ee 221-mutually coupled circuits
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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?
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
It is the ability of one inductor to induce a voltage across a neighboring inductor,
Mutual inductance is measured in henrys (H).
4
dt
diMv 1
212 dt
diMv 2
121 dt
diLv 1
11 dt
diLv 2
22
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13.2 Mutual Inductance
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.
5 Illustration of the dot convention.
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13.2 Mutual Inductance
)connection aiding-(series
221 MLLL
)connection opposing-(series
221 MLLL
6
Series connection of two mutually coupled inductors; (a) series-aiding connection, (b) series-opposing connection.
)()( 21dt
diM
dt
diL
dt
diM
dt
diLv )()( 21
dt
diM
dt
diL
dt
diM
dt
diLv
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Problem 1:
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13.2 Mutual Inductance
8
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.
A149.2I A;4.49.13I 21
9
Ans:
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13.3 Energy in a Coupled Circuit (1)
The coupling coefficient, k, is a measure of the magnetic coupling between two coils; 0≤ k ≤1.
21LL
Mk
10
• The instantaneous energy stored in the circuit is given by
• (+) if both currents enter or leave the dotted sides
• (-) if one current enters the dotted side and the other current leaves dotted side.
21
2
22
2
112
1
2
1iMiiLiLw
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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 = 1sec if v = 60cos(4t +30°) V.
11
Ans: k=0.56; w(1)=20.73J
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Problem 20:
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13.4 Linear Transformer – Reflected Impedance
13
Linear Transformer refers to a case where the
inductances L1 and L2 and mutual inductance M are
constant.
1V
2V
ref
L
in
L
ZLjRZRLj
MLjRZ
I
V
MIjILjIZR
MIjILjIRV
11
22
2
11
1
12222
21111
)(
)(
)()(
)(
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13.4 Linear Transformer
A1.1135.0I ;1.5314.100Z 1in
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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:
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13.4 Linear and Ideal Transformer
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Voltage Ratio (ignore resistances)
Current Ratio
In case of unity coupling (i.e., k =1) and very large
inductances,
2
211
2
)()( MZLjLj
MZj
V
V
L
L
LZLj
Mj
I
I
21
2
1
2
1
21
1
2
L
L
ZLj
ZLLj
V
V
L
L
2
1
2
12
1
2
L
L
Lj
LLj
I
I
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13.4 Ideal Transformer
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The inductance L is proportional to the number of turns
squared.
In case of an ideal Transformer,
nN
N
aN
aN
L
L
ZLj
ZLLj
V
V
1
2
2
1
2
2
1
2
21
221
1
2
nN
N
L
L
Lj
LLj
I
I 1
2
1
2
1
2
12
1
2
L=10-3N2r2/(228r+254l)
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13.5 Ideal Transformer
nN
Nn
N
N 1
I
I
V
V
2
1
1
2
1
2
1
2
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An ideal transformer is a unity-coupled, lossless transformer in which the primary and secondary coils have infinite self-inductances.
(a) Ideal Transformer
(b) Circuit symbol
V2>V1 → step-up transformer
V2<V1 → step-down transformer
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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.
18
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
Transformer as an isolation device to isolate ac supply from a rectifier
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13.6 Applications
Transformer as an isolation device to isolate dc between two amplifier stages.
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13.6 Applications
22
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.
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Ans: n = 0.2; VL = 3V
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13.6 Practical Electric Utility transformers
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• Used to step-up or step-down the voltage
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Ideal Auto-Transformer
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)
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3-Phase Transformer