litar elektrik ii eet 102/4. silibus litar elektrik ii mutual inductance two port network...

LITAR ELEKTRIK LITAR ELEKTRIK IIII

EET 102/4EET 102/4

SILIBUS LITAR SILIBUS LITAR ELEKTRIK IIELEKTRIK II

Mutual InductanceMutual Inductance Two port NetworkTwo port Network Pengenalan Jelmaan LaplacePengenalan Jelmaan Laplace Kaedah Jelmaan Laplace Dlm Analisis Kaedah Jelmaan Laplace Dlm Analisis

LitarLitar Sambutan Frekuensi Litar ACSambutan Frekuensi Litar AC Siri FourierSiri Fourier Jelmaan FourierJelmaan Fourier

MUTUAL INDUCTANCEMUTUAL INDUCTANCE

Self inductanceSelf inductance Concept of mutual inductanceConcept of mutual inductance Dot conventionDot convention Energy in a coupled circuitEnergy in a coupled circuit Linear transformer Linear transformer Ideal transformerIdeal transformer

MUTUAL INDUCTANCEMUTUAL INDUCTANCE

INTRODUCTIONINTRODUCTION magnetically coupledmagnetically coupled When two loops with or without When two loops with or without

contacts between them, affect each contacts between them, affect each other through magnetic field other through magnetic field generated by one of them – they are generated by one of them – they are said to be magnetically coupled.said to be magnetically coupled.

Example of device using this Example of device using this concept-transformer.concept-transformer.

TransformerTransformer

Use magnetically coupled coils to Use magnetically coupled coils to transfer energy from one circuit to transfer energy from one circuit to another.another.

Key circuit element where it is used Key circuit element where it is used for stepping down or up ac voltages for stepping down or up ac voltages or currents.or currents.

Also used in electronic circuits such Also used in electronic circuits such as radio and tv receiver.as radio and tv receiver.

Consider a single inductor with N turns.When current i, flow through coil, magnetic flux is produced around it.

Faraday’s LawFaraday’s Law

Induced voltage, v in the coil Induced voltage, v in the coil is proportional to number of is proportional to number of turns N and the time rate of turns N and the time rate of change of magnetic flux, change of magnetic flux, . .

dt

dNv

But we know that the flux But we know that the flux is is produce by current produce by current i,i, thus thus any change in the current any change in the current will change in flux will change in flux as well. as well.

dt

di

di

dNv

dt

diLv @

The inductance L of the The inductance L of the inductor is thus given byinductor is thus given by

di

dNL

Self-Inductance

Self InductanceSelf Inductance

Inductance that relates the Inductance that relates the induced voltage in a coil with induced voltage in a coil with a time-varying current in the a time-varying current in the same coil.same coil.

Mutual InductanceMutual Inductance When two inductors or coils When two inductors or coils

are in close proximity to are in close proximity to each other, magnetic flux each other, magnetic flux caused by current in one caused by current in one coil links with the other coil links with the other coil, therefore producing coil, therefore producing the induced voltage.the induced voltage.

Mutual Inductance Mutual Inductance

Magnetic flux Magnetic flux 11 originating from originating from coil 1 has 2 components:coil 1 has 2 components:

Since entire flux Since entire flux 11 links coil 1, links coil 1, the voltage induced in coil 1 is:the voltage induced in coil 1 is:

12111

dt

dNv 1

11

Only flux Only flux 1212 links coil 2, so the links coil 2, so the voltage induced in coil 2 is:voltage induced in coil 2 is:

As the fluxes are caused by As the fluxes are caused by current icurrent i11 flowing in coil 1, flowing in coil 1, equation vequation v11 can be written as: can be written as:

dt

dNv 12

22

dt

diL

dt

di

di

dNv 1

11

1

111

Self inductance of coil 1

Similarly for equation vSimilarly for equation v22::

dt

diM

dt

di

di

dNv 1

211

1

1222

Mutual inductance of coil 2With respect to coil 1

Coil 2

Magnetic flux Magnetic flux 22 comprises of 2 comprises of 2 components:components:

The entire flux The entire flux 22 links coil 2, so links coil 2, so the voltage induced in coil 2 is:the voltage induced in coil 2 is:

22212

dt

diL

dt

di

di

dN

dt

dNv 2

22

2

22

222

Self-inductance of coil 2

Since only flux Since only flux 21 21 links with links with coil 1, the voltage induced in coil 1, the voltage induced in coil 1 is:coil 1 is:

dt

diM

dt

di

di

dN

dt

dNv 2

122

2

211

2111

Mutual inductance of coil 1 with respect to coil 2

For simplicity, M12 and For simplicity, M12 and M21 are equal:M21 are equal:

MMM 2112

Mutual inductance between two coils

ReminderReminder

Mutual coupling exists when Mutual coupling exists when inductors or coils are in close inductors or coils are in close proximity and circuit are driven by proximity and circuit are driven by time-varying sources.time-varying sources.

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

Dot ConventionDot Convention A dot is placed in the circuit A dot is placed in the circuit

at one end of each of the two at one end of each of the two magnetically coupled coils to magnetically coupled coils to indicate the direction of indicate the direction of magnetic flux if current enters magnetic flux if current enters that dotted terminal of the that dotted terminal of the coil.coil.

Dot convention is stated as Dot convention is stated as follows:follows:

If a current If a current entersenters the dotted terminal the dotted terminal of one coil, the reference polarity of of one coil, the reference polarity of mutual voltage in second coil is mutual voltage in second coil is positivepositive at dotted terminal of second coil.at dotted terminal of second coil.

If a current If a current leavesleaves the dotted terminal the dotted terminal of one coil, the reference polarity of of one coil, the reference polarity of mutual voltage in second coil is mutual voltage in second coil is negativenegative at dotted terminal of second at dotted terminal of second coil.coil.

Dot convention for coils Dot convention for coils in seriesin series

MLLL 221

MLLL 221

Example 1Example 1

Example 1Example 1

dt

diM

dt

diLRiv 21

1111

dt

diM

dt

diLRiv 12

2222

Coil 1:

Coil 2:

In frequency domain..In frequency domain..

22212

21111

)(

)(

ILjRMIjV

MIjILjRV

Example 2Example 2

Example 2Example 2

221

2111

)(0

)(

ILjZMIj

MIjILjZV

L

Example 3Example 3

Solution..Solution.. For coil 1, we use KVL:For coil 1, we use KVL:

123

03)54(12

21

21

IjjI

IjIjj

For coil 2, For coil 2,

0)612(3 21 IjIj

22

1 )42(3

)612(Ij

j

IjI

Substitute equation 1 into 2:Substitute equation 1 into 2:

12)4()342( 22 IjIjj

Aj

I o04.1491.24

122

Solve for I1:Solve for I1:

A

IjI

o

oo

39.4901.13

)04.1491.2)(43.63372.4(

)42( 21

Energy in a coupled Energy in a coupled circuitcircuit

Energy stored in an inductor is given Energy stored in an inductor is given by:by:

Now, we want to determine energy Now, we want to determine energy stored in magnetically coupled coils.stored in magnetically coupled coils.

2

2

1Liw

Circuit for deriving energy Circuit for deriving energy stored in a coupled circuitstored in a coupled circuit

Power in coil 1:Power in coil 1:

Energy stored in coil 1:Energy stored in coil 1:

dt

diLiivtp 1

11111 )(

2110 11111 2

11

ILdiiLdtpwI

Maintain iMaintain i1 1 and we increase iand we increase i22 to Ito I22. So, the power in coil 2 is:. So, the power in coil 2 is:

dt

diLi

dt

diMI

vidt

diMi

vivitp

222

2121

222

121

22112 )(

Energy stored in coil 2:Energy stored in coil 2:

2222112

0 2220 2112

22

2

1

22

ILIIM

diiLdiIM

dtpw

II

Total energy stored in the coils Total energy stored in the coils when both i1 and i2 have when both i1 and i2 have reached constant values is:reached constant values is:

2112222

211

21

2

1

2

1IIMILIL

www

Since MSince M1212=M=M2121=M, thus=M, thus

21222

211 2

1

2

1IMIILILw

Generally, energy stored in Generally, energy stored in magnetically coupled circuit magnetically coupled circuit is:is:

21222

211 2

1

2

1IMIILILw

Coupling coefficient, kCoupling coefficient, k

A measure of the magnetic A measure of the magnetic coupling between two coils; 0 coupling between two coils; 0 ≤ k ≤ 1≤ k ≤ 1

21LL

Mk

Linear TransformerLinear Transformer

Transformer is generally a four-Transformer is generally a four-terminal device comprising two or terminal device comprising two or more magnetically coupled coils.more magnetically coupled coils.

Coil that is directly connected to Coil that is directly connected to voltage source is voltage source is primary winding.primary winding.

Coil connected to the load is called Coil connected to the load is called secondary winding.secondary winding.

R1 and R2 included to calculate for R1 and R2 included to calculate for losses in coils.losses in coils.

Linear TransformerLinear Transformer

Primary winding Secondary winding

Obtain input impedance, Zin Obtain input impedance, Zin as seen from source because as seen from source because Zin governs the behaviour of Zin governs the behaviour of primary circuit.primary circuit.

Apply KVL to the two loops:Apply KVL to the two loops:

2221

2111

)(0

)(

IZLjRMIj

MIjILjRV

L

Input impedance ZInput impedance Zinin::

L

in

ZLjR

MLjR

I

VZ

22

22

11

1

RZ

Equivalent circuit of Equivalent circuit of linear transformerlinear transformer

Equivalent T circuit Equivalent T circuit

Equivalent ∏ circuitEquivalent ∏ circuit

Voltage-current Voltage-current relationship for primary relationship for primary and secondary coils give and secondary coils give

the matrix equation:the matrix equation:

2

1

2

1

2

1

I

I

LjMj

MjLj

V

V

By matrix inversion, this By matrix inversion, this can be written as:can be written as:

2

1

221

12

21

221

221

2

2

1

)()(

)()(V

V

MLLj

L

MLLj

MMLLj

M

MLLj

L

I

I

Matrix equation for Matrix equation for equivalent T circuit:equivalent T circuit:

2

1

2

1

)(

(

I

I

LLjLj

LjLLj

V

V

cbc

cca

If T circuit and linear circuit If T circuit and linear circuit are the same, then:are the same, then:

MLLa 1

MLLb 2

MLc

For ∏ network, nodal For ∏ network, nodal analysis gives the terminal analysis gives the terminal

equation as:equation as:

2

1

2

1

111

111

V

V

LjLjLj

LjLjLjI

I

CBC

CCA

Equating terms in admittance Equating terms in admittance matrices of above, we obtain:matrices of above, we obtain:

ML

MLLLA

2

221

ML

MLLLB

1

221

M

MLLLC

221

IDEAL TRANSFORMERIDEAL TRANSFORMER

Properties of ideal Properties of ideal transformer:transformer:

Coils have very large reactances (LCoils have very large reactances (L11, , LL22, M, M→∞)→∞)

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

Primary and secondary winding are Primary and secondary winding are lossless (Rlossless (R11=0=R=0=R22))

Ideal transformer is a unity-coupled, losslesstransformer where primary and secondary coils have infinite self-inductance.

Transformation ratioTransformation ratio

We know that:We know that:

Divide v2 with v1, we get: Divide v2 with v1, we get:

dt

dNv

11

dt

dNv

22

nN

N

v

v

1

2

1

2

Energy supplied to the primary Energy supplied to the primary must equal to energy absorbed by must equal to energy absorbed by secondary since no losses in ideal secondary since no losses in ideal transformer.transformer.

Transformation ratio is: Transformation ratio is:

2211 iviv

nV

V

I

I

1

2

2

1

Types of transformer:Types of transformer:

Step-down transformerStep-down transformer

One whose secondary voltage is less One whose secondary voltage is less than its primary voltage.than its primary voltage.

Step-up transformerStep-up transformer

One whose secondary voltage is One whose secondary voltage is greater than its primary voltage.greater than its primary voltage.

Typical circuits in ideal Typical circuits in ideal transformertransformer

Complex PowerComplex Power

From:From:

Complex power in primary winding for Complex power in primary winding for ideal txt:ideal txt:

212

1 @ nIIn

VV

2*

22*

22*

111 )( SIVnIn

VIVS

Input impedanceInput impedance

We know that:We know that:

Since VSince V2 2 / I/ I2 2 = Z= ZLL , thus , thus

2

22

1

1 1

I

V

nI

VZ in

2n

ZZ Lin Reflected

impedance

Example Example

Find IFind I11 dan I dan I2 2 for given circuit:for given circuit:

Solution…Solution…

11stst: Find input impedance: Find input impedance

2

210

nZ in

3

1n

RZ

Therefore, solve for Therefore, solve for II1 1

281810inZ

AIo

5.028

0141

An

II 5.1

)(

5.0

31

12

THE ENDTHE END