1109
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-30, NO. 7, JULY 1982
[5]
L, A, Weinstein, “The Theory of Diffraction Metiod~ in Generalized WzenerHopf Technique.
and the Factorization
Boulder, CO: Golem, 1969. S. W. Lee, W. R. Jones, and J. J. Campbell, “Convergence of numericrd solutions of iris-type discontinuity problems,” IEEE Trans. Microwaoe Theory Tech., vol. MTT- 19, pp. 528–536, June 1971.
[6]
(a)
(b)
Exact Derivation of the Nonlinear Negative-Resistance Oscillator Pulling Figure J. OBREGON Abstract figure
— Only approximate
of an oscillator.
AND A. P. S. KHANNA relations
are so far available
An exact derivation
of the pnlling
here, tafdng fufly into account the nonlinearity Effect
of the oscillator
nonlinearity
for the pufling
of the oscillator
on the asymmetry
(c)
figure is presented Fig,
admittance.
of the pulling
1.
Simulation
is presented.
and I.
INTRODUCTION
dBTo
ABL+ Oscillator
frequency
represented been
either
with
the
approximately the oscillator
taking
voltage
into
derivation
the nonlinear
relation
between Pulling
The pulling
the transferred perturbation figure
admittance
particular
calculated
by
admittance
in
fully
admittance.
of the oscillator
certain
has so far
AYL=AGL+ABL and
pulling
Y~o = G~o + BTO.
into The
From
cases are also pre-
FREQUENCY VARIATION WITH THE LOAD
The oscillation
condition
any load perturbation
at the oscillator-output
is represented
AGL% Au=
dGTo dBTo _ dGTO dBTo —. —— — dV do do “ dV
Y~ is the oscillator
nonlinear
dG ABL. ~
CHANGES
—
plane without
dGTo dBTo dGTo dBTo —— .— .— dv da dw dV
by
YTO=YT+YO=O where
(4) and (5)
range
has been de-
sented. 11.
(1)
output
admittance
and YO is
If the oscillations output
ing frequency
dG ABL. ~ exist with
and RF voltage
of A YL in the
Au and A~ as the correspond-
changes,
the oscillation
condition
Av=
do
Au
+
dY -#
AGL. ~
Av=o.
—
(2)
real and imaginary
dGTo
AGL + ~Aco+
III. From
parts
LOAD VARIATION
(4)
tion of AG by suitably line between
the oscillator
of the exact derivation Manuscript received November 18, 198 1; revised February 9, 1982. The authors are with Laboratoire d%fectronique des Microndes, E.R.A. au C.N.R.S., U.E.R. des Sciences, 123 rue Albert-Thomas, 87060 Limoges, Cedex, France.
0018-9480/82/0700-1
selecting
the reference figure,
For the purposes
we simulate
the load
the transmission-line
For any value of b’ = I?l, the transferred
IEEE
load perturba-
plane in the output
and the perturbation.
perturbation by AG (Fig. 1(b)) with variable between O and ~ /2.
109$00.7501982
load perturbation
by a nonreactive
of the pulling
(7)
SIMULATION
Fig. 1 it can be noted that any reactive
of value jA B cm be represented
dGTo ~Av=O
dGTo _dBTo “
dGTo dBTo dV”dudm”dV
(1) and (2)
into
dV dBTo
“~o
(3) separating
dGTo . _dBTo
dGTo — dBr-o —. dV do
by
Y~o+AYL+z. From
a load perturbation
plane and writing
can be represented
(6)
and
the load admittance. oscillator
(5)
where
admittance
taking
dBTO ~v=o
~Au+
is often
[2]. We present
of the oscillator
of the oscillator for
figure
the oscillator
of the pulling
the asymmetry
figures
changes
[1] or has been
account
behavior
the nonlinearity
the load
neglecting
for a small-load
account
rived.
with
figure.
by
RF
plane
here an exact
and
variation
by its pulling
calculated
variation
of load variation.
range
load admittance
length
1
YL at
1110
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-30, NO. 7, JULY 1982
the oscillator
output
plane can be written
f-y=~
41
YO+ AC + jYOtan 19 YL=YO.
(8) YO+j(YO
+ AG)tanO
=ys(l+tan*e)
+ jYO.
~ is the VSWR
output
line given by
of
(1–r)tane
1
2
I
(9)
1 + ~2 tan2t7
the perturbating
1
admittance
in
s-
the
0 l“
~=l+x The change in the load admittance, AYL( = AGL + jABL) following
----
3
0 1+ Sztan*o where
..
5
as
-? I
(lo)
Y. “ at the oscillator
= YL – YO, can now
output
be represented
I
-3
plane, by the
-4
two equations:
AC =Yo(s–l)(l–stan%) L
LO
2
Maximum
Substituting
“
OF THE PULLING
(11) and (12) into
can be written
(12)
1+ Sztan%
DERIVATION
Substituting
Fig
frequency
various
=Y (1–s*)tano
AB
IV.
(11)
l+sztan%
A@
FIGURE
(6) the frequency
values
nonlinearity
of load VSWR
constant
2s
S for
a.
(13)
+l(2a2+s+l)–2a(
F~
=YKS–l
l+a2)
(19)
~2-~G+l
Ati
as
as a function
oscillator
(17) and (18) into
10
variation
variation
of
and
dBTo AU=YO.
~G
Au =YK1–
dV TO dBTo
S w(2~2+~+l)+2@+~2)
20
dGTo dBTo
2s
az+a
(20)
Fa-+1+1
dV”dm–dm”dV where . (S-1)
(1-~tan20)
dGTo/dV
1+ ~2 tan28
K=
dGTo . (1-
dV
–Yo.
dGTo
dGTo — dBTo
dBTo
The
1+ ~2tan26’
maximum
found
maximum
frequency
derivation
as a function
giving
This figure
total
it
brings
around
us (15)
dBTo/dV a z dGTo/dV where a is a nonlinear following two solutions values of Au,
constant
shown
the
of VSWR
values
with increasing
the
same
way,
I
s
the
pulling
range,
as a function
Fig.
2 represents
S of the perturbating noted
that
of Aa I
admittance for a=
O the
more and more asym-
(7) the RF
voltage
variation
AV can be
of
dGTo/da
and asymmetrical
(17)
of
values of a.
from
to be a function
the pulling
and (20) results, is that
and becomes
b= (a2+l-a
rigorously
frequency,
of a. It maybe
and AU2:
~=tm_,
be
oscillator.
asymmetry
oscillator
range is symmetrical
metrical
represents
a of the oscillator.
and AU2 as a function
In
constant of the oscillator. (15) has the of d which correspond to the extreme
can now
(22)
of the above(19)
the free-running
pulling
(16)
which
evidence
(10) for various
with
AOM = Au, – Aaz
negative-resistance
feature
into
nonlinearity -1=0
variation
is the relation of a nonlinear
An interesting
(14)
5’2tan26+2~atan0
(21)
dBTo
ACOM=YOK~(~-;}
of 13can be
from
dAu = o dO
dGTo
to be
(13) Now
dBTo
dV”dco-da”dV
S2)tan6’
dV”dcodco”dV
calculated
dGTo
The unknown by dividing
dBTo\dm
in the pulling constants
range.
a and K of relation
(22) can be found
(19) and (20) and arranging
and
AO, _kY2+l(S+l)+a (18)
Auz –
a(s–1)–(m(s+l)
(S-1)
(23)
IEEE TRANSACTIONS
ON MICROWAVE
THEORY
AND
TECHNIQUES,
VOL.
MTT-30,
NO.
~, JULY
1111
1982
or
TABLE I with
without :,pproximmt ion
/-(1+::2 ‘)+’’(’-2)=0
~
’24)
f
Y“, K,
“(”M
where ‘ represents the reflection coefficient of perturbed load equals S – 1/S+ 1. Ati, and Auz being opposite in sign, Ati ~/AQ2 is always negative. .. Substituting
W1
;) Y{, K
,,
#&42T{s.
)
~
we have x always positive. From (24) (26)
From (26), a can now be calculated for the three possible cases as
~
%xt
Qext
<l,
(30)
==. P
=
$
%
and a and K are defined by (16) and (21), respectively. It maybe noted that
“=-A forl<x<S,
(yl
Knowing AuM, S, and a AuMo can be determined. A number of Particular cases can be derived from the general solution presented above. Table I lists expressions for the pulling figure with and without approximations where
~=’
forx=l,
{s.;)
.X
+yy}
““M=A”MO
Im(l-x)+dl+x)=o.
+~*
From (22) and (29)
(25)
for~<x
{s-;) ( Q
with / dGTo l~=~=o
‘GTo T=O
—
(27)
““b with’
where
2<<1 1
~=r(l+x)
(l-x)
s–1
‘–3=4
“
.,
Knowing a, S, Aw,, and AW2, the value of K can be determined from (19) or (20). A frequency deviation AWMOcan be defined [3] for the condition when the real part of the perturbed load admittance at the oscillator plane is Y., i.e., AGL = O. This corresponds to Wm — tin where m and n are the common points on G = 1 and the perturbed load impedance locus (Fig. (1c)). From ( 11), this condition gives
and AOM, for example, for the approximation be written as given in [2]
ACKNOWLEDGMENT
REFERENCES [1]
F.
L.
Gmm
From (13), the total frequency deviation AWMOis now given by
dG~o /dw = Q can
Thanks are due to an unknown reviewer, for his comments on the determination of a and K.
(28)
(–) ‘wM0=2y0K ‘&l“
{–)S+l
p]
G.
Warner
S. Hobson, using
pp.
191-193,
G.
84-86.
S. Hobson,
“Measurement
frequency
S. Hobson,
G.
“Loaded
Microwave Jaarnal, vol.
oscillators:
tors
pp.
and
pulting
of external or frequency
Q-factor 13,
measurements
pp. 46-52,
Q-factor locking;
of
Feb.
on
1970.
microwave
Electron. I.dt.,
oscillavol.
8,
ch.
7,
1973. The
Gunn
Effect. ..
Oxford:
Clarendon
Press,
1974,