m N m A N m2 d I m
E Nm
A Nm
A Nm
dt
dt
dt
dt
• Modified Induced voltage
d Im
EL
dt
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Transformers
• Inductance
L
AN
2
m
• Magnetic energy
LI
Energy
2
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2
m
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Transformers
Acore
5.22 Magnetic circuit
analysis
•
Dimension of the core
w = 3in, h = w, a = 1in, b = 1.5in
Im = 2A, Nm = 20, f = 60Hz,
•
Magnetic path length and area
Lm = 2 (w + a) + 2 (h + a)
Acore = a b
•
Magnetic field intensity
Hm
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Im Nm
Lm
A
Hm 98.425
m
b
a
Im
m
Nm
Vm
h
a
a
w
a
Fig 5.13 Magnetic circuit
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Transformers
Acore
b
5.22 Magnetic circuit analysis
•
,
•
•
Magnetic flux density
Bm o r Hm
m
Nm
Vm
Permeability of free space
7 H
o 4 10
m
h
a
a
w
a
Magnetic flux density from B(H)
A
Hm 98.425
m
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a
Im
Bm 1.237T
Fig 5.13 Magnetic circuit
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Transformers
5.22 Magnetic circuit analysis
•
,
•
•
Magnetic flux
m Bm A core
m 0.001 Wb
After substitution
A core Nm
m o r
Im
Lm
Magnetizing curve is sinusoidal)
Imag( t) 2Imcos t
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Transformers
5.22 Magnetic circuit analysis
•
Magnetic flux time function
Acore Nm
mag ( t) o r
2 Im cos t
Lm
,
•
Maximum flux is:
A co re Nm
max 2 o r
Im
Lm
•
Flux time function
mag( t) max cos t
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Transformers
5.22 Magnetic circuit analysis
•
Induced voltage
d
Eind( t) Nm mag ( t)
dt
d
Eind( t) Nm max cos t
dt
Eind( t) Nm max sin t
•
rms. voltage value
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Erms 9.025V
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Erms
Nm max
2
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Transformers
5.22 Magnetic circuit analysis
•
Simplified induced voltage equation
Erms 4.443f Nmmax
•
Derivation of inductance
Eind( t)
d
Lind Imag ( t)
dt
d
Nm mag ( t)
dt
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d
Lind Imag ( t)
dt
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Transformers
5.22 Magnetic circuit analysis
•
Inductance equations
Lind
•
Nm mag( t)
Imag( t)
Lind
Nm max cos t
2 Im cos t
Derivation of inductance
Lind
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Nm rms
Im
Li nd 11.969
mH
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Transformers
5.22 Magnetic circuit analysis
•
Inductance equations
Li nd
•
A co re Nm
Nm 2 o r
Im
Lm
2 Im
Derivation of inductance
2
Li nd o r
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A co re Nm
Lm
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Transformers
5.23 Magnetic energy
Instantaneous power
p(t) Imag(t)Eind(t)
p( t) 15.631 W
d
p( t) Imag( t) Lind Imag( t)
dt
t
Energy(t )
Energy
L
ind
I mag (t )
d I mag (t )
dt
dt
I mag ( t )
1
2
Energy( t) LindImag ( t)
2
Lind
I
mag
I mag ( )
Lind I
1
2
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dI mag
I (t )
2
mag I ( )
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Transformers
5.2.4 Magnetization curves B (H)
•
•
•
The linear region, where the permeability of
the material is a constant
The transition or knee region, where the
material permeability reaches saturation; and
The saturation region.
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Transformers
M45 Semiprocessed 26 Gage Steel
12000
1.8
10000
1.6
1.4
8000
1.2
B (T)
1
6000
µr
0.8
4000
0.6
0.4
Realtive Permeability
Magnetic Flux Density, B (T)
2
2000
0.2
0
0
0.1
1
10
100
Magnetic Field Intensity, H (oersteds)
Figure 5.14 B(H) Magnetization curves and relative permeability
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Transformers
Figure 5.15 Magnetization curve regions and relative
permeability of a high-silicon transformer steel.
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Transformers
5.2.5 Magnetic circuit with air
gap
c
1
2
b
a = 3cm, b = 2cm, c = 4cm, g = 0.2cm, w = 10cm
h = 11 m, Bgap = 0.8T, Im = 8A
a
a
3
h
•
Dimensions of iron core
Lvert = (h –b) + (h-b-g)
Lvert = 17.8 cm
Avert = c a
Avert = 12 cm2
Lhorz = 2 (w –a)
Lhorz = 14 cm
Ahorz = c b
Ahorz = 12 cm2
c
g
4
b
5
6
w
Fig 5.16 Magnetic circuit with air gap
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Transformers
5.2.5 Magnetic circuit with air gap
•
Gap flux = Total flux
gap BgapAg
•
4
gap 9.6 10
Wb
Flux density in the horizontal and vertical sections
Bv ert
Bh orz
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g ap
Av ert
g ap
Ah orz
Bv ert 0.8 T
Bh orz 1.2 T
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Transformers
5.2.5 Magnetic circuit with air gap
•
Magnetic field intensity in the gap
Hg ap
•
Bg ap
5A
Hg ap 6.366 10
o
m
Magnetic field intensity in the vertical and horizontal sections from
Fig 5.17
Hv ert 140
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A
Hh orz 400
m
360 Chapter 5 Transformers
A
m
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Transformers
1.6
Magnetic Flux Density, B (T)
1.4
1.2
1
0.8
0.6
140 A/m
0.4
400 A/m
0.2
0
0
500
1000
1500
2000
2500
Magnetic Field Intensity, H (A-turn/m)
Figure 5.17 B(H) Magnetization curves
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Transformers
5.2.5 Magnetic circuit with air gap
•
Number of turns using Ampere law
Hg ap g Hv ert Lv ert Hh orz Lh orz
Nm
Im
•
Nm 169.27
Inductance calculation
Lw ith_ iron
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g ap Nm
Lw ith_ iron 20.312mH
Im
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