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: [email protected]
**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, Cdif f, Cdepl, and gm. It shows that nonlinearities of rπ and
Cdif f are almost completely cancelled bygm
nonlinear-ity at all frequencies. The residualgmnonlinearity are highly degenerated by the input impedances. There-fore, rπ, Cπ and gm nonlinearities generate less IM3 thanCbc. IfCbc is linearized,Cdepl andgmare the main nonlinear sources of HBT, andCdepl becomes very im-portant at a high frequency. It was also found that the degeneration resistor,RE, is more effective thanRB 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 ofZS,2ω2 = 0 andZS,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]. Cbcis 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. CBCis 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 µ v BE ηBVT ¶ − 1 ¸ (1) ib = IB ηBVT vbe+ IB 2η2 BVT2 vbe2 + IB 6η3 BVT3 vbe3 (2) ≡ g1vbe+ g2vbe2 + g3vbe3 (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 µ v BE 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 ZL iB qBE iC vBE +
-Fig. 1. HBT nonlinear equivalent model
ic = IC ηCVT vbe+ IC 2η2 CVT2 vbe2 + IC 6η3 CVT3 v3be (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 = µ I C ηCVT τB+ IB ηBVT τE+ CjE ¶ vbe + Ã IC 2η2 CVT2 τB+ IB 2η2 BVT2 τE+ CjE 4(Vbi− VBE) ! vbe2 + Ã IC 6η3 CVT3 τB+ IB 6η3 BVT3 τE+ CjE 8(Vbi− VBE)2 ! vbe3 (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 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 q q² E²BNEPB 2(²ENE+²BPB)(Vbi−VBE), and Cdif f is IC ηCVTτB + IB
ηBVTτE. For simplicity, depletion approximation
has been used. Vbi is built-in potential of base-emitter junc-tion and VBEis the base-emitter dc voltage. c1is 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 = g1+jω1qc1, 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 ZL gm1vbe vbe,2 + -Ic,2 Ib,2 Iq,2 Cdiff+Cdepl rπ ZIN Iq,3 Ib,3 Ic,3 IO
Fig. 2. HBT equivalent circuit for the 2nd- and 3rd-order in-termodulation analysis. Iq,2ω2 = 1 2j(2ω2)c2V 2 be,ω2= jω2c2V 2 be,ω2, (11) Ib,2ω2 = 1 2g2V 2 be,ω2, (12) Ic,2ω2 = 1 2gm2V 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,2ω2Ic,2ω2 is
can-celled out by the gm1REfeedback 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) whereZAis (ZS,2ω2+RB+RE)Zπ,2ω2 ZIN,2ω2+ZS,2ω2 . IO,2ω2 = ZA gm 2rπ µ 1 2VTηC − 1 2VTηB ¶ Vbe,ω2 2 + ZAjω2gm Cdif f (βac+ 1) µ 1 2ηCVT − 1 2ηBVT ¶ Vbe,ω2 2 + ZAjω2gmCjE µ 1 2ηCVT − 1 4(Vbi− VBE) ¶ Vbe,ω2 2 + Zπ,2ω2 ZIN,2ω2+ ZS,2ω2 gm 4ηCVT Vbe,ω2 2 (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
Cdif f and gmnonlinearities. 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 Cdif f 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 "à 1 8η2 CVT2 − 1 8η2 BVT2 ! + A µ 1 2ηCVT − 1 2ηBVT ¶# B + (ZS,2ω2−ω1+ RB+ RE)Zπ,2ω2−ω1 ZIN,2ω2−ω1+ ZS,2ω2−ω1 gmj(2ω2−ω1) Cdif f (1 + βac) "à 1 8η2 CVT2 − 1 8η2 BVT2 ! + A µ 1 2ηCVT − 1 2ηBVT ¶# B + (ZS,2ω2−ω1+ RB+ RE)Zπ,2ω2−ω1 ZIN,2ω2−ω1+ ZS,2ω2−ω1 gmj(2ω2−ω1)CjE "à 1 8η2 CVT2 − 3 32(Vbi−VBE)2 ! + A µ 1 2ηCVT − 1 4(Vbi−VBE) ¶# B + Zπ,2ω2−ω1 ZIN,2ω2−ω1+ ZS,2ω2−ω1gm à 1 8η2 CVT2 + 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 = Vbe,ω22 Vbe,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−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 by
ZS,2ω2−ω1+RB+RE
ZIN,2ω2−ω1+ZS,2ω2−ω1Ic,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 ou r te rm s c o m po si ng o f I O ,2 w 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 rπ
and gm, and Cdif f and gmnonlinearities is quite good for all frequencies. Therefore, the dominant nonlinear sources for all frequencies are Cdepland 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 Cdepland gmnonlinearities. At higher frequencies, the dominant source is Cdepl. Complete cancellation of Cdif f and rπ nonlinearities by gmnonlinearity 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
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, Cdif f, Cdepl, and gm. The analysis we present reveals that the cancellations between rπ and gm, and Cdif f and gm nonlinearities are quite good for all frequencies. Further, Cdepl and gmare the main nonlinear sources of HBT. gmcan 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.
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