IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 3, MARCH 2006
1043
Time-Constant Control of Microwave
Integrators Using Transmission Lines
Ching-Wen Hsue, Senior Member, IEEE, Lin-Chuan Tsai, and Yi-Hsien Tsai
Abstract—A model describing the time constant of a transmission-line integrator is presented. By representing the formulations
of integrators in the discrete-time (or ) domain, we implement the
integrators with equal-length transmission lines. Three integrators
with different time constants and frequency bands are built and
tested. The experimental results are in good agreement with theoretical values.
Index Terms—Equal-length line, microwave integrator, time
constant, transforms.
I. INTRODUCTION
T
HE integrator is an instrumental tool to estimate the time
integral of measured signals. It has been used extensively
in many areas such as coherent detection, correlation estimation,
accumulator analysis, and waveform shaping [1]. The integrator
can also be employed to measure the delay times of microwave
transistors [2] or it can be used to implement high-frequency
active filters [3]. In other words, not only does the integrator
plays an important role in determining the inter-relation among
various signals, but it can also detect the history of the signal itself. In the Fourier spectral analysis, the spectral of a measured
signal is the output of an integrator that takes the time integration of the multiplication of the measured signal by harmonic
signals [1]. Thus far, the integrators are mainly employed in circuits for low-speed applications. Therefore, the implementation
of integrators for high-frequency applications has been largely
ignored.
A serial R–C circuit, in conjunction with an operational
amplifier, has been widely employed to form an integrator
[2]–[4]. However, the configuration of such a circuit is good
only for low-speed applications. Many other techniques have
been developed to design integrators using finite impulse response (FIR) or infinite impulse response (IIR) methods in the
study of discrete-time signal processing (DSP) [5]–[7]. Among
various techniques, trapezoidal rule and Simpson’s rule in the
-domain are two popular methods used for integrators. The
trapezoidal-rule integrator produces a zero at the normalizing
frequency [1], while the Simpson-rule integrator yields a
quasi-zero lying between dc and the normalizing frequency.
Manuscript received June 12, 2005; revised November 5, 2005. This work
was supported by the National Science Council, R.O.C., under Grant NSC932218-E011-001.
C.-W. Hsue and Y.-H. Tsai are with the Department of Electronic
Engineering, National Taiwan University of Science and Technology, Taipei,
Taiwan 106, R.O.C. (e-mail: cwh@et.ntust.edu.tw).
L.-C. Tsai is with the Department of Electronic Engineering, Lunghwa University of Science and Technology, Taoyuan, Taiwan 333, R.O.C.
Digital Object Identifier 10.1109/TMTT.2006.869722
Fig. 1. Electronic integrator.
The existence of zeros causes the performance of these two
integrators largely deviates from that of the ideal integrator.
Therefore, both trapezoidal- and Simpson-rule integrators are
not adequate to be employed as a wide-band integrator. To overcome the limitation, we propose a new discrete-time integrator
whose transfer function fits well with that of an ideal integrator
for the frequency band extending from dc to the normalizing
frequency. In particular, the time constant is proposed to characterize the performance of the integrator and it serves as an
important factor that determines the amplitude response of an
integrator. Instead of taking the time constant as the multiplication of resistance by capacitance, the time constant is dictated
by both signal frequency and transfer function of the integrator
so that the time constant is accessible in the microwave circuit.
It has been shown that the scattering characteristics of equalelectrical-length transmission lines can be represented with the
variable in the discrete-time domain [8]. Therefore, the transmission-line configuration can emulate the characteristics of an
integrator developed in the discrete-time study and the operating
frequency band of the integrator is extended further into the microwave range. To verify the theoretical study, we implement
three integrators in the microstrip format that have the operating
frequencies up to 10 GHz. Each of three integrators has a distinct
time constant. The experimental results, except for the lower frequency band, are in good agreement with the theoretical values.
II. TIME CONSTANT OF AN INTEGRATOR
Fig. 1 shows an integrator formed by an inverted operational
and
amplifier and a serial resistor-capacitor circuit, where
are the input and output of the circuit, respectively, and
is the signal angular frequency. The transfer function
of the integrator in the frequency domain is defined as the ratio
to
and is given as follows:
of
(1)
where is the resistor and is the capacitor. Notice that the
transfer function is inversely proportional to the angular frequency of the signal. As a result, an integrator is treated as
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 3, MARCH 2006
a low-pass filter. In particular, the multiplication of by is
the time constant of the circuit. We define the time constant
of an integrator as follows:
(2)
The time constant determines the transient behavior of an integrator in the time-domain consideration. From Fig. 1, it is
is related to the input
easy to show that the output voltage
voltage
through the following relation:
(3)
where is the time. Equation (3) reveals that the output
of
an integrator is inversely proportional to the time constant. Since
no appropriate method can be employed to obtain the equivalent capacitor and resistor of an integrator implemented by
using transmission lines, we assume that (2) is valid to get the
time constant of an integrator.
Fig. 2. Amplitude responses of H (z ); H (z ), ideal integrator.
III. TRANSFER FUNCTIONS OF INTEGRATORS IN THE -DOMAIN
Many techniques have been developed to design integrators
using FIR or IIR methods in the study of DSP [6], [7]. The
trapezoidal-rule integrator in the -domain is as follows [1]:
(4)
is the unit of time delay. Notice that the trapezoidalwhere
rule integrator in (4) represents a bilinear transformation, which
also transforms the system function in the frequency domain
into the corresponding system function in the discrete-time do. When the fremain [1]. In particular, a zero exits at
quency-domain response of the integrator is concerned, in (4)
is replaced with the following relation:
(5)
is the frequency angle (or normalized frequency) and
. An integrator can also be obtained by inverting the
transformation of a wide-band differentiator in [9]. This gives
us the following:
where
(6)
Fig. 2 shows the amplitude responses of both (4) and (6) as a
function of normalized frequency. The amplitude response of an
ideal integrator is also shown in Fig. 2, which is inversely proportional to the normalized frequency. Both (4) and (6) deviate
from the values of an ideal integrator in the upper frequency
band. To obtain an integrator that fits better the ideal integrator
over the entire normalized frequency band, a new integrator is
set as follows:
(7)
With such a selection, the zero occurring at the normalizing frein the trapezoidal-rule integrator is removed.
quency
Fig. 3 shows the amplitude responses of both
and the
Fig. 3. Amplitude responses of both H (z ) and ideal integrator.
ideal integrator. Apparently, the integrator
can well represent the ideal integrator in the entire frequency band of
. As the frequency is changed over the entire band, as
shown in (5), it is equivalent to moving along the unit circle
in the complex -plane. Equation (7) reveals that a zero
and this zero is not feasible by using a
occurs at
transmission-line element. It has been shown that zeros occurcan be implemented by using
ring on the unit circle
shunted transmission-line elements [8]. In particular, the zero at
is far from the unit circle and it has little effect on
in the entire frequency range of
. To facilitate the design procedure, we use the parametric method [1] to
convert the autoregression moving average (ARMA) process of
(7) into the autoregression (AR) process. As a result, we obtain
(8), shown at the bottom of the following page. To implement
an integrator, the next step is to obtain an equal-length transmission-line configuration so that its transmission coefficient fits the
.
transfer function
IV. IMPLEMENTATION OF INTEGRATORS
Equation (2) indicates that the multiplication of the transfer
function by angular frequency of an integrator is a constant
value. Therefore, in order to vary the time constant of the integrator, it is required to change its transfer function. To obtain
integrators with different time constants, it is simply to multiply
HSUE et al.: TIME-CONSTANT CONTROL OF MICROWAVE INTEGRATORS USING TRANSMISSION LINES
1045
propagation delay time of the finite line. The chain-scattering
. The reparameter matrix is obtained by setting
lation between the frequency angle and the transmission-line
parameters is
.
serial secIf a transmission-line configuration consists of
of
tions, the overall chain scattering parameter
such a circuit is obtained by the sequential multiplication of the
chain-scattering parameter matrices of all transmission-line elements. We have
(9)
Fig. 4.
Amplitude responses of H (z ); 2H (z ); and 3H (z ).
where all are real and are determined by the characteristic impedances of all transmission-line elements. If the output of the
transmission-line circuit is loaded with a matched termination,
, is
the transfer function of the overall circuit, denoted as
as follows:
TABLE I
CHAIN-SCATTERING PARAMETER MATRIX OF A SERIAL LINE
(10)
where
is a function of characteristic impedances of all serial transmission-line elements. If we
in (8) to approximate the transfer function
of
use
the transmission-line circuit and neglect the propagation factor
, we obtain
(11)
in (7) by corresponding constants. Fig. 4 shows the amplitude responses of
and
. The transfer
.
functions of all three integrators become infinite at
This reveals that the integrator is very difficult to implement
if the amplitude response is to be met in the low-frequency
band. In addition, it is pertinent to point out that
and
have the value of unity at the normalized frequenand
, respectively. From (2), it is known
cies
and
have time constants, which
that
are 3.18, 1.59, and 1.06 s, respectively. If an integrator is implemented by using transmission lines, the maximum value of
the transfer function of the integrator is unity. To facilitate the
as 1 for
design, we set the amplitude of transfer function
the frequency range
. The rest part of the transfer
satisfies (7). Under such a
function in the range
circumstance, the circuit thus obtained behaves as an integrator
. Similar situations hold
over the frequency range
for the integrators of
and
.
Table I shows the chain-scattering parameter matrix of a seand
rial transmission line in the -domain, where
are the propagation constant, physical length, and characteristic
impedance, respectively. Notice that
is the reference characteristic impedance, which is assumed to be 50 unless men, where is the
tioned otherwise. It is assumed that
The next step is to compare the coefficients of the denominator
to the coefficients of the denominator in
so that
in
is as close to
as possible. Notice that
in (11) is
determined by the characteristic impedances of all transmission
lines. Upon using the optimization method [8] in the sense of
minimum square error for the coefficients of denominators on
both sides of (11), we obtain the characteristic impedances of
transmission lines. We may employ the same procedure from
(8)–(11) to attain the characteristic impedance profiles for both
and
.
To implement an integrator with equal-length transmission
lines, the electrical length of each transmission line is set to 90
at the normalizing frequency
. We thus have
,
where is the physical length of each transmission-line section
is the wavelength at the normalizing frequency.
and
V. EXPERIMENTAL RESULTS
We assume that the normalizing frequency of three discretetime integrators is 10 GHz, i.e., the normalized frequency at
in Fig. 4 is corresponding to 10 GHz. For the integrator
of
having a time constant of 3.18 s in the normalized
frequency scale, the corresponding time constant in the regular
frequency scale is 1.6 10
s. The time constants for
(8)
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 3, MARCH 2006
Fig. 5. Layouts of three integrators. (a) 14-section line for H (z ) with =
s. (b) Ten-section line for 2H (z ) with = 0:8 10
s.
1:6
10
(c) Six-section line for 3H (z ) with = 0:53 10
s.
2
2
2
and
in the regular frequency scale are 0.8 10
and
0.53 10
s, respectively. In particular, microstrips are used
to implement three integrators. Fig. 5(a)–(c) shows the physical
layouts of microstrips, which are built on Duroid substrates with
dielectric constant
, loss tangent
, and a
thickness of 30 mil (0.762 mm). Notice that Fig. 5(a)–(c) shows
the integrators having time constants 1.6 10
s, 0.8 10
s,
and 0.53 10
s, respectively. The impedance profiles of the
14-section line in Fig. 5(a) are 73.2, 27, 130, 20, 130, 114.9,
44.3, 20, 20.7, 37.6, 64.6, 75.4, 65.2, and 54.1 . Fig. 5(b) is
a ten-section line and its impedance values are 53.4, 140, 17.5,
86.8, 71.2, 68.5, 62.6, 57.1, 53.1, and 50.8 . The impedance
profiles of the six-section line in Fig. 5(c) are 10, 23, 100, 67,
56.6, and 51.4 . All characteristic impedances of transmission lines are obtained by using an optimization process [8]
that involves the comparison between two AR processes on both
sides of (11). The propagation delay time of each finite line is
25 ps, which produces the normalizing frequency of 10 GHz.
Fig. 6 shows the experimental results
of three integrators
shown in Fig. 5. For convenience, Fig. 6 also shows the theoretical values of each integrator. Apparently, the experimental results are in good agreement with the theoretical values. The experimental results of the circuit in Fig. 5(a) fit well with the theoretical values for the frequency range GHz
GHz.
As mentioned in Section IV, the magnitude of
has a
value of unity for the frequency range GHz
GHz.
Therefore, the circuit in Fig. 5(a) behaves as an integrator for
the frequency band GHz
GHz. It has a time constant of 1.6 10
s. The experimental results of the circuits
in both Fig. 5(b) and (c) also fit well with the theoretical values
over the respective frequency ranges. In particular, the time constants of the circuits in both Fig. 5(b) and (c) are 0.8 10
and
s, respectively. To illustrate the circuit behavior of
0.53 10
Fig. 6. (a)–(c) Experimental results of S (f ) for three integrators shown in
Fig. 5(a)–(c), respectively.
integrators in the time domain, we show in Fig. 7 the responses
of integrators with a square wave as the input signal. The integrators turn the square wave into the triangular wave. In order
HSUE et al.: TIME-CONSTANT CONTROL OF MICROWAVE INTEGRATORS USING TRANSMISSION LINES
1047
REFERENCES
Fig. 7. Time-domain responses of three integrators with a square wave as the
input signal.
to make a good comparison, the time delays are adjusted so that
three integrators attain the same delay values.
Notice that the device sizes in Fig. 5 are different depending
on the time constants of the integrators. The number of sections
of integrators are determined by the optimization process that
involves the curve fittings of transfer functions of transmission
lines to the amplitude responses of the ideal integrators in Fig. 4
. The
for the frequency range with
transfer functions of transmission lines are set to unity for the
frequency range with
.
As a result, the characteristics of the integrators shown in
Fig. 6 are different from those of the ideal integrators in the
lower frequency band. This is due to the fact that the maximum value of the transfer function of the transmission line is
unity. In order to show the impedance profiles of a conventional
low-pass filter, we consider the transfer function of a Butterworth low-pass filter [1] in the -domain as follows:
(12)
It is pertinent to point out that both the Butterworth low-pass
filter in (12) and the integrator in Fig. 5(a) have the 3-dB point
at the same frequency in the attenuation band. Employing the
same synthesis method shown in Section IV and [8], we obtain
the impedance profiles of a 12-section line as 106.3, 45, 150,
117.4, 39.4, 20, 21.5, 36.2, 58, 30.2, 64.5, and 56 . The serial
line is shunted with two stubs with characteristic impedances of
48.8 and 35 at the first and second sections of the serial line.
Apparently, the impedance profiles of the Butterworth low-pass
filter are different from that of the integrator shown in Fig. 5(a).
VI. CONCLUSION
A model representing the time constant of a microwave integrator has been presented. Three integrators formed by cascade connections of equal-length serial transmission lines have
been implemented to verify the feasibility of integrators with
different time constants. Except for the lower frequency bands,
the experimental results were in good agreement with the theoretical values of integrators.
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Ching-Wen Hsue (S’85–M’85–SM’91) was born
in Tainan, Taiwan, R.O.C. He received the B.S. and
M.S. degrees in electrophysics and electronic from
the National Chiao-Tung University, Hsin-Chu,
Taiwan, R.O.C., in 1973 and 1975, respectively, and
the Ph.D. degree from the Polytechnic University
(formerly the Polytechnic Institute of Brooklyn),
Brooklyn, NY, in 1985.
From 1975 to 1980, he was a Research Engineer
with the Telecommunication Laboratories, Ministry
of Communication, Taiwan, R.O.C. From 1985 to
1993, he was a Member of Technical Staff with Bell Laboratories, Princeton, NJ.
In 1993, he joined the Department of Electronic Engineering, National Taiwan
University of Science and Technology, Taipei, Taiwan, R.O.C., as a Professor,
and from August 1997 to July 1999, he was the Department Chairman. His
current interests are pulse-signal propagation in lossless and lossy transmission
media, wave interactions between nonlinear elements and transmission lines,
photonics, high-power amplifiers, and electromagnetic inverse scattering.
Lin-Chuan Tsai was born in Taipei, Taiwan, R.O.C.,
in 1968. He received the M.S. and Ph.D. degrees
in electronic engineering from the National Taiwan
University of Science and Technology, Taipei,
Taiwan, R.O.C., in 1998 and 2004, respectively.
From October 2002 to August 2005, he was a
Project Engineer with the Mobile Business Group,
Chunghwa Telecom, Taipei, Taiwan, R.O.C., where
he was involved with wide-band code division
multiple access (WCDMA) network planning. He is
currently an Assistant Professor with the Department
of Electronic Engineering, Lunghwa University of Science and Technology,
Taoyuan, Taiwan, R.O.C. His current interests are discrete time signal processing, wireless communications, microwave planar filter design, and passive
circuit design.
Yi-Hsien Tsai was born in Taipei, Taiwan, R.O.C.,
in 1977. He received the B.S. and M.S. degrees in
electrical engineering from the National Taiwan University of Science and Technology, Taipei, Taiwan,
R.O.C., in 2003 and 2005, respectively, and is currently working toward the Ph.D. degree in electronic
engineering at the National Taiwan University of Science and Technology.