Magnetic Circuits
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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
FARADAY AMPERE
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 nonzero !
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 timevarying magnetic field
inside the coil
Moving a magnet towards a coil produces a timevarying 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 [Ampturn/m]
due to macroscopic & microscopic
s)
due to macroscopic currents
Faradays Law
( H
)B = o +M
)= o
(H + mH = orH
demf = mag
dt
emf =
ENC dl and mag =
B ndA
8
Example: Magnetic Write Head
Bit density is limited by how well the field can be localized in write head
Horizontal Magnetized Bits
Ring Inductive Write Head
Shield
GMR Read Head
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
Thin Film Write Head
How do we apply Amperes Law to this geometry (low symmetry) ?
Recording Current
Magnetic Head Coil
Magnetic Head Core
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)= Magnetomotive 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 Nturn 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 = crosssection area
Electric current
Magnetic flux
18
Fields from a Toroid
A = crosssection 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 = crosssection 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
Magnetic Circuit for Write Head
A = crosssection area
+ 
core gap
l =A
21
Parallel Magnetic Circuits
A = crosssection 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
Flux linkage
Step 2: Estimate voltage v2 due to timevarying flux
+ + sinusoidal v1 v2
 
A = crosssection area
Step 1: Estimate voltage v1 due to timevarying flux
N1 N2
+ 
i1 i2
Load
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
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