0  This is the author’s version of an article that has been published in this
journal. Changes were made to this version by the publisher prior to
publication.
The final version of record is available at
http://dx.doi.org/10.1109/TAP.2020.3016422

# Compact Uniform Circular Quarter-Wavelength Monopole Antenna Arrays with
Wideband Decoupling and Matching Networks

Jonas Kornprobst, , Thomas J. Mittermaier, , Raimund A. M. Mauermayer, ,
Gerhard F. Hamberger, Matthias G. Ehrnsperger, , Bernhard Lehmeyer, Michel T.
Ivrlač, Ulrik Imberg, Thomas F. Eibert, , and Josef A. Nossek Manuscript
received December 21, 2019; revised June 14, 2020; accepted June 30, 2020;
date of this version July 6, 2020. This work was supported in part by the
Huawei Technologies Sweden AB, 164 94 Kista, Sweden. _(Corresponding author:
Jonas Kornprobst.)_ J. Kornprobst, T. J. Mittermaier, R. A. M. Mauermayer, G.
F. Hamberger, M. Ehrnsperger and T. F. Eibert are with the Chair of High-
Frequency Engineering, Department of Electrical and Computer Engineering,
Technical University of Munich, 80290 Munich, Germany (e-mail:
<EMAIL_ADDRESS>hft@ei.tum.de).M. Lehmeyer, M. T. Ivrlač and J. A. Nossek
are with the Associate Professorship of Signal Processing Methods, Department
of Electrical and Computer Engineering, Technical University of Munich, 80290
Munich, Germany.U. Imberg is with Huawei Technologies Sweden AB, 164 94 Kista,
Sweden.Color versions of one or more of the figures in this article are
availableonline at https://ieeexplore.ieee.org.Digital Object Identifier
10.1109/TAP.2020.3016422

###### Abstract

Two novel decoupling and matching networks (DMNs) in microstrip technology for
three-element uniform circular arrays (UCAs) are investigated and compared to
a more conventional DMN approach with simple neutralization lines. The array
elements are coaxially-fed quarter-wavelength monopole antennas over a finite
groundplane. Three-element arrays are considered since UCAs with an odd number
of elements are able to provide an almost constant maximum array factor over
the whole azimuthal angular range. The new designs are explained from a
theoretical point of view and their implementations are compared to four- and
three-elements UCAs without DMN in terms of decoupling and matching bandwidth
as well as beamforming capabilities. In addition to excellent decoupling and
matching below $\bm{-}\bm{1}\bm{6}$ dB, a broader bandwidth is obtained by the
two DMNs. The reasons for the enhanced bandwidth are similar in both cases: By
introducing several circuit elements offering additional degrees of freedom,
matching of the monopole input impedances at different frequencies becomes
feasible. One of the presented designs offers a larger bandwidth, while the
other design is able to provide a better total efficiency. Scattering
parameters, radiation patterns, beamforming capabilities, and enhanced gain
are all verified by measurements over the operating bandwidth.

###### Index Terms:

5G, antenna arrays, beamforming, decoupling and matching network (DMN),
quarter-wavelength monopole.

##  I Introduction

Future wireless communication systems — such as the next generation mobile
wireless network 5G — strive for ever-increasing data rates for more and more
users which are served simultaneously. In addition to higher radio frequencies
(up to mm-waves) and more bandwidth (e.g., carrier aggregation across
frequency bands), one key technology will be to increase the number of
antennas in (massive) multiple-input multiple-output (MIMO) systems, and to
employ advanced beamforming techniques to serve multiple users with spatial
multiplexing [1, 2, 3, 4, 5, 6, 7]. In particular for the lower GHz range with
wavelengths comparable to the size of the mobile devices, it is of utmost
importance to keep the size of the mobile antenna array systems reasonably
small. On the side of the base station, size is not that critical and it might
be more beneficial to match and decouple the array elements as good as
possible. In the mm-wave bands, an increasing number of array elements
together with beamforming can be an approach towards a sufficiently large gain
to overcome the path loss [7, 8, 9, 10, 11, 12, 13, 14, 15, 16].

In the following, we will focus on compact antenna arrays for the S-band
(around $3.6$ GHz), which is of interest for LTE and 5G and which is already
under investigation for various antenna designs [17, 18, 19, 20, 21, 22, 23,
24, 25, 26]. Compact arrays are, by their literal meaning, based on very
closely spaced antenna elements. In particular for a separation distance
smaller than half a wavelength, the radiating elements need to be small enough
to allow for the narrow spacing. Furthermore, super-directive effects might
occur, which make efficiency and gain considerations more important than the
bare look on the radiation patterns [27, 28, 29, 30]. Very important is that
full beamforming capabilites are still available and no gain loss occurs due
to a poor radiation efficiency [31].

An important topic of the presented work is to investigate the coupling
effects of compact arrays and possibly take advantage of them [32, 33, 34].
The coupling effects are tackled by the design and realization of decoupling
and matching networks (DMNs). Matching commonly means (complex-conjugate)
power matching to a given reference impedance, where 50 $\Omega$ is a typical
widely employed value which is taken throughout this paper. Other matching
strategies are of course also possible and meaningful, for instance noise
matching for low-noise amplifiers (LNAs) in the receive case, where the
optimal impedance may even assume complex values [35].

For the considered DMN realizations, we have transmit scenarios in mind.
Therefore, matching as well as decoupling levels of $-16$ dB, i.e., for all
scattering parameters $S_{ij}$, are requested to reduce the influence of
scattered waves onto the transmit power amplifiers (PAs), which may be
disturbed in their operating point leading in particular to excitation
dependent nonlinearities. Moreover, good matching and decoupling is essential
for a good efficiency of the system. Thereby, decoupling of the array does not
mean to cure mutual coupling effects such as modified current distributions on
the antenna elements or shadowing effects. The goal is rather to provide
decoupled and matched ports to the transmit power amplifiers.

Good decoupling is often achieved by orthogonal geometrical configurations
[36, 37, 38]. Other possibilities include electromagnetic bandgap and
metamaterial structures [39, 40], modified ground structures [41, 42, 43, 21,
44], and coupled parasitic elements [45]. Also, a lot of literature focuses on
two-element (super-directive) arrays [46, 47, 48, 49, 50, 51], which are
interesting but not sufficient for many applications. For more elements, the
eigenmode excitations and conjugate-matching techniques are well-known [52,
53, 54]. Tunable DMNs can eliminate the necessity of broadband antenna arrays
[55].

In this paper, we focus on compact uniform circular arrays (UCAs). Considering
super-directivity, a maximum directivity on the order of
$\mathcal{O}(N^{3/2})$ is possible with UCAs comprising $N$ elements [56]. We
investigate three and four element UCAs with quarter-wavelength monopole
radiators over a finite ground plane, and demonstrate that three elements are
preferable. If the maximum beamforming gain in any azimuthal direction is
determined for ideal omnidirectional antenna elements, i.e., the array factor,
it is found that UCAs with an even number of elements show gain fluctuations
while odd-numbered UCAs do not show this behavior and reach a rather constant
maximum gain in all azimuthal directions. Interesting to note is that the
directivity into the minimum-gain directions of an even-numbered UCA can be
realized in all directions using an odd-numbered UCA with just one element
less. Related results are found in literature, but the behavior has not been
studied in detail [57, 58, 59].

In our study, the employed monopole antenna elements are fed from the rear
side of the ground plane via a coaxial transmission line, offering the
opportunity to implement decoupling and matching networks in microstrip
technology.

The paper is organized as follows. In Section II, the performance differences
of UCAs with even and odd numbers of elements are investigated with focus on
the three and four element cases. Then, three different DMNs are analyzed in
Section III from a theoretical point of view for the three-element scenario:
The first one is a state-of-the-art approach with rather simple neutralization
transmission lines. The second one is a rather complicated two-stage network
with the largest decoupling and matching bandwidth, but with the downside of a
slightly reduced radiation efficiency. The second one features several
decoupling paths in a star-triangle configuration and offers a superior
efficiency with a slightly increased bandwidth. The specific design procedure
for the simulation, the fabrication, and scattering parameter measurements for
these three DMNs is discussed in Section IV. Finally, all simulations of the
DMN designs are verified by measurements in Section V, for radiation patterns,
gain, and beamforming vectors.

##  II UCAs with Even and Odd Numbers of Elements

Towards the design of a compact UCA with an optimal number of elements, we
investigate some general properties of ideal UCAs first. In the following
array factor considerations, it becomes clear why a UCA with an even number of
elements should be replaced by a UCA with one element less. After introducing
an antenna setup based on monopoles, which is used throughout this work, the
claims are also validated by full-wave simulations and measurements.

###  II-A Theoretical Investigations with Ideal Radiators

We consider a scenario with ideal radiators for an array-factor description,
where we assume a suppressed $\mathrm{e}^{\,\mathrm{j}\omega t}$ time
dependence with angular frequency $\omega$ throughout this paper. An ideal
$N$-element UCA on a circle with radius $k_{0}r_{\mathrm{uca}}=0.2\uppi$ is
considered, see Fig. 1,

Figure 1: Sketch of a UCA in the xy-plane.

and we obtain the array factor [60]

$\mathit{AF}=\sum\nolimits_{n=0}^{N-1}a_{n}\mathrm{e}^{\,\mathrm{j}k_{0}r_{\mathrm{uca}}\sin\vartheta(\cos\varphi\cos({2\uppi}n/{N})+\sin\varphi\sin({2\uppi}n/{N}))}$
(1)

with the complex element excitations $a_{n}$, the free-space wavenumber
$k_{0}=\omega\sqrt{\varepsilon_{0}\mu_{0}}=2\uppi/\lambda_{0}$, and the
corresponding free-space wavelength $\lambda_{0}$. The array elements are
assumed to be quarter-wavelength monopole antennas oriented along the
$z$-axis. Their radiation characteristic — idealized without mutual influence
— reads [60]

$C(\vartheta,\varphi)=\begin{cases}\cos(\uppi/2\cos\vartheta)/\sin\vartheta&0\leq\vartheta<\uppi/2\\\
0&\uppi/2\leq\vartheta\leq\uppi\\\ \end{cases}\,.$ (2)

With the knowledge of this setup, we can calculate optimal beamsteering
vectors for each direction in space according to [61]. Considering the
$\vartheta=$70\text{\,}\mathrm{\SIUnitSymbolDegree}$$ cut, we determine the
maximum array gain for each azimuthal direction $\varphi_{0}$ as the product
of single-element gain and array factor. These gain graphs are shown in Fig. 2

Figure 2: Array gain of UCAs with different numbers of idealized quarter-
wavelength monopoles with optimal azimuthal beamsteering in the
$\vartheta=70^{\circ}$ cut.

for UCAs with 1 to 8 elements and radius $r_{\mathrm{uca}}=0.1\lambda_{0}$. We
observe, as announced in the introduction that the achievable array
gains111Speaking of gain, we refer to the realized gain throughout this paper
both for simulation and measurement results. For the theoretical
investigations with uncoupled radiators, the realized gain equals the IEEE
gain and the directivity. with an odd number $N$ of elements are independent
of the desired steering angle. However, for even numbers $N$, the achievable
gain fluctuates between the values achievable with $N-1$ and $N+1$ elements.

So far, gain is equal to directivity. Major differences will be observed not
only with non-ideal radiation efficiencies and with non-ideal non-
omnidirectional patterns, but also if the input impedances of the array
antennas are considered together with the active reflection coefficients
dependent on the different beamforming vectors. Direction-dependent gain drops
may be observed due to the varying mismatch.

The last non-ideality is integrated into the theoretical analysis, bringing
the array model one step closer to reality. The individual monopole antennas
are now canonical minimum scattering antennas. The azimuthal gain pattern is
still omnidirectional, even though coupling impedances between the antenna
elements are observed [62]. Next, we implement power matching with a DMN,
which is assumed to be perfect for now. The gain in the $\vartheta=70^{\circ}$
cut with and without DMN is shown in Fig. 3. Other $\vartheta$-cuts show the
same behavior, but are influenced by the varying single-radiator gain.

(a)  (b)

Figure 3: Array gain of UCAs with different numbers of minimum-scattering
quarter-wavelength monopoles. (a) No matching circuit. (b) Ideal DMN for power
matching.

The gain (or better directivity) curves from Fig. 2 are exactly reproduced
with the DMN. Without DMN, the mismatch loss reduces the gain to small values,
where the different numbers of UCA-elements are almost indistinguishable. The
model employed for Fig. 3 does not consider reduced radiation efficiencies and
distorted radiation patterns due to closely placed radiators.

###  II-B Setup of Real-World Implementations

In order to justify the three-element approach for the subsequent designs also
by full-wave simulations and measurements, we describe the design,
fabrication, and measurement of three-element and four-element compact antenna
arrays without decoupling network. The described monopole configuration
exhibits horizontally omnidirectional beamforming capabilities, but a fixed
directivity and beamwidth in the vertical patterns.

The UCAs consisting of quarter-wavelength monopoles are located on a circle of
a certain radius $r_{\mathrm{uca}}$, see Fig. 4.

(a)  (b)

(c)  (d)

Figure 4: Sketch of UCAs built with monopole antennas over a finite circular
groundplane. (a) Three elements. (b) Four elements. (c) Back side with
feeding, no DMN. (d) Photograph of built three-element array.

The distance between the array elements

$d=r_{\mathrm{uca}}\sqrt{2-2\cos\upDelta\varphi}$ (3)

is known from geometrical considerations, where $\upDelta\varphi$ is the angle
between the antennas. The monopoles on the circular metallic ground plane with
diameter $d_{\mathrm{gp}}=100\,\textrm{mm}$ and thickness of
$t_{\mathrm{gp}}=5\,\textrm{mm}$ exhibit a radius
$r_{\mathrm{m}}=1.5\,\textrm{mm}$ and a height $h\approx 0.25\lambda_{0}$,
with the free-space wavelength $\lambda_{0}$, where $h$ is carefully adjusted
in each realization for a resonance frequency of $3.6\,$GHz. The coordinate
system is defined by the positive $z$-axis pointing into the direction of the
monopoles, away from the ground plane. The feeding through the groundplane is
realized in a coaxial fashion: the outer coaxial radius is
$r_{\mathrm{d}}=2.5\,\textrm{mm}$ and the dielectric is Teflon with relative
permittivity $\varepsilon_{\mathrm{r}}=2.2$. This results in a characteristic
line impedance of about 20$\,\Omega$, which matches the input impedance of the
monopoles and does not reduce the impedance bandwidth. The theoretically
expected monopole input impedance of $36.5$ $\Omega$ [60] is reduced because
of the monopole thickness and the array coupling. On the rear side of the
groundplane, the feed transitions into a microstrip structure on a low-loss
RO3003 substrate [63] of thickness $t_{\mathrm{s}}$ and
$\varepsilon_{\mathrm{r}}=3$ via a pin of radius
$r_{\mathrm{pin}}=0.5\,\textrm{mm}$, which is soldered to a copper matching
disk. The capacitance of this disk can compensate for the inductive behavior
of the coaxial-microstrip transition and is tuned for each realization of the
UCA. Except otherwise mentioned, $t_{\mathrm{s}}=0.508\,$mm. For the following
UCAs without DMN, a linear taper matches the impedance to 50$\,\Omega$.
Measurements are conducted at an SMA connector, which is placed with a defined
distance of $g=$1\text{\,}\mathrm{m}\mathrm{m}$$ above the microstrip line.

###  II-C Characterization of the Three-Element Prototype

The three-element monopole array was built with a distance $d=0.17\lambda_{0}$
(i.e., $r_{\mathrm{uca}}=0.1\lambda_{0}$) and $h=0.22\lambda_{0}$. A
photograph of this prototype during S-parameter measurement is shown in Fig.
4(d). The S-parameters of this array are given in Fig. 5, where full-wave
simulation results and measurement results from a vector network analyzer
(VNA) show excellent agreement.

Figure 5: Scattering parameters of three-element monopole array.

All simulations were performed with the frequency domain solver of Computer
Simulation Technology Microwave Studio (CST MWS) [64]. The input matching of
all three ports reaches well below $-20$ dB at $3.6$ GHz, and the $-16$-dB
bandwidth is larger than 150 MHz. The coupling between the array feeding ports
reaches up to about $-9$ dB; a value considered too large for practically used
antenna arrays.

The radiation and total efficiencies are $-0.01$ dB and $-1.28$ dB in the
simulation. The poor total efficiency is explained by the mutual array element
interactions. The simulated single element patterns show a realized gain of
$5.09$ dBi. The realized gain is enhanced to values between $6.5$ dBi and $7$
dBi if optimal beamforming based on radiation patterns and S-parameters is
performed, i.e., the azimuthal gain is maximized in the
$\vartheta=70^{\circ}$-cut. The far fields were measured in the anechoic
chamber of the Chair of High-Frequency Engineering, Technical University of
Munich and the resulting gain radiation patterns are shown in Fig. 6 [65, 66].

(a)  (b)

(c)  (d)

Figure 6: Comparison of simulated and measured active element and array
patterns at a $\vartheta=70^{\circ}$ cut. (a) Meas. three-element UCA
individual patterns. (b) Meas. four-element UCA individual patterns. (c)
Three-element UCA for beamsteering in $\varphi_{0}=300^{\circ}$ direction. (d)
Four-element UCA for beamsteering in $\varphi_{0}=300^{\circ}$ direction.

Optimal beamforming weights for the $\vartheta=70^{\circ}$ cut were calculated
for an exemplary azimuthal beamsteering angle of $\varphi_{0}=300^{\circ}$ and
fed into the array via an analog beamforming network, see Fig. 7. The
measurement results are corrected for the measured efficiency of the feeding
network to obtain the realized gain of the array with the beamforming vector.
The attained patterns agree very well with the corresponding patterns from CST
MWS simulation, see Fig. 6(c); the gain deviation is explained by the
measurement uncertainties and also by deviations of the measurement setup as
compared to the simulation model, e.g., the imperfect modeling of the
absorbers behind the ground plane as displayed in Fig. 7(b).

(a)

(b)

Figure 7: UCA measurement setup. (a) Analog beamforming network during VNA
measurement. (b) Four-element UCA mounted in anechoic chamber.

###  II-D Characterization of the Four-Element Prototype

The four-element UCA was designed to have the monopoles placed on a circle
with the same radius $r_{\mathrm{uca}}=0.1\lambda_{0}$ as for the one with
three elements. Therefore, the element distance is now slightly smaller with
$d=0.14\lambda_{0}$ and $h$ is still $0.22\lambda_{0}$.

A photograph of the fabricated prototype is shown in Fig. 7(b), where the
prototype is mounted on the positioner in the antenna measurement chamber. The
scattering parameters of the prototype are seen in Fig. 8, where full-wave
simulation results and measurement results are compared with good agreement.
The input matching of the four ports is again very good, but the coupling
$S_{ij}$ between the ports is a bit stronger (both in simulation and
measurement) than in the case of the three-element array. Also, three coupling
curves are observed for each fed port: two identical ones for neighboring
elements and one for elements across the diagonal.

Figure 8: Scattering parameters of four-element monopole array. The first,
blue $S_{ij}$ crosstalk curve is for neighbouring-element crosstalk, e.g.,
$S_{12}$, while the second, green one is for opposite array elements, e.g.,
$S_{14}$, see Fig. 7(b).

The radiation and total efficiencies are $-0.06$ dB and $-2.46$ dB in
simulation, which is worse than for the three-element array. Hence, the
simulated realized gain drops to $3.84$ dBi. The measured gain patterns are
given in Fig. 6(b). In Fig. 6(d), the beamformed gain pattern for the
exemplary direction $\varphi_{0}=300^{\circ}$ goes up to a very similar level
as for the three-element UCA.

###  II-E Beamforming Gain for Azimuthal Scans with Three- and Four-Element
UCAs

To summarize the investigations of Section II, the azimuth angle dependent
beamforming behavior for a fixed polar angle cut of $\vartheta=70^{\circ}$ is
investigated for all so-far presented UCAs in Fig. 9.

Figure 9: Optimum array gain dependent on the scan angle for simulations and
measurements of UCAs with $N=3$ and $4$ elements and with
$r_{\mathrm{uca}}=0.1\lambda_{0}$.

The comparison involves simulations and measurements of the three- and four-
element UCAs with radius of $0.1\lambda_{0}$. For the angle $300^{\circ}$,
the displayed gains agree with those shown in Figs. 6(c) and (d).

Dependent on the radiation angle, the optimized array gains exhibit some minor
oscillations even for the three-element UCA, which is due to the variations in
the mismatch and in the single-element patterns of the array (influenced by
the finite groundplane, the backlobes, and the mainlobe direction). This
effect should be considered separately from the array factor oscillations
studied in this section. The array gain of all UCAs is on the same level just
below 7 dBi, which is a bit lower than observed in Fig. 3. It remains to study
how real DMNs influence the gain, in particular towards the achivable
bandwidth and efficiency.

As outlined in Section IIA, UCAs with odd numbers of array elements should be
preferred over even numbered ones, if each possible beamsteering direction is
equally important for the realizable array gain.

##  III Theoretical Analysis of Decoupling and Matching Networks

In the following, three DMNs for the presented three-monopole UCA are
discussed. Since the environment of each monopole is the same, the three
antenna excitation ports A1–A3 form a symmetric three-port, see Fig. 10(a),
which is for instance modeled with input impedances $Z_{\mathrm{in}}$ and
coupling impedances $Z_{\mathrm{c}}$. We describe three designs of a
decoupling and matching network (DMN) that can be used in conjunction with
such a symmetric three-port; two of them show the very same three-port
symmetry and the third exhibits a rotational symmetry. In principle, all of
them match $Z_{\mathrm{in}}$ to $Z_{0}=50\,\Omega$ and minimize the influence
of $Z_{\mathrm{c}}$. For the theoretical analysis, fixed input and coupling
impedances are assumed.

(a)  (b)

Figure 10: Schematic of the neutralization lines DMN. For the sake of
readability, the ground connections of all ports are omitted in this Figure.
(a) Equivalent circuit of the symmetric three-element UCA; this circuit
matches an impedance-matrix description of the UCA. (b) Neutralization lines
DMN. The equivalent circuit of (a) is connected at the ports A1–A3.

###  III-A DMN with Neutralization Transmission Lines

Realizing a DMN with neutralization lines is a well-known technique [54]; we
briefly summarize the concept in the following and explain the special
considerations of our implementation. In Fig. 10(b), the basic structure of
this kind of DMN is shown. First, the antenna modelled as in Fig. 10(a) is
connected to the ports A1–A3. The coupling is eliminated by the transmission
lines $\mathit{TL}_{\mathrm{dec}}$ connecting the antenna ports. Additionally,
the lines $\mathit{TL}_{\mathrm{ant}}$ allow for impedance and phase
adjustments. Second, the connection and matching to the 50-$\Omega$ ports
P1–P3 is realized by the transmission line $\mathit{TL}_{\mathrm{port}}$.
Exploiting the symmetry of the UCA, the DMN is also symmetric.

###  III-B Two-Stage DMN

For the next design, we start with a lumped-element decoupling circuit which
shows a rotational symmetry. We will demonstrate the ability of this network
to decouple and match a symmetric three-port and derive an equivalent circuit
with transmission lines for a single frequency. The proposed DMN circuit is a
linear, reciprocal, and (ideally) loss-less DMN six-port, see Fig. 11. If the
ports A1–A3 are connected to the antenna array excitation ports, the ports
P1–P3 are decoupled and present a prescribed impedance, i.e., they are
matched. The design of the DMN can be realized using planar micro-strip
technology: only connections which do not lead to cross-overs of transmission
lines are involved.

Figure 11: Network structure of the proposed decoupling and matching six-port.
All ports have ground connections which are not shown. A1–A3 are the antenna
ports and P1–P3 are the decoupled and matched feeding ports.

The DMN has already been presented in [67].

The proposed DMN circuit exhibits five real-valued degrees of freedom — the
susceptances $B_{1}$ to $B_{5}$. In total, it consists of 15 loss-less
reactances. Numbering the ports A1–A3 and P1–P3 from 1 to 6, the DMN is
described by the admittance matrix

$\bm{Y}_{\mathrm{DMN}}=\mathrm{j}\begin{bmatrix}\bm{A}&\bm{B}^{\mathrm{T}}\\\
\bm{B}&\bm{C}\end{bmatrix}\,,$ (4)

with the real-valued sub-matrices

$\bm{A}=\begin{bmatrix}2B_{1}+\sum\nolimits_{k=2}^{4}B_{k}&-B_{1}&-B_{1}\\\
-B_{1}&2B_{1}+\sum\nolimits_{k=2}^{4}B_{k}&-B_{1}\\\
-B_{1}&-B_{1}&2B_{1}+\sum\nolimits_{k=2}^{4}B_{k}\end{bmatrix}\,,$ (5)
$\bm{B}=\begin{bmatrix}-B_{2}&0&-B_{3}\\\ -B_{3}&-B_{2}&0\\\
0&-B_{3}&-B_{2}\end{bmatrix}\,,$ (6) $\bm{C}=(B_{2}+B_{3}+B_{5})\bm{1}_{3}\,,$
(7)

where $\bm{1}_{3}$ is the $3\times 3$ identity matrix.

Connecting the antenna array at A1–A3 (modeled by the admittance matrix
$\bm{Y}_{\\!\mathrm{A}}$, see Fig. 10(a)), we obtain the admittance matrix

$\bm{Y}=\mathrm{j}\bm{C}+\bm{B}\,(\bm{Y}_{\\!\mathrm{A}}+\mathrm{j}\bm{A})^{-1}\bm{B}^{\mathrm{T}}\,.$
(8)

Since the antenna array is symmetric, $\bm{Y}_{\\!\mathrm{A}}$ has the
structure

$\bm{Y}_{\mathrm{A}}=\bm{Z}_{\mathrm{A}}^{-1}=\left[\begin{array}[]{ccc}\alpha&\beta&\beta\\\
\beta&\alpha&\beta\\\
\beta&\beta&\alpha\end{array}\right]\,,\qquad\textrm{with
}\alpha,\leavevmode\nobreak\ \beta\leavevmode\nobreak\
\in\mathds{C}\,\Omega^{-1}\,,$ (9)

which depends on the symmetric antenna array input and coupling impedances as

$\bm{Y}_{\mathrm{A}}=\frac{1}{Z_{\mathrm{in}}Z_{\mathrm{c}}}\left[\begin{array}[]{ccc}\\!\\!2Z_{\mathrm{in}}\\!+\\!Z_{\mathrm{c}}\\!\\!&-Z_{\mathrm{in}}&-Z_{\mathrm{in}}\\\
-Z_{\mathrm{in}}&\\!\\!2Z_{\mathrm{in}}\\!+\\!Z_{\mathrm{in}}\\!\\!&-Z_{\mathrm{in}}\\\
-Z_{\mathrm{in}}&-Z_{\mathrm{in}}&\\!\\!2Z_{\mathrm{in}}\\!+\\!Z_{\mathrm{c}}\\!\\!\end{array}\right].$
(10)

Choosing the matrix $\bm{A}$

$\bm{A}=-\mathrm{Im}\\{\bm{Y}_{\mathrm{\\!\mathrm{A}}}\\}$ (11)

as the negative imaginary part of $\bm{Y}_{\\!\mathrm{A}}$ and using (5) and
(9), we obtain the conditions

$B_{1}=\mathrm{Im}\\{\beta\\}\,,$ (12)
$B_{2}+B_{3}+B_{4}=-\mathrm{Im}\\{\alpha+2\beta\\}\,.$ (13)

Furthermore, (8) reduces to

$\bm{Y}=\mathrm{j}\bm{C}+\bm{B}\,(\mathrm{Re}\\{\bm{Y}_{\\!\mathrm{A}}\\}^{-1})\bm{B}^{\mathrm{T}}\,.$
(14)

Expressing the inverse of the real part of $\bm{Y}_{\mathrm{A}}$ leads to

$\left(\mathrm{Re}\\{\bm{Y}_{\\!\mathrm{A}}\\}\right)^{-1}=\begin{bmatrix}a&b&b\\\
b&a&b\\\ b&b&a\end{bmatrix}\,,\quad\textrm{with }a,b\in\mathds{R}\,\Omega\,.$
(15)

It follows that

$\bm{B}\left(\mathrm{Re}\\{\bm{Y}_{\\!\mathrm{A}}\\}\right)^{-1}\bm{B}^{\mathrm{T}}=\begin{bmatrix}\gamma&\xi&\xi\\\
\xi&\gamma&\xi\\\ \xi&\xi&\gamma\end{bmatrix}$ (16)

with the two entries $\gamma$ and $\xi$ with

$\xi=B_{2}B_{3}a+\left(B_{2}^{2}+B_{2}B_{3}+B^{2}_{3}\right)b\,.$ (17)

For port decoupling, $\xi=0$ must hold and gives a condition for $B_{3}$ in
the form of

$B_{3}=-B_{2}\frac{a+b\pm\sqrt{a^{2}+2ab-3b^{2}}}{2b}\,.$ (18)

Since $B_{3}$ must be real-valued, the expression under the square-root must
be positive, i.e.,

$a^{2}+2ab-3b^{2}\geq 0\,.$ (19)

This is fulfilled for compact arrays: In our investigations, this was the case
for $d\gtrsim 0.2\lambda$. With (18) and (16), we get

$\gamma=B_{2}^{2}\frac{\left(a^{2}+ab-2b^{2}\right)\left(a+b\pm\sqrt{a^{2}+2ab-3b^{2}}\right)}{2b^{2}}\,.$
(20)

In order to obtain a matched circuit, the desired form of (14) looks like the
diagonal matrix222In this work, we only consider matching to a real impedance.
Nevertheless, the DMN is easily adapted by changing (21) for any complex PA
output or LNA input impedance if desired.

$\bm{Y}=Z_{0}^{-1}\bm{1}_{3}\,,$ (21)

where $Z_{0}$ is the prescribed value of the impedance of the decoupled ports.
This gives the two further conditions

$\gamma=Z_{0}^{-1}\,,$ (22) $B_{5}=-B_{2}-B_{3}\,.$ (23)

Eq. (20) is rewritten as

$B_{2}=\pm\sqrt{\frac{2b^{2}Z_{0}^{-1}}{\left(a^{2}+ab-2b^{2}\right)\left(a+b\pm\sqrt{a^{2}+2ab-3b^{2}}\right)}}\,\,,$
(24)

where the $+$ or $-$ signs in the two $\pm$ symbols can be chosen
independently. Hence, there are up to four solutions for $B_{2}$, and we
choose whichever is more convenient with respect to the practical realization
of the circuit. Having $B_{2}$, $B_{3}$ is obtained from (18). $B_{1}$ is
already known from (12), and the final two values are given by (13) and (23).

The next step is to find an equivalent transmission line circuit for the
proposed lumped-element DMN. Therefore, we replace the floating lumped
elements as depicted in Fig. 12(a) by a

(a)  (b)

Figure 12: Modeling reactive lumped elements with transmission lines. (a)
Series reactance. (b) Equivalent model with $\lambda/4$ transmission lines.

the circuit in Fig. 12(b). We find that both are equivalent at the single
frequency where the length of the transmission lines is equal to $\lambda/4$.
The remaining task is to replace all shunt reactances by transmission lines,
which is easily done by open stubs with the well-known input admittance

$Y_{\textsc{tl}}=\mathrm{j}Z_{\textsc{tl}}^{-1}\tan(k_{\textsc{tl}}l_{\textsc{tl}})\,,$
(25)

with the wavenumber $k_{\textsc{tl}}=2\uppi/\lambda$ and the physical line
length $l_{\textsc{tl}}$. The final step to calculate the electrical line
length and impedances, and subsequently microstrip dimensions, was carried out
for the inital designs before further optimization. Without going into detail
about all transmission line values, the final DMN is given in Fig. 13.

Figure 13: Two-stage DMN from Fig. 11 realized with (possibly planar)
transmission lines without intersections. Ground connections are omitted for
readability. Stubs represent susceptances at the design frequency and serial
lines are quarter-wavelength transformers.

It features 33 pieces of transmission lines, each with a length of about a
quarter wavelength.

###  III-C Star-Triangle DMN

We follow the same procedure as in Section III _B_ , starting with a
symmetric, lossless and reciprocal sixport consisting of 12 ideal
susceptances, see Fig. 14. The DMN has already been presented in [67].

Figure 14: Star-triangle DMN, port ground omitted for readability.

We have again for the admittance matrix the same expression given in (4) and
(14) but now with

$\bm{A}=\begin{bmatrix}2B_{\mathrm{t}}+B_{\mathrm{s}}+B_{\mathrm{b}}&-B_{\mathrm{t}}&-B_{\mathrm{t}}\\\
-B_{\mathrm{t}}&2B_{\mathrm{t}}+B_{\mathrm{s}}+B_{\mathrm{b}}&-B_{\mathrm{t}}\\\
-B_{\mathrm{t}}&-B_{\mathrm{t}}&2B_{\mathrm{t}}+B_{\mathrm{s}}+B_{\mathrm{b}}\end{bmatrix}\,,$
(26)
$\bm{B}=-B_{\mathrm{b}}\bm{1}_{3}\,,\quad\bm{C}=(B_{\mathrm{a}}+B_{\mathrm{b}})\bm{1}_{3}\,.$
(27)

Figure 15: Augmented antenna array.

To add a degree of freedom for easier design, we augment the admittance matrix
of the antenna array as shown in Fig. 15 and get

$\bm{Y}_{\\!\mathrm{B}}=\left({\bm{Y}}_{\mathrm{\\!A}}^{-1}+\frac{1}{\mathrm{j}B_{\mathrm{c}}}\bm{1}_{3}\right)^{-1}=\begin{bmatrix}e&c&c\\\
c&e&c\\\ c&c&e\end{bmatrix}=\begin{bmatrix}\alpha&\beta&\beta\\\
\beta&\alpha&\beta\\\ \beta&\beta&\alpha\end{bmatrix}$ (28)

with $e=\chi-\mathrm{j}B_{\mathrm{c}}^{-1}$. The susceptance $B_{\mathrm{c}}$
is chosen to make the $\mathrm{Re}\\{\bm{Y}_{\\!\mathrm{B}}\\}$ diagonal,
because thsi allows us to decouple the ports P1, P2 and P3 by choosing

$\bm{C}=\bm{0}\,,\quad\bm{A}=-\mathrm{Im}\\{\bm{Y}_{\mathrm{\\!\mathrm{A}}}\\}\,,$
(29)

leading to

$B_{\mathrm{t}}=\mathrm{Im}\\{\beta\\}\,,$ (30)
$B_{\mathrm{b}}+B_{\mathrm{s}}=-\mathrm{Im}\\{\alpha+2\beta\\}\,,$ (31)
$B_{\mathrm{a}}+B_{\mathrm{b}}=0\,.$ (32)

For the admittance of the decoupled ports, we get

$\bm{Y}=B_{\mathrm{b}}^{2}\,\big{(}\mathrm{Re}\\{\bm{Y}_{\mathrm{B}}\\}\big{)}^{-1}=\frac{B_{\mathrm{b}}^{2}}{\mathrm{Re}\\{\alpha\\}}\bm{1}_{3}$
(33)

with

$B_{b}^{-2}\mathrm{Re}\\{\alpha\\}=Z_{0}\,,$ (34)

The condition

$\mathrm{Re}\\{\beta\\}=\mathrm{Re}\left\\{\frac{c(c-e)}{e^{3}+2c^{3}-3c^{2}e}\right\\}=0$
(35)

leads to a fourth-order equation for $e$, resulting in

$B_{\mathrm{c}}=\frac{1}{\mathrm{Im}\\{\chi\\}-\mathrm{Im}\\{e\\}}\,.$ (36)

From the four solutions for $e$, we choose a real-valued on which leas to the
most convenient $B_{\mathrm{c}}$ value. Finally, with the knowledge of
$B_{\mathrm{c}}$, (30), (31), and (32), we have completed the design.

Next, an implementation with transmission lines is discussed. The matching
step can be realized in many different (and well-known) fashions. Later on, we
employ transmission lines of suitable lengths and impedances instead of
$B_{\mathrm{a}}$ and $B_{\mathrm{b}}$.

Replacing the inner star-triangle part of Fig. 14 (consisting of the floating
susceptances $B_{\mathrm{t}}$ and the grounded susceptances $B_{\mathrm{s}}$)
by the star-triangle of the transmission lines in Fig. 16,

Figure 16: Core part of star-triangle DMN realized with transmission lines.

we have to make sure that the admittance matrix $\bm{A}$ of the lumped circuit
has to be equal to the admittance matrix

$\bm{Y}_{\mathrm{tl}}=\begin{bmatrix}2Y_{\mathrm{t}}+Y_{\mathrm{s}}-Y_{\mathrm{s}}^{\prime}&-Y_{\mathrm{t}}-Y_{\mathrm{s}}^{\prime}&-Y_{\mathrm{t}}-Y_{\mathrm{s}}^{\prime}\\\
-Y_{\mathrm{t}}-Y_{\mathrm{s}}^{\prime}&2Y_{\mathrm{t}}+Y_{\mathrm{s}}-Y_{\mathrm{s}}^{\prime}&-Y_{\mathrm{t}}-Y_{\mathrm{s}}^{\prime}\\\
-Y_{\mathrm{t}}-Y_{\mathrm{s}}^{\prime}&-Y_{\mathrm{t}}-Y_{\mathrm{s}}^{\prime}&2Y_{\mathrm{t}}+Y_{\mathrm{s}}-Y_{\mathrm{s}}^{\prime}\end{bmatrix}$
(37)

of the threeport depicted in Fig. 16, with

$Y_{\mathrm{t}}=\frac{-\mathrm{j}}{Z_{\mathrm{t}}\tan
k_{\mathrm{t}}l_{\mathrm{t}}}\,,\leavevmode\nobreak\ \leavevmode\nobreak\
Y_{\mathrm{s}}=\frac{-\mathrm{j}}{Z_{\mathrm{s}}\tan
k_{\mathrm{s}}l_{\mathrm{s}}}\,,\leavevmode\nobreak\ \leavevmode\nobreak\
Y_{\mathrm{s}}^{\prime}=\frac{-2\mathrm{j}}{3Z_{\mathrm{s}}\sin
2k_{\mathrm{s}}l_{\mathrm{s}}}\,.$ (38)

We get a condition for the diagonal matrix entries

$2\mathrm{j}B_{\mathrm{t}}+\mathrm{j}B_{\mathrm{s}}=2Y_{\mathrm{t}}+Y_{\mathrm{s}}-Y_{\mathrm{s}}^{\prime}$
(39)

and one for the off-diagonal elements

$-\mathrm{j}B_{\mathrm{t}}=Y_{\mathrm{t}}-Y_{\mathrm{s}}^{\prime}\,.$ (40)

In addition to the four possible choices of $B_{\mathrm{c}}$, there is some
design freedom hidden in (39) and (40). For the two prescribed lumped element
values, six design variables (two line impedance, two line lengths, and two
wave numbers) are to be chosen. However, this freedom is indeed necessary for
any real-world design: The transmission line lengths

$l_{\mathrm{t}}\geq d,\quad l_{\mathrm{s}}\approx d/\sqrt{3}=r_{\textrm{uca}}$
(41)

should be feasibly chosen as indicated by the antenna array element distance.
For microstrip realizations as discussed later, the substrate material and
height basically determine the propagation constants. Then, it is
straightforward to calculate the line impedances from (39) and (40).

It is worth noting that the circuit complexity of this DMN is reduced compared
to the two-stage DMN with 33 pieces of transmission lines. In Fig. 16, we have
6 pieces, where matching can be achieved by 3 additional serial lines.
Furthermore, $l_{s}\approx r_{\textrm{uca}}$ can be much shorter than half a
wavelength. Compared to the decoupling-lines DMN in Fig. 10(b), the star-
triangle DMN exhibits additional decoupling paths by the star configuration.
Implications of these additional design freedoms are revealed when discussing
the design.

##  IV DMN Implementation, Fabrication, and S-Parameter Measurement

The simulations preceding the DMN realizations have been performed on an
equivalent circuit level in Keysight Advanced Design System (ADS) [68] and as
full-wave simulations in CST MWS. The design process involves several steps
discussed in the following. Basically, this procedure was followed for each of
the DMN designs.

The two starting ingredients for the initial ADS simulations are i)
S-parameter data sets obtained by full-wave CST MWS simulations of the UCA
with microstrip feed: These data sets comprise parameter variations of the UCA
distance $d$ (with individually optimized monopole length) and of the coax-
microstrip transition. ii) Furthermore, the ideal equivalent circuit of the
DMN under consideration: First, the DMN is constructed with ideal elements in
ADS and initial parameter values are calculated according to the analytical
derivations in Section III. This DMN of course shows excellent decoupling and
matching at the single design frequency. Subsequently, optimizations for a
wider bandwidth are carried out based on microstrip transmission line models.
Different parameters of the UCA implementation are studied.

At this point, practical implications begin to play an important role: Some of
the transmission line lengths are heavily restricted by the dimensions of the
UCA. For instance, the minimum length of lines connecting two antenna elements
is given by the distance between the antennas. The maximum length should not
be much larger. The line impedances should also be restricted to reasonable
choices of the microstrip line width. Hence, the line length and width
variables are optimized in ADS for a feasible solution; simultaneously,
broadband matching and decoupling goals are considered. This procedure
constitutes optimization cycles executed until all microstrip lengths and
widths seem feasible and the bandwidths are satisfyingly large. Going back to
the first step, i.e., choosing a UCA with different dimensions, is still
possible if no satisfying solution is found.

Eventually, the microstrip circuit is implemented in CST MWS and further full-
wave optimizations of all parameters are carried out. On the one hand, this is
necessary since a simultaneous UCA optimization is not possible on a circuit
level; on the other hand, the circuit functionality is changed to a small
extent since some effects (e.g., the behavior of microstrip junctions, bends,
and line-to-line coupling) are not fully included in ADS. The tuned parameters
include the antenna parameters such as array spacing or monopole lengths
together with all DMN parameters. Overall, only very few design parameters
were fixed, see the setup description in Section IIB. The goal was to achieve
the best possible UCA/DMN performance for each DMN approach.

###  IV-A Design of the Neutralization-Lines DMN

This DMN exhibits a quite simple structure, see the circuit model in Fig.
10(b) and the final design in Fig. 17.

Figure 17: CST MWS simulation model of neutralization-lines DMN, rear side.

The radius is $r_{\textrm{uca}}=8.6\,\mathrm{mm}=0.103\lambda_{0}$, which was
subject to the optimization. The design process is easily feasible; however,
it turns out that the bandwidth is limited. The widths and lengths of all
microstrip lines are attained via the described optimization procedure of
subsequent ADS and CST MWS simulations.

For this particular array, the circular groundplane was modified with quarter-
wavelength grooves at its edge to support the wave detachment and to suppress
radiation behind the ground plane caused by edge diffraction. This is
visualized in Fig. 18 and results in a larger realized gain above the
groundplane, i.e., in $+z$-direction.

Figure 18: Electric near field evaluations for neutralization-lines DMN.

Due to the corrugated groundplane, single-element radiation patterns and gains
of the array are only qualitatively comparable to the other arrays since the
corrugated groundplane causes an increase in gain. Nevertheless, the
corrugations are an interesting means for modifying the radiation behavior of
the array, which can be useful for many applications.

In Fig. 19, photographs of the fabricated prototype show the DMN microstrip
circuit on the rear side and the monopole UCA and the corrugated ground plane
on the front side.

(a)

(b)

Figure 19: Photographs of the 3-element UCA with neutralization-lines DMN and
corrugated ground plane. (a) Mounted in anechoic chamber, front view. (b) PCB
of DMN, rear view.

The scattering parameters in Fig. 20 demonstrate a good agreement between
measurement and full-wave simulation (neglecting the minimal frequency shift),
and the $-16$ dB bandwidth is about $100-110$ MHz in the measurement and 120
MHz in the simulation, limited only by the input matching due to the excellent
decoupling.

Figure 20: Simulated scattering parameters $S_{\mathrm{s}}$ and measured ones
$S_{\mathrm{m}}$ of three-element UCA with corrugated groundplane and
neutralization-lines DMN.

To obtain these simulation results, a little adjustment had to be applied: The
corrugations, which behave kind of resonant, have been filled with slightly
lossy material of $\varepsilon_{\mathrm{r}}=1$ and $\tan\delta=0.01$ due to
numerical problems in the frequency-domain solver of CST MWS. These additional
losses influence the gain by about $-0.3$ dB, which is corrected for in the
pattern and gain discussion for a valid comparison of measurements and
simulation.

###  IV-B Design of the Two-Stage DMN

During the discussed optimization procedure, it became clear that this two-
stage DMN circuit allows a realization with a larger bandwidth since it is not
completely symmetric; it exhibits rotational symmetry only. As seen for
instance in Fig. 11, the feed port P1 is connected to the two antennas A1 and
A3 via separate susceptances $B_{2}$ and $B_{3}$. The ADS-optimized design was
realized in a way that, for a lower frequency range, one antenna (A1 or A3) is
matched and the two others are only excited for port decoupling purposes of P2
and P3. For the upper part of the bandwidth, the other antenna (A3 or A1) is
then matched and, again, the other ports P2 and P3 are decoupled. Over the
whole bandwidth, a fluent transition in the matching and later on also in the
far-field patterns is observed. In other words, the antenna A1 is matched to
either P2 or P1 dependent on frequency.

The CST MWS simulation model in Fig. 21 adapts the circuit of Fig. 13.

(a)

(b)

Figure 21: The implemented version of the two-stage DMN connected to a three-
element UCA. (a) Simulation model in CST MWS. (b) Photograph of fabricated
PCB.

The transmission lines can of course not be straight lines but have to be bent
to fulfill the constraints given by the UCA with
$r_{\mathrm{uca}}=11.08\,\mathrm{mm}=0.136\lambda_{0}$. The major difference
to Fig. 13 is that there are (quite broad) microstrip lines from the antenna
ports A1–A3 to the entry point of the discussed DMN; introducing this
additional antenna feed line allows for more freedom in the DMN/UCA
optimization process and leads to a better solution.

In Fig. 22,

Figure 22: Scattering parameters of three-element UCA with two-stage DMN,
simulated is $S_{\mathrm{s}}$ and measured $S_{\mathrm{m}}$.

the measured and simulated S-parameters agree very well. The $-16$-dB matching
bandwidth is 185 MHz in simulation and almost $200$ MHz in measurement. The
coupling stays below the $-16$-dB threshold above $3.5$ GHz in simulation and
shows a bandwidth of over $200$ MHz in the measurement. This DMN can achieve
approximately twice the bandwidth of the neutralization-lines DMN since it
exhibits more design freedom.

###  IV-C Design of the Star-Triangle DMN

In the same way as for the other two designs, the original single frequency-
design was first optimized in ADS for a broadband matching and decoupling and
afterwards modeled in CST MWS. This optimized structure is shown in Fig. 23.

(a)

(b)

Figure 23: The implemented version of the star-triangle DMN connected to a
three-element UCA. (a) Simulation model in CST MWS. (b) Photograph of
fabricated PCB.

There is one design difference to all other discussed DMNs: The substrate
thickness $t_{\mathrm{s}}=0.762\,\mathrm{mm}$ was chosen instead of $0.508$ mm
to minimize loss; this means a trade-off between dielectric and microstrip
radiation loss. Other design choices include the UCA radius
$r_{\mathrm{uca}}=12.7\,\mathrm{mm}=0.152\lambda_{0}$, the matching with two
cascaded transmission lines at the port instead of lumped susceptances and an
optimized antenna connection instead of a parallel susceptance at the antenna
feed points.

In Fig. 24, the S-parameters are compared, where the decoupling is below $-16$
dB for a very broad bandwidth and the $-16$-dB input impedance bandwidth is
120 MHz in simulation and $105-110$ MHz in the measurement.

Figure 24: Simulated scattering parameters $S_{\mathrm{s}}$ and measured ones
$S_{\mathrm{m}}$ of three-element UCA with star-triangle DMN.

###  IV-D Total Efficiencies of Decoupled Arrays

We consider the total efficiencies $\eta_{\mathrm{tot}}$ in Fig. 25, including
loss and mismatch.

Figure 25: Simulated total efficiencies of the realized DMNs.

Every DMN offers a better efficiency than the UCA without DMN between $3.5$ to
$3.7$ GHz. Interesting to note is that the star-triangle DMN offers the best
total efficiency at any frequency even given that its S-parameters bandwidth
was the same as for the neutralization-lines DMN and much smaller than the
bandwidth of the two-stage DMN. Furthermore, it becomes obvious that the
larger operating bandwidth of the two-stage DMN with respect to input matching
is bought by a larger loss in its complicated microstrip structure. Analyzing
the per-material loss in CST MWS reveals that the worse total efficiency of
the two-stage DMN solely originates from around 80$\mathrm{\char 37\relax}$
copper conductivity loss and 20$\mathrm{\char 37\relax}$ dielectric substrate
loss.

In a real scenario, one would choose the design according to the importance of
design goals: If loss should be kept as low as possible, the star-triangle DMN
offers the better solution; if wide impedance bandwidth is important, the two-
stage DMN is preferable.

###  IV-E A Four-Element Neutralization-Lines DMN

For the evaluation of the differences between even- and odd-numbered UCAs, a
DMN with neutralization lines was realized in simulation only for a four-
element UCA. The radius was adjusted to $r_{\mathrm{uca}}=0.16\lambda_{0}$.
The DMN and its S-parameters are shown in Fig. 26.

(a)  (b)

Figure 26: Four-element array with a DMN consisting of simple neutralization
lines. (a) Simulation model. (b) Simulated S-parameters.

Since the UCA radius is larger than for the four-element model in Section IID,
the coupling without DMN is a bit lower and goes up to $-9$ dB. With the DMN,
this is reduced to maximally $-15$ dB. The total efficiency varies between
$-0.1$ dB and $-0.5$ dB, dependent on the beamsteering direction.

(a)  (b)

(c)  (d)

(e)  (f)

Figure 27: Comparison of simulated and measured single excitation (i.e.,
active element) and array excitation realized gain patterns. (a)
Neutralization-lines DMN, single element patterns, $\vartheta=40^{\circ}$ cut.
(b) Neutralization-lines DMN, array gain, $\vartheta=40^{\circ}$ cut. (c) Two-
stage DMN, single element patterns, $\vartheta=70^{\circ}$. (d) Two-stage DMN,
measured patterns for optimized $\varphi_{0}$ beamsteering,
$\vartheta=70^{\circ}$. (e) Star-triangle DMN, single element patterns,
$\vartheta=70^{\circ}$. (f) Star-triangle DMN, measured patterns for optimized
$\varphi_{0}$ beamsteering, $\vartheta=70^{\circ}$.

##  V Gain Measurements For Single-Port Excitation and Beamforming Capability
Verification

###  V-A Far-Field Patterns

Realized gain measurements of the three-element UCAs with DMN have been
performed in the anechoic chamber of TUM. The measurement results of
individual port patterns and beamformed patterns are summarized in Fig. 27.

Due to the corrugations at the ground plane edge, the pattern of the UCA with
neutralization-lines DMN has the maximum at around $\vartheta=40^{\circ}$, not
at $\vartheta=70^{\circ}$ as all other UCAs. Hence, the individual-port
patterns are given for this $40^{\circ}$-cut in Fig. 27(a), where an increase
in gain is observed due to the suppressed radiation behind the ground plane.
Furthermore, this radiation pattern is evaluated at $3.65$ GHz, the frequency
with the best matching in the measured S-parameters. The measurements agree
quite well with the simulation around the main beam. The beamforming weights
for the analogously-fed and measured optimum excitation coefficients have been
computed in the $\vartheta=30^{\circ}$ cut; the beamsteering effect is
nevertheless obvious in Fig. 27(b) for the main-beam directions
$\varphi_{0}=0^{\circ}$, $\varphi_{0}=300^{\circ}$, and
$\varphi_{0}=330^{\circ}$.

The same comparisons of measured and simulated single-excitation patterns are
performed for the two-stage DMN in Fig. 27(c) and the comparison of the
respective beamsteered gain pattern measurements are shown in Fig. 27(d). Both
measurements are analyzed in the $\vartheta=70^{\circ}$ cut. The lower gain
values of this UCA are expected due to the corrugated ground plane in the
first UCA. The agreement with the simulated pattern and the beamsteering
capabilities are well confirmed.

The very same comparisons are shown for the the star-triangle DMN in Fig.
27(e) and (f). The gain values of the simulation and the beamforming behavior
are demonstrated.

###  V-B Azimuthal Beamforming

Section II focused on even- and odd-numbered UCAs: In a theoretical
investigation based on array factors, we postulated that the achievable gain
of odd-numbered UCAs is rather constant over the beamsteering angle, whereas
even-numbered UCAs with one element more show the same maximally possible gain
in certain directions and a larger gain in between. Another insight was that
DMNs are needed to realize these effects and achieve the maximum beamforming
performance.

We have in mind that the gain in the $\vartheta=70^{\circ}$ cut for both
three- and four-element UCAs was about $6.7$ dBi on average for azimuthal
beamsteering, see Fig. 9. Similar gain curves are presented in Fig. 28

Figure 28: Optimum array gain dependent on the scan angle for simulations and
measurements of three- and four-element UCAs with DMNs. The three-element
neutralization-lines DMN is evaluated at $\vartheta=40^{\circ}$, all others at
$\vartheta=70^{\circ}$.

for arrays with the two-stage and star-triangle DMNs as well as for the array
with the four-element neutralization-lines DMN. The beamforming results in
Fig. 28 are based on array realized gain patterns for each beamsteering angle,
as shown for exemplary angles in Fig. 27. We want to emphasize two findings.
First, the gain fluctuations of the four-element UCA with DMN are larger than
for the three-element UCAs, confirming the investigations in Section II.
Second, the star-triangle DMN indeed achieves the same array gain as the
minimum gain value of the four-element UCA with DMN. The two-stage DMN
exhibits a bit lower gain due to its worse total efficiency.

The neutralization-lines DMN with the corrugated ground plane is also included
in Fig. 28, again for $\vartheta=40^{\circ}$ as a special case. Since the
radiation behind the groundplane is effectively suppressed, the gain of almost
9 dBi comes very close to the ideally expected values of almost $10$ dBi seen
in in Figs. 2 and 3.

Finally, we summarize the performance of all three-element arrays and their
different DMNs in Tab. I, which is based on simulation results. The proposed
array configurations are compared regarding relative $-16$ dB bandwidth BW,
efficiency, realized gain, UCA radius, and number of radiators to designs from
literature.

Table I: Comparison of UCA/DMN properties to corresponding designs from literature regarding center frequency $f_{c}$, relative $-16$ dB bandwidth BW in relation to $f_{c}$, maximum coupling $S_{ji}$ withing the bandwidth, maximum total efficiency $\eta_{\mathrm{tot}}$, realized gain $G$, UCA radius $r_{\mathrm{uca}}$, and number of UCA elements $N$. | $f_{\mathrm{c}}\nicefrac{{}}{{\mathrm{GHz}}}$ | $\mathit{BW}$ | $S_{ji}\nicefrac{{}}{{\mathrm{dB}}}$ | $\eta_{\mathrm{tot}}\nicefrac{{}}{{\mathrm{dB}}}$ | $G\nicefrac{{}}{{\mathrm{dBi}}}$ | $r_{\mathrm{uca}}\nicefrac{{}}{{\lambda_{0}}}$ | $N$
---|---|---|---|---|---|---|---
no DMN | $3.6$ | — | $-9$ | $-1.28$ | $6.8$ | $0.1$ | $3$
dec.-lines | $3.6$ | $3.3$\mathrm{\char 37\relax}$$ | $<-20$ | $-0.09$ | $9.7$ | $0.1$ | $3$
two-stage | $3.6$ | $5.1$\mathrm{\char 37\relax}$$ | $-16$ | $-0.32$ | $7.6$ | $0.14$ | $3$
\-  | $3.6$ | $3.3$\mathrm{\char 37\relax}$$ | $<-20$ | $\approx\,0$ | $8.0$ | $0.15$ | $3$
[54] | $2.45$ | $2.9$\mathrm{\char 37\relax}$$ | $-18$ | $-0.96$ | $5.5$ | $0.05$ | $2$
[54] | $2.55$ | $1.5$\mathrm{\char 37\relax}$$ | $<-20$ | $-3.28$ | $3.0$ | $0.041$ | $4$
[69] | $2.4$ | $1.0$\mathrm{\char 37\relax}$$ | $-16$ | $-0.46$ | $4.9$ | $0.035$ | $2$
[69] | $2.4$ | — | $<-20$ | $-0.9$ | — | $0.246$ | $3$

The $-16$ dB matching bandwidths and the decoupling performances within this
bandwidth have already been discussed in detail. The total efficiencies at the
design frequency reveal that there is an improvement of $-1$ dB to $-1.3$ dB.
This is reflected in the realized-gain increase, too, where the averaged
realized gain $G_{\mathrm{real}}$ in the cut of Fig. 28 is considered. The UCA
with neutralization-lines DMN, which shows a gain of $9.7$ dBi in the
$\vartheta=40^{\circ}$ cut, is a special case. At $\vartheta=70^{\circ}$, the
gain drops to only $5.1$ dBi. In the comparison in Tab. I, we have to keep in
mind that each of our proposed designs (DMN plus UCA) was optimized for the
its best performance as a system. Here, it was in particular found that a
varying element distance helped to improve the system performance for some of
the designs. Therefore, our comparison relates to properties of the systems
consisting of DMN and UCA and it should not just be seen as a comparison of
UCAs with a certain geometric arrangement.

##  VI Conclusion

Two new DMN circuits for three-element UCAs have been introduced, analyzed,
designed, fabricated, and verified in measurements. The proposed DMNs were
analyzed and design formulas for arbitrary antenna impedances have been
provided. In the practical validation, simple monopole radiators have been
utilized to realize compact UCAs. The DMNs were built in microstrip
technology. All of them are realized without cross-overs which is very
convenient for planar PCBs. In this respect, it was demonstrated that odd-
numbered UCAs are capable of offering an approximately constant gain in
azimuthal cuts whereas even-numbered ones (with one element more) show gain
fluctuations, where the minimum has about the same gain as odd-numbered UCAs.
We conclude that in general the same array performance is achieved with odd-
numbered UCAs, which was investigated in detail for arrays with three and four
elements. This was investigated in theory, in simulation, and by measurement.

In addition to the theoretical insights, one of the presented DMNs provides an
excellent total efficiency over a large bandwidth, while the other one
exhibits an excellent matching and decoupling bandwidth bought by reduced
efficiency. As compared to plain UCAs without DMNs, the decoupled and matched
UCAs offer a gain and efficiency improvement of about $1$ to $1.5$ dB. These
improvements have been achieved at the expense of more complex DMNs, which
have a relatively large size in our realizations. However, compact
realizations of the DMNs with lumped elements (as discussed theoretically in
the paper), with transmission lines in multilayer PCBs, or even with shielded
transmission lines are also possible.

## References

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|  Jonas Kornprobst (S’17) received the B.Eng. degree in electrical
engineering and information technology from the University of Applied Sciences
Rosenheim, Rosenheim, Germany, in 2014, and the M.Sc. degree in electrical
engineering and information technology from the Technical University of
Munich, Munich, Germany, in 2016. Since 2016, he has been a Research Assistant
with the Chair of High-Frequency Engineering, Department of Electrical and
Computer Engineering, Technical University of Munich. His current research
interests include numerical electromagnetics, in particular integral equation
methods, antenna measurement techniques, antenna and antenna array design, as
well as microwave circuits.
---|---
|  Thomas J. Mittermaier (S’12$-$M’17) received the Dipl.-Ing.(FH) degree from
Fachhochschule Regensburg, Regensburg, Germany, and the M.Sc. degree from the
Technical University of Munich, Munich, Germany, in 2009 and 2011,
respectively, both in electrical engineering and information technology. From
2011 to 2017 he was a Research Assistant at the Chair of High-Frequency
Engineering of the Technical University of Munich, Munich, Germany. His
doctoral research focused on signal processing and tracking algorithms for
extremely short-range radar in industrial machine tools as well as microwave
circuits. In May 2017, he joined the RF360 Europe GmbH, Munich, as a
Development Engineer for bulk acoustic wave devices. He is currently involved
in linear and nonlinear modeling, in power durability and life-time testing of
acoustic resonators and filters for future mobile communications.
---|---
|  Raimund A. M. Mauermayer (S’14) received the Dipl.-Ing. degree in
electrical engineering and information technology from the Technical
University of Munich, Munich, Germany, in 2012. From 2012 to 2018, he was a
Research Assistant with the Chair of High-Frequency Engineering, Technical
University of Munich. In 2019, he joined the Department of Antenna Design and
Measurement, Mercedes-Benz AG, Sindelfingen, Germany. His research interests
are near-field far-field transformation and antenna measurement techniques.
---|---
|  Matthias G. Ehrnsperger (S’19) received the B.Eng. degree in electrical
engineering and information technology from the Technical University of
Applied Sciences Regensburg, Regensburg, Germany, in 2015, and the M.Sc.
degree in electrical engineering and information technology from the Technical
University of Munich, Munich, Germany, in 2017. Since 2017, he has been a
Research Assistant with the Chair of High-Frequency Engineering, Department of
Electrical and Computer Engineering, Technical University of Munich. His
current research interests include radar systems, in particular antenna and
antenna array design, rapid prototyping methodologies, statistical- and
machine learning based signal processing, as well as microwave circuits.
---|---
|  Gerhard F. Hamberger received the B.Sc., the M.Sc., and the Dr.-Ing. degree
in electrical and computer engineering from the Technical University of
Munich, Munich, Germany, in 2012, 2014, and 2019, respectively. In 2018, he
had joined the Systems and Projects Department of Rohde & Schwarz, Munich,
Germany, where he was working on various state-of-the-art 5G antenna and
device test systems and field transformation software. In 2019, Dr.-Ing.
Hamberger has become a Senior Development Engineer in the Microwave Imaging
Department of Rohde & Schwarz, Munich, Germany, focusing on automotive radar
and radome test systems
---|---
|  Bernhard Lehmeyer received the Dipl.-Ing. and Dr.-Ing. degrees in
electrical and computer engineering from the Technical University of Munich
(TUM), Munich, Germany, in 2010 and 2018. In 2010, he joined Rohde & Schwarz
GmbH & Co. KG Germany as circuit design engineer. From 2012 to 2018, he was a
research assistant at the institute of circuit theory and signal processing at
the Technical University of Munich. From 2018 to 2019, he was a postdoctoral
researcher at the High Voltage Lab/ETHZ, Swiss Federal Institute of Technology
in Zurich, Zurich, Switzerland. Since 2019, he is the CEO of Tronika GmbH
Germany. His current research interests are RF circuitry, optical detector
design and power inverter optimization.
---|---
Michel T. Ivrlač, biography not available at time of publication.
---

|  Ulrik Imberg received the M.Sc. degree in applied physics and electrical
engineering from The Institute of Technology at Linköping University in 1997.
He is currently leading the Huawei Technologies Sweden AB Wireless Technology
Planning team. His research interests cover 5G & 6G telecommunication systems
& technologies, especially active antennas, antenna-near electronics and
photonics.
---|---
|  Thomas F. Eibert (S’93$-$M’97$-$SM’09) received the Dipl.-Ing. (FH) degree
from Fachhochschule Nürnberg, Nuremberg, Germany, the Dipl.-Ing. degree from
Ruhr-Universität Bochum, Bochum, Germany, and the Dr.-Ing. degree from
Bergische Universität Wuppertal, Wuppertal, Germany, in 1989, 1992, and 1997,
all in electrical engineering. He is currently a Full Professor of high-
frequency engineering at the Technical University of Munich, Munich, Germany.
His current research interests include numerical electromagnetics, wave
propagation, measurement and field transformation techniques for antennas and
scattering as well as all kinds of antenna and microwave circuit technologies
for sensors and communications.
---|---
|  Josef A. Nossek (S’72$-$M’74$-$SM’81$-$F’93$-$LF’13) received the
Dipl.-Ing. and Dr. techn. degrees in electrical engineering from the Vienna
University of Technology, Vienna, Austria, in 1974 and 1980, respectively. In
1974, he joined Siemens AG, Munich, Germany, where he was engaged in filter
design for communication systems. From 1987 to 1989, he was the Head of the
Radio Systems Design Department, where he was instrumental in introducing
highspeed VLSI signal processing into digital microwave radio. From 1989 to
2016, he was a Full Professor for circuit theory and signal processing with
the Technical University of Munich (TUM), Munich. Since 2016, he has been an
Emeritus of Excellence of TUM and a Full Professor with the Federal University
of Ceará, Fortaleza, Brazil. Dr. Nossek was the President-Elect, President,
and Past President of the IEEE Circuits and Systems Society in 2001, 2002, and
2003, respectively. He was the President of Verband der Elektrotechnik,
Elektronik, und Informationstechnik (VDE) from 2007 to 2008 and the President
of the Convention of National Associations of Electrical Engineers of Europe
(EUREL) in 2013. In 2009, he became a member of the National Academy of
Engineering in Germany (acatech). He was a recipient of the ITG Best Paper
Award in 1988, the Mannesmann Mobilfunk (currently Vodafone) Innovations Award
in 1998, and the Award for Excellence in Teaching from the Bavarian Ministry
for Science, Research and Art in 1998. From the IEEE Circuits and Systems
Society, he was a recipient of the Golden Jubilee Medal for Outstanding
Contributions to the Society in 1999 and the Education Award in 2008. He was
also a recipient of the Order of Merit of the Federal Republic of Germany
(Bundesverdienstkreuz am Bande) in 2008, the IEEE Guillemin-Cauer Best Paper
Award in 2011, the honorary doctorate (Dr. h. c.) from the Peter Pazmany
Catholic University, Hungary, in 2013, and the VDE Ring of Honor in 2014.
---|---