Digital Signal Processing Reference
In-Depth Information
v
1
+
−
i
1
L
0
L
M
i
2
L
0
+
−
v
2
Figure 4-2
Coupled inductor circuit.
Looking next at the opposing current flow case, with the assumption that the
transition time of current
i
1
is equal to that of
i
2
, we have
di
1
/dt
=−
di
2
/dt
=
di/dt
, and derive
di
dt
=−
v
2
=
(L
0
−
L
M
)
v
1
(4-6)
When the two inputs are equal, we say that the system is being driven in
even mode
; in the case of opposite polarity inputs, we say that the system is
driven in
odd mode
. Notice that the
effective inductance
of the system as seen
by our input signals is changed by the mutual inductance and is a function
of the switching pattern. In particular, the even-mode inductance is increased
relative to the self-inductance by an amount equal to the mutual inductance. Cor-
respondingly, the odd-mode inductance is decreased by the mutual inductance,
giving us a relative comparison of self-, even-mode, and odd-mode inductances,
L
even
>L
0
>L
odd
.
.
As a first step toward developing a general expression for the inductance, we
write the equation for the voltage across the inductive elements as a function of
inductances and input currents in matrix form:
v
1
v
2
L
0
di
1
/dt
di
2
/dt
L
M
=
(4-7)
L
M
L
0
If we add a third line to our system, as shown in Figure 4-3, we can extend
equation (4-7):
v
1
v
2
v
3
L
11
L
12
L
13
di
1
/dt
di
2
/dt
di
3
/dt
=
L
21
L
22
L
23
(4-8)
L
31
L
32
L
33
In equation (4-8), the diagonal elements
L
11
,
L
22
, and
L
33
represent the
self-inductances on lines 1, 2, and 3, respectively. The mutual inductances are
represented by
L
ij
, where
i
and
j
correspond to the lines coupled by the mutual
inductance. The inductance matrix is symmetric. In other words, the mutual
inductances between lines do not depend on the direction, so that
L
ij
=
L
ji
.
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