IEEE Solid-State Circuits Magazine - Winter 2017 - 8
and to assume that this wall can be
removed instantly, corresponding
to turning the valve on.
As shown in Figure 2, water is initially trapped on the left side of the
wall. When we instantly remove the
wall, the water rushes from left to
right, gaining kinetic energy while
losing some of its potential energy
on the path, then losing the kinetic
energy to the potential energy again
and moving back to the left, then to
right, and so on. In fact, if there is
no resistance in the path of water
motion that turns kinetic energy to
heat, we would expect the energy
to oscillate back and forth between
potential energy and kinetic energy.
Readers can verify experimentally at
home that this behavior is not limited to the ideal situation we have
imagined. In fact, even with a simple
U-tube and a valve, as shown in Figure 3, turning the valve on quickly will
create oscillations in the water level,
albeit damped because we could not
remove friction completely.
Why is there such a difference in
behavior? Why can we not observe
a similar behavior in capacitors? Or
can we? Readers are encouraged to
ponder these questions before considering the answers that follow.
The energy stored in the two
glasses can take the form of either
potential energy (when the water
is still) or kinetic energy (when the
water is moving). The stored energy
changes form as we open the valve
and allow water to move (i.e., store
energy in kinetic form). However,
at any moment in time, part of the
stored energy is in kinetic form, and
part is in potential form. After the
water settles, due to friction, only
potential energy will be left in the
two glasses.
In our capacitor example, however, we h ave on ly consider ed
potential energy, and that is the
energy stored in the capacitor. How
about the kinetic energy? What is
the equivalent of kinetic energy in
our capacitor example? The answer
is the magnetic energy in an inductance that we have totally ignored so
far in our capacitor circuit. A more
8
W i n t e r 2 0 17
t<0
V
0
C
C
t >> 0
0
V/2
2C
Figure 2: (a) An imaginary wall (in red)
separates two glasses. At time zero, the
wall disappears instantly, allowing the
water to move to the right glass without
much resistance. (b) After some time, the
water level settles to V/2.
t≥0
Valve Is
Open
t<0
Valve Is
Closed
Figure 3: Upon opening the valve, water
will oscillate between the two branches of
a frictionless tube.
R
L t=0
+
V
-
+
C
C
0
-
Figure 4: Inductance L is included as part
of the charge-sharing circuit between two
capacitors.
accurate representation of our circuit must include an inductance L
in series with the resistance R, as
shown in Figure 4. Once we con-
IEEE SOLID-STATE CIRCUITS MAGAZINE
sider L, we can see that L stores the
equivalent of kinetic energy (which
is the energy stored in the magnetic
field), and this gives the possibility
of oscillation similar to the case of
the two glasses. Let us now revisit
the two-capacitor problem and see
what happens when R approaches
zero, while we assume a nonzero L
in the circuit.
When the switch is open, all the
energy is stored in the capacitor
on the left. There is zero current in
the circuit; therefore, there is zero
energy stored in the inductor. As
we turn on the switch, there will be
a current in the circuit. Note that,
initially, the inductor impedes the
rise of the current and will force the
current waveform to be continuous.
However, as the current begins to
increase, there will be more energy
stored in the inductor, and some
energy is wasted in the resistor. If
the resistance is small enough, corresponding to an under-damped
behavior (R 1 2 2L/C ), the stored
energy will oscillate back and forth
between the capacitor and the inductor until the current goes to zero,
storing the remaining energy in the
capacitor. Again, during this process,
half the initial stored energy is lost
to heat in the resistor, no matter how
small the resistor is or how fast the
oscillations settle. In case the resistor
is large enough (R $ 2 2L/C ), there
will be no oscillation, corresponding
to either an over-damped or a critically damped behavior. In this case,
a portion of the energy does move to
the inductor, but it will be wasted in
the process in the resistor, never to
return to the capacitor. In this case,
too, half the initial energy is wasted
in the resistor
Finally, we reconsider the question of what happens when R is
exactly zero, while we have a nonzero inductance in the circuit. In
this case, the energy stored initially
in the capacitor will swing back and
forth between the capacitor and the
inductor and, because it will have
no place to be consumed, will result
(continued on p. 51)
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