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EE 70 Winter Quarter 2008 Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang February 28, 2008

Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

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Page 1: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Magnetic Circuits and Transformers(Chapter 15)

EE 70Clifford Hwang

February 28, 2008

Page 2: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Agenda• Physical model for resistors• Physical model for capacitors• Physical model for inductors• Mutual inductance• Transformers• Diodes

Page 3: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Physical Model of Resistors

• ρ: resistivity (fundamental property of the material)

• Resistance depends on how long the object is andits cross-sectional area

• See p. 30-31

AL

=

Page 4: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Capacitance of a Parallel-Plate Capacitor

WLAdA

C == ε

mF 1085.8 120

−×=ε

0εεε r=

(derived using Gauss’ Law)

(vacuum permittivity)

Dielectric constant (value is material dependent)

Page 5: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Sample Calculation of Capacitance

( ) nF77.1F10x1770m10x1

m02.0m

F10x854.8dA

C

m10x1mm1.0d

m02.0)m10x20)(m10x10()cm20)(cm10(WLA

124

212

0

4

222

==

==

==

====

−−

−−

ε

Fairly large

Page 6: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Inductance

• Wire coiled around a magnetic core• Current flow àMagnetic field / flux• Core improves magnetic flux and can

re-direct it

B

B

I

I

Page 7: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Flux Linkages

φλ

φ

N

BA

dA

=

=

⋅= ∫ AB

Magnetic flux passing through a surface area A:

For a constant magnetic flux density perpendicular to the surface:

The flux linking a coil with N turns:

Page 8: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Definition of Inductance

icurrentlinkagesFlux

==

φλ N=Substitute for the flux linkages using

iN

=

Page 9: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Reluctance

RNi

• The magnetic flux can also be defined in terms of a quantity called reluctance– Magnetic equivalent of resistance

RN

iN

L2

==φ

Permeability (material dependent)

(constant, time independent)

Page 10: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Faraday’s Law

Faraday’s law of magnetic induction:

The voltage induced in a coil whenever its flux linkages are changing. Changes occur from:

• Magnetic field changing in time

• Coil moving relative to magnetic field

dtdi

Ldt

)Li(ddtd

V ===λ

Page 11: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Mutual Inductance• Current flowing through one coil can produce a magnetic

field that can induce a current in a different coil

1

11

1

111

i

iL

λ

λ

=

= ←

2

22

2

222

i

iL

λ

λ

=

= ←

2

12

2

21

1

21

1

12

iiiiM

λλλλ==== ←←

Self inductance for coil 1

Self inductance for coil 2

Mutual inductance between coils 1 and 2:

i1 i2

v1 v2

+ +

––

Page 12: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Circuit Equations for Mutual Inductance

dtdi

Ldtdi

Mdt

dv

dtdi

Mdtdi

Ldt

dv

iLMiMiiL

22

122

211

11

2212

2111

+±==

±==

+±=±=

λ

λλλ

• The mutual inductance can increase or decrease the voltage• Depends on the direction of the field produced

Page 13: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Mutual Inductance• To determine whether or not the mutual inductance increases

or decreases the voltage, one of the terminals is “dotted”• Currents entering the dotted terminals produce aiding fluxes (based

on right hand rule)

Fields are aiding Fields are opposing

Page 14: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Example 15.8

mHN

L

mHN

L

410

200

110

100

7

222

2

7

221

1

===

===

R

R

WeberturnsampereR /)(107 −=

Self inductance:

Page 15: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Example 15.8

15

7111

1 1010

100i

iiN −===R

φ

mH2i

M

i10200N

1

21

15

1221

==

×== −

λ

φλ

Mutual inductance:

(Magnetic flux from left coil) (Flux linkage on right coil, due to flux from left coil)

Page 16: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Example 15.8

dtdi

Mdtdi

Le

dtdi

Mdtdi

Le

1222

2111

−=

−=

• Flux produced by i2 opposes the flux produced by i1• Proof: use right hand rule• If there’s a dot on the lower terminal of the left coil, then

add a dot to the upper terminal of the right coil

Page 17: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Transformers• An application of mutual inductance• Used to step up or step down AC voltages

Page 18: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Ideal Transformers

01

m10

1

m101

1

1m11

)tsin(NV

dt)tcos(NV

dt)t(vN1

(t)

Law sFaraday'by dtd

N)tcos(V)t(v

φωω

φωφφ

φω

+=+=+=

==

∫ ∫

Page 19: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Ideal Transformers

[ ]

)t(vNN

)tcos(VNN

)tsin(dtd

NV

Ndtd

N)t(v

11

2m1

1

2

o1

m1222

==

+==

ω

φωω

φ

Step up voltage: N2 > N1Step down: N2 < N1

Page 20: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Lenz’s Law• Why is i2 going from left to right?

• Unlike Example 15.8, there is no applied voltage / current on the right side

• Polarity of the induced voltage is such that the voltage would produce a current (through an external resistance) that opposes the original change in flux linkages

Top view

Page 21: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Ideal Transformers

)()(

0 since0

12

12

2211

2211

tiNN

ti

iNiNiNiN

=

=≈≈=−= RRF φ

The magneto-motive force (mmf) applied to the core:

If the voltage is stepped upthe current is stepped down

Page 22: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Ideal Transformers

)t(i)t(v)t(p

)t(i)t(vNN

)t(i)t(v)t(p

111

211

2222

=

==

( ) ( )tptp 12 =

• Since an ideal transformer is a purely inductive circuit (no resistance), there should be no power lost• This means that the AC current will be stepped down if the AC

voltage is stepped up• Net power is neither generated nor consumed by an ideal transformer

)()( 12

12 ti

NN

ti =

Page 23: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Impedance Transformations

'L

2

1

2'L

21

12

121

112

2

2L

ZNN

Z)N/N()N/N(

)N/N()N/N(

Z

==

==IV

IV

• Goal: transform the circuit into one where the load impedance (ZL) and transformer are merged into a single impedance (ZL’)• Circuit elements on the secondary side are reflected to the primary side

1

1LZ

IV

=′

L

2

2

1'L Z

NN

Z

=

Page 24: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Example 15.11

45282845220002000j2000

2000j10001000ZRZ

2000j1000)20j10()10(ZNN

Z

20j10Z

'L1s

2L

2

2

1'L

L

∠=∠=+=

++=+=

+=+=

=

+=

• Impedance transformations are used to find the voltages across the transformer

Zs

Page 25: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

o

oo

o

oo

o

43.186.790

)43.632236)(453536.0(

)2000j1000)(453536.0(Z

A453536.045282801000

Z'L11

s

s1

∠=

∠−∠=

+−∠==

−∠=∠∠

==

IV

VI

Example 15.11

Page 26: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Example 15.11

oo

oo

43.1860.79)43.186.790(101

45536.3)453536.0(10

11

22

12

12

∠=∠==

−∠=−∠==

VV

II

NNNN

We can now calculate the current and voltage phasors on the secondary side using the turns ratio:

Page 27: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Example 15.12

Ω=

=

=

∠==

10)1000(101

01000101

2

1

2

1

2'1

1

2'

RNN

R

VNN

V sso

• Circuit elements and sources can also be reflected from the primary side to the secondary side of the transformer

Reciprocal of ZL’ equation

Same as ideal voltage transformation

Page 28: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Example 15.12

ooo

o

oo

o

43.1806.79)0100(45220

43.63510

)0100(45220

)10/20(400100

)0100(20202010

2010102010

1

'2

∠=∠∠

∠=

∠∠

∠+=

∠++

=++

+=

−Tan

jj

Vj

jV s

Same result as Example 15.11

Page 29: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

• Resistivity is in between a conductor and an insulator– Conductor: ρ < 10-2 Ω-cm (ex.

aluminum, copper)– Insulator: ρ > 105 Ω-cm (ex.

silicon dioxide)

• Resistivity can be controlled precisely and even electrically modified

• Resistivity, doping concentration, and carrier mobility can be obtained from graphs

What is a Semiconductor?

pqnq11

pn µµσρ

+==

Page 30: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Why do different materials have different ρ?

• Different materials have different energy band diagrams

An electron in a single atom can only exist at certain energy levels

Energy levels split as atoms are brought together, which may result in gaps

Ec

Ev

Page 31: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

• The allowed energy levels are so close to one another that they can be modeled as a continuous band

• Gaps can occur between continuous bands– The size of the energy gap between the filled and unfilled levels

determines whether you have a conductor, insulator, or semiconductor

Band Diagrams

Conductor

Insulator (large Eg) or Semiconductor (Eg ~ 1eV)

Page 32: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Page 33: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

• To adjust the number of carriers in a semiconductor, impurities can be added– Use a nearby element in the periodic table

• Modeled in the band diagram as an Ed (donor/electron) or Ea (acceptor/hole) level

Doping

Want small gap

Page 34: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Page 35: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Page 36: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Page 37: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Page 38: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008

Page 39: Magnetic Circuits and Transformers (Chapter 15) · Magnetic Circuits and Transformers (Chapter 15) EE 70 Clifford Hwang ... • The magnetic flux can also be ... Circuit Equations

EE 70Winter Quarter 2008