Transcript Ch13_PPT_Fund_Elec_Circ_5e

Fundamentals of
Electric Circuits
Chapter 13
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
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:
vN
d
dt
• Or in terms of changing current:
vN
d di
di dt
• Solved for the inductance:
LN
d
di
• 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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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:
1  11  12
• 11 links the coil to itself, 12 links both coils.
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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:
2M 2
ZR 
R2  j L2  Z L
• Here you see that the secondary impacts Zin
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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:
V1   j L1
V    j M
 2 
j M   I1 
j L2   I 2 
• Transforming to the 
network the inductors are:
L1 L2  M 2
LA 
L2  M
L1 L2  M 2
LB 
L1  M
L1 L2  M 2
LC 
M
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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:
I 2 N1 1


I1 N 2 n
• 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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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:
V2
*
S1  V I   nI 2   V2 I 2*  S 2
n
*
1 1
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Reflected Impedance
• The input impedance that appears at the
source is:
ZL
Z in  2
n
• 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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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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