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Chapter 13
Magnetically Coupled Circuits
Chapter Objectives:
 Understand magnetically coupled circuits.
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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.
Be familiar with ideal autotransformers.
Learn how to analyze circuits involving three-phase transformers.
Be able to use PSpice to analyze magnetically coupled circuits.
Apply what is learnt to transformer as an isolation device and power
distribution
Huseyin Bilgekul
Eeng224 Circuit Theory II
Department of Electrical and Electronic Engineering
Eastern Mediterranean University
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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
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Mutual Inductance
 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.
v1  N1
d1
d (11  21 )
 N1
dt
dt
v2  N 2
d 2
d (12  22 )
 N2
dt
dt
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Mutual Inductance
v2  M 21
a) Magnetic flux produced by a single
coil.
v1  M 12
di1
dt
b) Mutual inductance M21 of coil 2
with respect to coil 1.
di2
dt
c) Mutual inductance of M12 of coil 1
with respect to coil 2.
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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 Henry’s.
 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.
M12  M 21  M
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Dot Convention
 If the 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 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.
v2  M
di1
dt
v2   M
di1
dt
v1   M
v1  M
di2
dt
di2
dt
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Dot
Convention
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Coils in Series
 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.
a)
Series-aiding connection.
L=L1+L2+2M
b) Series-opposing connection.
L=L1+L2-2M
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Time-domain and Frequency-domain Analysis
jM
V1
I1
jL1
jL2
I2
V2
b) Frequency-domain circuit
a) Time-domain circuit
Time Domain
di1
di2
v1  i1 R1  L1
M
dt
dt
di
di
v2  i2 R2  L2 2  M 1
dt
dt
Frequency Domain
V1  ( R1  j L1 ) I1  j MI 2
V2  j MI1  ( R2  j L2 ) I 2
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Induced mutual voltages
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Induced mutual voltages
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P.P.13.2 Determine the phasor currents
Mesh 1 1260=(5+j2+j6-j3  2)I1  j 6I2  j3I2
j3I1
+
-
+
-
j3I2
+
-
Mesh 2 0=(j6-j4)I2  j 6I1  j3I1
j3I1
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Mutually Induced Voltages
 To find I0 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.
-j50
Io
I3
j20Ic
j40
+ 
j10Ib
 +
+

Ia
j60
j30Ic

+
 +
Ic
j30Ib
j20Ia
500 V
+

j80
I1
Ib

+
I2
100 
I a = I1 – I3
I b = I2 – I1
I c = I3 – I2
Io = I3
Blue Voltage due to Ia
Red Voltage due to Ic
Green Voltage due to Ib
j10Ia
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Mutually Induced Voltages
 To find I0 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.
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Energy in a Coupled Circuit
 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.
1 2 1
w  L1i1  L2i2 2  Mi1i2
2
2
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Coupling Coefficient
 The Coupling Coefficient k is a measure of the magnetic coupling between two coils
0  k 1
k  1 Perfect Coupling
k  0.5 Loosly Coupling
k  0.5 Tightly Coupling
a)
Loosely coupled coil
b) Tightly coupled coil
0  k 1
k
M
L1 L2
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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
 Let us obtain the input impedance as seen from the source,
ZR
V  ( R1  j L1 ) I1  j MI 2
0   j MI1  ( R2  j L2  Z L ) I 2
V
2M 2
Zin   R1  j L1 
 R1  j L1  Z R
I1
R2  j L2  Z L
 2M 2
ZR 
R2  j L2  Z L
REFLECTED IMPEDANCE
• Secondary impedance seen from the primary side is the Reflected Impedance.
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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.
a)
Transformer circuit
b) Equivalent T circuit of the transformer
La  L1  M , Lb  L2  M , Lc  M
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Equivalent П Circuit for Linear Transformers
 The coupled transformer can equivalently be represented by an EQUIVALENT П
circuit using uncoupled inductors.
a)
Transformer circuit
b) Equivalent Π circuit of the transformer
L1L2  M 2
L1L2  M 2
L1L2  M 2
LA 
, LB 
, LC 
L2  M
L1  M
M
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La  L1  M
Lb  L2  M
Lc  M
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Homework 2
Problem 13.79
X23

Submit your results by giving in the following results similar to the form
given below by May 2, 2007.
a) Your origiinal schematic diagram
b) The print page of your results
c) Repeat the calculation for 2 other values of X23=j0, j10, j15 Ω
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Homework 2
Schematic Diagram
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