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Chapter 13
Magnetically
Coupled Circuits
1
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The magnetic flux created in one inductive coil
can couple into a nearby coil, a process called
mutual inductance.
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2
di1 (t)
v2 (t)  M 21
dt
di2 (t)
v1 (t)  M 12
dt
The double headed arrow indicates that these inductors are coupled.
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3
A current entering the dotted terminal of one
coil produces an open circuit voltage with a
positive voltage reference at the dotted
terminal of the second coil.
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4
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5
di1
di2
v1  L1
M
dt
dt
di2
di1
v2  L2
M
dt
dt
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6
di1
di2
v1  L1
M
dt
dt
di2
di1
v2  L2
M
dt
dt
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7


The assumed currents
i1 and i2 produce
additive fluxes.
Dots may be placed
either on the upper
terminal of each coil
or on the lower
terminal of each coil.
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8
Show that V2/V1 =6.88e -j16.70◦
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9
1
1
2
w(t)  L1[i1 (t)]  L2 [i2 (t)]2  M i1 (t)i2 (t)
2
2
This equation implies a limit on M:
M  L1 L2
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10
The coupling coefficient k measures how tightly
coupled the two inductors are:
k
M
L1 L2
where 0 ≤ k ≤ 1

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11
Transformers have a
primary (source side)
and a secondary (load
side).
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12
The impedance Zin seen by the source is
( j )2 M 2
Z in  Z11 
Z 22
Z11  R1  jL1
Z 22  R2  jL2  Z L
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13
Consider the mesh-current
equations to show that these
circuits are equivalent.
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14
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15
In an ideal transformer, k=1 and the
inductances are assumed large in comparison
to the other impedances.
2
L
N
The turns ratio a is defined as a 2  2  22
L1 N1
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16
Step Up
Step Down
Electronics
Power
Step Down to
Household
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17
Impedance matching:
Current adjustment:

Voltage adjustment:

ZL
Z in  2
a
I2 1

I1 a
V2
a
V1
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18
Determine the average power dissipated in the
10 kΩ resistor.
Answer: 6.25 W
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19
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