Digital Signal Processing Reference
In-Depth Information
I
1
I
2
R
w
R
w
dI
1
+
−
+
−
dI
2
L
1
L
2
dt
dt
+
−
V
1
R
L
dI
2
+
−
+
−
dI
1
L
12
L
21
dt
dt
Figure 2-21
Circuit for Example 2-5, showing magnetically coupled loops.
loops, assuming that the second loop is terminated in a resistor
R
L
and each
wire has a resistance of
R
w
. Kirchhoff's voltage relations can be written for each
loop:
Loop 1:
+
L
1
dI
1
dt
+
L
12
dI
2
v
1
=
R
w
I
1
dt
Loop 2:
+
L
2
dI
2
dt
+
L
21
dI
1
(R
w
+
R
L
)I
2
dt
=
0
Note that inductance is always a positive quantity that can be compared to
a mass in a mechanical system. Large masses are difficult to move, making it
difficult to accelerate in any direction. Similarly, the greater the inductance, the
more difficult it is to change the current because of the back emf (back voltage)
generated in a direction to oppose the current, which is enforced by the negative
sign in (2-98). This is called
Lenz's law
.
2.5.3 Energy in a Magnetic Field
If a circuit has a finite amount of inductance, it takes energy to make a current
flow because it requires work to overcome the back EMF voltage described by
Lenz's law. The work done on a charge against this back EMF is
—v
, from
(2-98). The negative sign dictates that it is work being done against the EMF,
not work done by the EMF. Since current is charge flow per unit time, the work
done per unit time is derived from (2-58) and given by
L
I
dW
dt
=−
vI
=−
dI
dt
W
=−
vq
→
(2-99)
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