Non-linear transmission limes (NLTL) are used for pulse shaping. We developed the theory of pulse propagation through the NLTL. The problem of a wide pulse degenerating into multiple pulses rather than a single pulse is solved by using a novel gradually scaled NLTL. We exploit certain favorable properties of accumulation mode MOS varactors to design an NLTL that can sharpen both rising and falling edges, simultaneously. There is a good agreement among the theory,

simulations, and measurements

Afshari, Ehsan

Hajimiri, Ali

English

Caltech Authors - Main

Non-Linear Transmission Lines for Pulse Shaping in Silicon 
Ehsan Afshari and Ali Hajimii 
California Institute of Technology (Caltech) 
Pasadena, CA, USA 
Caltech MC 136-93,1200 E. California blvd., Pasadena, CA 91 125 
Abstract 
Non-linear transmission limes (NLTL) are used for pulse shaping. We 
developed the theory of pulse propagation through the NLTL. The 
problem of a wide pulse degenerating into multiple pulses rather than 
a single pulse is solved by using a novel gradually scaled NLTL. We 
exploit certain favorable properties of accumulation mode MOS 
vaiactors to design an NLTL that can sharpen both rising and Falling 
edges, simultaneously. There is a good agreement among the theory, 
simulations, and measurements. 
Introduction 
The wncept of a solitary wave was introduced to science by John 
Swtt Russell more than a centwy ago [I]. In 1834 he observed a 
wave formed when a boat which was rapidly drawn along a narrow 
channel by a pair of horses suddenly stopped. According to his diary, 
this wave wntinued, “at great velocity, assuming the form of a large 
solitary elevation, a well-defined heap of water that wntinued its 
course along the channel apparently without change of form or 
diminution of speed”. These solitary waves, now called ‘solifom’, 
have become important items of research in diverse fields of physics 
and engineering. There is a wnsiderable body of work on solitons in 
applied mathematics (e.g., [ZJ), applied physics, especially in optics 
(e.g. [3J and [4]), and few works in electronics [ 5 ] .  The ability of 
solitons to propagate with small dispersion can be used as an 
effective means to transmit data modulated as short pulses over long 
distances. 
An imponant related application is pulse sharpening for the more 
traditional non-rem-to-zero (NRZ) data transmission in digital 
circuits by improving the rise and fall times of pulses. Improving the 
bansitions by shrinking the rise and fall times of pulses can be useful 
in other applications, such as high-speed sampling and timing 
systems. Non-hear transmission lines (NLTLs) sharpening either of 
the rising or the Falling edge of a pulse have been demonstrated on a 
GaAs technology 1-51, However, to the best of our knowledge, there 
has been no demonstration of simultaneous reduction of both rise and 
fall times in an NLTL to this date. Neither are we aware of any 
demonstration of such NLTLs in silicon-based CMOS process 
technologies. 
In this work, we first show a soliton line on a wnventional siliwn 
technology which can achieve very narrow pulses, with a bandwidth 
in excess of the cut-off frequency, fT, of the fastest transistor in this 
process. Next, we demonstrate that using a favorable characteristic of 
MOS varactors leading to a different kind of non-linearity, we can 
improve both the rise and fall times, simultaneously. This is not 
possible with the nonlinear elements commonly used in NLTLs (e.g., 
reverse bias PN junctions). Neither can it be done using transistors, 
as they are limited by their fT. 
In this paper, the propagation of soliton waves in a non-linear 
transmission line will be studied. Then, we introduce two different 
types of non-linear transmission lines to generate narrow pulses and 
to sharpen pulse bansitions, respectively. Finally, we show the 
experimental results verifying the agreement between the theory and 
the measurement. 
The Theory of Non-linear Transmission Line 
In this section, we review the basic theory behind non-linear 
transmission lines and their use for pulse narrowing and edge 
sharpening in subsections A and B, respectively. 
Fig. 1 shows an example of a non-linear transmission line using 
inductors, I ,  and voltage dependent (and hence non-linear) 
capacitors, C(V. 
v i  T I  1 ., I 
= 
Figure I ,  A non-linear transmission line 
By applying KCL at node n, whose voltage with respect to ground is 
V,,, and applying KVL across the two inductors connected to this 
node, one can easily show the voltages of adjacent nodes on this 
NLTL are related via [ 5 ] :  
The right-hand side of (I) can be approximated with partial 
derivatives with respect to distance, x, eom the beginning of the line, 
assuming that the spacing between two adjacent sections is 6 (i.e., 
x.=n6.) An approximate continuous partial differential equation can 
be obtained by using the Taylor expansions of V(x-6). I+), and 
V(x+s) to evaluate the right hand side of (1). Assuming a small 6, 
and ignoring the high order terms, we obtain: - . .  
(2) a av aLv s’ a4v L - [C( V) -1 = - + - - 
at at &2 12 & 4  
where C and L are the capacitance and inductance per unit length, 
respectively. It is noteworthy that for a continuous transmission line 
(8 + O), (2) reduces to: 
(3) 
A.  Pulse Narrowing Non-Linear Transmission Lines 
In this subsection, we approximate the capacitor’s voltage 
dependence using the following first-order linear relationship 
C ( Y ) = C , ( I - b Y )  (4) 
where CO and b are constants. In this case, (2) reduces to 
a2v  I a 2 v  s2 I a4v b a 2 ( v 2 )  = _-- +-- ( 5 )  
a,2 Le,, ,2 12 Le, &4 2 at2 
where the left-hand side is the classic wave equation and the terms 
on the right-hand side represent dispersion and non-linearity, 
respectively. 
If the effect of the dispersive and non-linear terms in ( 5 )  are on the 
same order of magnitude it is wssible to have a single pulse solution 
~ . .  
for ( 5 )  with a profile that does not change as it propagates with 
velocity, Y. A propagating mode solution can be obtained by 
6-2-1 converting the partial differential equation @‘DE) of ( 5 )  to an 
91 0-7803-7842-3/03/$17.00 0 2003 IEEE IEEE 2003 CUSTOM INTEGRATED CIRCUITS CONFERENCE 
ordinary differential equation (ODE) by a simple change of variable: 
where v is the propagation velocity of the pulse and vo = I m. 
This solution is shown in Fig. 2 for three different values ofL and C, 
and hence different 6. Note that this solution is not a function of the 
input waveform, and thus any arbitrary input will eventually tums 
into (6) going through a line which is long enough, if it has enough 
enera.  
L=lnH and 0 InF (b) L=ZnH and C=ZnF (c) L4nH and M n F  
As can be seen from (6). the peak amplitude is a f ic t ion  of the 
velocity. Defining an effective capacitance, C ,  so that 
= I l a ,  the pulse height is given by: 
2 2  
v,, 
b v z  b CO 
(7) 
Using (4), we can relate Cefto an effective voltage Vep It is easy to 
show that 
So it is the capacitance at one-third the peak amplitude that 
determines the effective propagation velocity. Using (6)-(8) we can 
easily calculate the half-height width of the pulse to be: 
WE- 6 "0 (9) qpq 
As can be seen, in a weakly dispersive and non-linear transmission 
line, the non-linearity can counteract the normally present dispersive 
properties of the line maintaining solitary waves that propagate 
without dispersion. This behavior can be explained using the 
following intuitive argument: The instantaneous propagation velocity 
at any given point in time and space is given by 1 I m. In the 
presence of a non-linear capacitor with a characteristic given by (4), 
the instantaneous capacitance is smaller for higher voltages. 
Therefore, the p in ts  closer to the crest of the voltage waveform 
experience a faster propagation velocity and produce a shock-wave 
front due to the nonlinearity, as shown symbolically in the upper part 
of Fig. 3. Note that this is not a real waveform and more a fictitious 
representation of how each p i n t  on the curve tends to evolve. On the 
other hand, dispersion of the l i e  causes the waveform to spread out. 
as shown in the lower half of Fig. 3. For a proper non-linearity 
determined by (9, these two effects can cancel each other out. 
Non linearity n Dispersion 
Input pulse 
Figure 3. Dispenion and non-linear effects in he NLTZ 
A few important observations in this line are: 1) the velocity of the 
solimy wave increases with its amplitude, 2) pulse width decreases 
with increasing pulse velocity, 3) the width shrinks for higher 
amplitudes, 4) the sign of solution depends on s i 5  of non-linearity 
factor, b; For a capacitor with a positive voltage dependence (e.g., an 
nMOS varactor in accumulation mode) we have 
resulting in upside down pulses. 
Based on these results, to achieve large-amplitude narrow pulses, 
inductance and capacitance of the NLTL must be as small as possible 
and non-linearity factor, b, should be large enough to compensate the 
dispersion ofthe line. 
It is also important to know the characteristic impedance of these 
l i e s  (for impedance matching, etc.) As in a NLTL the capacitance is 
a function of voltage, we can only define an effective semi-empirical 
value for the characteristic impedance. Simulations results indicate 
that one can approximate 2,using the capacitance at Vefdefined in 
C(v) = C o ( l + b V )  (IO) 
(E), i.e., 
= 
B. Edge Sharpening Lines 
It is possible to design NLTLs to sharpen the pulse transitions. This 
is particularly useful for digital transmission (e.g., NRZ data). 
Unfortunately, all the efforts in the past [6] have resulted in 
sharpening of only one of the rising and falling edges. This, however, 
has very little practical value, as both hansitions are equally 
important in common NRZ digital systems. This problem can be 
traced back to the monotonic dependence of the non-linear capacitive 
elements used in NLTL on the voltage (e.g., reverse biased PN 
junction, or the ideal behavior of (4) and ( I  0)). 
Fortunately, CMOS processes offer different characteristics for non- 
linear capacitors that can be exploited to achieve simultaneous edge 
sharpening for both rising and falling edges. More specifically, 
accumulation mode MOS varactors [7] (an nMOS capacitor in an n- 
well) offer non-monotonic voltage dependence. Particularly, the 
secondary reduction of capacitance shown in Fig. 4 due to poly- 
silicon depletion [S, 91 and short-channel charge quantization [9] 
effects can be used for edge sharpening. 
' V  v, v, v, 
Figure 4. Capacitance versus voltage for a MOSVAR 
Fig. 5 shows symbolically how one can use the behavior of Fig. 4 to 
sharpen both edges. First, let us focus on the risptime reduction. 
92 6-2-2 
Consider the rising edge shown in the upper part of Fig. 5. Initially 
the voltage is low, which corresponds to a smaller capacitance per 
Fig. 4, and hence a faster instantaneous propagation velocity for the 
lower end of the pulse. As the voltage goes up, the capacitance 
increases, resulting in an increase in the instantaneous propagation 
velocity. This pushes the lower end of the transition forward in time 
and results in sharpening of the rising edge. This effect is 
symbolically shown in the fictitious middle waveform of Fig. 5 .  The 
fall time reduction can be explained using the lower part of Fig. 5 .  
This is where the non-monotonic behavior of Fig. 4 plays its mle. 
The upper part of the transition (voltages above V,) will be 
accelerated due to the reduction of the capacitance and will create an 
advancing front as symbolically shown in the middle waveform of 
Fig. 5 .  The lower capacitance at the very low voltages can generate a 
leading !ail, which will be partially dissipated by the line'. 
Figure 5 .  How rise and fall time vary within the NLTI. 
while the above explanation based on a simplified memory-less 
description of the line provides a basic intuition for its operation, a 
complete description can only be obtained by solving the differential 
equation in (2) to account for the memory of the system. Our 
numerically solution of (2) also confirms that as long as the input 
voltage range exceeds voltages, V, and V,, for a range of L's and Cs, 
the line sharpens both rising and falling edges, simultaneously. 
Gradually Scaled NLTL 
One problem in pulse "owing NLTLs is that if the input pulse is 
wider than a certain minimum related to the natural pulse width of 
the line in (9), it is incapable of concentrating all that energy into one 
pulse and instead the input pulse degenerates into multiple soliton 
pulses, as shown in the simulated upper waveforms of Fig.7. This is 
an undesirable effect that cannot be avoided in a standard line. 
We have found a solution for this problem by using gradually scaled 
non-linear transmission lines. We notice that the characteristic pulse 
width of the line is controlled by the node spacing 6, and the 
propagation velocity, Y ,  which is in hum contmlled by, L and C. 
Thus, we use a gradual l i e  consisting of several sections that are 
gradually scaled to have smaller characteristic pulse width, as shown 
in Fig. 6. 
The first few sections have the widest characteristic pulse, meaning 
that their output is wider and has smaller amplitude. As a result, the 
input pulse will cause just one pulse at the output of these sections. 
The following sections have a narmwer response and the last section 
has the narmwest one. This will guarantee the gradual narrowing of 
the pulses and avoids degeneration. Each section has to be long 
enough so that the pulse can reach the section's steady-state response 
' We hypothesize that other dynamic effects in the MOS varactor help edge 
sharpening, e.g., the processes of charge being amacted from the n+ 
diffusions to the channel and repelling them are not exact inverses of each 
other over shon time intervals. Some of the repelled accumulation charges 
will be absorbed inside the well. This changes the response time of the 
capacitor and keeps it higher for a longer period of time for the falling 
e d m  
before entering next section. 
Figured Schematic ofthe gradually scaled non-linear transmission 1;"; 
The waveforms of this gradually scaled NLTL are shown in lower 
prut of Fig. 7, demonstrating the effectiveness of this technique. It is 
notewotihy that this gradually scaling technique is also applicable to 
the edge sharpening lines and does improve their performance, tm. 
I 
Figure 7 Output waveform of the normal and d u a l  d i t o n  line 
Simulation 
A Pulse Narrowing Lines 
We have designed a pulse narrowing NLTL using MOS varactors 
and metal mino-ship transmission lines in a 0.181" BiCMOS 
pmcess. We simulated the transmission lines in Sonnet and the 
complete NLTL in ADS. The output waveform of the line to a 65ps 
input pulse is shown in Fig. 8. This silicon-based NLTL produces 
pulses as narrow as 2 . 5 ~ ~  (half ampliNde width) with a 0.8V 
amplitude. Active devices in this process are incapable of producing 
pulses nearly as narrow as these. 
V 1 
O O " . ' . ' ' , .  $ 8 ,  1 ' 1 ' 1 '  
%a 370 SW 
t". Paec 
Figure 8 Simulated output waveform afthe pulse m w n g  line 
B. Edge ShDrpening Lines 
We have also designed an edge sharpening non-linear transmission 
line using MOS varactors and gradual scaling of the lines in the same 
process. Fig. 9 shows the simulated input and the output waveforms 
of this line. 
6-2-3 93 
- 
- 9: 
3 -  
Fig& IO. Output ofpulse mwing'line 
' 
The rise time of two cascaded systems can be estimated using [IO]: 
:rise = 7'. trrse~ +Irise2 (12) 
Using this equation and the simulated rise and fall times, we should 
expect 9.lps and 22ps at the output of a fust-order system with a 
bandwidth of 40GHz. Measurement waveforms of Fig. 1 I show 
measured output rise and fall times of I Ips and 25ps. respectively. 
These numbers art in agreement with the simulated results 
considering the additional bandwidth degradation due to the pads and 
the multi-pole nature of the system. The rise and fall times do not 
change with the input amplitude which verifies the non-linear 
behavior of the line. 
*", 
:Ir-Wk, - 
Conclusion 
We have analyzed pulse narmwing and edge sharpening passive 
non-linear tm"ission lines, using MOS varactors and the novel 
gradual scaling lines, showing simultaneous edge sharpening for 
both rising and falling edges in a silicon subsbate. The experimental 
results show considerable improvement in the rise and fall times of 
b e  (66PSddiV) 
Figure I I .  hput and output waveforms of the edge m i n g  line 
Acknowledgements 
Authors would like to thank helpful discussions with D. Ham, H. 
Wu, A. Komijani, C. White, J. Buckalter, M. Taghivand, H. 
Hashemi, S. Kee, B. Analui, and A. Natarajan of Caltech and Pmf. 
M. Homwitz of Stanford University. lhey also acknowledge the 
generous suppon of Lee Center, IBM Corp., Agilent Tech., and NSF. 
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94 6-2-4 


Non-linear transmission lines for pulse shaping in silicon

J o h n , Solitom, Cambridge Univnsity PES,

Nonlineor W w , Solitom and Chm,

Nonlineor Ww, Solitom and Chm,

Repon on Waves,&quot; Repon of the founeenth

Repon on Waves,&quot; Repon of the founeenth reetin8 ofthe Britirh Associationfor the A h c e m e n t ofSeience,

Solitom, Cambridge Univnsity PES,

Tailor, 0pt;colSolitom - ThemyondErperiment,

The Der@ of CMOS Rado-Frquency Integrated Cimiu,

Validity of Mobility Universality for Scaled Metal-Oxide-Semiconductor Field-Effm Transistors Down to 100 nm Gate Length&quot;

W m died Solitom: Compts ond E q w h &quot; , Spn'nger-Veda&

Wm died Solitom: Compts ond Eqwh&quot;, Spn'nger-Veda&

Wow Hwmnic Genermion, Ph.D. disertation. University ofcalifomis

http://authors.library.caltech.edu/10814/1/AFScicc03.pdf

Non-linear transmission lines for pulse shaping in silicon

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