2012/11/29
Magnetically Coupled Circuits
•Introduction
•Mutual Inductance
•Energy in a Coupled Circuit
•Linear Transformers
•Ideal Transformers
•Applications
Introduction
•Conductively coupled circuit means that one loop
affects the neighboring loop through current
conduction.
•Magnetically coupled circuit means that two loops,
with or without contacts between them, affect each
other through the magnetic field generated by one of
them.
•Based on the concept of magnetic coupling, the
transformer is designed for stepping up or down ac
voltages or currents.
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Magnetic Flux
Φ
B
dA

S


BdA
S
where
Φ is the magnetic flux
B is the magnetic field
S is the surface area
denotes dot product
dA is the infinitesimal vector
Self Inductance
An inductor :
inductance L


 N turns
For each turn, the induced volatge is
d
v1T 
(Faradays' s Law)
dt
For N turns, the induced volatge is
d
ddi
di
v N
N
L
dt
di dt
dt
d
 L N
(self - inductance)
di
+
+
v_1T
+
v
v_1T
+
_
v_1T
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Mutual Inductance (1/5)
self - inductances L1

Coil 1 : 
N1 turns

self - inductances L2

Coil 2 : 
N 2 turns

Assuming no current in coil 2,
the flux generated by coil 1 is



1 
11 (only coil 1) 
12 (both coils)
 v1 N1
d
d di
di
1
N1 1 1 L1 1
dt
di1 dt
dt
where L1 N1
 v2 N 2
d
1
di1
d
d di
di
12
N 2 12 1 M 21 1
dt
di1 dt
dt
The mutual - inductance of
coil 2 with respect to coil 1 is
d
M 21 N 2 12
di1
The open - circuit mutual voltage is
v2 M 21
di1
dt
Mutual Inductance (2/5)
self - inductances L1

Coil 1 : 
N1 turns

self - inductances L2

Coil 2 : 
N 2 turns

Assuming no current in coil 1,
the flux generated by coil 2 is
2 22 (only coil 2) 21 (both coils)
 v2 N 2
d2
d di
di
N 2 2 2 L2 2
dt
di2 dt
dt
where L2 N 2
d2
di2
d
d di
di
 v1 N1 21 N1 21 2 M 12 2
dt
di2 dt
dt
The mutual - inductance of
coil 1with respect to coil 2 is
d
M 12 N1 21 (M 21 )
di2
The open - circuit mutual voltage is
v1 M 12
di2
dt
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Mutual Inductance (3/5)
•We will see that M12 = M21 = M.
•Mutual coupling only exists when the inductors or
coils are in close proximity, and the circuits are driven
by time-varying sources.
•Mutual inductance is the ability of one inductor to
induce a voltage across a neighboring inductor,
measured in henrys (H).
•The dot convention states that a
current entering the dotted terminal
induces a positive polarity of the
mutual voltage at the dotted terminal
of the second coil.
i1
+
di
v2 M 1
dt
_
Mutual Inductance (4/5)
i1 induces 
11 and 
12 ,
d
1
i2 induces 21 and 22 . v1 N1 dt N1


21
1 (
11 
12 ) 
2 
22 )
12 (
21 
v2 N 2
d (

d
di
di
11 
12 )
N1 21 L1 1 M 12 2
dt
dt
dt
dt
d2
d (21 22 )
d
di
di
N 2
N 2 12 L2 2 M 21 1
dt
dt
dt
dt
dt
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Mutual Inductance (5/5)
i1
i1
+
+
di
v2 M 1
dt
v2 M
_
_
i2
i2
+
v1 M
di2
dt
di1
dt
+
v1 M
_
di2
dt
_
Series-Aiding Connection
11
12
21
+ v1 _
di
di
M 12
dt
dt
di
di
v2 L2 M 21
dt
dt
v v1 v2
+ v2 _
22
v1 L1
di
di
di
di
M 12 L2 M 21
dt
dt
dt
dt
di

L1 L2 M 12 M 21 
dt
L1
But M 12 M 21 M ,
di
 v 
L1 L2 2 M 
dt
 Leq L1 L2 2 M
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Series-Opposing Connection
11
12
21
+ v1 _
di
di
M 12
dt
dt
di
di
v2 L2 M 21
dt
dt
v v1 v2
+ v2 _
22
v1 L1
di
di
di
di
M 12 L2 M 21
dt
dt
dt
dt
di

L1 L2 M 12 M 21 
dt
L1
But M 12 M 21 M ,
di
 v 
L1 L2 2M 
dt
 Leq L1 L2 2M
Circuit Model for Coupled Inductors
di1
di
M 2
dt
dt
di
di
v2 M 1 L2 2
dt
dt
v1 L1
V1 jL1I1 jMI 2
V2 jMI1 jL2I 2
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Example 1
_
+
_
+
_
+
Applying KVL to mesh 1 gives
Substituting (1b) into (1a) gives
12 (j 4 j 5)I1 j 3I 2 0 (1a)
12
 I2 
2.9114.04
4 j
Applying KVL to mesh 2 gives
 I1 (2 j 4)I 2 13.0149.39
( j 6 12)I 2 j 3I1 0
12 j 6
I1 
I 2 (2 j 4)I 2
j3
(1b)
Example 2
+
+
_
_
+
From (1a) and (1b) we get
Applying KVL to mesh 1 gives
100 (4 j 3)I1 j 6(I1 I 2 ) j 2I 2 0
 (4 j3)I1 j8I 2 100
(1a)
Applying KVL to mesh 2 gives
j 6(I 2 I1 ) j 2I 2 j8I 2 5I 2 j 2(I 2 I1 ) 0
 j8I1 (5 j18)I 2 0
-
(1b)
I1  
4 j3 j8 
100


 


j8 5 j18
I 2  0 



 I1 20.33.5
 I 2 8.69319
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Energy in a Coupled Circuit (1/4)
To find the stored energy as i1 I1 and i2 I 2 :
Step I : i2 0, i1 increases from 0 to I1.
p1 (t ) i1v1 i1 L1
di1
dt
I1
1
 w1 
p1dt L1 i1di1  L1 I12
0
2
Step II : i1 I1 , i2 increases from 0 to I 2 .
p2 (t ) i1v1 i2v2 I1M 12
di2
di
i2 L2 2
dt
dt
I2
I2
0
0
 w2 
p2 dt M 12 I1 di2 L2 i2 di2
I
i2
II
i1
1
M 12 I1 I 2  L2 I 22
2
1
1
w w1 w2  L1 I12  L2 I 22 M 12 I1 I 2
2
2
I2
I1
t
Energy in a Coupled Circuit (2/4)
The analysis process can be changed as
Step I : i1 0, i2 increases from 0 to I 2 .
1
 w1  L2 I 22
2
Step II : i2 I 2 , i1 increases from 0 to I1.
1
 w2 M 21 I1I 2  L1I12
2
1
1
w w1 w2  L1I12  L2 I 22 M 21I1 I 2
2
2
But the total energy must equal to the
former case.
 M 12 M 21 M
I
i2
II
i1
I2
I1
t
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Energy in a Coupled Circuit (3/4)
1
1
w  L1i12  L2i22 Mi1i2
2
2
1
1
w  L1i12  L2i22 Mi1i2
2
2
Energy in a Coupled Circuit (4/4)
For any current assignments, the
instantaneous energy stored is given as
1
1
w  L1i12  L2i22 Mi1i2 0
2
2
i1
w 1
1
Let x  , f ( x)  2  L1 x 2  L2 Mx 0
i2
i2 2
2
To find the minimum f ( x),
df ( x)
M
L1 x M 0  xmin 
dx
L1
1 M2 1
M2
 f ( xmin )  L1 2  L2 
2
L1 2
L1
1 M2
 
L2  0
2
L1 
Alternative proof :

d (

11 
12 )
L1 N1
di
1


d (21 22 )
L2 N 2
di2


d
d
M 12 N1 21 M 21 N 2 12
di
di1

2

M 12 M 21
d
d21
12

L1L2
d (

22 )
11 
12 ) d (
21 

21
 12
1




22
11
12 21 
 M 2 L1L2
 M 2 L1 L2
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Coupling Coefficient
The coupling coefficient k is
defined as
k
M
L1 L2
(0 k 1)
or M k L1 L2


k  12  21


21 22
11 
12
k 1 means perfect coupling.

22 0
11 
Coupling vs. Winding Style
Loosely coupled
k < 0.5
Tightly coupled
k > 0.5
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Example
Q : Find k and the energy stored in the
coupled inductors at t 1 s.
Sol : k 
M
2.5

0.56
L1 L2
20
For mesh 1,
(10 j 20)I1 j10I 2 6030 (1a)
For mesh 2,
j10I1 ( j16 j 4)I 2 0
v 60 cos(4t 30
)V
4 rad/s
(1b)
I 3.90519.4

 1
I 2 3.254160.6
)
i1 3.905 cos(4t 19.4

i2 3.254 cos(4t 160.6
)

 i1 (1) 3.389, i2 (1) 2.824
1
1
w  L1i12  L2i22 Mi1i2 20.73 J
2
2
Linear Transformers
R1 and R2
are winding
resistances.
Zin
Applying mesh analysis gives
V ( R1 jL1 )I1 jMI 2 (1)
jMI1 ( jL2 R2 Z L )I 2 0 (2)
V
But Z in 
I1
2 M 2
 Z in R1 jL1 
R2 jL2 Z L

ZP

ZR
Z : primary impedance
where  P
Z R : reflected impedance
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T (or Y) Equivalent Circuit
V1  jL1



V2 

 jM
jL1
jM

V1  j( La Lc )



V2 

  jLc
jM 
I1 



jL2 
I2 


jM  j( La Lc )

jL2 
  jLc
jLc 
I1 



j( Lb Lc )
I2 


La L1 M
 
 Lb L2 M
j( Lb Lc )
 L M
c
jLc
П(or ) Equivalent Circuit
M 
L2
I1  jK jK 
V1 







L1 
I 2  M
V2 




jK jK 

where K L1 L2 M 2
 L2
jK

M



j

 K
1
1


I
1  jLA jLC


I2 

   1

jLC

1
M  1


jK  jLA jLC

L1  
1


jK 
j

LC
 
1



V1 

jLC


1
1 
V2 



jLB jLC 

K

L 
1
  A L2 M

 
K
jLC

 L 
1
1   B L1 M


jLB jLC 
 L K
C

M

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Ideal Transformers (1/3)

2 
1 
v1 N1
d
dt
v2 N 2
d
dt
1. Coils have very large reactance. (L1, L2, M ~ )
2. Coupling coefficient is equal to unity. (k = 1)
3. Primary and secondary are lossless.
(series resistances R1= R2= 0)
Ideal Transformers (2/3)
V1 jL1I1 jMI 2

V2 jMI1 jL2 I 2
(1a)
(1b)
From (1a),
I1 
V1 jMI 2 jL1 (1c)
Substituting 1(c) into (1b) gives
 M2 
M
V2  V1 
L2  L 
jI 2
L1
1 

For perfect coupling,
k 1 or M  L1 L2
 V2 
L1 L2
L
V1  2 V1 nV1
L1
L1
where n is called the turns ratio.
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Ideal Transformers (3/3)
d
d
, v2 N 2
dt
dt
v
N
V2 N 2
 2  2 n or
 n
v1 N1
V1 N1
v1 N1
The transformer is lossless  v1i1 v2i2
i2 v1 1
I
V 1
   2  1 
i1 v2 n
I1 V2 n
More Comments (1/2)
d
d
, v2 N 2
dt
dt
L1 , L2    N1 , N 2  
v1 N1
For finite voltages, we have
d v1
 0
dt N1
 The net stored energy approaches zero.
Absorbed power Supplied power
 v1i1 v2i2 (Lossless)
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More Comments (2/2)
The stored energy must be finite.
1
1
w  L1i12  L2i22  L1L2 i1i2
2
2
1 2 L2 2
L2 
 L1 
i

i

2
i1i2 
1
2

2 
L
L
1
1


2
1 
L 
 L1 
i1  2 i2 

2 
L1 

 i1 

L2
2w
i2 
0 (L1  )
L1
L1
i2
L
1
 1 
i1
L2 n
Types of Transformers
•When n = 1, we generally call the transformer an
isolation transformer.
•If n > 1 , we have a step-up transformer (V2 > V1).
•If n < 1 , we have a step-down transformer (V2 < V1).
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Impedance Transformation
V2 N 2
 n
V


V1 N1

V  2
 1 n
I
N
1
2  1 

I1 nI 2

I1 N 2 n
The complex power in the primary is
V
*
S1 V1I1*  2 
nI 2 V2 I *2 S 2
n
The complex power supplied to the
primary is delivered to the secondary
without loss.
 The transformer is lossless!
Zin
The input impedance as seen
by the source is
V V n 1 V
Z in  1  2  2 2
I1
nI 2
n I2
Z
 Z in  2L (reflected impedance)
n
Useful for impedance matching!
How to make a transformer ideal?
Zin
2 M 2
Zin R1 jL1 
R2 jL2 Z L
If R1 R2 0 and M  L1 L2
2 M 2
Zin jL1 
jL2 Z L

jL1Z L 2 L1 L2 2 L1 L2
Z L jL2

jL1Z L
jL2 Z L
L2  
 L2 Z L for 0
 Z in 
jL1Z L L1Z L Z L

 2
jL2
L2
n
where n 
L2
: the turns ratio
L1
L
For a desired n , L1  22  
n
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Impedance Matching
Linear network
The condition for maximum
power transfer is :
Z L
*
n 2 Z Th : complex Z L
R
 2L Z Th : Z L RL j 0
n
Ideal Transformer Circuit (1/3)
Linear network 1
Linear network 2
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Ideal Transformer Circuit (2/3)
1
I1 nI 2


V
V1  2

n

V V n 1 V
 Z Th  1  2  2 2
I1
nI 2
n I2
I1 0 I 2
V
 VTh V1  2
n
V
 s2
n
Z
 22
n
Ideal Transformer Circuit (3/3)
c
Equivalent 1:
c
c
Equivalent 2:
c
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Applications of Transformers
•To step up or step down voltage and current (useful
for power transmission and distribution).
•To isolate one portion of a circuit from another.
•As an impedance matching device for maximum
power transfer.
• Frequency-selective circuits.
Applications: Circuit Isolation
When the relationship between
the two networks is unknown,
any improper direct connection
may lead to circuit failure.
This connection style can
prevent circuit failure.
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Applications: DC Isolation
Only ac signal can pass, dc signal is blocked.
Applications: Load Matching
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Applications: Power Distribution
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