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Synthesis of Broadband Negative Group Delay Active
Circuits
Blaise Elysée Guy Ravelo, André Pérennec, Marc Le Roy
To cite this version:
Blaise Elysée Guy Ravelo, André Pérennec, Marc Le Roy. Synthesis of Broadband Negative Group
Delay Active Circuits. IEEE MTT-S International Microwave Symposium, Jun 2007, Hawaï, United
States. pp.2177-2180, �10.1109/MWSYM.2007.380357�. �hal-00468141�
Synthesis of Broadband Negative Group Delay Active Circuits
Blaise Ravelo, André Pérennec, Marc Le Roy and Yann G. Boucher
LEST– UMR CNRS 6165 – UBO/ENSTBr- CS93837, F-29238 Brest cedex 3, France.
RESO – EA3380 – CS72862, F-29238 Brest cedex 3, France.
Abstract — This paper deals with the design and synthesis of
active circuits that produce simultaneously negative group delay
and gain at microwave frequencies or for baseband signals. For a
single cell, analytical equations show that the proposed topology
meets these objectives while also satisfying active device
requirements. Then, a synthesis approach is extracted and
applied to design a two stage microwave circuit which is
validated by a comparison between simulation and experimental
results. This method is extended to design a four-stage baseband
active circuit that provides gain and negative group delay up to 1
GHz. Frequency and time-domain simulations illustrate the
topology ability to achieve high relative time-advance.
Index Terms — Active device, FET circuits, negative group
delay, superluminal group velocity.
I. INTRODUCTION
Materials with simultaneously or independently negative
permittivity and permeability, initially introduced in the late
1960s by Veselago [1], are now validated in 3-D artificial
media [2] and in 1-D or 2-D microwave circuits [3]. Much
recently, such metamaterials also know as Left-Handed Media
(LHM), has been applied, in microstrip and coplanar
waveguide technologies, to design antennas, filters, couplers
[4], phase shifters[5]. In general, in these devices, the LHM
properties are used either to reduce the circuit size or to
enlarge the initial operating bandwidth.
In 1960, the classical analysis of Brillouin and Sommerfeld
[6] showed that in a frequency band of anomalous dispersion,
e.g. obtained from electromagnetic Bragg gratings [7] or
metamaterials, the group velocity can become greater than the
speed of light in vacuum c, or even be negative. Since 1999, a
renewed interest for the “superluminal” group velocity has
emerged and this counter-intuitive effect has been widely
debated [8]. It is well confirmed now that this behavior is not
at odd with causality, and most points of controversy have
been settled down. However, to illustrate and establish the
properties of this intriguing phenomenon, several
configurations or experimental circuits exhibiting Negative
Group Delay (NGD) or superluminal phenomenon have been
proposed [9]-[10]. These devices can be merged in two major
categories. The first one essentially devoted to microwave
applications consists in LHM built with the passive left-
handed cell [3], i.e. a series capacitance and a shunt
inductance, associated with a RLC parallel-resonant network.
To achieve a significant NGD value, many cells have to be
cascaded resulting in a very low transmission level.
The second category brings together experiments with NGD
or Negative Refractive Index (NRI) with an amplification of
the input signal or, at least, loss compensation. In the most
debated one [8], a region of anomalous dispersion is created
in a kind of optical amplifier, and the authors measured a
group velocity of – c/310. All the other active NGD circuits
only operate for baseband signals under a few hundred of
KHz and are based on a simple electronic circuit built with an
operational amplifier, a resistor and a capacitor [9]-[10]. Gain
and high relative advances of a Gaussian baseband signal can
be achieved by cascading several stages. Such simple
electronics circuits are very useful to analyze the properties of
superluminal group velocity; but nevertheless, they are fairly
narrowband and restricted to low frequencies.
This assessment led us to investigate for a more general
approach that should bring of course NGD, gain, thus
necessarily active, and should operate for broad and base-
band signals as for modulated ones over the microwave
domain. Section II will describe the proposed topology and
the theoretical analyze of its S-parameters. Section III
compares the simulated and measured frequency results of a
two stages NGD active device. In section IV, a broadband
synthesis method is developed from the initial cell equations.
Then, frequency and time-domain simulations of a broadband
and baseband NGD active circuit of four stages are presented
and we discuss on limitations and improvements of this
circuit.
II. P
ROPOSED NGD TOPOLOGY AND SYNTHESIS EQUATIONS
Here, the more global notion of group delay is to be used
instead of group velocity because it is well defined both for
lumped circuits, and for distributed ones. Generally, the group
delay,
τ
= -d
ϕ
/d
ω
, is extracted from experimental S-
parameters, where
ϕ
is the frequency-dependent transmission
phase shift and ω is the angular frequency. In the case of a
frequency limited bandwidth input signal, the group delay
corresponds to the time shift of its envelope maximum. We
though that a Field Effect Transistor (FET) associated with
passive components could bring promising possibilities to
achieve our severe objectives; i.e. NGD, gain, input and
output matching, low noise and stability.
A. Study of the Proposed Unit Cell
2
Intuitively, we first tested several configurations of the
classical LHM cell [3] which provides by itself NGD, in
cascade or in parallel with a FET. But finally, the topology
depicted in Fig. 1, compared to those with LHM cells,
proposes simultaneously higher significant NGD and gain
values, a higher simplicity, and moreover a possibility of
input/output matching. It consists in a series RLC resonant
circuit in shunt at the FET drain output.
FET
C
L
R
Fig. 1. Series RLC resonant network in cascade with a FET.
To allow an analytical description of the cell, the FET is
modeled by a voltage-controlled current generator and R
DS
,
the drain-source resistor. The group delay,
τ
, is negative at the
RLC network resonance frequency
ω
0
=1/
LC
and at this
particular frequency:
)(
2
)(
0
0
021
dsds
dsm
RRZRR
RgRZ
S
++
−
=
ω
(1)
)](.[
2
)(
0
0
0
dsds
ds
RRZRRR
RLZ
++
−
=
ωτ
(2)
)(
)(
)(
0
0
022
dsds
dsds
RRZRR
RRZRR
S
++
+−
=
ω
(3)
where Z
0
is the input impedance of the reference ports
(50 Ω in practice) and g
m
is the FET transconductance. A few
remarks must be underlined:
- S
12
is null and S
11
is close to one at low frequency;
- the group delay is always negative at the cell resonance;
- (3) shows that an output matching is possible;
- a significant gain value can be obtained even if it is
minimum at the resonance; R must be small to get a
significant NGD value but this implies a low transducer gain;
- S
21
,
τ
, and
ω
0
do not depend of exactly the same parameters,
so it means that
τ
can be controlled independently from S
21
through L and that the resonance frequency can be changed
via C without modifying the level of
τ
; and thus, this gives us
degrees of freedom to find a satisfying compromise.
B. The whole circuit synthesis
Surprisingly, this topology is that of a resistive amplifier,
excepted for the components values. To match the circuit
input, a shunt resistor, R
m
, is placed between the first cell and
the generator (Fig. 2).
Z
0
C
n
L
n
R
n
F
n
C
k
L
k
R
k
F
k
C
1
L
1
R
1
F
1
Z
0
E
R
m
1
st
cell
Fig. 2. Structure of the n-stages cascaded ideal circuit.
Then, considering this resistor and the following unit cell as a
new first cell, the group delay (2) and S
22
(3) expressions are
not modified, but the other significant parameters are now
defined by:
0
0
111
)(
ZR
ZR
S
m
m
+
−
=
ω
(4)
))((
2
)(
11000
01
121
dsdsm
dsmm
RRRZRZZR
RgZRR
S
+++
−=
ω
(5)
where
ω
1
is the R
1
L
1
C
1
resonant angular frequency. For a
relatively high value of R
m
, the first stage masks the noise
brought back by a possible low value of R
1
. To synthesize the
network component values, we define the objectives for the
synthesis of the first cell and R
m
as followed:
aS =)(
111
ω
,
bS =)(
121
ω
and
11
)(
τ
ω
τ
=
. Then, by simply
inverting (3), (4) and (5), we obtain:
a
a
ZR
m
−
+
=
1
1
0
(6)
)])((2[
)(
0
2
00
0
dsmdsmdsmm
mds0
1
RRRRZZbRRZg
ZRRb.Z
R
+++−
+
=
(7)
ds
dsds
RZ
RRZRRR
L
0
10111
1
2
)]([. ++−
=
τ
and
2
11
1
1
ω
L
C =
. (8)-(9)
For a unit cell k, and at the series resonance frequency
ω
k
,
only the expression of R
k
is different:
)](2[
00 dsdsm
ds0
k
RZbRZg
RbZ
R
+−
=
. (10)
Obviously, R
k
plays also the part of inter-stage matching
impedance. To choose an appropriate transistor, particularly
through g
m
, and to obtain the S
22
(
ω
1
) = c objective, the four
following equations may also be used for the first cell:
a
a
ZR
m
−
+
=
1
1
0
)1)(1(
2
0
caZ
b
g
m
++
=
(11)-(12)
)(
)1(
00 dsds
ds0
1
RZcZR
cRZ
R
+−−
+
=
2
00
2
01
1
)]([
)1(
dsds
ds
RZcRZ
cRZ
L
++−
+−
=
τ
(13)-(14)
3
C
1
is still defined as in (9). Obviously, these synthesis
equations are only valid for the simple microwave low
frequency model, but they provide component values really
close to final values achievable through optimization with a
more complete modelization.
III.
EXPERIMENTAL VALIDATION AND NGD CIRCUIT
IMPLEMENTATION
In order to validate the theory, we chose to build a circuit
working around 1 GHz. Moreover, we initially limited
ourselves to a two stages design with identical resonance
frequencies. In that case, from either the classical transducer
gain formulas or by cascading chain matrices of cells defined
in the previous section, we obtain for the total insertion loss:
()()( )
200201
021
22
1122
2121
21
2
1
21
21
RRRZZRRZRR
RZRRRg
SS
SS
S
dsdsmds
mdsm
T
++++
=
−
=
(15)
As the FETs are unilateral, S
11T
can be identified as the return
loss of the first cell (4) and S
22T
to the output return loss of the
second one (3). The objectives were that S
11T
and S
22T
were
below -10 dB, a theoretical total gain around 6 dB and a NGD
better than -2 ns. We searched for a FET with high g
m
and R
DS
values. Then, we got R
2
from (10) and Rm from (11). For the
specified frequency, initially 1.45 GHz, C
1
and C
2
were
chosen identical and small to allow L
1
and L
2
to be sufficiently
high to achieve a significant NGD. Thanks to R
1
, a -1.66 ns
NGD is achieved for the first cell and due to the R
2
low value,
only -0.33 ns for the second stage and then each S
21
stage
parameter can be evaluated, i.e. 10.84 dB and 0.06 dB for the
first and second respectively. Finally, we achieve, in theory, a
total transmission gain of 6 dB and a matching of -14 dB at
the input and -11 dB at the output with a -2.3 ns group delay.
The total group delay is not simply equal to the sum of each
stage one due to the Z
0
reference of the S-parameters at their
interface. All the circuit references are summarized in Fig. 5.
The S-parameters and the group delay remain practically
unchanged when a high frequency equivalent circuit model is
introduced. On the other hand, when the connection
microstrip lines and the real passive components models are
introduced in the schematic or electromagnetic simulations,
the operating frequency is shifted down to 1.07 GHz and S
21T
is reduced to 3 dB. Fig. 5 shows the layout of the final circuit.
Two different bias networks has been implemented, a
conventional one (V
GS1
and V
DS1
for gate and drain biases),
and a second network (V
GS2
and V
DS2
) which is connected
through the resonant cells and they both give identical
measurement results. Fig 6-a and 6-b show a good agreement
between electromagnetic simulations and experimental results
(obtained with no adjustment). The measured group delay is
of -2.3 ns at 1.03 GHz while the S
21T
parameter is 1.68 dB. At
this specific frequency, S
11T
is around -14 dB and S
22T
below -
12 dB.
C
1
R
1
F
1
R
m
C
2
L
2
R
2
F
2
L
b
L
b
L
b
L
b
L
b
L
b
C
b
C
b
C
DCB
Input Output
C
DCB
C
DCB
L
1
V
GS1
V
GS2
V
DS1
V
DS2
R
m
= 75
Ω
R
1
= 11 Ω
R
2
= 36 Ω
L
1
= L
2
= 12 nH
C
1
= C
2
= 1 pF
FET: F
1
= F
2
PHEMT
g
m
= 98.14 mS
R
DS
= 116.8
Ω
DC blocking
capacitor
C
DCB
= 300 pF
Bias networks
L
b
= 470 nH
C
b
= 1 nF
Substrate:
RF-35,
ε
R
= 3.5
h = 0.508 mm
Fig. 5. Microstrip circuit layout in black, bias networks layout in
hatched gray (White circles indicate ground via-holes) and
component references.
0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.60.7 1.7
-18
-16
-14
-12
-10
-8
-20
-6
Frequency (GHz)
S11T (dB), S22T (dB)
Simulations
Measurements
S
22T
S
11T
0.80.91.01.11.21.31.41.51.60.7 1.7
5
10
15
20
25
30
35
0
40
-5
-4
-3
-2
-1
0
1
-6
2
Frequency (GHz)
S21T (dB)
Group Delay (ns)
Simulations
Measurements
Fig. 6. Comparison of simulated and measured S-parameters and
group delay. (a) Input and output return losses. (b) S
21
parameters
and group delay.
Compared to previous studies with passive components [3],
we achieve slightly higher NGD and bandwidth values for an
equivalent number of cells. The possibility of each stage
characteristics to be controlled independently and particularly
the resonance frequencies brings opportunity to propose the
synthesis of a broadband NGD circuit.
(a)
(b)
4
IV. D
ESIGN OF A BROADBAND NGD ACTIVE CIRCUIT FOR
BASEBAND SIGNALS
In digital transmission domain and for high data rates, the
ideal characteristic of a NGD active circuit would be a
constant phase advance and gain over the entire signal wide
bandwidth. Fortunately, in many applications, the digital
signal is filtered to frequency limit its spectrum. So, we will
restrict our goal to the bandwidth that minimize the
intersymbol interferences of a gaussian baseband signal and
thus keep as low as possible signal distortions, ripples and
pulse reshaping. With the topology presented in section II, a
NGD broadband has been achieved with a 4 stage circuit.
Each stage component values have been calculated separately
for resonant frequency regularly spread from 0 to 600 MHz
and the whole circuit behavior has been predicted from the
sum of the stage transfert functions for the gain and by the
sum of the time delay of these transfert functions (instead pof
S-parameters) for the total group delay. A final optimization
and electromagnetic simulations have been performed in the
same conditions as in section IV (Fig. 7-a), i.e. with the
component S-parameters, the interconnection lines and the
bias network, and with the same FET and substrate. For such
a broadband, an active bias network had to be implemented.
Time-domain simulations (Fig. 7-b) are also presented and
exhibit a relative time advance of 82 % compare to the 0.7 ns
standard deviation. A pulse compression is also evidenced,
particularly in the inset of fig.7-b where the input and output
pulses are displayed on a same ordinate scale.
Frequency (MHz)
100 200 300 400 500 600 7000 800
5
10
15
20
25
30
35
0
40
-1.2
-0.8
-0.4
0.0
0.4
0.8
-1.6
1.2
S21 (dB)
Group Delay (ns)
23456718
0.30
0.70
1.10
1.50
1.90
2.30
-0.10
2.70
0.175
0.375
0.575
0.775
-0.025
0.975
Time
(
ns
)
Input (V)
Output (V)
Fig. 7. Simulated S
21
parameter and group delay (a). Time-
domain simulations (b). R
m
= 75 Ω, R
1
= R
2
= 10 Ω, R
3
= 18 Ω, R
4
=
30 Ω, C
1
= 150 nF, C
2
= 3.9 nF, C
3
= 11 pF, C
4
= 2.7 pF, L
1
= L
2
=
3.6 nH, L
3
= 8.7 nH, L
4
= 8.2 nH.
V. C
ONCLUSION
We report on a new NGD active topology that provides
simultaneously gain and NGD. To validate our theoretical
results, we have designed and implemented a two stage circuit
for microwave frequency modulated signals. Analysis and
synthesis equations are proposed and are sufficiently general
to be applied to several frequency bands. Moreover, a few
numbers of cells lead to significant gain and relative time-
advance values. Indeed, we simulated in frequency and in
time-domain a four-stage circuit that provided gain and a
negative delay even for a broadband and baseband Gaussian
input signal. This proposal brings some prospects for
applications in the communication domains. Time-domain
measurements are scheduled to verify the time advance
sensibility versus noises and mm-waves distributed circuits
are also planed.
R
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(a)
(b)
