Magnetic Ck Ts

7/31/2019 Magnetic Ck Ts

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1ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu

Magnetically Coupled Circuits

Objectives:

Understand magnetically coupled circuits.

Learn the concept of mutual inductance.

Be able to determine energy in a coupled circuit.

Learn how to analyze circuits involving linear and ideal transformers.


Mutual Inductance Transformers are constructed of two coils placed so that the charging

flux developed by one will link the other.

The coil to which the source is applied is called the primary coil.

The coil to which the load is applied is called the secondary coil.

Three basic operations of a transformer are:

Step up/down

Impedance matching

Isolation


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Mutual Inductance Devices


Mutual Inductance

1 11 211 1 1

( )d dv N N

dt dt

+= =

2 12 222 2 2

( )d dv N N

dt dt

+= =

When two coils are placed close to each other, a changing flux in one coil will cause

an induced voltage in the second coil. The coils are said to have mutual inductance M,

which can either add or subtract from the total inductance depending on if the fields are

aiding or opposing.

Mutual inductance is the ability of one inductor to induce a voltage across a

neighboring inductor.


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b) Mutual inductance M21 of coil 2

with respect to coil 1.

Mutual Inductance

a) Magnetic flux produced by a single

coil.

c) Mutual inductance of M12 of coil 1

with respect to coil 2.

2

1 12

di

v M dt=

12 21

div M

dt=


Mutual Inductance Mutual inductances M12 and M21 are equal.

They are referred as M.

We refer to M as the mutual inductance between two coils.

M is measured in Henrys.

Mutual inductance exists when two coils are close to each other.

Mutual inductance effect exist when circuits are driven by time varying sources.

Recall that inductors act like short circuits to DC.

12 21M M= =


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Dot Convention If the current ENTERS the dotted terminal of one coil, the reference polarity of themutual voltage in the second coil is POSITIVE at the dotted terminal of the second coil.

If the current LEAVES the dotted terminal of one coil, the reference polarity of the

mutual voltage in the second coil is NEGATIVE at the dotted terminal of the second coil.

12

div M

dt=

12

div M

dt=

21

div M

dt=

21

div M

dt=


DotConvention


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Coils in Series

a) Series-aiding connection.

L=L1+L2+2M

b) Series-opposing connection.

L=L1+L2-2M

The total inductance of two coupled coils in series depend on the placement of

the dotted ends of the coils. The mutual inductances may add or subtract.


Time-domain and Frequency-domain Analysis

1 21 1 1 1

2 12 2 2 2

1 1 1 1 2

2 1 2 2 2

TimeDomain

FrequencyDomain

( )

( )

di div i R L M

dt dt

di div i R L M

dt dt

V R j L I j MI

V j MI R j L I

= + +

= + +

= + +

= + +

V1 V2I1 I2jL1 jL2

jM

a) Time-domain circuit b) Frequency-domain circuit


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Induced mutual voltages


Induced mutual voltages


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Mutually Induced Voltages To findI0 in the following circuit, we need to write the mesh equations.

Let us represent the mutually induced voltages by inserting voltage sources in

order to avoid mistakes and confusion.

+

+

+

I1 I2

Io

j20Ic

100

500 V

I3

+

++

+

j10Ibj40

j30Ic

j80

j10Ia

j20Ia

j60

j30Ib

-j50

Ia

Ib

IcIa = I1 I3Ib = I2 I1Ic = I3 I2

Io = I3Blue Voltage due to IaRed Voltage due to Ic

Green Voltage due to Ib


Energy in a Coupled Circuit

2 2

1 1 2 2 1 2

1 1

2 2w L i L i Mi i= +

The total energy w stored in a mutually coupled inductor is:

Positive sign is selected if both currents ENTER or LEAVE the dotted terminals.

Otherwise we use Negative sign.


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Coupling Coefficient

a) Loosely coupled coil b) Tightly coupled coil

1 2

0 1k

Mk

L L

=

The Coupling Coefficient kis a measure of the magnetic coupling between two coils

0 1k 1 Perfect Coupling

0.5 Loosly Coupling

0.5 Tightly Coupling

k

k

k

=


Linear Transformers A transformer is generally a four-terminal device comprising two or more

magnetically coupled coils.

The transformer is called LINEAR if the coils are wound on magnetically linear

material.

For a LINEAR TRANSFORMER flux is proportional to current in the windings.

Resistances R1 and R2 account for losses in the coils.

The coils are named as PRIMARY and SECONDARY.


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Reflected Impedance for Linear Transformers

1 1 1 2

1 2 2 2

( )

0 ( )L

V R j L I j MI

j MI R j L Z I

= +

= + + +

2 2

1 1 1 11 2 2

in RL

V M

Z R j L R j L ZI R j L Z

= = + + = + +

+ +

2 2

2 2

REFLECTED IMPEDANCER

L

MZ

R j L Z

=

+ +

Secondary impedance seen from the primary side is the Reflected Impedance.

Let us obtain the input impedance as seen from the source,

ZR



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Equivalent T Circuit for Linear Transformers

The coupled transformer can equivalently be represented by an EQUIVALENT T

circuit using UNCOUPED INDUCTORS.

1 2, ,a b cL L M L L M L M= = =

a) Transformer circuit b) Equivalent T circuit of the transformer


Equivalent Circuit for Linear Transformers The coupled transformer can equivalently be represented by an EQUIVALENT circuit using uncoupled inductors.

2 2 2

1 2 1 2 1 2

2 1

, ,A B C

L L M L L M L L ML L L

L M L M M

= = =

a) Transformer circuit b) Equivalent circuit of the transformer


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1

2

a

b

c

L L M

L L M

L M

=

=

=


Ideal Transformers

Objectives: Understand magnetically coupled circuits.

Learn the concept of mutual inductance. Be able to determine energy in a coupled circuit.

Learn how to analyze circuits involving linear and ideal transformers.


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Ideal Transformers A Ideal Transformer is a unity Coupled, lossless transformer in which the primary

and secondary coils have infinite self inductances.

A Transformer is ideal if:

1.) Large reactance coils;

2.) Unity Coupling k=1.

3.) Coils are lossless (R1=R2=0)

1 2, ,L L M

Ideal transformer

Circuit symbol for the Ideal transformer


Ideal Transformers


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Non Ideal Transformers

An ideal transformer has no power loss; all power applied to the primary is all

delivered to the load. Actual transformers depart from this ideal model. Some

loss mechanisms are:

Winding resistance: Causing power to be dissipated in the windings.

Hysteresis loss: Due to the continuous reversal of the magnetic field.

Core losses: Due to circulating current in the core (eddy currents).

Flux leakage: Flux from the primary that does not link to the secondary.

Winding capacitance: It has a bypassing effect for the windings.

The ideal transformer does not dissipate power. Power delivered from the source

is passed on to the load by the transformer.

The efficiency of a transformer is the ratio of power delivered to the load (Pout)

to the power delivered to the primary (Pin).


Input-Output Variables of an Ideal Transformer

2 2

1 1

Turns RatioV N

nV N

= = =

1 21 1 1 2 1

1

2

1 22 1 2 2 2 2 2

1 1

1 2

1 2 1 1 2 2 22 2 1 1

1 1 1

22 1

1

Perfect Coupling 1, Thus we have Substitute

V j MI V j L I j MI I j L

MV j M IV j MI j L I V j L I

L L

k M L L

L L V j L L I Lj L I V nVL L

NV VNL

= + =

= + = +

= =

= + = = =

The input and output voltages and currents

of an ideal transformer are related only by the

turns ratio.


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A Ideal Transformer is called:

1.) Step-up transformer if n > 1.2.) Step-down transformer if n < 1.

3.) Isolation transformer if n=1.

Input-Output Variables of an Ideal Transformer

2 1 2

1 2 1

V I Nn

V I N= = =


Transformer DOT convention is needed to assign the polarity of the output variables.

1.) IfV1 and V2 are BOTH + or BOTH at the dotted terminals use +n, otherwise n.

2.) IfI1 andI2 BOTH ENTER or BOTH LEAVE the dotted terminals use n, otherwis

+n.

2 1 2

1 2 1

V I Nn

V I N

= = =

Transformer Dot Convention

In phase Out of phase

Dot convention indicating the phase relationship between the input and the output.


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Typical circuits illustrating polarity for voltages and direction of currents of an ideal

transformer

Dot Convention for Ideal Transformers


Conservation of the Complex Power

21 1 1 2 2 2 2( )n

n

= = = =

VS V I I V I S

An ideal transformer absorbs no power.

The complex power in the primary winding is equal to the complex power

delivered to the secondary winding.

Transformer absorbs no power. We assume a lossless transformer.

S1 S2


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Reflected Impedance of Ideal Transformers The ability of a transformer to transform a given impedance to another value is

very useful in IMPEDANCE MATCHING.

a) Obtaining the VTh. b) Obtaining the ZTh.

21

2s

n n= = =Th

V VV V

2 2 2

1

1 2

2

2

2

Z

n nZZ

n n n= = = =Th

V I

V

I I I

Zth


Reflected impedance Equivalent circuit of reflection of the secondary to primary side.

Equivalent circuit of reflection of the primary to secondary side.

21 2

Reflected to PrimaryR

ZZ

n=

2

2 1

Reflected to Secondary

RZ n Z=


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ZR


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I3


Example 13.15 Determine the voltage across the load.

Apply superposition principle.

Load voltage due to DC is zero (No induction without change in time)

O O-DC O-AC

1200 cos 40 cos

3V V V t t = + = + =

DC Source only AC source only


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Problem 13.2Determine the inductance of the three series-connected inductors.

Consider the polarities of the coupled inductances.

M12 is series adding while M23 and M31are series opposing .

L = L1 + L2 + L3 + 2M12 2M23 2M31

= 10 + 12 +8 + 2x6 2x6 2x4

= 22H


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Problem 13.9 Find Vx

in the network shown.


Problem 13.21Find I1 and I2 in the circuit. 13.90. Calculate the power absorbedby the 4- resistor.


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Problem 13.28 find the value of X that will give maximum power transfer to

the 20- load.



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I3

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