SJTU1 Chapter 13 Magnetically coupled circuits. SJTU2 Mutual inductance A single inductor:
Chapter 13 Magnetically Coupled Circuits Chapter...
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Chapter 13 Magnetically Coupled Circuits Huseyin Bilgekul Eeng224 Circuit Theory II Department of Electrical and Electronic Engineering Eastern Mediterranean University Chapter Objectives: Understand magnetically coupled circuits. 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 Payam zarbakhsh EElE301 Circuit Theory II Department of Electrical and Electronic Engineering Cyprus International university
Transcript of Chapter 13 Magnetically Coupled Circuits Chapter...
‹#› Eeng 224
Chapter 13 Magnetically Coupled Circuits
Huseyin Bilgekul
Eeng224 Circuit Theory II
Department of Electrical and Electronic Engineering
Eastern Mediterranean University
Chapter Objectives: Understand magnetically coupled circuits.
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
Payam zarbakhsh
EElE301 Circuit Theory II
Department of Electrical and Electronic Engineering
Cyprus International university
‹#› Eeng 224
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
Payam zarbakhsh
EElE301 Circuit Theory II
Department of Electrical and Electronic Engineering
Cyprus International university
‹#› Eeng 224
Mutual Inductance
1 11 211 1 1
( )d dv N N
dt dt
2 12 22
2 2 2
( )d dv N N
dt dt
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.
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b) Mutual inductance M21 of coil 2
with respect to coil 1.
Mutual Inductance
a) Magnetic flux produced by a single
coil.
c) Mutual inductance of M12 of coil 1
with respect to coil 2.
21 12
div M
dt
12 21
div M
dt
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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.
12 21M M M
Payam zarbakhsh
EElE301 Circuit Theory II
Department of Electrical and Electronic Engineering
Cyprus International university
‹#› Eeng 224
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.
12
div M
dt
12
div M
dt
21
div M
dt
21
div M
dt
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Dot Convention
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Coils in Series
a) Series-aiding connection.
L=L1+L2+2M
b) Series-opposing connection.
L=L1+L2-2M
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.
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Time-domain and Frequency-domain Analysis
1 21 1 1 1
2 12 2 2 2
1 1 1 1 2
2 1 2 2 2
TimeDomain
Frequency Domain
( )
( )
di div i R L M
dt dt
di div i R L M
dt dt
V R j L I j MI
V j MI R j L I
V1 V2 I1 I2 jL1 jL2
jM
a) Time-domain circuit b) Frequency-domain circuit
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Induced mutual voltages
Payam zarbakhsh
EElE301 Circuit Theory II
Department of Electrical and Electronic Engineering
Cyprus International university
‹#› Eeng 224
Induced mutual voltages
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+ - +
-
+ -
j3I1
j3I1 j3I2
P.P.13.2 Determine the phasor currents
1 2 2
2 1 1
Mesh 1 12 60 =(5+j2+j6-j3 2)I 6I 3I
Mesh 2 0=(j6-j4)I 6I 3I
j j
j j
Payam zarbakhsh
EElE301 Circuit Theory II
Department of Electrical and Electronic Engineering
Cyprus International university
‹#› Eeng 224
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.
+
+
+
I1 I2
Io
j20Ic
100
500 V
I3
+
+ +
+
j10Ib j40
j30Ic
j80
j10Ia
j20Ia
j60
j30Ib
-j50
Ia
Ib
Ic Ia = I1 – I3
Ib = I2 – I1
Ic = I3 – I2
Io = I3
Blue Voltage due to Ia
Red Voltage due to Ic
Green Voltage due to Ib
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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
2 2
1 1 2 2 1 2
1 1
2 2w L i L i Mi i
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.
Payam zarbakhsh
EElE301 Circuit Theory II
Department of Electrical and Electronic Engineering
Cyprus International university
‹#› Eeng 224
Coupling Coefficient
a) Loosely coupled coil b) Tightly coupled coil
1 2
0 1k
Mk
L L
The Coupling Coefficient k is a measure of the magnetic coupling between two coils
0 1k 1 Perfect Coupling
0.5 Loosly Coupling
0.5 Tightly Coupling
k
k
k
Payam zarbakhsh
EElE301 Circuit Theory II
Department of Electrical and Electronic Engineering
Cyprus International university
‹#› Eeng 224
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
1 1 1 2
1 2 2 2
( )
0 ( )L
V R j L I j MI
j MI R j L Z I
2 2
1 1 1 1
1 2 2
in R
L
V MZ R j L R j L Z
I R j L Z
2 2
2 2
REFLECTED IMPEDANCER
L
MZ
R j L Z
• Secondary impedance seen from the primary side is the Reflected Impedance.
Let us obtain the input impedance as seen from the source,
ZR
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Payam zarbakhsh
EElE301 Circuit Theory II
Department of Electrical and Electronic Engineering
Cyprus International university
‹#› Eeng 224
Equivalent T Circuit for Linear Transformers
The coupled transformer can equivalently be represented by an EQUIVALENT T
circuit using UNCOUPED INDUCTORS.
1 2, ,a b cL L M L L M L M
a) Transformer circuit b) Equivalent T circuit of the transformer
Payam zarbakhsh
EElE301 Circuit Theory II
Department of Electrical and Electronic Engineering
Cyprus International university
‹#› Eeng 224
Equivalent П Circuit for Linear Transformers The coupled transformer can equivalently be represented by an EQUIVALENT П
circuit using uncoupled inductors.
2 2 2
1 2 1 2 1 2
2 1
, ,A B C
L L M L L M L L ML L L
L M L M M
a) Transformer circuit b) Equivalent Π circuit of the transformer
Payam zarbakhsh
EElE301 Circuit Theory II
Department of Electrical and Electronic Engineering
Cyprus International university
‹#› Eeng 224
1
2
a
b
c
L L M
L L M
L M
