IEEE Microwave Magazine - June 2015 - 89

optimization, either with the conversion matrix
approach (large-signal, small-signal analysis) or with
a two-tone HB. In the case of a flip bifurcation [5], [6],
the AG will operate at the frequency ~ AG = ~ o /2, and
the additional unknowns will be h b and the AG phase
(or input source phase) z AG . The condition to be solved
will be YAG (h b, z AG) = 0, with the pure HB system at the
fundamental frequency ~ o /2 as the inner tier.
The bifurcation analysis enables a direct determination of the stability boundaries and, therefore, avoids
the need for successive stability analyses, considering all the possible parameter values in an exhaustive
manner. As an example, a bifurcation analysis will be
applied to the example power amplifier in Figure  17.
The parameters considered are the gate bias voltage
VGS and Pin . The bifurcation analysis in the plane
defined by VGS and Pin will provide a map from which
we can predict what kind of operation (stable or unstable) the circuit will exhibit. This particular choice of
parameters (VGS and Pin ) will also allow us to relate
the small- and large-signal stability properties of the
demonstrator amplifier.
A small-signal stability analysis of this amplifier
(under Pin = 0 W) shows that it should oscillate for bias
voltage -2.8 V < VGS < - 1.2 V. This oscillation should
be extinguished at a certain input power Pin due to the
natural reduction of negative resistance with the signal
amplitude. To analyze this, the Hopf bifurcation locus
will be calculated using an AG of the voltage type
(Figure 17). The generator at the frequency ~ AG = ~ a,
nonrationally related to ~ in, is connected to the drain
terminal of one of the two transistors [24]. The AG
amplitude is made arbitrarily small, solving the nonperturbation condition in terms of ~ a and the analysis
parameters, that is, YAG (VGS, Pin, ~ a) = 0, with a pure HB
system as an inner tier. This complex equation in three
unknowns provides a curve in the plane VGS , Pin . The
locus has been represented in Figure 19 with a solid
line. The locus is open on the right-hand side. For all
the Hopf locus points, there are a pair of complex conjugate poles ! j~ a located on the imaginary axis. The
frequency ~ a is expressible either as ~ a = ~ in /2 - D~
or as ~ a = ~ in /2 + D~ and varies through the locus due
to its autonomous nature. Approaching the two edge
points FH and FH', the frequency of the poles located
in the imaginary axis tends to ! j~ in /2, (D~ tends to 0) .
Because the dimension of the critical subspace must be
preserved, at the two edge points, there are two independent pairs of poles at the subharmonic frequency
~ in /2 located in the imaginary axis.
In view of the previously described degeneration of
the pole frequency to the subharmonic value ! j~ in /2,
there must be a flip bifurcation locus, completing the
map in Figure 19. In fact, two flip bifurcations had
already been detected with the pole locus of Figure
18, marked with "x" in Figure 19. The flip bifurcation
locus is traced with a small amplitude AG at the

June 2015

frequency ~ AG = ~ in /2, solving the complex equation
YAG (VGS, Pin, z AG) = 0, which provides a curve in the
plane VGS , Pin . The flip locus is closed and composed
of two curves: a physical one (green) and an unphysical
one (red). To understand this, one must take into account
the following. At the two edge points of the Hopf
locus, there are two different pairs of poles at ! j~ in /2.
Through the flip locus, one of these two pairs stays on
the imaginary axis ! j~ in /2 and the other v ! j~ in /2
shifts from this axis either to the left v < 0, in the green
section of the flip locus, or the right ^v > 0 h, in the
red section. When crossing the red section, the solution
instability persists due to the presence of the pair of
poles v ! j~ in /2, where v > 0. The crossing of the green
section does have a physical effect. The subharmonic
component ~ in /2 is either generated or extinguished
when crossing the green section, depending on the
sense of the variation of the parameter. All these
predictions have been validated with measurements,
which are superimposed on Figure 19. In these
measurements, Pin has been increased in small steps,
obtaining at each step the VGS values that delimit the
stable operation interval.
Figure 20 compares the predictions of the bifurcation
loci [Figure 20(a)] with the results of pole-zero identification at the constant Pin = 18 dBm, when increasing VGS . In Figure 20(b), the real part of the dominant
poles has been traced versus Pin . A single value of
real part is obtained when the dominant poles are
complex conjugate at an incommensurable frequency
v ! j (~ in /2 ! D~) . The merging of these poles (D~ " 0)
at the point M gives rise to two independent pairs of
poles at the subharmonic frequency: v ! j~ in /2 and
v' ! j~ in /2. Note that from the merging point, the real
part of the poles splits into two different values.
For a very low VGS , the amplifier is stable, as
expected. When increasing VGS , the dominant pair of
complex conjugate poles v ! j~ a crosses the imaginary
axis (v = 0) , which gives rise to the Hopf bifurcation
H 1, well predicted by the Hopf locus in Figure 20(a).
From H 1, the amplifier periodic solution is unstable and
behaves instead as a self-oscillating mixer. At the Hopf
bifurcation H 2, the complex conjugate poles v ! j~ a
cross again to the LHP and the amplifier periodic solution becomes stable. At the point M, the incommensurate poles split into two pairs v ! j~ in /2 and vl ! j~ in /2
on the LHP, and one of these pairs crosses the axis to the
RHP at the flip bifurcation F1, giving rise to a frequency
division by  2. The same pair of subharmonic poles
crosses again to the LHP at the flip bifurcation F2, from
which the subharmonic regime is extinguished. The
amplifier periodic solution becomes stable again at F2 .

Conclusions
Stability analysis methods have been presented in a
self-contained manner. The main procedures have
been described with an analytical insight, enabling a

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