IEEE Microwave Magazine - June 2015 - 82

The described oscillation will enable
the implementation of parametric
frequency divider.

would be formally identical in terms of s. The pole-
zero identification should be performed between ~ , 0
and the frequency ~ max . Accurate identification in a
very wide frequency interval may require a high-order
n of the polynomial in the denominator of (19), so the
quality of the identification may degrade. As a matter
of fact, the required order would be theoretically inficonnected to the circuit, which behaves linearly with
nite in the case of circuits containing distributed elerespect to this generator.
ments. However, because the transfer function Z in ( j~)
As already stated, whatever input is connected to
a linear system, all the possible transfer functions
is linear, the total frequency interval 0 - ~ max may be
share the same denominator, which depends only on
divided into subintervals, which will allow for accurate
the system itself instead of the particular location or
identification using a smaller-order n in the denominanature of the input and output signals. This can be
tor of (19).
gathered from the inspection of (17). The calculation
As already stated, all the closed-loop transfer funcof any possible transfer function S out /I n, where S out
tions that can be defined in a given linear system share
the same denominator [28]-[31]. In contrast, the numerais an arbitrary output, requires the inversion of the
tor depends on the particular definition of the transfer
matrix on the left-hand side. This matrix is formally
function. Thus, it will depend on the node selected for
identical to the one analyzed in (7) for the determinathe connection of the current source. Because of this,
tion of the system singularities. Actually, the matrix
cancellations of unstable poles with zeros in the RHP
in (17) is obtained by simply replacing s = j~ in (7).
may take place at some particular locations. If unstable
For the definition of the transfer function, a very conpoles are canceled, an incorrect conclusion about the
venient output (for reasons of numerical accuracy) is
stability of the solution may be obtained. This is why
the voltage Vn at the node where the current generathe pole-zero identification should be performed for
tor is connected [28]-[31]. Thus, the transfer function
different locations of the current source. The terminal
considered is
nodes of the active devices are the most convenient
Vn ( j~)
for this analysis due to their proximity to the potential
Z in ( j~) =
.
(18)
I n ( j~)
sources of instability. Although the case of a current
source connected in parallel has been considered, the
technique is equally applicable with a voltage source
Using pole-zero identification, the complex function
Z in ( j~) is fitted through a least-squares technique,
Vn in a series connection, using an admittance transwith a quotient of polynomials of the form [28], [31]
fer function Yin = I n /Vn . As shown in [30], the admittance type transfer function is more convenient if there
~
~
(
j
z
)
f
(
j
z
)
1
m
is a passive impedance of small value dominating the
Z in ( j~) = Vn ( j~) = A
.
(19)
In
( j~ - p 1) f ( j~ - p n)
impedance coming from the active part where the instability occurs. Note that the possible existence of zeros in
the RHP does not have any implication for the system
Note that the zeros and poles of Z in (s) depend on the
stability. If a transfer function has poles and/or zeros in
constants z 1 fz m and p 1 fp n, as the Z in expression
the RHP, then the system shows nonminimum phase behavior [12].
The transfer function (19) exactly
0.5 pF
50 X
50 X
agrees with the inverse of the total
90°
90°
V2
V1
input admittance at the node n where
+
the current source is introduced,
100 X V
10 pF
70
X
Yin ( j~) = I n ( j~) /Vn ( j~) . Assuming
50 X
50 X
0.2 pF
180°
180°
that Yin (~) is evaluated at a sensitive
i = gmv-0.9v3
location, it is possible to relate the oscillation startup conditions [see (1)] derived
0.5 pF
50 X
50 X
50 X
50 X
by Kurokawa, G in (~ c) < 0, B in (~ c) = 0,
90°
90°
V3
V4
and 2B in (~ c) /2~ > 0, to the existence
+
of a pair of unstable complex-conju10
pF
100 X V
gate poles in the closed-loop transfer
70 X
0.2 pF
function Z in ( j~) [inverse of Yin ( j~)],
i = gmv-0.9v3
as shown in [5]. The dual is true of an
analysis based on the series connection
of a voltage generator in terms of the
Figure 11. A circuit containing two active devices and feedback elements in a
power-combining topology. Transmission lines are calculated at 5 GHz.
input impedance.

82

June 2015



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