RADIOENGINEERING, VOL. 6, NO., APRIL 2007 Moeling Delay of Microwave Tranitor an Tranmiion Line by the 2 n -Orer Beel Function Joef DOBEŠ, Karel ULOVEC Dept. of Raio Electronic, Czech Technical Univerity, Technická 2, 66 27 Praha, Czech Republic obe@fel.cvut.cz, xulovec@fel.cvut.cz Abtract. At preent, mot of imulation program can characterize gate elay of microwave tranitor. However, the elay i motly approximate by mean of firtorer ifferential equation. In the paper, a more accurate way i uggete which i bae on an appropriate econorer ifferential equation. Concerning the tranmiion line elay, majority of the imulation program ue both Branin (for lole line) an LCRG (for loy line) moel. However, the firt caue extreme imulation time, an the econ caue well-known puriou ocillation in the imulation reult. In the paper, an unuual way for moeling the tranmiion line elay i efine, which i alo bae on the econ-orer Beel function. The propoe moel oe not create the puriou ocillation an the imulation time are comparable with thoe obtaine with the claical moel. Propertie of the implementation of the econ-orer Beel function are emontrate by analye of both igital an analog microwave circuit. Keywor Beel function, orinary ifferential equation, group elay, MESFET, gate elay, tranmiion line.. Introuction The econ-orer Beel function [] appear to be an efficient mathematical tool for a general-purpoe elay moeling. In the paper, an appropriate mathematical form ha been choen for an approximation of both MESFET an tranmiion line elay. The moel have been implemente into an checke by our oftware tool CIA (Circuit Interactive Analyzer). Propertie of both moel have been checke by the analyi of microwave analog an igital circuit. 2. General Propertie of the 2 n -Orer Beel Function Ditribute phenomena are ecribe by the partial ifferential equation in the exact way. However, we houl ue an appropriate approximation of the exact olution of the partial ifferential equation in both MESFET an tranmiion line moeling ue to convenient moel complexity. In characterizing the bipolar junction tranitor [2], the econ-orer Beel function i efficiently ue for moeling the elay of collector current at high frequencie. However, that function can alo be ue for moeling the MESFET an tranmiion line in a imilar precie way. Conier a linear econ-orer ifferential equation () (which i ue in [] for the moel of collector current of the microwave bipolar junction tranitor) to be olve for a a reult the parameter will be efine later. Uing the Laplace tranform of (), we get, (2) where i the Laplace operator an i the econorer Beel function. (3) To examine the frequency propertie of (3), let u et, an formulate the argument of, i.e.,. The group elay i the value of the erivative of with repect to, i.e., it i poible to write. (4) ()
2 J. DOBEŠ, K. ULOVEC, MODELING DELAYS OF MICROWAVE TRANSISTORS AND TRANSMISSION LINES BY THE 2 ND -ORDER Fig.. Claical Branin moel of the lole tranmiion line, which will be approximate by mean of the 2 n -orer Beel function. New gate Fig. 2. Novel way of MESFET or phemt gate elay moeling. Conier now to be appreciably leer than. For example, for, therefore, the lat member in the enominator in () (i.e., ) may be neglecte, reulting to the imple formula. (6) Thu, the group elay i approximately contant, an () can therefore be coniere to be a very accurate tool for the elay moeling. If the phae elay i etermine in egree at, where repreent, e.g., a new MESFET moel parameter, the time elay at thi frequency can be expree by the formula. (7) Comparing (6) an (7), we obtain the final equation for the moel parameter (in fact, i not a primary moel parameter; it i erive from the primary parameter an, the phae elay in egree an reciprocal limit frequency, repectively): however, if the phae elay i etermine in raian at, we obtain the more natural form (8). (9) 2. Limit of Approximating by the 2 n -Orer Beel Function If the function (3) i ue for the elay moeling, the magnitue of ha to be alo checke. For, the magnitue of can eaily be expree a Fig. 3. New CIA built-in ub-element calle ϕ-hift. Fig. 4. One element of the new tranmiion line moel create by the four ϕ-hift ub-element.. (0) Thi magnitue ecreae to. for. Therefore, the angular frequency houl not be greater than in other cae, the ϕ-hift element alo change the amplitue of the ignal (not only the phae). Subtituting in (4), we obtain the hift. () therefore, the phae hift greater than approximately cannot be moele by one equation (() or (3)) only. 3. Moeling the MESFET Gate Delay During the tranient operation in a FET, the electron epletion with uner the gate mut be change by charge tranport [2]. Hence, a change in the gate voltage oe not caue an intantaneou change in the rain-ource current.
RADIOENGINEERING, VOL. 6, NO., APRIL 2007 3 V3/0.6 G/fet 2 V/pule(t,0p,0p,300p,0p,600p,09,0.0) 3 I/equal(IS+) + S/0G,2.2 G2/efet 4 V2/0. G3/efet R/20k C/0.06p Fig.. Digital circuit with both enhancement- an epletion-moe MESFET ue a a tet of the moel with the 2 n -orer Beel function. V(2), V(3) (V).6.4.3.2. without elay with elay 0 0.E-9.2E-9.3E-9.4E-9 E-9.6E-9 time () Fig. 6. Aitional elay given by the 2 n -orer Beel function. Thi elay can be accurately efine by mean of an appropriate ifferential equation (ee () an Fig. 2) (2) with the following natural initial conition:. (3) A the ource i unilateral an DC conition cannot change, the i alo neceary (ee Fig. 2 an [3], e.g.). (4) The olution of (2) can be performe in a numerical way by mean of a circuit analyzer integration algorithm. However, a mixe analytical-numerical olution i alo poible, becaue the affiliate homogenou ifferential equation to (2) i linear. 4. Moeling the Tranmiion Delay Let u conier the claical Branin moel in Fig. [2], where i the characteritic impeance of the tranmiion line. The main problem of thi moel i introuce by the incompatibility of the elay with the numerical integration metho. Therefore, another approximation of the elay moel i neceary [4]. An unuual uggetion can alo be create by the ifferential equation, (). (6) The olution of () an (6) are ue a the approximation of the elaye voltage an current, i.e.,. (7) In fact, a novel unuual moel i create in thi way. Firt, a new circuit element calle ϕ-hift i implemente with the ymbol, which i rawn in the left part of Fig. 3. Secon, one element of the new tranmiion line moel i create in the way, which i hown in Fig. 4 thi i an unuual analogy of the Branin moel. Let u emphaize again that the phae hift mut not be greater than approximately ee (). Therefore, everal Branin ubcircuit (the circuit in Fig. 4 with the Beel approximation of the elay) mut be erially connecte in the cae of greater phae hift. The moel of tranmiion line bae on () (7) ha quite ifferent propertie than LCRG one. The LCRG moel generate puriou ocillation on the contrary, the new moel generate moothe hape of all the ignal.
4 J. DOBEŠ, K. ULOVEC, MODELING DELAYS OF MICROWAVE TRANSISTORS AND TRANSMISSION LINES BY THE 2 ND -ORDER p Vg/-0.3 Vg2/-3.k.k 0p.k Vg2/-3 Vg3/-3.k p 20n V/2 20n w=0.6 l=8 20p w=0. l=.8 w=0.27 l=8 2xatf3376 2x20p w=0.36 l=8 w=0.8 l=.8 atf3376 w=0. w=.8 w=0.4 l=4 w=0.4 l= w= l=0 20p w=0.67 l=.8 atf3376 w=0.3 l=0.3 w=2 l=2 w=0.4 l=2.9 w=0.0 l=2 0 0 output 0p w=0.37 l=8 p w=0.33 l=8 p w=0. l=4 w=2 l= w= l=0 Fig. 7. Microwave voltage-tunable itribute ocillator ue a a tet of the tranitor gate elay moel bae on the 2 n -orer Beel function. V(output) (V) 0 - - without elay with elay - 0.0E-9.E-9.E-9.2E-9.2E-9.3E-9.3E-9 time () Fig. 8. Comparion of the two output voltage for the zero an nonzero gate elay, veru GHz etecte.. Teting the MESFET Gate Delay. Teting Digital Circuit with MESFET Conier a fat igital circuit in Fig. [7] with the trapezoial pule at the input ee Fig. 6. A ubtantial part of the elay i caue by the MESFET nonlinear capacitance, of coure. However, another part of the elay i caue by the itribute circuit phenomena, which i alo hown in Fig. 6. Thi aitional part of the elay ha been moele in the way ecribe in Fig. 2. In thi imple circuit, the aitional elay i not funamental. However, for more complicate igital circuit, accurate moeling i neceary..2 Teting Ditribute Microwave Ocillator A another more ophiticate example, conier a itribute tunable ocillator in Fig. 7 []. The ocillator ha been evelope in our univerity an the analye with CAD tool have been performe by the claical Berkeley SPICE 3 ytem an our CIA program, becaue the tool bae on harmonic balance i not converge for thi circuit. Formerly, the gate elay ha not been moele (the SPICE 3 moel i unable to efine it) the importance of the gate elay moeling i emontrate here. The reult are hown in Fig. 8. The curve marke without elay can be obtaine by any of the tanar circuit imulator; the curve marke with elay ha been compute by the CIA program the tranitor ha ha the gate phae hift. at GHz (i.e., the limit wa not exceee). A oberve, the perio etermine by the new moel i lightly greater, which i expecte. However, the phae hift inuce by the tranitor elay caue omewhat greater magnitue, which i alo important. Note that the teay-tate ha been etecte by the CIA extrapolation algorithm monitoring the interection of the output voltage with the level V (automatically). A the ocillator i tunable, a number of analye ha to be performe therefore, the automatic perio etection i very important. The meaure reult [] confirm the imulate one. For the bia voltage hown in Fig. 7, the ocillation frequency GHz ha been meaure. In the tuning range, the output power Bm ha been meaure (the level Bm i relate to the output voltage V).
RADIOENGINEERING, VOL. 6, NO., APRIL 2007 L0/0uH,ohm V/-0.4 V2/ L9/0uH,ohm Feeback C6/0pF L8/nH 6 Tranmiion line L/0.03nH R2/4 C3/0.0pF R3/4 L6/0.03nH Tranmiion line 2 4x.7eg at 2GHz 4x.7eg at 2GHz R/0 3 C4/0.02pF GaAFET C/0.02pF R4/3 L7/0.0nH R/67 C7/0.07pF Fig. 9. Feeback microwave ocillator with great amplitue of output ue a a tet of both Beel an LC tranmiion line moel. 0 0 2 2 amplitue (V).2 amplitue (V).2...0.0.02.02.0 0 2 3 4 6 7 8 9 0 2 3 4 Fourier component.0 0 2 3 4 6 7 8 9 0 2 3 4 Fourier component Fig. 0. Harmonic analyi of the output waveform create by LC moel (etecte perio i GHz). 6. Teting the Tranmiion Delay Conier a feeback microwave ocillator in Fig. 9 [6]. The ocillator ha a number of harmonic component, an therefore it i very appropriate for teting the tranmiion line moel. The pin an mark the exterior gate an rain, repectively; the pin mark the output. The firt analyi ha been performe by the claical LC moel of the two tranmiion line (4 LC ection were ue, the inuctance an capacitance per ection were. nh an. pf, repectively). The econ analyi ha been performe with the 2 n -orer Beel moel of the two tranmiion line (a efine in Fig. 9, four ection ha been ue, each of them with the phae hift. at GHz for the accuracy purpoe, it i better to ue more ection with leer value of hift to protect the amplitue of the ignal). Fig.. Harmonic analyi of the output waveform create by Beel moel (etecte perio i GHz). The two reult are compare uing the Fourier component of the output ee Fig. 0 an. The propoe moel uppree the puriou ocillation (ee the higher harmonic component of the ignal). Moreover, the new moel ha claime a leer number of integration tep in comparion with for the LC moel. 7. Concluion The econ-orer Beel function ha been propoe a an efficient tool for moeling the elay in microwave an RF circuit. A practical way for uing thi function for moeling the MESFET gate an tranmiion line elay wa outline. The moel have been checke uing both igital an analog microwave circuit. The reult how that the moeling of the elay caue by the itribute circuit phenomenon coul be very important for the overall
6 J. DOBEŠ, K. ULOVEC, MODELING DELAYS OF MICROWAVE TRANSISTORS AND TRANSMISSION LINES BY THE 2 ND -ORDER preciion of the imulation, an therefore it houl be implemente into the CAD tool. Acknowlegement Thi paper ha been upporte by the Grant of the European Commiion TARGET (6 th Framework Programme), by the Grant Agency of the Czech Republic, grant No 02/0/0277, an by the Reearch Program of the Czech Technical Univerity in Prague MSM684077004. Reference [] MASSOBRIO, G., ANTOGNETTI, P. Semiconuctor Device Moeling With SPICE. 2 n e. New York: McGraw-Hill, 993. [2] VLADIMIRESCU, A. The Spice Book. t e. New York: John Wiley & Son, 994. [3] MADJAR, A. A fully analytical AC large-ignal moel of the GaA MESFET for nonlinear network analyi an eign. IEEE Tranaction on Microwave Theory an Technique, 988, vol. 36, no., p. 6-67. [4] GUO, Y.-S. Tranient imulation of high-pee interconnect bae on the emiicretization of telegrapher' equation. IEEE Tranaction on Computer-Aie Deign, 2002, vol. 2, no. 7, p. 799-809. [] DIVINA, L., SKVOR, Z. The itribute ocillator at 4 GHz. IEEE Tranaction on Microwave Theory an Technique, 998, vol. 46, no. 2, p. 2240-2243. [6] TAJIMA, Y., WRONA, B., MISHIMA, K. GaA FET large-ignal moel an it application to circuit eign. IEEE Tranaction on Electron Device, 98, vol. 28, no. 2, p. 7-7. [7] SUSSMAN-FORT, S. E., HANTGAN, J. C., HUANG, F. L. A SPICE moel for enhancement- an epletion-moe GaA FET. IEEE Tranaction on Microwave Theory an Technique, 986, vol. 34, no., p. - 9. About Author... Joef DOBEŠ receive the Ph.D. egree in microelectronic at the Czech Technical Univerity in Prague in 986. From 986 to 992, he wa a reearcher of the TESLA Reearch Intitute, where he performe analye on algorithm for CMOS technology imulator. Currently, he work at the Department of Raio Electronic of the Czech Technical Univerity in Prague. Hi reearch interet inclue the phyical moeling of circuit element for raio engineering, epecially RF an microwave tranitor an tranmiion line, creating or improving pecial algorithm for the circuit analyi an optimization, uch a time- an frequency-omain enitivity, pole-zero intermoulation, or teay-tate analye, an creating a comprehenive CAD tool for the analyi an optimization of RF an microwave circuit CIA (Circuit Interactive Analyzer). Karel ULOVEC wa born in 977 in Prague, Czech Republic. He receive hi mater' egree in electrical engineering from the Faculty of Electrical Engineering, Czech Technical Univerity in Prague, in 200. At preent, he i an aitant profeor at the ame faculty an he i working on the Ph.D. thei. Hi reearch interet inclue meaurement of raio tranmitter an receiver an raio ignal proceing.