Summary
14.27
Let us denote the instant of switching as
t
=
0. Then, the
flux linkages in the windings
just before switching are
y
y
p
p p
s
s
s s
p
L I
MI
L I
MI
( )
( )
0
0
−
−
=
−
= −
+
and the flux linkages at
t
=
0
+
are
Fig. 14.8-1
Breaking the
primary current
in a transformer
L
p
L
s
r
p
r
s
k =
1
–
+
S
v
o
(
t
)
R
I
p
I
s
y
y
p
p
s
s
s
s s
s s
L
Mi
Mi
L i
M
L i
( )
( )
( )
( )
( )
( )
0
0
0
0
0
0
0
0
+
+
+
+
+
+
=
−
= −
= −
+
= −
Since there is no impulse is the closed secondary loop, the flux linkage in the secondary winding
will have to be continuous. Therefore,
y
y
s
s
s s
p
s s
L I
MI
L i
( )
( )
( )
0
0
0
+
−
+
=
⇒ −
+
= −
Therefore,
i
I
M
L
I
I
I
n
s
s
s
p
s
p
( )
0
+
= −
= −
This current is usually in a
direction opposite to that of i
s
( )
0
-
.
The corresponding flux linkage in primary winding at
t
=
0
+
is
y
p
s
s
s
p
Mi
MI
M
L
I
( )
( )
0
0
2
+
+
= −
= −
+
.
The initial flux linkage in this coil was
y
p
p p
s
L I
MI
( )
0
−
=
−
. Therefore, the instantaneous decrease in flux
linkage that took place in this coil is (
)
(
)
L I
MI
MI
M
L
I
I
L
M
L
k L I
p p
s
s
s
p
p
p
s
p p
−
−
+
=
−
= −
2
2
2
1
.
Hence, an impulse voltage of area content equal to (1
-
k
2
)
L
p
I
p
V-s will appear across the primary
winding and will have to be supported by the switch. In fact, there will be arcing across the switch due
to this very large voltage trying to establish across it.
However, arcing involves energy. Where does the energy come from? It is possible to show that
the energy that gets dissipated across the switch is 0 5 1
2
2
. (
)
-
k L I
p p
J. This comes from the magnetic
energy storage in the coupled-coil system.
The sudden change of secondary current from
I
s
to
I
I
n
s
p
-
results
in a sudden change across
the load voltage from
R I
s
to
RI
RI
n
s
p
-
. There are practical applications in which the second term
dominates and makes the load voltage a negative voltage with enough magnitude to destroy the load
if the load happens to be a sensitive electronic equipment.
This large negative voltage decays exponentially with a time constant of
L
s
/
R s. By the time the
transient is over, the remaining initial magnetic energy stored in the coil system would have got
dissipated in
R.
14.9
Summary
• Self-inductance of coil is the flux linkage in it when 1A flows in it with all other coils in the
vicinity kept open. Mutual inductance between two coils is the flux linkage in one coil when it is
kept open with 1A flowing in the other coil.
14.28
Magnetically Coupled Circuits
• For two coupled coils with self-inductance
L
1
and
L
2
, the mutual inductances
M
12
and
M
21
are equal.
The maximum value mutual inductance can have is
L L
1 2
H. The ratio between actual value of
mutual inductance and its maximum possible value is defined as the
magnetic coupling coefficient.
• A system of two coils with constant values of
L
1
,
L
2
and
M, with two pairs of terminals identified for
application of excitation and/or measurement of response, is called a
two-winding linear transformer.
• Secondary voltage magnitude in a transformer with unity coupling
coefficient is turns ratio
times the primary voltage magnitude. A transformer with unity coupling coefficient
reflects the
secondary load impedance
Z
L
onto the primary side as a
turns-ratio transformed impedance of
value
Z
L
/
n
2
where
n
=
secondary turns/primary turns.
• An
ideal transformer is a two-winding transformer with perfect coupling (
k
=
1) and infinite self- and
mutual inductance values (
L
1
→
∞
,
L
2
→
∞
,
M
→
∞
). The input impedance of such a transformer is
Z
L
/
n
2
. Complex power is conserved in an ideal transformer. Practical transformers that approach ideal
transformer are employed in voltage/current level conversion and impedance matching applications.
• The input impedance function of a
passively terminated two-winding transformer is independent
of relative polarity of windings.
• The secondary voltage of a transformer with unity coupling coefficient (
k
=
1) and zero winding
resistance is turns ratio times primary voltage, quite independent of the waveshape of input.
•
A transformer with k
=
1 reflects an impedance
Z(
s) connected in the secondary side to the primary
side as
Z(
s)/
n
2
where
n is the ratio of secondary turns to primary turns, quite independent of the
waveshape of input.
• A practical two-winding transformer with finite inductance values, imperfect coupling (
k < 1) and
non-zero winding resistance will be a band-pass system with a lower cut-off frequency decided by
winding resistance and upper cut-off frequency decided by coupling coefficient.
• The flux linkage in a shorted coil with zero resistance remains constant in time. If the shorted coil
has non-zero resistance, DC and low-frequency flux will penetrate into the coil. However, high-
frequency flux will be expelled from the coil.
• There can be instantaneous change in coil currents in a coupled-coil system even in a circuit that
does not apply or support impulse voltage if the coils are perfectly coupled. However, there cannot
be an instantaneous change in flux linkage of any coil, whether coupled or uncoupled, unless the
circuit applies or supports impulse voltage.
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