1 Magnetic circuits Summary for EM ELEC 3105 BASIC EM AND POWER ENGINEERING.

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Magnetic circuits

Summary for EM

ELEC 3105 BASIC EM AND POWER ENGINEERING

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MAGNETIC CIRCUIT PRELIMINARIES

I

�⃗�

Iron core µrc=5000 Air µra=1.000

Want to examine how B and H are related:Through the side wall of the coreThrough the end face of the core

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MAGNETIC CIRCUIT PRELIMINARIESI

�⃗�


Through the side wall of the core

�⃗�𝑐𝑜𝑟𝑒

�⃗�𝑎𝑖𝑟

Use boundary conditions for tangential components𝐻𝑐𝑡=𝐻𝑎𝑡

𝐵𝑐𝑡=𝜇𝑟𝑐 (𝜇𝑜𝐻¿¿𝑎𝑡 )¿

𝐵𝑐𝑡=5000(𝐵¿¿𝑎𝑡 )¿ B primarily in the iron core

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MAGNETIC CIRCUIT PRELIMINARIESI

�⃗�


Through the end face of the core

�⃗�𝑐𝑜𝑟𝑒 �⃗�𝑎𝑖𝑟Use boundary conditions for normal components𝐵𝑐𝑛=𝐵𝑎𝑛

B preserved through end face

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I

�⃗�


In core region all B field lines are inside the core

At the end all B field lines are continuous and exit into air

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I

�⃗�

�⃗�

�⃗�

D

L

Φ𝑚=𝐵𝐿𝐷L

L

D/2

D/2

Φ𝑚𝑢=𝐵𝐿𝐷 /2

Φ𝑚𝑑=𝐵𝐿𝐷 /2+

Conservation of magnetic flux at a branch point

Similar to conservation of current at a node 𝐼=𝐼 1+𝐼 2

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I

�⃗�

The magnetic flux is produced by the current in the wires

D

L

Φ𝑚=𝐵𝐿𝐷

Φ𝑚∝𝐵 and 𝐵∝𝑁𝐼

Φ𝑚∝𝐼

𝐼⇒Φ𝑚 Just like 𝑉⇒ 𝐼 in electrical circuits

Q? What plays the role of resistance in magnetic circuits?

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MAGNETIC CIRCUITS

I

M

H

B

We can build a torus of ferromagnetic material and wrap N turns of wire carrying a current I around it.

The iron core magnetizes, giving a large bound current on the surface of the torus, thousands of times larger than I.

𝐼

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MAGNETIC CIRCUITS

I

M

H

B

The B field inside the torus is much larger than that outside, so most of the field is inside the core. Boundary Conditions

Since magnetic field lines form closed loops, the fields cannot vary as we go around the torus core. There may be minor variations at different radii within the core though.

M

H

B

constant inside the core

Gaussian volumeFlux through top equalsflux through bottom.

𝐼

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MAGNETIC CIRCUITS

For closed loop

Closed loop inside the core material and encloses a current NI, the number of wires passing through the loop times the current in each

Approximate length of core

N NI

core

enclosedIdH

H

d

core

NIH

HB

core

core

𝐼

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MAGNETIC CIRCUITS

I

N HB

????

Can read B from B versus H curve for the core material.

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MAGNETIC CIRCUITS

I

M

H

B

We now introduce a small gap in the core material. Assuming minimal spreading of the field lines, B in the core is the same as B in the gap.

N

Small gapB

continuous across gap

gap

coregap

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MAGNETIC CIRCUITS

I

M

H

B

We now introduce a small gap in the core material. Assuming minimal spreading of the field lines, B in the core is the same as B in the gap.

N

Small gap

H

not continuous across gap

core

core

BH

o

gap

BH

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MAGNETIC CIRCUITS

I

For closed loop

Substituting for Hgap gives an expression for B and Hcore which are also related by the magnetization curve of the core. NI

loop

enclosedIdH

H

d

HB

core

gap

coregap

NIHHgapgapcorecore

Thus

o

gap

BH

NIB

Hgap

o

corecore

Solving for B

gap

corecore

o

HNIB

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MAGNETIC CIRCUITS

gap

corecore

o

HNIB

Hcore

B

Straight line

core

NI

gap

o

NI

Magnetization curve

The intersection gives B and H fields in the core, and the H field in the gap can be found from B.

NIBB

gap

o

core

core

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MAGNETIC CIRCUITS

gap

corecore

o

HNIB

Hcore

B

Straight line

core

NI

gap

o

NI

Magnetization curve

The intersection gives B and H fields in the core, and the H field in the gap can be found from B.

NIBB

gap

o

core

core

Linear approximation to magnetization curve

Some error introduced using a linear fit for B-H curve.

HB constant

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MAGNETIC CIRCUITS

Then

core

NI

NIBB

gap

o

core

core

Hcore

B

gap

o

NI

o

gap

core

core

NIB

In the denominator, the two terms may be of the same order even thoughsince

coregap

coreo

o

gap

core

core

NIB

21RR

VI

Similar to the formula

for a current through two resistors in series

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MAGNETIC CIRCUITS

o

gap

core

core

NIB

21RR

VI



reluctance

mmfflux

resistance

emfcurrent

AA

NIBA

o

gap

core

core

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MAGNETIC CIRCUITS

21RR

VI



reluctance

mmfflux

resistance

emfcurrent

Magnetic circuit Electric circuit

Flux Current

MMF EMF

Reluctance Resistance

Permeance Conductance

Permeability Conductivity

INI V

A

A

A

A

AA

NIBA

o

gap

core

core

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ELECTRIC AND MAGNETIC CIRCUITS “COMPARISON”

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MAGNETIC CIRCUITS

21RR

VI

Similar but only

approximately

reluctance

mmfflux

resistance

emfcurrent

AA

NIBA

o

gap

core

core

Flux tends to leak out of a magnetic circuit since 0o

Magnetic fields take the path of least reluctance.

Current takes the path of least resistance.

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MAGNETIC CIRCUITS

Magnetic shielding

Cavity has very smallB inside

B

B

B

Iron core

Magnetic field bypasses the cavity

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MAGNETIC CIRCUITSMagnetic shielding

Long thin cavity

B

B

magnetized iron core

0

dH

iron

o

cavity BB OR iron

o

cavityBB

iron

r

cavityBB

1 with 10000

r or higher

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�⃗�

EXAMPLE 1: MAGNETIC CIRCUIT

I

µ1,L1,A1

µ2,L2,A2

µ3,L3,A3

N

Determine electrical equivalent circuit

Example completed in class

25

�⃗�

EXAMPLE 1: MAGNETIC CIRCUIT

I

µ,L1,A1

µ,L2,A2

µ,L3,A3

N

Determine electrical equivalent circuit

Example completed in class

�⃗�

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Summary for EM

ELEC 3105 BASIC EM AND POWER ENGINEERING

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1 Magnetic circuits Summary for EM ELEC 3105 BASIC EM AND POWER ENGINEERING.

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