Impact of Correlated RF Noise on SiGe HBT
Noise Parameters and LNA Design Implications
by
Xiaojia Jia
A thesis submitted to the Graduate Faculty of
Auburn University
in partial fulfillment of the
requirements for the Degree of
Master of Science
Auburn, Alabama
December 14, 2013
Keywords: SiGe HBT, RF noise, LNA design, technology scaling
Copyright 2013 by Xiaojia Jia
Approved by
Guofu Niu, Alumni Professor of Electrical and Computer Engineering
Fa Foster Dai, Professor of Electrical and Computer Engineering
Stuart Wentworth, Associate Professor of Electrical and Computer Engineering
Abstract
This work presents analytical models of SiGe HBT and LNA noise parameters accounting
for high frequency noise correlation. The models are verified using measurement data and circuit
simulation. The impact of noise correlation is shown to be a strong function of base resistance r
b
which acts as both a noise source and an impedance. Correlation and r
b
as impedance have oppo
site e?ects on minimum noise figure NF
min
, which explains why a widely used NF
min
model that
neglects correlation and r
b
as impedance agreed with measurements. The agreement, however,
does not hold for noise matching source resistance R
opt
, an important parameter for LNAs. With
correlation, noise matching condition is better met for impedance matched LNAs.
ii
Acknowledgments
Foremost, I would like to express my sincere gratitude to my advisor Dr. Guofu Niu, for his
patience, assistance and guidance throughout the process of writing this thesis. Without his help
this work would not haven been possible. I appreciate his immense knowledge and skills that
helped me in all the time of research and this thesis.
Besides my advisor, I would like to thank the rest of my thesis committee members, Dr.
Fa Dai, and Dr. Stuart Wentworth, for their encouragement, insightful comments, and helpful
suggestions.
I would like to extend my sincere gratitude to former and current labmates in our SiGe HBT
research group. I would like to thank Zhen Li for his patient instructions and constant help that
are important to my academic growth. Many thanks also go to Ruocan Wang, Jingshan Wang,
WeiChung Shih, Jingyi Wang, Zhenyu Wang, Pengyu Li, and Rongchen Ma, for the days and
nights we were working together, and for all the fun we have had in the last two years. Also, I
would like to thank my friends and everyone else who helped me in completion of this work.
Last but not the least, I would like to thank my parents, for their selfless support, under
standing and love that protect me and give me courage to face any di?culties throughout my
life.
iii
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 SiGe HBT fundamental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Noise in Semiconductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Thermal Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Terminal Current Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1 SPICE Noise Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.2 Correlation Noise Model . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Analytical Derivation of Noise Parameters . . . . . . . . . . . . . . . . . . . . . . . 9
3.1 Noise Parameters of LNA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Device Noise Parameters and Relations to LNA Noise Parameters . . . . . . . . 16
4 Impact of Correlation on Device Noise Parameters . . . . . . . . . . . . . . . . . . 17
4.1 Analytical Model Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2 Correlation?s Impact on R
Device
opt
. . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.3 Correlation?s Impact on F
Device
min
. . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.4 Correlation?s Impact on X
Device
opt
. . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.5 Correlation?s Impact on R
Device
n
. . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.6 Two Roles of r
b
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
iv
4.7 Correlation and r
b
Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5 LNA Design Implication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6 Technology Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
A Derivation of Intrinsic Noise Sources? Contributions to i
out
: i
c
out
, i
b
out
, i
rb
out
, and i
Rs
out
. . . 39
A.0.1 i
c
out
: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
A.0.2 i
b
out
: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
A.0.3 i
rb
out
: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
A.0.4 i
Rs
out
: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
B Matlab Code for Noise Parameters of Device Calculation . . . . . . . . . . . . . . . 43
C Matlab Code for Noise Parameters of Matched LNA . . . . . . . . . . . . . . . . . 48
v
List of Figures
1.1 Simplified schematic of the LNA consisting of a single SiGe HBT. . . . . . . . . . . 2
1.2 Energy band diagrams of a gradedbase SiGe HBT and an Si BJT [25][27]. . . . . . 3
2.1 RF noise sources of a transistor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 1D bipolar transistor with collectorbase space charge region (CB SCR) e?ect [8]. . 7
2.3 PSDofi
b
andi
c
usingcorrelationmodelandSPICEmodelversusFrequency,J
C
=0.414
mA/?m
2
, ?
n
=0.651E12 sec, f
T
=50GHz. . . . . . . . . . . . . . . . . . . . . . . . 8
3.1 Simplified small signal equivalent circuit of LNA. . . . . . . . . . . . . . . . . . . . 10
3.2 Equivalent circuit of simplified LNA without power source. . . . . . . . . . . . . . . 14
3.3 Equivalent small signal circuit of simplified LNA without noise sources. . . . . . . . 15
4.1 Analytical,simulatedandmeasurednoiseparametersofSiGeHBTwithA
E
=0.8?20?3
?m
2
versus J
C
at V
CE
=3.3 V, f=5 GHz. . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2 Device noise resistance R
n
normalized by 50?. . . . . . . . . . . . . . . . . . . . . 21
4.3 Analytical and measured noise parameters of SiGe HBT device versus frequency at
V
CE
=3.3 V, I
c
=3.47 mA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.4 r
b
?s impact on NF
min
of device versus frequency at V
CE
=3.3 V, I
c
=3.47 mA. . . . . 23
4.5 r
b
?s impact on R
opt
of device versus frequency at V
CE
=3.3 V, I
c
=3.47 mA. . . . . . . 24
vi
4.6 r
b
?s impact on X
opt
of device versus frequency at V
CE
=3.3 V, I
c
=3.47 mA. . . . . . 24
4.7 (a)NF
Device
min
versus J
C
at 5 GHz; (b)R
Device
opt
versus J
C
at 5 GHz. . . . . . . . . . . . 25
4.8 F
min
1 versus frequency at J
C
=6.62 mA/?m
2
. . . . . . . . . . . . . . . . . . . . . 26
5.1 R
LNA
opt
,X
LNA
opt
,NF
LNA
min
,andNF
LNA
ofimpedancematchedLNAversusL
E
atJ
C
=0.158
mA/?m
2
, f=5 GHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.2 NF
min
, NF, R
opt
, X
opt
, and corresponding L
E
and I
C
of LNAs using SPICE model
and correlation model versus J
C
at 5GHz. . . . . . . . . . . . . . . . . . . . . . . . 29
6.1 f
T
, NF
Device
min
, and r
b
?g
m
versus J
C
of three technology devices. . . . . . . . . . . . 32
6.2 NF
Device
min
and R
Device
opt
? L
E
versus r
b
?g
m
of three technology devices. . . . . . . . . 33
6.3 In LNA design, L
E
, I
C
, and J
C
versus lithography node. . . . . . . . . . . . . . . . 33
A.1 Simplified small signal equivalent circuit of LNA . . . . . . . . . . . . . . . . . . . 39
A.2 Simplified small signal equivalent circuit of LNA with intrinsic current noise source i
c
. 40
A.3 Simplified small signal equivalent circuit of LNA with intrinsic current noise source i
b
. 40
A.4 Simplified small signal equivalent circuit of LNA with thermal noise source v
rb
. . . . 41
A.5 Simplified small signal equivalent circuit of LNA with thermal noise source v
Rs
. . . . 42
vii
List of Tables
4.1 S
ibib
?, S
icic
?, S
icib
?, and S
ibic
? of Correlation model and SPICE model . . . . . . . . . 17
6.1 Main Features of Three Technologies . . . . . . . . . . . . . . . . . . . . . . . . . 31
viii
Chapter 1
Introduction
1.1 Motivation
High frequency noise correlation has been shown to be important for accurate modelling of
transistor noise parameters, in particular, minimum noise figure NF
min
is smaller with correlation
[1][2][3][4][5][6][7][8][9]. Recently the impact of noise correlation on LNA design was investi
gated using ADS simulation [10]. However, accurate analytical equations that express transistor
and LNA noise parameters in terms of small signal equivalent circuit parameters like cuto? fre
quency f
T
, transconductance g
m
, and base resistance r
b
have yet to be developed using correlated
noise model.
The most used analytical transistor noise parameter expressions [11][12][13] are derived
with varying degree of approximations using the socalled SPICE noise model, which models
base and collector current noises as uncorrelated shot noises. These equations have been widely
used in both device technology development [14][15][16] and circuit design [12][17]. Transistor
noise parameter expressions were derived in [1] and [7] by introducing correlation. However, the
base current noise was still assumed to be shot like, while more recent experimental extractions
[6][18][19] and impedance field based analysis of modern SiGe HBTs [20] show a strong and
clear frequency dependence in base current noise. The 2qI
B
shot noise component is a result
of emitter minority hole velocity fluctuations, and is not correlated to the collector current noise
[21][22]. It is the frequency dependent component of base current noise that correlates with
collector current noise.
This work aims to develop new expressions using a recent transistor correlation noise model
[8][23] that accounts for frequency dependent base current noise. As the primary application is
1
lownoise amplifier (LNA) design, derivation is made directly on a LNA as shown in Fig. 1.1.
Transistor results are then obtained as a special case by setting the matching inductances to zero.
This allows an easier inspection of the relationship between transistor and LNA noise parameters,
as well as how it is a?ected by noise correlation.
In particular, the new expressions will be used to investigate how correlation a?ects tran
sistor noise parameters and the nice transistor property of being able to approximately achieve
simultaneous LNA noise and impedance matching, which was obtained using the SPICE noise
model [12][24]. The Z
s
in Fig. 1.1 is in general equal to R
s
, 50?, with a reactance X
s
=0. Tran
sistor size can be optimized for noise matching source resistance R
opt
=50?, or real part noise
matching. Emitter and base inductors L
e
and L
b
can then be adjusted for R
in
=50?, X
in
=0, or
real part and imaginary part impedance matching. The resulting noise matching source reactance
X
opt
?X
s
= 0, or imaginary part noise matching. A logical question is if such property still holds
with correlation, and if so, how transistor size and LNA noise figure are a?ected.
??
??
???
?????
Figure 1.1: Simplified schematic of the LNA consisting of a single SiGe HBT.
1.2 SiGe HBT fundamental
SiGe heterojuction bipolar transistor (HBT) technology is extensively used to develop elec
tronics for low noise operation due to its excellent analog and RF performance. The SiGe HBT
2
technology is the first practical bandgap engineering device realized in silicon and can be inte
grated with the modern CMOS technology. As a result of introducing the graded Ge layer into
the base of bipolar junction transistor (BJT), SiGe HBT technology has better performance than
traditional Si BJT in DC, RF, and noise performance [25][13]. To illustrate the di?erence be
tween the SiGe HBT and the Si BJT, Fig. 1.2 shows the energyband diagrams for both the the
SiGe HBT and the Si BJT biased identically in forwardactive mode. The Ge profile linearly
increases from zero near emitterbase (EB) junction to some maximum value near collectorbase
(CB) junction, and then rapidly ramps down to zero. This gradedGe results in an extra drift
field in the neutral base. The induced drift field accelerates minority carrier transportation, thus
it minimizes transit time and increases cuto? frequency [26]. Because of this advantage and
low noise performance, SiGe HBTs have been widely used in commercial and military wireless
communication applications.
Figure 1.2: Energy band diagrams of a gradedbase SiGe HBT and an Si BJT [25][27].
3
1.3 Thesis Structure
The work is organized as follows. Chapter 1 gives the motivation of this work as well as an
overview of SiGe HBT technology. Chapter 2 presents thermal noise and intrinsic current noises
in semiconductor. Derivations of noise parameters are made in Chapter 3. Small signal equivalent
circuit analysis is used instead of linear noise two port analysis to better track how each noise
source or equivalent circuit parameter, e.g. r
b
, enters the final noise parameter expressions. While
r
b
as a thermal noise source is well appreciated, its role as an impedance is often neglected, e.g. in
[12][13][28]. In Chapter 4, we verify the model equations with measurement and simulation, and
examine how correlation a?ects transistor noise parameters, as well as the two roles of r
b
. The
role of r
b
as impedance is shown to be important. Chapter 5 discusses LNA design implications,
followed by technology scaling discussions in Chapter 6. Chapter 7 concludes the work.
4
Chapter 2
Noise in Semiconductor
The operation of semiconductor devices is based on free carriers transportation [29]. From
the equivalent circuit and compact modeling stand point, the velocity fluctuation caused by major
ity carrier thermal motion can be expressed by the thermal noises of resistance, and the velocity
fluctuation caused by minority carrier thermal motion can be equivalently expressed by the in
trinsic terminal current noises [26][30]. Fig. 2.1 shows the thermal noise sources of resistances
at base, emitter and collector terminals respectively as well as the terminal current noises of base
and collector.
Figure 2.1: RF noise sources of a transistor.
5
2.1 Thermal Noise
As described by the Nyquist theorem, the power spectral density (PSD) of thermal noise
voltage source of a resistance R is usually given by
S
vr,vr
? = 4KTR,
(2.1)
and the PSD of thermal noise current source is
S
ir,ir
? =
4KT
R
,
(2.2)
where K is the Boltzmann constant and T is the standardized noise source temperature 290 K.
2.2 Terminal Current Noise
2.2.1 SPICE Noise Model
The base and collector terminal noises are defined as shot noise with the PSDs,
8
>
>
>
<
>
>
>
:
S
i
c
,i
c
?
= 2qI
C
,
S
i
b
,i
b
?
= 2qI
B
,
S
i
c
,i
b
?
= 0,
(2.3)
where I
B
and I
C
are base and collector DC currents.
2.2.2 Correlation Noise Model
Considering various noise physics mechanisms, a correlation noise model was developed in
[8]. Fig.2.2illustratesthemodelincludingterminalcurrentnoisesduetominoritycarriervelocity
fluctuation and due to additional noises from the collectorbase space charge region (CB SCR)
transport e?ect. i
b1
is the base noise current resulting from minority hole velocity fluctuation at
emitter, with a PSD of 2qI
B
. i
c1
is the collector noise current resulting from minority electron
velocity fluctuation at base, with a PSD of 2qI
C
. Both of them are independent with frequency
6
Figure 2.2: 1D bipolar transistor with collectorbase space charge region (CB SCR) e?ect [8].
and uncorrelated with each other. i
c1
noise current transfer through the CB SCR becomes i
c2
and
leads to an extra base current noise i
b2
=i
c1
i
c2
.
Through a series of derivation, the final PSDs of correlation model are [8]:
8
>
>
>
<
>
>
>
:
S
i
c2
,i
c2
?
= 2qI
C
,
S
i
b2
,i
b2
?
= 2qI
B
+2qI
C
!
2
?
2
n
,
S
i
c2
,i
b2
?
= j2qI
C
!?
n
,
(2.4)
where !=2?f, ?
n
is noise transit time which approximately equals to collector transit time
[23][8].
Fig. 2.3 shows calculated PSDs of i
b
and i
c
versus frequency using correlation and SPICE
model. It illustrates that PSDs of i
b
and i
c
using SPICE model are constant over frequency, while
correlation leads to S
ibib
? increase with frequency, and imaginary part of S
icib
? decrease with
frequency.
7
0 10 20 30
1
0.5
0
0.5
1
freq (GHz)
Sicic*
0 10 20 30
0.5
1
1.5
2
2.5
x 10
23
freq (GHz)
Sibib*
0 10 20 30
1
0.5
0
0.5
1
freq (GHz)
real Sicib*
0 10 20 30
1.5
1
0.5
0
x 10
22
freq (GHz)
imag Sicib*
Correlation
SPICE
2qI
B
2qI
B
+2qI
C
(wt
n
)
2
j2qI
C
(wt
n
)
Figure 2.3: PSD of i
b
and i
c
using correlation model and SPICE model versus Frequency,
J
C
=0.414 mA/?m
2
, ?
n
=0.651E12 sec, f
T
=50GHz.
8
Chapter 3
Analytical Derivation of Noise Parameters
To obtain insight into device and LNA noise performance for design and optimization, ana
lytical expressions of noise parameters are required. We develope analytical expressions of noise
parameters for the LNA in Fig. 1.1. Z
s
is source impedance that consists of resistance R
s
and
reactance X
s
. In general, R
s
=50 ? and X
s
=0 ?. However, to examine how close noise match
ing is to impedance matching, we need to include X
s
in Z
s
to find optimal noise reactance X
opt
.
For general applicability, arbitrary L
e
and L
b
are used. Setting L
b
=0 and L
e
=0 leads to noise
parameters of the transistor.
3.1 Noise Parameters of LNA
Fig. 3.1 shows a small signal equivalent circuit of Fig. 1.1. Basecollector capacitance C
bc
and r
?
between base and emitter are neglected for simplicity. This circuit includes two main kinds
of RF noise sources of bipolar transistor pointed in the previous chapter, e.g. the terminal resis
tance thermal noise and the intrinsic terminal current noise. v
Rs
and v
rb
are thermal noise voltage
sources due to R
s
and r
b
, respectively. i
b
and i
c
are intrinsic terminal current noise sources due to
base current and collector current, respectively. C
be
is capacitance between base and emitter. g
m
is transconductance. i
out
is output noise current due to all noise sources: v
Rs
, v
rb
, i
b
, and i
c
.
LNA noise factor, F
LNA
, is given by:
F
LNA
= 1+
Noise output due to LNA
Noise output due to source
(3.1)
The LNA has noise sources including the thermal noise source v
rb
due to r
b
, and the terminal
noise currents i
b
and i
c
. The power source has a noise source v
Rs
whose source impedance is Z
s
.
9
?? ?
??
???
??
????
??
?? ??
???
? ?
??
????????
Figure 3.1: Simplified small signal equivalent circuit of LNA.
The noise currents i
b
and i
c
are assumed to be correlated with each other, and no correlation is a
special condition when i
b
i
?
c
and i
c
i
?
b
terms are zero. The thermal noise v
b
is independent to i
b
and
i
c
. Therefore, (3.1) is rewritten as:
F
LNA
= 1+
D
i
ic
out
+i
ib
out
,
i
ic
out
+i
ib
out
?
E
+
?
i
rb
out
,i
rb?
out
?
?
i
Rs
out
,i
Rs?
out
? ,
(3.2)
? = 1!
2
C
be
(L
b
+L
e
) +j!C
be
(Z
s
+r
b
) +j!g
m
L
e
, (3.3)
i
c
out
= V
c
be
? g
m
+i
c
= j!g
m
i
c
L
e
?
1
?
+i
c
,
(3.4)
i
b
out
= V
b
be
? g
m
= g
m
i
b
[(Z
s
+r
b
) +j!(L
b
+L
e
)] ?
1
?
,
(3.5)
i
rb
out
= V
rb
be
? g
m
= g
m
v
rb
?
1
?
,
(3.6)
i
Rs
out
= V
Rs
be
? g
m
= g
m
v
Rs
?
1
?
,
(3.7)
where i
ic
out
, i
ib
out
, i
rb
out
, and i
Rs
out
are output noise currents due to i
c
, i
b
, r
b
(v
rb
), and R
s
(v
Rs
), respec
tively, and can be obtained using circuit analysis. The detailed analysis is discussed in Appendix
A. Noise figure NF relates to noise factor F by NF=10logF.
10
The output noise power produced by i
c
and i
b
is:
?
i
ic
out
+i
ib
out
,
i
ic
out
+i
ib
out
?
?
= Dhic,ic
?
i(!C
be
)
2
Z
s
+r
b

2
+Dhib,ib
?
ig
2
m
?
(R
s
+r
b
)
2
+ (X
s
+
1
!C
be
)
2
Dhic,ib
?
ig
m
!C
be
?
(Z
s
+r
b
)
?
j
!C
be
[j(Z
s
+r
b
)]
+Dhib,ic
?
ig
m
!C
be
?
(Z
s
+r
b
) +
j
!C
be
[j(Z
s
+r
b
)]
(3.8)
where
D =
1
??
?
=
1
(!C
be
)
2
Z
s
+r
b

2
+2!
T
L
e
(R
s
+r
b
) +!
2
T
L
2
e
(3.9)
where !
T
=g
m
/C
be
=2?f
T
, f
T
is cuto? frequency.
hic,ic
?
i=S
icic
??f, hib,ib
?
i= S
ibib
??f, hic,ib
?
i=S
icib
??f, and hib,ic
?
i= S
ibic
??f. ?f is a
very narrow bandwidth, in which the noise spectral component have a mean square value [13].
S
icic
?, S
ibib
?, S
icib
?, and S
ibic
? are PSDs of i
b
and i
c
. Substituting hic,ic
?
i, hib,ib
?
i, hic,ib
?
i, and
hib,ic
?
i into (3.8):
?
i
ic
out
+i
ib
out
,
i
ic
out
+i
ib
out
?
?
= DS
icic
??f(!C
be
)
2
Z
s
+r
b

2
+DS
ibib
??fg
2
m
?
(R
s
+r
b
)
2
+ (X
s
+
1
!C
be
)
2
D2g
m
?f<(S
icib
?)(R
s
+r
b
)
+D2g
m
?f=(S
icib
?)
?
!C
be
Z
s
+r
b

2
+X
s
?
,
(3.10)
where < stands for real part, and = stands for imaginary part.
The output noise power produced by r
b
is:
?
i
ib
out
,i
ib?
out
?
= g
2
m
D4kTr
b
?f, (3.11)
where k is Boltzmann constant, and T is temperature, which is assumed to be equal to standard
ized noise source temperature 290 K here for simplicity.
The output noise power produced by R
s
is:
D
i
R
s
out
,i
R
s
?
out
E
= g
2
m
D4kTR
s
?f. (3.12)
11
Substituting (3.10), (3.11), and (3.12) into (3.2), we get noise figure of an LNA:
F
LNA
=1+
r
b
R
s
+
S
icic
?
4kTR
s
?
!
!
T
?
2
"
?
!
T
!g
m
`
?
2
+ (R
s
+r
b
)
2
#
+
S
ibib
?
4kTR
s
?
(R
s
+r
b
)
2
+`
2
?
<(S
icib
?)
2g
m
kTR
s
(R
s
+r
b
)
+
=(S
icib
?)
2kTR
s
?
!
!
T
??
(R
s
+r
b
)
2
`
?
!
T
!g
m
`
?
,
(3.13)
where `=X
s
+!(L
b
+L
e
). In both SPICE model and correlation model, <(S
icib
?)=0, thus we set
<(S
icib
?)=0 in below.
To find out the optimum X
s
, X
LNA
opt
to minimize the noise figure, we solve @F
LNA
/@X
s
= 0:
@F
LNA
@X
s
=
S
icic
?
2kTR
s
?
!
!
T
?
2
?
!
T
!g
m
`
?
+
S
ibib
?
2kTR
s
`
=(S
icib
?)
skTR
s
?
!
!
T
??
!
T
!g
m
2`
?
= 0,
(3.14)
`
"
S
ibib
? +S
icic
?
?
!
!
T
?
2
+2=(S
ibib
?)
?
!
!
T
?
#
+S
icic
?
?
!
!
T
g
m
?
=(S
icib
?)
g
m
= 0, (3.15)
where !
T
=g
m
/C
be
. Substituting `=X
s
+!(L
b
+L
e
) in (3.15), we obtain the expression of X
LNA
opt
:
X
LNA
opt
=
1
N
!
T
g
m
!
h
S
icic
? +=(S
icib
?)
?
!
T
!
?i
! (L
b
+L
e
), (3.16)
where
N = S
icic
? +S
ibib
?
?
!
T
!
?
2
+2=(S
icib
?)
?
!
T
!
?
. (3.17)
To find the optimum R
s
that minimizes F
LNA
, R
LNA
opt
, we solve @F
LNA
/@R
s
= 0. The result
is:
R
LNA
opt
=
p
A
N
, (3.18)
12
A = r
2
b
N
2
+
?
!
T
!
?
2
4kTr
b
N
 {z }
v
rb
contribution
+
1
g
2
m
?
!
T
!
?
4
S
icic
?S
ibib
? =(S
icib
?)
2
,
(3.19)
Substituting (3.16) and (3.18) into (3.13) leads to the minimum noise factor of LNA, F
LNA
min
:
F
LNA
min
=1+
r
b
R
LNA
opt
{z}
v
rb
contribution
+
R
LNA
opt
+r
b
2
4kTR
LNA
opt
?
!
!
T
?
2
N
+
1
4kTR
LNA
opt
1
g
2
m
?
!
T
!
?
2
S
icic
?S
ibib
? =(S
icib
?)
2
N
,
(3.20)
where the r
b
/R
LNA
opt
term is due to v
rb
which represents the e?ect of r
b
as a noise source. The
remaining terms of (3.20) are due to i
c
, i
b
, and v
Rs
, and depend on r
b
because r
b
a?ects how i
c
,
i
b
, and v
Rs
are transferred to i
out
. Here r
b
?s e?ect is manifested as an impedance element. While
r
b
as a noise source is generally understood to be important, r
b
as an impedance element is much
more important in a?ecting F
min
, as detailed below in Chapter 4.
Noise resistance R
LNA
n
could be obtained the following equation which is from linear noisy
twoport theory [13][31]:
R
n
=
S
v
a
,v
?
a
4KT
,
(3.21)
where S
v
a
,v
?
a
is chain representation equivalent input noise voltage [26][31], and could be calcu
lated by:
S
v
a
,v
?
a
=
?
i
out
,i
?
out
?
Rn
Y
21

2
,
(3.22)
where Y
21
is the forward transfer admittance with output short circuit, and
?
i
out
,i
?
out
?
Rn
is the total
output noise power without power source, as shown in Fig. 3.2:
?
i
out
,i
?
out
?
Rn
=
?
(i
ic
out,Rn
+i
ib
out,Rn
),(i
ic
out,Rn
+i
ib
out,Rn
)
?
?
Rn
+
?
i
r
b
out,Rn
,i
r
b
?
out,Rn
?
.
(3.23)
We find out i
ic
out,Rn
, i
ib
out,Rn
,i
r
b
out,Rn
using equivalent circuit in Fig. 3.2:
?
Rn
= 1!
2
C
be
(L
b
+L
e
) +j!C
be
r
b
+j!g
m
L
e
, (3.24)
13
??
?
??
?
????
??
? ??
????
Figure 3.2: Equivalent circuit of simplified LNA without power source.
i
c
out,Rn
=
i
c
?
Rn
?
1!
2
C
be
(L
e
+L
b
) +j!C
be
r
b
?
, (3.25)
i
b
out,Rn
=
g
m
i
b
?
Rn
[r
b
+j!(L
b
+L
e
)], (3.26)
i
rb
out,Rn
=
g
m
v
rb
?
Rn
. (3.27)
The output noise power produced by i
c
and i
b
is:
?
(i
ic
out
+i
ib
out
),(i
ic
out
+i
ib?
out
)
?
Rn
= D
Rn
hic,ic
?
i
h
1!
2
C
be
(L
e
+L
b
)
2
+!
2
C
2
be
r
2
b
i
+D
Rn
hib,ib
?
ig
2
m
?
r
2
b
+!
2
(L
e
+L
b
)
2
?
2D
Rn
?f<(S
icib
?)g
m
r
b
+2D
Rn
?f=(S
icib
?)g
m
?
!C
be
r
2
b
+! (L
e
+L
b
)
1!
2
C
be
(L
e
+L
b
)
?
,
(3.28)
where D
Rn
is:
D
Rn
=
1
?
1!
2
C
be
(L
e
+L
b
)
?
2
+!
2
(C
be
r
b
+g
m
L
e
)
2
. (3.29)
The output noise power produced by r
b
is:
?
i
r
b
out
,i
r
b
?
out
?
= g
2
m
D
Rn
?4kTr
b
?f. (3.30)
14
?
??
?
?
??
?? ????
?
Figure 3.3: Equivalent small signal circuit of simplified LNA without noise sources.
Y
21
is calculated using Fig. 3.3:
Y
21
=
i
out
V
s
=
i
c
r
b
i
c
+V
be
+j!L
b
i
b
+j!(i
b
+i
c
)L
e
, (3.31)
1
Y
21
=
r
b
RF
+
1
g
m
+
j!
RF
(L
b
+L
e
) +j!L
e
, (3.32)
where
RF
=
g
m
j!C
be
=
!
T
j!
. Therefore,
1
Y
21

2
=
?
1
g
m
!
2
!
T
(L
b
+L
e
)
2
+
?
r
b
!
!
T
+!L
e
?
2
=
1
g
m
h
?
1!
2
C
be
(L
e
+L
b
)
?
2
+!
2
(C
be
r
b
+g
m
L
e
)
2
i
=
1
g
2
m
D
Rn
.
(3.33)
From (3.21)(3.33), we obtain the noise resistance of LNA, R
LNA
n
:
R
LNA
n
=r
b
+
r
2
b
+!
2
(L
e
+L
b
)
2
4KT
S
ibib
?
+
?
1!
2
C
be
(L
e
+L
b
)
?
2
+!
2
C
2
be
r
2
b
4KTg
2
m
S
icic
?
+
r
b
2KTg
m
<(S
icib
?)
+
!C
be
r
2
b
! (L
e
+L
b
)
1!
2
C
be
(L
e
+L
b
)
2KTg
m
=(S
icib
?).
(3.34)
15
Setting <(S
icib
?)=0 in (3.34), the noise resistance of LNA is:
R
LNA
n
=r
b
+
r
2
b
+!
2
(L
e
+L
b
)
2
4KT
S
ibib
?
+
?
1!
2
C
be
(L
e
+L
b
)
?
2
+!
2
C
2
be
r
2
b
4KTg
2
m
S
icic
?
+
!C
be
r
2
b
! (L
e
+L
b
)
1!
2
C
be
(L
e
+L
b
)
2KTg
m
=(S
icib
?).
(3.35)
3.2 Device Noise Parameters and Relations to LNA Noise Parameters
An inspection of (3.16), (3.18), and (3.20) shows that L
b
and L
e
only enter (3.16), expres
sion of X
LNA
opt
, and 3.35, expression of R
LNA
n
. Therefore, transistor R
opt
and F
min
are as the same
as LNA R
opt
and F
min
:
R
Device
opt
= R
LNA
opt
, (3.36)
F
Device
min
= F
LNA
min
. (3.37)
Setting L
b
and L
e
to zero in (3.16), we obtain transistor X
Device
opt
:
X
Device
opt
= X
LNA
opt
+! (L
b
+L
e
)
=
1
N
!
T
!g
m
h
S
icic
? +=(S
icib
?)
?
!
T
!
?i
.
(3.38)
Setting L
b
and L
e
zero in (3.35), we obtain noise resistance of device R
Device
n
:
R
Device
n
= r
b
+
r
2
b
4KT
S
ibib
? +
2
6
4
1
4KTg
2
m
+
?
!
!
T
?
2
r
2
b
4KT
3
7
5
S
icic
? +
?
!
!
T
?
r
2
b
2KT
=(S
icib
?). (3.39)
These relations between device and LNA noise parameters can also be obtained using two
port combining techniques [13][31], and hold with or without correlation.
16
Chapter 4
Impact of Correlation on Device Noise Parameters
In the SPICE model, i
b
and i
c
noise currents are described as shot noise of majority carriers
passing through the EB and CB junction, which relate to the corresponding DC currents by 2qI.
At higher frequency,i
b
increases with frequency and is correlated withi
c
, due to both base and CB
SCR transport, with the later dominant in modern HBTs [23]. For analytical analysis, we use the
correlation model in [8]. Table 4.1 shows S
ibib
?, S
icic
?, S
icib
?, and S
ibic
? of correlation model in [8]
and SPICE noise model. ?
n
is noise transit time which approximately equals to collector transit
time [23][8]. Substituting expressions ofS
ibib
?, S
icic
?, and=(S
icib
?) into (3.36), (3.37), (3.38), and
(3.39), we obtain device noise parameters with and without correlation, i.e. the SPICE model.
Table 4.1: S
ibib
?, S
icic
?, S
icib
?, and S
ibic
? of Correlation model and SPICE model
SPICE Correlation
S
icic
? 2qI
C
2qI
C
S
ibib
? 2qI
B
2qI
B
+2qI
C
(!?
n
)
2
S
icib
? 0 j2qI
C
(!?
n
)
S
ibic
? 0 j2qI
C
(!?
n
)
4.1 Analytical Model Verification
Fig. 4.1 compares analytical model, Agilent ADS simulation and measurement of F
Device
min
,
R
Device
opt
, and X
Device
opt
versus J
C
at 5 GHz. The device used is from a commercial SiGe HBT
BiCMOS technology, with an emitter area A
E
of 0.8?20?3 ?m
2
, a peak f
T
of 36 GHz, and a
peak f
max
of 65 GHz. Measurement data were obtained using a commercial system, and have
been deembedded. A modified HICUM model with correlation is used [8]. ?
n
is extracted by
fitting measured noise parameters [19]. Other compact model parameters are extracted by fitting
17
DC IV curves and Yparameters. For calculation, i
b
, i
c
, r
b
, g
m
, !
T
, and ?
n
are generated from
operation point information using the ddx operator in the VerilogA device model. The Matlab
codes for noise parameters calculation are shown in Appendix B.
0 0.5 1 1.5
0
5
10
15
Jc(mA/um
2
)
NF
Device min
(dB)
0 0.5 1 1.5
0
20
40
60
Jc(mA/um
2
)
R
Device opt
(ohm)
0 0.5 1 1.5
0
50
100
Jc(mA/um
2
)
X
Device opt
(ohm)
Analytical,correlation
Analytical,SPICE
Simulation,correlation
Simulation,SPICE
Measurement
(b)
(a)
(c)
Figure 4.1: Analytical, simulated and measured noise parameters of SiGe HBT with
A
E
=0.8?20?3 ?m
2
versus J
C
at V
CE
=3.3 V, f=5 GHz.
Analytical model agrees fairly well with simulation. Correlation model is much closer to
measured data than SPICE [10], especially at higher J
C
. Correlation leads to smaller NF
min
and
larger R
opt
as detailed below.
18
4.2 Correlation?s Impact on R
Device
opt
Since emitter length is first decided in LNA design for R
Device
opt
=50 ?, and NF
min
requires
R
opt
, we discuss the impact of correlation on R
Device
opt
first. In Fig. 4.1 (b), the R
Device
opt
with cor
relation is larger. To see if this can be generalized, we further simplify R
Device
opt
by substituting
expressions of PSDs from Table 4.1 into (3.36), and approximating g
m
with (qI
C
)/(kT). The N
in (3.17) can be rewritten as 2qI
c
M, with M being:
M =
8
>
<
>
:
M
Spice
= 1+
1
!
T
!
2
, SPICE
M
Cor
= 1+
1
!
T
!
2
+ (!
T
?
n
)
2
!
T
?
n
, Correlation
(4.1)
where =I
C
/I
B
, M
Spice
and M
Cor
are M for SPICE model and correlation model. Then,
R
Device
opt
=
v
u
u
u
u
u
t
r
2
b
+
?
!
T
!
?
2
2
6
6
4
2r
b
g
m
M
{z}
v
rb
contribution
+
1
M
2
1
g
2
m
?
!
T
!
?
2
3
7
7
5
.
(4.2)
(4.2) showsthattheonlydi?erencebetweenR
Device
opt
withandwithoutcorrelationisfactorM.
As f
T
<1/(2??
f
), and ?
n
R
Device
opt,Spice
, the second term r
b
/R
Device
opt
is smaller
with correlation. Since M
cor
R
Device
opt,cor
<
>
:
L
e
=
R
s
r
b
!
T
,
L
b
=
1
!
2
C
be
R
s
r
b
!
T
.
(5.1)
Then we will examine how correlation a?ects X
LNA
opt
, which is supposed to approximately be zero
using SPICE model [12][24].
Fig. 5.1 shows analytical R
LNA
opt
, X
LNA
opt
, NF
LNA
min
, and NF
LNA
of impedance matched LNA
versus L
E
at J
C
=0.158mA/?m
2
, f=5GHz. R
LNA
opt
, X
LNA
opt
, and NF
LNA
min
are plotted by substituting
(5.1) into (3.16), (3.18), and (3.20) that impedance matches at each L
E
. NF
LNA
are plotted
by setting R
s
=50? and X
s
=0 in (3.13). The markers on R
LNA
opt
curves are firstly decided at L
E
required for R
LNA
opt
=50?. Then X
LNA
opt
, NF
LNA
min
, and I
C
at corresponding L
E
are also marked.
At L
E
required for R
LNA
opt
=50 ?, X
LNA
opt
equals to 2.924 ? for correlation model and 
8.286 ? for SPICE model. The closeness between X
LNA
opt
and zero is better with correlation.
Even though X
LNA
opt
slightly deviates from zero, NF
LNA
is very close to NF
LNA
min
. Also, noise
27
20 40 60 80 100 120
0
50
100
150
200
X: 42.33
Y: 50.08
Emitter Length L
E
(um)
R
LNA opt
(ohm)
X: 64.46
Y: 50.12
20 40 60 80 100 120
20
15
10
5
0
5
X: 64.27
Y: 2.924
Emitter Length L
E
(um)
X
LNA opt
(ohm)
X: 42.09
Y: 8.286
20 40 60 80 100 120
1
1.5
2
2.5
3
X: 42.56
Y: 2.015
Emitter Length L
E
(um)
NF
LNA
& NF
LNA min
(dB)
X: 64.74
Y: 1.39
20 40 60 80 100 120
0
5
10
15
20
X: 64.95
Y: 8.202
Emitter Length L
E
(um)
I
c
(mA)
X: 42.32
Y: 5.344
Correlation
SPICE
Correlation
SPICE
NF
min,cor
NF
cor
NF
min,SPICE
NF
SPICE
X
opt
=0
SPICE
Correlation
(a) (b)
(c) (d)
Figure 5.1: R
LNA
opt
, X
LNA
opt
, NF
LNA
min
, and NF
LNA
of impedance matched LNA versus L
E
at
J
C
=0.158 mA/?m
2
, f=5 GHz
correlation provides smaller distance between NF
LNA
and NF
LNA
min
. In this case, correlation
leads to more closeness between noise and impedance matching.
A larger L
E
required using correlation model results in a higherI
C
: 8.202mA for correlation
model versus 5.344mA for SPICE model. To constrain power consumption, L
E
does not have
to be chosen exactly at R
LNA
opt
=50?, since slight deviation of X
LNA
opt
barely increases NF
LNA
min
, so
does R
LNA
opt
. One can choose a smaller L
E
like 50?m using correlation model for lower I
C
.
X
LNA
opt,Spice
=
!
T
!
g
m
1
!
T
!
2
h
1+
1
!
T
!
2
i, (5.2)
28
0.1 0.2 0.3
1
2
3
Jc(mA/um
2
)
NF
LNA min
(dB)
0.1 0.2 0.3
1
2
3
Jc(mA/um
2
)
NF
LNA
(dB)
0.1 0.2 0.3
46
48
50
52
Jc(mA/um
2
)
R
LNA opt
(ohm)
0.1 0.2 0.3
10
5
0
5
10
Jc(mA/um
2
)
X
LNA opt
(ohm)
Analytical,Correlation
Simulation,Correlation
Analytical,SPICE
Simulation,SPICE
0.1 0.2 0.3
20
40
60
80
Jc(mA/um
2
)
L
E
for R
opt
=50ohm (um)
0.1 0.2 0.3
0
5
10
15
Jc(mA/um
2
)
Ic for R
opt
=50ohm (mA)
(a) (b)
(d)(c)
(e) (f)
Figure 5.2: NF
min
, NF, R
opt
, X
opt
, and corresponding L
E
and I
C
of LNAs using SPICE model
and correlation model versus J
C
at 5GHz.
X
LNA
opt,cor
=
!
T
!
g
m
1
!
T
!
2
+ (!
T
?
n
)
2
(!
T
?
n
)
h
1+
1
!
T
!
2
+ (!
T
?
n
)
2
2(!
T
?
n
)
i. (5.3)
Then we repeat LNA design at di?erent J
C
(V
be
). Fig. 5.2 shows NF
LNA
min
, NF
LNA
, R
LNA
opt
,
and X
LNA
opt
, and corresponding L
E
and I
C
with and without correlation versus J
C
at 5GHz. The
simulation data are generated using a cascode LNA [10][34]. The analytical curves agree well
with simulation. In Fig. 5.2, X
LNA
opt
with correlation is relatively closer to zero. X
LNA
opt
without
correlation is negative, i.e. (5.2), while X
LNA
opt
with correlation is positive at small J
C
, which
agrees with analytical expressions, i.e. (5.3).
29
At every J
C
bias, L
E
is required to be rescaled. Although L
E
could be chosen by optimizer
in some simulator like Agilent ADS, it takes a plenty of time to generate one curve. By contrast,
the L
E
can be easily scaled for R
LNA
opt
=50? in calculation. Thus, analytical approach improves
the e?ciency of LNA design.
30
Chapter 6
Technology Scaling
To investigate e?ect of technology scaling, we compare three lithography nodes 0.5?m,
0.24?m, and 0.13?m. Parameters of the reference device are given in Table 6.1.
Table 6.1: Main Features of Three Technologies
Lithography Node (?m) 0.50 0.24 0.13
Peak f
T
(GHz) 36 60 212
Peak f
max
(GHz) 100 120 265
Working f (GHz) 6 10 35
Normalized r
b
*L
E
(??m) 412.9 341.4 210.8
Emitter area A
E
(?m) 0.8?20?3 0.24?20?1 0.12?12?1
With increasing frequency, the contribution of
1
in (4.2) and (4.3) is small [11]. (4.2) and
(4.3) are simplified by neglecting contribution of
1
except that in M:
R
Device
opt
= r
b
s
1+
2
r
b
g
m
M
?
!
T
!
?
2
, (6.1)
F
Device
min
= 1+r
b
g
m
M
?
!
!
T
?
2
2
4
1+
s
1+
2
r
b
g
m
M
?
!
T
!
?
2
3
5
, (6.2)
where term r
b
g
m
is independent of L
E
for a given J
C
, because r
b
/ 1/L
E
and g
m
/L
E
.
Fig. 6.1 shows f
T
, NF
Device
min
, and r
b
? g
m
versus J
C
of three technologies. Because of dif
ferent f
T
, for a fair comparision, the J
C
at which f
T
=f
T,peak
/2 in each technology is used. Since
working frequency f is chosen as f
T,peak
/6, !
T
/!=3 in (6.1) and (6.2).
Fig. 6.2 shows NF
Device
min
and R
Device
opt
? L
E
versus r
b
g
m
. The trends of NF
Device
min
are consistent
with r
b
g
m
: smaller r
b
g
m
leads to smaller NF
Device
min
. In (6.1), R
Device
opt
? L
E
is related with not only
31
10
3
10
2
10
1
10
0
10
1
10
2
0
100
200
X: 1.463
Y: 106.1
Jc(mA/um
2
)
f
T
(GHz)
X: 0.1643
Y: 30.95
X: 0.05966
Y: 18.07
0.5 um
0.24 um
0.13 um
10
3
10
2
10
1
10
0
10
1
10
2
1
2
3
4
5
6
Jc(mA/um
2
)
NF
Device min
(dB)
10
3
10
2
10
1
10
0
10
1
10
2
0
2
4
6
Jc(mA/um
2
)
r
b
?
g
m
dash: SPICE
solid: Correlation
Figure 6.1: f
T
, NF
Device
min
, and r
b
?g
m
versus J
C
of three technology devices.
r
b
g
m
, but also r
b
? L
E
. The 0.13?m technology has much smaller R
Device
opt
? L
E
than the others,
which will lead to smaller L
E
in LNA design.
Fig. 6.3 shows L
E
for noise matching and corresponding I
C
and J
C
versus lithography node
in LNA design. Although J
C
is highest at 0.13?m node, its smaller L
E
requested for R
LNA
opt
=50?
and smaller W
E
lead to less I
C
since I
C
=L
E
W
E
J
C
. Despite the increasing J
C
required to enable
higher frequency design, the smaller L
E
and W
E
required for noise matching help keeping power
consumption of LNA low.
32
0.2 0.4 0.6 0.8
1
1.5
2
2.5
r
b
? g
m
NF
Device min
(dB)
0.2 0.4 0.6 0.8
500
1000
1500
2000
2500
3000
r
b
? g
m
R
Device opt
L
E
(ohm*um)
SPICE
Correlation
0.5 um
0.13 um
0.5 um
0.13 um
Correlation
SPICE
0.24 um
0.24 um
Figure 6.2: NF
Device
min
and R
Device
opt
? L
E
versus r
b
?g
m
of three technology devices.
0 0.13 0.24 0.5
25
30
35
40
45
50
55
60
Lithography Node (um)
Emitter Length L
E
(um)
0 0.13 0.24 0.5
0
1
2
3
4
5
LNA I
C
(mA)
Lithography Node (um)
0
0.1
0.2
0.3
0.4
0.5
Jc(mA/um
2
)
Ic,SPICE
Ic,Correlation
Jc
Correlation
SPICE
Figure 6.3: In LNA design, L
E
, I
C
, and J
C
versus lithography node.
33
Chapter 7
Conclusion
The general analytical expressions of SiGe HBT and LNA noise parameters have been de
veloped using small signal circuit analysis and verified by simulation and measurement data.
The analytical expressions show that correlation leads to smaller NF
min
and larger R
opt
at a fixed
emitter size. This impact depends on base resistance r
b
, which plays more important role as an
impedance than as a noise source for NF
min
. Thus, r
b
as impedance could not be neglected in
analytical models. In LNA design, noise correlation leads to smaller NF
min
and NF at a given
bias, and better closeness of noise matching and impedance matching. Although a larger R
opt
using correlation model leads to a larger L
E
for LNA design, and a relatively higher I
C
, L
E
can
be slightly adjusted for a good tradeo?that provides low power consumption, as well as low noise
figure. Scaling of technology suggests that NF
min
depends on r
b
g
m
at the same f
T
/f. Despite
technology scaling required higher J
C
for higher working frequency, shrunken W
E
and L
E
of
transistor for noise matching keep I
C
of LNA low.
34
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37
Appendices
38
Appendix A
Derivation of Intrinsic Noise Sources? Contributions to i
out
:
i
c
out
, i
b
out
, i
rb
out
, and i
Rs
out
?? ?
??
???
??
????
??
?? ??
???
? ?
??
Figure A.1: Simplified small signal equivalent circuit of LNA
Fig. A.1 is small signal equivalent circuit of LNA with all noise sources. The transistor noise
sources include the terminal current noises i
c
and i
b
, and the thermal noises v
b
of r
b
. Power source
has a noise source v
s
of Z
s
. i
c
out
, i
b
out
, i
rb
out
, and i
Rs
out
are denoted as contributions of i
c
, i
b
, v
b
, and v
s
to total output noise current i
out
, respectively. Each could be calculated using circuit analysis by
removing the other noise sources in Fig. A.1.
A.0.1 i
c
out
:
i
c
out
could be calculated by circuit analysis using Fig. A.2, which includes only one noise
source, i.e. i
c
.
V
c
be
? j!C
be
[Z
s
+r
b
+j!(L
b
+L
e
)]+V
c
be
+j!(i
c
+g
m
V
c
be
)L
e
= 0. (A.1)
Then,
39
?
??
?
??????
??
??
??
?????
Figure A.2: Simplified small signal equivalent circuit of LNA with intrinsic current noise source
i
c
.
V
c
be
= j!i
c
L
e
?
1
?
, (A.2)
where
? = 1!
2
C
be
(L
b
+L
e
) +j!C
be
(Z
s
+r
b
) +j!g
m
L
e
. (A.3)
Therefore,
i
c
out
= V
c
be
? g
m
+i
c
= j!g
m
i
c
L
e
?
1
?
+i
c
,
(A.4)
A.0.2 i
b
out
:
?
??
?
??????
??
?
???
? ?
Figure A.3: Simplified small signal equivalent circuit of LNA with intrinsic current noise source
i
b
.
40
i
b
out
could be calculated by circuit analysis using Fig. A.3, which includes only one noise
source, i.e. i
b
:
V
b
be
= i
b
[(Z
s
+r
b
) +j!(L
b
+L
e
)] ?
1
?
, (A.5)
i
b
out
= V
b
be
? g
m
= g
m
i
b
[(Z
s
+r
b
) +j!(L
b
+L
e
)] ?
1
?
.
(A.6)
A.0.3 i
rb
out
:
??
?
??
?
??????
??
???
? ??
Figure A.4: Simplified small signal equivalent circuit of LNA with thermal noise source v
rb
.
i
rb
out
could be calculated by circuit analysis using Fig. A.4, which includes only one noise
source, i.e. v
rb
:
V
rb
be
= v
rb
?
1
?
, (A.7)
i
rb
out
= V
rb
be
? g
m
= g
m
v
rb
?
1
?
.
(A.8)
A.0.4 i
Rs
out
:
i
Rs
out
could be calculated by circuit analysis using Fig. A.2, which includes only one noise
source, i.e. v
Rs
:
V
Rs
be
= v
Rs
?
1
?
, (A.9)
i
Rs
out
= V
Rs
be
? g
m
= g
m
v
Rs
?
1
?
.
(A.10)
41
??
??
???
??
?
??
????
?? ????
Figure A.5: Simplified small signal equivalent circuit of LNA with thermal noise source v
Rs
.
42
43
Appendix B
Matlab Code for Noise Parameters of Device Calculation
close all;
clear all;
format long;
datapath =
'C:\Users\Xiaojia\Documents\Iccap\Noise\Noise5PAE_Hicum\7WLNoise_wrk\export_d
ata\';
OPinfo = sprintf('%s7WL_p24_OPinfo.csv',datapath);
OPinfo_wt = sprintf('%s7WL_p24_OPinfo_wt.csv',datapath);
[OP] = textread(OPinfo,'','delimiter',',','headerlines',46);
%'Vbe','0Hz','1','OP_betadc','OP_ib','OP_ic','OP_rb','OP_Cbe','OP_taun'
[OP_wt] = textread(OPinfo_wt,'','delimiter',',','headerlines',36);
%'','','','wt_gm','','','wt_fT'
[Ic] = OP(:,6)';
[rb] = OP(:,7)';
[Cbe] = OP(:,8)';
[beta] = OP(:,4)';
[taun] = OP(:,9)';
[wt] = OP_wt(:,7)';
freq = 10e9;
Length_e0 = 1 * 20;
Width_e = 0.24;
Ae = Width_e * Length_e0;
q = 1.602189e19;
twoq = 2 * q;
w = 2 * pi * freq;
Temp = 290;
44
k = 1.380662e23;
kt = k * Temp;
m = size(Ic,2);
for n = 1:m
Cbe_bc(n) = Cbe(n);
Ib(n) = Ic(n)/beta(n);
gm(n) = wt(n)*Cbe_bc(n);
wCbe_inv(n) = 1/(w * Cbe_bc(n));
wt_w(n) = wt(n)/w;
Jc(n) = Ic(n)/Ae;
fg1 = 0.757;
T(n) = taun(n)*fg1;
%%correlation
Sib(n) = twoq * Ib(n) + twoq * Ic(n) * (w * T(n)).^2;
Sic(n) = twoq * Ic(n);
Bu(n) = twoq * Ic(n) * w * T(n);
AA(n) = Sic(n) + Sib(n) * wt_w(n).^2 + 2 * Bu(n) * wt_w(n);
Xopt(n) = wCbe_inv(n) * (Sic(n) + Bu(n) * wt_w(n))/AA(n);%  wCbe_inv(n);
% " wCbe_inv(n)" for LNA
Ropt2(n) = 4*kt * rb(n) * wt_w(n).^2/AA(n) + rb(n).^2 + wCbe_inv(n).^2 *
Sic(n)/AA(n)  wCbe_inv(n).^2 * (Sic(n) + Bu(n) * wt_w(n)).^2/(AA(n).^2);
Ropt(n) = sqrt(Ropt2(n));
Fmin(n) = 1 + rb(n)/Ropt(n) + 1/(4 * kt * Ropt(n)) * (1/wt_w(n)).^2 *
(Ropt(n) + rb(n)).^2 * AA(n) + 1/(4 * kt * Ropt(n)) * wCbe_inv(n).^2 *
(Sic(n) * Sib(n)  Bu(n).^2) / AA(n) ;
NFmin(n) = 10*log10(Fmin(n));
R(n) = (50+rb(n)).^2+0.^2;
F(n) = 1+rb(n)/50+Sic(n)*(1+(w*Cbe_bc(n)).^2*R(n)
2*w*Cbe_bc(n)*0)/(gm(n).^2*4*kt*50)+Sib(n)*R(n)/(4*kt*50)
2*Bu(n)*0/(gm(n)*4*kt*50)+2*Bu(n)*w*Cbe_bc(n)*R(n)/(gm(n)*4*kt*50);
45
NF(n) = 10*log10(F(n));
C(n) = 1 + (w * Cbe_bc(n) * rb(n)).^2;
Iout_icib(n) = twoq * Ic(n) + (twoq * Ib(n) + twoq * Ic(n) * (w *
T(n)).^2) * gm(n).^2 * rb(n).^2/C(n) + 2 * Bu(n) * gm(n) * w * Cbe_bc(n) *
rb(n).^2/C(n);
Iout_rb(n) = 4*kt * rb(n) *gm(n).^2/C(n);
Iout(n) = Iout_icib(n) + Iout_rb(n);
Y21_inv(n) = (1 + j * w * Cbe_bc(n) * rb(n))/gm(n);
Sva(n) = Iout(n)*(abs(Y21_inv(n))).^2;
Rn(n) = Sva(n)/(4*kt);
%%spice
Sib_spice(n) = twoq * Ib(n);
Sic_spice(n) = twoq * Ic(n);
Bu_spice(n) = 0;
AA_spice(n) = Sic_spice(n) + Sib_spice(n) * wt_w(n).^2 + 2 * Bu_spice(n)
* wt_w(n);
Xopt_spice(n) = wCbe_inv(n) * (Sic_spice(n) + Bu_spice(n) *
wt_w(n))/AA_spice(n);%  wCbe_inv(n); % " wCbe_inv(n)" for LNA
Ropt2_spice(n) = 4*kt * rb(n) * wt_w(n).^2/AA_spice(n) + rb(n).^2 +
wCbe_inv(n).^2 * Sic_spice(n)/AA_spice(n)  wCbe_inv(n).^2 * (Sic_spice(n) +
Bu_spice(n) * wt_w(n)).^2/(AA_spice(n).^2);
Ropt_spice(n) = sqrt(Ropt2_spice(n));
Fmin_spice(n) = 1 + rb(n)/Ropt_spice(n) + 1/(4 * kt * Ropt_spice(n)) *
(1/wt_w(n)).^2 * (Ropt_spice(n) + rb(n)).^2 * AA_spice(n) + 1/(4 * kt *
Ropt_spice(n)) * wCbe_inv(n).^2 * (Sic_spice(n) * Sib_spice(n) 
Bu_spice(n).^2) / AA_spice(n) ;
NFmin_spice(n) = 10*log10(Fmin_spice(n));
R_spice(n) = (50+rb(n)).^2+0.^2;
46
F_spice(n) = 1+rb(n)/50+Sic_spice(n)*(1+(w*Cbe_bc(n)).^2*R_spice(n)
2*w*Cbe_bc(n)*0)/(gm(n).^2*4*kt*50)+Sib_spice(n)*R_spice(n)/(4*kt*50)
2*Bu_spice(n)*0/(gm(n)*4*kt*50)+2*Bu_spice(n)*w*Cbe_bc(n)*R_spice(n)/(gm(n)*4
*kt*50);
NF_spice(n) = 10*log10(F_spice(n));
C(n) = 1 + (w * Cbe_bc(n) * rb(n)).^2;
Iout_icib_spice(n) = twoq * Ic(n) + (twoq * Ib(n)) * gm(n).^2 *
rb(n).^2/C(n) + 2 * Bu_spice(n) * gm(n) * w * Cbe_bc(n) * rb(n).^2/C(n);
Iout_rb(n) = 4*kt * rb(n) *gm(n).^2/C(n);
Iout_spice(n) = Iout_icib_spice(n) + Iout_rb(n);
Y21_inv(n) = (1 + j * w * Cbe_bc(n) * rb(n))/gm(n);
Sva_spice(n) = Iout_spice(n)*(abs(Y21_inv(n))).^2;
Rn_spice(n) = Sva_spice(n)/(4*kt);
end
figure(1);
subplot(4,1,1); hold on;
plot(Jc*1e3,NFmin,'r','LineWidth',2);
plot(Jc*1e3,NFmin_spice,'r','LineWidth',2);
xlabel('Jc(mA/um^2)');ylabel('NF^{Device}_{min} (dB)');
subplot(4,1,2); hold on;
plot(Jc*1e3,Ropt,'r','LineWidth',2);
plot(Jc*1e3,Ropt_spice,'r','LineWidth',2);
xlabel('Jc(mA/um^2)');ylabel('R^{Device}_{opt} (ohm)');
subplot(4,1,3); hold on;
plot(Jc*1e3,Xopt,'r','LineWidth',2);
plot(Jc*1e3,Xopt_spice,'r','LineWidth',2);
xlabel('Jc(mA/um^2)');ylabel('X^{Device}_{opt} (ohm)');
subplot(4,1,4); hold on;
47
plot(Jc*1e3,Rn./50,'r','LineWidth',2);
plot(Jc*1e3,Rn_spice./50,'r','LineWidth',2);
xlabel('Jc(mA/um^2)');ylabel('R^{Device}_{n}/50');
48
Appendix B
Matlab Code for Noise Parameters of Matched LNA
close all;
clear all;
format long;
datapath =
'C:\Users\Xiaojia\Documents\Iccap\Noise\Noise5PAE_Hicum\7WLNoise_wrk\export_d
ata\';
OPinfo = sprintf('%s7WL_p24_OPinfo.csv',datapath);
OPinfo_wt = sprintf('%s7WL_p24_OPinfo_wt.csv',datapath);
[OP] = textread(OPinfo,'','delimiter',',','headerlines',46);
%'Vbe','0Hz','1','OP_betadc','OP_ib','OP_ic','OP_rb','OP_Cbe','OP_taun'
[OP_wt] = textread(OPinfo_wt,'','delimiter',',','headerlines',36);
%'','','','wt_gm','','','wt_fT'
[Ic] = OP(:,6)';
[rb] = OP(:,7)';
[Cbe] = OP(:,8)';
[beta] = OP(:,4)';
[taun] = OP(:,9)';
[wt] = OP_wt(:,7)';
freq = 10e9;
Length_e0 = 1 * 20;
Width_e = 0.24;
Ae = Width_e * Length_e0;
q = 1.602189e19;
twoq = 2 * q;
w = 2 * pi * freq;
T = 290;
k = 1.380662e23;
49
kt = k * T;
rb_per_um = rb * Length_e0;
Ic_per_um = Ic/Length_e0;
Cbe_per_um = Cbe/Length_e0;
m = size(Ic,2);
for n = 1:m
Jc(n) = Ic(n)/Ae;
Ropt(n) = 100;
Length_e(n) = 1;
while (Ropt(n) >= 50)
Length_e(n) = Length_e(n) + 0.01;
rb_2(n) = rb_per_um(n) / Length_e(n);
Ic_2(n) = Ic_per_um(n) * Length_e(n);
Cbe_2(n) = Cbe_per_um(n) * Length_e(n);
Ae_2(n) = Length_e(n) * Width_e;
Ib(n) = Ic_2(n)/beta(n);
gm(n) = wt(n)*Cbe_2(n);
wCbe_inv(n) = 1/(w * Cbe_2(n));
wt_w(n) = wt(n)/w;
fg1 = 0.757;
T(n) = taun(n)*fg1;
Sib(n) = twoq * Ib(n) + twoq * Ic_2(n) * (w * T(n)).^2;
Sic(n) = twoq * Ic_2(n);
Sicib(n) = j * twoq * Ic_2(n) * w * T(n);
Bu(n) = twoq * Ic_2(n) * w * T(n);
50
AA(n) = Sic(n) + Sib(n) * wt_w(n).^2 + 2 * Bu(n) * wt_w(n);
Ropt2(n) = 4*kt * rb_2(n) * wt_w(n).^2/AA(n) + rb_2(n).^2 +
wCbe_inv(n).^2 * Sic(n)/AA(n)  wCbe_inv(n).^2 * (Sic(n) + Bu(n) *
wt_w(n)).^2/(AA(n).^2);
Ropt(n) = sqrt(Ropt2(n));
end
Xopt(n) = wCbe_inv(n) * (Sic(n) + Bu(n) * wt_w(n))/AA(n)  wCbe_inv(n);
Fmin(n) = 1 + rb_2(n)/Ropt(n) + 1/(4 * kt * Ropt(n)) * (1/wt_w(n)).^2 *
(Ropt(n) + rb_2(n)).^2 * AA(n) + 1/(4 * kt * Ropt(n)) * wCbe_inv(n).^2 *
(Sic(n) * Sib(n)  Bu(n).^2) / AA(n) ;
NFmin(n) = 10*log10(Fmin(n));
R(n) = (50+rb_2(n)).^2+0.^2;
F(n) = 1+rb_2(n)/50+Sic(n)*(1+(w*Cbe_2(n)).^2*R(n)
2*w*Cbe_2(n)*0)/(gm(n).^2*4*kt*50)+Sib(n)*R(n)/(4*kt*50)
2*Bu(n)*0/(gm(n)*4*kt*50)+2*Bu(n)*w*Cbe_2(n)*R(n)/(gm(n)*4*kt*50);
NF(n) = 10*log10(F(n));
C(n) = 1 + (w * Cbe_2(n) * rb_2(n)).^2;
Iout_icib(n) = twoq * Ic_2(n) + (twoq * Ib(n) + twoq * Ic(n) * (w *
T(n)).^2) * gm(n).^2 * rb_2(n).^2/C(n) + 2 * Bu(n) * gm(n) * w * Cbe_2(n) *
rb_2(n).^2/C(n);
Iout_rb(n) = 4*kt * rb_2(n) *gm(n).^2/C(n);
Iout(n) = Iout_icib(n) + Iout_rb(n);
Y21_inv(n) = (1 + j * w * Cbe_2(n) * rb_2(n))/gm(n);
Sva(n) = Iout(n)*(abs(Y21_inv(n))).^2;
Rn(n) = Sva(n)/(4*kt);
end
51
figure(1);
subplot(2,2,1); hold on;
plot(Jc*1e3,NFmin,'r','LineWidth',2);
xlabel('Jc(mA/um^2)');ylabel('NFmin(dB)');
subplot(2,2,2); hold on;
plot(Jc*1e3,NF,'r','LineWidth',2);
xlabel('Jc(mA/um^2)');ylabel('NF(dB)');
subplot(2,2,3); hold on;
plot(Jc*1e3,Ropt,'r','LineWidth',2);
xlabel('Jc(mA/um^2)');ylabel('Ropt(ohm)');
subplot(2,2,4); hold on;
plot(Jc*1e3,Xopt,'r','LineWidth',2);
xlabel('Jc(mA/um^2)');ylabel('Xopt(ohm)');