IEEE Microwave Magazine - June 2015 - 87

Drain Voltage (V)

Imaginary Part
(Hz # 108)

As shown in the next paragraph, this relevant property
the poles of a periodic solution at ~ o are distributed
enables a straightforward detection of the boundaries of
in sets of equivalent poles, having the form c m + jp~ o
the stable interval.
or v m ! j~ m + jp~ o, where p is an integer number.
Because of this pole distribution, the frequency sweep
in ~ may be limited to (0, ~ o). Division into subinterBifurcations
vals of smaller length and using a lower-order n will
A bifurcation is a qualitative change in the stability of
generally increase the accuracy.
a steady-state solution or in the number of solutions
Pole-zero identification has been applied to an
when a parameter h is varied continuously [1]-[3], [10].
example power amplifier in Figure 17, which operThe most relevant bifurcations in a microwave circuit
ates at fin = 1.5 GHz [24]. The circuit poles have been
are those giving rise to oscillations and those giving
rise to turning points, such as the points T1 and T2 in
analyzed versus the input power Pin at the particular
gate bias voltage VGS =-1 V [Figure 18(a)]. To obtain
the closed-loop transfer function, a small-signal cur8.5
rent source I d (~) has been connected to the drain
Pin = -30 dBm
node of one of the two transistors. The transfer func8 dBm
fin/2
tion considered is Z (~) = Vd (~) /I d (~), which is calcu7.5
lated with the conversion-matrix approach. For each
7 dBm 21 dBm
Pin , two analyses are carried out: 1) an HB analysis to
Pin = -30 dBm
obtain the periodic large-signal steady state and 2) a
6.5
4
-10 -8
-6
-4
-2
0
2
conversion-matrix analysis, sweeping ~ , to obtain the
Real Part (Hz # 107)
function Z (~) . Pole-zero identification is applied to
each of the functions Z (~) resulting versus Pin .
(a)
Transformations between poles of periodic solutions,
2
fin
taking into account (21), have been analyzed in depth in
[5]. An example of these transformations is explained
1.5
in the following. At a small signal ( Pin =-30 dBm), the
amplifier in Figure 17 exhibits two pairs of dominant
1
complex-conjugate poles, associated with a same
fin/2
complex multiplier, located on the LHP. These two
0.5
pairs of poles can be expressed as v ! j (~ in /2 ! D~) .
fin
As Pin increases, the two pairs of poles approach
0
each other and merge ^D~ " 0 h at Pin = 7 dBm. Since
0
5
10
15
20
25
the dimension of the near-critical subspace must be
Input Power (dBm)
preserved, after this merging, they will split into two
(b)
independent pairs: 1) v ! j~ in /2 and 2) vl ! j~ in /2,
each corresponding to one negative real multiplier, as
Figure 18. A stability analysis of the circuit in Figure 15 for
derived from (21). One of these two pairs crosses to the
VGS =-1V: (a) the pole-zero identification versus Pin and (b)
RHP at Pin = 8 dBm and to the LHP at Pin = 21 dBm.
the validation with HB by tracing the solution curves at fin
Within the interval (8 dBm, 21 dBm), the circuit exhibits
and fin /2 with the aid of an AG [5], [6]. The components of
a subharmonic oscillation at ~ in /2. The results of pole-
the subharmonic solution are represented in red.
zero identification have been validated by
obtaining the subharmonic steady-state
Table 4. The main types of bifurcation phenomena.
solution in HB [Figure 18(b)]. This analysis
has required the use of an auxiliary
Critical Poles (Poles
generator (AG) [5] at the subharmonic
Original
Crossing Through
Generated Regime or
frequency to avoid undesired convergence
Bifurcation
Regime
the Imaginary Axis) Phenomenon
to the coexistent nondivided solution. The
details of this technique are given in the
Primary Hopf dc
Periodic oscillation at ~ o
v ! j~ o
following. As gathered from the figure, the
subharmonic solution arises when the pair
Secondary
Periodic at ~ o v ! j~ a
Self-oscillating mixing
Hopf
regime ~ o and ~ a
of poles at ~ in /2 crosses to the RHP and is
extinguished when it crosses again to
Flip
Periodic at ~ o v ! j~ o /2
Divided by 2 regime at
the LHP. The steady-state subharmonic
~ o /2
component degenerates to zero amplitude
Hysteresis (and
Turning point dc or periodic c
at the two points at which the pair of
synchronization in
poles at ~ in /2 is exactly located on the
oscillators)
imaginary axis, as shown in Figure 18(b).

June 2015

87



Table of Contents for the Digital Edition of IEEE Microwave Magazine - June 2015

IEEE Microwave Magazine - June 2015 - Cover1
IEEE Microwave Magazine - June 2015 - Cover2
IEEE Microwave Magazine - June 2015 - 1
IEEE Microwave Magazine - June 2015 - 2
IEEE Microwave Magazine - June 2015 - 3
IEEE Microwave Magazine - June 2015 - 4
IEEE Microwave Magazine - June 2015 - 5
IEEE Microwave Magazine - June 2015 - 6
IEEE Microwave Magazine - June 2015 - 7
IEEE Microwave Magazine - June 2015 - 8
IEEE Microwave Magazine - June 2015 - 9
IEEE Microwave Magazine - June 2015 - 10
IEEE Microwave Magazine - June 2015 - 11
IEEE Microwave Magazine - June 2015 - 12
IEEE Microwave Magazine - June 2015 - 13
IEEE Microwave Magazine - June 2015 - 14
IEEE Microwave Magazine - June 2015 - 15
IEEE Microwave Magazine - June 2015 - 16
IEEE Microwave Magazine - June 2015 - 17
IEEE Microwave Magazine - June 2015 - 18
IEEE Microwave Magazine - June 2015 - 19
IEEE Microwave Magazine - June 2015 - 20
IEEE Microwave Magazine - June 2015 - 21
IEEE Microwave Magazine - June 2015 - 22
IEEE Microwave Magazine - June 2015 - 23
IEEE Microwave Magazine - June 2015 - 24
IEEE Microwave Magazine - June 2015 - 25
IEEE Microwave Magazine - June 2015 - 26
IEEE Microwave Magazine - June 2015 - 27
IEEE Microwave Magazine - June 2015 - 28
IEEE Microwave Magazine - June 2015 - 29
IEEE Microwave Magazine - June 2015 - 30
IEEE Microwave Magazine - June 2015 - 31
IEEE Microwave Magazine - June 2015 - 32
IEEE Microwave Magazine - June 2015 - 33
IEEE Microwave Magazine - June 2015 - 34
IEEE Microwave Magazine - June 2015 - 35
IEEE Microwave Magazine - June 2015 - 36
IEEE Microwave Magazine - June 2015 - 37
IEEE Microwave Magazine - June 2015 - 38
IEEE Microwave Magazine - June 2015 - 39
IEEE Microwave Magazine - June 2015 - 40
IEEE Microwave Magazine - June 2015 - 41
IEEE Microwave Magazine - June 2015 - 42
IEEE Microwave Magazine - June 2015 - 43
IEEE Microwave Magazine - June 2015 - 44
IEEE Microwave Magazine - June 2015 - 45
IEEE Microwave Magazine - June 2015 - 46
IEEE Microwave Magazine - June 2015 - 47
IEEE Microwave Magazine - June 2015 - 48
IEEE Microwave Magazine - June 2015 - 49
IEEE Microwave Magazine - June 2015 - 50
IEEE Microwave Magazine - June 2015 - 51
IEEE Microwave Magazine - June 2015 - 52
IEEE Microwave Magazine - June 2015 - 53
IEEE Microwave Magazine - June 2015 - 54
IEEE Microwave Magazine - June 2015 - 55
IEEE Microwave Magazine - June 2015 - 56
IEEE Microwave Magazine - June 2015 - 57
IEEE Microwave Magazine - June 2015 - 58
IEEE Microwave Magazine - June 2015 - 59
IEEE Microwave Magazine - June 2015 - 60
IEEE Microwave Magazine - June 2015 - 61
IEEE Microwave Magazine - June 2015 - 62
IEEE Microwave Magazine - June 2015 - 63
IEEE Microwave Magazine - June 2015 - 64
IEEE Microwave Magazine - June 2015 - 65
IEEE Microwave Magazine - June 2015 - 66
IEEE Microwave Magazine - June 2015 - 67
IEEE Microwave Magazine - June 2015 - 68
IEEE Microwave Magazine - June 2015 - 69
IEEE Microwave Magazine - June 2015 - 70
IEEE Microwave Magazine - June 2015 - 71
IEEE Microwave Magazine - June 2015 - 72
IEEE Microwave Magazine - June 2015 - 73
IEEE Microwave Magazine - June 2015 - 74
IEEE Microwave Magazine - June 2015 - 75
IEEE Microwave Magazine - June 2015 - 76
IEEE Microwave Magazine - June 2015 - 77
IEEE Microwave Magazine - June 2015 - 78
IEEE Microwave Magazine - June 2015 - 79
IEEE Microwave Magazine - June 2015 - 80
IEEE Microwave Magazine - June 2015 - 81
IEEE Microwave Magazine - June 2015 - 82
IEEE Microwave Magazine - June 2015 - 83
IEEE Microwave Magazine - June 2015 - 84
IEEE Microwave Magazine - June 2015 - 85
IEEE Microwave Magazine - June 2015 - 86
IEEE Microwave Magazine - June 2015 - 87
IEEE Microwave Magazine - June 2015 - 88
IEEE Microwave Magazine - June 2015 - 89
IEEE Microwave Magazine - June 2015 - 90
IEEE Microwave Magazine - June 2015 - 91
IEEE Microwave Magazine - June 2015 - 92
IEEE Microwave Magazine - June 2015 - 93
IEEE Microwave Magazine - June 2015 - 94
IEEE Microwave Magazine - June 2015 - 95
IEEE Microwave Magazine - June 2015 - 96
IEEE Microwave Magazine - June 2015 - 97
IEEE Microwave Magazine - June 2015 - 98
IEEE Microwave Magazine - June 2015 - 99
IEEE Microwave Magazine - June 2015 - 100
IEEE Microwave Magazine - June 2015 - 101
IEEE Microwave Magazine - June 2015 - 102
IEEE Microwave Magazine - June 2015 - 103
IEEE Microwave Magazine - June 2015 - 104
IEEE Microwave Magazine - June 2015 - 105
IEEE Microwave Magazine - June 2015 - 106
IEEE Microwave Magazine - June 2015 - 107
IEEE Microwave Magazine - June 2015 - 108
IEEE Microwave Magazine - June 2015 - 109
IEEE Microwave Magazine - June 2015 - 110
IEEE Microwave Magazine - June 2015 - 111
IEEE Microwave Magazine - June 2015 - 112
IEEE Microwave Magazine - June 2015 - 113
IEEE Microwave Magazine - June 2015 - 114
IEEE Microwave Magazine - June 2015 - 115
IEEE Microwave Magazine - June 2015 - 116
IEEE Microwave Magazine - June 2015 - 117
IEEE Microwave Magazine - June 2015 - 118
IEEE Microwave Magazine - June 2015 - 119
IEEE Microwave Magazine - June 2015 - 120
IEEE Microwave Magazine - June 2015 - 121
IEEE Microwave Magazine - June 2015 - 122
IEEE Microwave Magazine - June 2015 - 123
IEEE Microwave Magazine - June 2015 - 124
IEEE Microwave Magazine - June 2015 - 125
IEEE Microwave Magazine - June 2015 - 126
IEEE Microwave Magazine - June 2015 - 127
IEEE Microwave Magazine - June 2015 - 128
IEEE Microwave Magazine - June 2015 - 129
IEEE Microwave Magazine - June 2015 - 130
IEEE Microwave Magazine - June 2015 - 131
IEEE Microwave Magazine - June 2015 - 132
IEEE Microwave Magazine - June 2015 - 133
IEEE Microwave Magazine - June 2015 - 134
IEEE Microwave Magazine - June 2015 - 135
IEEE Microwave Magazine - June 2015 - 136
IEEE Microwave Magazine - June 2015 - 137
IEEE Microwave Magazine - June 2015 - 138
IEEE Microwave Magazine - June 2015 - 139
IEEE Microwave Magazine - June 2015 - 140
IEEE Microwave Magazine - June 2015 - 141
IEEE Microwave Magazine - June 2015 - 142
IEEE Microwave Magazine - June 2015 - 143
IEEE Microwave Magazine - June 2015 - 144
IEEE Microwave Magazine - June 2015 - Cover3
IEEE Microwave Magazine - June 2015 - Cover4
https://www.nxtbook.com/nxtbooks/ieee/microwave_201903
https://www.nxtbook.com/nxtbooks/ieee/microwave_201902
https://www.nxtbook.com/nxtbooks/ieee/microwave_201901
https://www.nxtbook.com/nxtbooks/ieee/microwave_20181112
https://www.nxtbook.com/nxtbooks/ieee/microwave_20180910
https://www.nxtbook.com/nxtbooks/ieee/microwave_20180708
https://www.nxtbook.com/nxtbooks/ieee/microwave_201806
https://www.nxtbook.com/nxtbooks/ieee/microwave_201805
https://www.nxtbook.com/nxtbooks/ieee/microwave_201803
https://www.nxtbook.com/nxtbooks/ieee/microwave_january2018
https://www.nxtbook.com/nxtbooks/ieee/microwave_november2017
https://www.nxtbook.com/nxtbooks/ieee/microwave_september2017
https://www.nxtbook.com/nxtbooks/ieee/microwave_july2017
https://www.nxtbook.com/nxtbooks/ieee/microwave_june2017
https://www.nxtbook.com/nxtbooks/ieee/microwave_may2017
https://www.nxtbook.com/nxtbooks/ieee/microwave_march2017
https://www.nxtbook.com/nxtbooks/ieee/microwave_january2017
https://www.nxtbook.com/nxtbooks/ieee/microwave_december2016
https://www.nxtbook.com/nxtbooks/ieee/microwave_november2016
https://www.nxtbook.com/nxtbooks/ieee/microwave_october2016
https://www.nxtbook.com/nxtbooks/ieee/microwave_september2016
https://www.nxtbook.com/nxtbooks/ieee/microwave_august2016
https://www.nxtbook.com/nxtbooks/ieee/microwave_july2016
https://www.nxtbook.com/nxtbooks/ieee/microwave_june2016
https://www.nxtbook.com/nxtbooks/ieee/microwave_may2016
https://www.nxtbook.com/nxtbooks/ieee/microwave_april2016
https://www.nxtbook.com/nxtbooks/ieee/microwave_march2016
https://www.nxtbook.com/nxtbooks/ieee/microwave_february2016
https://www.nxtbook.com/nxtbooks/ieee/microwave_january2016
https://www.nxtbook.com/nxtbooks/ieee/microwave_december2015
https://www.nxtbook.com/nxtbooks/ieee/microwave_november2015
https://www.nxtbook.com/nxtbooks/ieee/microwave_october2015
https://www.nxtbook.com/nxtbooks/ieee/microwave_september2015
https://www.nxtbook.com/nxtbooks/ieee/microwave_august2015
https://www.nxtbook.com/nxtbooks/ieee/microwave_july2015
https://www.nxtbook.com/nxtbooks/ieee/microwave_june2015
https://www.nxtbook.com/nxtbooks/ieee/microwave_may2015
https://www.nxtbook.com/nxtbooks/ieee/microwave_april2015
https://www.nxtbook.com/nxtbooks/ieee/microwave_march2015
https://www.nxtbook.com/nxtbooks/ieee/microwave_january2015
https://www.nxtbookmedia.com