1 electromechanical energy conversion chapter 3. 2 1. it can be transmitted easily for long...
Post on 21-Dec-2015
221 views
TRANSCRIPT
![Page 1: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/1.jpg)
1
ELECTROMECHANICAL ENERGY
CONVERSION
CHAPTER 3
![Page 2: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/2.jpg)
2
1. it can be transmitted easily for long distance, at high
efficiency and reasonable cost.
Electrical energy is the most popular form of energy, because:
2. It can be converted easily to other forms of energy such as
sound, light, heat or mechanical energy.
![Page 3: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/3.jpg)
3
Hidro power station, Kenyir Terengganu
Power consumers, JB Johor
![Page 4: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/4.jpg)
4
Electrical energy
Electrical energy
Electrical energy
Sound energy
Loud speaker
Light energy
Lamp
KettleHeat energy
![Page 5: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/5.jpg)
5
converts electrical energy into mechanical energy
or
converts mechanical energy into electrical energy.
Electromechanical energy conversion device:
![Page 6: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/6.jpg)
6
a. Small motion
- telephone receivers, loud speakers, microphones
There are various electromechanical conversion devices may categorized as under:
b. Limited mechanical motion
- electromagnets, relays, moving-iron instruments,
moving-coil instruments, actuators
c. Continuous energy conversion
- motors, generators
![Page 7: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/7.jpg)
7
According to the principle of conservation of energy, energy can neither be created nor destroyed,
Principle of Energy Conversion
it can merely be converted from one form into another.
The total energy in a system is therefore constant.
![Page 8: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/8.jpg)
8
In an energy conversion device, out of the total input energy, some energy is converted into the required form, some energy is stored and the rest is dissipated.
Energy conversion in electromechanical system
It is possible to write an equation describing energy conversion in electromechanical system:
Electrical
energy from source
= Mechanical
energy to load
+ Increase of
field energy
+ Energy
converted to heat
(losses)
3.1
![Page 9: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/9.jpg)
9
The last term on the right-hand side of Eq. 3.1 (the losses) may be divided into three parts:
Electrical
energy from source
= Mechanical
energy to load
+ Increase of
field energy
+ Energy
converted to heat
(losses)
3.1
Energy
converted to heat
(losses)
= Resistance
losses
+ Friction and
windage losses
Field losses+ 3.2
Then substitution from Eq. 3.2 in Eq. 3.1 yields
![Page 10: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/10.jpg)
10
Electrical
energy from source minus resistance losses
= Mechanical
energy to load plus friction and windage losses
+ Increase of
magnetic coupling field energy plus core losses
3.3
Now consider an electromechanical system (actuator) illustrated in Fig. 3.1.
![Page 11: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/11.jpg)
11
u
Fm
e
iR
v
Bahagian tak bolehgerak
Bahagian bolehgerak
x
dx
Figure 3.1
Fixed steel core
Moveable steel armature
g
SW
![Page 12: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/12.jpg)
12
dt
de
At any instant, the emf e induced in the coil by the change in the flux linkage is
Consider now a differential time interval dt, during which the current in the coil is changing and the armature is moving.
volt 3.4
![Page 13: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/13.jpg)
13
Therefore, the differential energy transferred in time dt from the electric source to the coupling field is given by the energy output of the source minus the resistance loss:
dtRividtdWe2
idtRiv )(
eidt
eidtdWe
Joule
3.5
![Page 14: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/14.jpg)
14
The coupling field forms an energy storage to which energy supplied by the electric system. At the same time, energy is released from the coupling field to the mechanical system.
The rate of release energy is not necessarily equal at any instant to the rate of supply of energy to the field, so that the amount of energy stored in the coupling field may vary.
![Page 15: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/15.jpg)
15
It’s like a pipe system in our house.
The water out from the tap will make water flow into the storage tank from the supply.
Water tank
![Page 16: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/16.jpg)
16
It’s like a pipe system in our house.
The water out from the tap will make water flow into the storage tank from the supply.
Water tank
![Page 17: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/17.jpg)
17
In time dt, let dWf be the energy supplied to the field and either stored or dissipated. Let dWm be the energy converted to mechanical form, useful or as loss, in the same time, dt.
Then, by the principle of conservation of energy, the following equation may be written for the field:
fme dWdWdW
3.6
![Page 18: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/18.jpg)
18
Field Energy
To obtain an expression for for dWf of Eq. 3.6 in terms of the system variables, it is first necessary to find an expression for the energy stored in the magnetic field for any position of the armature. The armature will therefore be clamped at some value of air-gap length g so that no mechanical output can be produced.
dWm = 0 3.6
![Page 19: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/19.jpg)
19
Field Energy (continue…..)
If switch SW in Fig. 3.1 is now closed, the current will rise to a value v/R, and the flux will be established in the magnetic system. Let the relationship between coil flux linkage and the current i for the chosen air-gap length be that shown in Fig. 3.2
i
dWf
i1 i2
Fig. 3.2
![Page 20: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/20.jpg)
20
Since core loss is being neglected, this will be a single-valued curve passing through the origin. In the absence of any mechanical output energy, all of the electric input energy must be stored in the magnetic field:
3.8dWe = dWf
Field Energy (continue…..)
Substitution from Eqs. 3.4 and 3.8 in Eq. 3.5 yields
dWf = dWe = i.edt = id J 3.9
![Page 21: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/21.jpg)
21
If now v is changed, resulting in a change in current from i1 to i2, there will be a corresponding change in flux linkage from 1 to 2 .
Field Energy (continue…..)
The increase in energy stored during the transition between these two states is
The area is shown in Fig 3.2. When the flux linkage is increased from zero to , the total energy stored in the field is
2
1
iddW f J 3.10
![Page 22: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/22.jpg)
22
Field Energy (continue…..)
This integral represents the area between the –i characteristic and the –axis, the entire shaded area of Fig. 3.2.
If it is assumed that there is no leakage flux, so that all flux in the magnetic system links all N turns of the coil, then
= N Wb 3.12
3.11
idW f 0
J
![Page 23: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/23.jpg)
23
From Eqs. 3.9 and 3.12,
dWf = id = Nid = F d J 3.13
Field Energy (continue…..)
where
F = Ni A 3.14
F is mmf (magneto-motive force)
The characteristic of Fig. 3.3 can also represent the relationship between and F .
![Page 24: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/24.jpg)
24
i
dWf
i1 i2F1 F2
2
1
F
![Page 25: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/25.jpg)
25
If the reluctance of the air gap forms a large part of the total reluctance of the magnetic system, then that of the steel may be neglected and the –i characteristic becomes the straight line through the origin shown in Fig. 3.3. For this system,
3.16
Field Energy (continue…..)
Where L is the inductance of the coil. Substitution in Eq. 3.11 gives the energy Wf in several useful forms:
= Li Wb 3.15
222
22
0
iLi
Ld
LW f J
![Page 26: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/26.jpg)
26Fig. 3.3
i
dW f
i1 i 2
![Page 27: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/27.jpg)
27
If the reluctance of the magnetic system (that is, of the air gap) as seen from the coil is S, then F = S , and from Eq. 3.13,
3.17
Field Energy (continue…..)
J
S
SdW f 22
22
0
FF
J
If A is the cross-section area of the core and l = 2g is the total length of air gap in a flux path, then from Eq. 3.16,
HBlAFi
W f 2
1
22
J 3.18
![Page 28: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/28.jpg)
28
Where B is the flux density in the air gaps. Since B/H=µ0 and lA is the total gap volume, it follows from Eq. 3.18 that the energy density in the air gaps is
3.19
Field Energy (continue…..)
J
J/m3
0
22
0 2
1
2
1
2
1
B
HBHlA
Ww f
f
0
22
0 2
1
2
1
2
1
B
HBHlA
Ww f
f
Equations 3.16, 3.17 and 3.19 represent three different ways of expressing the field energy.
![Page 29: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/29.jpg)
29
Example 3.1 The core and armature dimensions of the actuator of Fig. 3.1 are shown in Fig. 3.4. Both parts are made of mild steel, whose magnetization curve is given in Fig. 3.5. Given la = 160 mm, lb = 80 mm. The coil has 2000 turns. Leakage flux and fringing may be neglected. The armature is fixed, so that the length of the air gas, lu= 9 mm, and a direct current is passed through the coil, producing a flux density of 0.8 T in the air gap.
a) Determine the required coil current.
b) Determine the energy stored in the air gap.
c) Determine the energy stored in the steel.
d) Determine the total field energy.
![Page 30: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/30.jpg)
30
20 mm
lu
20 mmlb
la
ketebalan = 20 mm
angker
Fig. 3.4
The thickness
The armature
![Page 31: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/31.jpg)
31
500 1000 1500 2000 2500 30000
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
H (AT/m)
B (Tesla)
Besi tuang
Keluli tuang
Keluli keping
Fig. 3.5
Sheet steel
mild steel
cast iron
![Page 32: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/32.jpg)
32
Solution
(a) Area, A = (20 10-3)(20 10-3) = 4 10-4 m2.
Ni = Htlt + Hulu
lt = 160 + 80 = 240 mm = 240 10-3 m lu = 2 9 mm = 18 mm = 18 10-3 m
Given Bu = 0.8 T Bu = Bt = 0.8 T
From Fig. 3.6, magnetic field intensity in the steel is, Ht = 450 A/m
![Page 33: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/33.jpg)
33
37
1062.636104
8.0
o
uu
BH
2000
)1018)(1062.636()10240)(450( 333 i
78.52000
16.11567
37
2
1065.254)104(2
B
w fu
For the air gaps
=
(b) Energy density in the air gaps is
J/m3
A/m
A
![Page 34: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/34.jpg)
34
Volume of air gaps = length of air gaps area of air gaps
= 0.018 0.02 0.02 = 7.2 10-6 m3
Energy stored in the air gaps, Wfu = the volume of air gaps wfu
= (7.2 10-6) 254.65 103
= 1.834 Joule.
(c) Energy density in the steel,
8.0
0 HdBw ft
![Page 35: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/35.jpg)
35
Energy density in the steel is given by the area enclosed between the characteristic and the B axis in Fig. 3.6 up to value of 0.8 T.
wft ½ 0.8 450 = 180 J/m3 (straight-line
approximation)
Volume of steel= length of steel area of steel = (240 10-3) (0.02 0.02) = 9.6 10-5 m3
Energy stored in the steel, Wft = 9.6 10-5 180 = 0.01728 Joule
(d) Total field energy, Wf = Wft + Wfu
= 0.01728 + 1.834= 1.851 Joule.
![Page 36: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/36.jpg)
36
The proportion of field energy stored in the steel is, therefore, seen to be negligibly.
![Page 37: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/37.jpg)
37
Coenergy, Wf’ is the area enclosed between the -i characteristic and the i axis of Fig.3.2.
Wf
Wf'
i
Fig. 3.6 Field energy and coenergy
Coenergy
For linear -i characteristic, Wf’ = Wf.For nonlinear -i characteristic, Wf’ > Wf.
![Page 38: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/38.jpg)
38
Mechanical Energy in a Linear System
It will be assumed that the armature of the actuator in Fig. 3.1 may move from position x1 to position x2, as a result, the length of air gaps is reduced. The –i characteristics for the two extreme positions of the armature may be assumed to be the two straight lines (linear).
![Page 39: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/39.jpg)
39
u
Fm
e
iR
v
Bahagian tak bolehgerak
Bahagian bolehgerak
x
dx
Figure 3.1
Fixed steel core
Moveable steel armature
g
SW
![Page 40: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/40.jpg)
40
i
λ
x1
x2
![Page 41: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/41.jpg)
4141
Consider a very slow armature displacement. It may assumed that it takes place at essentially constant current as illustrated in Fig. 3.7 (as d/dt is negligible). The operational point has changed from a to b.
Mechanical Energy in a Linear System
)( 1221 oi
At the moment of armature movement,
The change of field energy,
3.20
3.21
)( 12
2
1
oe iideidtW
121
221 oof iiW
![Page 42: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/42.jpg)
42
c
d
b
a
io
x = x2
x = x1
io
Fig. 3.7 Current is fixed
![Page 43: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/43.jpg)
43
From Eq. (3.6),
Mechanical Energy in a Linear System
fme WWW
fem WWW
)()( 1221
12 oo ii
)( 1221 oi
= ΔWf
= ΔWf’ = the change of coenergy
![Page 44: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/44.jpg)
44
Mechanical Energy in a Linear System
where dWm = Fmdx
Fm = mechanical force on moving part (armature)
Fmdx = dWf’ 3.21
For small change of x or dx,
dWm = dWf’
![Page 45: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/45.jpg)
45
),( xix
WF f
m
Eq. 3.21 can be written as,
Eq. 3.22 is partial differential since Wf is function of more than one variable.
N
i = constant
3.22
Mechanical Energy in a Linear System
![Page 46: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/46.jpg)
46
Consider now a very rapid differential armature displacement dx. It may be assumed that it takes place at essentially constant flux linkage o, as illustrated in Fig. 3.8. At the instant, the current is changed from i1 to i2 , where i1 > i2.
Mechanical Energy in a Linear System
io i2 i1
dW f
Fig. 3.8 Flux linkage is fixed
![Page 47: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/47.jpg)
47
121
221 iidW oof
)( 1221 iio
Refer to Fig. 3.8, the change of field energy is
Since does not change, no emf is induced in the coil , and
dWe = 0
From Eq.3.6,
Mechanical Energy in a Linear System
3.23
3.24
![Page 48: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/48.jpg)
48
fme dWdWdW
)( 1221 iidxF om
),( xx
WF f
m
-Fmdx = dWf
Eq. 3.26 can be written as,
= constant
Mechanical Energy in a Linear System
3.27
3.25
3.26
Since the electrical input energy is zero, the mechanical output energy has been supplied entirely by the coupling field.
= the change of field energy
![Page 49: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/49.jpg)
49
Mechanical Energy in a Linear System
)(2)(
2
00 xLd
xLidW f
2
21
22
)()(2
)(ixL
xL
ixL
221 )( ixLWW ff
For a linear electromagnetic system,
= L(x) i
where L(x) = the inductance of the coil which dependent on length of the air gaps.
From Eqs. 3.11 and 3.28,
3.28
3.29
3.30
3.31
![Page 50: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/50.jpg)
50
Mechanical Energy in a Linear System
From Eqs. 3.22 and 3.31,
),( xix
WF f
m
))(( 221 ixL
x
dx
xdLi
)(
2
1 2
dx
xdLiFm
)(
2
1 2
i = constant
i = constant
3.32
3.33
![Page 51: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/51.jpg)
51
Mechanical Energy in a Linear System
From Fig. 3.1 (for linear system),
gHNi u 2
gB
o
u 2
o
uB
2
2
o
uu
BgA
22
2
From Eq. 3.18
Wf = volume of air gaps
where Au = cross section area of air gap
3.34
3.35
![Page 52: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/52.jpg)
52
Mechanical Energy in a Linear System
From Fig. 3.1, it is seen that a positive displacement dx will correspond to a reduction dg in the air gap length. Thus,
dx = dg m 3.36
o
uum
BgA
gF
22
2
o
uu
BA
22
2
From Eqs. 3.27, 3.35 and 3.36 yield,
Fm
where2Au = The total cross-section area of air gaps
3.37
![Page 53: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/53.jpg)
53
Mechanical Energy in a Linear System
o
um
Bf
2
2
The force per unit area of air gaps, fm is
N/m2 3.38
![Page 54: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/54.jpg)
54
Example 3.2An electromagnet system is shown in Fig. 3.9.
luN
i
Given that N = 600, i = 3 A, cross section area of air gap is 5 cm2 and air gap length is 1.5 mm. By neglecting core reluctance, leakage flux and fringing effects, find:(a) Force between the electromagnetic surfaces.(b) Energy stored in the air gap.
Fig. 3.9: linear system
![Page 55: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/55.jpg)
55
o
uum
BAF
2
2
o
uuuu
lBlHNi
Solution
(a) The total cross-section area of air gap = Au, Eq. 3.37
becomes,
For linear system,
3.39
u
ou l
NiB
3.40
![Page 56: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/56.jpg)
56
2
22
2
u
oum
l
iNAF
23
2274
)105.1(2
)3()600)(104)(105(
Substitution from Eq. 3.40 in Eq. 3.39 yields
= 452.39 N
(b) Since the system is linear, the entire field energy is stored in the air gap,
![Page 57: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/57.jpg)
57
= lu Fm
= (1.5 10-3) 452.39 Nm = 0.6789 Nm = 0.6789 Joule
o
uf
BW
2 gapair of volume
2
o
uuu
BAl
2
2
![Page 58: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/58.jpg)
58
Example 3.3 Electromagnet system in Fig. 3.10 has cross-section
area 25 cm2. The coil has 350 turns and 5 ohm resistance. Magnetic core reluctance, fringing effects and leakage flux can be neglected. If the length of air gap is 4 mm and a 110 V DC supply is connected to the coil, find
(a) Stored field energy(b) Lifting force
lu
Fig. 3.10
![Page 59: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/59.jpg)
59
225
110i
uo
uuu l
BlHNi
u
ou l
NiB
2
)104(2
)22)(350)(104(3
7
SolutionCoil current,
Since the electromagnet system is linear, core reluctance is neglected,
= 1.2095 Tesla
A
![Page 60: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/60.jpg)
60
o
uf
BW
2 gapair of volume
2
7
234
1042
2095.1)104()1025(2
o
um
BF
2 gapsair of area total the
2
)104(2
2095.1)1025(2
7
24
Field energy,
= 11.6413 Joule
(b) Applying Eq. 3.37 to obtain lifting force,
= 2910.33 N
![Page 61: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/61.jpg)
61
Mechanical Energy in a Saturable System
Figure 3.11 shows a diagram illustrating the -i characteristics for the actuator in Fig. 3.1 when the effect of the ferromagnetic material is taken into account. It is no longer a linear system due to saturation of the steel. Wf is smaller than coenergy Wf’.
io
x + dx
x
di
i
dWm
Fig. 3.11: At constant flux linkage.
![Page 62: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/62.jpg)
62
However the areas of this diagram may be interpreted in exactly the same way as were those of Fig. 3.8 for the ideal linear system. Field energy is still given by Eq. 3.11.
Mechanical Energy in a Saturable System
If an analytical expression is available that gives the coil current as a function of and x, then the force on the armature for a given value of x can readily be determined. Fig. 3.11 illustrates a differential movement of the operating point in the -i diagram corresponding to a differential displacement dx of the armature made at high speed; that is, at constant flux linkage.
![Page 63: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/63.jpg)
63
Mechanical Energy in a Saturable System
By integrating to obtain an expression for the area between the -i curve for any x and the –axis, Wf is obtained as a function of and x can be written as
Wf = Wf (,x) J 3.41
For the movement, the electrical energy input is zero, since does not change and the emf is zero. Consequently,
dWm = dWf (,x) 3.42
![Page 64: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/64.jpg)
64
Mechanical Energy in a Saturable System
and
xx
W
dx
dWF fm
m ,
xx
WF f
m ,
= constant
= constant
3.43
This corresponds to the expression for a linear system in Eq. 3.27
![Page 65: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/65.jpg)
65
Mechanical Energy in a Saturable System
More usually, however, it is convenient to express as a function of x and i and to employ different approach. Figure 3.12 illustrates a differential movement of the operating point in the -i diagram corresponding to a differential displacement dx of the armature made at low speed; that is, at constant current. For this displacement, during flux linkage changes, the emf is not zero, and therefore dWe is not zero.,
2
1
ideidtdWe
= area defg
dWf = area oef – area odg
3.44
3.45
![Page 66: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/66.jpg)
66
Mechanical Energy in a Saturable System
f
g
e
d
io
x + dx
xd
dWf
i
Fig. 3.12: At constant current
![Page 67: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/67.jpg)
67
Mechanical Energy in a Saturable System
xidx
WF f
m ,
dWm = dWe – dWf
= area defg + area odg – area oef = area ode
The differential mechanical energy associated with movement dx is given by the shaded area, it is equal to the increase of coenergy.
dWm = dWf’Fmdx = dWm = dWf’
From Eq. 3.6,
3.46
3.47
i = constant
![Page 68: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/68.jpg)
68
Mechanical Energy in a Saturable System
i
f dixiW
0 ),(
where
Wf’ is the function of i
and x.
Joule 3.47
Also since = NΦ, and i = F /N, substitution in Eq. 3.47 yields the coenergy as a function of mmf and displacement
F
0 ),( FF dxW f 3.48
![Page 69: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/69.jpg)
69
Mechanical Energy in a Saturable System
and
xx
WF f
m ,F
3.49
F = constant
![Page 70: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/70.jpg)
70
Example 3.4
The flux linkage and current relationship for an actuatorcan be expressed approximately by,
2
2
gi
Between the limits 0 < i < 3 A and 3 < g < 9 cm. If the current is maintained at 2 A, what is the force on the armature for g = 4 cm?
![Page 71: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/71.jpg)
71
Solution
The -i relationship is nonlinear, and thus the force must be determined using Eq. 3.11.
32322
00 12
1
342
xx
dx
idλW f
12
23 x
From Eq. 3.43,
xx
WF f
m ,
= constant
=
![Page 72: 1 ELECTROMECHANICAL ENERGY CONVERSION CHAPTER 3. 2 1. it can be transmitted easily for long distance, at high efficiency and reasonable cost. Electrical](https://reader031.vdocuments.site/reader031/viewer/2022020117/56649d5d5503460f94a3ce8c/html5/thumbnails/72.jpg)
72
71.7004.0
222 21
21
x
i
12
04.0271.70 3 mF
For x = 0.04 m and i = 2 A,
= 2356.95 N
Wb-turn