IEEE Solid-State Circuits Magazine - Fall 2017 - 11

t=0

θ = ωt
+

θ
N

A

C

B

FIGURE 3: An LC circuit with no resistive
loss and an initial voltage V0 across the
resistor is known to produce a sinusoidal
voltage across the capacitor.

FIGURE 2: A coil rotating at a constant speed in the presence of a magnetic field produces a
sinusoidal voltage at its terminals.

v C = L di L
dt
i L =-C

dv C
.
dt

(1)
(2)

We can combine the two equations to find a second-order differential equation for the voltage across
the capacitor:
LC

d2 vC
+ v C = 0.
dt 2

(3)

This equation is known to have a
sinusoid as its solution. In fact, it
turns out that the voltage across the
capacitor and the current through
the inductor will be of the forms:
v C = V0 cos ^~t h
i L = I 0 sin ^~t h,

(4)
(5)

where ~ = 1/ LC and I 0 = C/L V0 .
Clearly, the voltage across the capacitor and the current through the
inductors are sinusoids, as found by
solving the differential equation, but
where is the circle in this solution?
If we multiply the opposite sides
of (1) and (2), we will have
Cv C

dv C
+ Li L di L = 0.
dt
dt

(6)

In other words:
1 d ^Cv 2 + Li 2h = 0,
C
L
2 dt

(7)

L

-

S

R

opposite direction. This description
is consistent with the basic equations that govern the capacitor and
the inductor behavior:

iL (t )

vc (t )

iL(t )

or simply

ωt

1/2Cv C2 + 1/2Li L2 = constant.

vC (t )

(8)

This equation tells us that the
sum of the stored energy in the
capacitor and the inductor is constant, i.e., it does not change with
time. This energy is equal therefore to the initial energy we stored
in the capacitor (1/2CV02) . In addition, (8) tells us that the voltage
across the capacitor and the current
of the inductor form an ellipse, or
a circle w ith proper scaling factors, as shown in Figure  4(a) and
(b), respectively. In other words, the
voltage across the capacitor and the
current in the inductor (with proper
scaling factors) will always lie on a
circle, such that the energy in the
system remains constant. What we
really observe as the voltage across
the capacitor is only the projection
of the state of the system (voltagecurrent pair) along the voltage axis
(for the voltage across the capacitor)
and along the current axis (for the
inductor current).
L et us now e x plor e wh at h ap pens if we add a resistor in parallel
with the LC circuit, as shown in Figure 5(a). We assume the resistor is
large enough (R > 0.5 L/C ) to create an underdamped behavior. In
this case, the resistor will begin to
deplete the stored energy by turning it into heat. What happens to our
state variables v C and i L ? The gen-

(a)
√L /2 iL(t )
√C /2 V0

ωt
√C /2 vC (t )

(b)
FIGURE 4: (a) The trajectory of the voltage
across the capacitor and the current through
the inductor is an ellipse. (b) When properly
scaled, the trajectory becomes a perfect
circle. The square of the radius of this circle
represents the energy stored in this circuit.

eral derivation of v C and i L can be
found in most introductory textbooks such as in [1]. Here, we simply plot, in Figure 5(b), the energy
stored in the capacitor, the energy
stored in the inductor, and the sum of
the two energies. The energy moves
back and forth between the capacitor
and the inductor but loses its value
in each cycle. The total energy in the
system is monotonically decreasing
as time progresses.
It is also helpful to plot a trajectory of the scaled versions of the
voltage across the capacitor and the

IEEE SOLID-STATE CIRCUITS MAGAZINE

FA L L 2 0 17

11
Table of Contents for the Digital Edition of IEEE Solid-State Circuits Magazine - Fall 2017
