Transcript Ideal Transformers File - Eastern Mediterranean University Open

Chapter 13
Ideal Transformers
Chapter Objectives:
 Understand magnetically coupled circuits.
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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
Huseyin Bilgekul
Eeng224 Circuit Theory II
Department of Electrical and Electronic Engineering
Eastern Mediterranean University
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Ideal Transformers
 A Ideal Transformer is a unity Coupled, lossless transformer in which the primary
and secondary coils have infinite self inductances.
A Transformer is ideal if:
L1 , L2 , M  
1.) Large reactance coils;
2.) Unity Coupling k=1.
3.) Coils are lossless (R1=R2=0)
Ideal transformer
Circuit symbol for the Ideal transformer
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Ideal Transformers
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Non Ideal Transformers

An ideal transformer has no power loss; all power applied to the primary is all
delivered to the load. Actual transformers depart from this ideal model. Some
loss mechanisms are:
Winding resistance: Causing power to be dissipated in the windings.
Hysteresis loss: Due to the continuous reversal of the magnetic field.
Core losses: Due to circulating current in the core (eddy currents).
Flux leakage: Flux from the primary that does not link to the secondary.
Winding capacitance: It has a bypassing effect for the windings.

The ideal transformer does not dissipate power. Power delivered from the source
is passed on to the load by the transformer.

The efficiency of a transformer is the ratio of power delivered to the load (Pout)
to the power delivered to the primary (Pin).
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Input-Output Variables of an Ideal Transformer
 The input and output voltages and currents
of an ideal transformer are related only by the
turns ratio.
V1  j L1 I1  j MI 2
V1  j MI 2
I1 
j L1
V2  j MI1  j L2 I 2
MV1 j M 2 I 2
V2  j L2 I 2 

L1
L1
Perfect Coupling k  1, Thus we have M  L1 L2
L1 L2 V1 j L1 L2 I 2
V2  j L2 I 2 


L1
L1
Substitute
L2
N2
V1  nV1 
V1
L1
N1
V2 N 2

 n  Turns Ratio
V1 N1
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Input-Output Variables of an Ideal Transformer
V2 I1
N2
 
 n
V1 I 2
N1
 A Ideal Transformer is called:
1.) Step-up transformer if n > 1.
2.) Step-down transformer if n < 1.
3.) Isolation transformer if n=1.
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Transformer Dot Convention
Transformer DOT convention is needed to assign the polarity of the output variables.
1.) If V1 and V2 are BOTH + or BOTH – at the dotted terminals use +n, otherwise –n.
2.) If I1 and I2 BOTH ENTER or BOTH LEAVE the dotted terminals use –n, otherwise
+n.
V2 I1
N2
 
 n
V1 I 2
N1
In phase
Out of phase
Dot convention indicating the phase relationship between the input and the output.
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Dot Convention for Ideal Transformers
 Typical circuits illustrating polarity for voltages and direction of currents of an ideal
transformer
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Conservation of the Complex Power
 An ideal transformer absorbs no power.
 The complex power in the primary winding is equal to the complex power
delivered to the secondary winding.
 Transformer absorbs no power. We assume a lossless transformer.
S1
S2
V2
S1  V I 
(nI2 )  V2 I 2  S 2
n

1 1
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Reflected Impedance of Ideal Transformers
 The ability of a transformer to transform a given impedance to another value is
very useful in IMPEDANCE MATCHING.
Zth
b) Obtaining the ZTh.
a) Obtaining the VTh.
V2 Vs 2
VTh  V1 

n
n
Z Th
V2
Z 2I 2
V1
Z2
n
n



 2
I1 nI 2
nI 2
n
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Reflected impedance
 Equivalent circuit of reflection of the secondary to primary side.
Z2
Z R1  2
n
Reflected to Primary
Equivalent circuit of reflection of the primary to secondary side.
Z R 2  n 2 Z1
Reflected to Secondary
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Autotransformers
 An auto transformer is a transformer in which both the primary and secondary
are in a single winding.
 Autotransformers are smaller and lighter than an equivalent two winding
transformer.
 Electrical isolation is lost between the primary and secondary windings.
a) step-down autotransformer
V1 I 2 N1  N 2
N1
 
 1
V2 I1
N2
N2
b) step-up autotransformer
V1 I 2
N2
 
V2 I1 N1  N 2
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ZR
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I3
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Example 13.15
 Determine the voltage across the load.
Apply superposition principle.
DC Source only
AC source only
• Load voltage due to DC is zero (No induction without change in time)
VO  VO-DC  VO-AC  0 
120
cos t  40 cos t
3
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