Multi-Port Storage Elements Energy storage elements can have multiple ports Power into/out of element via multiple ports Examples:

o Actuators: motors1, relays, generators o Sensors: probes o Distributed elastic devices: beams o Piezoelectricity & Magnetostriction

Energy stored in field, accessible to multiple “ports”

1 Multiport energy storage devices are more accurate models for motors and generators than simple GY models.

Example: Capacitative Actuator or Sensor

x+

-

V

+ + + + + +

+ + + + + ++

+

++

++

-

- - - - - -

-

-

-

-

R

Parallel plates: area A, separation x, charge q Capacitance: C = C(x) = εA/x Voltage drop (effort) across plates: Vc = q/C Oppositely charged plates attract => forces F Potential energy stored in electric field:

o E = ∫ Vc dq = q2/[2C(x)] = E(q, x) o Changing x changes E: work ∫ F dx done,

electric field acts like “spring” o Nature wants to minimize potential energy

Two efforts from energy: o Vc = ∂E/∂q = q/C o F =∂E/∂x = (-q2/2C2)(∂C/∂x)

= (-q2/2C2)(−εA/x2) F = q2/(2εA)

2-port capacitor has:

o electrical port: effort Vc = Vc (q, x), displacement q, flow dq/dt

o mechanical port: effort F = F(q, x), displacement x, flow dx/dt

CVc

q.

x.F

System with 2-port capacitor:

CVc

q.

x.F

1Se:VV(t)

R:R

1

I:Mplate

p.

p/Mplate

Extract state equations:

!

˙ q =1

RV (t)"V

c(q, x)[ ] =

V (t)" qx /#A

R

!

˙ x =p

Mplate

!

˙ p = F(q, x) =q

2

2A"

Multi-port Capacitance

C

e1

ke

em

...

...

fm

f1

kf

m-Port Capacitance Cm o Flows on C via kinematics:

fk = qk.

o Displacements: qk

o Power: P = ∑k = 1

m Pk = ∑

k = 1

m ek fk

o o Potential energy in capacitance’s field:

E = ∫ P dt

Total potential energy stored:

E = ⌡⌠

∑k = 1

m Pk dt =

⌡⌠

∑k = 1

m ek fk dt

= ⌡⌠

∑k = 1

m ek qk

. dt = ⌡⌠

∑k = 1

m ek dqk

E = E(q1, q2, ..., qm) depends on all displacements qk

Energy & Power: dEdt = P = ∑

k = 1

m ek fk

Derivative of E = E(q1, q2, ..., qm) via chain rule:

dEdt = ∑

k = 1

m ∂E∂qk

dqkdt = ∑

k = 1

m ∂E∂qk

fk Compare blue terms’ coefficients of fk:

ek = ek(q1, q2, ..., qm) = ∂E∂qk

Effort on kth bond = partial of energy with respect to displacement on kth bond.

Example: 3-Port C Electrical port (V, q) Translational port (F, x) Rotational port (T, θ) E = E(x, θ, q) = 2xq2 + 4qx3θ + 3θ2

Torque: T = ∂E/∂θ = 4qx3 + 6θ Voltage: V = ∂E/∂q = 4xq + 4x3θ Force: F = ∂E/∂x = 2q2 + 12qx2θ

Example: Electromechanical Relay

Magnetic energy produced by coil Magnetic field energy stored in iron core & air gap Relay minimizes magnetic potential energy by

adjusting displacements (flux φ and gap size x).

GY:N

Cleak

Cleft

1 0Se:VV(t)

1

R

Cbottom

Cair:1/!M

".F

x. 1 C: k

I: Marm

R: FµRleft Ctop

1

Rtop

Rbottom

Cbar

Rbar

electrical magnetic mechanical

2-Port C

C : 1/!air

M

".

F

x.

Magnetic & Mechanical Ports Potential energy

Eair = Eair(φ , x) =

φ2←(x)2 Efforts

o F = ∂Eair∂x =

φ22

∂←(x)∂x =

φ22 µair A

o M = ∂Eair∂φ = φ ℜ(x) = φ

xµair A

Example: Electric Motors

•rotor: squirrel cage with solid bars & steel laminations •stator: 3 – phase AC coils

0

0

0

0

0

1

1

1

1

1

:mm

:Rr1

:r1

R

R

R

R

R

MGY

MGY

MGY

MGY

MGY

1

TF

1 I

R

:r2

:r3

:r4

:r5

:Rr2

:Rr3

:Rr4

:Rr5

:c

:J

:mr1MTF

:mr2MTF

:mr3MTF

:mr4MTF

:mr5MTF

:mr6MTF

:mr7MTF

:mr8MTF

:mr9MTF

:mr10MTF

:nr10

:nr9

:nr8

:nr7

:nr6

:nr5

:nr4

:nr3

:nr2

:nr1

:1/m1

TF

:1/m2TF

:1/m3TF

:1/m4TF

:1/m5

TF

GY

GY

GY

GY

GY

GY

GY

GY

GY

GY

1

1

0

0

C

C

0

0

1

1

:R sc

:R sb

:R sa

R

1

1

1

R

R

:ns2

:ns3

:ns1

GY

GY

GY

:Rsc

:Rsb

:Rsa

. .va

. .vb

. .vc

MSe

MSe

MSe

R

1

1

1

R

R

MechanicalMagnetic Mathematical ElectricalMathematicalElectrical Magnetic

MechanicalMagnetic Mathematical ElectricalMathematicalElectrical Magnetic

Energy stored in magnetic field of air gap between stator and rotor

Multiport C’s couple magnetic energy stored by stator and rotor

Example: Electrostatic Micro-motor

MEMS (Micro Electro Mechanical Systems)

device Operation

o Voltage applied to electrodes o Very sharp tips (curvature ~ 2 µm) o Intense electric field ionizes air molecules o Electrodes repel like charged ions o Ions drift to rotor, deposit charge on rotor o Electrode repels rotor

Generates torque Generates bearing action: pushes rotor

center to stator center

1

Rotor Surface

Tilt angle

ElectrodeRotor

Stator electrodes

(a) Motor configuration (b) Stator electrode’s tilt angle

Rotor Surface

Tilt angle

ElectrodeRotor

Stator electrodes

Rotor

Stator electrodes

Rotor

Stator electrodes

(a) Motor configuration (b) Stator electrode’s tilt angle

Figure 1 Rotor and stator configuration and stator electrode's tilt angle

0 250 5000.1

0.25

0.4

Printing Gap (um)

MD

E (u

m)

Proximity Printing

Contact Printing

< Simulation Condition > T = 300 um r = 4 wavelength = 1 nm

Figure 2 Maximum diffraction error (MDE) versus normal proximity printing gaps (gp)

Figure 1 Stator and electrodes of a corona motor

Figure 2 PMMA rotor and shaft/rotor assembly

Figure 3 Assembled motor and size of a stator

Bond Graph Model

0Se:VV(t)

C:Cfield

V.q

T

x.

.!

1

I: Mrotor

I: Jrotor

R:Rgas bearing

R:Relectrode

1

electrical

rotational

R:Rcathode

R:Ranode

R:Relectrode

F1

translational

Example: Magnetostriction

• Ferromagnetic materials iron, nickel, cobalt, rare earths magnetostrictive strains s ≈ 10-5

• Transformer hum: Fe core vibrates (extends/contracts) under AC • Rare earth alloy, terfenol D: strains ≈

10-3 to 10-2

Constitutive Equations: Linear Magnetostriction

H = H(s,B) = d s + Bµs

T = T(s,B) = YB s + d B T: normal stress in rod s: normal strain in rod B: magnetic flux density H: magnetic field intensity Parameters

o Young’s modulus YB measured with B = 0 o permeability µs measured with s = 0 o magnetically induced stress (d B) o strain-induced magnetic field (d s)

Multiport capacitance:

o displacements (s, B) o efforts (T,H)

Magnetostrictive Actuator

25.4mm

Aluminum Attachment

BodyTop Cover

Belleville Washers

Coil 2

Plunger Rod

Terfenol Rod

Coil 1

Permanent Magnet

Steel SpacerBottom Cover

Aluminum Spool

Nylon Bearing

Bond Graph

1V

I:Lair

R:Relec

Se

!•

!/Lair

GY:N 1 CM(",x)

1

•pn

1

p1/m1

I:m1•p1

0 OUTfOUT

1

I:M

•P P/M

R:Bb

xp•

eOUT.

.

...

..

.

R:b1

C:1/#ret

R:Rmag

Fn(",x)#ret " TF:nn

TF:n1

1 TF:nr

C:1/kp

kpxp

I:mn

R:bnpn/mn

F1(",x)

•xn

•x1

Fr(", x)

•xr

"•

electrical magnetic mechanical translational

"•

Multi-port Inertances

I

e1= p1

.

...

fm = pm

.

em

f1

ek = pk

.

fk

Stores kinetic energy

E = E(p1, p2, ..., pm)

Multiple ports with momenta pk

Flows: fk = ∂E/∂pk = fk(p1, p2, ..., pm)

IC Device

Stores kinetic & potential energies in same “field” E = E(p1, p2, ..., pm, q1, q2, ..., qn)

Ports with momenta pk & displacements ql Flows on I bonds:

fk = ∂E/∂pk = fk(p1, p2, ..., pm, q1, q2, ..., qn) Efforts on C bonds:

el = ∂E/∂ql = el(p1, p2, ..., pm, q1, q2, ..., qn)

IC