2012/11/29

1

Magnetically Coupled Circuits•Introduction

•Mutual Inductance

•Energy in a Coupled Circuit

•Linear Transformers

•Ideal Transformers

•Applications

Introduction•Conductively coupled circuit means that one loop

affects the neighboring loop through currentconduction.

•Magnetically coupled circuit means that two loops,with or without contacts between them, affect eachother through the magnetic field generated by one ofthem.

•Based on the concept of magnetic coupling, thetransformer is designed for stepping up or down acvoltages or currents.

2012/11/29

2

Magnetic Flux

vectormalinfinitesitheisproductdotdenotes

areasurfacetheisfieldmagnetictheisfluxmagnetictheis

where

dA

SBΦ

dAB

dABΦ

S

S

Self Inductance

)inductance-(self

isvolatgeinducedtheturns,For

Law)s(Faradays'

isvolatgeinducedtheeach turn,For

1T

did

NL

dtdi

Ldtdi

did

Ndtd

Nv

Ndtd

v

turnsinductance

:inductorAn

NL

v1T+_

v1T+_

v1T+_

v

+

_

2012/11/29

3

Mutual Inductance (1/5)

dtdi

Mv

did

NM

1212

1

12221

isvoltagemutualcircuit-openThe

is1coilrespect towith2coilofinductance-mutualThe

dtdi

Mdtdi

did

Ndt

dNv

did

NL

dtdi

Ldtdi

did

Ndtd

Nv

NL

NL

121

1

1

122

1222

1

111

11

1

1

11

111

12111

2

2

1

1

where

coils)(both1)coil(onlyis1coilbygeneratedfluxthe2,coilincurrentnoAssuming

turnssinductance-self

:2Coil

turnssinductance-self

:1Coil

Mutual Inductance (2/5)

dtdi

Mdtdi

did

Ndt

dNv

did

NL

dtdi

Ldtdi

did

Ndt

dNv

NL

NL

212

2

2

211

2111

2

222

22

2

2

22

222

21222

2

2

1

1

herew

)coilsboth()2coilonly(is2coilbygeneratedfluxthe1,coilincurrentnoAssuming

turnssinductance-self

:2Coil

turnssinductance-self

:1Coil

dtdi

Mv

Mdi

dNM

2121

212

21112

isvoltagemutualcircuit-openThe

)(

is2coilrespect to1withcoilofinductance-mutualThe

2012/11/29

4

Mutual Inductance (3/5)•We will see that M12 = M21 = M.

•Mutual coupling only exists when the inductors orcoils are in close proximity, and the circuits are drivenby time-varying sources.

•Mutual inductance is the ability of one inductor toinduce a voltage across a neighboring inductor,measured in henrys (H).

i1

+

_ dtdi

Mv 12

•The dot convention states that acurrent entering the dotted terminalinduces a positive polarity of themutual voltage at the dotted terminalof the second coil.

Mutual Inductance (4/5)

)(

)(

.andinduces,andinduces

2221122

2112111

22212

12111

ii

dtdi

Mdtdi

Ldt

dN

dtd

Ndt

dNv

dtdi

Mdtdi

Ldt

dN

dtd

Ndtd

Nv

121

22

122

22212

222

212

11

211

12111

111

)(

)(

2012/11/29

5

Mutual Inductance (5/5)i1

+

_dtdi

Mv 12

i1

+

_dtdi

Mv 12

i2

+

_dtdi

Mv 21

i2

+

_dtdi

Mv 21

Series-Aiding Connection

dtdi

MMLL

dtdi

Mdtdi

Ldtdi

Mdtdi

L

vvvdtdi

Mdtdi

Lv

dtdi

Mdtdi

Lv

211221

212121

21

2122

1211

MLLLdtdi

MLLv

MMM

2

2

,But

21eq

21

2112

+ _v1 + _v2

11

12

21

22

2012/11/29

6

Series-Opposing Connection

dtdi

MMLL

dtdi

Mdtdi

Ldtdi

Mdtdi

L

vvvdtdi

Mdtdi

Lv

dtdi

Mdtdi

Lv

211221

212121

21

2122

1211

MLLLdtdi

MLLv

MMM

2

2

,But

21eq

21

2112

+ _v1 + _v2

11

12

21

22

Circuit Model for Coupled Inductors

dtdi

Ldtdi

Mv

dtdi

Mdtdi

Lv

22

12

2111

2212

2111

IIV

IIV

LjMj

MjLj

2012/11/29

7

Example 1

(1b))42(3

61203)126(

gives2meshtoKVLApplying(1a)03)54(12

gives1meshtoKVLApplying

221

12

21

III

II

II

jj

jjj

jjj

39.4901.13)42(

04.1491.2412

gives(1a)into(1b)ngSubstituti

21

2

II

I

jj

+

_

_

+

+

_

Example 2

(1b)0)185(80)(2582)(6

gives2meshtoKVLApplying(1a)1008)34(

02)(6)34(100gives1meshtoKVLApplying

21

1222212

21

2211

IIIIIIIII

IIIIII

jjjjjj

jjjjj

19693.85.33.20

0100

1858834

getwe(1b)and(1a)From

2

1

2

1

II

II

jjjj

+

_

_

+

+ -

2012/11/29

8

Energy in a Coupled Circuit (1/4)

2112222

21121

2222112

0 2220 211222

222

212122112

2211

2110 11111

111111

112

2211

21

21

21

)(

.to0fromincreases,:IIStep21

)(

.to0fromincreases,0:IStep:andasenergystoredthefindTo

22

1

IIMILILwww

ILIIM

diiLdiIMdtpw

dtdi

Lidtdi

MIvivitp

IiIi

ILdiiLdtpw

dtdi

Livitp

IiiIiIi

II

I

i1

i2

I1

I2I II

t

Energy in a Coupled Circuit (2/4)

MMM

IIMILILwww

ILIIMw

IiIi

ILw

Iii

2112

2121222

21121

21121212

1122

2221

221

case.formerthetoequalmustenergytotalBut the

21

21

21

.to0fromincreases,:IIStep21

.to0fromincreases,0:IStepaschangedbecanprocessanalysisThe

i1

i2

I1

I2I II

t

2012/11/29

9

Energy in a Coupled Circuit (3/4)

21222

211 2

121

iMiiLiLw 21222

211 2

121

iMiiLiLw

Energy in a Coupled Circuit (4/4)

212

1

2

2

1

2

221

2

1min

1min1

22

1222

1

21222

211

021

21

21

)(

0)(

,)(minimumthefindTo

021

21

)(,Let

021

21

asgivenisstoredenergyousinstantanethes,assignmentcurrentanyFor

LLM

LM

L

LM

LL

MLxf

LM

xMxLdx

xdfxf

MxLxLiw

xfii

x

iMiiLiLw

212

2221

21

1211

12

2221

21

1211

12

21

2112

1

12221

2

21112

2

222122

1

121111

1

)()(

)(

)(

:proofeAlternativ

LLM

dd

dd

LLMM

did

NMdid

NM

did

NL

did

NL

2012/11/29

10

Coupling Coefficient

0coupling.perfectmeans1

or

)10(

asdefinedisThe

2211

2221

21

1211

12

21

21

k

k

LLkM

kLL

Mk

koefficientcoupling c

Coupling vs. Winding Style

Loosely coupledk < 0.5

Tightly coupledk > 0.5

2012/11/29

11

Example

J73.2021

21

824.2)1(,389.3)1(

)6.1604cos(254.3)4.194cos(905.3

6.160254.34.19905.3

(1b)0)416(102,meshFor

(1a)306010)2010(1,meshFor

56.0205.2

:

s.1atinductorscoupledin thestoredenergytheandFind:

21222

211

21

2

1

2

1

21

21

21

iMiiLiLw

ii

titi

jjj

jj

LLM

kSol

tkQ

II

II

II

rad/s4

V)304cos(60 tv

Linear Transformers

Zin

impedancereflectedpedanceprimary im

ZLjRM

LjR

R

P

RP

L

::

where

22

22

11in

ZZ

ZZ

Z

1in

2221

2111

But

(2)0)(

(1))(givesanalysismeshApplying

IV

Z

II

IIV

LZRLjMj

MjLjR

R1 and R2are windingresistances.

2012/11/29

12

T (or Y) Equivalent Circuit

2

1

2

1

2

1

II

VV

LjMjMjLj

2

1

2

1

)()(

II

VV

cbc

cca

LLjLjLjLLj

)(

)(2

1

2

1

MLMLLMLL

LLjLjLjLLj

LjMjMjLj

c

b

a

cbc

cca

П(or ) Equivalent Circuit

221

2

1

1

2

2

1

where MLLK

KjL

KjM

KjM

KjL

VV

II

2

1

2

1

111

111

VV

II

CBC

CCA

LjLjLj

LjLjLj

111

111

1

2

1

2

MK

L

MLK

L

MLK

L

LjLjLj

LjLjLj

KjL

KjM

KjM

KjL

C

B

A

CBC

CCA

2012/11/29

13

Ideal Transformers (1/3)

1. Coils have very large reactance. (L1, L2, M ~ )

2. Coupling coefficient is equal to unity. (k = 1)

3. Primary and secondary are lossless.

(series resistances R1= R2= 0)

21 dtd

Nvdtd

Nv

2211

Ideal Transformers (2/3)

.thecallediswhere

or1

coupling,perfectFor

gives(1b)into1(c)ngSubstituti(1c)

(1a),From

(1b)(1a)

111

21

1

212

21

21

2

211

2

1211

2212

2111

oturns ratin

nLL

LLL

LLMk

jL

ML

LM

LjMj

LjMjMjLj

VVVV

IVV

IVI

IIVIIV

2012/11/29

14

Ideal Transformers (3/3)

nnvv

ii

iviv

nNN

nNN

vv

dtd

Nvdtd

Nv

11losslessisertransformThe

or

,

2

1

1

2

2

1

1

2

2211

1

2

1

2

1

2

1

2

2211

VV

II

VV

More Comments (1/2)

(Lossless)powerSuppliedpowerAbsorbed

zero.approachesenergystorednetThe

0

havewevoltages,finiteFor,,

,

2211

1

1

2121

2211

iviv

Nv

dtd

NNLLdtd

Nvdtd

Nv

2012/11/29

15

More Comments (2/2)

nLL

ii

LLw

iLL

i

iLL

iL

iiLL

iLL

iL

iiLLiLiLw

1

)(02

21

221

21

21

finite.bemustenergystoredThe

2

1

1

2

11

21

21

2

21

211

211

222

1

2211

2121222

211

Types of Transformers•When n = 1, we generally call the transformer an

isolation transformer.

•If n > 1 , we have a step-up transformer (V2 > V1).

•If n < 1 , we have a step-down transformer (V2 < V1).

2012/11/29

16

Impedance Transformation

lossless!isertransformTheloss.without

secondarythetodeliveredisprimarythetosuppliedpowercomplexThe

isprimaryin thepowercomplexThe

1

2*22

*2

2*111

21

21

2

1

1

2

1

2

1

2

SIVIV

IVS

II

VV

IIVV

nn

nn

nNN

nNN

matching!impedanceforUseful

)(

1issourceby the

seenasimpedanceinputThe

2in

2

22

2

2

1

1in

impedancereflectedn

nnn

LZZ

IV

IV

IV

Z

Zin

How to make a transformer ideal?

Zin

L

L

L

L

L

L

LjLj

LjLLLLLj

LjM

Lj

LLMRR

LjRM

LjR

ZZ

ZZ

ZZ

ZZ

2

1

2

212

212

1

2

22

1in

2121

22

22

11in

and0If

22

1

1

2

22

1

2

1in

2

2

,desiredaFor

the:where

0for

nL

Ln

oturns ratiLL

n

nLL

LjLj

L

L

LLL

L

ZZZZ

Z

2012/11/29

17

Impedance Matching

Linear network

0:

complex:

:issferpower tranmaximumforconditionThe

Th2

*Th2

jRnRn

LLL

LL

ZZ

ZZZ

Ideal Transformer Circuit (1/3)Linear network 1 Linear network 2

2012/11/29

18

Ideal Transformer Circuit (2/3)

n

n

s2

21Th

21 0

V

VVV

II

22

2

22

2

2

1

1Th

21

21

1

n

nnn

n

n

ZIV

IV

IV

Z

VV

II

1

Ideal Transformer Circuit (3/3)

cc

c c

Equivalent 1: Equivalent 2:

2012/11/29

19

Applications of Transformers•To step up or step down voltage and current (useful

for power transmission and distribution).

•To isolate one portion of a circuit from another.

•As an impedance matching device for maximumpower transfer.

• Frequency-selective circuits.

Applications: Circuit Isolation

When the relationship betweenthe two networks is unknown,any improper direct connectionmay lead to circuit failure.

This connection style canprevent circuit failure.

2012/11/29

20

Applications: DC Isolation

Only ac signal can pass, dc signal is blocked.

Applications: Load Matching

2012/11/29

21

Applications: Power Distribution