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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