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### Transcript of Magnetic Circuits

• Magnetic Circuits

Outline Amperes Law Revisited Review of Last Time: Magnetic Materials Magnetic Circuits Examples

1

• Electric Fields Magnetic Fields

dA =

oE dV B

S V

dA = 0

S

= Qenclosed

GAUSS GAUSS

dE dl = B dA H dl

C dt

(S

)

C

=

dJ dA+

E dA

dtd S S

emf = v =dt

2

• Amperes Law Revisited

In the case of the magnetic field we can see that our old Amperes law can not be the whole story. Here is an example in which current does not gives rise to the magnetic field:

Consider the case of charging up a capacitor C which is connected to very long wires. The charging current is I. From the symmetry it is easy to see that an application of Amperes law will produce B fields which go in circles around the wire and whose magnitude is B(r) = oI/(2r). But there is no charge flow in the gap across the capacitor plates and according to Amperes law the B field in the plane parallel to the capacitor plates and going through the capacitor gap should be zero! This seems unphysical.

Side View

B

BB

B = 0??

I

II I

3

• Amperes Law Revisited (cont.)

If instead we drew the Amperian surface as sketched below, we would have concluded that B in non-zero !

BB Side View

I

I I I

BMaxwell resolved this problem by adding a term to the Amperes Law. In equivalence to Faradays Law, the changing electric field can generate the magnetic field:

d l

C

COMPLETE H d =

AMPERES LAW S

J dA+dt

S

E d A4

• Faradays Law and Motional emf What is the emf over the resistor ?

demf = mag

In a short time t the bar moves a distance x = v*t, and the flux increases by mag = B (L v*t)

There is an increase in flux through the circuit as the bar of length L moves to the right

(orthogonal to magnetic field H) at velocity, v.

from Chabay and Sherwood, Ch 22

dt

emf =magt

= BLv

L

vt

vB

5

• Faradays Law for a Coil

from Chabay and Sherwood, Ch 22

Will the current run CLOCKWISE or ANTICLOCKWISE ?

Rotating a bar of magnet (or the coil) produces a time-varying magnetic field

inside the coil

Moving a magnet towards a coil produces a time-varying magnetic field inside the coil

6

The induced emf in a coil of N turns is equal to

N times the rate of change of the magnetic flux on one loop of the coil.

• Complex Magnetic Systems

DC Brushless Stepper Motor Reluctance Motor Induction Motor

We need better (more powerful) tools

Magnetic Circuits: Reduce Maxwell to (scalar) circuit problem

Energy Method: Look at change in stored energy to calculate force

H

C

dl = Ienclosed

BS

dA = 0 f = q(v B

)

7

• Magnetic Flux [Wb] (Webers) Magnetic Flux Density B [Wb/m2] = T (Tesla

Magnetic Field Intensity H [Amp-turn/m]

due to macroscopic & microscopic

s)

due to macroscopic currents

( H

)B = o +M

)= o

(H + mH = orH

demf = mag

dt

emf =

ENC dl and mag =

B ndA

8

Bit density is limited by how well the field can be localized in write head

Horizontal Magnetized Bits

Shield

Recording Medium

9

• Review: Ferromagnetic Materials

hysteresis

0

B

B,J

H

C

r H : coercive magnetic field strength S B : remanence flux density

B : saturation flux density

H 0

Initial Magnetization

Curve

Slope = i Behavior of an initially unmagnetized

material. Domain configuration during several stages of magnetization.

BsBr

HC HC

Br Bs

H > 0

10

How do we apply Amperes Law to this geometry (low symmetry) ?

Recording Current

Recording Magnetic Field

H

C

dl =

JS

dA

11

• Electrical Circuit Analogy

Charge is conserved Flux is conserved

Electrical

EQUIVALENT

CIRCUITS

Magnetic

i

ii

+ V

S

B d A = 0

V

i

R

12

• Electrical Circuit Analogy

Electromotive force (charge push)= Magneto-motive force (flux push)=

Electrical

Magnetic

EQUIVALENT CIRCUITS

V

i

i

C

H dl = Ienclosedv =

E dl

+ V

ii

R

13

• Electrical Circuit Analogy

Material properties and geometry determine flow push relationship

B = OHMs LAW oJ = E rHDCB

Recovering macroscopic variables:

I =

VJ d =

A E dA = H

lA

l lV = I = I = IR

A A Ni =

V

i

R

14

• Reluctance of Magnetic Bar

Magnetic OHMs LAW

A

l Ni =

l =A

15

• Flux Density in a Toroidal Core

N turn coil

(of an N-turn coil) i

Core centerline

H

NiR B =2R

2R

Ni = 2RB = lB

lBmmf = Ni =

=

l

A

mmf =

16

• Electrical Circuit Analogy

Electrical Magnetic

Voltage v Magnetomotive Force = NiCurrent i Magnetic Flux Resistance R Reluctance

1/Conductivity Permeability Current Density J Magnetic Flux Density BElectric Field E Magnetic Field Intensity H

17

• Toroid with Air Gap

Why is the flux confined mainly to the core ?

Can the reluctance ever be infinite (magnetic insulator) ?

Why does the flux not leak out further in the gap ?

A = cross-section area

Electric current

Magnetic flux

18

• Fields from a Toroid

A = cross-section area

Electric current

Magnetic flux

H

C

dl =

JS

dA

= Ienclosed

NiH =

B = o

( H +M

)Ni

B = 2R 2R

A = BA = Ni

2R

19

• Scaling Magnetic Flux

&

Same answer as Amperes Law (slide 9)

A = cross-section area

Electric current

Ni = Magnetic flux

l = 2R=A

A

A = BA = Ni

2R

20

• i N=500

2cm 8cm

2cm

0.5cm

Core Thickness = 3cm

A = cross-section area

+ -

core gap

l =A

21

• Parallel Magnetic Circuits

A = cross-section area

10cm 10cm

0.5cm 1cm i

Gap a Gap b

22

• A Magnetic Circuit with Reluctances in Series and Parallel

Shell Type Transformer

+ + N1 turns vv 2 N2 turns 1 - -

Depth A l2 l1 l=l1+l2

N1i1

N2i2

a b

c

c

Magnetic Circuit

1

3 + 2

- +

-

l 11 =A

2 = l2A

23

• Faraday Law and Magnetic Circuits

Step 2: Estimate voltage v2 due to time-varying flux

+ + sinusoidal v1 v2

- -

A = cross-section area

Step 1: Estimate voltage v1 due to time-varying flux

N1 N2

+ -

i1 i2

Primary Secondary

Laminated Iron Core

d = N v =

dt

v2v1

=

24

• Complex Magnetic Systems

DC Brushless Stepper Motor Reluctance Motor Induction Motor

Powerful tools

Magnetic Circuits: Reduce Maxwell to (scalar) circuit problem

Makes it easy to calculate B, H,

Energy Method: Look at change in stored energy to calculate force

25

• Stored Energy in Inductors

In the absence of mechanical displacement

For a linear inductor:

Stored energy

dWS =

Pelecdt =

ivdt =

i =dt

i () d

i () =

L

WS =

2d =

0 L 2L

26

• Relating Stored Energy to Force

Lets use chain rule

This looks familiar

Comparing similar terms suggests Wfr = S

r

WS (, r) WS d WS dr= +

dt dt r dt

dWS dr= i

dt v fr

dtdi dr

= iLdt

frdt

27

• Energy Balance

l dWSdt

heat

electrica mechanical

For magnetostatic system, d=0 no electrical power flow

dWS dr=

dtfr

dt

i v drfrdt

dWS (, r) WS d WS dr= + neglect heat

dt dt r dt

28

• Linear Machines: Solenoid Actuator

Coil attached to cone

If we can find the stored energy, we can immediately compute the force

lets take all the things we know to put this together

W S 1 2

r| WS (, r) =

2 Lfr =

x

l

29

• KEY TAKEAWAYS COMPLETE AMPERES LAW

RELUCTANCE

Electrical Magnetic

Voltage Magnetomotive Force Current Magnetic Flux Resistance Reluctance Conductivity Permeability Current Density Magnetic Flux Density Electric Field Magnetic Field Intensity

R

i

1/

J

E

v = Ni

B

H

C

H dl =S

J d A+ ddt

S

E d A

= lA

30

• MIT OpenCourseWare http://ocw.mit.edu

6.007 Electromagnetic Energy: From Motors to Lasers Spring 2011