Amplification of short pulses in transmission lines

IEICE Electronics Express, Vol.6, No.16, 1199–1204

Amplification of short pulses in transmission lines periodically loaded with Schottky varactors Koichi Naraharaa) Graduate School of Science and Engineering, Yamagata University, 4–3–16 Jonan, Yonezawa, Yamagata 992–8510, Japan a) [email protected]

Abstract: Pulse progagation on nonlinear transmission lines (NLTLs), which are transmission lines with regularly spaced Schottky varactors, is investigated for the amplification of short pulses. It is found that the soliton developed in an NLTL experiences an exponential amplitude growth, when it couples with a co-existing voltage edge. Keywords: solitons, nonlinear transmission lines (NLTLs), pulse amplification Classification: Science and engineering for electronics 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] Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys., vol. 61, no. 4, pp. 763–915, 1989. [4] T. Taniuti, “Reductive perturbation method and far fields of wave equations,” Prog. Theor. Phys. Suppl., vol. 55, pp. 1–35, 1974.

1

c 

IEICE 2009

DOI: 10.1587/elex.6.1199 Received July 24, 2009 Accepted July 24, 2009 Published August 25, 2009

Introduction

It is well-known that a nonlinear transmission line (NLTL) defined as a lumped transmission line containing a series inductor and shunt Schottky varactor in each section succeeds in 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 sub-picosecond electrical shock generator [2]. The resulting short pulse can be applied to high-resolution measurement and high-speed communication systems. An NLTL is useful for more than just short-pulse generation. We found that it enables the amplification of short pulses. This

1199

IEICE Electronics Express, Vol.6, No.16, 1199–1204

article discusses the method of pulse amplification in an NLTL, together with design criteria obtained analytically and several results of numerical evaluations that validate the method.

2

Operating principle

Figure 1 shows the representation of an NLTL. L and C represent the series inductor and shunt Schottky varactor of the unit cell, respectively. Ψn and In show the line voltage and current at the nth cell. For later convenience, we consider the case where each cell of the line is individually biased. Φn shows the terminal voltage of the nth Schottky varactor. The capacitance–voltage relationship of a Schottky varactor is generally given by C(V ) = 

C0 1−

V VJ

m ,

(1)

where C0 , VJ and m are the optimizing parameter. Note that V < 0 for reverse bias. Figure 2 shows the operating principle of the amplification method. Figure 2 (a) shows the initial setup of signal application. The red short pulse part, supported by a blue step-like part, is to be amplified. The top voltage level is set to −V0 . The voltage level first decreases up to −V1 , forming an edge pe . The pulse to be amplified is set up after the voltage reaches −V1 . First, the red pulse becomes solitonic due to the presence of varactors. Because the voltage levels of pe are greater than those forming the solitonic pulse, the velocity of pe has to be smaller than that of the solitonic pulse, so that the solitonic pulse overtakes pe (Fig. 2 (b)) and finally leaves pe behind (Fig. 2 (c)). During this process, the solitonic pulse will be amplified due to the effect of its coupling with pe , yielding the final amplitude of the pulse to be Af (>> Ai ) of Fig. 2 (c). Although the pulse propagation in Fig. 2 cannot be characterized rigorously owing to the presence of varactors’ nonlinearity, it is still possible to examine the effect of pe on the short pulse by using the soliton’s perturbation theory [3], when we assume that the degree of voltage variation of pe is much smaller than that of the short pulse. In order to apply the perturbation theory, we first derive the Kortewegde Vries (KdV) equation from the transmission equation of an NLTL with varactors modeled by eq.(1), being followed by a modulation term that corresponds to the effect caused by the coupling with pe . We will see that the

c 

IEICE 2009

Fig. 1. Equivalent representation of NLTLs.

DOI: 10.1587/elex.6.1199 Received July 24, 2009 Accepted July 24, 2009 Published August 25, 2009

1200

IEICE Electronics Express, Vol.6, No.16, 1199–1204

Fig. 2. Operation principle of pulse amplification with an NLTL. The temporal voltage waveforms at (a) the input, (b) mid point and (c) output. modulation term results in the exponential growth of the soliton’s amplitude in eq.(18). This observation establishes our method. The transmission equation of an NLTL is given by dIn−1 = Ψn−1 − Ψn, dt dΨn = In−1 − In. C(Ψn − Φn ) dt

(2)

L

(3)

When the pulse spreads over many cells, the discrete spatial coordinate n can be replaced by a continuous one x, series-expanding Ψn±1 up to the fourth order of the cell length d as Ψn±1 ∼ Ψn ±

d2 ∂ 2 Ψ(x, t) d3 ∂ 3 Ψ(x, t) d4 ∂ 4 Ψ(x, t) ∂Ψ(x, t) d+ ± + . (4) ∂x 2 ∂x2 6 ∂x3 24 ∂x4

Applying this long-wavelength approximation to eqs.(2) and (3), we obtain the evolution equation of the line voltage: ∂2Ψ 1 ∂4Ψ dc(Ψ − Φ) ∂2Ψ + −l lc(Ψ − Φ) 2 = 2 4 ∂t ∂x 12 ∂x dΨ c 

IEICE 2009

DOI: 10.1587/elex.6.1199 Received July 24, 2009 Accepted July 24, 2009 Published August 25, 2009



∂Ψ ∂t

2

,

(5)

where l and c are the line inductance and capacitance per unit length defined as l = L/d and c = C/d, respectively. In order to derive the soliton equation

1201

IEICE Electronics Express, Vol.6, No.16, 1199–1204

from eq.(5), the voltage variables are series-expanded as Ψ(x, t) =

∞ 

i ψ (i) (x, t),

i=1

Φ(x, t) = V0 +

∞ 

(6)

i φ(i) (x, t),

(7)

i=1

for  0 for biasing Schottky varactors reversely. Moreover, the following transformations are applied. 1/2



ξ=

τ = 3/2 t,



1 x− √ t , c1 l

(8) (9)

where c1 = c(−V0 ). By evaluating eq.(5) for each order of , we can extract the equation that describes the developing solitonic pulses [4]. It has been shown that O() and O(2 ) terms give trivial identities, and O(3 ) terms result in the following modulated KdV equation: m 1 ∂ 3 ψ (1) ∂ψ (1) ∂ψ (1) − √ + √ ψ (1) ∂τ ∂ξ 2 c1 l(V0 + VJ ) 24 c1 l ∂ξ 3 ∂ψ (1) m = 0. φ(1) + √ ∂ξ 2 c1 l(V0 + VJ ) Scaling ψ (1) , τ and ξ as ψ (1) = 18(V0 + VJ )ψ/m, τ = respectively, eq. (10) becomes

(10) √

c1 lτ  /9 and ξ = ξ  /6,

∂ψ ∂3ψ ∂ψ − 6ψ + = R(ξ  , τ  ), ∂τ  ∂ξ  ∂ξ 3 ∂ψ R(ξ  , τ  ) = βφ(1)  , ∂ξ

(11) (12)

where β = −m/3(V0 + VJ ). For convenience, we briefly review the fundamental properties of KdV solitons. The standard KdV equation, ∂τ  ψ − 6ψ∂ξ ψ + ∂ξ3 ψ = 0, has the one-soliton solution which is explicitly described as ψ = −2κ2 sech2 z, 

(13)

z = κ(ξ − η),

(14)

η = 4κ2 τ  ,

(15)

where κ is the parameter that corresponds to the soliton’s amplitude, which is time-invariant for unmodulated ones. In the framework of the soliton’s perturbation theory, the soliton’s amplitude becomes time-dependent by the presence of R(ξ  , τ  ), and is described by the following evolution equation of κ [3]: 1 dκ =− dτ  4κ c 

IEICE 2009

DOI: 10.1587/elex.6.1199 Received July 24, 2009 Accepted July 24, 2009 Published August 25, 2009

 ∞

−∞

dzR(z, τ  )sech2 z .

(16)

In order to investigate the situation shown in Fig. 2, we consider the case √ where Φ(x, t) is given by V0 + 3/2 Vr (x − t/ c1 l). Using eq.(8), it results in φ(1) = Vr ξ  /6.

(17) 1202

IEICE Electronics Express, Vol.6, No.16, 1199–1204

Originally, both the floor step-like and short pulses are applied to the NLTL, while Φ is set to identically zero. At this point, we think of the floor step-like pulse as applied as Φ. This corresponds to the situation where pe travels at √ a speed of 1/ c1 l with a spatial gradient of 3/2 Vr . Using eqs.(12), (13) and (17), the right-hand side of eq.(16) is calculated be −βVr κ/18, thus giving 

βVr τ  κ = κ0 exp − 18



(18)

,

where κ0 corresponds to the unperturbed soliton’s amplitude. Because β < 0, κ grows if Vr > 0. The gradient of pe shown in Fig. 2 satisfies this condition. Moreover, the growth rate increases when the gradient of pe becomes larger; therefore, the pulse gain in an NLTL can be controlled by the input waveform.

3

Numerical evaluation

We numerically solve eqs.(2) and (3) using standard finite-difference timedomain method for an NLTL with Schottky varactors having C0 = 64.77 pF, VJ = 3.561 V and m = 1.259. L, V0 , V1 and the fall-time of pe are set to 100 μH, 0.0 V, −1.0 V, and 30 μs, respectively. Moreover, the total cell size is 3000. The input pulse to be amplified has a form of V0 + V J sech2 Ψ(t) = −A m

c 

IEICE 2009

DOI: 10.1587/elex.6.1199 Received July 24, 2009 Accepted July 24, 2009 Published August 25, 2009







A A 1+ t , LC(−V1 ) 6

(19)

which represents the one-soliton waveform given by eq.(13) using the n and t coordinates. For the present analyses, A is kept fixed at 0.3. The numerically obtained waveforms monitored at three distinct points on an NLTL are shown in Fig. 3 (a), (b) and (c). The short pulse really travels faster than the step-like one, and starts coupling with pe at 300 cell distances from the input, and then completes the coupling with pe at 2100 cells from the input. As a result of the shock formation, the fall-time of pe decreases as it propagates along the line. The floor step-like pulse has sufficiently large voltage variation, so that its effect to the solitonic short pulse cannot be treated perturbatively. However, the exponential growth of the pulse amplitude is successfully demonstrated. To see the growth of amplitude more clearly, the variations of the pulse peak are shown in Fig. 3 (d). The black curve corresponds to the analysis of the present loss-less NLTL, and the red one corresponds to the case where the NLTL has 0.2 Ω series resistance at each cell. The exponential increase in the peak height is explicitly observed for both cases. From eq.(13), the pulse width decreases as the amplitude increases. During the amplification process, the pulse width is reduced so much that the discrete line structure influences the waveform, giving oscillatory temporal variations observed in Fig. 3 (d). When the spatial pulse extent is reduced to a few cells, it will be relaxed by decreasing the amplitude. Then the pulse is amplified again and its width is reduced. This cycle explains the oscillations. Although it requires further investigations to quantify how much resistance an NLTL allows, an NLTL 1203

IEICE Electronics Express, Vol.6, No.16, 1199–1204

Fig. 3. Numerical evaluations of pulse amplification using an NLTL. The waveforms monitored at (a) 300, (b) 1500 and (c) 2100 cell distant from input, and (d) the temporal variations of the peak voltages. The temporal range sandwiched by two red circles in (a), (b) and (c) corresponds to pe of Fig. 2 (a). succeeds in the pulse amplification even at the finite presence of electrode resistance. Note that the method does not need any sophisticated semiconductor process technologies, because a unique nontrivial device is a Schottky varactor, the simplest and fastest semiconductor device. Although the present analysis deals with MHz pulses, we believe that it is equally possible to amplify very short pulses with picosecond durations by using monolithically integrated devices.

c 

IEICE 2009

DOI: 10.1587/elex.6.1199 Received July 24, 2009 Accepted July 24, 2009 Published August 25, 2009

1204