Comparison of Distributed and Lumped Element
Models for Analysis of Filtering Properties of
Nonlinear Transmission Lines
F. Martı́n, X. Oriols, J. Garcı́a-Garcı́a
Departament d’Enginyeria Electrònica, Universitat Autònoma de Barcelona, 08193 Barcelona, Spain
Received 3 December 2001; accepted 24 April 2002
ABSTRACT: The filtering properties of periodic loaded lossy transmission lines are studied
from the point of view of microwave network theory. Under the condition that the per-section
capacitance of the line is small compared to that of the loading capacitors, it is shown that the
distributed circuit can be described accurately by means of a lumped element ladder network.
The effects of transmission line losses on this approximation are also analyzed. © 2002 Wiley
Periodicals, Inc. Int J RF and Microwave CAE 12: 503–507, 2002. Published online in Wiley InterScience
(www.interscience.wiley.com). DOI 10.1002/mmce.10050
Keywords: nonlinear transmission lines; frequency multipliers; phase shifters; Bragg frequency
I. INTRODUCTION
frequency products can be eliminated if the Bragg
frequency is selected properly. As phase shifters,
NLTLs are designed such that the cutoff frequency is
separated from that of the source signals to avoid
dispersion effects. Within this framework of applications, the analysis of the transmission properties of
NLTLs is of great interest and to this end two main
approaches can be used: the lumped element model
and the distributed model (Fig. 1). The latter is expected to give a more accurate description of the
behavior, but it uses a complex formulation that cannot be applied easily to study nonlinear effects. In this
work a detailed analysis of lossy NLTLs, based on
microwave network theory, is performed. The main aim
is to determine the conditions under which the circuit
can be treated as a lumped element network. The results
of the analysis are supported by computer-based calculation of the small signal response of a 10-section NLTL.
Transmission lines periodically loaded with voltage
dependent capacitances, also known as nonlinear
transmission lines (NLTLs), are of interest because of
their applications in several fields. For example, in the
linear regime, NLTLs can be used as phase shifters in
phased antenna arrays [1], where time delay can be
controlled by means of a DC bias applied to Schottky
diodes acting as variable reactances. Under large signal conditions, NLTLs can serve as impulse compressors [2] or frequency multipliers (e.g., from microwave to terahertz frequencies [3]). The main
advantage of NLTLs in all of these applications is
their broad-band nature, which arises from the small
sensitivity of the characteristic impedance to the frequency. However, signal propagation above the cutoff
(or Bragg) frequency (except at certain passbands) is
inhibited because of the periodicity of the structure
[4]. From the viewpoint of multiplier design, this low
pass filter response is of interest because undesired
II. MICROWAVE NETWORK THEORY
OF NLTLs
Correspondence to: Dr. F. Martı́n; e-mail: Ferran.Martin@
uab.es.
Contract grant sponsor: DGES; contract grant number:
BFM2001-2001.
Following the analysis of Wang and Hwu [4], signal
propagation in a distributed, lossless NLTL is possible for frequencies satisfying
© 2002 Wiley Periodicals, Inc.
503
504
Martı́n, Oriols, and Garcı́a-Garcı́a
Figure 1. The elementary section of the NLTL according to the (a) distributed and (b) lumped
element model. CD(V) is the capacitance of the nonlinear devices. In (b) each transmission line
section between nonlinear reactances is modeled by a series inductance (L) and shunt capacitance
(C0). The variables R and G account for the transmission line losses.
␻ C lsZ0 ⬍ 2 cot ␪,
(1)
where Cls is the linearized capacitance of the loading
devices for small signal operation or the large signal
capacitance [defined as the average value of CD(V)
over the voltage excursion of the feeding signal] in the
nonlinear regime; Z0 ⫽ (L/C0)1/2 is the characteristic
impedance of the unloaded line; and ␪ ⫽ [␻(LC0)1/2]/2
is half the electrical length between nonlinear reactances, where L and C0 are the per-section inductance
and capacitance, respectively. Expression (1) determines the passbands of the structure and implicitly
gives the Bragg frequency as the cutoff point of the
first passband.
In the lumped element model the Bragg frequency
is given by
␻ Ble ⫽
2
冑L共C0 ⫹ Cls兲
.
(2)
Let us now analyze the influence of Cls on the
Bragg frequency for the two models. In the distributed
model, for large values of Cls, the electrical length
between the nodes becomes very short and the Bragg
frequency can be approximated by
␻ Bd ⫽
2
冑LCls
,
(3)
which coincides with the Bragg frequency given by
the lumped element model under the condition that Cls
Ⰷ C0. For small values of Cls (Cls Ⰶ C0), both
members of eq. (1) are very small at the Bragg frequency, and the following approximation can be applied:
␻ Bd ⫽
␲
冑LC0
,
(4)
where the condition that ␪(␻B) ⬇ ␲/2 has been implicitly assumed. In this case, the Bragg frequency
differs from that given by the lumped element model
by a factor of ␲/2. In Figure 2 the variation of the
cutoff frequency with the Cls shows that for values
significantly larger than C0, the lumped element and
distributed models give similar results. To further
examine the validity of the lumped element model, we
obtained the T-network equivalent circuit of the elementary section of the NLTL, including the effects of
transmission line losses. To this end, we first obtained
the transmission matrix (A) and hence the impedance
matrix using standard transforms. The series Zs and
shunt Zp impedances of the T-network are then given
by
Zs ⫽
2 sinh2 ␪ ⫹ j␻ClsZ0 sinh ␪ cosh ␪
,
sinh 2␪/Z0 ⫹ j␻Clscosh2 ␪
(5a)
Analysis of Filtering Properties of NLTLs
505
Figure 2. A comparison of the Bragg frequency obtained by means of the two models as a
function of Cls; L ⫽ 0.108 nH and C0 ⫽ 43.5 fF.
Zp ⫽
1
.
sinh 2␪/Z0 ⫹ j␻Clscosh2 ␪
(5b)
Due to losses, ␪ and Z0 are now given by (1/2)[(R ⫹
j␻L)(G ⫹ j␻C0)]1/2 and [(R ⫹ j␻L)/(G ⫹ j␻C0)]1/2,
respectively. If Cls is larger than C0 (the per-section
capacitance of the line), then for the frequencies of
interest (i.e., below the Bragg frequency) ␪ is small
and (5) can be approximated by
Zs ⬇
Zp ⬇
␪ 共2 ␪ ⫹ j ␻ C lsZ0 兲 R
L
⬇ ⫹ j␻ ,
2␪
2
2
⫹ j␻Cls
Z0
1
2␪
⫹ j ␻ C ls
Z0
⬇
1
.
G ⫹ j␻共C0 ⫹ Cls兲
(6a)
(6b)
According to these results, under the condition Cls Ⰷ
C0, the distributed model can be approximated by the
lumped element network. We also assumed that transmission line losses are small enough such that R␻C Ⰶ
1 and G␻L Ⰶ 1, otherwise the Taylor series approximation used to obtain eq. (6) is not possible. These
conditions lead us to R Ⰶ Z0l(Cls/C0)1/2 and G Ⰶ
Z0l⫺1(Cls/C0)1/2, where Z0l is the impedance of the
lossless unloaded line [Z0l ⫽ (L/C0)1/2]. The latter
inequalities are easily satisfied if Cls is larger than C0,
even if the losses are not negligible. (In actual structures the skin effect losses can lead to high values of
R at high frequencies.) It is therefore clear that the
lumped element network is a good approximation to
the behavior of periodic loaded lossy transmission
lines under the assumption that Cls/C0 Ⰷ 1. If this
condition is not satisfied, the lumped element network
is only valid at moderate and low frequencies (compared to the Bragg frequency), and the differences
between the lumped element and distributed models
are sensitive to transmission line losses.
III. SIMULATION RESULTS
To support the previous comments, we obtained the
small signal response of a 10-section transmission line
(lumped element and distributed model) by means of
computer simulation. For this we developed our own
code, because it allows us to express the electrical
parameters of the distributed model (Z0 and ␪) in
terms of the parameters of the lumped model (L, C0,
R, and G) and frequency. In this way a realistic
comparison between both models can be done. To
obtain the small signal response, the transmission (A)
matrix is first calculated from the contribution of each
individual section, and then the scattering parameter
S21 is inferred from standard transformation. Two
situations were considered: Cls Ⰷ C0 [Fig. 3(a)] and
Cls comparable to C0 [Fig. 3(b)]. In order to analyze
the effect of losses, different values of R were considered. (G is assumed to be negligible in both cases.)
As can be seen in Figure 3(a), the distributed and
lumped element models give similar results up to the
Bragg frequency (⬇12 GHz); apart from a degradation of the transmission characteristic, losses do not
affect this agreement. In Figure 3(b) it can be seen
506
Martı́n, Oriols, and Garcı́a-Garcı́a
ther limit the validity of the lumped element network
to analyze NLTL performance.
IV. CONCLUSION
We showed that there is good agreement between the
lumped element and distributed models of NLTLs
when the values of the nonlinear loading capacitors
are greater than the per-section capacitance of the
line. When Cls ⬇ C0, good agreement only occurs
well below the Bragg frequency and depends on the
losses. In practice, it is desirable for Cls to be the
dominant capacitance, and the losses are low. Therefore, the lumped element analysis can be used to
design NLTL delay lines and frequency multipliers.
ACKNOWLEDGMENT
The authors are grateful to DGES for the support of this
work. Thanks also to Dr. R.E. Miles (University of Leeds)
for his suggestions.
REFERENCES
Figure 3. A comparison of the linear response (|S21|2) of
a 10-section NLTL obtained by means of the (—) distributed and (- - -) lumped element models. (a) L ⫽ 1.25 nH,
G ⫽ 0 S, Cls ⫽ 0.5 pF, and C0 ⫽ 0.043 pF; and (b) C0 ⫽
0.5 pF.
that the lumped element model underestimates the
cutoff frequency and provides a reasonable approximation to the distributed model response at low frequencies only. As R increases, the divergence of responses between the two models begins at lower
frequencies. Therefore, transmission line losses fur-
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Domier, and N.C. Luhman, Jr., Novel low-loss delay line
for broadband phased antenna array applications, IEEE
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Impulse compression using soliton effects in a monolithic GaAs circuit, Appl Phys Lett 58 (1991), 173–175.
3. E. Lheurette, M. Fernández, X. Melique, P. Mounaix, O.
Vanbésien, and D. Lippens, Non linear transmission line
quintupler loaded by heterostructure barrier varactors,
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Analysis of Filtering Properties of NLTLs
507
BIOGRAPHIES
Ferran Martı́n was born in Barakaldo (Vizcaya), Spain in 1965. He received the B.S.
degree in physics from the Universitat Autònoma de Barcelona in 1988 and the Ph.D.
degree in 1992. Since 1994 he has been an
Associate Professor of Electronics in the Departament d’Enginyeria Electrònica (Universitat Autònoma de Barcelona). His current
research interests include modeling and simulation of electron devices for high frequency applications, millimeter wave and terahertz generation systems, and the application of
electromagnetic bandgaps to microwave and millimeter wave circuits.
Joan J. Garcia was born in Barcelona in
1971. He received the B.S. degree in physics
in 1994 and the Ph.D. degree in electrical
engineering in 2001, both from the Universitat Autònoma de Barcelona. Afterward he
became a Postdoctoral Research Fellow at
the Institute of Microwaves and Photonics of
Leeds University in the INTERACT European project. Dr. Garcia is currently working
at the Universitat Autònoma de Barcelona as a Postdoctoral Research Fellow in the Ramon y Cajal project of the Spanish government.
Xavier Oriols received his B.S. and M.S.
degrees in physics and Ph.D. in electronic
engineering from the Autonomous University of Barcelona in 1993, 1994, and 1999,
respectively. During 1997 he worked on the
simulation of dynamic properties of GaAs
resonant tunneling diodes at the Institute of
Electronic and Microelectronic du Nord
(France). In 2001 he was a Visiting Professor
in the Physics Department of the State University of New York,
working on dual-gate nanometric metal oxide semiconductor field
effect transistors. He is currently an Associate Professor in the
Electronic Engineering Department, Autonomous University of
Barcelona. Dr. Oriols’ present interests include the study of noise
and high frequency properties of nanometric devices.