Acoplamento Magnético em Circuitos Elétricos

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Lecture 10
Chapter 3: Magnetically Coupled Circuits
Electric Circuits II
Diego Mej́ıa Giraldo
August 9, 2019
Mutual Inductance
I When two loops with or without contacts between them affect
each other through the magnetic field generated by one of
them, they are said to be magnetically coupled.
I The transformer uses the concept of magnetic coupling.
I Uses of transformers: power systems, electronics, impedance
matching.
I What we will cover in this chapter:
I Mutual inductance,
I Convention for determining the voltaje polarities of inductively
coupled components,
I The linear transformer, the ideal transformer.
Mutual Inductance
L
+ −v
Consider an inductor, a coil with N turns.
If a current i flow through the coil, a
magnetic flux is produced, the voltage
induced is
v =?
The self inductance is therefore
L = N
dφ
di
Mutual Inductance
L1
+ −v1
L2
+ −v2M
Now consider two coils with self-inductances L1
and L2.
v1 = N1
dφ11
dt
= L1
di1
dt
This induces a voltage across terminals of coil 2
given by Faraday’s law:
v2 = N2
d φ12
d t
=?
where φ12 is tha magnetic flux produced by i1
induced in coil 2.
M21 is the mutual inductance of coil 2 with
respect to coil 1.
If we instead apply a current i2 to coil 2, there
will be an induced voltage in coil 1. So, what is
the expression for the induced v1?
Mutual Inductance
L1
+ −v1
L2
+ −v2M
Now consider two coils with self-inductances L1
and L2.
v1 = N1
dφ11
dt
= L1
di1
dt
This induces a voltage across terminals of coil 2
given by Faraday’s law:
v2 = N2
d φ12
d t
=?
where φ12 is tha magnetic flux produced by i1
induced in coil 2.
M21 is the mutual inductance of coil 2 with
respect to coil 1.
If we instead apply a current i2 to coil 2, there
will be an induced voltage in coil 1. So, what is
the expression for the induced v1?
Mutual Inductance
Definition
Mutual inductance is the ability of one inductor to induce a voltage
across a neighboring inductor, measured in henrys (H).
Q: How to determine the induced voltage polarity?
We need to use the dot convention. What is it about?
Dot convention
If a current enters (leaves) the dotted terminal of coil 1, the
reference polarity of the mutual voltage in coil 2 is positive
(negative) at the dotted terminal of the second coil.
Mutual Inductance
Definition
Mutual inductance is the ability of one inductor to induce a voltage
across a neighboring inductor, measured in henrys (H).
Q: How to determine the induced voltage polarity?
We need to use the dot convention. What is it about?
Dot convention
If a current enters (leaves) the dotted terminal of coil 1, the
reference polarity of the mutual voltage in coil 2 is positive
(negative) at the dotted terminal of the second coil.
Mutual Inductance
Definition
Mutual inductance is the ability of one inductor to induce a voltage
across a neighboring inductor, measured in henrys (H).
Q: How to determine the induced voltage polarity?
We need to use the dot convention. What is it about?
Dot convention
If a current enters (leaves) the dotted terminal of coil 1, the
reference polarity of the mutual voltage in coil 2 is positive
(negative) at the dotted terminal of the second coil.
Mutual Inductance
Definition
Mutual inductance is the ability of one inductor to induce a voltage
across a neighboring inductor, measured in henrys (H).
Q: How to determine the induced voltage polarity?
We need to use the dot convention. What is it about?
Dot convention
If a current enters (leaves) the dotted terminal of coil 1, the
reference polarity of the mutual voltage in coil 2 is positive
(negative) at the dotted terminal of the second coil.
Fill the blanks...
i1
L1
+
−
v1
+
−
v2
i2
L2
M (
v1
v2
)
=
(
? ?
? ?
)
× d
dt
(
i1
i2
)
i1
L1
+
−
v1
+
−
v2
i2
L2
M (
v1
v2
)
=
(
? ?
? ?
)
× d
dt
(
i1
i2
)
Fill the blanks...
i1
L1
+
−
v1
+
−
v2
i2
L2
M (
v1
v2
)
=
(
? ?
? ?
)
× d
dt
(
i1
i2
)
i1
L1
+
−
v1
+
−
v2
i2
L2
M (
v1
v2
)
=
(
? ?
? ?
)
× d
dt
(
i1
i2
)
Fill the blanks...
i1
L1
+
−
v1
+
−
v2
i2
L2
M
(
v1
v2
)
=
(
? ?
? ?
)
× d
dt
(
i1
i2
)
i1
L1
+
−
v1
+
−
v2
i2
L2
M
(
v1
v2
)
=
(
? ?
? ?
)
× d
dt
(
i1
i2
)
Fill the blanks...
i1
L1
+
−
v1
+
−
v2
i2
L2
M
(
v1
v2
)
=
(
? ?
? ?
)
× d
dt
(
i1
i2
)
i1
L1
+
−
v1
+
−
v2
i2
L2
M
(
v1
v2
)
=
(
? ?
? ?
)
× d
dt
(
i1
i2
)
Where do the dots come from?
Lenz’s law
The induced current must create a magnetic flux such that it
opposes the flux change that produced it.
Example
Determine the dots in the following coil configurations:
Where do the dots come from?
Example
Obtain the KVL equations for the following circuit. Assume Lp, Ls ,
and M are given.