eec principles
TRANSCRIPT
-
8/11/2019 EEC Principles
1/49
Nguyn Cng Phng
ELECTROMECHANICAL ENERGY
CONVERSION
Electromechanical Energy
Conversion Principles
-
8/11/2019 EEC Principles
2/49
Contents
I. Magnetic Circuits and Magnetic MaterialsII. Electromechanical Energy Conversion
Principles
III. Introduction to Rotating Machines
IV. Synchronous MachinesV. Polyphase Induction Machines
VI. DC Machines
VII.VariableReluctance Machines and Stepping
MotorsVIII.Single and TwoPhase Motors
IX. Speed and Torque Control
sites.google.com/site/ncpdhbkhn 2
-
8/11/2019 EEC Principles
3/49
ElectromechanicalEnergy
Conversion Principles
1. Forces and Torques in Magnetic Field Systems2. Energy Balance
3. Energy in SinglyExcited Magnetic FieldSystems
4. Determination of Magnetic Force and Torquefrom Energy and Coenergy
5. MultiplyExcited Magnetic Field Systems
6. Forces and Torques in Systems with Permanent
Magnets7. Dynamic Equations
8. Analytical Techniques
sites.google.com/site/ncpdhbkhn 3
-
8/11/2019 EEC Principles
4/49
sites.google.com/site/ncpdhbkhn 4
Forces and Torques in Magnetic
Field Systems (1)
( )q F E v B
qF Ev
B
F
( )q F v B
( )v F v B
J v
v F J B
-
8/11/2019 EEC Principles
5/49
sites.google.com/site/ncpdhbkhn 5
Forces and Torques in Magnetic
Field Systems (2)Ex.A nonmagnetic rotor contains a single
turn coil, it
is in a uniform magnetic field. The rotor is of radius
R and of lengthl. Find thedirected torque as a
function of?
0 B y
x
I
I
v F J B
1( )S F J B
I B
0
sinin
F IB l
0 sinoutF IB l
0(2 ) 2 sin (Nm)T R F RIB l
-
8/11/2019 EEC Principles
6/49
sites.google.com/site/ncpdhbkhn 6
Forces and Torques in Magnetic
Field Systems (3)
Lossless magnetic
energy storage system
i
,e
fldf
x
Electrical
terminal
Mechanical
terminal
Lossless winding
i1
e
Magnetic core
v
Winding
resistance x fld
f
Movable
magnetic
plunger
-
8/11/2019 EEC Principles
7/49
sites.google.com/site/ncpdhbkhn 7
Forces and Torques in Magnetic
Field Systems (4)
Lossless magnetic
energy storage system
i
,e
fldf
x
Electrical
terminal
Mechanical
terminal
fld
fld
dW dxei f
dt dt
d
e dt
fld flddW id f dx
-
8/11/2019 EEC Principles
8/49
ElectromechanicalEnergy
Conversion Principles
1. Forces and Torques in Magnetic Field Systems2. Energy Balance
3. Energy in SinglyExcited Magnetic FieldSystems
4. Determination of Magnetic Force and Torquefrom Energy and Coenergy
5. MultiplyExcited Magnetic Field Systems
6. Forces and Torques in Systems with Permanent
Magnets7. Dynamic Equations
8. Analytical Techniques
sites.google.com/site/ncpdhbkhn 8
-
8/11/2019 EEC Principles
9/49
sites.google.com/site/ncpdhbkhn 9
Energy Balance
Energy is neither created or destroyed, it is merely changed in form
electrical mechanical field dW eidt dW dW
-
8/11/2019 EEC Principles
10/49
ElectromechanicalEnergy
Conversion Principles
1. Forces and Torques in Magnetic Field Systems2. Energy Balance
3. Energy in SinglyExcited Magnetic FieldSystems
4. Determination of Magnetic Force and Torquefrom Energy and Coenergy
5. MultiplyExcited Magnetic Field Systems
6. Forces and Torques in Systems with Permanent
Magnets7. Dynamic Equations
8. Analytical Techniques
sites.google.com/site/ncpdhbkhn 10
-
8/11/2019 EEC Principles
11/49
-
8/11/2019 EEC Principles
12/49
sites.google.com/site/ncpdhbkhn 12
Energy in SinglyExcited
Magnetic Field Systems (2)
21 ( )
2
fldW L x i
2201 (1 / ) 1 J
2 2fld
N ld x d xW i K
g d
LosslessNturns coil
+
i
h
g
x g
d
l
x
d
d x g
gMagneticflux
Ex.
A relay is made of infinitely
permeable magnetic
material,h >>g. Compute the magnetic energy as
a function of plunger position?
02( )2
gS
L x Ng
( ) 1g
xS l d x ld
d
-
8/11/2019 EEC Principles
13/49
ElectromechanicalEnergy
Conversion Principles
1. Forces and Torques in Magnetic Field Systems2. Energy Balance
3. Energy in SinglyExcited Magnetic FieldSystems
4. Determination of Magnetic Force and Torquefrom Energy and Coenergy
5. MultiplyExcited Magnetic Field Systems
6. Forces and Torques in Systems with Permanent
Magnets7. Dynamic Equations
8. Analytical Techniques
sites.google.com/site/ncpdhbkhn 13
-
8/11/2019 EEC Principles
14/49
sites.google.com/site/ncpdhbkhn 14
Determination of Magnetic Force and
Torque from Energy and Coenergy (1)
( , )fld flddW x id f dx
const
const
( , )
( , )
fld
x
fld
fld
W x
i
W xf
x
const const
( , ) ( , )( , )
fld fld
fld
x
W x W xdW x d dx
x
21( , )
2 ( )fldW x
L x
2 2
2
const
1 ( )
2 ( ) 2 ( )fld
dL xf
x L x L x dx
( )L x i
2 ( )
2fld
i dL xf
dx
-
8/11/2019 EEC Principles
15/49
sites.google.com/site/ncpdhbkhn 15
Determination of Magnetic Force and
Torque from Energy and Coenergy (2)Ex. 1
Plot the force as a function of position for a current of 1A.
x (cm) 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
L (mH) 2.8 2.26 1.78 1.52 1.34 1.26 1.20 1.16 1.13 1.11 1.10
-
8/11/2019 EEC Principles
16/49
sites.google.com/site/ncpdhbkhn 16
Determination of Magnetic Force and
Torque from Energy and Coenergy (3)Ex. 2
Plot the force as a function of position for a flux of 2mWb.
x (cm) 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
L (mH) 2.8 2.26 1.78 1.52 1.34 1.26 1.20 1.16 1.13 1.11 1.10
-
8/11/2019 EEC Principles
17/49
sites.google.com/site/ncpdhbkhn 17
Determination of Magnetic Force and
Torque from Energy and Coenergy (4)
( , )fld flddW x id f dx
const
( , )fld
fld
WT
( , )fld flddW id T d
const const
( , ) fld fld
fld
W WdW d d
21
( , )2 ( )
fldW xL x
2 2
2
const
1 ( )
2 ( ) 2 ( )fld
dLT
L x L d
( )L i
2 ( )
2fld
i dLT
d
-
8/11/2019 EEC Principles
18/49
sites.google.com/site/ncpdhbkhn 18
Determination of Magnetic Force and
Torque from Energy and Coenergy (5)
Rotor
Air gap
i
0 2( ) cos(2 )L L L
2 ( )
2fld
i dLT
d
2
2( ) [ 2 sin(2 )]2
fld
iT L
-
8/11/2019 EEC Principles
19/49
sites.google.com/site/ncpdhbkhn 19
Determination of Magnetic Force and
Torque from Energy and Coenergy (6)
( , ) ( , )fld fldW i x i W x
( , ) ( ) ( , )fld fld
dW i x d i dW x
( )d i id di
( , )fld flddW x id f dx
( , )fld fld
dW i x di f dx
const const
( , ) fld fld
fld
x i
W WdW i x di dx
i x
const
const
( , )
( , )
fld
x
fld
fld
i
W i x
i
W i xf
x
-
8/11/2019 EEC Principles
20/49
sites.google.com/site/ncpdhbkhn 20
Determination of Magnetic Force and
Torque from Energy and Coenergy (7)
LosslessNturns coil
+
i
h
g
x g
d
l
Ex. 3
A relay is made of infinitely
permeable magneticmaterial,h >>g. Compute the magnetic energy as
a function of plunger position, ifi(x) =I0x/d(A)?
2
0 (1 / )
( )2
N ld x dL x
g
02( )2
gSL x N
g
( ) 1g xS l d x ld d
2 ( )
2fld
i dL x
f dx
2 2
0( )2 2
i N lL x
g
0( ) x
i x Id
22 2
0 0( )4
I N l xL x
g d
-
8/11/2019 EEC Principles
21/49
-
8/11/2019 EEC Principles
22/49
sites.google.com/site/ncpdhbkhn 22
Determination of Magnetic Force and
Torque from Energy and Coenergy (9)
2ag
gH Ni
g r
i
Ex. 4
Find the torque acting on the rotor as a function ofthe dimensions and the magnetic field in the two
air gap, suppose that the reluctance of the steel is
negligible ( ). The axial length is h.
1Energy density :2
BH
21Energy densityof the core : 0
2 2
steelsteel steel
BB H
2
0Energy densityof the air-gap :
2 2
ag ag agB H H
2
0[2 ( 0.5 ) ]
2
ag
ag
HW gh r g
2
0( ) ( 0.5 )
4
Ni h r g
g
const
( , )agfld
i
W iT
2
0( ) ( 0.5 )
4fld
Ni h r gT
g
-
8/11/2019 EEC Principles
23/49
sites.google.com/site/ncpdhbkhn 23
Determination of Magnetic Force and
Torque from Energy and Coenergy (10)
02( )2
gSL N
g
g r
i
Ex. 5
Find the inductance as a function of, then extractthe expression for the torque acting on the rotor as
a function ofi &. The axial length ish.
( 0.5 )gS h r g
2
0 ( 0.5 )
( )
2
N h r gL
g
2 ( )
2fld
i dLT
d
2
0( ) ( 0.5 )
4fld
Ni h r gT
g
-
8/11/2019 EEC Principles
24/49
ElectromechanicalEnergy
Conversion Principles
1. Forces and Torques in Magnetic Field Systems2. Energy Balance
3. Energy in SinglyExcited Magnetic FieldSystems
4. Determination of Magnetic Force and Torquefrom Energy and Coenergy
5. MultiplyExcited Magnetic Field Systems
6. Forces and Torques in Systems with Permanent
Magnets7. Dynamic Equations
8. Analytical Techniques
sites.google.com/site/ncpdhbkhn 24
-
8/11/2019 EEC Principles
25/49
sites.google.com/site/ncpdhbkhn 25
MultiplyExcited Magnetic
Field Systems (1)
Lossless magnetic
energy storage system
1i
1
fldT
Electrical
terminals
Mechanical
terminal
2i
2
1 2 1 1 2 2( , ) ( , , )fld fld fld flddW id T d dW i d i d T d
2 22 1
1 2 1 2 1 2
1 2
1 2 , are const, are const , are const
( , , ) ( , , ) ( , , ); ;
fld fld fld
fld
W W Wi i T
1 22 1
1 2 1 21 2 , are const, are const , are const
( , , )
fld fld fld
fld
W W W
dW d d d
-
8/11/2019 EEC Principles
26/49
sites.google.com/site/ncpdhbkhn 26
MultiplyExcited Magnetic
Field Systems (2)
Lossless magnetic
energy storage system
1i
1
fldT
Electrical
terminals
Mechanical
terminal
2i
2
1 11 1 12 2
2 21 1 22 2
L i L i
L i L i
22 1 12 2 22 1 12 21
11 22 12 21
21 1 11 2 21 1 11 22
11 22 12 21
L L L Li
L L L L D
L L L Li
L L L L D
2 2 12 010 20 0 11 0 20 22 0 10 10 20
0 0 0
1 1 ( )( , , ) ( ) ( )
2 ( ) 2 ( ) ( )fld
LW L L
D D D
1 1,e
-
8/11/2019 EEC Principles
27/49
sites.google.com/site/ncpdhbkhn 27
MultiplyExcited Magnetic
Field Systems (3)1 2 1 1 2 2( , , )fld fldW i i i i W
1 2 1 1 2 2( , , )fld flddW i i di di T d
2
1
1 2
1 2
1
1 , are const
1 2
2
1 , are const
1 2
, are const
( , , )
( , , )
( , , )
fld
i
fld
i
fld
fld
i i
W i i
i
W i i
i
W i iT
2 2
1 2 11 1 22 2 12 1 2
1 1( , , ) ( ) ( ) ( )
2 2fld
W i i L i L i L i i
1 2
2 21 2 1 11 2 22 12
1 2
, are const
( , , ) ( ) ( ) ( )
2 2
fld
fld
i i
W i dL i dL dLT i i
d d d
-
8/11/2019 EEC Principles
28/49
sites.google.com/site/ncpdhbkhn 28
MultiplyExcited Magnetic
Field Systems (4)
+
+
1 1,e 2 2,e
1i
2i
mechT
fldT
Ex.
L11 =a + cos2,L12 =bcos,L22 =c +ecos2. FindTfld()?
2 2
1 11 2 22 121 2
( ) ( ) ( )
2 2fld
i dL i dL dLT i id d d
2 2
1 2 1 2sin 2 sin 2 sini ei bi i
-
8/11/2019 EEC Principles
29/49
sites.google.com/site/ncpdhbkhn 29
MultiplyExcited Magnetic
Field Systems (5)2 2
1 2 1 2sin 2 sin 2 sinfldT i ei bi i
0 1 2 3 4 5 6-4
-3
-2
-1
0
1
2
3
4x 10
-3
(rad)
Torque(Nm)
Reluctance torque
Mutual - interaction torque
Total torque
-
8/11/2019 EEC Principles
30/49
sites.google.com/site/ncpdhbkhn 30
MultiplyExcited Magnetic
Field Systems (6)
1 2
1 2
1 2
, are const
2 21 2 1 11 2 22 12
1 2
, are const
2 2
1 2 11 1 22 2 12 1 2
( , , )
( , , ) ( ) ( ) ( )
2 2
1 1( , , ) ( ) ( ) ( )2 2
fld
fld
fld
fld
i i
fld
WT
W i i i dL i dL dLT i i
d d d
W i i L i L i L i i
1 2
1 2
1 2
, are const
2 21 2 1 11 2 22 12
1 2
, are const
2 2
1 2 11 1 22 2 12 1 2
( , , )
( , , ) ( ) ( ) ( )2 2
1 1( , , ) ( ) ( ) ( )
2 2
fld
fld
fld
fld
i i
fld
W xf
x
W i i x i dL x i dL x dL xf i ix dx dx dx
W i i x L x i L x i L x i i
-
8/11/2019 EEC Principles
31/49
ElectromechanicalEnergy
Conversion Principles
1. Forces and Torques in Magnetic Field Systems2. Energy Balance
3. Energy in SinglyExcited Magnetic FieldSystems
4. Determination of Magnetic Force and Torquefrom Energy and Coenergy
5. MultiplyExcited Magnetic Field Systems
6. Forces and Torques in Systems with
Permanent Magnets7. Dynamic Equations
8. Analytical Techniques
sites.google.com/site/ncpdhbkhn 31
-
8/11/2019 EEC Principles
32/49
sites.google.com/site/ncpdhbkhn 32
Forces and Torques in Systems
with Permanent Magnets (1)
x
Fictitious windingNf turns
x
fi
f
( , )fld flddW i x di f dx
( , )fld f f f flddW i x di f dx
const
( 0, )
f
fld f
fld
i
W i xf
x
0
0
( 0, ) ( , )f
fld f f f fI
W i x i x di
-
8/11/2019 EEC Principles
33/49
sites.google.com/site/ncpdhbkhn 33
Forces and Torques in Systems
with Permanent Magnets (2)
Fictitious
winding
x
f
i
fN
gW
0W
d
mW
0g
DepthDEx. 1
Find an expression for:a) the coenergy of the system as a function of plunger
positionx?
b) the force on the plunger as a function ofx?
0 0f f m gN i H d H x H g
0 0 0m g m m g gB S B S B S
0 0m m g gB W D B W D B W D
( )m R m cB H H
0
0 0
( )R f f cm
Rm
g
N i H dBx g
d WW W
f f m f m m f m mN N B S N B W D
0
0 0
( )f m R f f cf
Rm
g
N W D N i H d
x gd W
W W
-
8/11/2019 EEC Principles
34/49
sites.google.com/site/ncpdhbkhn 34
Forces and Torques in Systems
with Permanent Magnets (3)Ex. 1Find an expression for:a) the coenergy of the system as a function of plunger
positionx?
b) the force on the plunger as a function ofx?
Fictitious
winding
x
f
i
fN
gW
0W
d
mW
0g
DepthD
0
0 0
( )f m R f f cf
R
mg
N W D N i H d
x gd W
W W
00 c
f f f
f
H di I
N
0
0
( )f
fld f fI
W x di 2
0
0 0
( )
2
m R c
Rm
g
W D H d
x gd WW W
2 2
const 0
0 0
( 0, ) ( )
2f
fld f m R cfld
i Rg m
g
W i x W D H d f
x x gW d W
W W
-
8/11/2019 EEC Principles
35/49
sites.google.com/site/ncpdhbkhn 35
Forces and Torques in Systems
with Permanent Magnets (4)
External
magnetic
circuit
1
d
eF
S
mH
( )m R m cB H H
External
magnetic
circuit
2
d
eF
S
m R mB H
( )equivNi
0m eH d F
1 ( )
m R m cB S S H H
eR c
FS H
d
eNi F H
2 RBS HS
2
( )equiv eR
Ni FS
d d
If ( )equiv cNi H d
1 2
-
8/11/2019 EEC Principles
36/49
sites.google.com/site/ncpdhbkhn 36
Forces and Torques in Systems
with Permanent Magnets (5)Ex. 2a) Find thex
directed force on the plunger when the
current in the excitation winding is zero and x = 3 mm?
b) Find the current in the excitation winding required to
reduce the plunger force to zero? x
1i
1N
gW
W
d
W
0g
DepthD
( )equiv cNi H d
+
+
( )equiv
Ni
1 1N i
0gR
mR
xR
00
0 0
; ;m x
R R g
d d x gR R R
S WD W D WD
2
1
1
2fldW Li
2
1
total
NL
R
2
0
( )1
2
equiv
fld
m x
NiW
R R R
-
8/11/2019 EEC Principles
37/49
sites.google.com/site/ncpdhbkhn 37
Forces and Torques in Systems
with Permanent Magnets (6)Ex. 2a) Find thex
directed force on the plunger when the
current in the excitation winding is zero?
b) Find the current in the excitation winding required to
reduce the plunger force to zero? x
1i
1N
gW
W
d
W
0g
DepthD
+
+
( )equiv
Ni
1 1N i
0gR
mR
xR
2
0
( )1
2
equiv
fld
m x
NiW
R R R
2
2
0const
( )
( )equiv
fld equiv xfld
m xi
W Ni dRf
x R R R dx
2
2
0 0
( )
( )
equiv
g m x
Ni
W D R R R
1 1( ) 0equivNi N i 1
1
( )equivNi
iN
-
8/11/2019 EEC Principles
38/49
ElectromechanicalEnergy
Conversion Principles
1. Forces and Torques in Magnetic Field Systems2. Energy Balance
3. Energy in SinglyExcited Magnetic FieldSystems
4. Determination of Magnetic Force and Torquefrom Energy and Coenergy
5. MultiplyExcited Magnetic Field Systems
6. Forces and Torques in Systems with Permanent
Magnets7. Dynamic Equations
8. Analytical Techniques
sites.google.com/site/ncpdhbkhn 38
-
8/11/2019 EEC Principles
39/49
sites.google.com/site/ncpdhbkhn 39
Dynamic Equations (1)
+
Electromechanical
energyconversion
system
x
,e
0v
fldf
i
R
K
B
M
0f
0
dv Ri
dt
( )L x i0
( )( )
di dL x dxv Ri L x idt dx dt
-
8/11/2019 EEC Principles
40/49
sites.google.com/site/ncpdhbkhn 40
Dynamic Equations (2)
+
Electromechanical
energyconversion
system
x
,e
0v
fldf
i
R
K
B
M
0f0( )Kf K x x
D
dxf B
dt
2
2M
d xf M
dt
0 0
fld K D Mf f f f f
2
0 02( ) 0flddx d x
f K x x B M fdt dt
-
8/11/2019 EEC Principles
41/49
sites.google.com/site/ncpdhbkhn 41
Dynamic Equations (3)
+
Electromechanical
energyconversion
system
x
,e
0v
fldf
i
R
K
B
M
0f
0
2
0 0 2
( )( ) ( )
( ) ( ) ( , )fld
di dL x dxv t Ri L x i
dt dx dt dx d x
f t K x x B M f x idt dt
-
8/11/2019 EEC Principles
42/49
sites.google.com/site/ncpdhbkhn 42
Dynamic Equations (4)Ex.
Extract the dynamic equations of motion of theelectromechanical system?
Length of flux path in the direction of field
(area of flux path perpendicular to field)R
1
0
g
gR
dx
2
0
g
gR
da
1 2
0 0
1 1g g
g g a xR R R
d x a da x
2 2
0
2
0
( ) ,N daN x x
L x LR g a x a x
daNL
g
0l
1l
x a
a
h
d
g
CoilSpring,K
Cylindrical
steel plunger,
MApplied force,ft
-
8/11/2019 EEC Principles
43/49
sites.google.com/site/ncpdhbkhn 43
Dynamic Equations (5)Ex.
Extract the dynamic equations of motion of theelectromechanical system?
2 2
2
const
( , )
2 2 ( )
fld
fld
i
W i x i dL i aLf
x dx a x
( )d Li di dL di dL dxe L i L i
dt dt dt dt dx dt
2( )
x di ai dxL L
a x dt a x dt
2
0( ) ,x daN
L x L La x g
0l
1l
x a
a
h
d
g
CoilSpring,K
Cylindrical
steel plunger,
MApplied force,ft
-
8/11/2019 EEC Principles
44/49
sites.google.com/site/ncpdhbkhn 44
Dynamic Equations (6)
0l
1l
x a
a
h
d
g
CoilSpring,K
Cylindrical
steel plunger,
MApplied force,ft
Ex.
Extract the dynamic equations of motion of theelectromechanical system?
2 2
2
2
2 2 ( )
( )
fld
i dL i aLf
dx a x
x di ai dxe L L
a x dt a x dt
0
2
0 0 2
( )( ) ( )
( ) ( ) ( , )fld
di dL x dxv t Ri L x i
dt dx dt
dx d xf t K x x B M f x i
dt dt
0 2
2 2
0 0 2 2
( )( )
( ) ( )2 ( )
x di a dxv t Ri L L i
a x dt a x dt
dx d x i aLf t K x l B M
dt dt a x
-
8/11/2019 EEC Principles
45/49
ElectromechanicalEnergy
Conversion Principles
1. Forces and Torques in Magnetic Field Systems2. Energy Balance
3. Energy in SinglyExcited Magnetic FieldSystems
4. Determination of Magnetic Force and Torquefrom Energy and Coenergy
5. MultiplyExcited Magnetic Field Systems
6. Forces and Torques in Systems with Permanent
Magnets7. Dynamic Equations
8. Analytical Techniques
sites.google.com/site/ncpdhbkhn 45
-
8/11/2019 EEC Principles
46/49
sites.google.com/site/ncpdhbkhn 46
Analytical Techniques (1)
2
2 2
0 2 2
( )( )
( ) ( )2 ( )
x di a dxv t Ri L L i
a x dt a x dt
dx d x i aLf t K x l B M
dt dt a x
2
(0)If
,( )
v V
x di a dxRi L L i
a x dt a x dt
Vi
R
0If
0
f
M
2
02
1( ) ( )
2 ( )
dx a V B L K x l f x
dt a x R
0 ( )
X Bt dx
f x
-
8/11/2019 EEC Principles
47/49
sites.google.com/site/ncpdhbkhn 47
Analytical Techniques (2)
2
2 2
0 2 2
( )( )
( ) ( )2 ( )
x di a dxv t Ri L L i
a x dt a x dt
dx d x i aLf t K x l B M
dt dt a x
2
(0)If
,( )
v Vx di a dx
Ri L L ia x dt a x dt
Vi
R
0If
0
f
B
22
02 2
1( ) ( )
2 ( )
d x a V M L K x l f x
dt a x R
0
2( )
( )
xdx Bv x dx
dt M f x
-
8/11/2019 EEC Principles
48/49
-
8/11/2019 EEC Principles
49/49
sites.google.com/site/ncpdhbkhn 49
Analytical Techniques (4)
0 00 0 2
0 0
2 2
00 0 0
2 20
( ) ( )( )
( )
1 ( )( )
2 ( ) t
L X x di L a I i dxV v R I i
a X x dt a X x dt
L a I i dx d xK X x l B M f f
a X x dt dt
0 0
2
0 0
2 2
0 0
2 2 3
0 0
( )
( ) ( )
L X di L aI dxv Ri
a X dt a X dt
L aI dx d x L aIi B M K x f
a X dt dt a X