Transcript Lecture Notes EEE 360 - Warsaw University of Technology

EEE 360
Energy Conversion and
Transport
George G. Karady & Keith Holbert
Chapter 5
Transformers
7/17/2015
360 Chapter 5 Transformers
1
Lecture 9
7/17/2015
360 Chapter 5 Transformers
2
5.1 Construction
7/17/2015
360 Chapter 5 Transformers
3
Transformers
Primary winding
Supply
NP
Secondary winding
NS
Load
Laminated iron core
5.1 Basic components of single phase transformer
7/17/2015
360 Chapter 5 Transformers
4
Transformers
Primary
Winding
Multi-layer
Laminated
Iron Core
Secondary
Winding
H1 H2
X1
X2
Winding
Terminals
5.2 Single phase transformer arrangement
7/17/2015
360 Chapter 5 Transformers
5
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
7/17/2015
360 Chapter 5 Transformers
6
Transformers
Iron core
Insulation
Secondary
winding
Terminals
5.4 Small transformer construction a) Lamination, b) Iron core
with winding
7/17/2015
360 Chapter 5 Transformers
7
Transformers
Wires
Primary
Winding
Holder
Secondary
Winding
Insulation
5.5 Winding Construction
7/17/2015
360 Chapter 5 Transformers
8
Transformers
Figure 5.6
Dry-type
three-phase
transformer
7/17/2015
360 Chapter 5 Transformers
9
Transformers
Bushing
Steel
tank
Iron core
behind the steel
bar
Winding
Insulation
Radiator
Figure 5.7 Oil Insulated and cooled transformer
7/17/2015
360 Chapter 5 Transformers
10
Transformers
Metal bar
Exterior
porcelain tube
Flange
Interior
porcelain tube
Figure 5.8 Porcelain transformer
bushing
7/17/2015
360 Chapter 5 Transformers
11
Transformers
High voltage
bushing
Oil tank
Low voltage
bushing
Figure 5.9
Large Oil
cooled high
voltage
transformer
Cooling
radiators
7/17/2015
360 Chapter 5 Transformers
12
5.2 Magnetic circuit
7/17/2015
360 Chapter 5 Transformers
13
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
7/17/2015
360 Chapter 5 Transformers
14
Transformers
Figure 5.11 B(H) Magnetization loss curves
7/17/2015
360 Chapter 5 Transformers
15
Transformers
• Ampere law
A
I m Nm  H 
Im
• Permeability and flux density
E
Nm
B H

• Flux
BA
7/17/2015
Fig 5.12 Magnetic
circuit
360 Chapter 5 Transformers
16
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
EL
dt
7/17/2015
360 Chapter 5 Transformers
17
Transformers
• Inductance
L
 AN
2
m

• Magnetic energy
LI
Energy 
2
7/17/2015
360 Chapter 5 Transformers
2
m
18
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  
7/17/2015
Im Nm
Lm
A
Hm  98.425
m
b
a
Im
m
Nm
Vm
h
a
a
w
a
Fig 5.13 Magnetic circuit
360 Chapter 5 Transformers
19
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
7/17/2015
a
Im
Bm  1.237T
Fig 5.13 Magnetic circuit
360 Chapter 5 Transformers
20
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)  2Imcos t
7/17/2015
360 Chapter 5 Transformers
21
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
7/17/2015
360 Chapter 5 Transformers
22
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
7/17/2015
Erms  9.025V
360 Chapter 5 Transformers
Erms 
Nm  max 
2
23
Transformers
5.22 Magnetic circuit analysis
•
Simplified induced voltage equation
Erms  4.443f  Nmmax
•
Derivation of inductance
Eind( t)
d
Lind Imag ( t)
dt
d
Nm  mag ( t)
dt
7/17/2015
d
Lind Imag ( t)
dt
360 Chapter 5 Transformers
24
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
7/17/2015
Nm rms
Im
Li nd  11.969
mH
360 Chapter 5 Transformers
25
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
7/17/2015
A co re Nm
Lm
360 Chapter 5 Transformers
26
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)  LindImag ( t)
2
 Lind
I
mag
I mag (  )
 Lind I
1
2
7/17/2015
360 Chapter 5 Transformers
dI mag
I (t )
2
mag I (  )
27
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.
7/17/2015
360 Chapter 5 Transformers
28
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
7/17/2015
360 Chapter 5 Transformers
29
Transformers
Figure 5.15 Magnetization curve regions and relative
permeability of a high-silicon transformer steel.
7/17/2015
360 Chapter 5 Transformers
30
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
7/17/2015
360 Chapter 5 Transformers
31
Transformers
5.2.5 Magnetic circuit with air gap
•
Gap flux = Total flux
gap  BgapAg
•
4
gap  9.6  10
Wb
Flux density in the horizontal and vertical sections
Bv ert 
Bh orz 
7/17/2015
g ap
Av ert
g ap
Ah orz
Bv ert  0.8 T
Bh orz  1.2 T
360 Chapter 5 Transformers
32
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
7/17/2015
A
Hh orz  400
m
360 Chapter 5 Transformers
A
m
33
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
7/17/2015
360 Chapter 5 Transformers
34
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
7/17/2015
g ap Nm
Lw ith_ iron 20.312mH
Im
360 Chapter 5 Transformers
35