TMHP51 Servomechanisms (HT2012)
Lecture 08
Electro-magnetics
Electro-mechanics
Proportional Valve Control and Operation
Magnus Sethson
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1
Electro-Magnetics
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2
Lenz’s Law
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3
Faraday´s Law
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The Maxwell Equations (partial)
I
Source: Elektricitet och magnetism från början
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C
H̄ · d¯l = If
H 2⇡r = N i
Ni
H=
2⇡r
Source: UIUC Physics
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The Maxwell Equations (partial)
I
C
Ē · d¯l =
d
dt
Z
S
B̄ · n̄ dS
d
v=N
dt
d di
d
v=N
= N (Li)
di dt
dt
(v = ˙ )
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6
Magnetic Hysteresis
B = µ0 µr H
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Proportional Electro-Magnet
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8
Single Stage Servo Valve
Source:
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Magnetic Circuit
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Magnetic Circuit Model (Proportional Valve)
N i(t) = Hc (t)lc + Ha (t)la (t)
= N AB
B(t)
(t)
=
µc
N Aµc
B(t)
(t)
Ha (t) =
=
µ0
N Aµ0
µc = µr µ0
d
d
d
v(t) = i(t)R +
(L(t)i(t)) = i(t)R + L(t) i(t) + i(t) L(t)
dt
dt
dt
la (t) = x0 xv (t) 0  xv (t)  l0
✓
◆
lc
x0 xv (t)
N 2 Aµ0
N i(t) =
+
(t) ! (t) = lc
i(t)
N Aµc
N Aµ0
+
(x
x
(t))
0
v
µr
Hc (t) =
d
N 2 Aµ0
L(t) ⌘
= lc
di
( µr + x0 ) xv (t)
2
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d
!
L(t) = ⇣
dt
N 2 Aµ0
( µlcr + x0 )
xv (t)
⌘2
d
x(t)
dt
B(t)2 A
(t)
Fm (t) =
=
2µ0
2N 2 Aµ0
d2
d
mv 2 x(t) = Fm (t) Bv x(t) kv (x(t) + xs0 ) Fv (t)
dt
dt
✓
◆
1
d
d
i(t) =
v(t) Ri(t) i(t) L(t)
dt
L(t)
dt
d
x(t) = v(t)
dt
✓
◆
d
d
1
v(t) =
Fm (t) Bv x(t) kv (x(t) + xs0 ) Fv (t)
dt
mv
dt
0
1
( µlcr + x0 ) xv (t) B
d
N 2 Aµ0
C
i(t) =
u(t)
i(t)R
i(t)v(t)
@
A
⇣
⌘
2
dt
N 2 Aµ0
lc
( µr + x0 ) xv (t)
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Magnetic Circuit Model (Proportional Valve), cont.
States:
d
i(t) =
dt
( µlcr + l0 )
N 2 Aµ
d
xv (t) = vv (t)
dt
0
0
xv (t) B
@u(t)
0
i(t)R
d
1 B
N 2 Aµ0
2
vv (t) =
@ ⇣
⌘2 i(t)
dt
mv
2 µlcr + (l0 xv (t))
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i(t)v(t) ⇣
N 2 Aµ0
( µlcr + l0 )
Bv vv (t)
xv (t)
0
1
i(t)
@ xv (t) A
vv (t)
1
C
⌘2 A
kv (xv (t) + xs0 )
1
C
Fv (t)A
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Hysteresis
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Hysteresis
If a hysteresis model is included it
affects the state-space significantly. At
least one extra state-variable is
needed. Often the model is extended
to include internal thermal states as
well.
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Possible States:
0
1
i(t)
B
C
xv (t)
B
C
B
C
vv (t)
B
C
B B(t) (or (t), (t)) C
B
C
@
A
M (t),
Tc (t)
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Next Lecture:
Thursday 2012-11-29, 08:15, P34
Magnus Sethson
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