University of Birmingham
Substrate integrated waveguide filter-amplifier
design using active coupling matrix technique
Gao, Yang; Zhang, Fan; Lv, Xin; Guo, Cheng; Shang, Xiaobang; Li, Lei; Liu, Jiashan; Liu,
Yuhuai; Wang, Yi; Lancaster, Michael J.
DOI:
10.1109/TMTT.2020.2972390
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Citation for published version (Harvard):
Gao, Y, Zhang, F, Lv, X, Guo, C, Shang, X, Li, L, Liu, J, Liu, Y, Wang, Y & Lancaster, MJ 2020, 'Substrate
integrated waveguide filter-amplifier design using active coupling matrix technique', IEEE Transactions on
Microwave Theory and Techniques, vol. 68, no. 5, 9007486, pp. 1706-1716.
https://doi.org/10.1109/TMTT.2020.2972390
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1
Abstract—This paper presents a comprehensive active N+4
coupling matrix approach for the design of integrated filter-
amplifiers. The feedback between gate and drain which is
neglected in previously work is considered, which improves the
accuracy of the coupling matrix model for transistors. More
importantly, the relationship between the coupling matrix and the
noise figure is also established, which extends the coupling matrix
method to tackle noise related circuit functions. Substrate
integrated waveguide (SIW) filters are used to implement an
integrated X-band filter-amplifier design and to validate the
design approach in terms of return loss, gain and noise. Compared
with rectangular waveguide, SIW is utilized for its appealing
advantages such as lower production cost, easier fabrication, and
most importantly easier integration with active components. A
second-order filtering circuit is applied to match simultaneously
the input and output of the transistor. The integration reduces the
losses from the intermediate networks in conventional designs,
which is particularly important when the frequencies go higher.
The measurements agree very well with the simulations in terms
of S-parameters, gains and noise figures.
Index Terms—Active coupling matrix, transistor, resonator,
SIW filter amplifier.
I. INTRODUCTION
MPLIFIERS are essential components in a variety of
microwave communication and radar systems [1]-[3].
Amplifiers and filters are usually constructed separately refer to
50-Ω impedance interface, and then cascaded in the system. For
conventional amplifiers, matching networks are required to
transfer the complex impedance of the transistor to 50-Ω.
Widely used matching networks include stubs, quarter
wavelength transformers, and coupled lines [4]; and specific
examples can be found using single coplanar waveguide (CPW)
line stubs [5], step ridges [6], and antipodal fin-line arrays [7].
These designs are based on independent transistors and the
matching networks, which can suffer from additional loss, extra
cost, and large circuit footprint.
Manuscript received September 30, 2019; revised December 09, 2019;
accepted January 5, 2020. This work is supported by the UK Engineering and
Physical Science Research Council (EPSRC) under Contract EP/S013113/1 and
the National Natural Science Foundation of China under Grant No. 61176008.
(Corresponding author Yi Wang; Yang Gao).
Y. Gao and L. Li are with the School of Physics (Microelectronics),
Zhengzhou University, Zhengzhou, 450052, China. (e-mail:
gaoyang678@outlook.com).
F. Zhang is with School of Physics, University of Electronic Science and
Technology of China, Chengdu, 610054, China.
X. Lv and J. Liu are with Beijing Key Laboratory of Millimeter Wave and
Terahertz Technology, Beijing Institute of Technology, Beijing, 100089,
China.
To overcome the ohmic loss caused by the planar matching
circuit as frequencies go higher, various low-loss 3-D
waveguide structures have been used in millimeter-wave and
sub-millimeter wave applications. Among them, SIW
components are prevalently applied mainly due to their
advantages of high Q-factors (low loss). Besides, their
properties of low cost, compactness and easy manufacture have
also attracted considerable attention, leading to the widespread
use of SIW [8]-[11]. There has been extensive study of passive
SIW components, such as filters [9]-[11], antennas [12],
transitions [13], couplers [14], and power dividers [15].
Integration on of SIW technology with active devices, however,
has not progressed as rapidly. Most SIW components are
designed separately such as the antenna, filter, transition and
amplifier components in [16]-[20].
Here, the SIW filters and the amplifier are integrated in a
more compact manner to form the filter-amplifier, yielding
reduced loss, simplified circuit structure, and reduced device
volume. Fig. 1 illustrates the two approaches of filter-amplifier
integration: Fig. 1(a) is the conventional amplifier cascaded
with filters, where the input matching network (IMN) and
output matching network (OMN) are employed to attain the 50-
Ω ports. In Fig. 1(b), the matching networks (IMN and OMN)
are replaced by the input and output matching filters in the
amplifier. This integrated filter-amplifier can be schematically
represented using a coupling topology in Fig. 1(c). Here the
triangle denotes the transistor, the black circles denote the
resonators and the white circles denote the source and load. It
is very important to note that there exists feedback coupling
between the input and output of the transistor. This has been
neglected in all the previous treatment of filter-amplifier
integration [21]-[24].
A few attempts have been made on the co-design of SIW
filter-amplifiers [20]-[22]. In [20], SIW-based filter was
combined with the amplifier to suppress up to the fourth
harmonic. In [21] and [22], SIW filters were integrated at the
C. Guo is with the Department of Information and Communication
Engineering, Xi’an Jiaotong University, Xi’an 710049, China.
X. Shang is with the National Physical Laboratory, Teddington, TW11
0LW, U.K.
Y. Liu is with National center for International Joint Research of Electronic
Materials and Systems, School of Information Engineering, Zhengzhou
University, Zhengzhou, 450052, China.
Y. Wang, Y. Gao and M. J. Lancaster are with the Department of Electronic,
Electrical and Systems Engineering, University of Birmingham, Birmingham
B15 2TT, U.K. (y.wang.1@bham.ac.uk, m.j.lancaster@bham.ac.uk).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier XXXXXXXXXXX
Substrate Integrated Waveguide Filter Amplifier
Design using Active Coupling Matrix Technique
Yang Gao, Fan Zhang, Xin Lv, Cheng Guo, Xiaobang Shang, Lei Li, Jiashan Liu, Yuhuai Liu,
Yi Wang, IEEE, Senior Member, and Michael J. Lancaster, IEEE, Senior Member
A
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2
output of the transistor and Chebyshev filter response was
achieved. Integrated waveguide filter amplifiers were reported
in [23] and [24]. The filter was only coupled to the input of the
transistor in [23]. This work was followed by [24] where filters
are integrated at both the input and output of the amplifier. Most
previous work only incorporates filters to match transistors at
either the input or output. However, as a more general case,
transistors can be integrated with filters on both sides, which is
what we have achieved here.
In this paper, an N+4 active coupling matrix is developed and
applied in the design of SIW integrated filter-amplifiers.
Different from the previous filter-amplifiers, this work (i)
reports the use of a new N+4 active matrix, which, for the first
time, includes the feedback of the transistor (ii) also, for the first
time, this paper derives the equations to calculate noise figures
directly using the active coupling matrix; (iii) implemented the
design using SIW, which allows easier integration with active
circuits compared to the previous work using waveguides [23],
[24]. Primary specifications regarding return loss, gain, as well
as noise figure can be described and predicted by the newly
derived N+4 coupling matrix.
Compared with the previous coupling matrix approach in
[23] and [24], this new matrix formulation makes a more
accurate representation of the transistors, as the input and
output filter as well as the feedback coupling are considered as
an entity. Without the feedback coupling, classic passive
coupling matrix synthesis [25] and previous active coupling
matrix [23], [24] are not able to predict amplifier responses
precisely. For example, the simulated return loss from the initial
extracted physical dimensions is only about 5 dB in the
Fig. 1. (a) Conventional cascaded filters and amplifier with 50 interfaces
between them; (b) The proposed integrated filter-amplifier, with feedback
coupling between the drain and gate; (c) The coupling topology representation
of (b).
(a)
(b)
Fig. 2. Lumped equivalent circuit and coupling matrix of the transistor with input and output resonators matching. (a) Schematic representation of the circuit. (b)
Coupling matrix representation of the filter-amplifier.
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3
passband in [24], whereas in our new work this is about 12.5
dB, which is much closer to the 20 dB return loss specification.
These more accurate initial values of the model facilitate the
final EM optimization of the filter-amplifier. This greatly
improves the design efficiency. For the N+4 active coupling
matrix, a matrix optimization method based on local gradient
algorithm is applied. An effective procedure for filter
dimension extraction is implemented using an example of an X-
band filter-amplifier.
This paper is organized as follows. Section II introduces the
concept of the new N+4 active coupling matrix, where its
relationship with noise figure is derived. Section III presents the
optimization of N+4 matrix using a local gradient optimization
algorithm to produce Chebyshev response. Section IV describes
the physical design using the N+4 coupling matrix, with a
prototype X-band SIW amplifier-filter. This is followed by the
fabrication and the measurement results in Section V. The
conclusions and discussions are in Section VI.
II. THE ACTIVE N+4 COUPLING MATRIX
The lumped circuit representation of resonators matching the
input and output of the transistor is illustrated in Fig. 2(a). The
parallel LC resonators are coupled through J inverters. There
are N resonators in total: the k
th
resonator is coupled to the input
of the transistor at the gate and the (N-k)
th
resonator is coupled
to the drain at the output of the transistor. G
P1
and G
P2
are the
source and load conductance; Y
gs
, Y
gd
, and Y
ds
are the
admittances between the transistor’s gate, source and drain, and
g
m
is the transconductance. Y
S
is the input admittance seen by
the source and Y
L
is the input admittance seen by the load. By
applying Kirchoff’s laws in the lumped circuit in Fig. 2(a) and
the matrix scaling process described in [25], the active N+4
coupling matrix can be constructed [24] and is illustrated in Fig.
2(b). This N+4 active coupling matrix is normalized to a 1-Ω
system as the port admittance is taken as 1. p is the complex
frequency variable defined as [22]
0
0
1
pj
FBW
(1)
There are two submatrices in Fig. 2(b) corresponding to the
input and output matching filters. 𝑌
gs
, 𝑌
gd
, 𝑔
m
and 𝑌
ds
are the
normalized parameters. m
i,j
is the inter-resonator coupling and
m
P1,1
= m
1,P1
and m
P2,4
= m
4,L
are the external couplings from the
resonators to the source and load. The self-coupling m
i,i
results
from a frequency offset of the resonator i from the central
frequency of the circuit, m
2,in
= m
in,2
and m
out,3
= m
3,out
are the
couplings between the transistor’s gate or drain with its adjacent
resonator. m
in,out
and m
out,in
are the couplings between the
transistor’s input and output and are given in terms of the
normalized transistors parameters as [19],
,,
,
gd
gd
m
in out out in
Yg
Y
mm
jj
(2)
Matrix [A] in Fig. 2(b) can be further decomposed into three
matrices [26]
A T p U j m
(3)
where the [T] contains the filter’s port admittances and the input
and output admittance of the transistor. [U] is the identity
matrix except for entries U
P1,P1
, U
in,in
, U
out,out
and U
P2,P2
, which
are zero. m is the coupling matrix with elements discussed
above.
The S-parameters for the complete filter-amplifier circuit can
be calculated using [25], [27]
11
11 1, 1 12 1, 2
11
21 2, 1 22 2, 2
2[ ] 1 2[ ] 1
2[ ] 1 2[ ] 1
P P P P
P P P P
S A S A
S A S A
(4)
The noise figure can also be calculated using the matrix [A]
and the formulas are derived as follows.
Firstly, the matrix [A] is divided into submatrices
corresponding the input and output filters. We use subscript
sub1 to denote the submatrix 1 and sub2 as the submatrix 2, as
shown in Fig. 2(b). The corresponding lumped circuit sub-
network of the input filter, shown in Fig. 3, is extracted to help
with the noise figure derivation. A current i
S
is set up as an
excitation source. The voltages at the nodes P
in
and P
1
are
denoted as v
in
and v
P1
; the voltage at the n
th
resonator is denoted
by v
n
(n = 1 to k). Applying Kirchoff’s law at each node, the
following matrix form equations can be written:
1 1,1
1
1, 1 1,2
1
2,1
,
,
0 0 0
0
' 0 0
0
0 ' 0 0
00
0 0 0 '
0
0 0 0
PP
P
P
k in
k
k in gs gd
Pin S
G jJ
v
jJ p jJ
v
jJ p
p jJ
v
jJ Y Y
vi
L
L
L
M M M O M M
MM
L
L
(5)
or
1sub
Y v i
(6)
where
'p p C FBW
(7)
The voltage at P
in
can be calculated by
1
1
,
Pin S sub
Pin Pin
v i Y
(8)
Note that input admittance seen from the current source i
S
is
given by
S
S gs gd
in
i
Y Y Y
v
(9)
Substituting (8) for (9) yields
Fig. 3. Lumped circuit sub-network representation of the input filter.
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4
1
1
,
1
in in
S gs gd
sub
PP
Y Y Y
Y
(10)
The next step is to find the relationship between [Y
sub1
] and
[A
sub1
]. Note that in the lumped circuit in Fig. 2(a) and Fig. 3
the port admittances G
P1
and G
P2
are not normalized, whereas
in coupling matrix [A] of Fig. 2(b) the port admittances are
normalized to identity. Applying narrow band approximation,
[𝐴
𝑠𝑢𝑏1
]
𝑃𝑖𝑛,𝑃𝑖𝑛
−1
can be expressed by [25]
11
1 0 1
,,
in in in in
sub sub
P P P P
A Y Y
(11)
The input admittance seen into the source Y
S
can be
calculated by substituting (11) into (10), giving
0
1
1
,
1
in in
gs gd
S
sub
PP
Y Y Y Y
A
(12)
After Y
S
is obtained, the noise figure can be calculated using the
noise figure model [4],
2
Re
N
min S opt
S
R
NF NF Y Y
Y
(13)
resulting in
2
0
1
1
,
0
1
1
,
1
1
Re
in in
in in
gs gd
N opt
sub
PP
min
gs gd
sub
PP
R Y Y Y Y
A
NF NF
Y Y Y
A
(14)
Here, Y
opt
is the optimum source admittance that results in
minimum noise figure, NF
min
is the minimum noise figure of the
transistor that occurs when Y
S
= Y
opt
. R
N
is the equivalent noise
resistance of the transistor.
By incorporating the transistor parameters, the integrated
filter-amplifier can be described using the active N+4 coupling
matrix. Both the S-parameter response and the noise figure of
the filter-amplifier can be calculated using the N+4 matrix.
III. THE ACTIVE N+4 MATRIX OPTIMIZATION
Generally, the input and output impedances of the amplifier
are complex. Previous research [21]-[23] showed that by
adjusting the couplings strength and center frequency the
input/output resonators can be matched to the complex
impedance of the transistor. In [24], a more general case was
presented, where the amplifier was matched at the input and
output ports. However, the feedback between the gate and
drain, represented as Y
gd
in the models shown in Fig. 2 and Fig.
3, was neglected. In [24], the absence of Y
gd
results in the
inaccurate calculated gain and initial dimensions of the physical
construction. Inclusion of the feedback coupling offers more
accurate coupling matrix, nevertheless, presents a difficulty for
the matching of the transistor.
To make a filter-amplifier with desired filtering response at
both the input and output of the transistor, a local optimization
method based on the gradient algorithm can be utilized. The
coupling matrix [m] can be produced achieving target response.
The goal of the optimization is to minimize a scalar Cost
Function (CF), by modifying the values of elements of [m]. The
cost function is formulated to quantify the difference between
the optimized results and the desired filter response. In this
paper, we apply the Chebyshev filter as an example, but the
technique can be generalized to other filter types. Some critical
characteristic points are chosen to form the cost function,
including the reflection zeros (RZ), the reflection poles within
the pass-band (RP), and the equal-ripple passband edges (BE).
Fig. 5 On-chip transistor circuit schematic of the filter-amplifier.
Fig. 6 Change of the cost function value with each iteration.
Fig. 4. The critical points of a filter having Chebyshev response.
