Circuits II
EE221
Unit 10
Instructor: Kevin D. Donohue
Magnetically Coupled Circuits, Linear
Transformers, Transformer Circuits
Coupling Coefficient
The coupling coefficient is a measure of the
percentage of flux from one coil that links
another (mutual inductance) coil. The coupling
coefficient for 2 mutual inductors is given by:
k
M
L2 L1
If k > 0.5 , then most of the flux from the one
coil links the other and the coils are said to be
tightly coupled. If k < 0.5, then most of the flux
is not shared between the 2 coils and in this case
the coils are said to be loosely coupled.
Linear Transformers
A transformer is a 4-terminal device comprised of 2 or more
magnetically coupled coils.
In a typical application they are used the change the ratio of
current to voltage (maintaining constant power) from a source to a
load.
R1
+
Vs
R2
ZL
-
Primary Coil Secondary Coil
Linear Transformers Equivalent (T Network)
A z parameter model of a T (or Y) network of inductors is given by:
La
i1
+
Lb
Lc
v1
i2
+
v2
-
 v1   z11
v    z
 2   21
Vˆ1   j ( La  Lc )
ˆ 
jLc
V2  
z12   i1 
z22  i2 
jLc   Iˆ1 
 
j ( Lb  Lc )  Iˆ2 
Find the equivalent T network for a transform circuit given by a
mutual inductance pair. Show equivalent T network requires:
M
I1
+
V1
-
+
L1
L2
V2
-
I2
Lc  M
La  L1  M
Lb  L2  M
Linear Transformers Equivalent (T Network)
By setting the corresponding y-parameter model elements equal, the
transformer can be mapped to the following equivalent T network.
L1-M
i1
+
v1
M
-
i2
L2-M
+
v2
-
y-parameter equivalent model for transformer is given by:
L2

 Iˆ1   j ( L1 L2  M 2 )
ˆ   
M
I 2  
 j ( L1 L2  M 2 )
M

j ( L1 L2  M 2 )  Vˆ1 
 
L1
 Vˆ2 
j ( L1 L2  M 2 ) 
A similar equivalence can be obtained for the  or  network.
Ideal Transformers
For ideal transformers assume that coupling coefficient is 1. If this is
the case, then both coils have the same flux. Assume the polarities of
the voltages and currents result in flux going in the same direction.
Then the result is:
v1  N1
d
dt
and
v2  N 2
d
dt
Vˆ2 N 2

n
Vˆ1 N1
And for the ideal transformer assume no power is lost in the
magnetic circuit (infinite permeability) or the coils (no wire
resistance), which implies:
Vˆ1 Iˆ1*  Vˆ2 Iˆ2*
Iˆ1 Vˆ2 N 2
 
n
ˆI
ˆ
V1 N1
2
Proper Polarity for Ideal Transformers
Example
Find Vˆo with Z1 = 10, Z2 = -j50, and ZL=20
Z1
Z2
1:4
+
+
1200
-
ZL
Vo
-
Circuit Transformations
The primary circuit can be transformed into the secondary by
performing a Thévenin equivalent at the terminals of the transformer
secondary.
Z1
Z2
1:n
+
+
Vs2
Vs1
-
-
Primary Coil
Secondary Coil
Circuit Transformations
The result is:
n2Z1
+
nVs1
-
Z2
+
Vs2
-
Example
Find Vˆo with Z1 = 10, Z2 = -j50, and ZL=20
Z1
Z2
1:4
+
+
1200
-
ZL
Vo
-
Circuit Transformations
The result is:
Z1
+
Vs1
-
Z2 / n2
+
Vs2 / n
-
Circuit Transformations
Perform a Norton transformation on the above circuits to show how a
current source transforms to an equivalent on the primary or secondary
circuits.
In summary:
Reflecting the secondary circuit to the primary side requires impedance
division by n2, voltage source division by n, and current source
multiplication by n.
Reflecting the primary circuit to the secondary side requires impedance
multiplication by n2, voltage source multiplication by n, and current
source division by n.
Circuit Transformations
Use the formulae derived from Thévenin and Norton to compute the
equivalent impedance.
8
Zeq
1:5
24
4:1
6
j2
Show that Zeq = 12.8 + j5.6
-j10