1 Microwave Electronics Laboratory, Chalmers University of Technology, S-41296 Goteborg,Sweden2 Mitsubishi Electric Corporation, 4-1 Mizuhara Itami Hyogo, Japan
Keywords: nonlinear models for active devices; nonlinear circuit design; microwave circuits
I. INTRODUCTION
Heterojunction bipolar transistors (HBTs) have be-come very promising devices for different applica-tions at microwave and millimeter-wave frequencies.An important condition for any successful design isthe availability of an accurate large-signal model(LSM). In recent years, more than 100 papers in theliterature have been devoted on the creation of acompact model for use in CAD tools and proceduresfor extraction of parameters for these models [1–29].Despite the tremendous work already done on thesubject, we still do not have unified, accurate modelstandards in the industry, together with an automaticextraction procedure. The reason for this is that thephysics of the device is very complicated, the range ofcurrents is broad, and the power density at which thedevice operates is very high. All this, together withpoor thermal conductivity of the III-V materials,makes the problem of creating universal LSM forHBTs difficult. Many of the existing HBT models arebased on solid physical background, but again, be-
cause of the difficulties of the problem, they end upwith many empirical coefficients that are difficult toextract.
In some cases, the HBT models can be excessivelyaccurate and include effects that may not be particu-larly important; model simplification is appropriate insuch cases. When the model is complicated, an addi-tional difficulty is added to the extraction problems —such a model may exhibit problems with convergence.
II. DEVICE MODELING
The HBTs used in this study are mainly AlGaAs/GaAs HBTs with a 4 � 20 �m2 emitter [30]. SystematicDC and multi-bias S-parameter measurements were per-formed at different temperatures (�25–�125°C) in or-der to obtain a general picture of the device’s behav-ior. ICCAP software (Agilent) was used in themeasurement and extraction of basic parasitic param-eters such as parasitic resistances, parasitic capaci-tances, and bias dependencies of Cbe and Cbc.
The frequency range of the S-parameter measure-ments were determined by the frequency range inwhich the devices should operate and the availablemeasurement setup.
Correspondence to: I. Angelov; email: [email protected] online in Wiley InterScience (www.interscience.
Figures 1–3 show some results for the forwardGummel (FG) Ibe, Ice, and � obtained from thesedevices that are typical of many HBTs. As can be seenin Figure 4, when keeping the Ibe (or Ice) constant, thevoltage shift of the BE junction is almost linear andcan be used to monitor the device’s temperature.Increasing the temperature will decrease the base volt-age required to sustain the same current and the tem-perature coefficient is rather small, with TcVjbe ��0.002. A problem with this type of transistors is that� cannot be considered constant (see Fig. 3). Themodels that consider � constant, such as the Gum-mel–Poon model, produce significant error, and sep-arate equations for Ibe and Ice should be used in thismodel. In addition, it should be considered that thereis a bias dependence of the voltage Vbe � f(Vce) atwhich we have maximum of �. At the change of the
Vbe, we have maximum � and this needs to be con-sidered in the LSM. The logogriphic plots in Figures1 and 2 are close to a straight line (as should be thecase based on the device physics), but deviate at smallcurrents. This means that the main function describingthe device current should be exponential with a propercorrection.
The transistor can be described with a conventionalequivalent circuit shown in Figure 6. Current sourceIce with transconductance gm is nonlinear, and alsononlinear can be considered capacitances Cbe and Cbc.The remaining elements can be considered linear andthere is a significant amount of papers that describingextraction of the small-signal equivalent circuit [14–28]. For devices exhibiting substrate effects, an extraport should be added.
Base Current
We can significantly simply the self-heating modelingif we use the fact that, when keeping the diode current
Figure 1. Ibe FG.
Figure 2. Ice vs. Vbe.
Figure 3. FG � vs. Vbe & T.
Figure 4. Vjbe, Pbe1, Pcf 1 vs. T.
HBT Large-Signal Model for CAD 519
constant, the junction voltages vary linearly with thetemperature; however, this requires modification ofthe diode-current definition. It is a common practiceto describe the complicated Ibe dependence by severaldiodes (respective exponential functions) in order toimprove the accuracy and describe the different phys-ical phenomena occurring in the device [1–12]. Ac-cording to the device physics, we use an exponentialfunction to describe the semiconductor junction, butchange the reference points of the currents and re-spective voltages at which we normally operate thedevice to be closer to the currents (voltages) at which� is maximum (see Fig. 3). In addition to this changeof reference points, the argument of the equation forthe junction current is described as a power series.
This gives us the possibility to fit a variety of dopingprofiles; many factors and effects will influence thecurrents:
where qe� is the electron charge, Kb� is the Boltz-mann constant, and Nb1 is the ideality factor. Thecoefficient 38.695 � 1/Vt at room temperature.
This definition is equivalent to a conventional di-ode equation, but the extraction point is changed. In atextbook definition of the diode current, the extractionpoint in fact is at Vbe � ��. At small currents, themeasurement data are quite often noisy and, what ismost important, we usually do not operate the deviceat such small currents. With the reference and extrac-tion points selected in the operating region of voltagesand currents, we will describes the currents with high-est accuracy in the useful operating range.
Figure 5. Ibe vs. Vbe.
Figure 6. Transistor equivalent circuit.
520 Angelov, Choumei, and Inoue
When Vbe � Vje, the base current Ibe is equal to Ijbe
and when Vbe � 0, Ibe � 0. The terms Pbe0, Pbc0 areimportant to define the current to be zero when thevoltage at the junction is zero. The correction is small,but critical so that the HB will converge properly. Theequations for Pbe0 and Pbc0 are obtained by substitut-ing in equations for Pbe, Pbc, Vbe � 0, and Vbc � 0,respectively. Usually, one term in the power series isenough to provide accuracy of better then 5% infive-to-six decades of the diode current. If higheraccuracy is required, or for some reason is importantto keep this accuracy in a range of currents more thensix decades, then more terms can be used. Typically,three terms of the power series are enough to obtain2% global accuracy.
The approach of using power series in the argu-ment is equivalent to fitting the specific doping profileof the device which will determine currents Ibe and Ice.Using this approach, we can reduce the number of thediodes (exponents) and obtain very high accuracy.The reduction of the number of exponents in turn willimprove the convergence of the LSM.
The temperature dependence of Ibe, Ic, � versus Vbe
is one of the main factors responsible for the hightemperature sensitivity of these devices. Figure 4 andTable I show extracted parameters of the base part ofthe model, using one term in the power series of eqs.(1–3) at different temperatures. The changes of theparameters are almost linear with respect the temper-ature and this can be used to model self-heating in asimple way by using linear dependencies. In fact, thecurrent of the bipolar devices is controlled by theintrinsic diode voltage, but when the device is biasedwith a constant DC voltage source at the base, this can
produce a thermal runaway; the model should de-scribe this phenomenon.
Collector Current
In a method similar to that used with the base current,we can keep the basic form of the Gummel–Poonmodel [10], but shift the reference point at Vbep anddefine the collector current source argument as apower series:
Ic � Ipkc�exp�Pcf� � exp�Pcr���1 �Vbe
Var�
Vbc
Vaf�, (5)
where Pcf is a power series centered at Vpcs and witha variable Vbe, given by
When Vbe � Vbep, the collector current is equal to Ipkc,the transconductance is equal to gm � Ipkc � Pcf1.
The Vbep and Ipkc are taken directly from the mea-surements and there is some freedom for their selec-tion. Probably, the best choice is to select Vbep equalto the voltage at which � is maximum, but usuallyeven a simple fitting program will converge and findthe optimum values, because of the presence of an
TABLE I.
Current parametersIpkc
(mA) Ncf1 Ncf2
Vbep
(V) �r �s � � Bbe
25.0 1.3 0 1.39 0.3 0.15 0.035 105 6
Ijb
(mA) Nbe1 Nbe2 Vjbe Dvpk — — — —
0.24 1.2 0 1.39 0.01 — — — —
ParasiticsRe Rc Rbmin Rmax Rbex Rbc1 n m Vcmin
1.3 1.4 4 4.1 4.2 4 0.5 0.05 2.1
Cbep Cbe0 Cbcp Cbc0 CP10 CP11 CP20 CP21 —
5 ff 350 ff 6 ff 33 1.1 1.04 0.1 0.5 —
Temperature coefficientsTcIpkc TcVj Tc� TcCbe0 TcCbc0
0.002 �0.0018 �0.008 �0.001 0.001
HBT Large-Signal Model for CAD 521
inflection point. Typical parameters are Ibe � 0.2–2mA and Ipkc Ice (at �max) with corresponding Vbep.
Usually, the device is biased at positive Vce, theforward and reverse part can be considered equal, andthese will simply further the equations. The collectorcurrent Ice can be expressed in compact form:
Ic � Icf � tanh�� � Vce��1 � � � Vcb�
Icf � Ipkc�exp�Pcf�
� exp�Pcf 0��/cosh�Bbe�Vbe � Vbepm�� (7a)
In the case of one term in the power series the argu-ments Pcf, Pcf0 are transformed to :
Pcf � Pcf 1�Vbe � Vbepm�; Pcf 0 � Pcf 1��Vbepm�
Some of the parameters like � and Vbep are biasdependent due to the Kirk effect, and this should beaccounted for as follows:
� � �r � �s�exp�38.695
Nc1� Vce� � 1�; (7b)
Vbepm � Vbep � Vpk � �1 � tanh�Pcf 1 � Vce�
� Vsb2 � �Vcb � Vtr�2; (7c)
The parameters �r, �s reflect the dependence of thecollector current versus Vbe at small collector volt-ages, as Figure 7 and parameter Bbe in the term1/cosh(Bbe (Vce � Vbepm)) will predefine the bellshape of � dependence versus Vbe. Parameters �r and�s adjust precisely the slope at small collector volt-ages due to the Kirk effect and �, the output conduc-tance dependence at high Vce due to the Early effect.Parameter Vsb2 is used in the breakdown modeling.
The definition of Ice behaves well with eq. (7) froma numerical point of view; it converges better than eq.(5), because tanh is a limited function, with thestrictly defined solution Ice � 0 when Vce � 0. In
addition, the result will not depend on the correctchoice of the forward and reverse parameters of theGummel-type model.
Figure 8 shows a comparison between the calcu-lated results for the collector current using eqs. (5)and (7). In this figure, for the sake of simplification,one term in the power series was used. As can be seen,the results are almost identical. The model, given byeq. (7), is simpler, with only eight parameters takendirectly from measurements. Thus, we can describethe collector current with a good accuracy.
Bounded Current Models for Ibe and Ice
It is a common practice to limit the exponentialgrowth of the current using a modified exponent (softexponent). Above some certain value of the exponent,the exponential dependence is switched to a lineardependence and numerical overflow is avoided. Butthis does not completely solve the problem, because atthe connecting point the derivatives will be differentand this discontinuity will increase the simulationtime.
Taking the approach of using a power series in theargument, we are no longer required to use the softexponent, because we can limit the exponentialgrowth of the current by selecting the coefficients ofthe arguments of eq. (3) in a proper way. The deriv-atives of Ibe and Ice will not be influenced, because weretain the exponential envelope.
When using a power series in eqs. (1) and (7), theargument function shown in Figure 9 can be adjustedto track the individual properties of the device andbehave monotonically from �� to ���, but we canlimit the argument to a useful range.
We can tailor the argument by properly selectingthe coefficients to be fitted. For example, when the
Figure 7. Measured and modeled Ice vs. Vce at small Vce.
Figure 8. Comparison between the results from eqs. (5)and (7). [Color figure can be viewed in the online issue,which is available at www.interscience.wiley.com.]
522 Angelov, Choumei, and Inoue
second term Pbe2 is used, we should always add apositive 3rd term, Pbe3, in order to provide stablesolution from �� to ��. Fitting of the extractedargument is rather easy, because a polynomial func-tion is used for this purpose.
As shown in Figure 9, the magnitude of the ex-tracted argument is determined by the leakage and themaximum current which the devices can handle (usu-ally not higher then 100 Å). This means that it ispossible to use a bounded function to limit the argu-ment, which is important when working with expo-nential functions as in the case of BT. Simultaneously,it is important to keep a correct value of Pbe1, becausethis will determine the derivative (transconductance).
We can use tanh as a limiting function and split thePbe1 into two parts, Pbe1e and Pbe1i [see eq. (3a)]. Forlarge deviation from the Vj tanh � 1, Pbe1e willdetermine the maximum of the argument, respectiveleakage, and the maximum current.
The derivatives will be correct if Pbe1i is calculatedfrom:
Pbe1 � Pbe1e tanh�Pbe1i�.
The external parameter Pbe1e can be calculated as:
Pbe1e � ln�Ibe1/Ijbe�
Pbe � Pbe1e � tanh�Pbe1i � �Vbe � Vje�
� Pbe2�Vbe � Vje�2. . . ,
Pbe0 � Pbe1 � tanh�Pbe1i � ��Vje�
� Pbe2��Vje�2 . . . �
Pbc � Pbc1 � tanh��Pbc1i � �Vbc � Vjc�
� Pbc2�Vbc � Vjc�2 . . . �,
Pbe0 � Pbc1 � tanh��Pbc1i � ��Vjc�
� Pbc2��Vjc�2 . . . � (3a)
The leakage typically is in the order of I �10�9, butusually we do not operate the device at such a smallcurrents. If we fix the magnitude of the exponent, asfor example in eq. (3a), to Pbe � 1/2Vt � Nb1 �19.347/Nb1, this will bound the maximum possiblecurrent to Ibmax � 126260 Å and the minimum currentto Ibmin � 10�13, which are adequate for most prac-tical cases. If required, the bounding level can bechanged. Rewriting the current equations using theideality factors Nb and Nc leads to the followingequations for the base and collector currents.
With three terms in the power series model, there are15 parameters and in single term case we need a totalof 11 parameters for the collector and base current:Ipkc, Vbep, �max, Bbe, Nc1, �r, �2, �, Nb1, Vje � Vbep,Vpk, Ibej � Ipkc/�max.
The base junction current Ibej is calculated from�max, Ipkc; and Vj � Vbep voltage is taken directly fromthe measurement. The ideality factor Ne1 is extractedfrom the slope of the logarithmic plot. When wedefine the parameters by using eqs. (8) and (9) we willhave correct derivatives, and bounded minimum andmaximum currents. This in turn will improve theconvergence. This is valid, because the first derivativeof tanh is in fact P1*z, but it is limited at largedeviations as follows:
tanh�P1 � z� � P1 � z � �P13z3�/3
� �2P15z5�/15 . . . ) (9b)
Figure 10 shows some measured and simulated resultsfor Ibe, when the argument is limited by using eq. (8).With one term in the power series, eq. (8) will providea good fit in many decades.
When using eq. (9), the collector current will belimited from the tanh(� � Vce) for the collector voltageand from the argument for Ibe � Ibej � exp(19.347 �tanh(2(Vbe � Vje)) in the exponential part of Ibe and Ice
versus Vbee. This leads to a compact, bounded modelwith correct derivatives without discontinuities.
III. HBT CAPACITANCES
Devices can operate at high currents and voltages andtheir nonlinear capacitances need to include both de-
pletion and diffusion parts. The total Cbe capacitanceshows rapid increase and then decrease at high bias.This is also the case for homojunction transistors [7,14], but for HBTs this increase will be larger than theincrease we observe in homojunction transistors. It isa mater of tradition to treat the device capacitance asconsisting of two parts — depletion and diffusion —that are connected in parallel.
Integrating with the terminal voltages we can obtainthe Qbe and Qbc when using the following chargedefinition:
Qbedif � Cbe0
log�cosh�Cbe10 � Cbe11 � Vbe
Cbe11
� �Cbep � Cbe0�Vbe,
Qbcdif � Cbc0
log�cosh�Cbc20 � Cbc21 � Vbc
Cbc11
� �Cbcp � Cbc0�Vbc. (11)
The ordinary equation for the depletion part Cdep �Cdep0/(1 � Vbe/Vbi)
n can create convergence problemswhen Vbc Vbi. The accepted approach is to definethe capacitance with two definitions — below andabove Vbi. At the connection point, it is important tokeep the function and the derivatives equal; other-wise, the HB simulator will not converge. This prob-lem is not easy to solve and can increase the simula-tion time. For the simulators using charge definitions,some stable solutions, suitable for the capacitancemodelling, can be integrated in hypergeometricalfunctions that are not available in circuit simulators.In addition, the use of such functions will make theextraction difficult. To avoid these problems, theequation for the depletion capacitance equation ismodified as follows:
be associated with the parasitic elements. As is ex-pected, the standard depletion capacitance model hasa singularity at Vbci. Eq. (12) gives results identical tothe textbooks’ depletion capacitance definition, but itis well defined at Vbci — the function is positive and
without poles in the interval �� � Vbc � �. Thederivatives of the capacitance are also well defined,[see Fig. 11(b)]. In addition, this equation models thedecrease of the capacitance as it is in [7, 14] atvoltages larger then Vbci. This equation can be mod-ified to be completely equivalent to the standard de-pletion capacitances and used as a direct replacementin the Spice circuit simulators (if required). For thesimulators using a charge definition, a simple, closedform of the charge can be derived and combined withthe diffusion part [see eq. (11)]:
can be further extended in order to consider the dy-namic dependence of Cbc upon the collector currentand an approach similar to that in [5] can be used:
Cbc � Cdep � �1 � Icf/Icr�p. (14)
The Cbc dependence on the collector current can betransferred directly to the intrinsic voltages by using
Cbc � Cdep � �1 �1
exp�Vbe � Vmin�
� Cdbe1�exp�38.695
Netanh�Vbe � Vmin��, (15)
where Vmin is the voltage (with respective current) atwhich the capacitance is minimum and Cdbe1 willdetermine the slope of the Cbc increase at voltages Vbe
� Vmin. This approach corresponds to the devicephysics and is stable in the extraction, because alimited function, which provides a good fit to themeasured Cbc [see Fig. 11(c)], is used.
IV. SELF-HEATING MODELING
Figures 12–14 show extracted basic parameters Vjbe,Pbe1, and Ijbe at different temperatures. Since thetemperature coefficients of different model parame-ters are small (on the order of 10�3) [8], the changesof the model parameters versus temperature can beconsidered linear in a limited temperature range [seeeq.(16)]. If required, when the temperature range ofoperation is extended, more terms can be added. Thevalues of Ice and Vce are available from the harmonicbalance simulator and are used to calculate both thedissipated power Ptot and the junction temperature Tj
(calculated dynamically):
Figure 11. Cbc [eq. (12)], n � 0.3, m � 0.002, Vbei �1.25, Cdbe0 � 1; (a) Cbc vs. Vbc; (b) derivative of Cbc; (c)measured and modeled Cbc vs. Vbe.
where Ptot is the dissipated power and Rth is thethermal resistance. For higher accuracy, the thermal
resistance Rth can be made temperature dependent,that is, Rtht.
RthT � Rth�1 � TcRth�Tj � Tr�� (17)
Thermal parameters are normalized to the referencetemperature Tref at which we extract parameters.
V. BREAKDOWN MODELING
The breakdown is influenced by the effects existingon the base side (with the change of Vbcp) and on thecollector side. The Ice breakdown can be modeled byusing an idea similar to the breakdown model imple-mentation in [33]:
The value of KVbd2 is taking into account the changeof intrinsic base voltage Vbei, because as a result ofthis change, the collector current Ice will increase. Thecoefficient Lsb0 reflects the sharp increase of Ice,which is influenced by the collector voltage at volt-ages larger than the threshold Vtr, Figure 15.
The breakdown parameters are temperature depen-dent, and when this problem becomes important, ther-mal coefficients for the Vtr, Lsb0, and KVbd2 parame-ters should be extracted.
Figure 12. Extracted Vjbe vs. T.
Figure 13. Extracted Pbe vs. T.
Figure 14. Extracted Ijbe vs. T.
526 Angelov, Choumei, and Inoue
VI. DELAY MODELING
The importance of the proper delay modeling hasbeen very well described in classic papers [5, 13,26–28]. Figure 16 shows a typical bias dependence ofthe delay. It increases with the collector current andeventually saturates and decreases, as the collectorvoltage increases. Usually in the simulators the biasdependence of the delay is defined via Ice, but it canalso be defined by directly using the intrinsic control-ling voltages and implementing the following equa-tion using CAD:
Tff � Tf�1 � 0.5 � Xtf � �1 � tanh��38.76/Ne��Vbe
� Vje� � � exp�Vbc/�1.44 � Vtf � � (19)
where Tf is the independent part of the forward delay,Xtf is the bias-dependent part of the forward delay,and Vtf is a fitting coefficient, as they are defined in
other software packages such as ADS and ICCAP. InADS the bias-dependent delay can be implementedusing the delay function H of SDD.
VII. EXTRACTION OF THE MODEL
The number of parameters to be extracted using thisapproach for the nonlinear part of the large signalmodel is significantly reduced in comparison to othermodels with comparable accuracy. Many of the modelparameters are taken directly from simple measure-ments. Even when using one term in the power series,the model will be exact at the selected operating pointand provide a good global accuracy in 5–6 decades ofcurrents, as shown in Figure 16. When higher accu-racy is required (that is, more terms) the remainingfitting parameters can be extracted with simple fittingprograms. The reduced number of parameters and theuse of bounded functions will result in improvedconvergence in CAD tools. This is also beneficial forthe extraction procedure, because the same problemswith convergence are valid for the extraction pro-grams.
The extraction of parameters starts with the extrac-tion of the DC parameters. The basis of the proposedmodel is that the main parameters are taken directlyfrom the measurements — not extracted. In the simplecase of one term in the power series, which is accurateenough for most practical RF applications, the modelequations are as follows.
Ibej, Vje, Ipkc, Vbep, and �max are taken directly fromthe measurements for currents at which we have max-imum �; typically, at Vk � 0.6–0.8 V. The idealityfactors are calculated from the derivative of the Ibe,which is Ice at this point. The four remaining param-eters, Bbe, �r, �s, and �, can be found with a simple
Figure 15. An example of Ice breakdown model using eq.18: Lsb � 0.1, Vtr � 1.
Figure 16. Delay Tau vs. Jc density.
HBT Large-Signal Model for CAD 527
fitting program. The parameter Bbe is found from the� versus Vbe dependence, parameter �r will influenceIce at small currents, and low Vce � Vk and �s param-eters will influence the shape of Ice at the knee Vk. Theoutput conductance parameter � will fit the Ice atvoltages above Vk and this give the possibility to fit�r, �s, and � separately.
There are many fitting programs such as Math-ematica, MathCAD, Kaleidagraph, XLfit3, and so forththat can be used to fit basic parameters of the model.Figure 17 shows some results from measured and mod-eled Ibe, Ice (measured points and modeled lines) withone term in the equations using a simple fitting program(Kaleidagraph), and if more terms are used in the powerseries in the argument. In addition, we can use special-ized extraction programs such as ICCAP.
The peak current Icmax Ice is close to the currentsfor which we have maximum � with correspondingVce. Pcf1 can be extracted directly from the derivativeof the argument (ideality factor) at the selected biasvoltage Vbe (see Fig. 8), such that Pcf1 � gm/Icmax �38.695/Nc1. A typical value for the extracted basecurrent is Iib 0.2–2 mA with the corresponding Vbe.
There is a large amount of papers on the extractionof parasitics in the small-signal equivalent circuit [8,14–28] of HBTs. Some of the parasitics such as theresistances and capacitances used in this study areavailable directly from measured data. The Re extrac-tion was compared using both the open-collectormethod and the high-frequency extraction methodproposed by S. Maas [18]. The rest of the parasiticsare found from optimization by using the CAD toolset for multiple-bias S-parameter optimizations.
Generally, the base resistance Rb in these types oftransistors is rather low (this is one of their benefits)and, in order to simplify the model extraction, thebase and collector resistances in this case were con-sidered constant. The base resistance consists of threeparts as follows: Rb � rb � rbs � rb0, where rb isactive, rbs is spreading resistance, and rb0 is ohmicmetal contact resistance. The series-access compo-nents are bias independent and when the injection level
Figure 17. Measured and modeled Ibe and Ice: (a) Ibe; (b) Ice.
Figure 18. Ice: Ibe � 20–200 �A step 20 �A.
Figure 19. Ice, Vbe � 1.2–1.4 V.
528 Angelov, Choumei, and Inoue
is high, the changes of Rb are very small and can beneglected. If required, the bias dependence of Rb can bemodeled using approach similar to that in [6, 34]:
Rb �Rb max
1 � �Ib/Ib0�arb (20)
VIII. EVALUATION OF THE MODEL
The model was implemented as an SDD in MDS andADS (Agilent) with self-heating and delay, and wasexperimentally evaluated using DC, S-parameter andpower-spectrum measurements.
Figure 20. Measured and modeled magnitude and phase S11, S21, S12, S22, Vce � 3 V; Vbe �1.32, 1.34, and 1.36 V.
HBT Large-Signal Model for CAD 529
Figures 18 and 19 show measured and modeled Ice.In the example shown in Figure 18, the base currentIbe is a parameter and in Figure 19, Vbe is a parameter.The fit is good, even with such a simple definition ofthe model and using two terms in the power series forIce. Despite this simplicity, the model is able to de-scribe the thermal runaway of Figure 19, with anaccuracy similar to more complicated models in thepractical range of currents and voltages with a signif-icantly reduced number of parameters.
Figure 20 shows some results of S-parameter mea-surements and simulations. The model accurately de-scribes the small-signal behavior.
The large-signal properties of the model were eval-uated using a PS method [31]. Figures 21 and 22 showsome results of the measurements and simulations,and the fit is good. The models for currents andcapacitances facilitate the convergence in large-signalapplications, due to the use of well-defined functionsand symmetry.
The model definitions are quite general and can beapplied to other types of HBT, as well as ordinary Si
bipolar transistors. Figure 23 shows the results of themeasured and simulated Ice for InGaP HBT.
Figures 24 and 25 show measured and modeled Ibe
and Ice for a commercial Si microwave transistor AT42070 (Agilent). With one term in the power series,the fit is very good for the practical range of currents.
This approach can be extended with more features,which we will eventually need in order to model morecomplicated effects in HBTs and BJTs. For example,the bias dependence of the output conductance incertain HBTs and high-power Si BJTs can be stronglybias dependent and, in order to track this, the outputconductance parameter � can be made bias dependent:
Ic � Icf � tanh�� � Vce��1 � � � Vcb � �max� (21)
�max � �2 � Vcb � �1 � tanh�19,347
Ncf� �2�Vbe
� Vbep���� � �3/�1 � Vce2 � (22)
In this case, the additional term �2 provides depen-dence of the output conductance on the base voltageVbe and �3 on collector voltage Vce. The parameter�2 is effective at high Vce and �3 at low Vce. Evenin this case of complicated output conductance, thenumber of extracted parameters (11) is rather small,the model is stable in extraction & simulations,because functions used in the model are monotonicand limited.
Figure 26 shows some results modeling SiGe HBTand high-power Si BJT is shown in Figure 27. Thecorrespondence between the measured parameters andmodel is good, even using one term in the powerseries. In these examples of a complex output-con-ductance dependence, the total number of Ice and Ibe
Figure 22. Measured and modeled PS, at 5 GHz, Ibe �0.25 mA, Vce � 2 V.
Figure 23. Measured and modeled Ice for InGaP HBT,Ibe: 2 to 300 �A.
current parameters is 11, with one term in the powerseries; however, this is sufficient to provide goodaccuracy in the practical working range of currentsand voltages. If for some reason a higher globalaccuracy is required, then two more terms can beadded to the power series in the argument.
IX. CONCLUSION
A simple bias-dependent HBT model applicable toCAD was proposed and implemented. The model was
experimentally evaluated using DC, S-parameter, andpower-spectrum measurements, and good correspon-dence was obtained between the measurements andthe model. The model can be adopted for use as asimple BJT model.
ACKNOWLEDGMENT
The authors wish to acknowledge Agilent for the donationof high-frequency simulation software, S. Maas for valuablediscussions, and H. Zirath, E. Kollberg, A. Olaussen, andSSF for their help and support.
Figure 24. Measured and modeled (a) Ibe and (b) Ice forIce, AT42070 BJT.
Figure 26. Measured and modeled results for Ice, SiGeHBT.
HBT Large-Signal Model for CAD 531
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Figure 27. Measured and modeled results for Ice, BC141.
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BIOGRAPHIES
Iltcho Angelov was born in Bulgaria. Hereceived an M.Sc. degree in electronics(1969) and a Ph.D. in mathematics and phys-ics from Moscow State University (1973).From 1969–1991 he was with Institute ofElectronics, Bulgarian Academy of Sciences,Sofia, as a Researcher, Research Professor(1982), and Head of the Department of Mi-crowave Solid State Devices (1982). Since
1991 he has been with Chalmers University, Goteborg, Sweden andis a member of IEEE . His main interests are in device modellingand low noise and nonlinear circuit design.
Kenichiro Choumei was born in Osaka, Ja-pan in 1968. He received B.S. and M.S.degrees in electronics engineering fromDoshisha University, Japan, in 1992 and1994, respectively. Since joining the Opto-electronic & Microwave Devices Labora-tory, Mitsubishi Electric Corporation, Itami,Japan, in 1994, he has been engaged in theresearch and characterization of various
GaAs power devices for mobile communications. He also devel-oped microwave measurement systems. His present research inter-ests include characterization and device modeling of HBT as well
as circuit design of millimeter-wave MMICs and their modules,such as HBT-based and HFET-based VCOs. He is a member of theIEEE and the Institute of Electronics, Information and Communi-cation Engineers of Japan (IEICE).
Akira Inoue was born in Osaka, Japan in1961. He received B.S. and M.S. degrees inphysics from the University of Kyoto, Japan,in 1984 and 1986, respectively. In 1986 hejoined the LSI Laboratory, Mitsubishi Elec-tric Corporation, where he was engaged inthe design of GaAs MMIC. In 1988 hejoined the Optoelectronic & Microwave De-vices Laboratory, where he participated in
the characterization of GaAs transistors, evaluation, and the designof modules and MMICs. He developed microwave measurementsuch as on-wafer RF testers, harmonic load-pull system, and mi-crowave waveform measurement. Currently, he has been engagedin the development of microwave power transistors and poweramplifier modules for mobile communications. He has developedinverse class-F power amplifiers for mobile handsets. He is alsoworking on the developments of GaAs FET and HBT modeling. Heis a member of the IEEE and the Institute of Electronics, Informa-tion and Communication Engineers of Japan (IEICE).
