4.2 Second-Order Circuits
141
To fix ideas, suppose the nonlinear characteristic of the resistor is the cubic
function
v = ˆ
v(i) = −i +
1
3
i
3
as shown in Fig. 4.9b. The oscillator with a cubic nonlinearity satisfies the SEs
dv C
dt
= −
i L
C
di L
dt
=
v C + i L −
1
3 i 3
L
L
(4.4)
and is known as Van der Pol oscillator.
It is possible to reduce the number of parameters by writing the SEs of the Van
der Pol oscillator in adimensional form. To this end, define the dimensionless time
τ =
t
√
LC
so that
dv C
dt
=
1
√
LC
dv C
dτ
and
di L
dt
=
1
√
LC
di L
dτ
.
Substituting in (4.4) we obtain the following equivalent SE in terms of the
dimensionless time τ
dv C
dτ
= −
1
i L
di L
dτ
=
v C + i L −
1
3
i
3
L
(4.5)
where we let
=
C
L
.
Now, we have only one parameter , so the geometric features of the family of
trajectories in the phase plane (phase portrait) of the Van der Pol oscillator can be
effectively analyzed by varying only . Three such phase portraits corresponding