Chapter 13: Magnetically Coupled Circuit
13.1 What is a transformer?
13.2 Mutual Inductance
13.3 Energy in a Coupled Circuit 13.4 Linear Transformers
13.5 Ideal Transformers
13.6 Applications
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13.1 What is a transformer?
• Transformer: an electrical device designed on the basis of the concept of magnetic coupling.
• It uses magnetically coupled coils to transfer energy from one circuit to another
• It is the key circuit elements for stepping up or stepping down ac voltages or currents, impedance matching, isolation, etc.
• Maxwell’s equations:
(differential form)
B
,
t
D
,
H  J 
t
  D   ,
E  
  B  0.
James Clerk Maxwell (1831-1879) was a
Scottish mathematician and theoretical
physicist. His most significant achievement
was aggregating a set of equations in
electricity, magnetism, and inductance —
Maxwell’s equations — including an important
modification of Ampère's Circuital Law. It is
famous for introducing to the physics
community a detailed model of light as an
electromagnetic phenomenon, building upon the
earlier hypothesis advanced by Faraday.
[The work of Maxwell] ... the most profound
and the most fruitful that physics has
experienced since the time of Newton.2
—Albert Einstein, The Sunday Post.
13.2 Mutual Inductance (1)
• 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, .
 If the flux changes, the induced voltage is: v  N
 In terms of changing current: v  N
 Solved for the inductance: L  N
d
di
d di
di
L
di dt
dt
d
dt
• 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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13.2 Mutual Inductance (2)
• 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.
• Even though the two coils are physically not connected, we say they are magnetically coupled.
• The entire flux passes through coil 1, thus the induced voltage d1
in coil 1 is:
v N
1

1
dt
• In coil 2, only 12 passes through, thus the induced voltage is:
v2  N 2
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d12
dt
13.2 Mutual Inductance (3)
• Mutual inductance: is the ability of one inductor to induce a voltage across a neighboring inductor, measured in henrys (H).
The open‐circuit mutual voltage across coil 2:
di1
v1  L1
dt
di1
v2  M 21
dt
The open‐circuit mutual voltage across coil 1:
v1  M 12
di2
dt
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13.2 Mutual Inductance (4)
• If a current enters (leaves) the dotted terminal of one coil, the reference polarity of the mutual voltage in the second coil is positive (negative) at the dotted terminal of the second coil.
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13.2 Mutual Inductance (5)
Dot convention for coils in series; the sign indicates the polarity
of the mutual voltage; series‐aiding connection & series‐
opposing connection.
L  L1  L2  2 M
(series-aiding connection)
By Faraday's law,
d
d
v1  N1 1 & v2  N 2 12
dt
dt
L  L1  L2  2 M
(series-opposing connection)
Michael Faraday (1791-
1867), was an English
chemist and physicist who
contributed to the fields
of electromagnetism and
electrochemistry.
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13.2 Mutual Inductance (6)
• Time‐domain analysis:
Applying KVL to coil 1,
v1  i1 R1  L1
di1
di
M 2
dt
dt
Applying KVL to coil 2,
v2  i2 R2  L2
di2
di
M 1
dt
dt
Applying Phasor,
V1   R1  j L1  I1  j MI 2
V2  j MI1   R2  j L2  I 2
• Frequency‐domain analysis:
Applying KVL to coil 1,
V   Z1  j L1  I1  j MI 2
Applying KVL to coil 2,
0   j MI1   Z L  j L2  I 2
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13.2 Mutual Inductance (7)
Example: Determine the voltage Vo in the circuit.
Example: Determine the phasor currents I1 and I2 in the circuit.
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13.3 Energy in a Coupled Circuit (1)
• The instantaneous energy stored in the circuit is given by
w
1 2 1 2
L1i1  L2i2  Mi1i2
2
2
Note: The positive sign is selected for the mutual term if both currents enter or leave the dotted terminals of the coils; the negative sign is selected otherwise. • The coupling coefficient, k, is a measure of the magnetic coupling between two coils; 0≤ k ≤1.
Loosely coupled Tightly coupled k  0.5
k  0.5
M  k L1L2  0  M  L1L2
Note: k 
12
12

21

or k  21 
1 11  12
2 21  22
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13.3 Energy in a Coupled Circuit (2)
Example: Determine the coupling coefficient. Calculate the energy stored in the coupled inductors at time t = 1 s if v(t) = 60 cos (4t
+ 30° ) V.
Example: Determine the coupling coefficient and the energy stored in the coupled inductors at time t = 1.5 s.
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13.4 Linear Transformer (1)
• It is generally a four‐terminal device comprising 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.
V
2M 2
Z in   R1  j L1  Z R , where Z R 
is reflected impedance.
I1
R2  j L2  Z L
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13.4 Linear Transformer (2)
 V1   j L1
 V    j M
 2 
j M   I1 
j L2  I 2 
An equivalent circuit removes the mutual inductance. The goal is to match the terminal voltages and currents from the original network to the new network.
• Transforming to the T network the inductors are:
• Transforming to the  network the inductors are:
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13.4 Linear Transformer (3)
Example: Calculate the input impedance in Fig. and the current from the voltage source.
Example: Determine the T‐equivalent circuit of the linear transform in Fig. (a).
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13.5 Ideal Transformer (1)
• An ideal transformer is a unity‐coupled, lossless transformer in which the primary and secondary coils have infinite self‐
inductances. (k = 1)
V2 N 2

n
V1 N1
I 2 N1 1


I1 N 2 n
V2 > V1 (n >1) → step‐up transformer
V2 < V1 (n <1) → step‐down transformer
(a) Ideal Transformer
(b) Circuit symbol
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13.5 Ideal Transformer (2)
• Complex power supplied to the primary is delivered to the secondary without loss. The ideal transformer is lossless. • Input impedance is also called reflected impedance. Impedance matching ensures maximum power transfer.
• Equivalent circuit:
1. Secondary one to primary one
2. Primary one to secondary one
13.5 Ideal Transformer (3)
• Equivalent circuit:
1. Secondary one to primary one
To find VTh
I1  0  I 2 so that V2  Vs2
To find ZTh
I1  nI 2 and V1 
V2
n
13.5 Ideal Transformer (4)
Example: An ideal transformer is rated at 2400/120V, 9.6 kVA, and has 50 turns on the secondary side. Calculate: (a) the turns ratio, (b) the number of turns on the primary side, and(c) the current ratings for the primary and secondary windings.
Example: In the ideal transformer circuit, find (a) I1, (b) Vo, and (c) the complex power supplied by the source.
Example: Calculate the power supplied the 10‐Ω resistor in the ideal transformer circuit.
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13.6 Applications (1)
• Transformer as an Isolation Device to isolate ac supply from a rectifier.
• Isolation between the power line and the voltmeter.
• As an Isolation Device to isolate dc between two amplifier stages. 19
13.6 Applications (2)
• As a Matching Device
Example: Using an ideal transformer to match the loudspeaker to
the amplifier to achieve maximum power transfer.
Equivalent circuit
Example: Calculate the turns ratio of an ideal transformer required
to match a 400‐Ω load to a source with internal impedance of
2.5 kΩ. Find the load voltage when the source voltage is 30 V. 20
13.6 Applications (3)
• Power distribution system
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