Michael David Bryant
ME 344 Dynamic Systems & Control
1
9/18/07
University of Texas at Austin
Electromechanics, sensors & actuators
Magnetic Energy Domain
Magnetic Capacitance
Magnetic Resistance
Magnetic Domain Example: Inductor with Flux Return
Magnetic Circuit
Example: Variable Reluctance Actuators / Relay
Example: Loudspeaker
Example: Loudspeaker with Acoustic Tube
Bond Graphs for Magnetostrictive Actuators
Michael David Bryant
ME 344 Dynamic Systems & Control
2
9/18/07
University of Texas at Austin
Magnetic Energy Domain
Magnetic field intensity:
H [Amp m-1 ]
Magnetic induction (or flux density): B = µ H [Webers m-2 = Tesla]
⌠
⌡ B ∑ dS
Magnetic flux: φ =
[Webers = Volt sec.]
surface
Permeability: µ [Henry m-1 = Volt sec. Amp.-1 m-1
= Ohm m-1
Faraday's law defines magnetic flow:
magnetic flux rate !˙ [Webers sec.-1
⌡ E ∑ dL
V= ⌠
C urve
= -
= Volt]
⌠ ∂B

⌡ ∂t ∑ dS
S urface
dφ
dλ
= - n dt = - n !˙ = - dt
The minus sign is grouped into definition of a voltage drop:
dφ
V = n dt
dλ
= n !˙ = dt
= !˙
Ampere's law defines magnetic effort: magnetomotive force M [Amp]
M
⌠
⌡ H ∑ dL =
⌡ J ∑ dS
= ⌠
C urve
S urface
⌠ ∂D

+
⌡ ∂t ∑ dS
S urface
=Ic
Magnetic power:
+ Id
=I
P = M !˙
[ Watts ]
sec.]
Michael David Bryant
ME 344 Dynamic Systems & Control
3
9/18/07
University of Texas at Austin
Magnetic Capacitance
dE
Magnetic power dt
= P = M !˙
"Potential" Energy stored in magnetic field
EM
⌠
dφ
= 
M
⌡
dt dt
⌠ M (φ) dφ
=⌡
Energy variable = Magnetic flux (magnetic displacement): φ
Effort (magnetomotive force) dependence: M = M(φ)
dφ
Flow (magnetic flux rate) obeys kinematics: !˙ = dt
M = M(!)
.
!
C
Michael David Bryant
ME 344 Dynamic Systems & Control
4
9/18/07
University of Texas at Austin
Magnetic Capacitance
l
Define Reluctance ℜ: ℜ = µ A
M=φℜ
Analogous to Ohm's law for a resistor V = i R
R resistance of element to current flow
ℜ reluctance of element to flux flow
Differences from Ohm's law
magnetic circuit instead of electrical circuit
potential energy stored in ℜ instead of power dissipation in R
Potential energy storage
# M(" )d"
"#
= $ "#d" =
2
EM =
2
! magnetic capacitance
Generates linear
C = φ /Μ = 1/ℜ
!
Michael David Bryant
ME 344 Dynamic Systems & Control
5
9/18/07
University of Texas at Austin
Magnetic Resistance
Power dissipation in magnetic circuit from
eddy currents
time varying magnetic field H(t) in core ⇒
time varying induction B(t) = µ H(t) ⇒
time varying electric field E(t) via
∂B
Faraday's law ∇ x E = - ∂t ⇒ current density J = σE
via Ohm's law ⇒ resistive power dissipation in core
B
H
magnetic hysteresis:
B-H follows different paths, increasing & decreasing
shaded area between paths ⇒ energy (power) loss
prevalent in iron
Generates magnetic resistance R
Michael David Bryant
ME 344 Dynamic Systems & Control
6
9/18/07
University of Texas at Austin
Magnetic Domain Example: Inductor with
Flux Return
.
flux rate
!
!
from flux
i
+
magnetomotive
V
force M = n i
n turn coil
Coil: converts current i ⇔ magnetomotive force M
⇒ GY:r
Reluctance of flux return path: stores magnetic potential energy EM
Bond Graph Model
.
.
V= r !="
M = M( !)
GY: r
-1
C
!
i = r M( ! ) =i( " )
Equivalent bond graph
.
.
VL = !
i = i ( !)
I
Michael David Bryant
ME 344 Dynamic Systems & Control
7
Magnetic Circuit
9/18/07
University of Texas at Austin
.
!
Iron core
i
+
Ma
Mb
air gap
V
n turn coil
Mc
Mo " 0
.
!
n turn coil applies magnetomotive force ni = Ma
Iron core routes magnetic flux
Break in core: air gap
define reference Mo
≡0
- Mo
Michael David Bryant
ME 344 Dynamic Systems & Control
8
9/18/07
University of Texas at Austin
Bond Graph of Magnetic Circuit
R
C top
V
i
top
.
1 !
GY
C air
R
C bottom
1 junction for common flux rate
bottom
"˙
through circuit
Rtop & Rbottom magnetic losses
Ctop & Cbottom potential energy stored in magnetic field in iron
!
Cair potential energy stored in magnetic field in air
Potential energy
!
E = E air + E iron " E air since µiron >> µair
!
Michael David Bryant
ME 344 Dynamic Systems & Control
9
R
C top
Ma
9/18/07
University of Texas at Austin
top
1
0
0
.
Mb
!
V
i
GY
.
.
C air
! 1
1 !
.
!
Mo
1
0
C bottom
0
R
Mc
bottom
0's mark magnetomotive force values Ma , ...., Mo
in bond graph; 1's distribute power to elements (C' 's & R 's from core)
& mark flow !˙
no magnetic losses in air (only Cair)
Michael David Bryant
ME 344 Dynamic Systems & Control
10
9/18/07
University of Texas at Austin
Simplify Bond Graph: Eliminate 0 Junctions
R top
C top
1
.
!
V
i
.
.
GY
! 1
1 !
.
!
1
C bottom
Rbottom
0 junctions non-essential: conceptual aid for
constructing bond graph
2 bonds on 0 's ⇒ same (e, f) on input and output
bonds to 0 's
simplify bond graph: eliminate 0 's
⇒ equivalent bond graph
C air
Michael David Bryant
ME 344 Dynamic Systems & Control
11
9/18/07
University of Texas at Austin
Simplify Bond Graph: Merge Neighboring 1
Junctions
R top
C top
V
i
.
C air
1 !
GY
C bottom
All 1 's have same flow !˙ and are neighbors
collapse into single 1
⇒ equivalent bond graph
Rbottom
Michael David Bryant
ME 344 Dynamic Systems & Control
12
9/18/07
University of Texas at Austin
Leakage Flux from Coil: Bond Graph
.
!
Iron core
i
+
air gap
V
leakage
-
flux in air
n turn coil
.
!
Leakage circuit in parallel with iron
Leakage & !˙ add about coil
0 junction and energy storage in leakage field
Cleak
V
i
GY
R top
C top
.
0
C air
1 !
C
bottom
R
bottom
Michael David Bryant
ME 344 Dynamic Systems & Control
13
9/18/07
University of Texas at Austin
Example: Variable Reluctance Actuators / Relay
Magnetic
field
energy
E = E air + E iron " E air
in iron &
air gap
relay arm moves, reduces
potential energy
"E
F=
"x
!
!flux flow: magnetic energy storage E air
" 2#(x)
=
2
x
in air gap, ℜ(x) ≈ µ
air A
Mechanical force F alters!
x ⇒ changes reluctance ℜ(x) ⇒ Eair
x
.
.
!
!
Mb
resistor
Mc
Ma
+
relay arm
V
spring
-
Mo
.
.
!
!
Md
Michael David Bryant
ME 344 Dynamic Systems & Control
14
9/18/07
University of Texas at Austin
Relay Arm 2 Port C: Magnetic & Mechanical Ports
Potential energy
Eair
E = E air + E iron " E air since µiron >> µair
= Eair(φ , x)
" 2#(x)
=
2
!
∂E
Efforts on kth bond from ek = ∂q
k
!
!
:
"E air # 2 "$(x)
#2
F=
=
=
"x
2 "x
2µair A
∂Eair
M = ∂φ
!
M
.
"
x
= φ ℜ(x) = φ µ A
air
C : 1/ !
air
F
.
x
Michael David Bryant
ME 344 Dynamic Systems & Control
15
9/18/07
University of Texas at Austin
Bond Graphs of Relay
C leak
R: Fµ
R
R
C top
top
.
!
S e: V
V(t)
0
GY
1
M
.
1
C : 1/ "
!
air
F
.
TF
C: 1/k
1
x
R bar
C
bottom
C bar
R
I: J
bottom
R
C top
Ma
R
top
1
0
R: Fµ
0
.
Mb
M
!
.
1
S e: V
V(t)
1
GY
.
1 !
1
!
F
.
TF
1
x
I: J
1
!
R bar
1
0
Md
Mo
C bottom
air
Mc 0
C leak
.
0
C : 1/ "
C bar
R
bottom
Siimilar to magnetic circuit
Relay Arm replaces Air Gap: Cair becomes 2 port capacitance
BG elements off 2-port C accounts for relay arm mass,
pin friction & spring
Cleak accounts for leakage flux in air
C: 1/k
Michael David Bryant
ME 344 Dynamic Systems & Control
16
9/18/07
University of Texas at Austin
Loudspeaker Example
speaker cone
I
+ E(t)
-
permanent magnet
with flux density B
voice coil
motion x(t)
D
low reluctance iron
routes flux to coil
voice coil
assembly
(N-turns)
Input voltage E(t) & electric currents I to voice coil
Voice coil: N-turns, diameter D, resistive losses
Permanent magnet flux B, crosses voice coil radially, returns axially
Stiffness and damping in cone & air
Displacements x(t): voice coil/cone assembly
Michael David Bryant
ME 344 Dynamic Systems & Control
17
9/18/07
University of Texas at Austin
Magnetic Lorentz force on voice coil
⌡ I B x dL
F= - ⌠
Curve
= - IBNπD
(Curve: N turns, coil circumference)
Voice coil back emf
Eback
∂
= - ∂t
⌠
⌡ B ∑ dS
S urface
=-B
dx
dA
= - BNπD dt
dt
(Surface A: Outer shell, coil)
Generates GY: BNπD
Michael David Bryant
ME 344 Dynamic Systems & Control
18
9/18/07
University of Texas at Austin
speaker cone
I
+ E(t)
-
permanent magnet
with flux density B
voice coil
motion x(t)
D
low reluctance iron
routes flux to coil
voice coil
assembly
(N-turns)
R: R coil
I : M
•
P
Se
V
1
GY: BN!D
P/ M
1
R: B
kx
•
x
C: 1 / k
Michael David Bryant
ME 344 Dynamic Systems & Control
19
9/18/07
University of Texas at Austin
Example: Loudspeaker with
Acoustic Tube
speaker cone
(area A)
ACOUSTIC IMPEDANCE
MATCHING TUBE
SOUND WAVES
v(t) = A dx/dt
I
+ E(t)
permanent
magnet
plunger/cone
motions x(t)
voice coil
(N-turns &
diameter D)
Conversion: mechanical power into acoustic power
Tube for impedance matching
R: R coil
I : M
•
P
Se
V
1
GY: BN!D
P/ M
kx
R: B
TF: A
1
•
x
C: 1 / k
I : M air
• P/M
air
P
k
x
air air
1
C: 1 / k
•
air
x
air
R: B
air
Michael David Bryant
ME 344 Dynamic Systems & Control
20
9/18/07
University of Texas at Austin
Bass Reflex Speaker System
• Woofer
• Tweeter
• Enclosure with tuned port
• Crossover network (capacitor)
21
Bond Graph of Bass Reflex Speaker System
t
C:C
R:Rc
.
GY: rt
.
TF: At
1
0
.
R:ba
f
TF: Aw
I:mw
pw pw / mw
.
w
R:Rc
1
1
GY: rw
!w !w / Lw
w
I:Lw
ka va
R:bc
kw xw
.
pp
r
.
.
va
pt / mt
I:mt
I:Lt
0
.
1
I:Ia
pt
!t / Lt
!t
.
p a p a / Ia
xt
1
.
E(t)
C:k t
kt xt
q/C
q
Se:E(t)
t
R:bc
0
TF: Aw
xw
C:k w
ke ve
.
ve
C:ke
1
I:Ip
p p / Ip
R:bp
C:ka
22
Bond Graphs for Magnetostrictive Actuators
Magnetostriction
strain smn = smn(Hi, Tjk)
induced by stress Tjk
induced by magnetic field Hi
present in most ferromagnetic materials
iron, nickel, cobalt, rare earths
magnetostrictive strains ≈ 10-5
transformer hum: Fe core magnetostrictively extends/contracts under AC
special rare earth alloy, terfenol D: strains ≈ 10-3 to 10-2
23
Magnetostrictive variables: (uniaxial) fields & stresses aligned with z direction
magnetic induction B = B(H, T)
∂u
axial displacement u and strain s = ∂z
applied (axial) magnetic field H & magnetic induction B
Constitutive Equations / Linear Magnetostriction
B = B(H, T) = do T + µT
s = s(H, T) = SH
H
(1a)
T + do H
(1b)
elastic compliance SH , measured with magnetic intensity H = 0
magnetic permeability µT , measured with stress T = 0
magnetostrictive coefficient do
2
do
energy coupling coefficient k =
µT SH
24
Invert
B
H=ds+ s
µ
(2a)
T = YB
(2b)
d=
s+dB
- k2
(1 - k2) do
,
elastic modulus YB
permeability µs =
"stiffness" term (YB
1
, measured with inductance B = 0
SH(1 - k2)
µT(1 - k2) measured with strain s = 0
=
s)
magnetically induced stress (d B)
strain induces magnetic field component (d s)
Multiport capacitance with displacements (s, B) & efforts (T, H)
25
Magnetostrictive Actuator
25.4mm
Aluminum Attachment
Plunger Rod
Body
Terfenol Rod
Coil 1
Aluminum Spool
Bottom Cover
Nylon Bearing
Top Cover
Belleville Washers
Coil 2
Permanent Magnet
Steel Spacer
26
Bond Graph Structure / Actuator
I:m 1
R:b 1
•
p
p /m
1
1
R:R mag
R:R elec
R mag
Se
•
!
•
N !
V
1
i
•
#
#/L
I:L
air
air
GY:N
Ni
•
!
M(!,x)
1
•
!
C
.
.
.
.
C:1/ "
•
p 1 /m 1 TF:n1 kpxp xp
TF:nr
F(!,x)
n
.
xn
•
p
R:b n
n
.
.
.
.
0
I:M
•
P
P/M
eOUT
1
fOUT
TF:nn
p n /m n
1
ret
C:1/k p
1
F(!,x)
1
.
x
1
F(!,x)
r
. 1
xr
"ret !
1
R:B b
pn /m n
I:m
n
Excitation circuit: coil
input V
coil resistance Relec
leakage inductance Lair
electric power ⇒ magnetic power (Ni !˙ ) by coil GY:N
Magnetic Circuit
magnetic effort: magnetic potential H = Ni
magnetic flow: magnetic flux rate !˙
magnetic capacitance C:1/ℜret , reluctance ℜret
resistance R:Rmag incorporates eddy current & hysteresis losses
OUT
27
Multi-port C
I:m 1
R:b 1
•
p
p /m
1
1
R:R mag
R:R elec
R mag
Se
•
!
•
N !
V
1
i
•
#
#/L
I:L
air
air
GY:N
Ni
•
!
M(!,x)
1
•
!
.
.
.
.
C
C:1/ "
•
p 1 /m 1 TF:n1 kpxp xp
TF:nr
F(!,x)
n
.
xn
•
p
R:b n
n
p n /m n
pn /m n
I:m
single magnetic port from magnetic circuit
multiple mechanical ports
continuous rod
incorporates vibration modes
.
.
.
.
0
I:M
•
P
P/M
eOUT
1
fOUT
TF:nn
1
ret
C:1/k p
1
F(!,x)
1
.
x
1
F(!,x)
r
. 1
xr
"ret !
1
n
R:B b
OUT
28
Efforts on Multi-port C
Ampere's law over magnetic circuit (using eqn (2a))
⇒ magnetomotive force on 2 port C
M(φ, x) = H l = d x + ℜterf
φ
x : total end displacement
φ : flux in rod
ℜterf reluctance of magnetostrictive rod
(n + 1) ports /axial vibration displacement modes uj = Uj(z) xj(t)
modal forces
Fj(φ, x) = nj d φ + Kj xj
shape function Uj(z)
modal mass I:mj , stiffness C:1/Kj , damping R:bj
modal 1 junctions & transformers TF:nj
couple forces to modes
construct modal displacements
pj
0 junction sums modal flows nj m
j
29
final 1 junction
applies load to the end mass M
R: Bb bushing friction
external element(s) "OUT"
plunger rod stiffness generates capacitance C:1/kp
30
Multi-port Capacitance
power flows over multiport C = time derivative of the potential energy
dE
dt
∞
˙
= M(φ, x) ! + ∑
j=1
.
Fj(φ, x) xj
d
= dt
(5)
∂E
M(φ, x) = ∂φ
∂E
Fj(φ, x) = ∂x
j
∞
x = ∑ nj xj(t)
j=1
= d x + ℜterf
= nj
d φ + Kj xj
φ
1
{ 2
ℜterf
φ2
∞ 1
+d φx+ ∑ 2
j=1
Kj xj2
}
31
State Equations from Bond Graph
Relec
!˙ = - L
air
.
Rmag φ
.
pj
.
xj
.
xp
λ - N !˙ + V(t)
=NL
λ
air
- ℜret
pj
= - Fj(φ, x) - bj m
j
pj
=m
j
n
,
pj
= ∑ nj m
j
Bb
=- M
φ - M(φ, x)
- nj
(6b)
kp xp , ( j = 1, 2,..., n)
( j = 1, 2,..., n)
∞
+
j=1
.
P
(6a)
∑
.
nj xj
(6c)
(6d)
1
-M
P
(6e)
j = n+1
P - eOUT
+
kp xp
(6f)
32
Substitute equations (3) and (4) into (6) & rearrange into state equation form
R
˙! = - [ elec
Lair
N2
+ L R
air mag
←ret + ←terf - d2/Kr
]λ+N
Rmag
-NR
N
!˙ = L R
air mag
d kp
mag Kr
←ret + ←terf - d2/Kr
λRmag
Nd
φ+R
mag
xp + V(t)
n
d
φ-R
∑
mag j = 1
n
∑
nj xj
j=1
(7a)
d kp
n j xj + R
mag Kr
x
(7b)
.
pj
.
xj
.
xp
= - nj
1
=m
j
bj
d φ - Kj xj - m
j
pj ,
n
Kr
pj
= k + K ∑ nj m
p
r
j
n
d
-R
∑
mag j = 1
Bb
=- M
kp xp,( j = 1, 2,..., n)
( j = 1, 2,..., n)
j=1
.
P
pj - nj
P - eOUT
+
(7d)
d
-k +K
p
r
nj xj + R
kp xp .
(7c)
N
-{ L R
air mag
d kp
mag Kr
←ret + ←terf - d2/Kr
λRmag
Kr
1
xp} - M k + K
p
r
P
(7e)
(7f)
φ
33
Frequency Response
measured / actuator: solid line
predicted / BG model: dashed line
t
⌠ P
two modes X ≡ 
⌡ M dt
=
∞
∑
nj xj
j=1
0
d
- xp ≈ x1 - x2 - (1 + kp/Kr) xp - K
r
geometry: design
material properties: handbook
X/i (m A
- 1
)
- 1
)
-110
500
-120
300
-130
300
-140
100
-150
100
-160
-100
-170
-100
-180
-300
-190
-300
-500
-210
-200
10
100
1000
10000
Frequency (Hz)
100000
Magnitude (m A-1)
500
-500
10
100
1000
Frequency (Hz)
10000
100000
Phase (degrees)
-100
Phase (degrees)
Magnitude (m V-1)
X/V (m V
φ
20
Table 1: Parameter values for the bond graph model of the magnetostrictive actuator.
SYMBOL
Relec
DESCRIPTION
SOURCE
solenoid resistance
Lair
solenoid leakage inductance
N
ℜret
turns in excitation coil
measured
200
reluctance of flux return circuit
Appendix 2
1.16 x 108
Rmag
ℜterf
resistance of flux return circuit
1.2 x 105
reluctance of magnetostrictive rod
Appendix 2
l
= s
µ Aterf
H-1
Ω-1
1.19 x 108
H-1
l
length of magnetostrictive rod
measured
0.038
m
D
Aterf
diameter of magnetostrictive rod
measured
0.00699
m
cross sectional area of rod
= π (D/2)2
3.83 x 10-5
m2
k
do
YB
energy coupling coefficient
Butler (1988)
0.72
magnetostrictive coefficient
Butler (1988)
15 x 10-9
elastic modulus-coil open circuited
Butler (1988)
5.5 x 1010
µs
permeability - rod clamped
Butler (1988)
56.5 x 10-7
V s m -1
A-1
ρ
ω1 /2π
terfenol D mass density
Butler (1988)
9.25 x 103
kg m-3
1st mode natural frequency of
eq. (3c) & (3e) 16,000
Hz
eq. (3c) & (3e) 48,000
Hz
modal damping coefficient
= 2 ζj ωj mj
kg s-1
modal damping ratios
lightly damped 0.05, 0.0167
measured
VALUE
UNITS
0.36
Ω
0.2335 x 10-3
H
m A-1
N m-2
clamped-free rod
ω2 /2π
2nd mode natural frequency of
clamped-free rod
bj
ζ1 , ζ2
68
estimate
6.75 x 10-3
6.83 x 107
mj
modal mass
eq. (3d)
K1
K2
1st mode stiffness
eq. (3f)
2nd mode stiffness
eq. (3f)
Kr
Kp
residual stiffness
eq. (4b)
6.14 x 108
5.57 x 108
plunger rod stiffness
geometry and
8.60 x 107
N m-1
N m-1
kg
N m-1
N m-1
materials
M
Bb
attached end mass
measured
0.036
kg
damping of guide bearing
Appendix 2
650
kg s-1