Nonlinear traveling-wave field effect transistors for ... - Semantic Scholar
IEICE Electronics Express, Vol.7, No.16, 1188–1194
Nonlinear traveling-wave field effect transistors for amplification of short electrical pulses Koichi Naraharaa) and Shun Nakagawa Graduate School of Science and Engineering, Yamagata University, 4–3–16 Jonan, Yonezawa, Yamagata 992–8510, Japan a) [email protected]
Abstract: We investigated the properties of pulse propagation on nonlinear traveling-wave field effect transistors (TW-FET) to develop a method for amplifying short electrical pulses. TW-FETs are a special type of FET whose electrodes are employed not only as electrical contacts but also as transmission lines. Due to the presence of electromagnetic couplings between the gate and drain lines, two different propagation modes called the c mode and π mode are developed on a TW-FET. Moreover, the Schottky contact beneath the gate electrode creates an ideal source of nonlinearity for soliton-like propagation. We can design the TW-FET to amplify only soliton-like pulses carried by one of the two modes and attenuate the ones carried by the other mode. This paper discusses the fundamental properties of a nonlinear TWFET, including the width and velocity of a soliton-like pulse carried by c and π modes, and gives design criteria of amplification of soliton-like pulses. Keywords: solitons, nonlinear transmission lines (NLTLs), travelingwave FETs, pulse amplification Classification: Microwave and millimeter wave devices, circuits, and systems References [1] R. Hirota and K. Suzuki, “Studies on lattice solitons by using electrical networks,” J. Phys. Soc. Jpn., vol. 28, pp. 1366–1367, 1970. [2] M. J. W. Rodwell, S. T. Allen, R. Y. Yu, M. G. Case, U. Bhattacharya, M. Reddy, E. Carman, M. Kamegawa, Y. Konishi, J. Pusl, and R. Pullela, “Active and nonlinear wave propagation devices in ultrafast electronics and optoelectronics,” Proc. IEEE, vol. 82, pp. 1037–1059, 1994. [3] K. Narahara and T. Otsuji, “Compression of electrical pulses using traveling-wave field effect transistors,” Jpn. J. Appl. Phys., vol. 38, pp. 4688–4695, 1999. [4] K. C. Gupta, R. Garg, and I. J. Bahl, Microstrip Lines and Slotlines, Artech, 1979. c
IEICE 2010
DOI: 10.1587/elex.7.1188 Received June 04, 2010 Accepted July 16, 2010 Published August 25, 2010
1188
IEICE Electronics Express, Vol.7, No.16, 1188–1194
[5] T. Taniuti, “Reductive perturbation method and far fields of wave equations,” Prog. Theor. Phys. Suppl., vol. 55, pp. 1–35, 1974.
1
Introduction
A nonlinear transmission line (NLTL) is defined as a lumped transmission line containing a series inductor and a shunt Schottky varactor in each section. NLTLs are used for the development of solitons [1]. Moreover, the operation bandwidth of carefully designed Schottky varactors goes beyond 100 GHz; therefore, they are employed in ultrafast electronic circuits including the subpicosecond electrical shock generator [2]. However, the line resistance generally attenuates the pulse amplitude, making pulses very small for an NLTL to exhibit nonlinearity. To bring out the potential of NLTLs, we considered a traveling-wave field effect transistor (TW-FET) to compensate for the wave attenuation by utilizing the transistors’ gain. Because of the Schottky contact beneath the gate, the gate line of a TW-FET simulates an NLTL. In contrast, the drain line is modeled by a linear transmission line coupled with the gate line via the gate-to-drain capacitance, mutual inductance and transconductance. Two different propagation modes called the c mode and π mode develop on a coupled line. Each mode has its own velocity, characteristic impedance and voltage fraction between the gate and drain lines. It is found that the nonlinearity introduced by the gate-to-source Schottky capacitors succeeds in compensating for the distortions caused by dispersion for pulses carried by either c or π modes; although, the nonlinear pulses are greatly attenuated due to the presence of finite electrode resistances irrespective of the carrying mode. For gate voltages higher than the threshold, it is expected that the transconductance relaxes pulse attenuation. Possibly, only a pulse carried by one of the modes gains amplitude, while a pulse carried by another mode decays [3]. Elaborating the design method enables the amplification of pulses that are not affected by both dispersion and attenuation. In this paper, we first describe the fundamental properties of nonlinear pulses observed in a subthreshold TW-FET. Then, we discuss the design criteria for amplifying the nonlinear pulses in a TW-FET, together with several results of numerical evaluations that validate this method.
2
c
IEICE 2010
Nonlinear TW-FETs
Figure 1 shows the diagram of a TW-FET. The gate and drain lines are modeled by an NLTL and a linear line, respectively. These two are coupled via the gate-to-drain capacitance Cm , the mutual inductance Lm , and the drain-to-source current Ids . The per-unit-cell series inductance and shunt capacitance of the gate (drain) line are denoted by Lg (Ld ) and Cg (Cd ), respectively. Note that Cg shows the Schottky varactor whose capacitance-
DOI: 10.1587/elex.7.1188 Received June 04, 2010 Accepted July 16, 2010 Published August 25, 2010
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IEICE Electronics Express, Vol.7, No.16, 1188–1194
voltage relationship is generally given by Cg (VX ) =
C0 1−
VX VJ
m ,
(1)
where VX is the voltage between the terminals. C0 , VJ , and m are the optimizing parameter. Note that VX < 0 for reverse bias. We denote the line voltages of the gate and drain lines at the nth cell as Vn and Wn , respectively, and the line currents of the gate and drain lines at the nth cell by In and Jn , respectively. Then, the transmission equations of a TW-FET are given by dIn dJn + Lm = Vn−1 − Vn − Rg In, (2) dt dt dJn dIn + Lm = Wn−1 − Wn − Rd Jn, (3) Ld dt dt dVn dWn − Cm = In − In+1, (4) [Cm + Cg (Vn )] dt dt dWn dVn − Cm = Jn − Jn+1 − Ids (Vn ). (5) (Cm + Cd ) dt dt When the pulse spreads over many cells, the discrete spatial coordinate n can be replaced by a continuous one x, series-expanding Vn±1 , and Wn±1 up to the fourth order of the cell length δ, we then obtain the evolution equation of the line voltage: Lg
∂2W ∂2V + [l c (V ) + l c − l c ] m g m m m d ∂t2 ∂t2 ∂W dIds ∂V dcg ∂V 2 + ld −rd cm +lm + rd (cm +cd ) +rd Ids ∂t dV ∂t dV ∂t ∂ 2W δ2 ∂ 4W = + (6) , ∂x2 12 ∂x4 ∂2W ∂2V (lm cm + lm cd − lg cm ) 2 + [lg cg (V ) + lg cm − lm cm ] 2 ∂t ∂t ∂W dIds ∂V dcg ∂V 2 + lm +rg cm +rg cg (V ) +lg − r g cm ∂t dV ∂t dV ∂t 2 2 4 ∂ V δ ∂ V = + (7) , ∂x2 12 ∂x4 (ld cm + ld cd − lm cm )
c
IEICE 2010
DOI: 10.1587/elex.7.1188 Received June 04, 2010 Accepted July 16, 2010 Published August 25, 2010
Fig. 1. Unit cell of nonlinear TW-FETs.
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where V = V (x, t) and W = W (x, t) are the continuous counterparts of Vn and Wn . Moreover, lg,d and cg,d,m are the line inductance and capacitance per unit length defined as l = L/δ and c = C/δ, respectively. There are two different propagation modes, c mode and π mode, on a linear coupled line [4], and the same is true for a subthreshold TW-FET. Each mode has its own velocity and voltage fraction between the lines (= drain voltage/gate voltage). Hereafter, we denote the velocity of c mode, velocity of π mode, voltage fraction of c mode, and voltage fraction of π mode at long wavelengths as uc , uπ , Rc , and Rπ , respectively. These are explicitly written as:
uc,π = Rc,π =
(x1 + x2 )2 − 4x3 , 2x3 x1 − x2 ± (x1 + x2 )2 − 4x3 , 2x4 x1 + x2 ±
(8) (9)
where the upper (lower) signs are for c (π) mode. For concise notations, we define x1,2,3,4 as x1 = cg (V0 )lg + cm lg − cm lm, x2 = cd ld + cm ld − cm lm,
(10)
2 x3 = [cg (V0 )cd + cg (V0 )cm + cd cm ] lg ld − lm
x4 = cm lg − cd lm − cm lm,
(11)
,
(12) (13)
for the case where the gate line is biased at V0 . Moreover, the short-wavelength waves travel slower than the long-wavelength waves due to dispersion, which results in the distortion of the baseband pulses having short temporal durations. In a nonlinear NLTL, this distortion can be compensated for by the nonlinearity introduced using Schottky varactors, regardless of the propagation mode. To quantify the compensation of dispersion by nonlinearity, we apply reductive perturbation [5] to the transmission equation of a coupled NLTL. We first series expand the voltage variables as V (x, t) = V0 +
∞
i V (i) (x, t),
(14)
i=1
W (x, t) = W
(0)
(x) +
∞
i W (i) (x, t),
(15)
i=1
for ( uc , while the π-mode pulse is amplified when us < uπ for nonzero lm . Because uc is always greater than uπ , we can see that when the characteristic velocity us is less than both uc and uπ , the slower mode is the unique amplified mode; in contrast, when us is greater than both uc and uπ , the faster mode is the unique amplified mode. For lm = 0, ν is given by
gm cm lg ld u2c,π rg rd νc,π = ± + + 2 (x1 + x2 )2 − 4x3 4ld 4lg rg rd x1 − x2 − + , 4ld 4lg (x1 + x2 )2 − 4x3
(24)
where upper (lower) signs are for c (π) mode. As a result, when the electrode lines are only capacitively coupled, the π-mode pulse is always the unique mode to be amplified. As far as a pulse travels on a unique mode, it is free from distortions caused by the dispersive difference between modes; therefore, a TW-FET succeeds in amplifying short electrical pulses. We numerically solve eqs. (2)−(5) using a standard finite-difference timedomain method for TW-FETs. The total number of cells is 200. Throughout the calculations, we set β, VT O , C0 , VJ , m, V0 , Lg , Ld , Rg , Rd , Cd , and Cm to 1.5 mA/V2 , −1.8 V, 0.5 pF, 1.0 V, 0.5, −0.5 V, 2.0 nH, 1.0 nH, 0.15 Ω, 0.15 Ω, 0.5 pF, and 0.2 pF, respectively. Figure 2 (a) shows the calculated waveforms monitored at n = 1 (blue), 50 (red), 100 (green), and 150 (black) for Lm = 0.5 nH. For the present line parameters, the values of νπ and νc are
c
IEICE 2010
DOI: 10.1587/elex.7.1188 Received June 04, 2010 Accepted July 16, 2010 Published August 25, 2010
Fig. 2. Amplification of short electrical pulses. (a) Lm = 0.5 nH and (b) Lm =0.
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calculated to be 7.62 × 108 s−1 and 3.78 × 108 s−1 , respectively. Consistently, the pulses simply decay along the line in Fig. 2 (a). Next, we set Lm to zero; therefore, the π-mode pulse is potentially amplified irrespective of the reactive design parameters. Actually, νπ and νc are calculated to be −7.16 × 108 s−1 and 1.84 × 109 s−1 , respectively. Figure 2 (b) shows the calculated waveforms monitored at n = 1 (blue), 50 (red), 100 (green), and 150 (black). We can see that the π-mode pulse is successfully amplified as expected.
c
IEICE 2010
DOI: 10.1587/elex.7.1188 Received June 04, 2010 Accepted July 16, 2010 Published August 25, 2010
1194
Nonlinear traveling-wave field effect transistors for amplification of short electrical pulses Koichi Naraharaa) and Shun Nakagawa Graduate School of Science and Engineering, Yamagata University, 4–3–16 Jonan, Yonezawa, Yamagata 992–8510, Japan a) [email protected]
Abstract: We investigated the properties of pulse propagation on nonlinear traveling-wave field effect transistors (TW-FET) to develop a method for amplifying short electrical pulses. TW-FETs are a special type of FET whose electrodes are employed not only as electrical contacts but also as transmission lines. Due to the presence of electromagnetic couplings between the gate and drain lines, two different propagation modes called the c mode and π mode are developed on a TW-FET. Moreover, the Schottky contact beneath the gate electrode creates an ideal source of nonlinearity for soliton-like propagation. We can design the TW-FET to amplify only soliton-like pulses carried by one of the two modes and attenuate the ones carried by the other mode. This paper discusses the fundamental properties of a nonlinear TWFET, including the width and velocity of a soliton-like pulse carried by c and π modes, and gives design criteria of amplification of soliton-like pulses. Keywords: solitons, nonlinear transmission lines (NLTLs), travelingwave FETs, pulse amplification Classification: Microwave and millimeter wave devices, circuits, and systems References [1] R. Hirota and K. Suzuki, “Studies on lattice solitons by using electrical networks,” J. Phys. Soc. Jpn., vol. 28, pp. 1366–1367, 1970. [2] M. J. W. Rodwell, S. T. Allen, R. Y. Yu, M. G. Case, U. Bhattacharya, M. Reddy, E. Carman, M. Kamegawa, Y. Konishi, J. Pusl, and R. Pullela, “Active and nonlinear wave propagation devices in ultrafast electronics and optoelectronics,” Proc. IEEE, vol. 82, pp. 1037–1059, 1994. [3] K. Narahara and T. Otsuji, “Compression of electrical pulses using traveling-wave field effect transistors,” Jpn. J. Appl. Phys., vol. 38, pp. 4688–4695, 1999. [4] K. C. Gupta, R. Garg, and I. J. Bahl, Microstrip Lines and Slotlines, Artech, 1979. c
IEICE 2010
DOI: 10.1587/elex.7.1188 Received June 04, 2010 Accepted July 16, 2010 Published August 25, 2010
1188
IEICE Electronics Express, Vol.7, No.16, 1188–1194
[5] T. Taniuti, “Reductive perturbation method and far fields of wave equations,” Prog. Theor. Phys. Suppl., vol. 55, pp. 1–35, 1974.
1
Introduction
A nonlinear transmission line (NLTL) is defined as a lumped transmission line containing a series inductor and a shunt Schottky varactor in each section. NLTLs are used for the development of solitons [1]. Moreover, the operation bandwidth of carefully designed Schottky varactors goes beyond 100 GHz; therefore, they are employed in ultrafast electronic circuits including the subpicosecond electrical shock generator [2]. However, the line resistance generally attenuates the pulse amplitude, making pulses very small for an NLTL to exhibit nonlinearity. To bring out the potential of NLTLs, we considered a traveling-wave field effect transistor (TW-FET) to compensate for the wave attenuation by utilizing the transistors’ gain. Because of the Schottky contact beneath the gate, the gate line of a TW-FET simulates an NLTL. In contrast, the drain line is modeled by a linear transmission line coupled with the gate line via the gate-to-drain capacitance, mutual inductance and transconductance. Two different propagation modes called the c mode and π mode develop on a coupled line. Each mode has its own velocity, characteristic impedance and voltage fraction between the gate and drain lines. It is found that the nonlinearity introduced by the gate-to-source Schottky capacitors succeeds in compensating for the distortions caused by dispersion for pulses carried by either c or π modes; although, the nonlinear pulses are greatly attenuated due to the presence of finite electrode resistances irrespective of the carrying mode. For gate voltages higher than the threshold, it is expected that the transconductance relaxes pulse attenuation. Possibly, only a pulse carried by one of the modes gains amplitude, while a pulse carried by another mode decays [3]. Elaborating the design method enables the amplification of pulses that are not affected by both dispersion and attenuation. In this paper, we first describe the fundamental properties of nonlinear pulses observed in a subthreshold TW-FET. Then, we discuss the design criteria for amplifying the nonlinear pulses in a TW-FET, together with several results of numerical evaluations that validate this method.
2
c
IEICE 2010
Nonlinear TW-FETs
Figure 1 shows the diagram of a TW-FET. The gate and drain lines are modeled by an NLTL and a linear line, respectively. These two are coupled via the gate-to-drain capacitance Cm , the mutual inductance Lm , and the drain-to-source current Ids . The per-unit-cell series inductance and shunt capacitance of the gate (drain) line are denoted by Lg (Ld ) and Cg (Cd ), respectively. Note that Cg shows the Schottky varactor whose capacitance-
DOI: 10.1587/elex.7.1188 Received June 04, 2010 Accepted July 16, 2010 Published August 25, 2010
1189
IEICE Electronics Express, Vol.7, No.16, 1188–1194
voltage relationship is generally given by Cg (VX ) =
C0 1−
VX VJ
m ,
(1)
where VX is the voltage between the terminals. C0 , VJ , and m are the optimizing parameter. Note that VX < 0 for reverse bias. We denote the line voltages of the gate and drain lines at the nth cell as Vn and Wn , respectively, and the line currents of the gate and drain lines at the nth cell by In and Jn , respectively. Then, the transmission equations of a TW-FET are given by dIn dJn + Lm = Vn−1 − Vn − Rg In, (2) dt dt dJn dIn + Lm = Wn−1 − Wn − Rd Jn, (3) Ld dt dt dVn dWn − Cm = In − In+1, (4) [Cm + Cg (Vn )] dt dt dWn dVn − Cm = Jn − Jn+1 − Ids (Vn ). (5) (Cm + Cd ) dt dt When the pulse spreads over many cells, the discrete spatial coordinate n can be replaced by a continuous one x, series-expanding Vn±1 , and Wn±1 up to the fourth order of the cell length δ, we then obtain the evolution equation of the line voltage: Lg
∂2W ∂2V + [l c (V ) + l c − l c ] m g m m m d ∂t2 ∂t2 ∂W dIds ∂V dcg ∂V 2 + ld −rd cm +lm + rd (cm +cd ) +rd Ids ∂t dV ∂t dV ∂t ∂ 2W δ2 ∂ 4W = + (6) , ∂x2 12 ∂x4 ∂2W ∂2V (lm cm + lm cd − lg cm ) 2 + [lg cg (V ) + lg cm − lm cm ] 2 ∂t ∂t ∂W dIds ∂V dcg ∂V 2 + lm +rg cm +rg cg (V ) +lg − r g cm ∂t dV ∂t dV ∂t 2 2 4 ∂ V δ ∂ V = + (7) , ∂x2 12 ∂x4 (ld cm + ld cd − lm cm )
c
IEICE 2010
DOI: 10.1587/elex.7.1188 Received June 04, 2010 Accepted July 16, 2010 Published August 25, 2010
Fig. 1. Unit cell of nonlinear TW-FETs.
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IEICE Electronics Express, Vol.7, No.16, 1188–1194
where V = V (x, t) and W = W (x, t) are the continuous counterparts of Vn and Wn . Moreover, lg,d and cg,d,m are the line inductance and capacitance per unit length defined as l = L/δ and c = C/δ, respectively. There are two different propagation modes, c mode and π mode, on a linear coupled line [4], and the same is true for a subthreshold TW-FET. Each mode has its own velocity and voltage fraction between the lines (= drain voltage/gate voltage). Hereafter, we denote the velocity of c mode, velocity of π mode, voltage fraction of c mode, and voltage fraction of π mode at long wavelengths as uc , uπ , Rc , and Rπ , respectively. These are explicitly written as:
uc,π = Rc,π =
(x1 + x2 )2 − 4x3 , 2x3 x1 − x2 ± (x1 + x2 )2 − 4x3 , 2x4 x1 + x2 ±
(8) (9)
where the upper (lower) signs are for c (π) mode. For concise notations, we define x1,2,3,4 as x1 = cg (V0 )lg + cm lg − cm lm, x2 = cd ld + cm ld − cm lm,
(10)
2 x3 = [cg (V0 )cd + cg (V0 )cm + cd cm ] lg ld − lm
x4 = cm lg − cd lm − cm lm,
(11)
,
(12) (13)
for the case where the gate line is biased at V0 . Moreover, the short-wavelength waves travel slower than the long-wavelength waves due to dispersion, which results in the distortion of the baseband pulses having short temporal durations. In a nonlinear NLTL, this distortion can be compensated for by the nonlinearity introduced using Schottky varactors, regardless of the propagation mode. To quantify the compensation of dispersion by nonlinearity, we apply reductive perturbation [5] to the transmission equation of a coupled NLTL. We first series expand the voltage variables as V (x, t) = V0 +
∞
i V (i) (x, t),
(14)
i=1
W (x, t) = W
(0)
(x) +
∞
i W (i) (x, t),
(15)
i=1
for ( uc , while the π-mode pulse is amplified when us < uπ for nonzero lm . Because uc is always greater than uπ , we can see that when the characteristic velocity us is less than both uc and uπ , the slower mode is the unique amplified mode; in contrast, when us is greater than both uc and uπ , the faster mode is the unique amplified mode. For lm = 0, ν is given by
gm cm lg ld u2c,π rg rd νc,π = ± + + 2 (x1 + x2 )2 − 4x3 4ld 4lg rg rd x1 − x2 − + , 4ld 4lg (x1 + x2 )2 − 4x3
(24)
where upper (lower) signs are for c (π) mode. As a result, when the electrode lines are only capacitively coupled, the π-mode pulse is always the unique mode to be amplified. As far as a pulse travels on a unique mode, it is free from distortions caused by the dispersive difference between modes; therefore, a TW-FET succeeds in amplifying short electrical pulses. We numerically solve eqs. (2)−(5) using a standard finite-difference timedomain method for TW-FETs. The total number of cells is 200. Throughout the calculations, we set β, VT O , C0 , VJ , m, V0 , Lg , Ld , Rg , Rd , Cd , and Cm to 1.5 mA/V2 , −1.8 V, 0.5 pF, 1.0 V, 0.5, −0.5 V, 2.0 nH, 1.0 nH, 0.15 Ω, 0.15 Ω, 0.5 pF, and 0.2 pF, respectively. Figure 2 (a) shows the calculated waveforms monitored at n = 1 (blue), 50 (red), 100 (green), and 150 (black) for Lm = 0.5 nH. For the present line parameters, the values of νπ and νc are
c
IEICE 2010
DOI: 10.1587/elex.7.1188 Received June 04, 2010 Accepted July 16, 2010 Published August 25, 2010
Fig. 2. Amplification of short electrical pulses. (a) Lm = 0.5 nH and (b) Lm =0.
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IEICE Electronics Express, Vol.7, No.16, 1188–1194
calculated to be 7.62 × 108 s−1 and 3.78 × 108 s−1 , respectively. Consistently, the pulses simply decay along the line in Fig. 2 (a). Next, we set Lm to zero; therefore, the π-mode pulse is potentially amplified irrespective of the reactive design parameters. Actually, νπ and νc are calculated to be −7.16 × 108 s−1 and 1.84 × 109 s−1 , respectively. Figure 2 (b) shows the calculated waveforms monitored at n = 1 (blue), 50 (red), 100 (green), and 150 (black). We can see that the π-mode pulse is successfully amplified as expected.
c
IEICE 2010
DOI: 10.1587/elex.7.1188 Received June 04, 2010 Accepted July 16, 2010 Published August 25, 2010
1194
