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EEL5225: Principles of MEMS Transducers (Fall 2003)
Instructor: Dr. Hui-Kai Xie
Lumped-Element Modeling
;
Last lecture
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Today:
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Two-port element example: inductor
Transducers
– Classification
– Linear, Conservative
– General Two-Port Theory
Review of Electromagnetics
Electrodynamic transduction
EEL5225: Principles of MEMS Transducers (Fall 2003)
Lecture 19 by
H.K. Xie
10/10/2003
Review of Electromagnetics
Maxwell's Equations
Differential Form
1. ∇i D = ρ
2. ∇i B = 0
3. ∇× E = −
∂B
∂t
4. ∇× H = J +
2
Integral Form
∫
∫
S
S
ˆ =q
DindS
Gauss's Law
Binˆ dS = 0
Gauss's Law
∂B
dφ
idS =
S ∂t
dt
∂D
H
i
dl
=
I
+
∫C
∫S ∂t idS
− ∫ C E idl
∂D
∂t
=∫
EEL5225: Principles of MEMS Transducers (Fall 2003)
Faraday's Law
Ampere's Law
Review of Electromagnetics
Constitutive Relations
Ohm's Law
J =σ E
Permitivity
D =εE
B = µH
Permeability
where µ =µr µ0 and µ0
Continuity Equation
∇i J = -
∂ρ
∂t
Lorentz Force
F = Felectric + Fmagnetic
= q ( E + u × B)
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EEL5225: Principles of MEMS Transducers (Fall 2003)
H
= 4π × 10
m
−7
Magnetic Transduction
„
„
Magnetic Transduction
Motor/generator action are produced by variations of the
attractive force tending to close the air gap in a ferromagnetic
circuit.
Fundamental Definitions
Ê Magnetic field, H (Units: A/m)
Ê Scalar magnetic potential, M (Units: A)
Also known as Magneto Motive Force (MMF)
FMM =
x2
∫
H id l
x1
Ê
Ê
Magnetic flux, φ (Units: Weber=AH=V-sec)
Magnetic flux density, B (Units: Wb/m2)
ˆ
φ = ∫ BindS
S
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EEL5225: Principles of MEMS Transducers (Fall 2003)
Analogy
MMF is effort.
φ is displacement. dφ/dt is flow. MMF and dφ/dt are conjugate power
variables (Check!). No exact analog of q for magnetics.
Ref. R. W. Erickson, Fundamentals of Power Electronics, p. 456
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EEL5225: Principles of MEMS Transducers (Fall 2003)
Induced Voltage/Induced MMF
Faraday's Law (integral form) relates voltage, v(t),
induced in a loop of wire to the time derivative of
the total flux passing through the winding:
d φ (t )
v(t ) = −
('sign' is given by Lenz's law.)
dt
Ampere's Law (integral form) relates magnetomotive
force, M (t), induced in magnetic core to the total
current passing through interior of path:
FMM (t ) =
∫ H (t )idl
= H (t )l m = i (t )
C
if magnetic field is uniform.
Ref. R. W. Erickson, Fundamentals of Power Electronics, p. 457-458
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EEL5225: Principles of MEMS Transducers (Fall 2003)
Magnetic Circuits
Consider a magnetic element of length, , and area, A c . Given a uniform magnetic field, H,
over the length, , the induced MMF (scalar magnetic potential) is:
B
FMM = Hl =  
µ
FMM
φ 
 
A
= c
[using B = µ H ]
[using φ =BA c assuming uniform magnetic flux density]
µ


FMM = 
 φ = Rφ
 µ Ac 
[R is called reluctance (not to be confused with resistance!)]
Ref. R. W. Erickson, Fundamentals of Power Electronics, p. 463.
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EEL5225: Principles of MEMS Transducers (Fall 2003)
Magnetic Circuits
Kirchoff-like Laws apply.
(1) Divergence of magnetic flux is zero at a node.
∇iB=0 indicates total flux entering node must be
zero.
(2) KML (Kirchoff's Magnetomotive Force Law)
Sum of MMF:
∫ H (t )idl
= ni (t )
C
where n= # of turns of wire carrying current i.
Ref. R. W. Erickson, Fundamentals of Power Electronics, p. 461,464.
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EEL5225: Principles of MEMS Transducers (Fall 2003)
Electromagnetic Transduction
dφ
1
is flow, φ displacement. From φ = FMM , what is the reluctance???
dt
R
The magnetic reluctance, R, is analogous to the spring constant.
In the magnetic energy domain, the magnetic element stores potential energy.
In the electrical energy domain, the MMF is related to the electrical current by
Since
FMM = ni, and the inductor stores kinetic energy in the current flow.
The coupling equations between the electrical domain and magnetic domain are:
φ=
v
n
& FMM = ni where φ =
From these, we can calculate the electrical inductance:
di
n2
v(t ) = L
where L = .
dt
R
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EEL5225: Principles of MEMS Transducers (Fall 2003)
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FMM
R
df
φ
Electromagnetic Transduction
„
Circuit Representation:
Ê
Transformer: impedance analogy to admittance analogy
 φ   0 1/ n   i 

=
 V 
n
0
F
 
 MM  
Ê
1:n
n:1
φf2
ei1
-
+ T (Y to Z)
FeMM
2
n = # wire turns
-
Gyrator: impedance analogy to impedance analogy
 φ  0

 = 
 FM M   n
10
+
V
f1
1 / n  V 
0   i 
+
fi 1
eV1
-
EEL5225: Principles of MEMS Transducers (Fall 2003)
n
r
φf2
+
G (Z to Z)
eF2MM
-
Example: Inductor with Air Gap
KML: M core + M gap = ni and continuity of flux:
φcore = φ gap = φ
Rcoreφcore + Rgapφ gap = ni
φ ( Rcore + Rgap ) = ni
Faraday's Law for n turns: v(t ) = n

n2
v(t ) = 
R +R
gap
 core
 di

 dt


n2
where
⇒ L= 
 R + R 
gap 
 core
lc
g
R core =
and R gap =
µ Ac
µ0 Ac
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dφ
dt
Ref. R. W. Erickson, Fundamentals of Power Electronics, p. 465.
I
+
V
-
n
φ
+
FMM
1/R
-
Caution: φ is not a conjugate power variable!
EEL5225: Principles of MEMS Transducers (Fall 2003)
Magnetic Actuator
The stored potential energy in the inductor
(magnetic energy domain) is :
dW PE = edq = FMM d φ
*
= qde = φ dFMM
dW PE
Since FMM = Rφ ,
W PE =
*
W PE
Also , FMM = ni ,
12
2
2
FMM
=
2R
*
Therefore, WPE = W PE
*
=
So , W PE
φ2R
1 2
Li
2
EEL5225: Principles of MEMS Transducers (Fall 2003)
Electrodynamic Transduction
„
Electrodynamic: motor/generator action are produced by
the current in, or the motion of an electric conductor located in a
fixed transverse magnetic field (i.e., voice coil, solenoid, etc.).
Lenz's law: "relates motion of conductor
in a magnetic field to the induced opencircuit voltage across terminals 1-2". For
V
(
)
a differential element, dV = u × β ⋅ d ,
u
β
d = line element
of conductor
Ref., Beranek, “Acoustics”, p. 71.
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where β = magnetic flux density.
Total induced voltage is
∫ dV = V = β
u. Velocity is "upward".
“Left-hand rule”
EEL5225: Principles of MEMS Transducers (Fall 2003)
Electrodynamic Transduction
„
Electrodynamic: motor/generator action are produced by
the current in, or the motion of an electric conductor located in a
fixed transverse magnetic field (i.e., voice coil, solenoid, etc.).
i
Laplace's law: "relates force on a conductor
in a magnetic field to the current passing
through the conductor". For a differential
F
element, dFmag = id × β . The total induced
β
d = line element
of conductor
force is ∫ dFmag = Fmag = β i. Force is "upward".
Ref., Beraneck, “Acoustics”, pg71.
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“Left-hand rule”
EEL5225: Principles of MEMS Transducers (Fall 2003)
Electrodynamic Transduction
„
Characteristic Transducer Equations:
From Lenz's and Laplaces' laws we get the
characteristic questions:
V =β u
and
Fmag = β i
or in matrix form,
 V   0
F  = 
 mag  TME
TEM   i 
, where TEM = TME = β



0  u 
Note: Z EB = Z MO = 0, so there is direct coupling
between V and u or Fmag and i .
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EEL5225: Principles of MEMS Transducers (Fall 2003)
Electrodynamic Transduction
„
Circuit Representation:
Ê
Transformer: impedance analogy to admittance analogy
 U  1/ β l
F  = 
 mag   0
0  V 
β l   I 
I
+
1
1:
βl
V
F mag
+
U
-
Ê
1
β
T (Z to Y)
Gyrator: impedance analogy to impedance analogy
I
 Fmag   0
 U  = 1/ β l

 
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n=
β l  V 
0   I 
+
βl
V
-
EEL5225: Principles of MEMS Transducers (Fall 2003)
U
+
F mag
-
n=β
G (Z to Z)
Example: Loud Speaker
Admittance ÅÆ Impedance
Ref., Beranek,
“Acoustics”, p. 184.
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EEL5225: Principles of MEMS Transducers (Fall 2003)