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Mechatronic system design

Mechatronic system design wb2414‐2012/2013 Course part 3

Electromagnetic actuators

Prof.ir. R.H.Munnig Schmidt Mechatronic System Design

Delft 1 University of Technology WB2414-Mechatronic System Design 2013-2014

Learning goals. The student:

• Can select and calculate a single axis functional electromagnetic actuator for a given specification, working according to the Lorentz or reluctance force generation principle.

Delft 2 University of Technology WB2414-Mechatronic System Design 2013-2014

1 Contents

• and •Electromagnetic actuators •Lorentz actuator •Variable reluctance actuator •Hybrid actuator

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Ohm’s law, the definition of impedance

Current (I) V VIRI  R + (V) Power in Watt (W): Impedance V 2 (Z) PIVIR2  - Source R Complex impedance Z(f) Vf() Vf() IfZf ()() If() Z()f

Delft 4 University of Technology WB2414-Mechatronic System Design 2013-2014

2 Rules of Kirchhoff (network theory)

1. At any node of an electronic circuit all currents add to zero. • No charge storage in a node 2. Following any loop in an electronic circuit all add to zero.

III123 0

VVV123 0

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Combination of Impedances

Serial •Current is shared, •Voltage is divided

Vt VV1  2 Z   V VV s Z t12  II s tt IItt IZ IZ I Z  I Z  t1 t2Z Z tt12 I 12 Z12 Z t It

Parallel VV V 1 •Voltage is shared Z tt  t  p III VV 11 •current is divided t12 tt  ZZ12Z12Z

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3 A voltage divider

V I  i ZZ12

VZi2 VIZo2  Z12 Z

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One of the oldest electromotors The Elias motor of 1842

An actuator is an electromotor for limited movements

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4 Definition of terms in ’s equations

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Maxwell equations for magnetics

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5 Law, magnetic fieldlines are closed loops

• A has its origin in a dipole, North and South pole • Density of fieldlines is proportional to density

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Faraday’s law, a changing magnetic field causes an over a wire E, I dl nˆ L dS S The line‐integral of the electrical field over a closed loop L equals the change of the flux through the open surface S bounded by the loop L. This is a voltage source (EMF), B where the current is driven in the direction of the electric field.

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6 Ampere’s law, Current through a wire gives a magnetic field Not relevant for + electromagnetic actuators B

x

The line‐integral of the magnetic field I over a closed loop L is proportional _ to the current through the surface S  I Bx() 0 enclosed by the loop L 2 x Bx() I (plus a surface‐integral term related Hx() to electric fields) 0 2 x 7 0 410 [Vs/Am] in vacuum/air

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The magnetic field of more current carrying wires Current directed towards observer

The total magnetic field is a linear vectorial combination of the magnetic field from each separate winding.

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7 Average magnetic field from a coil

0nI Bw,av   hc

Bw nI H w,av  0ch 0.3 1

Infinitely long coil:   1

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Hopkinson’s law of magnetics vs Ohm’s law on electricity

Magnetomotive force (  m ) vs = Voltage ()e  V (Φ) vs (I) ( ) vs electrical resistance (R)

• The is the amount of windings times the current

• The reluctance is proportional to the   nI   average length of the flux‐path and m   inversely proportional to cross‐section _  A times the magnetic permeability.

• The magnetic flux follows Hopkinson’s law of magnetics  nI m  w 

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8 The magnetic flux generated inside a coil

 w m nI Bwav,   AAcc A c

ii o io     0.3 1 AAi0 o0 A i0 

This approximates to:

0nI BHw,av= 0 w,av    i nI H w,av    i

Righthand rule: North and Southpole

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Magnetic energy

The energy equals the integral of B*H , which are connected by 0

Bxyz EHBVJ d d[] m   V 0 B BBVt xyz HEBVJ d d [ ] m  0000

Applying some relations like VA  y and Hopkinson’s law:

 V yyB xyzw dVAH d d , d , d AdB  d   wm   00AA 0 0 0

Results in :

B xyz ww11 2 EHBVd dd d2  mmw   22 V 000

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9 Adding a ferromagnetic material reduces the reluctance

m  nI

 y ferromagnetic yoke   0ryA  AnI I m  y0r yw  I V y _  + V + y  nI B y 0r  n windings w Ayy _ Ay Bw nI H w   0r y

With a large μr the flux increases

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Magnetisation curve of ferromagnetic

material (μr > 1)

Saturation Bw

BHw0rw   BHw0w 

In vacuum or air H w ()I

With ferromagnetic core

Saturation

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10 A permanent is a ferromagnetic material with a high hysteresis

B Area of w B interest r

Hc

HIw () Virgin curve

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PM materials act like a current carrying coil

Without an external field the magnet Br creates its own field as if it was a coil in air

Brm B m nIequivalent H c m 0r,m

H Modern PM μr,m= 1 c H B  B = 1 T  nI H  rm r is equivalent to m equivalent c m 800000 Amp turns 0 per meter length of magnet

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11 Use a permanent magnet to create a magnetic field in an airgap.

Ferromagnetic (iron) yokecore/yoke

Ay The magnetic flux in A l m Ag g  y the magnet equals:

S N  m B rm m  t0 t

 m  g

 yg BA m    rm t AA A m AA m0 y0r g0 1my mg AAym r gm

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Practical approximation

BArm m AAmy mg Ferromagnetic (iron) yokecore/yoke 1 AAym r gm

Ay Am y A l m  m Ag g  y In practice r Ay m BA S N  rm m A  1 mg A   m  g gm

But the flux in the air gap is smaller because magnetic flux is lost outside the air gap by “fringe/stray flux”. Air is “conductive” for magnetic fields.

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12 The fringe flux takes in practice between 25 to 75% of the total flux!

17  0.33  1los s 50

gggBA  m 0.25   0.57

If Ag=Am and gm=

 mmAB r B r Bg  AA Br AAggmg g g B  1 g 2 AAgm m m

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Flat have less fringe flux and pole pieces are not efficient!

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13 Concentration of flux by iron parts

Am >> Ag Bg (~0,7T) > Br (~0,4T) Much fringe flux !

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Contents

•Basic electromagnetism •Electromagnetic actuators •Lorentz actuator •Variable reluctance actuator •Hybrid actuator

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14 With the field in the airgap an actuator can be made with a current wire inserted in the gap

Lorentz Force on a charged particle F(E+v×B)= q

where •F is the vectorial force (in Newton) •E is the electric field (in Volts per meter) •B is the magnetic field (in ) •q is the of the particle (in coulombs) •v is the instantaneous velocity of the particle (in meters per second) •and ×is the cross product.

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For the magnetic force on a current this relation leads to

The with qI 

FBI  w sin α = the angle between the current and the magnetic field   w= length of the wire in the field

lw

Corkscrew rule due to cross product

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15 Formulate the Lorentz force differently to avoid mistakes lw The Lorentz Force

FBI  w sin Can also be written as : (α = π/2) d x FI w dx For multiple windings this becomes Because d FnI w dw  B w dx dx n = number of windings

Φw =flux per winding

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A first example, a flat Lorentz actuator Permanent Magnets Ferromagnetic yoke F

Magnetic Field

Flat wound coil

A

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16 Efficiency of a flat Lorentz actuator Permanent Magnets Ferromagnetic yoke F

Magnetic Field

Flat wound coil

F A

Detail A

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Importance of dΦw/dx, risk of non functional actuator

Permanent Magnets Ferromagnetic yoke F=0

Magnetic Field

Flat wound coil There should be motion between coil and permanent magnetic field. The force acts between the current and the source of the magnetic field.

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17 Extending in the x direction

Permanent Magnets Coil y Ferromagnetic yoke

F x

Useless windings

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Extending in the y direction

y

x

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18 The loudspeaker (moving coil) motor

A Permanent Magnets Ferromagnetic yoke

F

Coil A A ‐ A

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Limited range

Stiffness!

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19 Commutation of the coils

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Three phase force with switching at zero-crossings (mechanical commutation)

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20 Three phase force with sinusoidal currents

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Contents

•Basic electromagnetism •Electromagnetic actuators •Lorentz actuator •Variable reluctance actuator •Hybrid actuator

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21 A Lorentz actuator also has some non-linear reluctance force

• Reluctance force is force on the Ferromagnetic (iron) part coil without the permanent magnets Flux from the coil Coil • The flux by the current in the coil will pull the coil in the iron

F • This force is unidirectional and depending on the position

 Stiffness!

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Ultimate solution. Avoid iron!

Non‐ferromagnetic material • Remove the iron and use more magnet material. (Higher cost!)

• Reluctance of outer path is same order of magnitude as the reluctance of the gap between the magnets (Larger flux path and larger cross‐section)

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22 Variable reluctance actuator

mover  y,m x dx A A g g  g Force based on energy balance F Energy stored in magnetic field

2  y,s stator nI Ag0 Fw F   g 4 I n

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Application of variable reluctance actuator, the current relay switch •Avalanche effect • Elastic hinge is pre‐strained Elastic hinge • Current rises until mover starts to move • Resulting higher force speeds up movement • Strong and fast connection of I electric contacts

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23 Force of magnetic field to ferromagnetic material  nI nI dx x w0 Bg   A Ag l g g AAggg 2 2B  F gg Φw nI 0

2 nI A  BA2 F g0  gg  4  I n g0 2 F Bg “Magnetic pressure” Pm  1T  0.4 Mpa (4bar) 22Ag0

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The relation for attraction force is only valid for reluctance force

BA2 F  gg Two permanent magnets will attract or repel each other. 0 This is linear force related to the Lorentz force

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