Modelling and simulation of ultrafast pulse propagation in microstrip lines

Math/ Comput. Modelling, Vol.14,pp.378-382, 1990

0895-7177/90 93.00+0.00 Pergamon Pressplc

Printed in GreatBritain

MODELLING MICROSTRIP

AND SIMULATION LINES

OF ULTRAFAST

M.K.Webb and P.O.Kwok O.R.Baiocchi, Department of Electrical/Electronic California State University, Chico CA 95929 USA Chico,

PULSE

PROPAGATION

IN

Engineering

Advances in the electro-optics technology have made possible the Abstract. fraction of a order of a generation of pulses of time-duration of the in conventional transmission of such pulses picosecond or less; propagation Since these lines is severely affected by both dispersion and attenuation. and fields of science application in many their finding pulses are propagation simulate their model and important to engineering, it is In this paper process and to find new ways to overcome these difficulties. pulses we describe our procedure for modelling and simulation of ultrafast and superconductive. conventional in microstrip transmission lines, both the for curve-fitted expressions This procedure combines the approximate, the based on with a" analytical approach characteristics dispersion described by a is In this way, the propagation Taylor-series expansion. physical well known cascade of linear filters, each of them related to a through a The output of the system is then obtained numerically process. with method, as compared The advantage of this Fast Fourier algorithm. is to make it possible for the designer to have entirely numerical methods, a prior estimate of the distortion to be expected in a given situation, and search variables in to be able to manipulate the different consequently, for a better system performance. Pulse

Keywords. lines.

propagation;

INTRODUCTION Ultrafast pulses than a picosecond

of

the

ultrafast

dispersion;

curve-fitted order

of

less

analytical

dispersion transmission

we

for CAD equations of characteristics these system;

expressions

As compared techniques, this physical process, obtain

for

the parameters the

with method

the

conventional provides more

insight into the and enables the estimates of the

Since

Firstly, existing

obtained

linear filter to each of the terms of the expansion, and generate outputs from these filters through a" FFT algorithm.

method introduce a In this paper we and simulation of for the modelling process, combining this propagation techniques. numerical analytical and of

approximations

derivatives of these dispersion equations which make it possible to represent the characteristic equation by a Taylor-series expansion around a frequency of choice, like the carrier frequency. Thirdly, we associate a

and bandwidth increased the from these provided resolution by problem, The signals. ultrashort is that propagation of pulses however, hard to of this order of duration is in efficiently achieve any OXelectronic system, transmission of amount the due to optical, extended distortion introduced by the both in terms of dispersion bandwidth, and attenuation.

advantage

transmission

from the numerical solution of the electromagnetic boundary-value problems. Secondly, we obtain

find application in many fields of science and technology. these with spectroscopy Time-domain in strides new open pulses may Signal-processing materials science. wel I as as systems, radar and benefit will systems, communication

take

pulses:

signal,

process

propagation designer distortion,

is characterized

related to dispersion

the duration coefficients

to

by of of

length. physical its and I ine the coefficients dispersion the Being central the critically dependent on choice of appropriate frequency, an variables can be this and the other minimize to order attempted in distortion.

the

the the are

378

Proc. 7th Int. Conf. on Mathematical and Computer Modelling we sections, following In the transmission the describe initially (like used commonly structures most the miCrOStriD and miCrOStrip COPlanar 01 conventional 1 ines, show how and superconductive) be can coefficients dispersion analytically or numerically, obtained, We equations. from the curve-fitted Taylor-series the describe then filters I inear and the approach terms of associated with each of the the Simulations using the expansion. presented are Fast Fourier technique situations. typical set of for a we comment on the results and Finally, and advantages the discuss we1 I disadvantages of our method, as this development of as the possible work.

MODELLING

MICROSTRIP

dispersion characteristics for The only be these transmission lines can numerically, through the obtained of the respective solution However, boundary-value problem. curve-fitted analytical expressions for these characteristics have been presented by many authors [Kobayashi, 19881. Usually, these approximations in form of a appear the frequency-dependent expression for the dielectric constant. For the line of microstrip Figure la, for example, Pramanick and Bartia c 19831 give the expression:

(f) =

cf. Er fr

(> L fi

zo @oh

(2)

this equation, Er is the dielectric constant of the substrate, E(J is the effective dielectric constant for the quasi-static condition, h is the height of the substrate, 20 is the characteristic impedance of the line and u. is the free-space permeability. Similar expression can be found for the coplanar line of Figure lb CWhitaker, et al., 19871. point from

Fig.

la

COPLINAR LINE

of the

view, presence

Fig.

lb

higher order modes associated with the high-frequency components of the signal. The difference between the conventional dielectric constant and the effective one arises from the fact that the structure is not homogeneous (air/substrate interface), the relation between them being a function of the geometry CGupta, et al., 19791. Once the relation between the effective dielectric constant and the frequency is established, the propagation phase function is obtained by (3)

V(z, t) = e +w(o,

In

physical arises

1

the

(1)

where

From the dependence

r-

and, neglecting losses, transmission process is characterized by the transfer function operation

60

2

l+L!l

ft =

LINE

DISPERSION

1 ines common transmission The most millimeter-wave used in microwave and are the transmission systems and the coplanar I ines microstrip and Figures la shown in lb, The same structures are respectively. also used in the sub-millimeter region of the spectrum.

Ereff

MICROSTRIP

319

this of

Figures relative constant) against microstrip

t)

(4)

2a

and 2b show the effective permittivity (dielectric and the phase function the frequency, for a typical line.

To take the losses into consideration an additional term is included in the transfer function, usually in the form of a real exponential function proportional to the root-square of the frequency, with constants again obtained from the CAD approximations. The above expressions are good for the conventional transmission lines, where the metallic strips are made of normal conductors. For superconductive lines the whole problem needs to be reformulated, since dispersion and attenuation introduced the by

380

Proc.

7th Int. Conf. on Mathematical

rrequency-dependent conductivity of the superconductor needs to be considered CHattis and Bardin, 18581; the corresponding effects need to be added to the effects or the norma 1 moda I dispersion and dielectric attenuation. The problem is difficult to analyze and only with the assumption of absolute-zero temperature it is possible to obtain closed-form expressions for both the phase and attenuation functions [Baiocchi, et al., iQ89al.

and Computer inflection characteristic possible to

Modeliing

or

frequency curve. write

It

the then

is

where

(6)

and wO isgathe 8;efe:;;ce ,;requ;;;:. Figures respectively: the first, second third order derivatives of characteristic function (phase) the same microstrip as above.

Fig.

the It is interesting to notice that the term of order zero is related to phase velocity, the second order term is related to the group delay, and the others account for different kinds of dispersion. The second order term, which vanishes at the inflection frequency, is related to the so-called “chirping” of the signal; for the case 0r a Gauss ian pulse. it leads to spreading without change of shape. The third order term is the most important because the equivalent dielectric constant for a conventional microstrip line jumps almost abruptly at the on-set frequency or the non-transverse modes. situation The the for different quite is superconductive line where the complex affected by propagation constant is dependence of the complex conductivity mentioned frequency, as the with before.

2a

A linear filter can the each term of with a transfer-function H k(Y)

Fig.

THE TAYLOR-SERIES

9

and the for

=

associated to be expansion, Taylor given by

-jp,k[LJ -

=P

1

w.gz

k!

1 ine. the length of where L is the combined The outup signal is then the output of the cascade of these filters FFT the obtained by can be and technique.

2b

EXPANSION

for the function With the transfer it is line as above, dispersive FFT the simply using possible by technique to obtain the output for any procedure, This desirable input. very little insight however, gives into the physical process and does not characterize offer any parameters to proposed in the distortion. What is the is to expand this work around a characteristic function carrier the central frequency, as the signal or frequency of the

a applications, most For good pulse is a Gaussian-shaped actual signal, representation of the non-dimensional following and the to defined can be parameters characterize the dispersion:

(8)

where Gaussian Higher although

T

is pulse. order usually

the

half-width

terms not

can be needed,

of

the

added, except

381

Proc. 7th Int. Conf on Mathematical and Computer Modeliing extremely for the baseband signals of the case, In this short duration. spectrum has to be divided in segments to allow the series expansion, and the needs to meaning of these parameters compared evaluated, as be carefully (RF presented here the case with signals).

THE

NUMERICAL

SlMULATlONS

pulse distortion simulation of The using the Taylor expansion approach is being done through the FFT technique, considered the effects of each filter used in The algorithm separate. in this work is a radix-2 FFT written in the IBM-PC and for Turbo Pasta 1 code is also The same compatibles. for available for use with mainframes the results and better accuracy of improved speed. are filters fully the Since the dispersion characterized by the simulations parameters B and D, microstrip the and apply equally to fact, to the coplanar lines and, in system. They other dispersive any the context have been used before in of optical systems CBaiocchi, 19891.

Fig.

3a

input Figure 4a shows an pulse of Gaussian shape, and Figures 4b and 4c the show the distortion of same in the situations where predominant effect is from the term in D and in 8, ment loned As before, respectively. second the the situation is most common for these transmission 1 ines; the the loss of symmetry of the output as well as the ringing of its signal, we1 1 confirmed tail, is by experiments. They are both associated the behavior of the Airy with is function, which the impulse response for the third order filter. the A comparison between performance of microstrip and coplanar lines is The done by Baiocchi, et al. C1989bl. difference obtained by the use of strips is also superconductive discussed elsewhere [Baiocchi, et al., 1989al.

LOG FREQUENCY

t tL Fig.

3b

t

LOG FREQUENCY

Fig.

3c

Fig.

4a

INPUT

PULSE

Proc. 7th Int. Conf. on Mathematical and Computer Modelling

382

CONCLUSIONS A method for modelling and simulation pulses in conventional of ultrafast and superconductive transmission lines been presented. It has the has and advantage of combining numerical analytical procedures, giving the designer of systems that use these pulses a convenient way of estimating dispersion and attenuation in terms of parameters. simple It has the disadvantage of carrier requiring a frequency as reference for these parameteres, and that is always not the case in practical situations. The analysis of baseband signals is, therefore, the most immediate extension of this work. This research work funded through a grant of the Chancellor of State University, California.

was from the Long

partially the Office California Beach,

REFERENCES

Fig. 2ND

ORDER

EFFECT

(D=5.0)

b

-1

Fig. 3RD

4b

ORDER

4c

EFFECT

(B=2.0)

Modelling of Baiocchi, 0. Cl9891. optical dispersion effects in Phenomena, systems. ln -Wave and Theoretical, Computational Aspects , Practical York, Springer-Verlag, New 268-255. Baiccchi, O., P.O. Kwok, M.K. Webb and Simulation K. Khojasteh C19891. in ultrafast pulses of transmission superconductive 1989 Proceedings of the lines. --Congress, European Simulation forthcoming. Baiccchi, O., M.K. Webb and P.O. Kwok Simulation of ultrashort 119891. coplanar propagation in pulse lines. microstrio transmission Summer 1989 Proceedings of the ___ -Conference, Computer SiZlation Simulation Council Inc., 227-231. Bahl and I.J. Gupta, K.C., R. Garg Lines and C19791. In Microstrip Slotlines , Artech House, Dedham, Chapter 7. MA, Kobayashi, M.[l9881. A dispersion forrecent mula satisfying CAD. microstrip requirements in MTT-36 Transactions, IEEE % -t 1246-1250. Cl9581. .I. Bardin Mattis, D.C., and erfect Theory of anomalous skin normal superconductive in and metals. Phys. Rev. u, 412-417. Pramanick, P., and P. Barthia Cl9831. of accurate description An microstrip. disoersion in 89-93. Microwave J. 26, Whitaker, J., T.B . Norrys, G. Mourou Pulse [19871. and T. Hsiang in distortion and shaping E microstrip lines. L, 41-47. Transactions, MTT-35, 7