Fundamentals of Electric CircuitsChapter 13

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Overview

• This chapter introduces the concept of mutual inductance.

• The general principle of magnetic coupling is covered first.

• This is then applied to the case of mutual induction.

• The chapter finishes with coverage of linear transformers.

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Inductance

• When two conductors are in close proximity to each other, the magnetic flux due to current passing through will induce a voltage in the other conductor.

• This is called mutual inductance.• First consider a single inductor, a coil with N

turns.• Current passing through will produce a

magnetic flux, .

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Self Inductance

• If the flux changes, the induced voltage is:

• Or in terms of changing current:

• Solved for the inductance:

• This is referred to as the self inductance, since it is the reaction of the inductor to the change in current through itself.

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dv N

dt

d div N

di dt

dL N

di

Magnetic Coupling

• Now consider two coils with N1 and N2 turns respectively.

• Each with self inductances L1 and L2.

• Assume the second inductor carries no current.

• The magnetic flux from coil 1 has two components:

11 links the coil to itself, 12 links both coils.

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1 11 12

Dot Convention

• If a current enters the dotted terminal of one coil, the reference polarity of the mutual voltage in the second coil is positive at the dotted terminal of the second coil.

• If a current leave 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.

• See the examples in the next slide:

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Dot Convention II

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Problem Solving• Mutually coupled circuits are often

challenging to solve due to the ease of making errors in signs.

• If the problem can be approached where the value and the sign of the inductors are solved in separate steps, solutions tend to be less error prone.

• See the illustration for the proposed steps.

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Linear Transformers• A transformer is a magnetic device that takes

advantage of mutual inductance.• It is generally a four terminal device comprised of

two or more magnetically coupled coils.• The coil that is connected to the voltage source is

called the primary.• The one connected to the load is called the

secondary.• They are called linear if the coils are wound on a

magnetically linear material.

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Transformer Impedance

• An important parameter to know for a transformer is how the input impedance Zin is seen from the source.

• Zin is important because it governs the behavior of the primary circuit.

• Using the figure from the last slide, if one applies KVL to the two meshes:

• Here you see that the secondary impacts Zin

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2 2

2 2R

L

MZ

R j L Z

Equivalent circuits• We already know that coupled inductors can

be tricky to work with.• One approach is to use a transformation to

create an equivalent circuit.• The goal is to remove the mutual inductance.• This can be accomplished by using a T or a

network.• The goal is to match the terminal voltages

and currents from the original network to the new network.

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Equivalent Circuits II

• Starting with the coupled inductors as shown here:

• Transforming to the T network the inductors are:

• Transforming to the network the inductors are:

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1 1 1

2 2 2

V j L j M I

V j M j L I

2 21 2 1 2

2 1

21 2

A B

C

L L M L L ML L

L M L M

L L ML

M

Ideal Transformers II• Iron core transformers are close to ideal.• The voltages are related to each other by the

turns ration n:

• The current is related as:

• A step down transformer (n<1) is one whose secondary voltage is less than its primary voltage.

• A step up (n>1) is the opposite

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2 1

1 2

1I N

I N n

Ideal Transformers III• There are rules for getting the polarity

correct from the transformer in a circuit:

• If V1 and V2 are both positive or both negative at the dotted terminal, use +n otherwise use –n

• If I1 and I2 both enter or leave the dotted terminal, use -n otherwise use +n

• The complex power in the primary winding is:

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** *21 1 1 2 2 2 2

VS V I nI V I S

n

Reflected Impedance• The input impedance that appears at the

source is:

• This is also called the reflected impedance since it appears as if the load impedance is reflected to the primary side.

• This matters when one considers impedance matching.

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2L

in

ZZ

n

Removing the transformer• We can remove the transformer from the circuit by

adding the secondary and primary together by certain rules:

• The general rule for eliminating the transformer and reflecting the secondary circuit to the primary side is: Divide the secondary impedance by n2, divide the secondary voltage by n, and multiply the secondary current by n.

• The rule for eliminating the transformer and reflecting the primary circuit to the secondary side is: Multiply the primary impedance by n2, multiply the primary voltage by n, and divide the primary current by n.

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Three Phase Transformer

• When working with three phase power, there are two choices for transformers:– A transformer bank, with one transformer per

phase– A three phase transformer

• The three phase transformer will be smaller and less expensive.

• The same connection permutations of Delta and Wye hold as discussed previously.

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