Design of pulsed waveform oscillators with a short nonlinear

transmission line

Mabel Ponton, Franco Ramirez, Almudena Suarez

Dpto. de Ingenieria de Comunicaciones, Universidad de Cantabria, Santander, Spain

Abstract - In this paper we present a new methodology for the

design of pulsed-waveform oscillators. The design is based on the

use of a short section of nonlinear transmission line (NLTL),

constituting the load of a transistor-based subnetwork exhibiting

negative resistance. The oscillation start-up conditions are

imposed at a transistor port different from the one at which the

NLTL is connected, to increase the optimization flexibility. The

NL TL is ended in a reactive termination which is optimized in

order to reduce the duty cycle of the voltage waveform.

Compared with previous works, the new methodology enables a

smaller duty cycle with less varactor diodes and no need for

adaptive gain in the transistor stage. The methodology has been

applied to obtain a pulsed-waveform oscillator at 1.2 GHz,

achieving a 5% duty cycle. Good agreement has been obtained in

the comparison with the experimental results.

Index Terms - Pulsed waveform, oscillator, soliton, nonlinear transmission line.

I. INTRODUCTION

Nonlinear transmission lines (NLTL) can be used to generate a train of short-duration pulses, which are applied in time­domain reflectometry, high speed sampling or ultrawideband radar [1-2]. An NLTL is composed by a certain number of inductance-varactor cells. The generation of the pulsed waveform is due to the combined effects of the nonlinear capacitance C(v) and the dispersion of the periodic LC line.

This should give rise to one or more robust localized waves with permanent profile, also known as solitons [3-4]. It has been demonstrated [5] that an input pulse with full width at half maximum T FWHM decomposes into N = 2fB TFWHM

individual pulses or solitons, with fB the Bragg frequency of

the NL TL, given approximately by fB = 1 /( 1t�LCeff), Ceff the

effective varactor capacitance and L the cell inductance . Each of the N pulses or solitons travels at its own velocity and the propagation fulfils some general rules. For higher pulse amplitude, the propagation velocity increases, whereas T FWHM decreases. Solitons pass through one another without losing their identities, though during the overlap time their joint amplitude decreases (increases) when travelling in the same (opposite) direction. For the generation of a short-duty cycle pulsed waveform, the aim is to obtain one single narrow pulse, which requires a careful selection of the cell number. According to the Fermi-Pasta-Ulam recurrence phenomenon

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[3], if we neglect dissipation, the original input waveform is recomposed after a certain number of cells. The recurrence period is smaller for a higher input frequency and higher amplitude. In order to avoid the need for a periodic input signal, some works have proposed design techniques for soliton oscillators. In [1] the authors use a 3D-cell NLTL as the parallel feedback network of an amplifier incorporating a threshold dependent gain-attenuation mechanism. The aim is to attenuate the smaller amplitude solitons and thus eliminate the risk of instabilities from the coexistence of multiple oscillation modes. The same feedback topology is considered in [6], where a near soliton signal is obtained at the NLTL input through the use of substitution generators in harmonic balance (HB). The reflection oscillator topology has been investigated in the work [2], where the NL TL constitutes the load of a reflection amplifier with an adaptative bias scheme, similar to the one in [1]. The NLTL is composed of 24 L­varactor cells and terminated in an open circuit, which gives rise to a significant increase of the pulse amplitude. The output signal is observed with a high impedance probe.

In this work, we consider a similar topology to [2], with a different design procedure. The oscillation start-up conditions are imposed at a transistor port different from the one at which the NL TL is connected, so as to increase the flexibility in the NLTL design. We use a short NLTL section, with a small number of L-varactor cells and optimize the reactive termination in order to reduce the duty cycle. As will be shown, no adaptive control is necessary because the waveform seen by the transistor network does not exhibit any low amplitude pulses. The technique will be applied to obtain a pulsed waveform with 5% duty cycle at 1.2 GHz.

II. DESIGN PROCEDURE

In this particular design, a single-ended oscillator topology will be considered. To enable a flexible optimization of the NL TL, we will impose the oscillation start-up condition at a transistor port different from the one at which the NL TL is connected. Here a bipolar transistor is used, and the NL TL will be connected to the collector terminal, whereas the oscillation start-up condition will be analyzed at the base terminal. In order to facilitate the extraction of the pulsed voltage waveform, with a 50 Ohm load, we will preset a low characteristic impedance of the NL TL. In small signal

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conditions this impedance is given by Zc = �Lline / Co , where

Lline is the series inductance and Co is the capacitance of the varactor at the bias point V bias' The impedance selected is Zc =10 Ohm. Next, we decide on the element values taking into account the Bragg frequency. For too high fB the evolution of the pulsed waveform along the cells of the NLTL will be too slow and a large number of cells may be required. The selected Bragg frequency is fB=II.4 GHz, so we have L=O.3 nH for a varactor diode with the parameters: Cjo=4.2 pF, �0=1.7, M=0.9 and Rs=1.2 Q. Initially, we consider eight L­varactor cells, which is one half the number of cells in the previous work by the authors [6]. The aim is to optimize the NL TL termination in order to obtain a low duty cycle waveform with this small number of cells.

To facilitate the optimization, we will ensure that for all the NL TL reactive terminations, negative resistance is obtained at the transistor input port (between the base terminal and ground). The transistor output is connected to the NL TL input (Fig. 1 a). A parallel inductance L2 is also introduced at this point, as an additional parameter. We will tune the inductance and the transistor series feedback, in order to obtain negative resistance Re(Zin)<O for all the possible reactive terminations at the desired oscillation frequency fo=l.2 GHz. The NL TL termination is defined in terms of the reflection coefficient

r = lejs. We sweep 8, which provides a locus of input impedances located outside the unit circle, corresponding to the circle A in Fig. 1 b. We observe that for all the possible r values the transistor exhibits negative resistance, so, with a proper selection of the base termination, the oscillation start­up conditions should be fulfilled. The corresponding circle of NL TL input impedance is located inside the Smith chart.

F L v, L V2" :rsL VB

� or ... ... r=pe

<E--- NLTL � T

(a) (b) Fig. I Oscillation start-up. (a) Technique to guarantee a negative

resistance at the transistor input. (b) Variation of the input impedance for an NL TL with fixed LC values (A) and a gradual NL TL (B).

Note that in order to get the desired pulsed waveform we will need sufficiently high amplitude at the NL TL input. This amplitude is provided by the circuit oscillation, which in tum depends on the NL TL parameters. As an additional case, we have also considered a gradual line [7], with inductance 2L in Cells 1-4 and L in Cells 5-8, which might be more convenient for the global circuit performance. When applying the same input impedance analysis (Fig. 1) to this NL TL, we also obtain a transformed circle located outside the Smith chart (circle B).

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Next, we optimize the NL TL in a separate simulation. The NL TL is excited with a voltage source of impedance

Zc = �Lline / Ceff , in parallel connection with the inductor L2.

We optimize the reactive termination of the NL TL in order to reduce the duty cycle. By performing this analysis, we obtain a satisfactory NL TL behavior for the inductance value Lend=2L. The optimum waveform is obtained at the last NL TL cell (Fig. 2), which is attributed to the significant reflection effects. Fig. 2 shows the evolution of the total voltage waveform when travelling from Cell 1 to Cell 8 of NL TL. At Cell 1, the waveform is decomposed into three pulses. The one with lowest amplitude seems to dissipate, whereas the other two pulses move in opposite direction at different velocities. They approach each other and merge at Cell 6, giving rise to a much higher amplitude pulse.

Cell 1

� \ �� \ � : 50 Cell 6

2 1

Lrlr'lr'LrLrv 0 1 2 3 4 50

Cell 4

1

50 C�II

1 �t.rl,r.il,r.i�\I

1 2 3 4 5 Time (ns)

CeliS

V V V V V Cell

v-rrv-rv " J 4

Fig. 2 Evolution of the wavefonn along the eight NL TL cells when the line is tenninated with the inductance 2L.

We note that the TFWHM at the last cell agrees with TFWHM = 1/(2f8) = 43.85ps, which should be the minimum

achievable with this particular line. For validation, in Fig. 3 we compare the waveform at Cell 8 with the one obtained with a 50 cell NL TL excited with the same input amplitude, at the cell where the waveform is optimum. The 50 cell NL TL has been analyzed both in open circuit and output matching conditions. The amplitude reduction in the 50 cell NL TL is attributed to the dissipation loss in this long line.

The duty cycle obtained at the last node of the 8-cell NL TL is 5%. The inclusion of a 50 Ohm impedance in parallel hardly affects the performance. This is because the line characteristic impedance is low. Note that it is a similar situation to the works [1-2], with an average characteristic impedance of 50 Ohm and an output load constituted by the high input impedance of the oscilloscope employed for the observation of the waveforms. The possibility to use a 50 Ohm load without a significant disturbance of the waveform is an advantage of this design procedure. We have also analyzed the behavior of the gradual NL TL. The waveform at the last cell is very similar, with amplitude 4.5 V and duty cycle of 7%.

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o

1 , , ,

:. ,. -, . , - .�\ � 0.5 1

, • '. , , ,

) ,

� ,

. i i i i I' , . I' I. ' . I. 0\ �, ,. �l /" �, ,. �I I. �,!,' . � ..... ' �� .. '. I �-;. \ , � .;. \ �. ,\ , ., ,. ' I, y, . II I' • '.,t Vt �-. i�' -.� J •

-8·cell �L �L.Short

-_.- 50-cell NLTL-open � ._._- 50-cell NL TL-matched r

1.5 2 2.5 3 3.5 4 4.5 5 Time (ns)

Fig. 3 Comparison between the pulsed waveform obtained at the 8th

cell in the proposed 8-cell line and the ones obtained with an open and matched NL TL of 50 cells.

Next we connect the NL TL (plus the inductor) to the transistor collector terminal and complete the oscillator design. Due to the described small-signal design procedure, negative resistance is obtained at the base terminal. We calculate the base termination network in order to fulfil of the oscillation start-up conditions at fo=I.2 GHz. This termination consists of a parallel L-C resonator (Fig. 4a). The fulfillment of the well known oscillation start-up conditions, in terms of the total admittance, is shown in FigAb. The more rigorous stability analysis of the dc solution with pole-zero identification [8] confirms the existence of a pair of complex conjugate poles at about 1.2 GHz, located on the right hand side of the complex plane.

O.01r-��-��-�--' 0.005

9: � -0.005 c .� -0.01 -5 ·0.015 «

..().02

Irnag(YT) ... .. ...

Re(YT)

O. 1. 1.1 1. 1. 1.4 1.5 Frequency (GHz)

(a) (b) Fig. 4 Oscillator circuit. (a) Schematic. (b) Variation of the total admittance at the base terminal, with an LC resonator termination

connected to this terminal.

The large-signal design is performed introducing an auxiliary generator (AG) at the oscillation frequency fAG=fo in harmonic balance [9]. The voltage AG, with amplitude VAG, is connected in parallel at a sensitive circuit node and must fulfill a non-perturbation condition of the oscillatory solution, given by the zero value of the ratio between the current through the generator lAG and the delivered voltage VAG: Y AG=IAGiV AG=O. Solving for Y AG=O in terms of fAG, and VAG, the resulting oscillation frequency is fAG=I.2 GHz and the fundamental­frequency amplitude at the collector terminal is Vc,=O.87 V. The objective will be to maximize this oscillation amplitude

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for a sufficiently high-level excitation of the nonlinear behavior of the varactor diodes. With this aim, we sweep the AG amplitude from the original value V AGo optimizing C] and

the feedback line, in order to fulfill Y AG = 0 at the desired

oscillation frequency fAG=I.2 GHz, which is kept constant during the entire procedure. The effect of this sweep can be seen in Fig. 5a, which shows the initial waveform and the waveform obtained after the AG amplitude sweep, with final value Vc=1.73 V (at the fundamental frequency). We note the reduction of the pulse duty cycle, associated with its amplitude increase. This duty cycle is now similar to the one obtained when simulating the NLTL only. The measured waveform is presented in Fig. 5b. For comparison, we have also applied an identical design procedure with the gradual transmission line. The results are shown in Fig 6. As can be seen, the amplitude obtained with the gradual line is slightly higher, with a similar duty cycle. Hereafter we will consider the fixed NL TL only.

N I" )( � . t:: � 0.3

)(

� r-------+------j ., .£-0.5 Cl ., E )( -·1.3 )(

·0.022 -0.14 -0.06 0.02 Real part (GHz)

(a)

0.10 ::J 0 5 10 0..

15 20 25 30 35 40 Time (ns)

(b) Fig. 5 (a) Increase of the pulsed waveform amplitude (with the associated reduction of the T FHWM) by using an auxiliary generator at the design stage. (b) Measured waveform.

> 3�---��-�-----�-------,

� 2. ::::l � 2

� 1. r. E !i .2 � 0.5 ro � "0 �-0.5

t I. '1 I. "

" II

Time (ns)

l l iI

, , , .' II .' I

Fig. 6 Comparison between the pulsed oscillator waveforms obtained

with the fixed and gradual NL TL.

III. STABILITY AND PHASE-NOISE ANALYSIS

As discussed in [1-2], the soliton oscillation has a strong tendency to in stabilization due to the soliton nonlinear dynamics. To cope with this problem the authors in [1-2] propose the use of level-dependent amplification such that small perturbations are rejected in steady state. The aim is to avoid variations in pulse amplitude and repetition rate that would lead to undesired collisions and chaotic behavior.

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In the design presented here no adaptive biasing is used. In spite of this, the steady state oscillation is stable as has been verified with the pole-zero identification technique [8] and with transient analysis. In the pole-zero locus, all the poles are located on the left hand side of the complex plane, except the pair of poles at the oscillation frequency, associated with the solution autonomy (Fig. 7a). The transient in Fig. 7b shows the clear build-up and establishment of the soliton oscillation. The reason why the oscillator is stable (even though no adaptive biasing is used) comes from the fact that at the transistor output terminal the waveform is not pulsed yet. Unlike the description in [2], it is not a high pulse coexisting with other low amplitude pulses, susceptible to give rise to instability through amplification.

N I11 X � . 1:: �O.3

x

� r-------t---� '" .£·0.5 OJ '" E X -·1.3 x

-0.022 -0.14 -0.06 0.02 0.10 Real part (GHz)

(a) Time (ns)

(b) Fig. 7 Stable behavior of the soliton oscillator. (a) Pole-zero locus. (b) Transient simulation.

One advantage of the low number of NLTL cells is the plausible phase-noise reduction due to the smaller number of varactor diodes and thus, of noise contributions. In Fig. 8 we compare the phase noise of the 8-cell oscillator with the one obtained in the previous work [10], with a 16-cell NL TL in feedback configuration. The phase noise is analyzed with the carrier modulation and conversion matrix approaches in HB. The noise contributions considered are the flicker, shot and thermal noise from the transistor and the varactor diodes. Measurements are superimposed in both cases with very good agreement. With the new oscillator design, there is significant phase noise reduction, which is also due to the higher quality factor of the oscillator resonance. We have also analyzed the phase noise spectrum obtained when the 8 cell NL TL is ended in a high output resistance, in similar manner to [2]. In this case, the waveform contains an undesired bump and higher phase noise is obtained, compared with the optimized NL TL.

IV. CONCLUSIONS

We have presented a pulsed-oscillator design using an NL TL with a small number of inductance-varactor cells. The design procedure enables a more flexible design of the NL TL, which can be optimized prior to its connection to the oscillator circuit. The oscillation start-up conditions are imposed at the transistor input port, whereas the NL TL is connected at the transistor output. The low value of the characteristic

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impedance allows extracting the output waveform with a 50 Ohm load, instead of a high impedance value. In contrast with previous works, the transistor does not need an adaptive bias, which simplifies the overall design procedure. The technique has been applied to the design of a pulsed waveform oscillator at 1.2 GHz with very good results.

-2 N I -4 U CD -6 2-

-8

_. _. Free-running oscillator (NL TL-openl - Free-running oscillator (NL TL-short � Measurements (NL TL-short) ------ Free-running oscillator 11 0] � Measurements [10]

Fig. 8 Comparison between the phase noise spectra obtained with conversion matrix approach for the 8-cell oscillator for two different terminations, open and short. The phase noise spectrum obtained in [10] is also represented. Measurements have been superimposed.

ACKNOWLEDGMENT

This work was supported by project TEC2008-06874-C03-01.

REFERENCES

[1] D.S. Ricketts, X. Li, N. Sun, K. Woo, D. Ham, "On the Self­Generation of Electrical Soliton Pulses", IEEE J. Solid-State

Circuits, vo1.42, no.8 , pp. 1657-1668 , August 2007. [2] 0.0. Yildirim, D.S. Ricketts, D. Ham, "Reflection Soliton

Oscillator", IEEE Trans. Microwave Theory and Tech., vol. 57, no. 10, pp. 2344-2353, Oct..2009.

[3] M. Remoissenet, Waves Called Solitons: Concepts and Experiments. New York: Springer, 1999.

[4] R. Hirota, K. Suzuki, "Theoretical and experimental studies of lattice solitons in nonlinear lumped networks", Proc. IEEE, vol. 61, no. 10, pp. 1483-1491, Oct. 1973.

[5] M. G. Case, "Nonlinear Transmission Lines for Picosecond Pulse, Impulse and Millimeter-Wave Harmonic Generation", PhD Thesis University of California, Santa Barbara, July 1993.

[6] M. Ponton, F. Ramirez, A. Suarez, J.P. Pascual, "Analysis and design of soliton oscillators using harmonic balance", IEEE­

MTT-S Int. Microw. Symp. 2009. [7] E. Afshari, A. Hajimiri, "Nonlinear Transmission Lines for

Pulse Shaping in Silicon", IEEE J. Solid-State Circuits, vo1.40, no.3 , pp. 744-752 , March 2005.

[8] Jugo, 1. Portilla, A. Anakabe, A. Suarez, J. M. Collantes, "Closed-loop stability analysis of microwave amplifiers," Electronics Letters, vol. 37, no. 4, pp. 226-228, Mar, 2001.

[9] A. Suarez, Analysis and Design of Autonomous Microwave

Circuits, Hoboken, New Jersey: Wiley - IEEE Press, 2009. [10] M. Ponton, F. Ramirez, A. Suarez, J.P. Pascual, "Applications

of Pulsed-Waveform Oscillators in Different Operation Regimes", IEEE Trans. Microw. Theory and Tech. vo1.57, no.12, pp. 3362-3372. Dec. 2009.

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