# Antenna Radiation Efficiency Estimation From Backscattering Measurement
Performed Within Reverberation Chambers

Wafa Krouka, François Sarrazin, Julien de Rosny, Adnane Labdouni, and Elodie
Richalot W. Krouka, F. Sarrazin, A. Labdouni and E. Richalot are with the
ESYCOM lab, Univ Gustave Eiffel, CNRS, F-77454 Marne-la-Vall e, France
(e-mail: francois.sarrazin@univ-eiffel.fr).J. de Rosny is with ESPCI Paris,
PSL University, CNRS, Institut Langevin, Paris, France
(julien.derosny@espci.fr).

###### Abstract

This paper presents a contactless measurement method for antenna radiation
efficiency estimation within reverberation chambers (RCs). This method does
not require to connect any analyzer to the antenna under test (AUT). The AUT
radiation efficiency is obtained from the differential measurement of the RC
composite $Q$-factor estimated for two AUT load conditions, namely an open
circuit and a $50\text{\,}\mathrm{\SIUnitSymbolOhm}$ load. Remotely-controlled
RF switches are used in order to keep the setup identical for both
measurements. The method is experimentally validated in two RCs of different
volumes (1 m3 and 19 m3) with a wideband patch antenna in the 1.8 GHz to 2.8
GHz frequency range. The estimated efficiency is found to be in very good
agreement with the one obtained through a conventional invasive RC measurement
technique. This new method is suitable for miniature and buried antenna
characterization without access to its port.

###### Index Terms:

Antenna radiation efficiency, backscattering measurement, contactless
measurement, quality factor, reverberation chamber.

##  I Introduction

Antenna radiation efficiency is defined as the ratio between the radiated
power and the incident power, i.e., the one that goes through the antenna port
[1]. Its measurement has been initially performed using a transmission-type
setup within an anechoic chamber (AC) by applying the gain-integration
technique. Since first studies in 2001 [2], reverberation chambers (RCs)
became very attractive to measure the antenna radiation efficiency [3]–[6].
Indeed, thanks to the statistically isotropic and homogeneous field within the
RC working volume, such measurement neither exhibits alignment constraints nor
require the antenna under test (AUT) rotation. Moreover, the use of a
reference antenna can be avoided by comparing the RC $Q$-factor estimated in
the time and frequency domains [7, 8]. However, all current methods require an
invasive setup, i.e., a setup where the AUT needs to be connected to the
analyzer implying connectors and long cables.

The presence of cables in the measurement setup leads to unwanted interactions
between the AUT and the cables [9, 10]. Two different phenomena occur: 1) the
feed cables are placed in the antenna near-field zone; therefore both
resistive and reactive parts of the input impedance are modified, as well as
the antenna radiation characteristics, and 2) The imperfect balance between
the radiating element and the ground plane causes current leaking on the outer
shielding of the feed cables, so that the cables themselves act as radiators
[11]–[13]. These two effects become critical when dealing with electrically
small antennas and lead to inaccurate results [14, 15].

Different ways have been investigated to overcome this issue in an AC:
limiting the cable effects by adding ferrites, quarter-wavelength sleeves [17]
or balun [16]; compensating for disturbancies introduced by the measurement
setup thanks to prior electromagnetic simulations [14] or post-processing [18]
; replacing the coaxial cables by optical links [19, 20]; performing antenna
backscattering measurement. The latter is of particular interest as it enables
a contactless measurement, i.e., without the need to connect the AUT to an
analyzer. It is based on several measurements with either a varying load [21]
or a few discrete ones [22]. First introduced in an AC, such techniques have
been recently applied within an RC [23]–[25] but only for antenna radiation
pattern estimation.

The insertion of lossy objects within an RC has been studied in terms of
absorption [26] and diffusion [27] whereas the specific case of antennas has
been first dealt with as a dissipative load impedance [28] before that a
refined model was proposed recently [29] in order to take into account the
reflections due to the load impedance mismatch. In this paper, we take benefit
from these studies to introduce an original contactless measurement technique
that allows retrieving the antenna radiation efficiency.

This paper is organized as follows: Section II introduces the theoretical
background for the contactless measurement method. Section III describes the
experimental setup within an RC as well as the radiation efficiency results
obtained using the contactless method and compared to conventional
measurements. The very good agreement between the two methods is confirmed by
a second measurement in a smaller RC in Section IV. Finally, a conclusion ends
this paper.

##  II Theory

The RC $Q$-factor is proportional to the ratio between the stored energy $U$
and the dissipated power $P_{\mathrm{d}}$ within the cavity as:

$Q=\frac{\omega U}{P_{\mathrm{d}}}.$ (1)

Let us consider an RC measurement setup where two arbitrary antennas are
connected to a VNA in order to measure the RC $Q$-factor (from transmission
coefficient). $N$ identical AUTs are located on masts within the working
volume and unconnected to any measurement device. In this configuration, the
RC $Q$-factor $Q_{\mathrm{L}x}$ can be decomposed as

$Q^{-1}_{\mathrm{L}x}=Q^{-1}_{\mathrm{c}}+NQ^{-1}_{\mathrm{a,L}x}$ (2)

where $Q_{\mathrm{a,L}x}$ is associated with the losses of a single AUT
terminated by a load impedance $Z_{\mathrm{L}x}$ and $Q_{\mathrm{c}}$ is
related to all other sources of losses (wall, transmitting and receiving
antennas, stirrer, masts.). In order to isolate $Q_{\mathrm{a,L}x}$, one could
perform an ”empty” measurement where the $N$ AUTs are removed from the RC.
Indeed, the difference between the $Q$-factors for the two measurements (with
and without AUTs) leads to $NQ_{\mathrm{a,L}x}$. However, this requires two
distinct measurements whose setup will eventually vary due to manual handling
to remove the $N$ AUTs. This effect will be highlighted in section IV. Thus,
we suggest in this paper to perform two measurements, both with the $N$ AUTs
located within the RC, but terminated by two different load impedances
$Z_{\mathrm{L1}}$ and $Z_{\mathrm{L2}}$. The loads are modified thanks to
remotely controlled switches and are identical for all AUTs. Hence, no manual
handling is performed in this case. From (2), the difference between the
$Q$-factors measured for two load impedances can be expressed as:

$Q^{-1}_{\mathrm{L1}}-Q^{-1}_{\mathrm{L2}}=N\left(Q^{-1}_{\mathrm{a,L1}}-Q^{-1}_{\mathrm{a,L2}}\right)$
(3)

The term $Q_{\mathrm{c}}$ vanishes in this difference as the setup remains
identical. A first expression of the antenna $Q$-factor within an RC
$Q_{\mathrm{a}}$, referred thereafter as eq1, has been derived by Hill et al.
in [28] such as:

$Q_{\mathrm{a,L}x}^{\mathrm{eq1}}=\frac{Q^{0}_{\mathrm{a}}}{\eta_{\mathrm{a}}(1-|\Gamma_{\mathrm{a,L}x}|^{2})}$
(4)

where $Q^{0}_{\mathrm{a}}=16\pi^{2}V/\lambda^{3}$ is the $Q$-factor of an
ideal lossless antenna, and $\Gamma_{\mathrm{a,L}x}$ is the reflection
coefficient between the AUT input impedance and its load impedance
$Z_{\mathrm{L}x}$. $\Gamma_{\mathrm{a,L}x}$ can be computed as [32]:

$\left|\Gamma_{\mathrm{a,L}x}\right|=\left|\frac{\Gamma_{\mathrm{L}x}-\Gamma^{{\dagger}}_{\mathrm{a}}}{1-\Gamma_{\mathrm{L}x}\Gamma_{\mathrm{a}}}\right|$
(5)

where $\Gamma_{\mathrm{a}}$ and $\Gamma_{\mathrm{L}x}$ are the reflection
coefficients of the AUT and the load impedance, respectively, according to
$50~{}\Omega$. The superscript † stands for the complex conjugate. In 2018,
Cozza [29] suggested modifying the $Q$-factor formula (4) in order to take
into account the power reflected by the antenna load which is re-radiated
towards the RC. The modified formula, referred then as eq2, is given as:

$Q_{\mathrm{a,L}x}^{\mathrm{eq2}}=\frac{Q^{0}_{\mathrm{a}}}{1-\eta^{2}_{\mathrm{a}}|\Gamma_{\mathrm{a,L}x}|^{2}}$
(6)

Replacing $Q_{\mathrm{a,L1}}$ and $Q_{\mathrm{a,L2}}$ in (3) by their
expressions from (4) and (6), the AUT radiation efficiency $\eta_{\mathrm{a}}$
can be retrieved through two relations as:

$\eta_{\mathrm{a}}^{\mathrm{eq1}}=\frac{Q^{0}_{\mathrm{a}}\left(Q^{-1}_{\mathrm{L1}}-Q^{-1}_{\mathrm{L2}}\right)}{N\left(\left|\Gamma_{\mathrm{a,L2}}\right|^{2}-\left|\Gamma_{\mathrm{a,L1}}\right|^{2}\right)}$
(7)

and

$\eta_{\mathrm{a}}^{\mathrm{eq2}}=\sqrt{\frac{Q^{0}_{\mathrm{a}}\left(Q^{-1}_{\mathrm{L1}}-Q^{-1}_{\mathrm{L2}}\right)}{N\left(\left|\Gamma_{\mathrm{a,L2}}\right|^{2}-\left|\Gamma_{\mathrm{a,L1}}\right|^{2}\right)}}.$
(8)

It can be noticed that $\eta_{\mathrm{a}}^{\mathrm{eq2}}$ is the square root
of $\eta_{\mathrm{a}}^{\mathrm{eq1}}$. The radiation efficiency estimation
$\eta_{\mathrm{a}}^{\mathrm{eq2}}$ using (8) is expected to be more accurate
as it is based on the revised $Q$-factor formula from [29], but the estimation
$\eta_{\mathrm{a}}^{\mathrm{eq1}}$ using (7) will be evaluated in the
following sections in order to highlight the limitations of the previous model
from [28].

##  III Measurement

###  III-A Experiment

Figure 1: Antenna efficiency measurement setup in the ESYCOM RC. A1 and A2 are
a horn antenna and a log-periodic antenna, respectively, and are connected to
a VNA. A1 is oriented towards an edge of the RC and 17 different positions are
considered. A2 is oriented towards the mode stirrer. The vertical mode stirrer
rotates around its axis and 72 positions are considered over a revolution. The
AUT is located on a mast and connected to a mechanical switch through a 30-cm
cable. Four different orientations are considered for the AUT.

Figure 2: Reference plane considered for measuring both the antenna and the
load reflection coefficients.

The setup is presented in Fig. 1. This experiment has been conducted at the
ESYCOM laboratory in an RC whose dimensions are 2.95 m $\times$ 2.75 m
$\times$ 2.35 m, equipped with a rotating metallic mode stirrer (72 equally
spaced positions are considered). Two antennas (A1 and A2) are connected to a
Rohde&Schwarz ZNB20 vector network analyzer (VNA) in order to measure their
scattering parameters in the 1.8 GHz to 2.8 GHz frequency range (10001
frequency points). A1 is oriented towards an edge of the RC whereas A2 is
oriented towards the mode stirrer. A1 is positioned on a vertically-rotating
mast in order to perform source stirring over 17 positions. Both antennas are
carefully positioned to avoid any direct coupling between them for all
configurations; indeed, A1 and A2 are orthogonally positioned and polarized.
The AUT is a wideband slot-based patch antenna (details in [8]) that is
connected, through a 30-cm long coaxial cable, to a remotely-controlled
mechanical switch that allows switching between two load impedances (Fig. 2).
The AUT and the switch are positioned on a rotating mast and four positions
are considered to perform the AUT stirring. In order to maximize the
difference between the two measured $Q$-factors, and thus, the measurement
sensitivity, an open circuit (OC) and a 50 $\Omega$ load are chosen as the two
load impedances $Z_{\mathrm{L1}}$ and $Z_{\mathrm{L2}}$, respectively. All
measurements are performed in a single run, without manual manipulation.

To assess the validity of (7) and (8), we first perform an invasive estimation
of $\left|\Gamma_{\mathrm{a,L}x}\right|$. With a VNA, we measure on the one
hand the AUT reflection coefficient $\Gamma_{\mathrm{a}}$ within an AC, and on
the other hand the reflection coefficients
$\Gamma_{\mathrm{L1}}=\Gamma_{\mathrm{OC}}$ and
$\Gamma_{\mathrm{L2}}=\Gamma_{\mathrm{50}}$, associated to the two load
impedances $Z_{\mathrm{L1}}$ and $Z_{\mathrm{L2}}$. As shown in Fig. 2, the
measurement reference plane is located at the antenna’s port so that the
losses associated to the cable and the switch are included in
$\Gamma_{\mathrm{OC}}$ and $\Gamma_{\mathrm{50}}$. From this 2-step
measurement, $\left|\Gamma_{\mathrm{a,OC}}\right|$ and
$\left|\Gamma_{\mathrm{a,50}}\right|$ are deduced from (5) and results are
presented in Fig. 3 as a function of the frequency.

Figure 3: AUT and load impedances reflection coefficients as a function of
frequency.

We can notice that the AUT is well matched to 50 $\Omega$ in the whole
frequency range ($\left|\Gamma_{\mathrm{a}}\right|$ less than -10 dB). Due to
the switch insertion losses as well as the losses due to the cable between the
AUT and the switch, $\left|\Gamma_{\mathrm{OC}}\right|$ is not equal to 0 dB
but its average over the frequency band is of $-0.51$ dB.
$\left|\Gamma_{\mathrm{50}}\right|$ is less than -20 dB; thus, this load acts
as a good matched load even though the presence of the switch and the cable.
As the AUT is well matched, $\left|\Gamma_{\mathrm{a,50}}\right|$ and
$\left|\Gamma_{\mathrm{a,OC}}\right|$ exhibit similar behaviors to
$\left|\Gamma_{\mathrm{a}}\right|$ and $\left|\Gamma_{\mathrm{OC}}\right|$,
respectively.

RC $Q$-factors for both load conditions are computed in the time domain using
the transmission coefficient $S_{\mathrm{21}}$ between both measurement
antennas (A1 and A2) by fitting the power delay profile
$\mathrm{PDP}=\left<\left|\mathrm{IFT}(S_{\mathrm{21}})\right|^{2}\right>$ [7]
where the average denoted $\left<.\right>$ is calculated over all the
configurations in addition to a sliding frequency window of 200 MHz. As the
two $Q$-factors are very close, we present also $\Delta
Q^{-1}/\overline{Q^{-1}}=2\left(Q^{-1}_{50}-Q^{-1}_{\mathrm{OC}}\right)/\left(Q^{-1}_{50}+Q^{-1}_{\mathrm{OC}}\right)$
in Fig. 4 as a function of frequency. This quantity is always positive,
meaning that $Q^{-1}_{\mathrm{OC}}<Q^{-1}_{50}$ for all frequencies. This is
coherent as more losses are expected within the RC when the AUT is connected
to the 50 $\Omega$ load. Also, it decreases as a function of frequency, which
is expected as the contribution of the AUT on the RC $Q$-factor becomes
smaller.

Figure 4: $Q$ as a function of frequency (left) and $\Delta
Q^{-1}/\overline{Q^{-1}}$ as a function of frequency (right).

###  III-B Results

Figure 5: Radiation efficiency of the patch antenna measured in the ESYCOM RC
using the invasive and the contactless approach, as a function of frequency.
Results are smoothed over a 30 MHz window.

The radiation efficiency measured using the contactless setup is presented in
Fig. 5 using eq1 (7) and eq2 (8) as a function of frequency. To evaluate the
radiation efficiency accuracy, results are compared to those obtained through
a conventional invasive measurement method within the same RC (two-antenna
approach in [7]).

This method is chosen as a reference as it is widely used and well known to
provide accurate and repeatable radiation efficiency estimations within an RC
[30, 31].

These results show that the newly introduced method using (8), i.e., eq2,
allows obtaining very similar results to the ones obtained using a
conventional invasive RC approach applied in the same measurement conditions
(same RC $Q$-factor and same stirring processes). Indeed, the averaged
relative error is equal to only 2.3 %.

As the estimated efficiency using the conventional method is almost constant
over the frequency range, we compute the mean and the standard deviation of
the three estimated efficiencies for comparison purposes; they are presented
in Table I. The radiation efficiency averaged over the whole frequency band is
equal to 81.9 % with the conventional method and 83.7 % with the contactless
approach in the case of eq2. However, the standard deviation is higher for the
contactless approach.

Indeed, the proposed method is based on the estimation of a small $Q$-factor
variation due to the AUT load modification and is therefore more sensitive to
the non-ideal behavior of the RC, leading to higher fluctuations as a function
of frequency.

It has to be noted that, as expected, the contactless approach using (7),
i.e., eq1, leads to a strongly underestimated radiation efficiency (61.6 % on
average over the frequency bandwidth). This is due to the fact that (7) does
not take into account that a part of the power arising on the AUT is actually
reflecting back into the RC (due to mismatches). This power is thus considered
as absorbed by the AUT, which artificially decreases its radiation efficiency.
These results demonstrate the possibility to estimate the antenna radiation
efficiency using a contactless approach, i.e., without connecting the AUT to
the VNA, by measuring the RC $Q$-factor for two AUT load conditions. Also, it
confirms that the antenna $Q$-factor formula introduced in [29] better models
the antenna absorption within an RC.

TABLE I: Radiation Efficiency Mean Value and Standard Deviation over the Frequency Range for the Experiments Conducted in the 19 m3 Reverberation Chamber. | Mean | Standard Deviation
---|---|---
Conventional |  $81.9$ % | $0.0102$
Contactless eq1 |  $61.6$ % | $0.0228$
Contactless eq2 |  $83.7$ % | $0.0163$
Contactless eq2 (ideally matched AUT) |  $82.3$ % | $0.0149$
Contactless eq2 (ideal loads) |  $78.6$ % | $0.0131$

###  III-C Reflection coefficient approximations

The contactless method requires the determination of
$\left|\Gamma_{\mathrm{a,L}x}\right|$ from (5) and thus the measurement of
both reflection coefficients $\Gamma_{\mathrm{a}}$ and $\Gamma_{\mathrm{L}x}$.
In this paper, they are measured directly using a VNA, i.e., using an invasive
measurement. Although the impedance is usually less sensitive than radiation
properties to the proximity of cables, impedance measurements might not be
possible from a practical point of view, especially if the AUT is specifically
designed including an integrated miniaturized switching device. Thus, we
evaluate in the following the impact of no a priori knowledge on these
reflection coefficients. Accordingly, two approximations are successively
done: (1) the AUT is assumed to be ideally matched to 50 $\Omega$ so that
$\Gamma_{\mathrm{a}}=0$ and thus
$\left|\Gamma_{\mathrm{a,L}x}\right|=\left|\Gamma_{\mathrm{L}x}\right|$, and
(2) we consider an ideal OC so that $\Gamma_{\mathrm{OC}}=1$ and thus
$\left|\Gamma_{\mathrm{a,OC}}\right|=1$ and an ideal 50 $\Omega$ load so that
$\Gamma_{50}=0$ and thus
$\left|\Gamma_{\mathrm{a,50}}\right|=\left|\Gamma_{\mathrm{a}}\right|$.
Radiation efficiencies obtained from these two approximations are presented in
Fig. 6 and compared with the conventional method and the contactless one
without approximation. Considering the AUT ideally matched to 50 $\Omega$
leads to very similar results to the ones without approximation (the average
radiation efficiency is equal to 82.3 %) which is due to the fact that the AUT
is well-matched in this frequency band (see Fig. 3). Indeed, when
$\Gamma_{\mathrm{a}}$ is very low (around 1.85 GHz and 2.5 GHz), the two
curves merge. However, if we consider ideal loads, a systematic
underestimation of the radiation efficiency is observed (the average radiation
efficiency is equal to 78.6 %). This is mainly due to the switch and the cable
insertion losses located between the calibration port and the load, which are
not taken into account and thus, not properly compensated.

Figure 6: AUT radiation efficiency as a function of frequency, measured using
the contactless approach considering different approximations regarding the
reflection coefficients.

###  III-D Stirring effect

This part is dedicated to the evaluation of the measurement uncertainties
according to the stirring process. All results shown in the previous parts
were obtained by applying all the stirring possibilities of the measurement
setup, including the mechanical stirring (72 configurations), the source
stirring (17 configurations) and the AUT stirring (4 configurations), leading
up to 4896 configurations. The objective of such stirring process is to lower
the measurement uncertainties, i.e., to lower the standard deviation of the
radiation efficiency estimation. However, the accuracy of any RC measurements
relies on the effective sample size generated by the stirring process, i.e.,
the number of uncorrelated configurations. A correlation coefficient $\rho$ is
first calculated using the 72 stirrer positions for a single source position
and a single AUT position, based on the transmission coefficient between the
two measurement antennas. The effective sample size $N_{\mathrm{eff}}$ is then
deduced from $\rho$ [33]. Since it presents fluctuations as a function of
frequency, its value is averaged in the 2 GHz to 2.4 GHz frequency band. This
estimation is made for each source and AUT position before being averaged and
leads to $N_{\mathrm{eff}}=59$. It has to be noted that $N_{\mathrm{eff}}$ has
been computed for the two AUT load conditions but it leads to very similar
results thus, only the results obtained for the OC case are presented
thereafter.

Once the effective sample size is known, it is possible to evaluate the
normalized standard deviation of the radiation efficiency estimation as [34]:

$\frac{\sigma_{\eta_{\mathrm{a}}}}{\left<\eta_{\mathrm{a}}\right>}\approx\sqrt{\frac{2}{N_{\mathrm{eff}}}}.$
(9)

In the present case ($N_{\mathrm{eff}}=59$), the normalized standard deviation
is estimated to be equal to 0.184. In order to validate this result, we now
aim at evaluating the normalized standard deviation of the radiation
efficiency estimation from the measurement results. As two types of stirring,
namely the source stirring and the AUT stirring, are performed in addition to
the conventional mechanical stirring, two different estimations of the
normalized standard deviation are computed: one using the source stirring and
the other one using the AUT stirring. First, we estimate
$\sigma_{\eta_{\mathrm{a}}}^{p_{\mathrm{src}}}$ from all source positions and
for one AUT position. The result is then averaged over all AUT positions
$\left<\sigma_{\eta_{\mathrm{a}}}^{p_{\mathrm{src}}}\right>_{p_{\mathrm{AUT}}}$.
The same process is made considering only the 4 AUT positions for one source
position, before averaging over all source positions
$\left<\sigma_{\eta_{\mathrm{a}}}^{p_{\mathrm{AUT}}}\right>_{p_{\mathrm{src}}}$.
Results are presented in Table II. Both estimations from the measurement
results are in very good agreement with the theoretical prediction (relative
error of 6 % for the source stirring estimation and 1 % for the AUT stirring).

This confirms that the normalized standard deviation can be predicted directly
from $N_{\mathrm{eff}}$. In the present case, we evaluated 10 uncorrelated
source positions out of the 17 using the same process than the one used for
the mechanical stirring. If we consider that the 4 AUT positions are
uncorrelated (4 positions being too low to accurately estimate the effective
sample size), it comes $4\times 10\times 59=2360$ uncorrelated configurations.
According to (9), this leads to a predicted normalized standard deviation that
is equal to 0.029.

TABLE II: Normalized Standard Deviations of the Estimated Radiation Efficiency According to the Mechanical Stirring Process.  | Normalized Standard Deviation
---|---
Prediction [34] | 0.184
Source stirring | 0.173
AUT stirring | 0.186

##  IV Validation in a Small RC

Figure 7: Contactless antenna efficiency measurement setup in the small RC.
The vertical mode stirrer is rotating around its axis and both measurement
antennas A1 and A2 are translated along linear stages.

In this section, we aim at validating the proposed measurement method using a
much smaller RC, thus exhibiting a much smaller $Q$-factor. These experiments
have been conducted in the 1 m3 RC of the Institut Langevin with the
experimental setup presented in Fig. 7. The AUT is the same wideband patch
antenna as in the previous section. The $Q$-factor being much smaller than in
the large RC (about 1700 at the central frequency), two identical AUTs have
been placed within the RC in order to enhance the AUT absorption strength so
$N=2$ in (8). They are both connected to their own mechanical switch (that can
switch between an OC and a 50 $\Omega$ load) through a 30-cm coaxial cable.
For practical reasons (especially the limited size of the RC), the two
antennas A1 and A2 are the same as the AUTs so that four identical patch
antennas are used: two as AUTs and two as measurement antennas (A1 and A2).
Scattering parameters are measured thanks to an Anritsu VNA in the same
frequency range (1.8 GHz to 2.8 GHz) with 10001 frequency points. A1 and A2
antennas are positioned on a translation stage in order to perform source
stirring over 9 configurations (3 positions for each antenna) in addition to
the mechanical rotating stirrer that provides 72 positions. Thus, all results
are averaged over 648 configurations.

The same procedure as in the previous section is conducted here regarding the
AUT and the load impedances reflection coefficients measurement, the results
are not presented for brevity. $Q$-factors are presented in Fig. 8 as a
function of frequency. As expected, $Q$ increases with frequency, and it is
slighty lower for the 50 $\Omega$ case as more energy is absorbed by this
load. In order to highlight the sensitivity of the $Q$-factor estimation, we
also computed the $Q$-factor for the empty case, i.e. when the AUTs are
removed, whereas everything else, including the cables and the switches, are
kept within the RC. We can see that this result is not consistent with the
previous ones. Indeed, due to the estimation fluctuations, $Q$ in the empty
case is sometimes higher and sometimes lower than the one with AUT connected
to the OC. This is precisely why we suggested in this approach the use of two
load conditions obtained without manual manipulation and no empty RC
measurement.

Figure 8: RC $Q$-factors of the small RC as a function of frequency, measured
without any AUT (blue) or when both AUTs are connected to an OC (black) and a
50 $\Omega$ load (red).

The radiation efficiency is measured using the conventional setup as well as
with the contactless setup and results are presented in Fig. 9 as a function
of frequency. The values of $\left|\Gamma_{\mathrm{a,L}x}\right|$ have been
computed by averaging the measurements performed on the 2 switches. Mean
values as well as standard deviations over the frequency range for each
estimation are presented in Table III. These results confirm those obtained in
section III-B: a very similar radiation efficiency estimation is obtained
between the conventional and the contactless (eq2) methods (2 % absolute
difference on average, and a relative difference of 6 % on average) although
the standar deviation as a function of frequency is higher for the contactless
case. It has to be noted that the mean number of uncorrelated stirrer
positions, between 2 GHz and 2.4 GHz, is evaluated to be 21, for a mean number
of uncorrelated measurement antenna configurations of 5.4. This leads to 113
uncorrelated samples, which is much smaller than the 2360 uncorrelated samples
of the ESYCOM RC; this explains why the standard deviations are larger than
the ones obtained previously. Again, the radiation efficiency estimated
through eq1 is not consistent with the one obtained using the conventional
method. Also, the same approximations as the ones made in III-C regarding the
reflection coefficients are made. Considering the AUT perfectly matched to 50
$\Omega$ leads to a slight underestimation of only 2 % on average whereas
considering ideal loads leads to a strong underestimation of about 10 %.

Figure 9: AUT radiation efficiency as a function of frequency, measured in the small RC using the invasive and the contactless approaches, considering different approximations regarding the reflection coefficients. The results are smoothed over a 30 MHz window. TABLE III: Radiation Efficiency Mean Value and Standard Deviation over the Frequency Range for the Experiments Conducted in the 1 m3 Reverberation Chamber. | Mean | Standard Deviation
---|---|---
Conventional |  $80.9$ % | $0.0254$
Contactless eq2 |  $78.9$ % | $0.0424$
Contactless eq2 (ideally matched AUT) |  $76.9$ % | $0.0387$
Contactless eq2 (ideal loads) |  $69.0$ % | $0.0278$

##  V Conclusion and Discussion

In this paper, we introduced an original measurement method to evaluate the
antenna radiation efficiency within an RC. It is contactless so that the
method does not require to connect the AUT to an analyzer. It is based on the
measurement of the RC $Q$-factor for two load conditions of the AUT. It is
emphasized that the two measurements are performed without modifying the
measurement setup (no manual handling) thanks to remotely-controlled
mechanical switches. An OC and a 50 $\Omega$ load have been considered here as
the two load impedances in order to enhance the measurement sensitivity. It
has been validated with a wideband patch antenna in the 1.8 GHz to 2.8 GHz
frequency range in two different RCs of different volumes: 1 m3 and 19 m3. The
retrieved radiation efficiency is very similar to the one obtained using a
conventional invasive approach [7] (average relative difference of 2.3 % in
the large RC and 6 % in the small one).

This method paves the way for non-invasive antenna radiation efficiency
measurement, which is highly suitable for electrically small antennas as an
invasive setup disturbs their impedance and radiation properties. However,
this method also exhibits limitations. The contactless approach is based on
the estimation of the RC $Q$-factor variation for two AUT load conditions.
Depending of the considered setup (AUT, loads, and RC $Q$-factor), this
variation may be small. Therefore, the contactless method is more sensitive to
the non-ideal behavior of the RC and its stirring capabilities than
conventional measurements. This has been seen in the second set of measurement
(Section IV) in an RC with a smaller $Q$-factor: the retrieved efficiency
exhibits higher fluctuations over the frequency range than in the RC of higher
$Q$-factor. This limitation can be counterbalance by performing additional
stirring as well as enhancing the number of AUTs (increasing $N$) in order to
raise the RC $Q$-factor variation.

In this paper, we also confirmed that the antenna $Q$-factor formulation
introduced in 2018 [29] describes the absorption introduced by an antenna
within an RC in a better manner than the former model introduced by D. Hill in
1998 [28]. Indeed, some of the energy reflected back by the antenna load
towards the RC was not taken into account by the original model, leading to a
strong radiation efficiency underestimation once using this model in our
method.

Future works will include miniaturization of the switching device in order to
integrate it into the AUT. Specific antenna design including the switching
device could be realized for testing purposes only.

## References

  * [1] C. A. Balanis, _Antenna theory: analysis and design_ , Wiley-Interscience, USA, 2005.
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