To understand the linear characteristics of HBT more accurately, an analytical nonlinear HBT model using Volterra Series analysis is developed. The model considers four nonlinear components: rπ, Cdiff , Cdepl, and gm. It shows that nonlinearities of rπ and Cdiff are almost completely cancelled by gm nonlinearity at all frequencies. The residual gm nonlinearity are highly degenerated by the input impedances. Therefore, rπ, Cπ and gm nonlinearities generate less IM3 than Cbc. If Cbc is linearized, Cdepl and gm are the main nonlinear sources of HBT, and Cdepl becomes very important at a high frequency. It was also found that the degeneration resistor, RE, is more effective than RB for reducing gm nonlinearity. This analysis also provides the dependency of the source second harmonic impedance on the linearity of HBT. The IM3 of HBT is significantly reduced by setting the second harmonic impedance of ZS,2ω2 = 0 and ZS,ω2-ω1 = 0

Chung, Minchul

Kang, Sanghoon

Kim, Bumman

Kim, Woonyun

Lee, Kyungho

English

AMS Acta

NONLINEAR BEHAVIOR OF POWER HBT
Woonyun Kim*, Sanghoon Kang**, Kyungho Lee**, Minchul Chung**, and Bumman Kim**
*RFIC Design Team, System LSI Division, Samsung Electronics Co., Ltd., San 24, Nongseo-Lee,
Kiheung-Eup, Yongin-Si, Kyungki-Do, 449-711, Korea.
Phone: +82-54-279-5584, Fax: +82-54-279-2903, E-mail: kwn@postech.ac.kr
**Dept. of E. E. Eng. and MARC, POSTECH, San 31, Hyoja-Dong, Pohang, Kyungbuk, 790-784, Korea.
Abstract— To understand the linear characteristics
of HBT more accurately, an analytical nonlinear HBT
model using Volterra Series analysis is developed. The
model considers four nonlinear components: rπ, Cdiff ,
Cdepl, and gm. It shows that nonlinearities of rπ and
Cdiff are almost completely cancelled by gm nonlinear-
ity at all frequencies. The residual gm nonlinearity are
highly degenerated by the input impedances. There-
fore, rπ, Cπ and gm nonlinearities generate less IM3
than Cbc. If Cbc is linearized, Cdepl and gm are the main
nonlinear sources of HBT, and Cdepl becomes very im-
portant at a high frequency. It was also found that
the degeneration resistor, RE , is more effective than RB
for reducing gm nonlinearity. This analysis also pro-
vides the dependency of the source second harmonic
impedance on the linearity of HBT. The IM3 of HBT
is significantly reduced by setting the second harmonic
impedance of ZS,2ω2 = 0 and ZS,ω2−ω1 = 0.
I. Introduction
The transmitter of the handset of digital mobile com-
munication systems requires highly efficient linear
power amplifiers [1–4]. HBTs are widely used for the
amplifiers and their nonlinear behavior has been exten-
sively measured and analyzed [4–12]. It is commonly
known that Cbc is the dominant nonlinear source and
should be linearized to reduce the intermodulation dis-
tortions [7,9–13]. Cbc is a depletion capacitance and it
is rather moderate nonlinear component. It is surpris-
ing in view of the fact that HBT has highly nonlinear
sources. The dependence of the HBT’s base current,
iB , on base-to-emitter voltage, vBE , is an exponential
function, one of the strongest nonlinearities found in
nature. So is its collector current, iC , which is basically
similar to iB . Furthermore, the junction capacitance,
primarily a diffusion capacitance, is also strongly non-
linear. These exponential nonlinear behaviors are not
seen in the nonlinear characteristics of HBTs. Thus,
the measured high linear characteristics of HBT has
motivated many researchers to study intermodulation
(IM) mechanism of HBT. The good linearity of HBT’s
was attributed to the partial cancellation between IM
currents generated from the exponential junction cur-
rent and the junction capacitance [5], or the partial
cancellation of IM currents from the total base-emitter
current and the total base-collector current [8]. Ac-
cording to reference [9], the high linear characteris-
tis of HBT resulted from the almost complete can-
cellation between the output nonlinear current com-
ponents generated by emitter-base current source and
base-collector current source. It was also reported that
the emitter and base resistances linearize the HBT out-
put [10]. Because of the different descriptions for the
linear characteristics, a more complete explanation is
needed.
To understand the linear characteristics of HBT more
accurately, we have developed an analytical nonlinear
HBT model using Volterra Series analysis [14]. The
presented analysis describes the fundamental nonlinear
behavior of HBT.
II. The Nonlinear Model of HBT
The simplified equivalent circuit of HBT used for the
analysis is shown in Fig. 1. The emitter and base re-
sistances are RE and RB , respectively. These resis-
tances are linear components. The base and collector
nonlinear current source are represented as iB and iC ,
respectively. And, the base-emitter nonlinear capaci-
tance is appeared as a charge, qBE . CBC is assumed to
be linearized for a constant capacitance and is omitted
for the nonlinear circuit analysis. ZS and ZL repre-
sent the source and load impedances, respectively. The
source is conjugate-matched for the maximum gain.
This model includes all important nonlinearities and is
good enough for representing essential nonlinear prop-
erties of HBT.
The nonlinear elements are represented by the third-
order expansion of Taylor series. Under a small-signal
condition, the base nonlinear current source can be
expanded at the vicinity of its bias point. The iB is
modeled as
iB = ISB
[
exp
(
vBE
ηBVT
)
− 1
]
(1)
ib =
IB
ηBVT
vbe +
IB
2η2BV
2
T
v2be +
IB
6η3BV
3
T
v3be (2)
≡ g1vbe + g2v
2
be + g3v
3
be (3)
where ISB represents the saturation current, ηB is the
ideality factor of the base current, IB is dc base cur-
rent, and VT is thermal voltage. ib and vbe are the
small-signal components of iB and vBE , respectively.
Also the coefficient g1 = 1/rπ is a linear junction con-
ductance.
iC is also modeled as
iC = ISC
[
exp
(
vBE
ηCVT
)
− 1
]
(4)
where ISC represents the saturation current, ηC is the
ideality factor of the collector current. It can be ex-
panded at the vicinity of its bias point, yielding
RB
RE
VS
ZS
ZLiB iCqBE
vBE
+
-
Fig. 1. HBT nonlinear equivalent model
ic =
IC
ηCVT
vbe +
IC
2η2CV
2
T
v2be +
IC
6η3CV
3
T
v3be (5)
≡ gm1vbe + gm2v
2
be + gm3v
3
be (6)
where IC is dc collector current, and ic is the small-
signal component of iC . The coefficient, gm1 is gener-
ally called gm. The common emitter current gain (βac)
can be calculated by gm1/g1.
The stored charge at the base-emitter junction, qBE ,
is the sum of diffusion charge and depletion charge.
qBE = τBiC + τEiB + qAEXNNE (7)
qbe =
(
IC
ηCVT
τB +
IB
ηBVT
τE + CjE
)
vbe
+
(
IC
2η2CV
2
T
τB +
IB
2η2BV
2
T
τE +
CjE
4(Vbi − VBE)
)
v2be
+
(
IC
6η3CV
3
T
τB +
IB
6η3BV
3
T
τE +
CjE
8(Vbi − VBE)2
)
v3be (8)
≡ c1vbe + c2v
2
be + c3v
3
be (9)
where τB is the base transit time of the minority car-
rier in base, and τE is the transit time of the minority
carrier in the emitter, and is assumed to be compara-
ble to τB . AE is the emitter area, XN is the depletion
width of the emitter, and NE is the doping concentra-
tion of the emitter. qbe is the small-signal component
of qBE , CjE = AE
√
qεEεBNEPB
2(εENE+εBPB)(Vbi−VBE)
, and Cdiff is
IC
ηCVT
τB +
IB
ηBVT
τE . For simplicity, depletion approximation
has been used. Vbi is built-in potential of base-emitter junc-
tion and VBE is the base-emitter dc voltage. c1 is so-called the
base-emitter junction capacitance, Cπ .
III. The Second Order Harmonic Analysis
The first-order equation for the base-to-emitter voltage Vbe,ωq
at the excitation frequency ωq can be expressed by
Vbe,ωq =
Zπ,ωq
ZIN,ωq + ZS,ωq
VS,ωq (10)
where Zπ,ωq =
1
g1+jωqc1
, ZIN,ωq (= RB + Zπ,ωq + RE(1 +
gm1Zπ,ωq )) is the input impedance of the HBT, and
the source impedance, ZS,ωq is conjugately matched
to ZIN,ωq . The harmonics can be found by means of
Volterra Series analysis. Fig. 2 shows the equivalent
circuit for the second- and third-order nonlinear anal-
ysis. The second-order intermodulation current gen-
erated by the nonlinear base-emitter junction capac-
itance, Iq,2, and by the nonlinear base and collector
current sources, Ib,2 and Ic,2 are given by
RB
RE
ZS
ZLgm1vbe
vbe,2
+
-
Ic,2Ib,2 Cdiff+CdeplIq,2rπ
ZIN
Iq,3Ib,3 Ic,3
IO
Fig. 2. HBT equivalent circuit for the 2nd- and 3rd-order in-
termodulation analysis.
Iq,2ω2 =
1
2
j(2ω2)c2V
2
be,ω2
= jω2c2V
2
be,ω2
, (11)
Ib,2ω2 =
1
2
g2V
2
be,ω2
, (12)
Ic,2ω2 =
1
2
gm2V
2
be,ω2
. (13)
Performing a linear analysis of this circuit, we find the
base-emitter junction voltage at the second harmonic;
Vbe,2ω2 =
−(ZS,2ω2 +RB +RE)Zπ,2ω2 (Ib,2ω2 + Iq,2ω2 )
ZIN,2ω2 + ZS,2ω2
−
REZπ,2ω2Ic,2ω2
ZIN,2ω2 + ZS,2ω2
(14)
The output current IO,2ω2 at this second-harmonic fre-
quency is
IO,2ω2 = gm1Vbe,2ω2 + Ic,2ω2 (15)
Substituting (14) into (15),
REgm1Zπ,2ω2
ZIN,2ω2
+ZS,2ω2
Ic,2ω2 is can-
celled out by the gm1RE feedback term in the last term
of (14). The remain terms of the above equation is sim-
plified as
IO,2ω2 = ZA{−gm1Ib,2ω2 + g1Ic,2ω2}
+ ZA{−gm1Iq,2ω2 + j2ω2c1Ic,2ω2}
+
Zπ,2ω2
ZIN,2ω2 + ZS,2ω2
Ic,2ω2 (16)
where ZA is
(ZS,2ω2+RB+RE)Zπ,2ω2
ZIN,2ω2
+ZS,2ω2
.
IO,2ω2 = ZA
gm
2rπ
(
1
2VT ηC
−
1
2VT ηB
)
V 2be,ω2
+ ZAjω2gm
Cdiff
(βac + 1)
(
1
2ηCVT
−
1
2ηBVT
)
V 2be,ω2
+ ZAjω2gmCjE
(
1
2ηCVT
−
1
4(Vbi − VBE)
)
V 2be,ω2
+
Zπ,2ω2
ZIN,2ω2 + ZS,2ω2
gm
4ηCVT
V 2be,ω2 (17)
The first term in (17) originates from the cancellation
of rπ and gm nonlinearities which have exponential
forms. If the ideality factors of the current sources
are identical, the second order IM distortion gener-
ated from the nonlinear base current source (rπ) can
be completely removed by a portion of the nonlinear
distortion generated from the collector current source
(gm). The second term shows the cancellation between
Cdiff and gm nonlinearities. The second term of (17)
has the nonlinear term from the diffusion capacitance,
which is reduced by a factor of 1/(βac + 1) and can
again be completely removed by matching the ideality
factors of the current sources. The third term gener-
ated from the base-emitter depletion capacitance can
be cancelled by some portion of that generated from iC ,
but can not be completely eliminated due to the differ-
ent origin of sources. The last term is the remained gm
nonlinear portion. The coefficient
Zπ,2ω2gm
ZIN,2ω2
+ZS,2ω2
of the
last term is identical to the degenerated gm nonlinear-
ity equation at a low frequency. In summary, the rπ
and Cdiff nonlinearities almost completely cancelled
by gm nonlinearity and Cdepl nonlinearity is partially
removed by gm nonlinearity. The remained gm non-
linear component is quite similar to the low frequency
case with large degeneration resistances.
IV. The Third Order Analysis
We have performed a similar analysis for the third or-
der Intermodulation. From a similar sequence to the
second order analysis, the third order output current,
IO,2ω2−ω1 is given by
IO,2ω2−ω1 =
(ZS,2ω2−ω1 + RB + RE)Zπ,2ω2−ω1
ZIN,2ω2−ω1 + ZS,2ω2−ω1
gm
rπ
[(
1
8η2
C
V 2
T
−
1
8η2
B
V 2
T
)
+ A
(
1
2ηCVT
−
1
2ηBVT
)
]
B
+
(ZS,2ω2−ω1 + RB + RE)Zπ,2ω2−ω1
ZIN,2ω2−ω1 + ZS,2ω2−ω1
gmj(2ω2 − ω1)
Cdiff
(1 + βac)
[(
1
8η2
C
V 2
T
−
1
8η2
B
V 2
T
)
+ A
(
1
2ηCVT
−
1
2ηBVT
)
]
B
+
(ZS,2ω2−ω1 + RB + RE)Zπ,2ω2−ω1
ZIN,2ω2−ω1 + ZS,2ω2−ω1
gmj(2ω2 − ω1)CjE
[(
1
8η2
C
V 2
T
−
3
32(Vbi − VBE)
2
)
+ A
(
1
2ηCVT
−
1
4(Vbi − VBE)
)
]
B
+
Zπ,2ω2−ω1
ZIN,2ω2−ω1 + ZS,2ω2−ω1
gm
(
1
8η2
C
V 2
T
+ A
1
2ηCVT
)
B (18)
where
A = −
(ZS,2ω2 + RB + RE)(b2 + j2ω2c2) + REg2
2[1 + (ZS,2ω2 + RB + RE)(b1 + j2ω2c1) + REg1]
−
(ZS,ω2−ω1 + RB + RE)(b2 + j(ω2 − ω1)c2) + REg2
1 + (ZS,ω2−ω1 + RB + RE)(b1 + j(ω2 − ω1)c1) + REg1
(19)
B = V
2
be,ω2
V
∗
be,ω1
The third-order output IM current in (18) has very
similar form to the second-order one. The first term
of (18) indicates that Ib,2ω2−ω1 is partially cancelled
by
ZS,2ω2−ω1
+RB+RE
ZIN,2ω2−ω1
+ZS,2ω2−ω1
Ic,2ω2−ω1 , and it can be per-
fectly cancelled out for a matched ideality factors.
The second term of (18) represents that the diffu-
sion charge part of Iq,2ω2−ω1 is partially cancelled by
ZS,2ω2−ω1
+RB+RE
ZIN,2ω2−ω1
+ZS,2ω2−ω1
Ic,2ω2−ω1 . The third term shows
that the distortion signal generated from the base-
emitter depletion capacitance is again incompletely
cancelled by some portion of that generated from
iC . The last one is non-cancelled gm portion of
Zπ,2ω2−ω1gm
ZIN,2ω2−ω1
+ZS,2ω2−ω1
. It can be expressed at a low fre-
quency as,
Zπ,2ω2−ω1
ZIN,2ω2−ω1 + ZS,2ω2−ω1
gm ∼=
rπgm
rπ + gmrπRE +RE +RB
∼=
gm
1 + gmRE
. (20)
Symbol Value Symbol Value
IE 16 mA βdc 40
IB 0.39 mA βac 68
Pin -60 dBm ZL 150 Ω
ηC 1.0 ηB 1.7
g1 0.0089 ∆f 1 MHz
g2 0.1006 τB 8.72 × 10
−13
g3 0.7620 CjE 4.3 × 10
−13
c1 9.63 × 10−13 gm1 0.6027
c2 1.42 × 10−11 gm2 11.6350
c3 2.05 × 10−10 gm3 149.7427
RB 8 Ω RE 2 Ω
Vbi 1.627 V VBE 1.60 V
Table 1
The Model Parameters of HBT
0.1 1 10 100
-320
-300
-280
-260
-240
-220
-200
F
o
u
r 
te
rm
s 
co
m
p
o
si
n
g
 o
f 
I O
,2
w
2
-w
1
Frequency [GHz]
 I
O,2w2-w1
 1st term of Eqn. (18)
 2nd term of Eqn. (18)
 3rd term of Eqn. (18)
 Last term of Eqn. (18)
Fig. 3. Four terms composing IO,2ω2−ω1 in (18).
It is identical to the low frequency equation and the
emitter degeneration resistance is effective to reduce
the last term.
For the analytical calculation of the nonlinear com-
ponents of HBT, we used the model parameters of
POSTECH built 3× 20 µm2 emitter HBT. This de-
vice can deliver about 13 dBm of power. The param-
eters are summarized in Table 1. For the calculation,
ZS is adjusted to have impedance matching at the fre-
quencies and input power is -60 dBm. Fig. 3 shows
the frequency dependent values of the above 4 terms
of IO,2ω2−ω1 in (18). The cancellations between rπ
and gm, and Cdiff and gm nonlinearities is quite good
for all frequencies. Therefore, the dominant nonlinear
sources for all frequencies are Cdepl and gm. The third
term becomes comparable to the last term at 2 GHz
and above. At a low frequency (below 2 GHz), gm non-
linearity with large degeneration resistor creates IM3.
Between 2 ∼ 20 GHz, IM3 are generated by Cdepl and
gm nonlinearities. At higher frequencies, the dominant
source is Cdepl. Complete cancellation of Cdiff and rπ
nonlinearities by gm nonlinearity using identical ideal-
ity factors does not have any significant impact because
they are rather small amount. We scanned ηC and ηB
for a maximum IP3, HBT with ηC = 1.0 and ηB = 2.0
has maximum IP3 at 2 GHz (seen in Fig. 4).
Degeneration effects of the emitter and base resistances
are also studied. Because ZIN,2ω2−ω1 does not have
any feedback term for an HBT with RE = 0, the
portion,
Zπ,2ω2−ω1
ZIN,2ω2−ω1
+ZS,2ω2−ω1
, becomes relatively large,
and IP3 level is degraded. The maximum IP3 of HBT
is obtained at RE = 5 Ω and RB = 0 Ω, which is 28.2
dBm. As RE increases, IP3 level increases because the
portion,
Zπ,2ω2−ω1
ZIN,2ω2−ω1
+ZS,2ω2−ω1
is reduced. But, larger
RE reduces not only the IM3 signal but also the fun-
damental signal, and a optimum value of RE is about
5 Ω in our case. The emitter degeneration resistance
is more effective than the base resistance.
As can seen from (18) and (19), the third order IM cur-
rents are dependent on the second harmonic impedance
of ZS,2ω2 and ZS,ω2−ω1 . We have calculated the IP3
of HBT using the source-termination at the second-
harmonic frequency. The linearity of HBT is remark-
ably promoted for ZS,2ω2 = 0 and ZS,ω2−ω1 = 0. We
also compute the IP3 levels of HBT with ZS,2ω2 = ∞
or ZS,ω2−ω1 = ∞, but it is degraded. Fig. 5 shows
contour lines of IP3 level of HBT with ZS,2ω2 = 0 and
ZS,ω2−ω1 = 0 for various ηC and ηB . The maximum
IP3 of HBT with the source-termination is about 31
dBm for ηC = 1.15 and ηB = 1.55 which is 3.5 dB
improvement.
V. Conclusions
To accurately understand the linear characteristics
of HBT, we developed an analytical nonlinear HBT
model using Volterra Series analysis. Our model con-
siders four nonlinearities: rπ, Cdiff , Cdepl, and gm.
The analysis we present reveals that the cancellations
between rπ and gm, and Cdiff and gm nonlinearities
are quite good for all frequencies. Further, Cdepl and
gm are the main nonlinear sources of HBT. gm can be
linearized using degeneration resistors. The degenera-
tion resistor, RE , is more effective than RB for reduc-
ing gm nonlinearity. The Cdepl nonlinearity becomes
important at a high freequency. This analysis also pro-
vides the dependency of the source second harmonic
impedance on the linearity of HBT. The IP3 of HBT
improves considerably by setting the second harmonic
impedance of ZS,2ω2 = 0 and ZS,ω2−ω1 = 0.
Acknowledgements
This work was supported in part by the Agency for
Defense Development and the Brain Korea 21 Project
of the Ministry of Education.
References
[1] Christopher T. M. Chang and Han-Tzong Yuan, “GaAs
HBT’s for High-Speed Digital Integrated Circuit Applica-
tions,” Proc.IEEE, vol. 81, no. 12, pp. 1727-1743, 1993.
[2] P. M. Asbeck, M. F. Chang, J. J. Corcoran, J. F. Jensen, R.
N. Nottenburg, A. Oki, and H. T. Yuan, “HBT Application
Prospects in the US: Where and when?,” IEEE GaAs IC
Symp. Tech. Dig., Monterey, CA., pp. 7-10, Oct. 1991.
[3] Guang-Bo Gao, David J. Roulston, and Hadis Morkoc, “De-
sign Study of AlGaAs/GaAs HBTs,” IEEE Trans. Electron
Devices, vol. ED-37, no.5, pp. 1199-1208, May 1990.
[4] M. E. Kim, A. K. Oki, J. B. Camou, P. D. Chow, B. L.
Nelson, D. M. Smith, J. C. Canyon, C. C. Yang, R. Dixit,
and B. R. Allen, “12-40 GHz Low Harmonic Distortion and
Phase Noise Performance of GaAs HBTs,” IEEE GaAs IC
Symposium Digest, pp. 117-120, Nov. 1988.
[5] Stephen A. Maas, Bradford L. Nelson, and Donald L. Tait,
Fig. 4. The contour lines of IP3 level of an HBT using our
model parameters with various ηC and ηB .
Fig. 5. The contour lines of IP3 level of an HBT with ZS,2ω2
= 0 and ZS,ω2−ω1 = 0 using our model parameters with
various ηC and ηB .
“Intermodulations in HBTs,” IEEE Trans. Microwave The-
ory and Tech., vol. MTT-40, no. 3, pp. 442-448, Mar. 1992.
[6] Taisuke Iwai, Shiro Ohara, Horoshi Yamada, Yasuhiro Ya-
maguchi, Kenji Imanishi, and Kazukiyo Joshin, “High Effi-
ciency and High Linearity InGaP/GaAs HBT Power Ampli-
fiers: Matching Techniques of Source and Load Impedance to
Improve Phase Distortion and Linearity,” IEEE Trans. Elec-
tron Devices, vol. ED-45, no. 6, pp.1196-1200, June 1998.
[7] K. W. Kobayashi, J. C. Cowles, L. T. Tran, A. Gutierrez-
Aitken, M. Nishimoto, J. H. Elliott, T. R. Block, A. K. Oki,
and D. C. Streit, “A 44-GHz-High IP3 InP HBT MMIC
Amplifier for low DC Power Millimeter-wave Receiver Ap-
plications,” IEEE Journal of Solid-State Circuits,” vol. 34,
no. 9, pp. 1188-1194, Sept. 1999.
[8] Apostolos Samelis and Dimitris Pavlidis, “Mechanisms De-
termining Third Order Intermodulation Distortion in Al-
GaAs/GaAs HBTs,” IEEE Trans. Microwave Theory and
Tech., vol. MTT-40, No. 12, pp. 2374-2380, Dec. 1992.
[9] J. Lee, W. Kim, T. Rho, and B. Kim, “Intermodula-
tion Mechanism and Linearization of AlGaAs/GaAs HBT’s,”
IEEE Trans. on Microwave Theory and Tech., vol. MTT-45,
no. 12, pp. 2065-2072, Dec. 1997.
[10] Nan Lei Wang, Wu Jing Ho, and J. A. Higgins, “Al-
GaAs/GaAs HBT Linearity Characteristics,” IEEE Trans.
Microwave Theory and Tech., vol. MTT-42, no. 10, pp. 1845-
1850, Oct. 1994.
[11] Peter Asbeck, “HBT Linearity and Basic Linearization Ap-
proaches,” IEEE MTT-S Int. Microwave Symp. Workshop,
Baltimore, Maryland, June 1998.
[12] M. Iwamoto, T. S. Low, C. P. Hutchinson, J. B. Scott, A.
Cognata, X. Qin, L. H. Camnitz, P. M. Asbeck, and D. C.
D’Avanzo, “Influence of Collector Design on InGaP/GaAs
HBT Linearity,” IEEE MTT-S Int. Microwave Symp. Dig.,
Boston, MA, pp. 757-760, Jun. 2000.
[13] W. Kim, S. Kang, K. Lee, M. Chung, Y. Yang, and B. Kim,
“The Effects of Cbc on the Linearity of AlGaAs/GaAs Power
HBT,” IEEE Trans. on Microwave Theory and Tech., vol.
MTT-49, no. 49, pp. 1270-1276, Jul. 2001.
[14] Stephen A. Maas, Nonlinear Microwave Circuits, Artech
House, Norwood, MA, 1988.


Non linear behavior of power HBT

AMS Acta - Alm@DL - Università di Bologna

NONLINEAR BEHAVIOR OF POWER HBT
Woonyun Kim*, Sanghoon Kang**, Kyungho Lee**, Minchul Chung**, and Bumman Kim**
*RFIC Design Team, System LSI Division, Samsung Electronics Co., Ltd., San 24, Nongseo-Lee,
Kiheung-Eup, Yongin-Si, Kyungki-Do, 449-711, Korea.
Phone: +82-54-279-5584, Fax: +82-54-279-2903, E-mail: kwn@postech.ac.kr
**Dept. of E. E. Eng. and MARC, POSTECH, San 31, Hyoja-Dong, Pohang, Kyungbuk, 790-784, Korea.
Abstract— To understand the linear characteristics
of HBT more accurately, an analytical nonlinear HBT
model using Volterra Series analysis is developed. The
model considers four nonlinear components: rpi, Cdiff ,
Cdepl, and gm. It shows that nonlinearities of rpi and
Cdiff are almost completely cancelled by gm nonlinear-
ity at all frequencies. The residual gm nonlinearity are
highly degenerated by the input impedances. There-
fore, rpi, Cpi and gm nonlinearities generate less IM3
than Cbc. If Cbc is linearized, Cdepl and gm are the main
nonlinear sources of HBT, and Cdepl becomes very im-
portant at a high frequency. It was also found that
the degeneration resistor, RE , is more effective than RB
for reducing gm nonlinearity. This analysis also pro-
vides the dependency of the source second harmonic
impedance on the linearity of HBT. The IM3 of HBT
is significantly reduced by setting the second harmonic
impedance of ZS,2ω2 = 0 and ZS,ω2−ω1 = 0.
I. Introduction
The transmitter of the handset of digital mobile com-
munication systems requires highly efficient linear
power amplifiers [1–4]. HBTs are widely used for the
amplifiers and their nonlinear behavior has been exten-
sively measured and analyzed [4–12]. It is commonly
known that Cbc is the dominant nonlinear source and
should be linearized to reduce the intermodulation dis-
tortions [7,9–13]. Cbc is a depletion capacitance and it
is rather moderate nonlinear component. It is surpris-
ing in view of the fact that HBT has highly nonlinear
sources. The dependence of the HBT’s base current,
iB , on base-to-emitter voltage, vBE , is an exponential
function, one of the strongest nonlinearities found in
nature. So is its collector current, iC , which is basically
similar to iB . Furthermore, the junction capacitance,
primarily a diffusion capacitance, is also strongly non-
linear. These exponential nonlinear behaviors are not
seen in the nonlinear characteristics of HBTs. Thus,
the measured high linear characteristics of HBT has
motivated many researchers to study intermodulation
(IM) mechanism of HBT. The good linearity of HBT’s
was attributed to the partial cancellation between IM
currents generated from the exponential junction cur-
rent and the junction capacitance [5], or the partial
cancellation of IM currents from the total base-emitter
current and the total base-collector current [8]. Ac-
cording to reference [9], the high linear characteris-
tis of HBT resulted from the almost complete can-
cellation between the output nonlinear current com-
ponents generated by emitter-base current source and
base-collector current source. It was also reported that
the emitter and base resistances linearize the HBT out-
put [10]. Because of the different descriptions for the
linear characteristics, a more complete explanation is
needed.
To understand the linear characteristics of HBT more
accurately, we have developed an analytical nonlinear
HBT model using Volterra Series analysis [14]. The
presented analysis describes the fundamental nonlinear
behavior of HBT.
II. The Nonlinear Model of HBT
The simplified equivalent circuit of HBT used for the
analysis is shown in Fig. 1. The emitter and base re-
sistances are RE and RB , respectively. These resis-
tances are linear components. The base and collector
nonlinear current source are represented as iB and iC ,
respectively. And, the base-emitter nonlinear capaci-
tance is appeared as a charge, qBE . CBC is assumed to
be linearized for a constant capacitance and is omitted
for the nonlinear circuit analysis. ZS and ZL repre-
sent the source and load impedances, respectively. The
source is conjugate-matched for the maximum gain.
This model includes all important nonlinearities and is
good enough for representing essential nonlinear prop-
erties of HBT.
The nonlinear elements are represented by the third-
order expansion of Taylor series. Under a small-signal
condition, the base nonlinear current source can be
expanded at the vicinity of its bias point. The iB is
modeled as
iB = ISB
[
exp
(
vBE
ηBVT
)
− 1
]
(1)
ib =
IB
ηBVT
vbe +
IB
2η2BV
2
T
v2be +
IB
6η3BV
3
T
v3be (2)
≡ g1vbe + g2v
2
be + g3v
3
be (3)
where ISB represents the saturation current, ηB is the
ideality factor of the base current, IB is dc base cur-
rent, and VT is thermal voltage. ib and vbe are the
small-signal components of iB and vBE , respectively.
Also the coefficient g1 = 1/rpi is a linear junction con-
ductance.
iC is also modeled as
iC = ISC
[
exp
(
vBE
ηCVT
)
− 1
]
(4)
where ISC represents the saturation current, ηC is the
ideality factor of the collector current. It can be ex-
panded at the vicinity of its bias point, yielding
RB
RE
VS
ZS
ZLiB iCqBE
vBE
+
-
Fig. 1. HBT nonlinear equivalent model
ic =
IC
ηCVT
vbe +
IC
2η2CV
2
T
v2be +
IC
6η3CV
3
T
v3be (5)
≡ gm1vbe + gm2v
2
be + gm3v
3
be (6)
where IC is dc collector current, and ic is the small-
signal component of iC . The coefficient, gm1 is gener-
ally called gm. The common emitter current gain (βac)
can be calculated by gm1/g1.
The stored charge at the base-emitter junction, qBE ,
is the sum of diffusion charge and depletion charge.
qBE = τBiC + τEiB + qAEXNNE (7)
qbe =
(
IC
ηCVT
τB +
IB
ηBVT
τE + CjE
)
vbe
+
(
IC
2η2CV
2
T
τB +
IB
2η2BV
2
T
τE +
CjE
4(Vbi − VBE)
)
v2be
+
(
IC
6η3CV
3
T
τB +
IB
6η3BV
3
T
τE +
CjE
8(Vbi − VBE)2
)
v3be (8)
≡ c1vbe + c2v
2
be + c3v
3
be (9)
where τB is the base transit time of the minority car-
rier in base, and τE is the transit time of the minority
carrier in the emitter, and is assumed to be compara-
ble to τB . AE is the emitter area, XN is the depletion
width of the emitter, and NE is the doping concentra-
tion of the emitter. qbe is the small-signal component
of qBE , CjE = AE
√
q²E²BNEPB
2(²ENE+²BPB)(Vbi−VBE)
, and Cdiff is
IC
ηCVT
τB +
IB
ηBVT
τE . For simplicity, depletion approximation
has been used. Vbi is built-in potential of base-emitter junc-
tion and VBE is the base-emitter dc voltage. c1 is so-called the
base-emitter junction capacitance, Cpi .
III. The Second Order Harmonic Analysis
The first-order equation for the base-to-emitter voltage Vbe,ωq
at the excitation frequency ωq can be expressed by
Vbe,ωq =
Zpi,ωq
ZIN,ωq + ZS,ωq
VS,ωq (10)
where Zpi,ωq =
1
g1+jωqc1
, ZIN,ωq (= RB + Zpi,ωq + RE(1 +
gm1Zpi,ωq )) is the input impedance of the HBT, and
the source impedance, ZS,ωq is conjugately matched
to ZIN,ωq . The harmonics can be found by means of
Volterra Series analysis. Fig. 2 shows the equivalent
circuit for the second- and third-order nonlinear anal-
ysis. The second-order intermodulation current gen-
erated by the nonlinear base-emitter junction capac-
itance, Iq,2, and by the nonlinear base and collector
current sources, Ib,2 and Ic,2 are given by
RB
RE
ZS
ZLg
m1vbe
vbe,2
+
-
I
c,2Ib,2 Cdiff+CdeplIq,2
rpi
ZIN
Iq,3Ib,3 Ic,3
IO
Fig. 2. HBT equivalent circuit for the 2nd- and 3rd-order in-
termodulation analysis.
Iq,2ω2 =
1
2
j(2ω2)c2V
2
be,ω2
= jω2c2V
2
be,ω2
, (11)
Ib,2ω2 =
1
2
g2V
2
be,ω2
, (12)
Ic,2ω2 =
1
2
gm2V
2
be,ω2
. (13)
Performing a linear analysis of this circuit, we find the
base-emitter junction voltage at the second harmonic;
Vbe,2ω2 =
−(ZS,2ω2 +RB +RE)Zpi,2ω2 (Ib,2ω2 + Iq,2ω2 )
ZIN,2ω2 + ZS,2ω2
−
REZpi,2ω2Ic,2ω2
ZIN,2ω2 + ZS,2ω2
(14)
The output current IO,2ω2 at this second-harmonic fre-
quency is
IO,2ω2 = gm1Vbe,2ω2 + Ic,2ω2 (15)
Substituting (14) into (15),
REgm1Zpi,2ω2
ZIN,2ω2
+ZS,2ω2
Ic,2ω2 is can-
celled out by the gm1RE feedback term in the last term
of (14). The remain terms of the above equation is sim-
plified as
IO,2ω2 = ZA{−gm1Ib,2ω2 + g1Ic,2ω2}
+ ZA{−gm1Iq,2ω2 + j2ω2c1Ic,2ω2}
+
Zpi,2ω2
ZIN,2ω2 + ZS,2ω2
Ic,2ω2 (16)
where ZA is
(ZS,2ω2+RB+RE)Zpi,2ω2
ZIN,2ω2
+ZS,2ω2
.
IO,2ω2 = ZA
gm
2rpi
(
1
2VT ηC
−
1
2VT ηB
)
V 2be,ω2
+ ZAjω2gm
Cdiff
(βac + 1)
(
1
2ηCVT
−
1
2ηBVT
)
V 2be,ω2
+ ZAjω2gmCjE
(
1
2ηCVT
−
1
4(Vbi − VBE)
)
V 2be,ω2
+
Zpi,2ω2
ZIN,2ω2 + ZS,2ω2
gm
4ηCVT
V 2be,ω2 (17)
The first term in (17) originates from the cancellation
of rpi and gm nonlinearities which have exponential
forms. If the ideality factors of the current sources
are identical, the second order IM distortion gener-
ated from the nonlinear base current source (rpi) can
be completely removed by a portion of the nonlinear
distortion generated from the collector current source
(gm). The second term shows the cancellation between
Cdiff and gm nonlinearities. The second term of (17)
has the nonlinear term from the diffusion capacitance,
which is reduced by a factor of 1/(βac + 1) and can
again be completely removed by matching the ideality
factors of the current sources. The third term gener-
ated from the base-emitter depletion capacitance can
be cancelled by some portion of that generated from iC ,
but can not be completely eliminated due to the differ-
ent origin of sources. The last term is the remained gm
nonlinear portion. The coefficient
Zpi,2ω2gm
ZIN,2ω2
+ZS,2ω2
of the
last term is identical to the degenerated gm nonlinear-
ity equation at a low frequency. In summary, the rpi
and Cdiff nonlinearities almost completely cancelled
by gm nonlinearity and Cdepl nonlinearity is partially
removed by gm nonlinearity. The remained gm non-
linear component is quite similar to the low frequency
case with large degeneration resistances.
IV. The Third Order Analysis
We have performed a similar analysis for the third or-
der Intermodulation. From a similar sequence to the
second order analysis, the third order output current,
IO,2ω2−ω1 is given by
IO,2ω2−ω1 =
(ZS,2ω2−ω1 + RB + RE)Zpi,2ω2−ω1
ZIN,2ω2−ω1 + ZS,2ω2−ω1
gm
rpi[(
1
8η2
C
V 2
T
−
1
8η2
B
V 2
T
)
+ A
(
1
2ηCVT
−
1
2ηBVT
)]
B
+
(ZS,2ω2−ω1 + RB + RE)Zpi,2ω2−ω1
ZIN,2ω2−ω1 + ZS,2ω2−ω1
gmj(2ω2 − ω1)
Cdiff
(1 + βac)[(
1
8η2
C
V 2
T
−
1
8η2
B
V 2
T
)
+ A
(
1
2ηCVT
−
1
2ηBVT
)]
B
+
(ZS,2ω2−ω1 + RB + RE)Zpi,2ω2−ω1
ZIN,2ω2−ω1 + ZS,2ω2−ω1
gmj(2ω2 − ω1)CjE
[(
1
8η2
C
V 2
T
−
3
32(Vbi − VBE)
2
)
+ A
(
1
2ηCVT
−
1
4(Vbi − VBE)
)]
B
+
Zpi,2ω2−ω1
ZIN,2ω2−ω1 + ZS,2ω2−ω1
gm
(
1
8η2
C
V 2
T
+ A
1
2ηCVT
)
B (18)
where
A = −
(ZS,2ω2 + RB + RE)(b2 + j2ω2c2) + REg2
2[1 + (ZS,2ω2 + RB + RE)(b1 + j2ω2c1) + REg1]
−
(ZS,ω2−ω1 + RB + RE)(b2 + j(ω2 − ω1)c2) + REg2
1 + (ZS,ω2−ω1 + RB + RE)(b1 + j(ω2 − ω1)c1) + REg1
(19)
B = V
2
be,ω2
V
∗
be,ω1
The third-order output IM current in (18) has very
similar form to the second-order one. The first term
of (18) indicates that Ib,2ω2−ω1 is partially cancelled
by
ZS,2ω2−ω1
+RB+RE
ZIN,2ω2−ω1
+ZS,2ω2−ω1
Ic,2ω2−ω1 , and it can be per-
fectly cancelled out for a matched ideality factors.
The second term of (18) represents that the diffu-
sion charge part of Iq,2ω2−ω1 is partially cancelled by
ZS,2ω2−ω1
+RB+RE
ZIN,2ω2−ω1
+ZS,2ω2−ω1
Ic,2ω2−ω1 . The third term shows
that the distortion signal generated from the base-
emitter depletion capacitance is again incompletely
cancelled by some portion of that generated from
iC . The last one is non-cancelled gm portion of
Zpi,2ω2−ω1gm
ZIN,2ω2−ω1
+ZS,2ω2−ω1
. It can be expressed at a low fre-
quency as,
Zpi,2ω2−ω1
ZIN,2ω2−ω1 + ZS,2ω2−ω1
gm ∼=
rpigm
rpi + gmrpiRE +RE +RB
∼=
gm
1 + gmRE
. (20)
Symbol Value Symbol Value
IE 16 mA βdc 40
IB 0.39 mA βac 68
Pin -60 dBm ZL 150 Ω
ηC 1.0 ηB 1.7
g1 0.0089 ∆f 1 MHz
g2 0.1006 τB 8.72 × 10
−13
g3 0.7620 CjE 4.3 × 10
−13
c1 9.63 × 10−13 gm1 0.6027
c2 1.42 × 10−11 gm2 11.6350
c3 2.05 × 10−10 gm3 149.7427
RB 8 Ω RE 2 Ω
Vbi 1.627 V VBE 1.60 V
Table 1
The Model Parameters of HBT
0.1 1 10 100
-320
-300
-280
-260
-240
-220
-200
Fo
u
r 
te
rm
s 
c
o
m
po
si
n
g 
of
 I O
,
2w
2-
w
1
Frequency [GHz]
 IO,2w2-w1
 1st term of Eqn. (18)
 2nd term of Eqn. (18)
 3rd term of Eqn. (18)
 Last term of Eqn. (18)
Fig. 3. Four terms composing IO,2ω2−ω1 in (18).
It is identical to the low frequency equation and the
emitter degeneration resistance is effective to reduce
the last term.
For the analytical calculation of the nonlinear com-
ponents of HBT, we used the model parameters of
POSTECH built 3× 20 µm2 emitter HBT. This de-
vice can deliver about 13 dBm of power. The param-
eters are summarized in Table 1. For the calculation,
ZS is adjusted to have impedance matching at the fre-
quencies and input power is -60 dBm. Fig. 3 shows
the frequency dependent values of the above 4 terms
of IO,2ω2−ω1 in (18). The cancellations between rpi
and gm, and Cdiff and gm nonlinearities is quite good
for all frequencies. Therefore, the dominant nonlinear
sources for all frequencies are Cdepl and gm. The third
term becomes comparable to the last term at 2 GHz
and above. At a low frequency (below 2 GHz), gm non-
linearity with large degeneration resistor creates IM3.
Between 2 ∼ 20 GHz, IM3 are generated by Cdepl and
gm nonlinearities. At higher frequencies, the dominant
source is Cdepl. Complete cancellation of Cdiff and rpi
nonlinearities by gm nonlinearity using identical ideal-
ity factors does not have any significant impact because
they are rather small amount. We scanned ηC and ηB
for a maximum IP3, HBT with ηC = 1.0 and ηB = 2.0
has maximum IP3 at 2 GHz (seen in Fig. 4).
Degeneration effects of the emitter and base resistances
are also studied. Because ZIN,2ω2−ω1 does not have
any feedback term for an HBT with RE = 0, the
portion,
Zpi,2ω2−ω1
ZIN,2ω2−ω1
+ZS,2ω2−ω1
, becomes relatively large,
and IP3 level is degraded. The maximum IP3 of HBT
is obtained at RE = 5 Ω and RB = 0 Ω, which is 28.2
dBm. As RE increases, IP3 level increases because the
portion,
Zpi,2ω2−ω1
ZIN,2ω2−ω1
+ZS,2ω2−ω1
is reduced. But, larger
RE reduces not only the IM3 signal but also the fun-
damental signal, and a optimum value of RE is about
5 Ω in our case. The emitter degeneration resistance
is more effective than the base resistance.
As can seen from (18) and (19), the third order IM cur-
rents are dependent on the second harmonic impedance
of ZS,2ω2 and ZS,ω2−ω1 . We have calculated the IP3
of HBT using the source-termination at the second-
harmonic frequency. The linearity of HBT is remark-
ably promoted for ZS,2ω2 = 0 and ZS,ω2−ω1 = 0. We
also compute the IP3 levels of HBT with ZS,2ω2 = ∞
or ZS,ω2−ω1 = ∞, but it is degraded. Fig. 5 shows
contour lines of IP3 level of HBT with ZS,2ω2 = 0 and
ZS,ω2−ω1 = 0 for various ηC and ηB . The maximum
IP3 of HBT with the source-termination is about 31
dBm for ηC = 1.15 and ηB = 1.55 which is 3.5 dB
improvement.
V. Conclusions
To accurately understand the linear characteristics
of HBT, we developed an analytical nonlinear HBT
model using Volterra Series analysis. Our model con-
siders four nonlinearities: rpi, Cdiff , Cdepl, and gm.
The analysis we present reveals that the cancellations
between rpi and gm, and Cdiff and gm nonlinearities
are quite good for all frequencies. Further, Cdepl and
gm are the main nonlinear sources of HBT. gm can be
linearized using degeneration resistors. The degenera-
tion resistor, RE , is more effective than RB for reduc-
ing gm nonlinearity. The Cdepl nonlinearity becomes
important at a high freequency. This analysis also pro-
vides the dependency of the source second harmonic
impedance on the linearity of HBT. The IP3 of HBT
improves considerably by setting the second harmonic
impedance of ZS,2ω2 = 0 and ZS,ω2−ω1 = 0.
Acknowledgements
This work was supported in part by the Agency for
Defense Development and the Brain Korea 21 Project
of the Ministry of Education.
References
[1] Christopher T. M. Chang and Han-Tzong Yuan, “GaAs
HBT’s for High-Speed Digital Integrated Circuit Applica-
tions,” Proc.IEEE, vol. 81, no. 12, pp. 1727-1743, 1993.
[2] P. M. Asbeck, M. F. Chang, J. J. Corcoran, J. F. Jensen, R.
N. Nottenburg, A. Oki, and H. T. Yuan, “HBT Application
Prospects in the US: Where and when?,” IEEE GaAs IC
Symp. Tech. Dig., Monterey, CA., pp. 7-10, Oct. 1991.
[3] Guang-Bo Gao, David J. Roulston, and Hadis Morkoc, “De-
sign Study of AlGaAs/GaAs HBTs,” IEEE Trans. Electron
Devices, vol. ED-37, no.5, pp. 1199-1208, May 1990.
[4] M. E. Kim, A. K. Oki, J. B. Camou, P. D. Chow, B. L.
Nelson, D. M. Smith, J. C. Canyon, C. C. Yang, R. Dixit,
and B. R. Allen, “12-40 GHz Low Harmonic Distortion and
Phase Noise Performance of GaAs HBTs,” IEEE GaAs IC
Symposium Digest, pp. 117-120, Nov. 1988.
[5] Stephen A. Maas, Bradford L. Nelson, and Donald L. Tait,
Fig. 4. The contour lines of IP3 level of an HBT using our
model parameters with various ηC and ηB .
Fig. 5. The contour lines of IP3 level of an HBT with ZS,2ω2
= 0 and ZS,ω2−ω1 = 0 using our model parameters with
various ηC and ηB .
“Intermodulations in HBTs,” IEEE Trans. Microwave The-
ory and Tech., vol. MTT-40, no. 3, pp. 442-448, Mar. 1992.
[6] Taisuke Iwai, Shiro Ohara, Horoshi Yamada, Yasuhiro Ya-
maguchi, Kenji Imanishi, and Kazukiyo Joshin, “High Effi-
ciency and High Linearity InGaP/GaAs HBT Power Ampli-
fiers: Matching Techniques of Source and Load Impedance to
Improve Phase Distortion and Linearity,” IEEE Trans. Elec-
tron Devices, vol. ED-45, no. 6, pp.1196-1200, June 1998.
[7] K. W. Kobayashi, J. C. Cowles, L. T. Tran, A. Gutierrez-
Aitken, M. Nishimoto, J. H. Elliott, T. R. Block, A. K. Oki,
and D. C. Streit, “A 44-GHz-High IP3 InP HBT MMIC
Amplifier for low DC Power Millimeter-wave Receiver Ap-
plications,” IEEE Journal of Solid-State Circuits,” vol. 34,
no. 9, pp. 1188-1194, Sept. 1999.
[8] Apostolos Samelis and Dimitris Pavlidis, “Mechanisms De-
termining Third Order Intermodulation Distortion in Al-
GaAs/GaAs HBTs,” IEEE Trans. Microwave Theory and
Tech., vol. MTT-40, No. 12, pp. 2374-2380, Dec. 1992.
[9] J. Lee, W. Kim, T. Rho, and B. Kim, “Intermodula-
tion Mechanism and Linearization of AlGaAs/GaAs HBT’s,”
IEEE Trans. on Microwave Theory and Tech., vol. MTT-45,
no. 12, pp. 2065-2072, Dec. 1997.
[10] Nan Lei Wang, Wu Jing Ho, and J. A. Higgins, “Al-
GaAs/GaAs HBT Linearity Characteristics,” IEEE Trans.
Microwave Theory and Tech., vol. MTT-42, no. 10, pp. 1845-
1850, Oct. 1994.
[11] Peter Asbeck, “HBT Linearity and Basic Linearization Ap-
proaches,” IEEE MTT-S Int. Microwave Symp. Workshop,
Baltimore, Maryland, June 1998.
[12] M. Iwamoto, T. S. Low, C. P. Hutchinson, J. B. Scott, A.
Cognata, X. Qin, L. H. Camnitz, P. M. Asbeck, and D. C.
D’Avanzo, “Influence of Collector Design on InGaP/GaAs
HBT Linearity,” IEEE MTT-S Int. Microwave Symp. Dig.,
Boston, MA, pp. 757-760, Jun. 2000.
[13] W. Kim, S. Kang, K. Lee, M. Chung, Y. Yang, and B. Kim,
“The Effects of Cbc on the Linearity of AlGaAs/GaAs Power
HBT,” IEEE Trans. on Microwave Theory and Tech., vol.
MTT-49, no. 49, pp. 1270-1276, Jul. 2001.
[14] Stephen A. Maas, Nonlinear Microwave Circuits, Artech
House, Norwood, MA, 1988.


NONLINEAR BEHAVIOR OF POWER HBTWoonyun Kim*, Sanghoon Kang**, Kyungho Lee**, Minchul Chung**, and Bumman Kim***RFIC Design Team, System LSI Division, Samsung Electronics Co., Ltd., San 24, Nongseo-Lee,Kiheung-Eup, Yongin-Si, Kyungki-Do, 449-711, Korea.Phone: +82-54-279-5584, Fax: +82-54-279-2903, E-mail: kwn@postech.ac.kr**Dept. of E. E. Eng. and MARC, POSTECH, San 31, Hyoja-Dong, Pohang, Kyungbuk, 790-784, Korea.Abstract— To understand the linear characteristicsof HBT more accurately, an analytical nonlinear HBTmodel using Volterra Series analysis is developed. Themodel considers four nonlinear components: rpi, Cdiff ,Cdepl, and gm. It shows that nonlinearities of rpi andCdiff are almost completely cancelled by gm nonlinear-ity at all frequencies. The residual gm nonlinearity arehighly degenerated by the input impedances. There-fore, rpi, Cpi and gm nonlinearities generate less IM3than Cbc. If Cbc is linearized, Cdepl and gm are the mainnonlinear sources of HBT, and Cdepl becomes very im-portant at a high frequency. It was also found thatthe degeneration resistor, RE , is more effective than RBfor reducing gm nonlinearity. This analysis also pro-vides the dependency of the source second harmonicimpedance on the linearity of HBT. The IM3 of HBTis significantly reduced by setting the second harmonicimpedance of ZS,2ω2 = 0 and ZS,ω2−ω1 = 0.I. IntroductionThe transmitter of the handset of digital mobile com-munication systems requires highly efficient linearpower amplifiers [1–4]. HBTs are widely used for theamplifiers and their nonlinear behavior has been exten-sively measured and analyzed [4–12]. It is commonlyknown that Cbc is the dominant nonlinear source andshould be linearized to reduce the intermodulation dis-tortions [7,9–13]. Cbc is a depletion capacitance and itis rather moderate nonlinear component. It is surpris-ing in view of the fact that HBT has highly nonlinearsources. The dependence of the HBT’s base current,iB , on base-to-emitter voltage, vBE , is an exponentialfunction, one of the strongest nonlinearities found innature. So is its collector current, iC , which is basicallysimilar to iB . Furthermore, the junction capacitance,primarily a diffusion capacitance, is also strongly non-linear. These exponential nonlinear behaviors are notseen in the nonlinear characteristics of HBTs. Thus,the measured high linear characteristics of HBT hasmotivated many researchers to study intermodulation(IM) mechanism of HBT. The good linearity of HBT’swas attributed to the partial cancellation between IMcurrents generated from the exponential junction cur-rent and the junction capacitance [5], or the partialcancellation of IM currents from the total base-emittercurrent and the total base-collector current [8]. Ac-cording to reference [9], the high linear characteris-tis of HBT resulted from the almost complete can-cellation between the output nonlinear current com-ponents generated by emitter-base current source andbase-collector current source. It was also reported thatthe emitter and base resistances linearize the HBT out-put [10]. Because of the different descriptions for thelinear characteristics, a more complete explanation isneeded.To understand the linear characteristics of HBT moreaccurately, we have developed an analytical nonlinearHBT model using Volterra Series analysis [14]. Thepresented analysis describes the fundamental nonlinearbehavior of HBT.II. The Nonlinear Model of HBTThe simplified equivalent circuit of HBT used for theanalysis is shown in Fig. 1. The emitter and base re-sistances are RE and RB , respectively. These resis-tances are linear components. The base and collectornonlinear current source are represented as iB and iC ,respectively. And, the base-emitter nonlinear capaci-tance is appeared as a charge, qBE . CBC is assumed tobe linearized for a constant capacitance and is omittedfor the nonlinear circuit analysis. ZS and ZL repre-sent the source and load impedances, respectively. Thesource is conjugate-matched for the maximum gain.This model includes all important nonlinearities and isgood enough for representing essential nonlinear prop-erties of HBT.The nonlinear elements are represented by the third-order expansion of Taylor series. Under a small-signalcondition, the base nonlinear current source can beexpanded at the vicinity of its bias point. The iB ismodeled asiB = ISB[exp(vBEηBVT)− 1](1)ib =IBηBVTvbe +IB2η2BV2Tv2be +IB6η3BV3Tv3be (2)≡ g1vbe + g2v2be + g3v3be (3)where ISB represents the saturation current, ηB is theideality factor of the base current, IB is dc base cur-rent, and VT is thermal voltage. ib and vbe are thesmall-signal components of iB and vBE , respectively.Also the coefficient g1 = 1/rpi is a linear junction con-ductance.iC is also modeled asiC = ISC[exp(vBEηCVT)− 1](4)where ISC represents the saturation current, ηC is theideality factor of the collector current. It can be ex-panded at the vicinity of its bias point, yieldingRBREVSZSZLiB iCqBEvBE+-Fig. 1. HBT nonlinear equivalent modelic =ICηCVTvbe +IC2η2CV2Tv2be +IC6η3CV3Tv3be (5)≡ gm1vbe + gm2v2be + gm3v3be (6)where IC is dc collector current, and ic is the small-signal component of iC . The coefficient, gm1 is gener-ally called gm. The common emitter current gain (βac)can be calculated by gm1/g1.The stored charge at the base-emitter junction, qBE ,is the sum of diffusion charge and depletion charge.qBE = τBiC + τEiB + qAEXNNE (7)qbe =(ICηCVTτB +IBηBVTτE + CjE)vbe+(IC2η2CV2TτB +IB2η2BV2TτE +CjE4(Vbi − VBE))v2be+(IC6η3CV3TτB +IB6η3BV3TτE +CjE8(Vbi − VBE)2)v3be (8)≡ c1vbe + c2v2be + c3v3be (9)where τB is the base transit time of the minority car-rier in base, and τE is the transit time of the minoritycarrier in the emitter, and is assumed to be compara-ble to τB . AE is the emitter area, XN is the depletionwidth of the emitter, and NE is the doping concentra-tion of the emitter. qbe is the small-signal componentof qBE , CjE = AE√q²E²BNEPB2(²ENE+²BPB)(Vbi−VBE), and Cdiff isICηCVTτB +IBηBVTτE . For simplicity, depletion approximationhas been used. Vbi is built-in potential of base-emitter junc-tion and VBE is the base-emitter dc voltage. c1 is so-called thebase-emitter junction capacitance, Cpi .III. The Second Order Harmonic AnalysisThe first-order equation for the base-to-emitter voltage Vbe,ωqat the excitation frequency ωq can be expressed byVbe,ωq =Zpi,ωqZIN,ωq + ZS,ωqVS,ωq (10)where Zpi,ωq =1g1+jωqc1, ZIN,ωq (= RB + Zpi,ωq + RE(1 +gm1Zpi,ωq )) is the input impedance of the HBT, andthe source impedance, ZS,ωq is conjugately matchedto ZIN,ωq . The harmonics can be found by means ofVolterra Series analysis. Fig. 2 shows the equivalentcircuit for the second- and third-order nonlinear anal-ysis. The second-order intermodulation current gen-erated by the nonlinear base-emitter junction capac-itance, Iq,2, and by the nonlinear base and collectorcurrent sources, Ib,2 and Ic,2 are given byRBREZSZLgm1vbevbe,2+-Ic,2Ib,2 Cdiff+CdeplIq,2rpiZINIq,3Ib,3 Ic,3IOFig. 2. HBT equivalent circuit for the 2nd- and 3rd-order in-termodulation analysis.Iq,2ω2 =12j(2ω2)c2V2be,ω2= jω2c2V2be,ω2, (11)Ib,2ω2 =12g2V2be,ω2, (12)Ic,2ω2 =12gm2V2be,ω2. (13)Performing a linear analysis of this circuit, we find thebase-emitter junction voltage at the second harmonic;Vbe,2ω2 =−(ZS,2ω2 +RB +RE)Zpi,2ω2 (Ib,2ω2 + Iq,2ω2 )ZIN,2ω2 + ZS,2ω2−REZpi,2ω2Ic,2ω2ZIN,2ω2 + ZS,2ω2(14)The output current IO,2ω2 at this second-harmonic fre-quency isIO,2ω2 = gm1Vbe,2ω2 + Ic,2ω2 (15)Substituting (14) into (15),REgm1Zpi,2ω2ZIN,2ω2+ZS,2ω2Ic,2ω2 is can-celled out by the gm1RE feedback term in the last termof (14). The remain terms of the above equation is sim-plified asIO,2ω2 = ZA{−gm1Ib,2ω2 + g1Ic,2ω2}+ ZA{−gm1Iq,2ω2 + j2ω2c1Ic,2ω2}+Zpi,2ω2ZIN,2ω2 + ZS,2ω2Ic,2ω2 (16)where ZA is(ZS,2ω2+RB+RE)Zpi,2ω2ZIN,2ω2+ZS,2ω2.IO,2ω2 = ZAgm2rpi(12VT ηC−12VT ηB)V 2be,ω2+ ZAjω2gmCdiff(βac + 1)(12ηCVT−12ηBVT)V 2be,ω2+ ZAjω2gmCjE(12ηCVT−14(Vbi − VBE))V 2be,ω2+Zpi,2ω2ZIN,2ω2 + ZS,2ω2gm4ηCVTV 2be,ω2 (17)The first term in (17) originates from the cancellationof rpi and gm nonlinearities which have exponentialforms. If the ideality factors of the current sourcesare identical, the second order IM distortion gener-ated from the nonlinear base current source (rpi) canbe completely removed by a portion of the nonlineardistortion generated from the collector current source(gm). The second term shows the cancellation betweenCdiff and gm nonlinearities. The second term of (17)has the nonlinear term from the diffusion capacitance,which is reduced by a factor of 1/(βac + 1) and canagain be completely removed by matching the idealityfactors of the current sources. The third term gener-ated from the base-emitter depletion capacitance canbe cancelled by some portion of that generated from iC ,but can not be completely eliminated due to the differ-ent origin of sources. The last term is the remained gmnonlinear portion. The coefficientZpi,2ω2gmZIN,2ω2+ZS,2ω2of thelast term is identical to the degenerated gm nonlinear-ity equation at a low frequency. In summary, the rpiand Cdiff nonlinearities almost completely cancelledby gm nonlinearity and Cdepl nonlinearity is partiallyremoved by gm nonlinearity. The remained gm non-linear component is quite similar to the low frequencycase with large degeneration resistances.IV. The Third Order AnalysisWe have performed a similar analysis for the third or-der Intermodulation. From a similar sequence to thesecond order analysis, the third order output current,IO,2ω2−ω1 is given byIO,2ω2−ω1 =(ZS,2ω2−ω1 + RB + RE)Zpi,2ω2−ω1ZIN,2ω2−ω1 + ZS,2ω2−ω1gmrpi[(18η2CV 2T−18η2BV 2T)+ A(12ηCVT−12ηBVT)]B+(ZS,2ω2−ω1 + RB + RE)Zpi,2ω2−ω1ZIN,2ω2−ω1 + ZS,2ω2−ω1gmj(2ω2 − ω1)Cdiff(1 + βac)[(18η2CV 2T−18η2BV 2T)+ A(12ηCVT−12ηBVT)]B+(ZS,2ω2−ω1 + RB + RE)Zpi,2ω2−ω1ZIN,2ω2−ω1 + ZS,2ω2−ω1gmj(2ω2 − ω1)CjE[(18η2CV 2T−332(Vbi − VBE)2)+ A(12ηCVT−14(Vbi − VBE))]B+Zpi,2ω2−ω1ZIN,2ω2−ω1 + ZS,2ω2−ω1gm(18η2CV 2T+ A12ηCVT)B (18)whereA = −(ZS,2ω2 + RB + RE)(b2 + j2ω2c2) + REg22[1 + (ZS,2ω2 + RB + RE)(b1 + j2ω2c1) + REg1]−(ZS,ω2−ω1 + RB + RE)(b2 + j(ω2 − ω1)c2) + REg21 + (ZS,ω2−ω1 + RB + RE)(b1 + j(ω2 − ω1)c1) + REg1(19)B = V2be,ω2V∗be,ω1The third-order output IM current in (18) has verysimilar form to the second-order one. The first termof (18) indicates that Ib,2ω2−ω1 is partially cancelledbyZS,2ω2−ω1+RB+REZIN,2ω2−ω1+ZS,2ω2−ω1Ic,2ω2−ω1 , and it can be per-fectly cancelled out for a matched ideality factors.The second term of (18) represents that the diffu-sion charge part of Iq,2ω2−ω1 is partially cancelled byZS,2ω2−ω1+RB+REZIN,2ω2−ω1+ZS,2ω2−ω1Ic,2ω2−ω1 . The third term showsthat the distortion signal generated from the base-emitter depletion capacitance is again incompletelycancelled by some portion of that generated fromiC . The last one is non-cancelled gm portion ofZpi,2ω2−ω1gmZIN,2ω2−ω1+ZS,2ω2−ω1. It can be expressed at a low fre-quency as,Zpi,2ω2−ω1ZIN,2ω2−ω1 + ZS,2ω2−ω1gm ∼=rpigmrpi + gmrpiRE +RE +RB∼=gm1 + gmRE. (20)Symbol Value Symbol ValueIE 16 mA βdc 40IB 0.39 mA βac 68Pin -60 dBm ZL 150 ΩηC 1.0 ηB 1.7g1 0.0089 ∆f 1 MHzg2 0.1006 τB 8.72 × 10−13g3 0.7620 CjE 4.3 × 10−13c1 9.63 × 10−13 gm1 0.6027c2 1.42 × 10−11 gm2 11.6350c3 2.05 × 10−10 gm3 149.7427RB 8 Ω RE 2 ΩVbi 1.627 V VBE 1.60 VTable 1The Model Parameters of HBT0.1 1 10 100-320-300-280-260-240-220-200Four terms composing of I O,2w2-w1Frequency [GHz] IO,2w2-w1 1st term of Eqn. (18) 2nd term of Eqn. (18) 3rd term of Eqn. (18) Last term of Eqn. (18)Fig. 3. Four terms composing IO,2ω2−ω1 in (18).It is identical to the low frequency equation and theemitter degeneration resistance is effective to reducethe last term.For the analytical calculation of the nonlinear com-ponents of HBT, we used the model parameters ofPOSTECH built 3× 20 µm2 emitter HBT. This de-vice can deliver about 13 dBm of power. The param-eters are summarized in Table 1. For the calculation,ZS is adjusted to have impedance matching at the fre-quencies and input power is -60 dBm. Fig. 3 showsthe frequency dependent values of the above 4 termsof IO,2ω2−ω1 in (18). The cancellations between rpiand gm, and Cdiff and gm nonlinearities is quite goodfor all frequencies. Therefore, the dominant nonlinearsources for all frequencies are Cdepl and gm. The thirdterm becomes comparable to the last term at 2 GHzand above. At a low frequency (below 2 GHz), gm non-linearity with large degeneration resistor creates IM3.Between 2 ∼ 20 GHz, IM3 are generated by Cdepl andgm nonlinearities. At higher frequencies, the dominantsource is Cdepl. Complete cancellation of Cdiff and rpinonlinearities by gm nonlinearity using identical ideal-ity factors does not have any significant impact becausethey are rather small amount. We scanned ηC and ηBfor a maximum IP3, HBT with ηC = 1.0 and ηB = 2.0has maximum IP3 at 2 GHz (seen in Fig. 4).Degeneration effects of the emitter and base resistancesare also studied. Because ZIN,2ω2−ω1 does not haveany feedback term for an HBT with RE = 0, theportion,Zpi,2ω2−ω1ZIN,2ω2−ω1+ZS,2ω2−ω1, becomes relatively large,and IP3 level is degraded. The maximum IP3 of HBTis obtained at RE = 5 Ω and RB = 0 Ω, which is 28.2dBm. As RE increases, IP3 level increases because theportion,Zpi,2ω2−ω1ZIN,2ω2−ω1+ZS,2ω2−ω1is reduced. But, largerRE reduces not only the IM3 signal but also the fun-damental signal, and a optimum value of RE is about5 Ω in our case. The emitter degeneration resistanceis more effective than the base resistance.As can seen from (18) and (19), the third order IM cur-rents are dependent on the second harmonic impedanceof ZS,2ω2 and ZS,ω2−ω1 . We have calculated the IP3of HBT using the source-termination at the second-harmonic frequency. The linearity of HBT is remark-ably promoted for ZS,2ω2 = 0 and ZS,ω2−ω1 = 0. Wealso compute the IP3 levels of HBT with ZS,2ω2 = ∞or ZS,ω2−ω1 = ∞, but it is degraded. Fig. 5 showscontour lines of IP3 level of HBT with ZS,2ω2 = 0 andZS,ω2−ω1 = 0 for various ηC and ηB . The maximumIP3 of HBT with the source-termination is about 31dBm for ηC = 1.15 and ηB = 1.55 which is 3.5 dBimprovement.V. ConclusionsTo accurately understand the linear characteristicsof HBT, we developed an analytical nonlinear HBTmodel using Volterra Series analysis. Our model con-siders four nonlinearities: rpi, Cdiff , Cdepl, and gm.The analysis we present reveals that the cancellationsbetween rpi and gm, and Cdiff and gm nonlinearitiesare quite good for all frequencies. Further, Cdepl andgm are the main nonlinear sources of HBT. gm can belinearized using degeneration resistors. The degenera-tion resistor, RE , is more effective than RB for reduc-ing gm nonlinearity. The Cdepl nonlinearity becomesimportant at a high freequency. This analysis also pro-vides the dependency of the source second harmonicimpedance on the linearity of HBT. The IP3 of HBTimproves considerably by setting the second harmonicimpedance of ZS,2ω2 = 0 and ZS,ω2−ω1 = 0.AcknowledgementsThis work was supported in part by the Agency forDefense Development and the Brain Korea 21 Projectof the Ministry of Education.References[1] Christopher T. M. Chang and Han-Tzong Yuan, “GaAsHBT’s for High-Speed Digital Integrated Circuit Applica-tions,” Proc.IEEE, vol. 81, no. 12, pp. 1727-1743, 1993.[2] P. M. Asbeck, M. F. Chang, J. J. Corcoran, J. F. Jensen, R.N. Nottenburg, A. Oki, and H. T. Yuan, “HBT ApplicationProspects in the US: Where and when?,” IEEE GaAs ICSymp. Tech. Dig., Monterey, CA., pp. 7-10, Oct. 1991.[3] Guang-Bo Gao, David J. Roulston, and Hadis Morkoc, “De-sign Study of AlGaAs/GaAs HBTs,” IEEE Trans. ElectronDevices, vol. ED-37, no.5, pp. 1199-1208, May 1990.[4] M. E. Kim, A. K. Oki, J. B. Camou, P. D. Chow, B. L.Nelson, D. M. Smith, J. C. Canyon, C. C. Yang, R. Dixit,and B. R. Allen, “12-40 GHz Low Harmonic Distortion andPhase Noise Performance of GaAs HBTs,” IEEE GaAs ICSymposium Digest, pp. 117-120, Nov. 1988.[5] Stephen A. Maas, Bradford L. Nelson, and Donald L. Tait,Fig. 4. The contour lines of IP3 level of an HBT using ourmodel parameters with various ηC and ηB .Fig. 5. The contour lines of IP3 level of an HBT with ZS,2ω2= 0 and ZS,ω2−ω1 = 0 using our model parameters withvarious ηC and ηB .“Intermodulations in HBTs,” IEEE Trans. Microwave The-ory and Tech., vol. MTT-40, no. 3, pp. 442-448, Mar. 1992.[6] Taisuke Iwai, Shiro Ohara, Horoshi Yamada, Yasuhiro Ya-maguchi, Kenji Imanishi, and Kazukiyo Joshin, “High Effi-ciency and High Linearity InGaP/GaAs HBT Power Ampli-fiers: Matching Techniques of Source and Load Impedance toImprove Phase Distortion and Linearity,” IEEE Trans. Elec-tron Devices, vol. ED-45, no. 6, pp.1196-1200, June 1998.[7] K. W. Kobayashi, J. C. Cowles, L. T. Tran, A. Gutierrez-Aitken, M. Nishimoto, J. H. Elliott, T. R. Block, A. K. Oki,and D. C. Streit, “A 44-GHz-High IP3 InP HBT MMICAmplifier for low DC Power Millimeter-wave Receiver Ap-plications,” IEEE Journal of Solid-State Circuits,” vol. 34,no. 9, pp. 1188-1194, Sept. 1999.[8] Apostolos Samelis and Dimitris Pavlidis, “Mechanisms De-termining Third Order Intermodulation Distortion in Al-GaAs/GaAs HBTs,” IEEE Trans. Microwave Theory andTech., vol. MTT-40, No. 12, pp. 2374-2380, Dec. 1992.[9] J. Lee, W. Kim, T. Rho, and B. Kim, “Intermodula-tion Mechanism and Linearization of AlGaAs/GaAs HBT’s,”IEEE Trans. on Microwave Theory and Tech., vol. MTT-45,no. 12, pp. 2065-2072, Dec. 1997.[10] Nan Lei Wang, Wu Jing Ho, and J. A. Higgins, “Al-GaAs/GaAs HBT Linearity Characteristics,” IEEE Trans.Microwave Theory and Tech., vol. MTT-42, no. 10, pp. 1845-1850, Oct. 1994.[11] Peter Asbeck, “HBT Linearity and Basic Linearization Ap-proaches,” IEEE MTT-S Int. Microwave Symp. Workshop,Baltimore, Maryland, June 1998.[12] M. Iwamoto, T. S. Low, C. P. Hutchinson, J. B. Scott, A.Cognata, X. Qin, L. H. Camnitz, P. M. Asbeck, and D. C.D’Avanzo, “Influence of Collector Design on InGaP/GaAsHBT Linearity,” IEEE MTT-S Int. Microwave Symp. Dig.,Boston, MA, pp. 757-760, Jun. 2000.[13] W. Kim, S. Kang, K. Lee, M. Chung, Y. Yang, and B. Kim,“The Effects of Cbc on the Linearity of AlGaAs/GaAs PowerHBT,” IEEE Trans. on Microwave Theory and Tech., vol.MTT-49, no. 49, pp. 1270-1276, Jul. 2001.[14] Stephen A. Maas, Nonlinear Microwave Circuits, ArtechHouse, Norwood, MA, 1988.

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