Circuits and Systems, 2013, 4, 89-96
http://dx.doi.org/10.4236/cs.2013.41014 Published Online January 2013 (http://www.scirp.org/journal/cs)
Classic Linear Methods Provisos Verification for Oscillator
Design Using NDF and Its Use as Oscillators Design Tool
Angel Parra-Cerrada, Vicente González-Posadas, José Luis Jiménez-Martín, Alvaro Blanco
Department Ingeniería Audiovisual y Comunicaciones, Universidad Politécnica de Madrid, Madrid, Spain
Received September 22, 2012; revised December 22, 2012; accepted December 30, 2012
ABSTRACT
The purpose of this paper is to show the conditions that must be verified before use any of the classic linear analysis
methods for oscillator design. If the required conditions are not verified, the classic methods can provide wrong solu-
tions, and even when the conditions are verified each classic method can provide a different solution. It is necessary to
use the Normalized Determinant Function (NDF) in order to perform the verification of the required conditions of the
classic methods. The direct use of the NDF as a direct and stand-alone tool for linear oscillator design is proposed. The
NDF method has the main advantages of not require any additional condition, be suitable for any topology and provide
a unique solution for a circuit with independence of the representation and virtual ground position. The Transpose Re-
turn Relations (RRT) can be used to calculate the NDF of any circuit and this is the approach used to calculate the NDF
on this paper. Several classic topologies of microwave oscillators are used to illustrate the problems that the classic
methods present when their required conditions are not verified. Finally, these oscillators are used to illustrate the use
and advantages of the NDF method.
Keywords: Oscillator; Provisos; NDF; RR; RRT
1. Introduction
The oscillators are one of the most important circuit
types on nowadays for communication systems and, due
to its non-linearity, they are one of the circuits that have
more problems on their design and optimization process.
The linear simulation of these circuits is really important
due to it is suitable for circuit optimization [1-3] and it
needs less computational resources than the non-linear
simulation. In any case, even if the designer is going to
use a non-linear simulation, it is necessary to perform a
linear simulation to have a good approximation of oscil-
lation frequency before starting the non-linear simulation.
But to be sure that the linear methods provide good solu-
tions, there are some provisos that must be verified.
2. Classic Methods Provisos
The classic methods for linear analysis of oscillators can
be classified into two groups: reference plane methods
[1,4] and the gain-loop method [5]. The provisos for each
group of methods are described in the following sections.
This paper describes the main conclusions and the
necessary provisos for the proper use of the classic
methods; they are defined and justified in detail by the
authors [6-8].
2.1. Reference Plane Methods
Any oscillator may be analyzed using Z, Y or Γ network
functions (Figure 1), in some cases it is more convenient
to use a specific method, but any of the reference plane
methods can be used. It is important to remember that all
the system poles are included on any network function;
however all the poles are not always included on general
transfer functions. The necessary condition for a circuit
to be a proper oscillator is that it must only have a pair of
complex conjugated poles in the Right Half Plane (RHP).
The traditional drawing way is conditioned in order to
Figure 1. Oscillator as two subsystems.
C
opyright © 2013 SciRes. CS

A. PARRA-CERRADA ET AL.
90
find the resonant structure as a dipole isolated from the
negative Z/Y/Γ generator. The reference plane can be any
(Figure 1), without being a real division between reso-
nator and generator; it is possible thanks to the denomi-
nator of any network function has all the information
about the system poles. But using one of the traditional
divisions simplifies the necessary conditions to assure a
correct linear analysis. These classic methods are really
the application of the Nyquist’s criteria for resolving the
location of poles in the RHP of the network functions.
2.2. Negative Conductance Analysis (Impedance
Network Function )
Figure 2 presents the conceptual diagram for a negative
conductance analysis. The impedance network function
is defined by Equation (1), where Ig is the external cur-
rent; V is the circuit response; and Z is the inverse of the
sum of the admittances of Figure 2. The circuit is a
proper oscillator if the network function has only a pair
of conjugated complex poles in the RHP. The poles of
the network function are defined by the zeros of Equation
(2), which is the characteristic function of the circuit.
Then, Equation (2) is analysed with the Nyquist criteria.
g
VZI

(1)
where:
1
T
YY 0
res osc
res osc
ZYY
Y



0
esosc
Y=

0Y
(2)
The classic oscillator start-up condition,

Tr
YY

YY
and
Tresosc
is a simplification of the
Nyquist analysis to a condition for a single frequency,
but it is not sufficient condition to guarantee the start-up
of the oscillator [6]. The additional condition before
analyzing an oscillator with the negative conductance
method (Impedance Network Function) is to assure that
Yosc does not have any poles (visible or hidden) in the
RHP. This verification uses the Normalized Determinant
Function (NDF) [9] of a network built with the active
sub-circuit terminated with a short-circuit, as it is de-
Figure 2. Negative conductance method conceptual dia-
gram.
scribed by Jackson based on Platzker and Ohtomo papers
[9-11]. Each clockwise turning circle around the origin of
the NDF analysis, for positive frequencies, confirms the
existence of a pair of conjugated poles. As NDF has an
asymptotic response with frequency to +1, the upper
analysis frequency is easily determined. So, the Nyquist
analysis of the NDF of the active subnetwork loaded with
a short-circuit must not encircle the origin, then the nega-
tive conductance analysis of the oscillator can be per-
formed with guarantee. With this condition the Nyquist
analysis of the Equation (2) will predict the oscillation if
it has a unique encircle of the origin.
The NDF can be calculated by means of the Return
Relations (RR) as it was described by Platzker and in a
most suitable way using the Transpose Return Relations
(RRT) [7], Equation (3). The formulation of the NDF which
uses the RRT is the one used by the authors to analyze the
examples. To use the RRT, it is necessary to replace the
active devices with their linear models, then the RRTi
term is the Transpose Return Relation for the i-depend-
ing generator while previous i-1 ones have been disabled.




00
0
11
nn
iT
i
ii
s
NDFRR RR


 

0
Tresosc
ZZ Z
(3)
Negative Impedance Analysis (Admittance Network
Function).
The oscillators to be analyzed with the Y network
function must also guarantee an additional proviso in the
same way as the ones to be analyzed with the Z network
function. This proviso also makes use of the NDF analy-
sis. The Y network function is analyzed by means of the
Z characteristic function, then the Nyquist analysis of the
NDF of the active sub-circuit loaded with an open-circuit
must not encircle the origin. With this condition the Ny-
quist analysis of the Equation (4) will predict the oscilla-
tion if it has a unique encircle of the origin [6].


 
res osc
(4)
Negative Conductance Analysis (Impedance Network
Function).
The Γ network function must satisfy that the Nyquist
analysis of the NDF of the Zo loaded active sub-circuit
must not encircle the origin.
Then, the oscillation condition can be determined by
the analysis of the Equation (5), so the oscillation is satis-
fied if the Nyquist analysis of 1resosc encircles
clockwise the 0 or if the Nyquist analysis of


10
Tresosc
encircles clockwise the +1 [6].
(5)


2.3. Loop-Gain Method
When the feedback path of the circuit can be identified,
the Loop-Gain is commonly used [3]. When it is possible

A. PARRA-CERRADA ET AL.
91
to define the feedback path, this method is preferred by
the designers because it is more intuitive and it can pro-
vide more useful information about the circuit. The start
point for the loop-gain analysis is the general function of
a loopback system (Figure 3), and the most important
equation used nowadays is the one defined by Randall
and Hock (Equation (6)) [5].
rent Source (ACCS) with a constant value for all fre-
quencies and with a voltage meter that measures the
voltage at the control point of the real linear transistor
model, Figure 4.
21 1221
11221211 22
21
L
ZZ S
G12
12 2112
2
S
Z
ZZ SS

 SSS

 
2 S SS 
(6)
The same process must be performed for each transis-
tor, but the ACCS of the already analyzed ones must be
disabled before starting the simulation of the RRT of the
next one.
Some commercial simulation software, as for example
AWR, provides NDF function, but it is also possible to
calculate RRT with any simulator. RR T is −RR and it is
“the true loop gain”. The authors did not use the AWR
NDF function for the simulations presented in this paper,
but they used a general function based on the simulation
of the RRT of each transistor.
The Nyquist analysis of the loop-gain (GL) must search
for +1 clockwise encirclement to assure a proper oscilla-
tion condition, but the poles of the system must be only
from the zeros of 1 − GL. To guarantee that the poles of
the system only come from the zeros of 1 − GL it is nec-
essary that [7]: 4. NDF as Oscillators Design Tool
None of the S parameters can have any poles in the
RHP. This condition can be verified with the Nyquist
analysis of the NDF of the open-loop circuit loaded with
Zo in both ports.
The main conclusion that can be obtained from the study
of provisos for the classic linear oscillator design meth-
ods is that it is necessary to use the NDF to guarantee the
applicability of any method. So, as it is described by the
authors [8] the NDF is itself an interesting linear design
method for oscillators design.
“Test function” 1122122112 must
not have any zeros in the RHP. This condition is verified
with the Nyquist analysis of TF when the condition of
the previous point has been satisfied.
1TFS S
The NDF is the quotient of the network determinant
and the normalized network determinant, Equation (3).
The normalized network determinant is obtained by dis-
abling all the active devices of the network, but it is eas-
ily solved by using the RRT, as it has been described in
the previous section.
3. Calculus Using the RRT
The NDF can be easily calculated with the RRT of each
active device using the Equation 3 as it was described by
Platzker [9]. The first step for the RRT simulation of a
transistor is to have a linear model of the device. If this
linear model is not available for the transistor the pa-
rameters can be extracted with an “annotate” from the
spice model with the AWR simulation software.
The Nyquist plot analysis of the NDF determines in a
single step the number of poles in the RHP of a network.
Each clockwise encirclement of the origin for positive
frequencies indicates the existence of a pair of conju-
gated complex poles in the RHP. So, the total phase evo-
lution for a proper oscillator of the NDF for positive fre-
quencies from 0 Hz up to ∞ Hz must be −360 deg. The
Once the linear model of the transistor is available, the
next step is to calculate its RRT. It can be calculated by
making the transistor work as an independent AC Cur-
Figure 3. Reference plane method (left) and Feedback scheme (right).

A. PARRA-CERRADA ET AL.
92
Feedback
Circuit
Figure 4. Simulation of the RRT of a transistor.
NDF has an asymptotic behavior towards +1 which is
useful for determining the analysis upper frequency limit
without ambiguity. As the NDF can be applied to any
oscillator topology, it is a universal tool for analyzing
oscillators on a single step.
Other useful characteristics of the NDF are that it can
predict the oscillation frequency without transistor com-
pression (gm) for Kurokawa’s first harmonic approxima-
tion; and that it is suitable for the calculus of the QL of
the circuit because it is directly related with the RRT [8].
These characteristics make it suitable to use it as an op-
timization tool for low noise oscillators [12]. In the same
way it is also suitable for estimating the start-up time.
These two parameters, phase-noise and start-up time, are
proper to the loop-gain method, but also to the NDF
method. This way the NDF method makes available for
any oscillator topology the parameters that until now
were only available for the topologies that can be ana-
lysed with the loop-gain method, and on a single step and
without ambiguity. This method is suitable for calculate-
ing the real QL factor of the circuit and the gain margin,
so it is possible to estimate the start-up time and the
phase noise of any oscillator without dependence with its
topology.
5. Examples
Two examples are presented on this paper, one oscillator
circuit which is usually analyzed with a reference plane
method and another that is usually analyzed with the
loop-gain method. These two examples have been chosen
to illustrate the importance of the verification of the pro-
visos and the advantage of using the NDF as an oscillator
linear design tool. The circuits use as active device a
BFR360F transistor biased with C and
CE
10 mAI
3 VV

. The AWR software has been used for all the
simulations shown in this paper.
5.1. Example A
The common collector oscillators, Figure 5, are usually
analyzed by negative resistance, due it has the behaviour
of a negative resistance generator for its first harmonic

A. PARRA-CERRADA ET AL. 93
approximation response. The Nyquist representations of
the total impedance and admittance of the example cir-
cuit are shown in Figures 6(a) and (b).
ooscosc o
ooscosco
YYZ Z
YYZZ



osc

1
osc
osc
Y
Z

Z
osc
,Y
osc,
Γ
osc
1
res
res
Y
Z
oresres o
res
oresres o
YYZ Z
YYZZ

 

Z
res
,Y
res,
Γ
res
Figure 5. Common collector oscillator.
(a)
(b)
Figure 6. (a) Common collector oscillator total impedance
and (b) total admittance.
The Nyquist analysis of ZT, Figure 6(a), predicts an
oscillation frequency of 1887 MHz, but the Nyquist
analysis of YT, Figure 6(b), does not predict any oscilla-
tion condition. It can be explained because Yosc has a
conjugated pair of poles in the RHP that hide the zeros
on the Nyquist analysis of Y
T. This discrepancy points
out the importance of the verification of the provisos
before performing any classic analysis to an oscillator.
The Nyquist plot of the NDF of the active sub-circuit
loaded with an open-circuit, Figure 7(a), does not encir-
cle the origin, so it is possible to analyze this oscillator
using the Y network function (Nyquist analysis of the ZT).
But the NDF of the short-circuit loaded active sub-circuit,
Figure 7(b), encircles the origin, Yosc has a pair of poles
that will hide the zeros when the Nyquist criteria is used
with the YT. Then the oscillator cannot be analyzed by the
Z network function (admittance analysis).
As an example of the use of the NDF, the NDF analy-
sis of this circuit, Figure 8 encircles the origin and pre-
dicts an oscillation at 1474 MHz. The difference of os-
(a)
(b)
Figure 7. NDF of the (a) open-circuit and (b) short-circuit of

A. PARRA-CERRADA ET AL.
94
the active sub-circuit.
cillation frequency is caused because the ZT method does
not provided the first harmonic approximation but the
NDF solves the first harmonic approximation without
transistor compression. If the transistor is compressed to
gm = 0.0125 then the ZT solution is the same that the NDF
one, Figure 9.
As it has been shown with this example, an important
advantage of using the NDF method is that it can be used
with all oscillator topologies. In this example an oscilla-
tor that is usually analyzed with a reference plane method
has been presented, so the main parameters that the ref-
erence plane methods cannot provide are now. The NDF
is a “loop-gain concept” and it provides the first har-
monic approximation without transistor compression.
The QL of the oscillator obtained with Equation (7) [12]
is 9.3, Figure 10, and the gain margin is defined by the
real part of NDF at the crossing point of the encircle of
the origin. The big difference between the frequency ob-
GainMargin
Figure 8. NDF of the common collector oscillator.
Figure 9. Common collector oscillator total impedance with
gm = 0.0125.
tained by the ZT without compression and the one ob-
tained by the NDF is due to the low QL of the oscillator.


2
LT
QArgRR



  (7)

5.2. Example B
An oscillator, Figure 11(a), to which different virtual
ground points [13] are applied, is used for this example.
Some resulting possibilities of this example are the well-
known classic topologies: common collector (also named
Colpitts), common emitter (also named Pierce) and
common base. Using virtual ground concept, it is dem-
onstrated that there is a unique oscillator topology and, as
it will be explained throughout this example that the
NDF/RRT is the best tool to analyze it. The circuits in
Figure 11 include the parasites of the package of all de-
vices, but these parasites have not been represented for
readability.
The open-loop analysis, Figure 12, predicts that only
the common emitter topology will oscillate, but “How
can it be possible if the three schematics represent the
same circuit?”. The problem appears due to the Nyquist
analysis of GL expression is not valid for common col-
lector and common base topologies. In these two cases
the denominators of GL have two hidden zeros which
make Nyquist analysis not encircle +1. It is interesting to
point out that the GL analysis of the common base circuit
complies the Barkhausen criteria at two different fre-
quencies, 1131 MHz and 3825 MHz. The first one
crosses from a positive to a negative phase, but the sec-
ond one crosses from a negative to a positive phase. The
+1 is not encircled on the common collector example,
neither on the common base, so they can be considered
as complementary to Nguyen examples [14].
On the other hand, the NDF (or the RRT ), analyses
have a unique solution for the three schematics, Figure
13. As all the NDF analyses are identical and they predict
a unique complex pair of poles in the RHP, so the re-
quired condition for proper oscillation is satisfied for the

A. PARRA-CERRADA ET AL.
CS
95
Figure 10. QL of the common collector oscillator.
(a) (b)
(c) (d)
Figure 11. (a) Ground-less oscillator; (b) Common emitter oscillator; (c) Common collector oscillator; (d) Common base os-
cillator.
Figure 12. GL Nyquist plot for common emitter, common
collector and common base.
three schematics. As it is expected, the solution for the
three examples is the same, because they are the same
circuit but drawn on three different ways.
Figure 13. NDF Nyquist plot for the three circuit topologies.
6. Conclusions

A. PARRA-CERRADA ET AL.
96
The NDF method is a suitable tool for direct analysis of
oscillators and it does not require any additional proviso
or conditions before using it. Another advantage of this
NDF method is that all oscillator topologies can be ana-
lyzed with a “loop-gain concept” and the main parame-
ters that the reference plane methods cannot provide are
now available for any oscillator topology.
The NDF solution is independent of the virtual ground
position and it provides the oscillation frequency at first
harmonic approximation without requiring transistor m
g
compression. This NDF independence is based on its
relation with the Return Relations (and T), as they
provide the “true open-loop-gain”. To sum up, the NDF/
RRT method is an optimum tool for the quasi-lineal os-
cillator analysis in a single step; it does not require any
additional proviso or verification; and it is suitable for
any oscillator topology.
RR
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