Smith chart formulation of performance
characterisation for a microwave transistor
F.Guneg
B.A.Cetiner
Indexing terms: Microwave trunsistors, Circuit theory, Stability analy.yis
Abstract: A scattering parameter theory of the
performance characterisation for a bilateral
transistor is developed, where mismatching at the
input port V, is considered as a degree of
freedom, and its combination with noise and gain
is mapped as circles in the r,-plane. Stability
analysis is based on the unconditionally stable
working area (USWA) concept, and all possible
USWA configurations are determined by the
necessary and sufficient conditions. For each
USWA configuration, the constrained maximum
stable gain GTmal and its termination couple (Ts,
r,) are expressed as functions of the input
VSWR Vi, noise figure F and the scattering S and
noise N parameter vectors. Furthermore, the
possible incompatible cases for the (F, V,, GT,nox)
triplets are determined by their necessary and
sufficient conditions. A computer program based
on this formulation is developed, and crossrelations among the (F, Vi, GT,nux)triplets have
been utilised in obtaining the performance
contours at an operating frequency and bias
condition.
1
Fig. 1
Blach-box representation of small-signal microivave transistor
The transducer power gain of a linear two-port circuit can be expressed as a function of the termination
couple, which are the source and load reflection coefficients (rs,
rL),for a fixed S as follows [3, 41:
- (1 - )rLI2)
pz1i2
(1 - lrs12)
(2)
11 - s22rL~211rznrsi2
where PL and PA,sare the load power and the available
power from the source, respectively.
Input VSWR of the two-port circuit is a function of
the source and load reflection coefficients via the input
reflection coefficient Tm,as follows [3, 41:
Description of the work
A small-signal microwave transistor can be characterised by a two-port circuit (Fig. l), where S and N are
scattering and noise parameter vectors, respectively, at
an operating frequency and bias condition modelled
through neural networks [l, 21. They are given as
[SIt= [ I ~ l ~ I ~ 1 l I ~ ~ 2
[NIt = [ F o p , P o p t I P o p t R N I
where Pri is the reflected power from the input port.
Note that the input and output reflection coefficients
Ti, and rout
are the linear fractional transformations of
rLand Ts, respectively, as below:
1 ~ ~ 2 1 ~ 2 l l ~ Z l l ~ 2 2 I ~ 2 2 1
(1)
S and N vectors are fixed at an operating frequency of
a bias condition. Using linear two-port circuit theory, a
scattering parameter theory of performance characterisation for a microwave transistor is developed, where
the three performance measure functions are taken into
consideration as the basic functions; the transducer
power gain (GT),noise figure (4and input VSWR ( V J .
The noise figure of the two-port circuit is defined as the
ratio of signal-to-noise power ratios available at the
input and output ports. The N vector describes the
dependence of the noise figure F on the source reflection coefficient Ts only [ 3 , 41:
0IEE, 1998
IEE Proceedings online no. 19982389
Paper first received 20th October 1997 and in revised form 12th March
1998
The authors are with the Yildiz Technical University, Electronic &
Communication Engineering Department, 80750, Be? ikta? , Istanbul,
Turkey
IE'E Proc.-Circuits Devices Syst
,
Vol. 145, No. 6, Decemh8r 1998
The problem we focus on here is the same as in previous work [SI, but with a different chaxacterisation
419-
method of the two-port circuit for both deterministic
and noise inputs; this is the scattering parameter
method. The problem is a mathematically constrained
maximisation to find the maximum value, the function
GATSr, Tsi, rLr,rLi)and corresponding terminations
(rs,TL), subject to @, = - F(Tsr, Tsi) = 0 and Q2 =
Virrq- Vi(Tsr,Tsi, rL,,rLi)
= 0; where Frfqand Vireqare
the required noise and input VSWR, respectively, and
Ts and rLare given by Ts = rsr+ jTSi,rL= rL,.+ jTLj.
The method used previously [5] is utilised in the formation of the theory, which is based on gathering the
geometrical representations of the stability and the
three performance measure functions, gain, input
VSWR, and noise given by eqns. 2, 3 and 5 in the rin
plane. They are then analysed to obtain the constrained
maximum stable gain GTmu\- and its terminations (rS,
rL).The work can be summarised in the following.
(i) Stability of a linear active two-port circuit can be
identified under 12 possible cases in the Tin plane.
These are determined by their necessary and sufficient
conditions so that, at a given frequency, the S vector is
sufficient for the USWA to be obtained straightforwardly in the Tin plane. Five of these possible cases are
absolutely unstable, two of them are absolutely stable,
and the rest are unconditionally stable cases.
(ii) For each USWA configuration in the Tin plane, the
gain GT circle family constrained by the input VSWR is
formed and its achievable maximum stable value G,,
is determined.
(iii) In the Ts plane, the position of the gain (input
VSWR) circle with respect to the noise circle is controlled via the passive load termination rL,and all the
resulting situations are mapped into the Tin plane as the
solution regions. These solution regions are bounded
by the two circles (TI, T2) that stay completely inside
the unit Smith chart (USC) and take the place of each
other.
(iv) The resulting possible configurations formed by the
stability constrained gain and T I , T2 circles carrying the
requirements (Freq, Vireq)are analysed in the rinplane,
so that the formulation for the maximum stable gain
GTn,ouconstrained by noise and input VSWR, and its
termination couples (Ts, TL) can be obtained.
(v) A computer program, based on the derived analytical expressions, is developed to produce the numerical
output in the form of Frrq, VireqrGTmus, Ts, rL,at the
given frequency and bias condition which enable the
performance contours of the transistor to be obtained
and utilised in the sub-network growth of the microwave multiports.
2
Stability
Here we aim to determine all possible (USWA) configurations in the Tin plane by their necessary and suffi-
cient conditions. The unconditional stability and the
mappings between the input (output) and load (source)
planes are presented briefly; these are used later in the
analysis. The essentials of the stability analysis for the
USWA configurations are also presented with an
example.
2. I
Unconditional stability
A linear two-port circuit is said to be absolutely or
unconditionally stable at a given frequency if there is
no passive source (lrsl<l)
and passive load (lrlL<l)
combination that can cause /Till(t 1 and Irourl
2 1. It is
well known that the combination of the Rollet condition
and any of the following auxiliary conditions is necessary and sufficient for unconditional stability [6]:
+ Is1112
= 1+
B1 = 1
B 2
-
IS2212
Js1112 1S22I2
la1 = I S 1 1 S 2 2
-
S12S2lI
- la12
-
>0
>0
<1
(7a)
(7b)
( 7c)
(7 4
2.2 Mappings
Since Tin = f(rL),
rout
= g(Ts) given by eqn. 4 and their
inverses rL=f-l(rin),
rS= g-l(rour)are the linear fractional transformations, they map circles into circles
between the related port and termination planes. From
eqn. 4 rL=&l(rjn)
and Ts = g-l(rour)are well defined,
provided that S12S21
# 0 as follows:
Using the transformations g , f l , g-I, four different
stability circles can be defined; two of which take place
in the termination planes and two are in the port
planes, as shown in Tables 1 and 2.
2.3 USWA configurations in Fin plane
Here the input and conjugate source stability circles are
formed in the rinplane. The intersection area of their
stable regions is determined as the USWA, together
with its necessary and sufficient conditions. However,
for the unconditional stability geometries given in
Figs. 2a and b only the input stability circle is sufficient to determine the boundary of the USWA, since
all the conditions given by eqns. 6-7e are symmetrical
with respect to 1 and 2 indices. Since the magnitude of
the reflection from either port is always greater than
Table 1: Stability circles in termination planes
Original circle
Mapped circle
Centre phasor
rouA
gl(lrouJ
= 1): source stability circle
is=IS12S211/11Sl,121A121
(S;, - S22A*)/(lS;12-lA12)
C, = (Slz- SllA")/lSz2I2IA12)
r, = lS12S2,1/llSzz12
- 1A121
=I
ir,"i= I
F1(Ir,"l= 1): load stability circle
Radius
C,=
Table 2:Stability circles in port planes
Original circle
Mapped circle
ir,i
g(lT,I
=
I
irLi
= I
420
=
1): output stability circle
A lTLl = 1): input stability circle
Centre phasor
CO, = (S,,
C,
=
-
Radius
S:lA)/(l-
(Sll- S;2A)/(I
-
1S1,l2)
rout= lS12S211/ll-lSll121
ISz2l2)
rin =
ISlZSz11/11
- lS22121
IEE Pro<.-Circuits Devices S y s t . , Vol. 145, No. 6 , December 1998
unity, K < 0 cases given by Figs. 5b-7b correspond to
absolute instability.
r;,
rLplane
plane
t
rLplane
rinplane
I
a
I
rLplane
Tin plane
0
rinplane
rL plane
t
t
Fig.4
Possible stability cuses and U S W A configurations in
Crescent USWA configurations
c, plane
a 0 < K < I , IS,,/AJ
7 1, /S221
< 1
b 0 < K c 1, JSI,/Al
< 1,
> 1
Fig.2 Possible Stability cases and U S W A conjigurations in
Full moon USWA configurations
U K > I, 0 < 1
S,,I2> IS12S211,
lSll/Al
> 1
b K > I , 1 - P I I>~lS12&1l,ISIIIAI < 1
c, plane
~
rinplane
rLplane
t
t
rLplane
rinpLone
I
I
0
rin plane
r;,
r,
plane
t
plane
Fig.5
rL
Ye
I
I
b
Possible stability cases and U S W A configurations in
a Spindle USWA configuration
0 < K < 1, lSlI/AI< I, 15‘221 < I
Fig.3
Possible stability cases and U S W A configurations in
Full moon USWA configurations
U K > 1, 0 < 1 b K > 1, 1 - IS221 < 0, <lSIS12S211,
iI/Al< 1ISl1iAl < 1
b Absolute unstable case
-1 < K < 0, JSIliAl> 1, IS221 > 1
c, plane
1
2.3.2 Geometry features:
To determine the USWA with its necessary and sufficient conditions for each stability geometry, two main
group features are applied, as described below.
2.3.1 Position of stability reference point
(SRP): The origin of the
plane
= 0) is chosen
as a reference point in determining the USWA. Using
transformations given by eqns. 4 and 8, it can therefore
be concluded that the origin is a stable point for both
the rLand rSterminations if and only if
cn
1%)
< 1 and
c, plane
(cn
IS221
<1
(9)
Otherwise, it is an unstable point for either termination
depending on which reflection is greater than unity.
IEE Proc-Circuits Devices Syst., Vol. 145,No. 6, December 1998
(i) 1
4 > 1 is geometrically equivalent to the non-intersection of any stability circle given in Table 1 or
Table 2 with the boundary of the USC [6].
(ii) Otherwise if (14 s 1) both the conjugate source and
input stability circles can be shown to intersect at the
same two points with the boundary of the USC
through which all the constrained gain circles also pass
~71.
(iii) Using the expressions of rin, C,, CS, rS from Tables
1 and 2, the following equality relation between the
ratios of ICS12/rS2and ICin12/ri2can be obtained:
PSI2 -
r;
ICin12 - 1 + (1 - l S 2 2 I 2 ) (1S11l2 - M2)
r,”,
IS21 s 1 2 12
(10)
42 1
rLplane
rinplane
I
only if, either of the following two conditions are satisfied:
4
Otherwise, both the stability circles do not simultaneously include the origin:
r,
Tin plane
t
plane
t
Examples
2.4
The seven sets of S parameters have been used to give
the numerical results of K, l,Sll/A1, ISI2Szll,
and 1 for the main stability cases, which are given in Table 3 .
Gain GTconstrained by input VSWR Vjin T i n
plane
3
Our aim here is to obtain the variation of the gain constrained by the input VSWR (Vi) in the Tin plane for
each USWA configuration. First, the gain function
given by eqn. 2 is generally investigated in the Tin plane
taking Vi as a parameter. The variation of the constained gain is then obtained specifically for each
USWA configuration, considering its necessary and
sufficient conditions given in Section 2 together with
general function properties.
rLptone
+
Tin plane
t
3. I Constrained gain circle family in rin
plane
The gain function G7(rS, r,) given by eqn. 2 can be
expressed in the form of a circle family in the Tin plane
a
5,
TL plane
plane
for a transistor with the given magnitude of the source
impedance mismatch factor Ipil, as follows [8]:
p.
lrzn- c,12 =
(13)
where C, and rg are the centre phasors and the radii of
the circle family, respectively:
Fig.7 Possihle strrhility cases unci US W A conjigurations in
Ahsolufe unsttrhle cases
c,,plane
K < 1. 0 c /Sz2f- 1 < IS,&l. ISll/Al> 1
h K < -1. 1 - ISz21- 0, ISI,/Al > 1
N
Comparing the magnitude of the centre phase with the
radius for each stability circle using eqn. 10 it can be
concluded that both the conjugate source and input
stability circles simultaneously include the origin if, and
Table 3: Worked example for stability geometry parameters
IS111
Vll
VlZ
IS211
V21
w
V22
K
SlJAI
IS12S21
l-IS2212
0.5
0.99 (FM Fig. 2a)
0.15
0.75 (FM Fig. 2b)
1.8
0.75 (FM Fig. 3a)
4.39
-4.29 (FM Fig. 3b)
1.44
-0.004(crs Fig. 4b)
27.02 -38.7 (Sp Fig. 5b)
0.15
0.75 (crs Fig. 4a)
FM = full moon USWA, Crs = crescent USWA, Sp = spindle USWA, q j j ( i , j =1,2)is measured in degrees
0.5
0.2
0.75
3.6
1.59
3.8
1.05
422
0
IS121
20
-60
-43
-88
-75
20
0.25
0.05
0.3
1.33
1.02
3.86
0.05
180
120
70
85
-62
-93
120
2
3
6
3.3
1.41
7
3
0
40
90
-44
-65
-56
40
0.1
0.5
0.5
2.3
1.02
6.3
0.5
0
-50
60
49
43
-73
-50
7.5
2.53
1.34
1.26
0.5
-0.8
0.34
5
0.83
0.35
0.68
0.79
1.22
1.56
IEE Proc.-Circuits Devices Syst., Vol. I45, No. 6 , Decerinher 1998
where Y~ can be rearranged in a well-known canonical
quadratic form, as follows:
where
(15)
Using eqns. 13-15, the following properties for the
function R(GT,) can be deduced [SI:
( a ) the input stability circle (C,,, Y,,) is the zero gain (G,
= 0) circle.
(b) the USC is the infinite gain (G,
circle.
(c) the conjugate source stability circle coincides with
the achievable maximum stable gain circle for the conditional stability cases, whose value is G,,, = 2cKjS2,1
'K-b
-+
S12L
(cl) the extremum gain values realisable with the passive
terminations can be given for the K > 1 stability cases
as follows:
(
G T ~
1,1
w
z
c K
7"
rt7,
l,lcjd
,11 7 ,L
=
f
7
K2 -
cglrLuJ
=
,1L1 I ,
1
1
c,,
1 - T,, ( K
JF?i)
(16)
where the upper (lower) sign corresponds to the unconditional (conditional) stability case.
Considering the above general properties, together
with the necessary and sufficient conditions for each
USWA configuration, the gain versus USWA configuration can be shown to be classified into five main
groups:
(i) K > 1 unconditional stability cases.
(ii) K > 1 conditional stability cases.
(iii) 0 < K < 1, ~ S l l / A ~ </S221<1
l,
conditional stability
case.
(iv) 0 < K < 1, ISIIIAl>l, ISJ21<1 conditional stability
case.
(v) 0 < K < 1, IS,,lAl<l, IS221>1conditional stability
case.
(Details of all these can be found elsewhere [SI). As an
example, the variations for the constrained gain is
given in Fig. 8 for the crescent USWA configuration
with the stability parameter 0 < K < 1, /SI1lAl > 1, IS221
< 1.
4
Control of positions of input VSWR circle with
respect to noise circle in rs plane from rinplane
Here we determine the possible solution regions, which
can be defined as the geometric places of the points in
the Tin plane satisfying the possible (Freq, Vjreq,G,) triplets. First, variations of all the performance measure
functions GATs, rL),V,(T,, r,) and F(Ts) given by
eqns. 2, 3 and 5 , respectively, are investigated for a
fixed rL termination in the rs plane. As a result, the
G, = const., Vi = const. and F = const. circles are
obtained in the same plane. Considering the relative
positions of these circles, we have the solution regions
plane.
in the rN1
IEE Proc.-Circuits Devices Swt., Vol. 145, No. 6, December 1998
< G U S C ~CO
Fig.8
'GUSC
-CO
Guin vciriutionfor spindle USWA
4. I
Noise figure, input VSWR and gain in Ts
plane
4.7.7 Noise(F) circles: F(Ts) given by eqn. 5 can be
expressed in the form of a circle family in the Ts plane
as follows [SI:
IFS - CI,
= rn
(17a)
where
and
Centre phasors C, occur on the ropr
phasor and radii
increase as the noise increases (Fig. 9).
4.2
Y,
Input VSWR (Vi) and gain (GT)circles
lpjl given by eqn. 3 can be expressed in the form of a
circle family in the
rs plane as follows:
where
(18b)
The lpil = const. circle given by eqns. 18a and b can be
shown to belong to a GT = const. circle, whose value
can be found by rearranging the gain function given by
eqn. 2 in terms of lpil and rL,rin,
as follows:
423
and
Thus, only the input VSWR and noise circles are sufficient to be taken into account in the rs plane for the
performance analysis (Fig. 9).
rsplone
t
The upper and lower signs correspond to the T I and T2
circles, respectively. From eqn. 22, the centre phasors
of the TI and T, circles are seen to lie on the same line,
which is To;[ phasor, and the following inequalities can
be proved to be satisfied for all admissible values of
Iropt1
IPil
3and N PI:
(Ttl - T t d 2
2 (let11- Ictal)
2
e N ( N + 1 - iroptI2)
ip,i2 (1 - hi2)
x (1 - lropt1)2
(1 + lroptl)2
20
(23)
r,
plone
r,
r,
pione
plone
I
I
rs plone
rsplone
I
i
Fig.9
Noise and VSWR circles in
USC
input VSWR
_ _ _ _ noise
gain
rs-plane
~
I region L
I region 5
~
4.3
a
Possible solution regions
i-in plane
t
In Fig. 10 the possible relative positions of the Vi and F
circles are given in the Ts plane, where the position of
the V , circle is changed via the load termination rLand
the position of the F circle is fixed. To obtain the solution sets in the Tin plane for each possible case of
Fig. 10, the equations of the internal and external tangential cases should be presented first as they are transition stages between the non-touching and intersection
positions:
IC, - C,12 = (T, f
(20)
where (+) and (-) signs correspond to the external and
internal cases, respectively.
By replacing the expressions for (C,, r,) and ( C y , rv)
from eqns. 17b and 18b, respectively, into eqn. 20 and
rearranging, the tangential positions in the Ts plane
can be mapped into a couple of circles (T1, T2) in the
Tin plane (Fig. lob). Thus, the rLcontrol parameters,
which ensure the tangential positions of the F and Vi
circles, can be found to correspond to the Tin values on
the Tl and T2 circles in the Tin plane. The T1 and T2
circles can be represented as follows:
lrin - G,,,I = T t l , 2
(21)
where C,l,2centre phasors and rt1,2radii are given as
b
Fig.10
Relative positions of the V, circle w.r.t the F circles, and T, and
T, circles and the possible solution regions
a Relative positions of the V, circle w.r.t the F circles
b T I and T, circles and the possible solution regions
~
~
USC
noise or input VSWR
The geometrical result of eqn. 23 is that the T, circle is
always situated inside the T I circle without touching,
and both these circles always take place inside the
USC. In eqn. 23 in the rinplane the equality can be
obtained and the T , and T2 circles become a single circle, which is the conjugate of the noise circle, when the
input port is matched to the source impedance (Ipil =
0). In this case, using eqn. 22
yr*
U1
424
= U2 = 1
et1 =
ct, = l +optN
~
IEE Proc.-Circuits Devices Syst., Vol. 145, No. 6 , December 1998
At the same time, r, and C,, are equal to zero and rii ,
respectively, which means that the required input
VSWR circle becomes a single point in the Ts plane.
The two-port circuit is matched for noise if Ts = rupt.
Using eqn. 22
(25)
and Y, = 0, C, = rupt,
which means that the noise circle
becomes a single point in the rs plane.
Five different regions in the Tin plane bounded by the
T I and T2 circles as given in Fig. lob, cause five different interactions of the required input VSWR circle with
the noise circle in the Ts plane. In Fig. lob no rinvalue
in regions 1 or 5 will cause a mutual point of both the
noise and input VSWR circles in the Ts plane. In the
otherwords there are no Ts, rLvalues which satisty the
noise and input VSWR requirements. Thus, regions 1
and 5 will not give the solution for the maximum gain
constrained by noise and input VSWR. The rinvalues
chosen in regions 2 and 4 result in external and internal
tangential positions of the circles, and Tin values in
region 3 cause intersection of the same circles in the Ts
plane. Thus, regions 2, 3 and 4 are the solution regions.
Performance (Freq,Vireq,GTmax)
triplets and
applications
5
Using the performance characterisation theory developed above, for a given transistor with the required
noise and input VSWR, the basic performance measure
functions given by eqns. 2, 3 and 5 can be represented
in a geometrical configuration in the Tm plane. This
configuration may be called the design configuration,
which consists of the USWA, the T , and T2 circles and
the constrained gain circle family. In this design configuration, all possible (Freq,VLreq,
GT) triplets of a microwave transistor must take place in the intersection
areas of the possible solution regions 2, 3 and 4 with
the USWA of the transistor. Therefore, no physical
solution can exist for the constrained gain when the
USWA takes place completely in region 1 or 5. The
resultant principle incompatible geometries are given in
Fig. 11 whose basic equalities and/or inequalities can
be utilised to obtain the existence conditions of the
physical solution.
Determination processes for the triplets with GTmax
among the infinite number of possible (Freq, Vlreq,GT)
triplets are based on geometrical analyses [11, which
depend on the type of the design configuration. All the
design configurations may be considered under five different cases with respect to gain versus USWA configuration. The five different analyses resulting from these
five cases are utilised in the output boxes of the performance characterisation program, whose flow chart is
given in Fig. 12. The ralvalue corresponding to GTTnanax
can be obtained straightforwardly from the TI, plane.
Using T L = f-l(rln),
we can then get its rLvalue. We
can obtain the Ts value by considering the positions of
Freqand Kreq circles in the Ts plane, resulting from the
mapping process of the design configuration.
Two examples are presented for a microwave transistor. This is a GaAs transistor known as NE72089A,
IEE Proc.-Circuits Devices Syst., Vol. J45,Nu. 6 , December 1998
whose S and N parameters are given at the bias condition VDs= 3 V , IDS = 10 mA and operating frequency
4GHz. According to Section 2, these S parameters
resulted in the crescent type of USWA configuration
identified by the parameters K = 0.609, ISll/Al = 1.55,
lS221= 0.59. Thus, the performance analysis in the output box numbered 5 in Fig. 12 should be followed,
which gives the design configuration in Fig. 13 with the
requirement of Vfreq= 1, Freq = 1.25dB. In Fig. 13,
according to eqn. 24, the required matched input
resulted in a point for the input VSWR circle in the Ts
plane and the coincidence of the Tl and T2 circles.
There are infinite possible (1, 1.25dB, GT) triplets in
the intersection areas of the solution regions 2 and 4
with the crescent USWA. From the gain analysis for
the crescent USWA, the triplet with the maximum constrained gain GTmaxwhich is the performance triplet,
must therefore be the one corresponding to the external
tangential point of the T I = T2 circle with the gain circle. The corresponding ri, phasor of this performance
triplet can be found immediately from the external tangential geometry of the Tl = T2 and GTmaxcircles, as
follows:
where yg, Cg and r t l , Ctl are given in eqns. 14 and 22,
respectively. Using rL= J'(ri,),
the load termination rL
can then be obtained. Since r;,,takes place in the solution region 2, the source termination should be the
external tangential point of the Vjreqand Freqcircles in
\-
I
/
I
a
'"i'""'
I
b
Fig. 11 Principle incompatible geometries
-USC
~
_ conjugate
_
~source or input stability
___ T , and T,
a Crescent USWA takes place completely in region 1 in either case (impossible
solution)
h Full moon USWA takes place completely in region (impossible solution)
425
start
PI,
Is
OCKCI, 11 / A
1522 1-=1
c o n d i t i o n a l l y s t a b i l i t y anolysis (5)
I
Fig. 12
I
Muinflow churl of performance churucteriscttion progrumnnie
I
cn p l o n e
I
rin p l o n e
-
-10
-5
G1 :-IO
G2: - 5
0
6.99 dB
10.00dB
11.76 d B
G1= 0
Gq 6.99 d B
G5 = 10.00 d B
G6 ~ 1 1 . 7 6d B
1 L .13L d B
Fig. 13
Design geometry f o r example I
1 = USC
2 = input stability
GI ._.Gb = gain
3 = conjugate source stability
S-parameters:
Noise parameters:
SI, Mag. = 0.76, Ang. = -95
r Mag. = 0.65 Ang. = 70
S2, Mag. = 2.34, Ang. = 90
F z i Mag. = 1 dB
R,,/50 Mag. = 0.42 P
S Mag. = 0.1 I , Ang. = 26
SI2Mag. = 0.59. Ang. = -66
&bility parameters: K = 0.609, ISIIIAI = 1.554. IS221 = 0.59
Noise and input VSWR requirements: Free = 1.25 dB, V,,',, = 1.00
Performance triplet: (F"',, Vscq, G T , ~ =~(1.25dB,
~ ~ ) 1.00. 6.63dB)
the
rSplane, which can be given as
where Cfl,rfl and C,, rv are given by eqns. 17b and 18b,
respectively.
426
Gm= 13.623 dB
Fig. 14
Design geometry for exumple 2
1 = USC
2 = input stability
G, ... G, = gain
3 = conjugate source stability
S-parameters:
Noise parameters:
SI, Mag. = 0.76. Ang. = -95
r Mag. = 0.65 Ang. = 70
S 2 , Mag. = 2.34. Ang. = 90
Fzt, Mag. = l d B
R,/50 Mag. = 0.42P
S Mag. = 0.11, Ang. = 26
Si; Mag. = 0.59, Ang. = -66
Stability parameters: K = 0.609, lSI1/AI
= 1.554, lS221= 0.59
Noise and input VSWR requirements: F,, = 1.85dB, V,,e<= 2.00
Performance triplet: (Frry,Vzrc,,,,G,,,,,,) = f1.85dB. 2.00, 15.623dB)
In example 2, the S parameters remain the same as in
example 1, and so the performance analyses should be
the same but with the different requirements of Freq =
1.84dB and Vireq= 2, which cause the T, and T2 circles
to be separate with a different value of the maximum
achievable stable gain Gnl. For this case, Gn,lLI.y
is equal
IEE Proc.-Circuits Devices Sysl.. Vol. 145. No. 6 , Decernhrr 1998
Table 4: Part of numerical output of performance characterisation for transistor NE72089A at V, = 3 V, IDS 10
mA, f = 4GHz
1L -
12 -
m
U
10'
1.15
1.378
0.5118
-0.6933
0.0561
0.6432
1.00
1.20
4.957
0.4221
-0.5560
0.0327
0.6531
1.00
1.25
6.637
0.3565
-0.4473
0.0130
0.6626*
1.00
1.30
7.707
0.3061
-0.3580
-0.0040
0.6717
1.00
1.35
8.481
0.2659
-0.2828
-0.0188
0.6805
0.6889
1.00
1.40
9.081
0.2329
-0.2182
-0.0320
1.00
1.45
9.569
0.2054
-0.161 8
-0.0438
0.6970
1.00
1.50
9.978
0.1819
-0.1121
-0.0545
0.7047
1.00
1.55
10.331
0.1618
-0.0679
-0.0641
0.7121
1.00
1.60
10.640
0.1442
-0.028 1
-0.0729
0.7193
; 8-
1.00
1.65
10.915
0.1287
0.0079
-0.0810
0.7261
6L-
1.00
1.70
11.164
0.1150
0.0406
-0.0884
0.7327
1.00
1.75
11.390
0.1028
0.0705
-0.0953
0.7390
1.00
1.80
11.598
0.0918
0.0980
-0.1016
0.7451
1.00
1.85
11.790
0.0818
0.1233
-0.1075
0.7510
1.00
1.90
11.970
0.0728
0.1468
-0,1129
0.7566
1.00
1.95
12.138
0.0646
0.1686
-0.1180
0.7620
1.00
2.00
12.297
0.0570
0.1890
-0.1228
0.7672
1.00
2.05
12.447
0.0501
0.2080
-0.1272
0.7723
1.00
2.10
12.590
0.0437
0.2258
-0.131 4
0.7771
1.00
2.15
12.726
0.0377
0.2425
-0.1353
0.7818
1.00
2.20
12.857
0.0322
0.2582
-0.1390
0.7863
2.00
1.05
10.298
0.0759
0.0258
0.1535
0.6141
2.00
1.10
10.961
0.0517
0.1062
0.1272
0.6152
E
(3
20,
6
1.00
.
I
I
I
t
1
Conclusions
The significance of this work can be discussed in the
following two aspects:
Contributions of the theory developed using the scattering parameters to the circuit theory.
Importance of the problem focused on here as far as
applications are concerned;
As far as the developed theory is concerned, a rigorous
mathematical analysis is used to formulate the maximum stable constrained gain GTmr,\directly in terms of
S and N parameters and the requirements Freq and
Vireq.The following concepts and approaches are the
key points of the theory, which can be considered as
developments in linear circuit theory.
Stability analysis is based on the concept of the
USWA, and all possible USWA configurations are
determined by their necessary and sufficient conditions
which are expressed in terms of S parameters.
Mismatching at the input port is considered as a degree
of freedom in the circuit and by its combination with
noise and gain is mapped as circles in the Tinplane.
Performance analysis is classified with respect to the
type of USWA configuration, and for each configuration the gain, noise and VSWR performance of the
transistor are analysed using a rigorous geometrical
approach.
Possible incompatible (F, Vi, G,) triplets are also determined by their necessary and sufficient conditions.
As far as the application is concerned, the importance of this work is given in detail elsewhere [l]. We
briefly summarise: the conditional formulation of the
maximum gain subject to noise and input VSWR can
be used on data sheets of microwave transistors to provide a higher level tool for the design of MMIC amplifiers with which to overview all possible designs.
However, a tool is generated for use in MMIC design
by accompanying the computer program based on this
IEE Proc.-Circuits Devices Syst., Vol. 145, Nu. 6 , Decenzher I998
2.00
1.15
11.424
0.0350
0.1627
0.1078
0.6162
2.00
1.20
11.791
0.0222
0.2067
0.0921
0.6172
2.00
1.25
12.098
0.0118
0.2428
0.0789
0.6182
2.00
1.30
12.365
0.0032
0.2732
0.0673
0.6191
2.00
1.35
12.603
-0.0042
0.2995
0.0571
0.6201
2.00
1.40
12.819
-0.0106
0.3225
0.0479
0.6210
2.00
1.45
13.018
-0.0163
0.3429
0.0395
0.6219
2.00
1.50
13.203
-0.0213
0.3612
0.0319
0.6227
2.00
1.55
13.376
-0.0258
0.3777
0.0249
0.6235
2.00
1.60
13.623
-0.1774
0.3662
0.0390
0.6380
2.00
1.65
13.623
-0.2429
0.3665
0.0426
0.6430
2.00
1.70
13.623
-0.2887
0.3706
0.0441
0.6462
2.00
1.75
13.623
-0.3247
0.3761
0.0447
0.6487
2.00
1.80
13.623
-0.3545
0.3822
0.0449
0.6506
2.00
1.85
13.623
-0.3798
0.3886
0.0448
0.6523t
2.00
1.90
13.623
-0.4019
0.3950
0.0445
0.6537
2.00
1.95
13.623
-0.4214
0.4013
0.0441
0.6549
2.00
2.00
13.623
-0.4388
0.4076
0.0437
0.6559
2.00
2.05
13.623
-0.4545
0.4137
0.0432
0.6568
2.00
2.10
13.623
-0.4687
0.4196
0.0427
0.6576
2.00
2.15
13.623
-0.4816
0.4254
0.0422
0.6584
* worked example 1, t worked example 2
formulation with a multi-dimensional signal-noise neural network model [9]. As the constrained gain contours cannot be improved, the performance of low
noise transistors with the corresponding terminations
will facilitate rapid design. It will also inform designers
of some incompatibility among gain, noise and input
VSWR for narrow and medium band amplifier design.
427
Since this theory is capable of supplying not only the
performance (Freq, Vireq,GTmax)triplets but also the
possible Freq, Vireq,rT) triplets, it can thus be used to
provide accurate performance data for the optimisation
process of broadband amplifiers.
7
References
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2 GuNES, F., TORPI, H., and GURGEN, F.: ‘A multidimensional signal-noise neural network model’, ZEE Proc. Circuits
Devices, Syst., 1998, 145, (2), pp. 111-117
3 GUPTA, K.C., GARG, R., and CHADRA, R.: ‘Computer-aided
design of microwave circuits’ (Artech Hous, Dedham, Massachusetts, 1981)
428
4 VENDELIN, G.D., PAVIO, A.M., and ROHDE, U.L.: ‘Microwave circuit design using linear and nonlinear techniques’ (John
Wjley & Sons, 1990)
5 GUNES, F., GUNES, M., and FIDAN, M.: ‘Performance characterisation of a microwave transistor’, ZEE Proc. Circuits
Devices, Syst., 1994, 141, (5), pp. 337-344
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IEEE Trans., 1992, MTT-40, (12), pp. 2303-2311
7 COLLIN, E.R.: ‘Foundations for microwave engineering’
(McGraw-Hill, 1992), pp. 71 3-798
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the Yildiz Technical University, 1995
9 C m E S , F., TORPI, H., and CETINER, B.A.: ‘Neural network
approach for the active device characterisation’. Proceedings of
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Budapest, Hungary, 1997, pp. 440445
IEE Proc.-Circuits Devices Sysl., Vol. 145. No. 6, Decemher 1998