Ampli_Schottky.pdf

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

Amplication of shortpulses in transmission linesperiodically loaded withSchottky varactors

Koichi Narahara a)

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 Schottkyvaractors, is investigated for the amplication of short pulses. It isfound that the soliton developed in an NLTL experiences an exponen-tial amplitude growth, when it couples with a co-existing voltage edge.Keywords: solitons, nonlinear transmission lines (NLTLs), pulse am-plicationClassication: Science and engineering for electronics

References

[1] R. Hirota and K. Suzuki, “Studies on lattice solitons by using electricalnetworks,” 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 electronicsand optoelectronics,” Proc. IEEE , vol. 82, pp. 1037–1059, 1994.

[3] Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly inte-grable systems,” Rev. Mod. Phys. , vol. 61, no. 4, pp. 763–915, 1989.

[4] T. Taniuti, “Reductive perturbation method and far elds of wave equa-tions,” Prog. Theor. Phys. Suppl. , vol. 55, pp. 1–35, 1974.

1 Introduction

It is well-known that a nonlinear transmission line (NLTL) dened as alumped transmission line containing a series inductor and shunt Schottkyvaractor in each section succeeds in the development of solitons [1]. More-over, the operation bandwidth of carefully designed Schottky varactors goesbeyond 100 GHz; therefore, they are employed in ultrafast electronic circuitsincluding the sub-picosecond electrical shock generator [2]. The resultingshort pulse can be applied to high-resolution measurement and high-speedcommunication systems. An NLTL is useful for more than just short-pulsegeneration. We found that it enables the amplication of short pulses. This

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

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article discusses the method of pulse amplication in an NLTL, together withdesign criteria obtained analytically and several results of numerical evalua-tions that validate the method.

2 Operating principle

Figure 1 shows the representation of an NLTL. L and C represent the seriesinductor and shunt Schottky varactor of the unit cell, respectively. Ψ n andI n show the line voltage and current at the nth cell. For later convenience, weconsider the case where each cell of the line is individually biased. Φ n showsthe terminal voltage of the n th Schottky varactor. The capacitance–voltagerelationship of a Schottky varactor is generally given by

C (V ) = C 01 − V

V J

m , (1)

where C 0, V J and m are the optimizing parameter. Note that V < 0 forreverse bias.

Figure 2 shows the operating principle of the amplication method. Fig-ure 2 (a) shows the initial setup of signal application. The red short pulsepart, supported by a blue step-like part, is to be amplied. The top voltagelevel is set to −V 0. The voltage level rst decreases up to −V 1, forming anedge pe . The pulse to be amplied is set up after the voltage reaches −V 1.First, the red pulse becomes solitonic due to the presence of varactors. Be-cause 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, sothat the solitonic pulse overtakes pe (Fig. 2 (b)) and nally leaves pe behind(Fig. 2 (c)). During this process, the solitonic pulse will be amplied due tothe effect of its coupling with pe, yielding the nal amplitude of the pulse tobe Af (>> A i ) of Fig. 2 (c). Although the pulse propagation in Fig. 2 cannotbe 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 thesoliton’s perturbation theory [3], when we assume that the degree of voltagevariation of pe is much smaller than that of the short pulse.

In order to apply the perturbation theory, we rst derive the Korteweg-de Vries (KdV) equation from the transmission equation of an NLTL withvaractors modeled by eq.(1), being followed by a modulation term that cor-responds to the effect caused by the coupling with pe. We will see that the

Fig. 1. Equivalent representation of NLTLs.c IEICE 2009DOI: 10.1587/elex.6.1199Received July 24, 2009Accepted July 24, 2009Published August 25, 2009

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Fig. 2. Operation principle of pulse amplication with anNLTL. The temporal voltage waveforms at (a) theinput, (b) mid point and (c) output.

modulation term results in the exponential growth of the soliton’s amplitudein eq.(18). This observation establishes our method.

The transmission equation of an NLTL is given by

LdI n − 1

dt = Ψ n − 1 −Ψn, (2)

C (Ψn −Φn )dΨn

dt = I n − 1 −I n. (3)

When the pulse spreads over many cells, the discrete spatial coordinate n can

be replaced by a continuous one x, series-expanding Ψ n ± 1 up to the fourthorder of the cell length d as

Ψn ± 1 ∼Ψn ±∂ Ψ(x, t )

∂x d +

d2

2∂ 2Ψ(x, t )

∂x 2 ±d3

6∂ 3Ψ(x, t )

∂x 3 + d4

24∂ 4Ψ(x, t )

∂x 4 . (4)

Applying this long-wavelength approximation to eqs.(2) and (3), we obtainthe evolution equation of the line voltage:

lc(Ψ −Φ)∂ 2Ψ∂t 2 =

∂ 2Ψ∂x 2 +

112

∂ 4Ψ∂x 4 −l

dc(Ψ −Φ)dΨ

∂ Ψ∂t

2, (5)

where l and c are the line inductance and capacitance per unit length denedas l = L/d and c = C/d , respectively. In order to derive the soliton equationc IEICE 2009

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

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from eq.(5), the voltage variables are series-expanded as

Ψ(x, t ) =∞

i=1

i ψ(i) (x, t ), (6)

Φ(x, t ) = V 0 +∞

i=1

i φ(i) (x, t ), (7)

for << 1. Note that V 0 > 0 for biasing Schottky varactors reversely.Moreover, the following transformations are applied.

ξ = 1/ 2 x −1

√ c1lt , (8)

τ = 3/ 2t, (9)

where c1 = c(−V 0). By evaluating eq.(5) for each order of , we can extractthe equation that describes the developing solitonic pulses [4]. It has beenshown that O( ) and O( 2) terms give trivial identities, and O( 3) termsresult in the following modulated KdV equation:

∂ψ (1)

∂τ −m

2√ c1l(V 0 + V J )ψ(1) ∂ψ (1)

∂ξ +

124√ c1l

∂ 3ψ(1)

∂ξ 3

+ m

2√ c1l(V 0 + V J )φ(1) ∂ψ (1)

∂ξ = 0 . (10)

Scaling ψ (1) , τ and ξ as ψ(1) = 18( V 0 + V J )ψ/m , τ = √ c1lτ / 9 and ξ = ξ / 6,respectively, eq. (10) becomes

∂ψ

∂τ −6ψ

∂ψ

∂ξ +

∂ 3ψ

∂ξ 3 = R(ξ , τ ), (11)

R(ξ , τ ) = βφ(1) ∂ψ∂ξ , (12)

where β = −m/ 3(V 0 + V J ).For convenience, we briey review the fundamental properties of KdV

solitons. The standard KdV equation, ∂ τ ψ −6ψ∂ ξ ψ + ∂ 3ξ ψ = 0, has theone-soliton solution which is explicitly described as

ψ = −2κ2sech2z, (13)

z = κ(ξ −η), (14)

η = 4κ2τ , (15)

where κ is the parameter that corresponds to the soliton’s amplitude, whichis time-invariant for unmodulated ones.

In the framework of the soliton’s perturbation theory, the soliton’s ampli-tude becomes time-dependent by the presence of R(ξ , τ ), and is describedby the following evolution equation of κ [3]:

dκdτ

= −1

4κ ∞

−∞dzR (z, τ )sech 2z . (16)

In order to investigate the situation shown in Fig. 2, we consider the casewhere Φ(x, t ) is given by V 0 + 3/ 2V r (x −t/ √ c1l). Using eq.(8), it results in

φ(1) = V r ξ / 6. (17)


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Originally, both the oor step-like and short pulses are applied to the NLTL,while Φ is set to identically zero. At this point, we think of the oor step-likepulse as applied as Φ. This corresponds to the situation where pe travels ata speed of 1/ √ c1l with a spatial gradient of 3/ 2V r .

Using eqs.(12), (13) and (17), the right-hand side of eq.(16) is calculatedbe −βV r κ/ 18, thus giving

κ = κ0 exp −βV r τ

18 , (18)

where κ0 corresponds to the unperturbed soliton’s amplitude. Because β < 0,κ grows if V r > 0. The gradient of pe shown in Fig. 2 satises 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 nite-difference time-domain method for an NLTL with Schottky varactors having C 0 = 64 .77pF,V J = 3 .561V and m = 1 .259. L, V 0, V 1 and the fall-time of pe are set to100 µH, 0.0V, −1.0V, and 30 µs, respectively. Moreover, the total cell sizeis 3000. The input pulse to be amplied has a form of

Ψ(t) = −AV 0 + V J

m sech2 A

LC (−V 1)1 +

A6

t , (19)

which represents the one-soliton waveform given by eq.(13) using the n andt coordinates. For the present analyses, A is kept xed at 0.3.

The numerically obtained waveforms monitored at three distinct pointson an NLTL are shown in Fig. 3 (a), (b) and (c). The short pulse reallytravels faster than the step-like one, and starts coupling with pe at 300 celldistances from the input, and then completes the coupling with pe at 2100cells from the input. As a result of the shock formation, the fall-time of pe decreases as it propagates along the line. The oor step-like pulse hassufficiently large voltage variation, so that its effect to the solitonic shortpulse 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 pulsepeak 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 wherethe NLTL has 0.2 Ω series resistance at each cell. The exponential increase inthe peak height is explicitly observed for both cases. From eq.(13), the pulsewidth decreases as the amplitude increases. During the amplication process,the pulse width is reduced so much that the discrete line structure inuencesthe 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 amplied again and its widthis reduced. This cycle explains the oscillations. Although it requires furtherinvestigations to quantify how much resistance an NLTL allows, an NLTL


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Fig. 3. Numerical evaluations of pulse amplication usingan 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 circlesin (a), (b) and (c) corresponds to pe of Fig. 2 (a).

succeeds in the pulse amplication even at the nite presence of electroderesistance.

Note that the method does not need any sophisticated semiconductor pro-cess technologies, because a unique nontrivial device is a Schottky varactor,the simplest and fastest semiconductor device. Although the present analysisdeals with MHz pulses, we believe that it is equally possible to amplify veryshort pulses with picosecond durations by using monolithically integrateddevices.


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