lecture 15 electromechanical systems (1) 1… · sauer’s ece 330 lecture notes. some slides are...
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Copyright © 2017 Hassan Sowidan
ECE 330
POWER CIRCUITS AND ELECTROMECHANICS
Acknowledgment-These handouts and lecture notes given in class are based on material from Prof. Peter
Sauer’s ECE 330 lecture notes. Some slides are taken from Ali Bazi’s presentations
Disclaimer- These handouts only provide highlights and should not be used to replace the course textbook.
10/8/2017
LECTURE 15
ELECTROMECHANICAL SYSTEMS (1)
Copyright © 2017 Hassan Sowidan
ELECTROMECHANICAL SYSTEMS
• Focus on magnetic circuits in which
there is a moving member.
• It can be either a translational or rotational system.
• Source: clrwtr.com Source: stackexchange.com
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Copyright © 2017 Hassan Sowidan
ELECTROMECHANICAL SYSTEMS
The purpose is to present techniques of systematic
derivation of mathematical models (differential
equations) for lumped-parameter electromechanical
systems. The theory is outlined for N electrical coils
and M mechanical variables.
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Copyright © 2017 Hassan Sowidan
ELECTROMECHANICAL SYSTEMS
• A set of electrical coils that interact to produce the
force or torque on the mechanical system.
• The force or torque of electric origin will be derived
from fundamental principles of stored energy in the
magnetic field.
• both the currents in the electrical coils and the force
or torque are time varying.
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Copyright © 2017 Hassan Sowidan
ELECTROMECHANICAL SYSTEMS
• A basic electromechanical system has an electrical
part, a mechanical part, and electromechanical
coupling. This coupling is assumed to be lossless.
• Systems with linear motion see a force f and a
displacement x, while rotational systems see a torque
T and a displacement θ.
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v, i f, x
Copyright © 2017 Hassan Sowidan
EXAMPLES
• Linear(translational) motion:
• Rotational motion:
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I
Ngxlp μ
μ
Xo
Core mean length: lc
Core area: Ac
Coil
Stator
Rotor
Copyright © 2017 Hassan Sowidan
FLUX LINKAGE AND VOLTAGE
• The flux linkage λ can be a function of the
displacement x and the current i.
• The voltage becomes:
• If λ=L(x) i,
• If L(x)=L= constant,
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.d di dx
vdt i dt x dt
( )( ) .
d di L x dxv L x i
dt dt x dt
.d di
v Ldt dt
Copyright © 2017 Hassan Sowidan
MULTI-PORT SYSTEMS
• The force of electric origin at the mechanical ports
can be functions of all the variables.
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v1, i1 f1, x1
vN , iN fM , xM
1 2 1 2,.....( , , ,..... , ), 1.....k k N Ni i i x x x k N
1 1
, 1,....NN M
j jk k kk
j jj j
di dxdv k
dt i dt x dt
1 2 1 2,( , ,..... , , ..... )e e
i i N Mf f i i i x x x
Copyright © 2017 Hassan Sowidan
LINEAR SYSTEM EXAMPLE
Basic configuration is used for tripping circuit
breakers, operating valves, etc. It consists of:
• Fixed structure made of highly permeable material.
• A movable plunger of non magnetic sleeve.
,
• H exist only in the air gap
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N
g
x
w
g
2w
H1
H3
H2
i
,
=B H =
= 0H in the iron
Copyright © 2017 Hassan Sowidan
LINEAR SYSTEM EXAMPLE
• Ampere’s loops:
d is the depth of the structure
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N
g
x
w
g
2w
H1
H3
H2
i
3 1
2 1
1 1 3 3 2 2
1 2
1 2
2 2
0
0
2 2 0
2(2 )
o o
oo
H g H g
H x H g N i
B A B A B A
H wd wd H
NiH H
g x
wd NiH wd
g x
Copyright © 2017 Hassan Sowidan
LINEAR MOTOR EXAMPLE
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2
2
2 2
2
2( )
( )( )
2 2
o
o o
N
L i
w d NL x
i g x
di dL x dxv L x i
dt dx dt
w d N w d Ndi dxv i
g x dt dtg x
Transformer
voltage
Speed
voltage
This problem can be easily
solved using the magnetic
equivalent circuit !
Copyright © 2017 Hassan Sowidan
READING MATERIAL
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• Reading material: Sections 4.1-4.3.
• Next time: Continue Section 4.3.