703 bommga #(w, zowof? se/ oa4, email: mutambd@vax. …applications of integrated circuits and...

Investigation of tapered microstrip transmission

lines for high speed detection circuits

D. Mutambo, D. Mukherjee

Physical Electronics Group, School of Electrical, Electronic and

Information Engineering, South Bank University,

703 BomMgA #(W, ZoWof? SE/ OA4,

Email: mutambd@vax. sbu. ac. uk

Abstract

The analysis of high speed digital interconnects using a generalised time domain scatteringparameters is presented. The scattering parameters can represent any arbitrary N-port passivenetwork under an impedance reference system, and can be obtained from measurements. Atime domain simulation of pulse propagation through a tapered line is performed, the resultsof which are shown and used in the preliminary evaluation of the performance enhancementof a low-noise satellite receiver amplifier. This work forms a general basis for theinvestigation of the exponential tapered line for use in high speed detection circuits.

1 Introduction

The transmission lines and their associated interconnections play an important

role at virtually all aspects of today's communication technology involving

applications of integrated circuits and printed circuit boards. With the design of

fast devices having switching times in the picosecond range, transmitting data

at high megabaud rates has become very commonplace in modern digital

computers and switching networks used for telecommunication. Signal delays

and rise times are more and more limited by interconnection lengths rather than

by device speed and represent a potential obstacle to the ultimate scaling on

VLSI technology. In recent years, modelling of interconnections has become a

major focus of interest in the implementation of digital and microwave circuits.

Shorter rise and fall times as well as higher frequency signals have compelled

most transmission lines to operate within ranges where dispersion is no longer

negligible. Skin effect and losses contribute to signal corruption leading to

waveform attenuation as well as pulse rise and fall time degradation. In wafer-

Transactions on the Built Environment vol 31, © 1997 WIT Press, www.witpress.com, ISSN 1743-3509

82 Simulation and Design of Microsystems and Microstructures

scale integration, these losses can become very significant and may lead to an

RC type time delays.

The implementation of high-density compatible packaging scheme is

essential for the design of high-speed digital systems such as gallium arsenide

integrated circuits. For microwave or digital applications, printed circuit

boards, chip carriers, and modelling of these networks represent the first step

towards implementing reliable design guidelines.

In this study, time domain approach is used to formulate the propagation

equations on a loss-less line with non-linear behaviour at the termination. A

time-domain flow-graph representation of the solution is also derived. In the

loss-less case, the solution reduces to very simple expressions which greatly

increases computational efficiency.

Recent advances in microwave circuit designs and the wide variety of

emerging applications have pushed for more accurate circuit simulation as a

necessary step towards high performance, low cost, and highly miniaturised

systems. The circuits containing non-uniform transitions are able to improve

the performance of common components like couplers, filters and matching

circuits. In such circuits, the conductor widths, the propagation constant, and

the characteristic impedance are not constant along the direction of

propagation. To simulate these circuits, reliable tools with accuracy and speed

suitable for CAD applications must be developed. Unfortunately, very few

practical modelling and simulation techniques for these circuits are available

today. One is, therefore, often forced to resort to costly fully three dimensional

field simulators for this purpose, the exception being the numerical approach

described in [1] or the analytical solution proposed in [2]. The analysis of these

lines have been a subject of interest for several decades and many authors have

contributed significantly to the study of non-uniform lines [2-8].

This work involves the design and modelling of a high speed signal (1-4

GHz) detection circuit, based on loss-less tapered (non-uniform) transmission

lines. Exponential tapered configurations is being investigated using the

coplanar strip-line (CPS) method. The CPS's inherent property of reducing

line-to-line coupling means more compact layouts can be achieved. For

increased functionality and lower cost it is important to minimise the size of

monolithic microwave integrated circuits (MMIC). Exponential transmission

lines (ETL) transform the characteristic impedance gradually from one value to


Simulation and Design of Microsystems and Microstructures 83

another for matching purpose. The detection circuit utilising tapered lines

involves the use of components having low resistance, typically in the range of

3 to 10 ohms. The most common type of semiconductor used for these devices

consists of the ffl-V compounds, i.e. GaAs and related compounds.

Semiconductor laser diodes and PIN photodiodes are useful in many circuit

applications in microwave circuits because of their low cost, very small size,

light weight, high speed, time and space coherence. Laser diodes and high

speed photodiodes are normally coupled to 10 ohm circuits. Such

arrangements, beside limiting the bandwidth of components, make it difficult to

handle short electrical pulses in time scale of pico or femto-second, as required

for high speed optoelectronics [3]. ETL is used to match the input resistance of

such components to 50 ohms, allowing considerable improvement of the

temporal response of semiconductor laser diode and high speed photodiodes, as

compared with conventional coupling. The ETL eliminates the need for a

matching network between the transmission line and the detection circuit. Its

gradual change of characteristic impedance along the propagation direction

makes it possible to connect the circuit with low input resistance directly.

There are several mathematical methods of analysing the non-uniform lines.

An important aspect of previous studies [1] is that, except for limited cases [4-

8], they have dealt almost exclusively with non-uniform lines in frequency

(steady state) domain. In other words, the transient behaviour of non-uniform

lines in the time domain has not been studied extensively. The transient

response of a tapered line is a rather complicated function that depends on the

impedances, the line length, the propagation velocities and the time. These are

the most important parameters that determine the efficiency. To the author's

knowledge, no literature had treated the time domain scattering parameters of

an exponential line analytically. This investigation is partly motivated by the

desire to study the interaction of non-uniform lines with non-linear loads. Here

we limit our intentions to the time-domain characteristics of an exponential

line. For analysis of linear circuits, frequency domain scattering parameters are

used to evaluate the circuit performance. However, when transmission lines are

terminated with non-linear loads, the above technique is no longer an adequate

approach. A more appropriate approach is to characterise the transmission line

by a set of time-domain scattering parameters (S-parameters), so that the

interaction between the transmission line and non-linear loads can be expressed



by the corresponding time-domain convolution integral [5]. Therefore, in this

investigation, the transient characteristics of an exponential tapered line will be

analysed by time domain, and based on the results obtained, the formulation of

the tapered lines will be presented.

2 Time-Domain Formulation

Linear and non-linear subsystems can be modelled using time domain

measurement data and the entire system consisting of the individual subsystems

can then be very efficiently analysed. The model of the subsystems use linear or

non-linear time-domain transfer functions, not conventional equivalent circuits.

This approach gives computationally efficient non-linear models which can be

identified from time domain measurement. This results in a much reduced

effort and volume of the measured data compared to frequency domain

measurements. In the frequency domain a large number of measurements have

to be carried out to determine all four scattering parameters, in amplitude and

phase, as functions of frequency and at a number of power levels. In the time

domain the response is a real function and usually much fewer power levels

and time points are required to model non-linear systems. Scattering functions

for these systems are modelled by non-linear functions in the time domain as

shown in fig. 1.

82, (t)

Port 1 - S,, (t) " 822(0 Port2

Vlr(t) Si2(t) V%(t)

Figure 1: Time domain scattering functions model of a linear or non-linear two

ports.

The model in fig. 1 is similar to that used in frequency domain analysis but the

equations describing the model are now written and solved in the time-domain.



This is an important difference between frequency domain scattering

parameters which are only functions of frequency and time domain scattering

functions which are functions of instantaneous values of the signal. The

advantage of solving the equations in the time domain is that the scattering

functions can be updated at each time step according to the signal strength.

Using the scattering functions signal flow-graph to represent the

interconnection of the subsystem, the whole system can be simulated in the

time domain explicitly and efficiently without the need to use iteration to solve

non-linear equations.

A non-uniform line can be described as a set of time domain scattering

parameters which relate two reflected waves and two incident waves [6]:

'* a z f , (1)

M * a,(f) , (2)

where t is the time, * denotes a convolution in the time domain, aj(t), b,(t), az

(t), b:(t) are incident and reflected waves for port 1 and 2, respectively, see fig.

2, S-j(t)(iJ = 1,2) are the time domain scattering parameters. The voltages at

ports 1 and 2 are the summation of the incident and reflected waves

(3)

(4)

and the expression for the currents are

(6)


x=0

Figure2: Scattering parameters representation of a non-uniform transmission

line.

The non-uniform line extends over a distance / and is terminated with twouniform reference lines at both ends. These are Z^, and Z^ , as the source

and load end reference impedances, which are equal to the characteristicimpedance of the exponential line at the left and right sides, respectively. Z^,

and Z^2 are used as the reference impedance in determining the scattering

parameters 5,-,-M of the non-uniform line.

To evaluate the S -parameters of the ETL, we consider a loss-less, non-

uniform line having a characteristic impedance as follows,

(7)

where Z , and Z %, [9] are the characteristic impedances at the left (source)

and right (load) of the exponential line, respectively, / is the length of the line

and x is the space variable (exponential line extends from x = 0 to x = /). By

using the values listed in section 2.1 in equation (7), Z(%) = 50£2exp (-0.2618%)

2.1 ETL synthesis

The length of the ETL is given as


(8)

where F is the maximum reflection coefficient, Z^ is the normalised ( with

respect to source impedance) value of the load impedance to be matched, and/I is the guide wavelength at the lowest frequency of interest. The normalised

impedance profile as a function of length / is give as

/ . (9)

The result obtained from the above equation can the be used to obtain the

characteristic impedance of the ETL by substituting it in (7).

The time domain S parameters of the ETL section were computed with the

use of the method [4] at frequencies of up to 2 GHz. Its length is calculated by

substituting these values, Zs = 50 Q , ZL=20Q , maximum reflection coefficient

of 0.1 at 2 GHz and effective dielectric constant of 9.8, in equations (8) and (9).

Giving us, L = 3.5 cm. The significance of this simulated line length is that, it

is the required length which will give ETL lossless properties for the above

conditions.

2.2 Matched Condition

We assume that both the load and source ends are terminated with matchingresistances, i.e., Z, =Z^, and Z =Z ,, where Z, and Z are the source

and load impedances. Under such conditions, a (t) vanishes and V (t) = b (t).

For a step voltage source u.j(t), as is evident from (2), the output voltage 1/2 (t)

is equal to

The output voltage 1/2 W is proportional to the integral of the scattering

parameter $2i(t), which can be found in [8].


Simulation and Design of Microsystems and Micro structures

(a) (b)

Figure 3a and b: Step response at the load end for a well matched condition (a)

and an ideal voltage at the input end, (b). A normalised line interval A(f) = 1

represents the propagation delay for the wave travelling across the exponential

line.

Fig. 3 shows the step responses at the load end. The step response starts with a

maximum first arriving wave and the decreases to a steady state value. The

decrease can be approximated by a decay time constant which is dependent on

the physical line length and the characteristic impedances of the exponential

line at both ends [5]. The magnitude of the first arriving wave is proportional to

the square root of the ratio of the load end reference impedance to the source

end reference impedance. The horizontal scale is normalised with respect to the

propagation delay in which the wave travels across the non-uniform line. For

the values, say Z /Z /y = 4 and 9,the output voltage Vi (t) reaches the steady

state value at t ~ 3. If we invert the impedance ratio, i.e., Z /Z y = 0.25 and

0.11, aside from the normalised impedance factor, the outputs appear to be the

same as those for Z«/2 /Zre/i = 4 and 9, respectively

When Zs = 0, we have an ideal voltage source. The time-domain reflection

and transmission coefficients at the source end are -S(i) and S(i), respectively.

S(t) represents the impulse delta function commencing at t = 0. Therefore the

reflected wave bj (t) will be totally re-reflected back to the non-uniform line

and affect the output voltage at the load end. We assume that the load is

terminated with a matching impedance, i.e., #2 (t) = 0. For such a case, we may

use recursive equations [6] to compute the incident waves aj (t) and #2 (t). The


SY/?7W/(/f/6V7 fW D 9/g/7 6>/ M/C/YASl ^ ZA f/W M/C/ .S7r ra/r ( 9

reflected waves b\ (t) and i>2 (t) can be obtained by means of convolution

integrals in (1) and (2).

Figure 4: Step response of exponential and uniform lines terminated with a

FET

Figure 5: Pulse signal response of exponential and uniform lines loaded with a

FET.

The performance obtained by using the ETL as shown in fig. 4 compared to

uniform stub matching lines are by far better. The figure shows the response of

the exponential and uniform line to the first arriving pulse. The magnitude of

the pulse is 5 V and the pulse width is 1.2. The output of the exponential line

circuit reach the steady state value 4.3 V, while the output of the uniform line

configuration reach 3.6 V. Furthermore, the exponential line generates smaller

trailing ringing signal than the uniform line. Fig. 5 indicates that the falling

time of the pulse responses is smaller than the rise time of the pulse responses.



and the overall size of the unit reduced, achieving the goal for MMIC size. This

improvement is likely because the exponential line provides maximum first

arriving voltage wave to the load. The first arriving wave improves the rise

time of the output voltage when the circuit incurs a logic low to a logic high

transition. The time variation of b](t) due to the incident wave over time, the

non-uniform line, as compared to a uniform line ,appears to have a continuous

interactions with non-linear load which cause the output voltage to reach the

steady state in a relatively shorter period of time. It should be noted that the

settling down time of step response of an ETL depends on the signal line length

and impedance ratio [8]. Furthermore, to get a good matching result the

capacitive time constant at the load should be comparable to the settling down

time of the exponential line. Touchstone software was used to simulate the S

parameter of the amplifier, fig. 7, shows the response of the amplifier.

DBEGMAX]AMP

.DBCMF]AMP

32.60

38.88

28.88

1.788

1.588

1.3881.287 1.388 FREQ-GHZ 1.313

Figure 7 Touchstone simulation results for the amplifier (1.287 - 1.313 GHz).

3 Conclusion

In this paper, a technique was described for analysing exponential lines as used

for interconnects in high speed digital circuits. First, the theory was developed

and tested, and then it was extended to testing of the low noise amplifier. The S

parameters for the ETL and uniform microstrip lines for the amplifier were

calculated, and comparisons were made with result s by other methods. Good

agreement was found between this technique and others.

A method of analysing digital pulse propagation through the tapered

microstrip lines terminated with non-linear loads was described. Again, the



This is due to the different terminal resistances when the output is entering the

logic high from the logic low or the reverse.

~>lfe£

&

W?-Input

Stage

Input — )port

matchin

network

/!\

; / TL

*1

r,

Hi

\ Short\Stub

h-

ed

•*

]

1 1 ,LJn

device +, H, matchi ig ^

Output Stage

r r1 * 2 * LVdd

\ ^ to next

-jHVrET ?

/MRS

{A

RL

Figure 6a Low noise amplifier matching block network with its equivalent

schematic matching diagram.

Void

Rd

to nextstage

D (drain)G (gate)

(from preceeding'L-iS (source)stage) -L-

Figure 6b Circuit arrangements of ETL replacing the stub matching network.

These uniform lines (stub matching) have been used in the design and

building of a two stage FET satellite receiver low noise amplifier. See fig. 6a.

From the above analysis, we can predict that by replacing the uniform line

matching networks with the ETL, fig. 6b, the performance will greatly improve


92 Simulation and Design of Microsystems and Micro structure 8

results were compared with previously published results, and agreement was

good. To confirm the validity of the technique, the calculated results were

compared with experimental results. Measurements were taken to determine

the S parameters of the microstrip structures, and calculated values showed

good agreement with measured values. Note, these results were for the low

noise amplifier with stub-matching. As for the ETL, the analysis and evaluation

of the system is underway.

4 References

[1] O. A. Palusinki and A. Lee, "Analysis of Transients in Non-uniform and

Uniform Multiconductor Lines," IEEE Trans. Microwave Theory Tech., Jan.

79% /%;. 727-7J&

[2] C. Nwoke, "An Exact Solution for the Non-uniform Transmission Line

Problem," IEEE Trans. Microwave Theory Tech., July 1990, pp. 944-946.

[3] M. Cristina, R. Carvalho, W. Margulis and J. R. Souza, "A new, small-

sized Transmission Line impedance transformer, with applications in high

speed optoelectronics," IEEE Microwave and Guided Wave Letters, vol. 2, No.

11, Nov. 1992.

[4] J. F. Lee, R. Mittra and J. Joseph, "Time-domain scattering parameters of

an exponential transmission line," IEEE Trans. Microwave Theory Tech., vol.

39, No. 11, pp. 1891-1895, Nov. 1991.

[5] Hsue and Hectman, "Transient response of an exponential transmission line

and its applications," IEEE Trans. Microwave Theory Tech., vol. 42, No. 3,

March 1994, pp. 458-462.

[6] J. E. Schutt-Aine and R. Mittra, "Scattering parameter transient analysis of

transmission lines loaded with non-linear terminations," IEEE Trans.

Microwave Theory Tech., vol. MTT-36, pp. 529-536, arch 1988.

[7] C. W. Hue, " Time-domain Scattering parameters of an exponential

transmission line," IEEE Trans. Microwave Theory Tech., vol. 39, pp. 1891-

1895, Nov. 1991.

[8] C. W. Hue and C.D. Hechtman, "Transient analysis of non-uniform, high

pass transmission line," IEEE Trans. Microwave Theory Tech., vol. 38 pp.

1023-1030, Aug. 1990.


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