EE 221
Circuits II
Chapter 13
Magnetically Coupled Circuits
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Magnetically Coupled Circuits
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?
y
y
y
It is 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 element for stepping up or
stepping down ac voltages and currents,
impedance matching, isolation, etc….
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13.2 Mutual Inductance
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It is the ability of one inductor to induce a voltage across
a neighboring inductor,
Mutual inductance is measured in henrys (H).
di
v1 = L1 1
dt
di1
v2 = M 21
dt
V2: Open-circuit voltage
across coil 2
di
v2 = L2 2
dt
v1 = M 12
V1: Open-circuit voltage
across coil 1
4
di2
dt
13.2 Mutual Inductance
y
If a current enters the dotted terminal of the first
coil, the reference polarity of the voltage in the
second coil is positive at the dotted terminal.
Illustration of the dot convention.
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13.2 Mutual Inductance
Series connection of two mutually coupled inductors;
(a) series-aiding connection, (b) series-opposing
connection.
L = L1 + L2 + 2 M
(series - aiding connection)
L = L1 + L2 − 2 M
(series - opposing connection)
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Problem 1:
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13.2 Mutual Inductance
Time-domain
analysis of a circuit
containing coupled
coils.
Frequency-domain
analysis of a circuit
containing coupled
coils
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13.2 Mutual Inductance
Example 1
Calculate the phasor currents I1 and I2 in the circuit shown
below.
Ans: I1 = 13.01∠ − 49.39°A; I 2 = 2.91∠14.04°A
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13.3 Energy in a Coupled Circuit (1)
y
The coupling coefficient, k, is a measure of the magnetic
coupling between two coils; 0≤k≤1.
k=
•
M
L1 L2
The instantaneous energy stored in the circuit is given by
1 2 1 2
w = L1i1 + L2i2 ± Mi1i2
2
2
• (+) if both currents enter or leave the dotted sides
• (-) if one current enters the dotted side and the other
current leaves dotted side.
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13.3 Energy in a Coupled Circuit
Example 2
Consider the circuit below. Determine the coupling
coefficient. Calculate the energy stored in the coupled
inductors at time t = 1s if v=60cos(4t +30°) V.
Ans: k=0.56; w(1)=20.73J
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Problem 20:
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13.4 Linear Transformer
y
+
+
V1
V2
−
−
Let V1 and V2 be the voltages across the transformer
terminals:
V 1 = j ω L 1 I 1 − j ω MI
2
( Z 2 + j ω L 2 ) I 2 − j ω MI
1
= 0
where Z 2 = R 2 + Z L
(ω M ) 2
V1
⇒
= Z 1 = j ω L1 +
= j ω L 1 + Z reflected
( jω L 2 + Z 2 )
I1
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13.4 Linear Transformer
Example 3
In the circuit below, calculate the input impedance and current
I1. Take Z1=60-j100Ω, Z2=30+j40Ω, and ZL=80+j60Ω.
Ans:
Z in = 100.14∠ − 53.1°Ω; I1 = 0.5∠113.1°A
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13.4 Linear Transformer
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Voltage Ratio
V2
jωMZ 2
=
V1 jωL1 ( jωL2 + Z 2 ) + (ωM ) 2
y
Current Ratio
I2
j ωM
=
I1 jωL2 + Z 2
y
In case of unity coupling (i.e., k =1) and very large
inductances,
V2 jω L1 L2 Z 2
=
=
V1
jωL1Z 2
I 2 jω L2 L1
=
=
I1
jωL2
L2
L1
L1
L2
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13.4 Linear and Ideal Transformer
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The inductance L is proportional to the number of turns
squared.
L=10-3N2r2/(228r+254l)
y
In case of an ideal Transformer,
V2 jω L1 L2 Z 2
=
=
V1
jωL1Z 2
I 2 jω L2 L1
=
=
I1
jωL2
L2
aN 22 N 2
=
=
=n
2
L1
aN1
N1
L1 N1 1
=
=
L2 N 2 n
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13.5 Ideal Transformer
y
An ideal transformer is a unity-coupled, lossless transformer
in which the primary and secondary coils have infinite selfinductances.
V2 N 2
=
=n
V1 N1
I 2 N1 1
=
=
I1 N 2 n
V2>V1 → step-up transformer
V2<V1 → step-down transformer
(a)
(b)
Ideal Transformer
Circuit symbol
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13.5 Ideal Transformer
Example 4
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.
Ans:
(a) This is a step-down transformer, n=0.05
(b) N1 = 1000 turns
(c) I1 = 4A and I2 = 80A
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Problem 36
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13.6 Applications
y
Transformer as an isolation device to isolate ac supply
from a rectifier
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13.6 Applications
y
Transformer as an isolation device to isolate dc
between two amplifier stages.
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13.6 Applications
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Transformer as a matching device
Using an ideal transformer to match
the speaker to the amplifier
Equivalent circuit
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13.6 Applications
Example 5
Calculate the turns ratio of an ideal transformer
required to match a 100Ω load to a source with
internal impedance of 2.5kΩ. Find the load
voltage when the source voltage is 30V.
Ans: n = 0.2; VL = 3V
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13.6 Practical Electric Utility transformers
• Used to step-up or step-down the voltage
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Ideal Auto-Transformer
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Auto-transformers are used in cases where the voltage
ratio is less than 2.
Note that there is only one winding, the primary and
secondary side share part of this winding.
There is no electrical isolation between the primary and
secondary sides.
The apparent power rating of an auto-transformer is often
much higher than a two-winding transformer of the same
size (see example 13.10)
3-Phase Transformer