EEL5225: Principles of MEMS Transducers (Fall 2003 ... EEL5225: Principles of MEMS Transducers (Fall...

EEL5225: Principles of MEMS Transducers (Fall 2003)1

EEL5225: Principles of MEMS Transducers (Fall 2003)Instructor: Dr. Hui-Kai Xie

Lumped-Element ModelingLast lecture

Two-port element example: inductorTransducers

– Classification– Linear, Conservative– General Two-Port Theory

Today:Review of ElectromagneticsElectrodynamic transduction

Lecture 19 by H.K. Xie 10/10/2003


Review of Electromagnetics

S

ˆ

ˆ 0

Maxwell's EquationsDifferential Form Integral Form1. Gauss's Law

2. 0 SD ndS q

B n dS

D

B

ρ =

=

∇ =

∇ =∫ i

i

i

i

C

C

dS

Faraday's Law

4. Ampere's Law

Gauss's Law

3.

S

S

B B dE dl dSt t dtD DH dl It t

E

H J

φ∂ ∂= =

∂ ∂∂ ∂

= +∂ ∂

∇× = − −

∇× = +

∫

∫ ∫

∫ ∫

i i

i i


Review of Electromagnetics

70 0

and 4 10

Ohm's Law

Constitutive Relations

Permitivity

Permeability where =

Continuity Equation

rHm

J ED E

B H π

σε

µ µ µ µ µ −= ×

==

=

( )

-

Lorentz Force

electric magnetic

t

E u B

J

F F Fq

ρ∂∂

+ ×

∇ =

= +=

i


Magnetic TransductionMagnetic TransductionMotor/generator action are produced by variations of the attractive force tending to close the air gap in a ferromagneticcircuit.Fundamental Definitions

Magnetic field, H (Units: A/m)Scalar magnetic potential, M (Units: A)Also known as Magneto Motive Force (MMF)

Magnetic flux, φ (Units: Weber=AH=V-sec)Magnetic flux density, B (Units: Wb/m2)

2

1

x

M Mx

F H d l= ∫ i

ˆS

B ndSφ = ∫ i


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


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:

( )( ) ('sign' is given by Lenz's law.)d tv tdtφ

= −

Ampere's Law (integral form) relates magnetomotiveforce, (t), induced in magnetic core to the totalcurrent passing through interior of path:

( ) ( ) ( ) ( )

if magnetic field is uniform

MM mC

M

F t H t dl H t l i t= = =∫ i

.

Ref. R. W. Erickson, Fundamentals of Power Electronics, p. 457-458


Magnetic Circuits

Ref. R. W. Erickson, Fundamentals of Power Electronics, p. 463.

cConsider a magnetic element of length, , and area, A . Given a uniform magnetic field, H,over the length, , the induced MMF (scalar magnetic potential) is:

[using ]MM

M

BF Hl B H

F

µµ

= = =

c

c

[using =BA assuming uniform magnetic flux density]

= [R is called reluctance (not to be confused with resistance!)]A

cM

MM

A

F R

φ

φµ

φ φµ

=

=


Magnetic Circuits

Kirchoff-like Laws apply.(1) Divergence of magnetic flux is zero at a node.

B=0 indicates total flux entering node must bezero.

(2) KML (Kirchoff's Magnetomotive Force Law)

Sum of MMF: ( ) (H t dl ni t

∇

=

i

i )

where n= # of turns of wire carrying current i.C∫

Ref. R. W. Erickson, Fundamentals of Power Electronics, p. 461,464.


Electromagnetic Transductiond 1Since is flow, displacement. From , what is the reluctance???dt

The magnetic reluctance, , is analogous to the spring constant.In the magnetic energy domain, the magnetic element stores po

MMFRR

φ φ φ =

tential energy.In the electrical energy domain, the MMF is related to the electrical current by

, and the inductor stores kinetic energy in the current flow.

The coupling equations between the ele

MMF ni=

2

ctrical domain and magnetic domain are:

From these, we can calculate the electrical inductance:

( ) where .di nv t L Ldt R

= =

1 & where MM MMv F ni Fn R

φ φ= = =


φdfφ

Electromagnetic Transduction

Circuit Representation:Transformer: impedance analogy to admittance analogy

Gyrator: impedance analogy to impedance analogy

T (Y to Z)

G (Z to Z)

n = # wire turns-

+

-

+e2

f2

e1

n:1f1

i

V 1:n

MMF

-

+

-

+e2

f2

e1

rf1

V

in

φ

MMF

0 1/0MM

n in VF

φ =

φ

0 1 /0MM

n Vn iF

φ =


Example: Inductor with Air Gap

core

2

2

core

KML: and continuity of flux:

( )

Faraday's Law for n turns: ( )

( )

L= where

R

core gap

gap

core core gap gap

core gap

core gap

core gap

M M ni

R R niR R ni

dv t ndt

n div tR R dt

nR R

φ φ φ

φ φ

φ

φ

+ =

= =

+ =

+ =

=

= +

⇒ +

= cgap

c 0 c

l g and RA Aµ µ

=

Ref. R. W. Erickson, Fundamentals of Power Electronics, p. 465.

Caution: φ is not a conjugate power variable!

-

+

-

+V FMM

In

φ

1/R


Magnetic Actuator

2

2* *

MM*

PE

MM

* 2

2

2

Since F ,

Therefore, W, F ,

1, 2

PE MM PE

MMPE MM PE

PE

PE

RdW edq F d W

FdW qde dF W

RR

WAlso ni

So W Li

φφ

φ

φ

= = =

= = =

=

==

=

The stored potential energy in the inductor (magnetic energy domain) is :


Electrodynamic Transduction

( )

Lenz's law: "relates motion of conductorin a magnetic field to the induced open-circuit voltage across terminals 1-2". For

a differential element, d ,

where magnetic flux density. Tota

V u dβ

β

= × ⋅

=l induced voltage is

d . Velocity is "upward".V V uβ= =∫

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.).

line element of conductor

d =

V

u

β

“Left-hand rule”Ref., Beranek, “Acoustics”, p. 71.



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.).

Laplace's law: "relates force on a conductorin a magnetic field to the current passingthrough the conductor". For a differential

element, dF . The total induced

force is dF F . mag

mag mag

id

i

β

β

= ×

= =∫ Force is "upward". line element

of conductord =

F

i

β

“Left-hand rule”Ref., Beraneck, “Acoustics”, pg71.



Characteristic Transducer Equations:

: and F

or in matrix form,

0, where

F 0

Note: 0, so there is direc

mag

EMEM ME

mag ME

EB MO

V u i

V T iT T

T u

Z Z

β β

β

= =

= = =

= =

From Lenz's and Laplaces' laws we get the characteristic questions

t coupling between and or F and .magV u i



Circuit Representation:Transformer: impedance analogy to admittance analogy

Gyrator: impedance analogy to impedance analogy

1nβ

=

-

+

-

+Fmag

U

V

β lI

-

+

-

+1:β l1

V

I

U

Fmag

T (Z to Y)

G (Z to Z)

n β=

1/ 00mag

U l VF l I

ββ

=

01/ 0

magF l VU l I

ββ

=


Example: Loud Speaker

Ref., Beranek, “Acoustics”, p. 184.

Admittance Impedance

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